UC-NI 
 
 S71 133 
 
UJ 
 
 < < 
 
 < z 
 
 g s 
 
 Q u 
 
 O 3 
 
 Z bu 
 
 E 
 
 UJ > 
 
 UJ t 
 
 Zen 
 S 
 
 o S 
 
 z 1 
 
 UI ^ 
 
 1 2 
 
 CO 
 
 g 
 a. 
 
 u 
 
 UJ 
 
 :a J 
 
HANDBOOK ;^, 
 
 OF 
 
 NATURAL PHILOSOPHY. 
 
 BY 
 
 DIONYSIUS LAEDNEE, D.C.L. 
 
 FORMERLY PROFESSOR OF NATURAL PHILOSOPHY AND ASTRONOMY IN UNIVERSITY COLLEGE, 
 
 LONDON. 
 
 MECHANICS. 
 
 WITH THREE HUNDRED AND FIFTY-SEVEN ILLUSTRATIONS. 
 
 EIGHTH THOUSAND. 
 
 LONDON : 
 WALTON AND MABERLY, 
 
 137 GOWEB STREET. 
 
 1866. 
 
LOXDOS 
 
 PRIKTED BY SPOTTISWOODB AND CO. 
 SEW-STKEET SQUARE 
 
PREFACE. 
 
 THIS work is intended for all who desire to attain an accu- 
 rate knowledge of Physical Science, without the profound 
 methods of Mathematical investigation. Hence the expla- 
 nations are studiously popular, and everywhere accompanied 
 by diversified elucidations and examples, derived from 
 common objects, wherein the principles are applied to the 
 purposes of practical life. 
 
 It has been the Author's especial aim to supply a manual 
 of such physical knowledge as is required by the Medical 
 and Law Students, the Engineer, the Artisan, the superior 
 classes in Schools, and those who, before commencing a 
 course of Mathematical Studies, may wish to take the 
 widest and most commanding survey of the field of inquiry 
 upon which they are about to enter. 
 
 Great pains have been taken to render the work complete 
 in all respects, and co-extensive with the actual state of the 
 Sciences, according to the latest discoveries. 
 
 Although the principles are here, in the main, developed 
 and demonstrated in ordinary and popular language, mathe- 
 matical symbols are occasionally used to express results 
 more clearly and concisely. These, however, are never 
 employed without previous explanation. 
 
vi PREFACE. 
 
 The present edition has been augmented by the introduc- 
 tion of a vast number of illustrations of the application 
 of the various branches of Physics to the Industrial Arts, 
 and to the practical business of life. Many hundred en- 
 gravings have also been added to those, already numerous, 
 of the former edition. 
 
 For the convenience of the reader, the series has been 
 divided into Four Treatises, which may be obtained sepa- 
 rately. 
 
 MECHANICS . . . One Volume. 
 
 HYDROSTATICS, PNEUMATICS, and HEAT . One Volume 
 
 OPTICS and ACOUSTICS . . . One Volume. 
 
 ELECTRICITY, MAGNETISM, and METEOROLOGY . One Volume. 
 
 The Four Volumes taken together will form a complete 
 course of Natural Philosophy, sufficient not only for the 
 highest degree of School education, but for that numerous 
 class of University Students who, without aspiring to the 
 attainment of Academic honours, desire to acquire that 
 general knowledge of these Sciences which is necessary 
 to entitle them to graduate, and, in the present state of 
 society, is expected in all well educated persons. 
 
CONTENTS. 
 
 BOOK I. 
 
 Properties of Matter. 
 
 CHAPTER I. 
 
 MAGNITUDE, FIGURE, AND STATE OF 
 AGGREGATION. 
 
 Sect. Page 
 
 I. Most important properties of 
 
 matter 1 
 
 z. Magnitude classified - 
 
 3. Linear magnitude - 
 
 4. Superficial magnitude 
 
 5. Solid magnitude - 
 
 6. Solid magnitude defined 
 
 7. On what the form of a sol d de 
 
 pends - - - - 
 
 8. Boundaries of solid bodies - ib. 
 
 9. Boundaries of surfaces - ib. 
 io. ii. Magnitude unlimited - - 2-3 
 
 12. Bodies consist of parts 3 
 
 13. Solid, liquid, and gaseous states - ib. 
 
 14. Solid state - - - - - ib. 
 
 15. Liquid state - - - - - ib, 
 
 16. Pulverised state 4 
 
 17. Gaseous state .... ib. 
 
 18. Atmospheric air an example - ib 
 
 19. Other examples - ib. 
 uo. Same body may exist in either 
 
 state .... - ib. 
 
 i. Water an example - ib. 
 
 CHAP. II. 
 
 IMPENETRABILITY DIVISIBILITY. 
 
 it All matter impenetrable 
 
 23 . Gaseous bodies impenetrable - i 
 
 24. Air an example - - - - i 
 Z5- Examples of apparent penetrability 
 
 explained 
 
 z6. Walking through the air - - ib. 
 
 Z7- Solid plunged in liquid - - ib. 
 
 z8. Divisibility unlimited - - - ib. 
 
 zg. Water, its ultimate atoms com- 
 pound - .... 6 
 
 jo. Water divisible without practical 
 
 limit - - - - - ib. 
 
 Sect. Pag 
 
 31. Other bodies likewise so divisible 6 
 
 32. Examples 
 
 Pulverised marble ... 
 
 4. Polished surfaces covered with 
 
 asperities Diamond 
 
 5. Gold visible on touchstone - - 
 
 6. Minuteness of tubular filaments of 
 
 J7. Such a filament might penetrate 
 
 the flesh without pain - - ib'. 
 
 38. Wollaston's micrometric wire - 8 
 
 39. Illustration of its extreme minute- 
 
 ness ------ ib. 
 
 40. Minutei^s of organised filaments 9 
 
 41. Thinness of a soap-bubble - - ib. 
 
 42. Thinness ol'insects' wings - - ib. 
 
 43. Thinness of leaf gold ... ib. 
 41. 45. Wire used in embroidery - io 
 
 46. Composition ot blood - - ib. 
 
 47. Magnrude and form of its cor- 
 
 puscles - - - - - ib. 
 
 48. -Number of corpuscles in a drop of 
 
 blood - - - - - ii 
 
 49. Minuteness of animalcules ; their 
 
 organisation and functions - ib 
 
 50. Unlimited divisibility by solution 
 
 of solids in liquids - - - iz 
 
 51. Minute divisibility proved by 
 
 colour - - ... H. 
 
 52. Divisibility of musk - ib. 
 
 53. Fineness of spiders' web - - ib. 
 
 54. Divisibility shown by the taste - ib. 
 
 55. Effect of strychnine dissolved in 
 
 water .... 13 
 
 56. Effect of salt of silver dissolved in 
 
 water - - - - - i/>. 
 
 57. Effect of sugar dissolved - - ib. 
 
 58. Is matter, therefore, infinitely divi- 
 
 sible ? ib. 
 
 59. Existence of ultimate molecules 
 
 may be inferred - - - ib. 
 
 60. Crystallisation indicates their ex- 
 
 istence - - - - - ib. 
 
 61. Process of crystallisation - - 14 
 
 62. Existence ot ultimate molecules 
 
 interred .... ib. 
 
 63. Planes ol' cleavage - 15 
 
CONTENTS. 
 
 Sect. Page 
 
 64. Planes of cleavage indicate the 
 
 lorms of the ultimate molecules 15 
 
 65. Inference that all bodies consist 
 
 of ultimate atom s ot determinate 
 figure - - - - fb. 
 
 66. Those molecules too minute to be 
 
 the subjects of direct observa- 
 tion ... ---,/. 
 
 67. Ultimate molecules of mat>r in- 
 
 destructible - - - -id. 
 
 68. No matter destroyed in com- 
 
 bustion - - - - ' - 16 
 
 69. No matter destroyed in evapo- 
 
 ration .... - fb. 
 
 70. Destructive distillation - - 17 
 
 71. General conclusion ... ib. 
 
 CHAP. III. 
 
 I'OKOSITY, DENSITY, COMPRESSIBILITY, AND 
 DILATABILITY. 
 
 99. 
 JOG. 
 101. 
 icz. 
 103. 
 104. 
 
 Component molecules of a body 
 
 not in contact ~ - - - 17 
 
 Spaces which separate them, 
 
 called pores - - - - 18 
 
 Proportion of mass to pores de- 
 termines density ... ib. 
 
 Porosity and density correlative 
 
 terms - - - - - ib. 
 
 Uniform diffusion of pores consti- 
 tutes uniform density - - ib. 
 
 Pores different from cells - - ib. 
 
 Pores sometimes occupied by 
 
 more subtle matter - ib. 
 
 Examples of porosity and den- 
 sity 19 
 
 Porosity of wood ... fb. 
 
 Buoyancy of wood ... ib. 
 
 Densest substances have pores - ib. 
 
 Filtration ..... 20 
 
 Petrifaction - - - - ib. 
 
 Porosity of mineral substances - ib. 
 
 Porosity of mineral strata - - ib. 
 
 Compression diminishes bulk and 
 
 augments density - - - ib. 
 
 All bodies compressible - - 21 
 
 Compressibility increases with 
 
 porosity - - - - - ib. 
 
 Compression of wood - - ib. 
 
 of stone - - fb. 
 
 of metal - - ib. 
 
 ofiiquds - - ib. 
 
 of gases - - 22 
 
 Contractibility of liquids - - ib. 
 
 Elastic and inelastic bodies ib. 
 
 Elasticity of gases ... jf,. 
 
 Elasticity of liquids ... ib, 
 
 Expansibility of gases - - ij 
 
 Elasticity of solids ... ib. 
 
 Examples of elasticity of solids - ib. 
 
 Examples Ivory bafls - - ib. 
 
 Caoutchouc balls - ib. 
 
 Elasticity of steel 
 
 springs - - 24 
 
 ! Sect. P.ijre 
 
 | 105. Limits of elastic force - - 24 
 I 106 Elasticity of torsion ... ib. 
 I 107. Dilatation by elevation of tem- 
 perature ... - 25 
 I 108. Dilatation of liquids in ther- 
 mometers .... ib. 
 109. Useful application of dilatation 
 
 and contraction ... fb. 
 lie. General effects of dilatation and 
 
 contraction - 26 
 
 CHAP. IV. 
 
 INERTIA. 
 
 in. All matter inert - - - - 26 
 
 112. Inertia, inability to change state 
 
 of rest or motion - ib. 
 
 113. Vis inertia?, term leading to erro- 
 
 neous conclusions - - - zj 
 
 114. Erroneous supposition that matter 
 
 more inclined to rest than mo- 
 tion - - - - - - ib. 
 
 115. Why motion of bodies in general 
 
 retarded ----- z8 
 
 116. Astronomy supplies proof of law 
 
 of inertia - 29 
 
 117. Examples of inertia ... ib. 
 
 118. Effect of sudden change 
 
 of speed on horseback ib. 
 
 119. Leaping from carriages 
 
 in motion - - 30 
 
 120. Coursing ... fb. 
 
 CHAP. V. 
 
 SPECIFIC PROPERTIES. 
 
 izr. Properties general and specific - 
 
 122. Elasticity and hardness - - 
 
 123. Relative hardness of metals - 
 
 124. Hardness of a metal may be mo- 
 
 dified ..... 
 
 125. Effects of elasticity - - 
 
 126. No body perfectly elastic or per- 
 
 fectly i'nelastic - - 
 
 127. Elasticity not proportional to 
 
 hardness - - - - - 
 
 128. Vibratory metals - 
 
 129. Hardness and elasticity of metals 
 
 affected by their combination - 
 
 130. Flexibility and briitleness - - 
 
 131. Malleability - 
 
 132. Malleability varies with tempera- 
 
 ture ...... 
 
 133. Process of annealing ... 
 
 134. Welding - - - - - 
 
 135. Ductility - - - - . - 
 
 137. Tenacity - - - - - 
 
 138. Table of relative tenacities of 
 
 metals - - - - - 
 
 139. Table of tenacity of fibrous tex- 
 
 tures ... 
 
 140. Chemical properties . 
 
 ib. 
 
 32 
 
 ib. 
 
 . 36 
 . ib. 
 
CONTENTS. 
 
 1 ix 
 
 BOOK II. 
 
 Of Force and Motion. 
 
 CHAPTER I. 
 
 COMPOSITION AND RESOLUTION OF FORCES. 
 
 Sect. Page 
 
 141. Force produces, destroys, or 
 
 changes motion - - - 38 
 
 142. Force expressed by weight - ib. 
 
 143. Direction of force ... ib. 
 
 144. Effect of forces acting in same 
 
 direction - ib. 
 
 145. Resultant offerees in same direc- 
 
 tion 39 
 
 146. Resultant of opposite forces - ib. 
 
 147. Resultant of forces in different 
 
 directions - 16. 
 
 148. Composition of forces - - 40 
 
 149. Resultant and components me- 
 
 chanically interchangeable - 41 
 
 150. Resultant of any number offerees ib. 
 
 151. Composition of forces applied to 
 
 different points ... ib. 
 
 152. Resultant of parallel forces - 42 
 
 153. Composition of parallel forces 
 
 acting in same direction - -43 
 
 154. Composition of parallel forces 
 
 acting in opposite directions - ib. 
 
 155. Couple ..... 44 
 
 156. Mechanical effect of couple - ib. 
 
 157. Equilibrium of couples - - 45 
 
 158. Condition under which two forces 
 
 admit a single resultant - - ib. 
 
 159. Mechanical effect of two forces in 
 
 different planes - 46 
 
 160. Suspension bridges - - 48 
 
 CHAP. II. 
 
 COMPOSITION AND RESOLUTION OF MOTION. 
 
 161. Direction and velocity of motion 50 
 
 162. Direction of motion in a curve - ib. 
 
 163. Velocity defined - - ib 
 
 164. Table of velocities of certain 
 
 moving bodies - - 51 
 
 165. Uniform velocity - - 52 
 
 166. Principles of composition and re- 
 
 solution of force equally appli- 
 cable to composition and resolu- 
 tion of motion - - ib. 
 
 167. Resultant of two motions in one 
 
 direction ----- 5? 
 
 168. Resultant of two motions in oppo- 
 
 site directions - ib. 
 
 169. Resultant of two motions in dif- 
 
 ferent directions - ib. 
 
 170. Examples of composition and reso- 
 
 lution of motion - 54 
 
 171. swimming across a 
 
 stream - - ib. 
 
 I7i. >. effect of wind and tide 
 
 on a ship - - 56 
 
 Sect. Page 
 
 173. Examples of rowing a boat - 57 
 
 174. motion of fishes, 
 
 birds, &c. - - ib. 
 
 175. effect of wind on sail- 
 
 ing vessels - - ib 
 
 176. tacking - 5 g 
 
 177. ball dropped from 
 
 ship's topmast 59 
 
 178. object let fall from 
 
 railway carriage - 60 
 
 179. billiard-playing - 61 
 
 180. example proving the 
 
 diurnal rotation of 
 the earth . - 62 
 
 181. Motion absolute and relative - 63 
 
 182. Example Person walking on 
 
 deck of ship - - ib. 
 
 183. Gymnastic and eques- 
 
 trian feats - - 64 
 
 184. Flying kite - -65 
 
 185. Force of moving mass - -66 
 180. Momentum of solid masses - ib. 
 
 187. liquids - - . ib. 
 
 188. air ... ,b. 
 
 189. Conditions which determine force 
 
 of moving mass - . - 67 
 
 190. Moving force augments with velo- 
 
 city ib. 
 
 191. When velocity same, moving force 
 
 augments with mass moved - ib. 
 
 192. Example Force necessary to 
 
 project stone by 
 hand ... 68 
 
 193. Force necessary to 
 
 row a skiff . 
 
 194. Arithmetical expression for mo- 
 
 mentum - 
 
 195. General condition of equality of 
 
 moving forces . 
 
 - ib. 
 o- 
 
 - ib. 
 
 CHAP. III. 
 
 COMMUNICATION OF MOMENTUM BETWEEN 
 BODY AND BODY. 
 
 196. Effects of collision - - 69 
 
 197. Case of inelastic masses - - ib. 
 
 198. Effect of moving body striking 
 
 body at rest - ... ib. 
 
 199. Example Boat towing ship - 71 
 
 200. Collision of two bodies moving in 
 
 same direction ... jl>. 
 
 201. Equality of action and reaction - 72 
 
 202. Example Mutual action of two 
 
 boats - > - - - ib. 
 
 203. Effect of collision of two bodies 
 
 moving in contrary directions - ib. 
 204- Effect verifies law of action and 
 
 reaction ----- 73 
 205. Collision of equal masses with 
 
 equal and opposite velocities - 74 
 
CONTENTS. 
 
 Sect. Page 
 
 106. Examples of railway trains and 
 
 steam-boats - - - - 74 
 
 207. Pugilism - - - - - ,b 
 
 208. Collision of masses moving in 
 
 different directions - - - 75 
 
 209. How action and reaction are mo- 
 
 dified by elasticity - ib. 
 
 210. Perfect and imperfect elasticity - 76 
 2.1 1. Rebound of an ivory ball - - ib. 
 
 212. Oblique impact of elastic body - ib. 
 
 213. When striking body perfectly 
 
 elastic, angles of incidence and 
 reflection equal - - - 77 
 
 214. When striking body imperfectly 
 
 elastic, angle of incidence less 
 than angle of reflection - - ib. 
 
 215. Apparatus to illustrate collision 
 
 experimentally - '- - 78 
 
 216. Laws of motion -. - - - 80 
 
 217. Meaning of these laws - - 81 
 
 218. Maxim that there is always the 
 
 same quantity of motion in the 
 world explained ... ib. 
 
 219. Cannot a living agent produce 
 
 new motions ? - 82 
 
 220. Spontaneous motion of an animal 
 
 explained - . - - - ib. 
 
 221. Motion of railway train - - ib. 
 
 222. Earth a great reservoir of mo- 
 
 mentum ----- 83 
 
 223. Would spontaneous progressive 
 
 motion be possible in absence 
 of a mass like the earth to react 
 upon? - - - - - ib. 
 
 CHAP. IV. 
 
 TERRESTRIAL GRAVITY. 
 
 224. Plumb line ----- 84 
 
 225. Vertical direction _ - - tb. 
 
 226. Level surface ... - ib. 
 
 227. Terrestrial gravity indicated by 
 
 these facts . ... fb. 
 
 228. Earth attracts all bodies towards 
 
 its centre - .... 85 
 
 229. Bodies fall in vertical lines - - ib. 
 
 230. What is reaction, corresponding 
 
 to action of falling body ' - - ib. 
 
 231. All bodies fall with the same 
 
 velocity - - - - . ib. 
 
 232. Effects incompatible with this ex- 
 
 plained 86. 
 
 233. Guinea mid feather experiment - ib. 
 
 234. Weight of bodies proportional to 
 
 their quantity of matter - - 87 
 
 235. Motion of a falling body accele- 
 
 rated ib. 
 
 236. Force of fall not proportional to 
 
 height 88 
 
 237. Analysis of the motion of a falling 
 
 body - ib. 
 
 238. Velocitv augments with time of 
 
 fall -* . - --- 89 
 
 239. Uniformly accelerated motion 
 
 and force ----- ib. 
 
 240. Body falling acquires speed which 
 
 would in same time carry it over 
 twice the space - - -90 
 
 241. Analysis of heights fallen through 
 
 in successive seconds - - ib. 
 
 242. Tabular analysis of motion of fall- 
 
 ing body ----- 91 
 
 Sect. Page 
 
 243. Atwood's machine for illustrating 
 
 phenomena of falling bodies - 92 
 
 244. Morin's apparatus - - - 97 
 
 245. Heights from which body falls 
 
 proportional to squares of times 
 of fall ib. 
 
 246. Formula?, expressing rate, heights, 
 
 velocities, and times - - - 98 
 
 247. Calculation of height from which 
 
 body falls in one second - - 100 
 
 248. Method of calculating all circum- 
 
 stances of descent of falling body il. 
 
 249. Force with which body falls a's 
 
 square root o 1 ' height - - - 101 
 
 250. Why fall from great height not so 
 
 destructive as might be expected ib. 
 
 251. Case of person falling down coal- 
 
 pit - - - - - 102 
 
 252. Retarded motion of bodies pro- 
 
 jected upwards - ... ib. 
 
 253. Motion down inclined plane - ib. 
 
 254. Motion of projectiles - - - 105 
 
 255. Case of projectile shot horizon* 
 
 tally ------ ib. 
 
 2.56. Case of oblique projection - - 106 
 
 257. Projectiles move in parabolic 
 
 curves - - - - . ib 
 
 258. These conclusions modified by re- 
 
 sistance of the air 108 
 
 259. Application of projectiles to gun- 
 
 nery - ib. 
 
 260. Effect of a hammer upon a nail - 109 
 Influence of time on the effect of 
 
 force - - - - - -in 
 
 CHAP. V. 
 
 CENTRE OP GRAVITY. 
 
 261. Weight of body is the aggregate 
 
 of weights of its molecules -112 
 
 262. Effect of cohesion on gravity of 
 
 molecules - - - - - ib. 
 
 263. Resultant of gravitating forces of 
 
 molecules - - - - - 113 
 
 264. Experimental method of deter- 
 
 mining resultant of gravitating 
 forces ef molecules - - - 114 
 
 265. Different resultant for every dif- 
 
 ferent point of suspension - 115 
 
 266. AH these resultants have common 
 
 point of intersection - - - ib. 
 
 267. Another experimental proof of 
 
 this - - - - - - 116 
 
 268. Common point of intersection 
 
 called centre of gravity - - 117 
 
 269. When centre of gravity supported, 
 
 body will remain at rest - - ib. 
 
 270. Body having regular figure, centre 
 
 of magnitude centre of gravity - ib . 
 
 271. Centre of gravity of sphere - ib. 
 
 272. Body having symmetrical axis, 
 
 centre of gravity will be on it - ib. 
 
 273. Centre of gravity not always 
 
 within body - - - - 118 
 
 274. Has same properties - ib. 
 
 275. Centre of gravity takes lowest 
 
 position - - - - - 119 
 
 276. Centre of gravity at rest must be 
 
 always above or below a point 
 of support ----- 16 
 
 277. Centre of gravity not supported, 
 
 oscil ates - - - 6. 
 
CONTENTS. 
 
 XI 
 
 Sect. Page 
 
 278. Pendulum - - - - 119 
 
 170 Conditions determining stability 
 
 of body 120 
 
 280. Stability of pyramid - 121 
 
 281. Case in which line of direction 
 
 falls outside base - - - tb. 
 
 282. Leaning towers of Pisa and Bo- 
 
 logna- - - - - -'/>. 
 
 28j. Case in which line of direction 
 
 falls upon edge of base - - 122 
 
 284. Stability of loaded vehicle - - tb. 
 
 285. Drays 123 
 
 286. Stability of table - - - - ib. 
 
 287. Gestures and motions of animals 
 
 governed by direction of centre 
 of gravity ----- 124 
 
 288. Motion of centre of gravity of 
 
 person walking . ih. 
 
 289. Use of knee-joint- - - - ib. 
 
 290. Position of centre of gravity 
 
 changes with change of posture 125 
 
 291. Porter carrying load - ih.. 
 
 292. Walking up or down hill - - tb. 
 
 293. Rising from chair - ih. 
 
 294. Case of quadruped ... ib. 
 
 295. Cylinder rolled on level plane - 126 
 
 296. Elliptic body on 'level plane - 127 
 
 297. Stable, unstable, and neutral 
 
 equilibrium .... 128 
 
 298. Criterion of stable equilibrium - ib. 
 
 299. Criterion of unstable equilibrium flu 
 
 300. Criterion of neutral equilibrium ih. 
 
 301. Example 1. Children's toys - ib. 
 
 302. II. Feats of public exhi- 
 
 bitors ... 129 
 
 303. III. Spinning top - - ib. 
 
 304. IV. Object spinning on 
 
 point of sword - 130 
 jO). V. Cases in which cen- 
 tre of gravity seems 
 to acend - - ib. 
 
 306. ,, VI. Case of globe rolled 
 
 up inclined plane - 131 
 
 307. Magic clock- - - - - 132 
 
 308. Centre of gravity of fluids - - ib, 
 
 309. Centre of gravity of two separate 
 
 bodies - - - - - 133 
 
 CHAP. VI. 
 
 CENTIMFL'GA-L fOKCE. 
 
 310. Force consequent on rotatory 
 
 motion - - - - - 133 
 
 311. Centrifugal force - ... ib, 
 
 312. Centrifugal ferce a consequence 
 
 of inertia ----- 134 
 
 313. Method of calculating centrifugal 
 
 force . - - - - - -ib. 
 
 314. Rule to calculate centrifugal 
 
 force, weight, velocity, and ra- 
 dius of rotation being given - 135 
 
 315. Rule to calculate centrifugal 
 
 force, weight, radius of rotation, 
 and number of revolution* per 
 second being given - - - 136 
 
 316. Application ef whirling-table to 
 
 illustrate theory ... ib. 
 
 317. Centrifugal force of bodies revolv- 
 
 ing in same time round com- 
 mon centre of gravity equal - 138 
 i3. Examples of centrifugal force - 159 
 
 Sect. page 
 
 319. Example I. Turning rapidly 
 
 round corner - 139 
 
 320. II. Horse moving round 
 
 circus - ih 
 
 321. Method of calculating 
 
 inclination of horse 
 towards centre - 140 
 
 322. III. Carriage turning cor- 
 
 ner - - - 141 
 
 323. IV. Stone in a sling -142 
 
 324. V. Glass of water in a 
 
 sling - - jb, 
 
 325. VI. Water whirling - 142 
 
 326. Centrifugal drying machine - 144 
 
 327. Case of body moving down con- 
 
 vex surface - - - - 146 
 
 328. Centrifugal forces of solid body 
 
 revolving on fixed axis - - ib. 
 
 329. First case, in which centrifugal 
 
 forces in equilibrium - - 147 
 
 330. Second case, in which they have 
 
 a single resultant - ib. 
 
 331. Third case, in which they are 
 
 equivalent to a couple - - ib. 
 
 332. Fourth case, in which they have 
 
 effect of single force and couple, 
 combined - - - - - ib. 
 
 333. Examples of application of these 
 
 principles - - - - - 148 
 
 334. Example I. Ring revolving roud 
 
 its centre in its 
 own plane - - ib. 
 
 335. IL Flat circular plate - 16. 
 330. 'III. Cylinder revolving 
 
 round its geome- 
 trical axis - - ib. 
 
 337. IV. Solid of revolution 
 
 revolving round its 
 geometrical axis - ib. 
 
 338. Case of solids which have a sym- 
 
 metrical axis - - . - 150 
 
 339. Case of rectangular prism - - 151 
 
 340. Case of pyramid - ib. 
 
 341. Case of axis of revolution passing 
 
 through centre of gravity - - ib. 
 
 342. Centrifugal forces round such an 
 
 axis are either in equilibrium or 
 equivalent to a couple - - 152 
 J4J. Axes of centrifugal equilibrium 
 
 called principal axes - ib. 
 
 344. Case in which all lines through 
 
 centre of gravity are principal 
 axes ------ ib. 
 
 345. Case of axis of revolution parallel 
 
 to principal axis through centre 
 of gravity 153 
 
 346. Such axis called also principal axis ib. 
 
 347. Three principal axes at right 
 
 angles to each other pass through 
 each point - - - - - ib. 
 
 348. Effect of revolution round a line 
 
 which is not a principal axis - ib. 
 
 349. Experimental illustration of these 
 
 properties - - - - - ib. 
 
 CHAP. VII. 
 
 MOLECULAR FORCES. 
 
 550. Pores of bodies the region of mo- 
 lecular forces - ... 155 
 }>i. Attraction of cohesion - - it>. 
 352. Attraction ot adhesion - - 156 
 
CONTENTS. 
 
 Sect. Page 
 
 353. Capillary attraction ... 156 
 
 354. Chemical affinity. - ib. 
 
 355. Atoms of bodies manifest attrac- 
 
 tion and repulsion - ib. 
 
 356. Cohesion manifested in solids 
 
 and liquids - - - -157 
 
 357. Example of cohesion - - - ib. 
 
 358. Shot manufacture ... ib. 
 
 359. Why liquids form spherical drops il>. 
 
 360. Earth and planets once fluid - 158 
 
 361. Mutual repulsion of aioms of 
 
 gases- - - - - - ib. 
 
 36z. Gases may he reduced to liquid 
 
 and solid states .... ib. 
 563. Existence of attraction of cohe- 
 
 Sect. Page 
 
 sion between their atoms in- 
 ferred - - - - - 158 
 
 364. Mutual repulsion ascribed to in- 
 
 fluence of heat .... ib. 
 
 365. Same body may exist in solid, 
 
 liquid, or gaseous state - - 159 
 
 366. Adhesion of solids ... ib, 
 
 367. Adhesion of wheels of locomotive 
 
 to rails - - - - - ib. 
 
 368. Effect of lubricants - - - 160 
 
 369. The bite in metal working ex- 
 
 plained - - - - ib. 
 
 370. Effect of glues, solders, and like 
 
 adherents .... - ib. 
 
 371. Silvering mirrors ... ib. 
 
 BOO] 
 
 Theory of 
 
 CHAPTER I. 
 
 GENERAL PRINCIPLES. 
 
 Sct. Pajre 
 372. What constitutes a machine - 161 
 
 K III. 
 Machinery. 
 
 Sect. Page 
 389. When the power is too small to 
 equilibrate, motion is retarded - 167 
 390. Proper functions of a machine - 168 
 391, No machine can really add to the 
 mechanical energy of the power 169 
 392. Method of expressing mechanical 
 energy of power and weight -170 
 393. Moments of power and weight - ib. 
 394. Relation between these moments 
 determines state of machine - ib. 
 395. Equality of these moments deter- 
 mines equilibrium - - - 171 
 396. When moment of power exceeds 
 that of weight, motion accele- 
 rated in direction of power - ib 
 397. When moment of power less than 
 moment of weight, motion re- 
 tarded ib. 
 
 373* Moving power 
 
 
 376. Various resistances and physical 
 conditions omitted provisionally 
 in exposition of theory of ma- 
 
 377. Physical science consists of a se- 
 ries of approximations to truth 163 
 378. This gradual approximation to 
 truth not peculiar to mechanical 
 
 379. How machine provisionally re- 
 
 380. Mechanical truths improperly in- 
 vested wilh the appearance of 
 paradox ----- 164 
 381. Effect of fixed points or props - ib. 
 382. An indefinitely small power rais- 
 ing an indefinitely great weight 
 involves no paradox ... ib. 
 383. Effects of a machine under differ- 
 ent relations of power and 
 
 398. In equilibrium, velocity of power 
 is to that of weight as weight is 
 to power - . - - - ib. 
 399. Power always gained at expense 
 
 400. All paradox thus removed from 
 mechanical theorems - - ib. 
 401. Utility of machines not limited 
 to make small power overcome 
 great resistance - - - 172 
 402. Case in which point of application 
 of power and working point do 
 not move in direction of power 
 and weight .... ib. 
 
 CHAP. II. 
 
 SIMPLE MACHINES. 
 
 403. Machines, simple and complex - 174 
 404. Classification of simple machines ib. 
 
 384. When power and weight in equi- 
 librium, rest not necessarily im- 
 
 385. Equilibrium infers either absolute 
 rest or uniform motion - - 166 
 386. When the power more than equi- 
 librates, accelerated motion en- 
 
 387. All machines are in this state 
 
 388. Analysis of the effect in this case ib. 
 
CONTENTS. 
 
 Sect. Page 
 405. Condition of equilibrium of ma- 
 chine having fixed axis - - 174 
 406. Condition of equilibrium of flexi- 
 
 Sect. Page 
 451. Rack and pinion - - - 197 
 452. Ratchet wheel - - - - 198 
 
 PULLEYS. 
 
 453. Ropes not perfectly flexible nor 
 perfectly smooth ... ib. 
 454. Hence necessity of sheaves - 199 
 455. Fixed pulley, useful to change 
 direction of power - lift. 
 456. Case in which" power and resist- 
 
 407. Condition of equilibrium of weight 
 upon hard inclined surface - ib. 
 408. Mechanic powers ... ib. 
 
 THE LEVER. 
 
 409 Levers, ist, and, and 3rd kinds - 176 
 410. Condition of equilibrium - - ib. 
 411. Effect of power or weight varies 
 as leverage - - - - 177 
 412 Power or weight oblique to lever ib. 
 413. Relation of power and weight in 
 levers of ist. znd, and jrd kinds 178 
 414. Examples of levers of 1st kind 
 Balance - - - - - ib. 
 415. Sensibility of balance - 179 
 
 457. Power and weight equal when a 
 single rope used with fixed 
 pulley - - - - - ib. 
 458. Mechanical convenience of chang- 
 ing direction of power - - ib. 
 459. Fire-escape - 201 
 460. Single movable pulley - - ib. 
 461. Case in which cords not parallel ib. 
 462. Movable block with several 
 sheaves ----- 202 
 46?. Condition of equilibrium of such 
 
 417. Letter balance - - - - 181 
 
 419. Crowbar, scissors, &c. - - ib. 
 410. Examples of znd kind, Oar, 
 rudder. &c. .... 182 
 421. Examples of jrd kind, Limbs of 
 animals, &c. - - - - i8j 
 422. How to determine pressure on 
 fulcrum of lever ... ib. 
 423. Rectangular lever ... 184 
 424. Effect of beam resting on two 
 
 464. Effect of attaching end of rope 
 to movable block - ib. 
 465. Smeaton's and White's pulleys - ib. 
 466. Systems of pulleys consisting of 
 several ropes and movable 
 blocks - - - - - 203 
 467. Practical effect of pulleys varies 
 considerably from their theore- 
 tical effect - - - -204 
 
 INCLINED PLANE. WEDGE AND SCREW. 
 
 468. Effect of inclined surface on 
 
 425. Condition of equilibrium in com- 
 pound lever - . - - 185 
 426. Weighing machines - - - 186 
 427. Knee lever ----- 187 
 428. Beautiful example of complex le- 
 verage in the mechanism which 
 connects the key and hammer in 
 Erard's pianoforte - ib. 
 429. Power of machine how expressed 189 
 430. Equivalent lever ... - ib. 
 431. Complex machine may be repre- 
 sented by equivalent compound 
 
 469. Inclined plane Condition of 
 equilibrium .... 206 
 470. Apparatus to illustrate experi- 
 mentally inclined plane - - ib. 
 471. Inclined roads - ib. 
 47Z. Inclined planes on railways - 207 
 473. Case in which power acts in direc- 
 tion inclined to plane - - ib. 
 474. Examples ----- 208 
 475. Case of double inclined plane - ib. 
 476. Case of self-acting planes on rail- 
 
 WHEEL-WORK. 
 
 433 Condition of equilibrium - 190 
 434 Various ways of applying power ib. 
 
 477. Wedge - - - - - ib. 
 478. Wedges consist of two inclined 
 planes - ... 210 
 479. Theory of wedge practically inap. 
 
 435 ma ass to. 
 
 437 Treadmill, &c. - - - -191 
 438. French quarry wheels ... ib. 
 439. Case in which power or resist- 
 ance variable .... 192 
 440. Method of augmenting ratio of 
 power v. ithout complicating ma- 
 
 480. Power applied to wedge princi- 
 pally percussion ... ib. 
 481. Practical use of wedge - - ib. 
 482. Practical examples cutting and 
 piercing instruments - -211 
 483. Utility of friction in application 
 
 441. Compound wheels and axles ana- 
 logous to compound levers - ib. 
 442. Various methods of communi- 
 cating force between wheels and 
 
 485. Power applied to screw by means 
 
 443. By endless bands or cords - - ib. 
 444. By rough surfaces in contact - 195 
 
 486. Methods of transmitting power to 
 resistance .... ib. 
 487. Condition of equilibrium - - 213 
 488. Great mechanical force of screw 
 explained ... - ib. 
 489. Various methods of connecting 
 
 446. Formation of teeth ... ib. 
 447. Methods of computing condition 
 of equilibrium in wheel work - ib. 
 
 449. Crown wheels - - - - 197 
 4,0. Bevelled whppls . _ A 
 
 490. Various examples of application 
 
CONTENTS. 
 
 49'- Manner of cu'ting screw 
 
 492. Method of augmenting force 
 
 screw - 
 
 493. Hunter's screw - 
 
 494. Micrometer screw 
 
 495. Endless screw 
 
 Page 
 
 or" 5 
 
 - fb. 
 
 - ib 
 -216 
 
 - ib. 
 
 45 6. 
 
 497- 
 498. 
 
 49> 
 
 500 
 
 501. 
 502. 
 
 503. 
 
 504. 
 505. 
 
 506 
 
 507. 
 508- 
 
 509. 
 
 510. 
 511. 
 512. 
 
 53- 
 
 5'4- 
 
 515- 
 516. 
 
 518- 
 
 CHAP. III. 
 
 PENDULUM, BALANCE, AND FLY. 
 
 The pendulum - . - 217 
 
 Oscillation of pendulous mass - ib. 
 
 Isochronism of the pendulum - 219 
 
 Time of oscillation independent 
 ot weight of pendulum - - ib. 
 
 How time of oscillation affected 
 by length - - - - - ib. 
 
 Experimental illustration - - 220 
 
 Time of vibration being given, to 
 find length of pendulum - -221 
 
 Time of oscillation varies with 
 attraction of gravity - - ib. 
 
 Law cf this variation - - -222 
 
 Pendulum indicates variation of 
 gravity in different latitudes - 16. 
 
 Analysis of motion of a pendulous 
 mas's of definite magnitude - ib. 
 
 Centre of oscillation ... 224 
 
 Centres of oscillation and suspen- 
 sion interchangeable - - ib. 
 
 Variations of pendulum conse- 
 quent on change of temperature 225 
 
 Compensation pendulums - ib. 
 
 Pendulum a measure of time - ib. 
 
 Pendulum measure of force of 
 gravity - - - - - ib. 
 
 Pendulum indicates form and 
 diurnal rotation of the earth - 226 
 
 Pendulum measures velocity of 
 falling bodies - - - - ib. 
 
 Curious properties of a swing - 227 
 
 Actual length of pendulum which 
 vibrates seconds ... 229 
 
 Balance wheel - ib. 
 
 Theflv ib. 
 
 CHAP. IV. 
 
 REGCLATION AND MODIFICATION OF FORCE 
 AND MOTION.' 
 
 519. Regularity of motion necessary in 
 
 machinery - - - - - 232 
 
 520. Causes of irregular motion - - 6. 
 
 521. When varying power is opposed 
 
 to uniform resistance - - 233 
 
 522. When uniform power opposed to 
 
 varying resistance ... ib. 
 
 523. When variation of power not pro- 
 
 portional to variation of resist- 
 ance ------ ib. 
 
 524. When effect of power unequally 
 
 transmitted to resistance - - 234 
 
 525. Regulators General principle of 
 
 their action - - - - ib. 
 
 526. The governor ... - ib. 
 
 527. Fusee in watchwork - - - 235 
 
 Sect. Page 
 
 528. When efficacy of machine to 
 
 transmit power to working point 
 varies ..... 236 
 
 529. Effect of crank ... - ib. 
 
 530. Regulating effect of fly wheel - 238 
 
 531. Methods of augmenting or miti- 
 
 gating energy of moving power 239 
 Case of watchwork- moved by 
 mainspring .... ib. 
 
 532. Case of clockwork moved by 
 
 weight .... 240 
 
 533. Cases in which energy of power 
 
 augmented .... fb. 
 
 534. Analysis of action of hammers, 
 
 sledges, &c. .... ib. 
 
 535. Inertia supplies means of accumu- 
 
 lating force - - - - 241 
 536 Effect of weapon called life pre- 
 server, flails, &c. - ib. 
 
 537. Loaded lever ot screw press - 242 
 
 538. This accumulation of force in- 
 
 volves no paradox ... ib. 
 
 539. Rolling and punching mills 
 
 Effect of fly wheel - - - ib. 
 
 540. Position of fly wheel - 243 
 
 541. Method of cutting open work in 
 
 metal - - - - - ib. 
 
 542. Modification of motion - - ib. 
 
 543. Continued rectilinear motion -245 
 
 544. Reciprocating rectilinear motion 240 
 
 545. Continued circular motion - - 248 
 546 Universal joints ... - 250 
 
 547. Alternate circular motion - - 251 
 
 548. Joints - - - - - - 257 
 
 549. Gimbals - - - - - ib. 
 550 Ball and socket - 258 
 
 551. Cradle joint - - - - 259 
 
 552. Hinge ... - - ib. 
 
 553. Trunnions ----- 260 
 
 554. Axles ------ ib. 
 
 555. Telescope joint - - - - ib. 
 
 556. Bayonet joint - - - - ib. 
 
 557. Clamps and adjusting screws - 261 
 
 558. Couplings - - - - - ib. 
 
 CHAP. V. 
 
 RESISTING FORCES. 
 
 559. Forces which destroy but cannot 
 
 produce motion ... 264 
 
 560. Effects of friction ... 265 
 
 561. Friction aids power in supporting 
 
 weight, but opposes power in 
 moving it ----- 266 
 
 562. How this modifies conditions of 
 
 equilibrium - - - - ib 
 
 563. Cases in which friction is the 
 
 whole power - - - - ib. 
 
 564. Great use of friction in economy" 
 
 of nature and art 267 
 
 565. Friction of sliding and rolling - ib. 
 
 566. Laws of friction empirical but still 
 
 useful - - - - - ib. 
 
 567. Methods of measuring sliding fric- 
 
 tion --..-- 268 
 
 568. Angle of repose ... - 269 
 
 569. Friction proportional to pressure ib. 
 
 570. Effect of cro?sing the grains - ib. 
 
 571. Pivots of wheels - ... fb. 
 
 572. Selection of lubricants - - ib. 
 
 573. Rolling friction .... 270 
 
CONTENTS. 
 
 S"Ct. Pa*e 
 
 Sect. Page 
 610. Examples in the structure of ani- 
 mals and plants .... 290 
 6lT. Tabular statements of the average 
 transverse streng'h of beams - 291 
 612. Tredgold's table of the transverse 
 strength of metals and woods - ib. 
 613. Barlow's table of the transverse 
 strength of woods ... 292 
 614. E ffect of distribution of the weight 
 
 CTC. Use of rollers and wheels - -271 
 c 7 6. Friction rollers .... 272 
 577. Friction wheels .... i/>. 
 178. Effect of magnitude of wheels - ib 
 579. Best line of draught - - -27? 
 580. Friction of wheel carriages on 
 
 
 582. Brakes . - - - - - 274 
 583. Friction a point of resistance - 275 
 584. Effects of imperfect flexibility of 
 
 615. Strength of abeam increased by 
 partially sawing it transversely 
 and inserting a wedge - - 293 
 616. Why strength of structure dimi- 
 nished as magnitude increased - ib. 
 617. Strength on the large scale not to 
 be judged by that of the model ib. 
 
 CHAP. VII. 
 
 MOVING POWERS. 
 
 618. Mechan'cal agents - 294 
 619. Dynamical unit ... ib. 
 620. Machines and tools ... ib. 
 62.1. Animal power - ... 295 
 622. Relation between the load and 
 
 585. Ttesi>tance of fluids - - -277 
 580. Resistance depends on frontage 
 of moving body - - - - 278 
 Also affected by form of front - fh. 
 587. Increases in high ratio with speed ib 
 588. Advantages of ponderous missiles 279 
 589. Resistance of air to motion of fall- 
 
 CHAP. VI. 
 
 STRENGTH OF MATERIALS. 
 
 590. Strength of solid bodies - - 280 
 591. Difficulty ot ascertaining its laws ib. 
 592. Several ways in which strength 
 may be manifested ... 281 
 593. Strength to resist direct pull - ib. 
 594. Method of experimentally mea- 
 suring it ----- 282 
 595. Table showing most recent results 
 of such experiments - - - ib 
 596. Iron the most tenacious Effect 
 of alloys - - - - - ib. 
 597. Effect of heat - - - - 283 
 598. Strength of timber ... ib. 
 599. Strength of cordage ... tb. 
 ooo. Animal substances - - - ib 
 631. Strength to resist pressure or 
 
 623. Force of a man varies with the 
 way it is applied - - - 096 
 624. M*n working at capstan - - ib. 
 625. Man working by his weight - ib. 
 62.6. Apparatus for this method - - 297 
 627. Spade labour ... - 298 
 628. Earthwork on railways - - 299 
 
 630. Horse power compared with that 
 of other animals - - -301 
 631. Various estimates of animal power ib. 
 632. Peron's estimate ... ib. 
 633. Tredgold's estimate of horse 
 
 602. Tables showing the strength of 
 
 634. Comparative force of different 
 animals ..... 302 
 
 633. Results of Hodgskinson's re- 
 searches - - - - - ib, 
 604. Strength to resist torsion - - 285 
 605. Strength to resist transverse 
 
 636. Water power ... 303 
 637. Wind power ... ib. 
 638. Steam power ... ib. 
 639. Consumption of fuel - 304 
 640. Mechanical effects of fuel - ib. 
 641. Springs and weights - - ib. 
 642. Forces produced by heat - 305 
 643. Bi congelation - ib. 
 644. Electro-magnetic force - - ib. 
 645. Chem ical agency - - 306 
 
 Case of a beam supported at one 
 
 606. Case of a beam supported at both 
 
 607. Case of rectangular beams - - 288 
 608. How strength "of beam affected by 
 form of its transverse section - 289 
 609. Transverse strength of solid and 
 hollow cylinders - - 290 
 
 647. Capillary attraction ... ib. 
 648. Perpetual motion - - - ib. 
 
CONTENTS. 
 
 BOOK IV. 
 
 Illustrations of the Application of Mechanical Principles in the 
 Industrial Arts. 
 
 CHAPTER I. 
 
 CRANES. 
 
 Sect. Page 
 649. Cranes in general ... 310 
 
 Sect. Page 
 673. Compensation pendulums - - 345 
 674. Harrison's gridiron pendulum - 347 
 675. Mercurial pendulum ... 348 
 676. Another compensation pendulum ib. 
 
 651. Cranes for building: - - - 311 
 652. Manufactory and railway cranes - 312 
 653. Fixed cranes .... jij 
 654. Drops for loading and unloading 
 
 and escapement ... 349 
 678. Motion of hand intermitting - 350 
 679. How the motion of the pendulum 
 is sustained - - - - 351 
 680. Its action upon the escapement - ib. 
 
 
 681. Motion of hands, how proportioned 352 
 682. Method of cutting wheels and 
 
 CHAP. II. 
 
 ENGINES WORKING BY IMPACT. 
 
 655. Pile engines - - - - 317 
 656. Stamping engines ... 320 
 657. Sledge hammer - ... 321 
 658 Coining press .... 323 
 
 683. Moving power, weight - -353 
 684. Mainspring - 354 
 685. Fusee ...... 356 
 686. Balance wheel - - - - 359 
 687. Isochronism .... 360 
 688. Compensation balances - - ib. 
 689. Movement imparted to the hands 
 
 CHAP. III. 
 
 BORING, PLANING, SAWING, AND SCREW 
 AND TOOTH-CT T TTING ENGINES. 
 
 659 Boring tools - 327 
 660. Planing machine - ... 329 
 661. Saw mills - ... -330 
 66z. ^crew-cutting engine - 335 
 663. Engine for cutting the teeth of 
 
 690. Movement of the hands of a clock 363 
 691. To regulate the rate - - 366 
 692. Recoil escapement - - 367 
 Cylindrical escapement - ib. 
 693. Duplex escapement - - 370 
 Lever escapement - 371 
 694. Detached escapement - - ib. 
 695. Maintaining power in clocks 373 
 690. Maintaining power in watches 374 
 697. Cases in which weights and 
 springs are used - - 376 
 698. Watches and chronometers, ma 
 
 CHAP. IV. 
 
 FLOUR MILLS. 
 
 664. Process of grinding - - 337 
 Description of flour mills - - ib. 
 Form and structure of mill 
 stones - - - - - ib. 
 Separation of the flour and bran ib. 
 
 CHAP. V. 
 
 CLOCK AND WATCH WORK. 
 
 665. Time measurer a necessity of civi- 
 
 699. Observatory clocks - - ib. 
 700. Striking apparatus - - ib. 
 
 CHAP. VI. 
 
 THE PRINTING PRESS. 
 
 701. Setting the types ... 381 
 702. Old method of printing - - ib. 
 703. Inking rollers - - - - 382 
 704. Modern printing presses - - ib. 
 705. Arrangement of the form - - 383 
 706. Single print. ng machines - - i/>. 
 707. Double printing machines - - 384 
 708. Applegath and Cowper's double 
 printing machine ... 385 
 709. Printing machines on this princi- 
 
 
 
 667. Position of gnomon - - - 341 
 
 machine, increasing its power - ib. 
 
 66j. Sand-glass - - - - - ib. 
 670. Mercurial time measurer - -343 
 671. Clocks and watches - - - ib 
 672. Pendulum .... - 344 
 
 712. Marinoni's newspaper printing 
 machine ..... 394 
 713. Marinoni's book printing ma- 
 chine ..... 400 
 
: ';;>' "'/"' ll * 
 
 ELEMENTARY COURSE 
 
 MECHANICS. 
 
 BOOK THE FIRST. 
 
 PROPERTIES OF MATTER. 
 
 CHAPTER I. 
 
 MAGNITUDE, FIGURE, AND STATE OF AGGREGATION. 
 
 I. THE material world, the bodies which compose it, and their 
 qualities and properties, form the subjects of contemplation and 
 inquiry to the natural philosopher. 
 
 Among these properties, the most important are the following : 
 
 1 . MAGNITUDE and FORM. 
 
 2. STATES OF AGGREGATION. 
 
 3. IMPENETRABILITY. 
 
 4. DIVISIBILITY. 
 
 5. POROSITY and DENSITY. 
 
 6. COMPRESSIBILITY and CONTRACTIBILITY. 
 
 7. ELASTICITY and DILATABILITY. 
 
 8. INACTIVITY. 
 
 9. SPECIFIC PROPERTIES. 
 
 We shall, therefore, in the present book, explain and illustrate 
 these qualities. 
 
 2. The idea of magnitude is so simple that any attempt to ex- 
 plain it by definition would only tend to obscurity. 
 
2 , PROPERTIES OF MATTER. 
 
 ; Mag, i! fade is eiAe w linear, superficial, or solid. 
 
 $. Lirecr magnitude Is length or distance. Length or distance 
 is expressed numerically by means of some unit ; the unit adopted 
 being, as a matter of convenience, small or great^ according to the 
 magnitude of the length or distance to be expressed. Thus, we 
 say. the side of this leaf is so many inches, the length of the room 
 is so many feet, the distance from London to York is so many 
 miles. 
 
 4. Superficial magnitude, sometimes called area, has length and 
 breadth. Its quantity is expressed by so many square inches, so 
 many square feet, or so many square miles. Thus, we say, the area 
 of this page of paper is so many square inches, that of the floor of 
 this room so many square feet, and the entire surface of the earth 
 is so many square miles. 
 
 5. Solid magnitude has length, breadth, and depth, height or 
 thickness. 
 
 Its quantity, sometimes called its volume, is expressed by so 
 many cubic inches, so many cubic feet, or so many cubic miles. 
 Thus, we say, the volume of this book is so many cubic inches, the 
 volume of this room is so many cubic feet, and the volume of the 
 terrestrial globe is so many cubic miles. 
 
 6. The solid magnitude or volume of a body, is the space which 
 that body occupies or fills. 
 
 7. Thz form, figv.re, or shape of a line, surface, or solid, depends 
 upon the relative position of its parts or limits. 
 
 Two lines may be of equal length, while they differ in form, 
 shape, or figure. Thus, the arc of a circle and a straight line may 
 both measure a foot in length, while their figure or shape is dif- 
 ferent. Again, two lines may have the same form or figure, but 
 very different lengths : thus, the two straight lines which form the 
 edges of this page have the same form, both being straight, but 
 they have different lengths. In like manner, two different arcs of 
 the same circle will have like forms but different lengths, and two 
 arcs of different circles may have equal lengths, and will both be 
 curved and both circular, but they will, nevertheless, have different 
 forms, inasmuch as the curvature of one will be greater than that 
 of the other. 
 
 8. Solid bodies are bounded by surfaces. Thus, a globe is in- 
 cluded within its spherical surface, a cube is included within six 
 plane square surfaces. 
 
 9. Surfaces are bounded by lines. Thus, the surface of this 
 page is bounded by four straight lines, forming its edges; the 
 surface of a circle is bounded by the line called its circumference. 
 
 I o. Magnitude is unlimited as well in its increase as in its dimi- 
 nution. There is no magnitude so great, that we cannot conceive 
 
SOLIDS AND LIQUIDS. 3 
 
 a greater ; and none so small, that we cannot conceive a smaller. 
 The diameter of the earth measures about 8000 miles ; but it is 
 very small compared to the diameter of the sun, which measures 
 nearly 900000 miles ; and this, again, is itself very small compared 
 with the distance between the earth and the sun, which measures 
 little less than 100,000000 of miles ; and even this last space, great 
 as it is, vanishes to nothing, compared with the distance between 
 the sun and the fixed stars. 
 
 1 1. In like manner^ there is no limit to the diminution of which 
 magnitude is susceptible. There is no line so small, that may not 
 be bisected, or halved, or reduced, in any desired proportion. Let 
 p I (Jig- I.) be a line of any proposed length, as, for example, the 
 tenth of an inch. 
 
 Draw P o at right angles to it, and a line I, 10 parallel to P o. 
 Take upon this line distances I, 2, 
 
 1 * a > s 6 ? 8 s 10 3^ ^ & c< successively equal to p o : 
 the line o 2 will cut off half p I ; 
 o 3 will cut off one -third of P I ; 
 o 4 will cut off one -fourth of it. 
 o 5 one-fifth, and so on. Now, as 
 there is no limit to the number of 
 rig. r. equal parts that may be taken suc- 
 
 cessively on the parallel I, 10, so 
 
 there is no fraction so minute, that a more minute one may not be 
 cut from the line P I . 
 
 1 2. All bodies consist of component parts, similar in their 
 qualities to the whole, and into which, as will presently be shown, 
 they are separable or divisible. Thus, a mass of metal may be 
 reduced to powder, by filing, grinding, and a variety of other ex- 
 pedients ,- each particle of this powder will have the same qualities 
 as the entire mass of metal of which it constituted a part. 
 
 13. These component parts of bodies are observed to exist in 
 three distinct states of aggregation, which are distinguished in 
 mechanical science by the terms solid, liquid, and gaseous. 
 
 14. Solid state. A solitJ-body is one of which the component 
 parts cohere with such force, that it maintains its figure, unless 
 submitted to some action more or less violent, by which it will be 
 fractured, bruised, or otherwise changed in form. Thus, a solid 
 body laid upon a plane surface will rest upon it in any position, 
 without dropping asunder by the tendency of the weight of its 
 component parts to separate them. 
 
 Stones, metals, wood, are examples of solids. 
 
 15. Liquid state. A liquid' body is one of which the com- 
 ponent parts do not cohere with sufficient force to prevent their 
 separation by the mere influence of their weight. Thus, a mass of 
 
4 PROPERTIES OF MATTER. 
 
 liquid placed upon a plane will separate by the separate weights 
 of its particles, and will spread itself in a film, more or less thin, 
 over the surface of the plane. 
 
 If placed in a vessel within which it is confined by a bottom and 
 sides, it will flow into all the inferior parts of the vessel, and its 
 surface will become level ; for if any part of such surface were 
 more elevated than another, the particles forming such elevation 
 would fall down by their own weight to the level of the others. 
 
 1 6. Solids in a state of pulverization must not be confounded 
 with liquids, in the properties of which they do not participate. 
 Sand or dust consists of a great number of small solids, each of 
 which has the qualities of a solid as definitely as the largest mass 
 of rock. If such particles be examined with a microscope, they 
 will be found to have different forms, to be bounded by surfaces 
 and lines, and to maintain their figure in virtue of cohesion. 
 
 1 7. Gaseous state. A body in the gaseous or aeriform state 
 is one whose component particles not only are not held together by 
 mutual cohesion, but have towards each other a repulsion, in virtue 
 of which they will separate, so that the whole mass has a power of 
 expansion to which there is no known limit. 
 
 1 8. Atmospheric air is the most familiar example of this state of 
 body. If we suppose a quantity of air to be confined in a cylinder, 
 under a piston which moves air-tight, such piston being drawn 
 upwards, the air will not rest in the bottom of the cylinder, as the 
 same volume of liquid would do, leaving vacant the space above it, 
 but it will expand in virtue of the repulsive force already men- 
 tioned, which prevails among its particles, and it will fill the entire 
 space under the piston. To this expansion there is no practical 
 limit; such a piston may be indefinitely raised, producing an 
 indefinitely-increased space in the cylinder under it, and the air 
 will still continue to expand, so as to fill this space to whatever 
 extent it be increased. 
 
 19. Atmospheric air, as will be seen hereafter, is by no means 
 the only example of bodies in the gaseous form. Innumerable 
 varieties of such bodies are found existing in the material world, 
 and still greater varieties result from the experimental operations 
 of the natural philosopher and the chemist. 
 
 20. There are numerous bodies which may exist in any of these 
 three states of solid, liquid, or gas, according to certain physical 
 conditions, which will be explained hereafter ; and analogy renders 
 it probable that all bodies whatever may assume, under different 
 conditions, these several states. 
 
 2 1 . Water affords a familiar example of this : as ice it exists 
 in the solid state, as water in the liquid state, and as steam in the 
 gaseous state. 
 
IMPENETRABILITY AND DIVISIBILITY. 
 
 CHAP. H. 
 
 IMPENETRABILITY. DIVISIBILITY. 
 
 22. IMPENETRABILITY is the quality in virtue of which a body 
 occupies a certain space, to the exclusion of all other bodies. This 
 idea is so inseparable from matter, that some writers affirm that it 
 is nothing but matter itself; that is, that when we say that a body 
 is impenetrable, we merely say that it is a body. 
 
 However this be, the existence of this quality of impenetrability 
 is so evident as to admit of no other proof than an appeal to the 
 senses and the understanding. No one can conceive two globes of 
 lead, each a foot in diameter, to occupy precisely the same place at 
 the same time. Such a statement would imply an absurdity, 
 manifest to every understanding. 
 
 23. Even bodies in the gaseous form, the most attenuated state 
 in which matter can exist, possess this quality of impenetrability 
 as positively as the most hard and dense substances. 
 
 24. If we invert a common drinking-glass, and plunge its mouth 
 in water, the water will be excluded from the glass, in spite of the 
 pressure produced by the weight of the external water ; because 
 the air which filled the glass at the moment of immersion is still in 
 it, and its presence is incompatible with that of any other body. 
 It is true, however, that in this case the water will rise a little 
 above the mouth of the glass ; but this effect arises not from the 
 penetrability of the air, but from another quality, viz. its compres- 
 sibility, which we shall presently explain. 
 
 25. The numerous examples of apparent penetration which will 
 present themselves to all observers are only cases of displacement. 
 
 26. If we walk through the atmosphere, our bodies may be said 
 in one sense to penetrate it ; but they do so only in the same 
 manner as they would penetrate a liquid in passing through it. 
 They displace, as they move, as much air as is equal to their own 
 bulk. 
 
 27. If a cambric needle be plunged in a glass of water, it might 
 appear that penetration took place; but it is evident that the 
 needle displaces a quantity of water precisely equal to its own 
 bulk ; and if we had the means of measuring with the necessary 
 precision the position of the surface of the water in the glass, we 
 should find that on plunging the- needle in the liquid the surface 
 rises through a space corresponding precisely to a quantity of 
 water equal in volume to the bulk of the needle. 
 
 B 3 
 
6 PROPERTIES OF MATTER, 
 
 28. Divisibility unlimited. As all bodies consist of parts 
 which are similar in their qualities to the whole, a question arises 
 whether there is any limit to their subdivision or comminution. 
 Are there, in short, any ultimate particles at which the process 
 of division must cease ? or, to speak more correctly, are there 
 any ultimate particles so constituted that any further division 
 would resolve them into parts differing in quality from the entire 
 mass? 
 
 29. To make our meaning understood, let us take the example 
 of the most common of all substances, water. The discoveries 
 of modern chemistry have disclosed the fact that water is a com- 
 pound body, formed by the combination of two gases, called 
 oxygen and hydrogen. These gases, in their sensible properties 
 and appearance, are similar to atmospheric air, and have never 
 separately assumed the liquid form ; but, by certain means which 
 will be explained in a future part of this work, they may be 
 made to combine and coalesce, and when so combined they form 
 water. 
 
 Hence it is certain that the liquid, water, consists of ultimate 
 particles, or molecules, as they are sometimes called ; which mole- 
 cules are composed of particles of the two gases above mentioned 
 combined together. 
 
 30. Now, the question is, can water be practically so minutely 
 divided that its particles admit of no further subdivision, save and 
 except that which would resolve them into their constituent gases, 
 oxygen and hydrogen ? 
 
 To this it is answered, that, although the process of subdivision 
 may be carried to an extent which has no practical limit, yet that 
 by no process of art or science have we ever even approached to 
 those ultimate constituent atoms which admit of no other division 
 save decomposition. 
 
 3 1 . Nor is this unlimited divisibility and comminution peculiar 
 to water. It is common to all substances, whether solid, liquid, 
 or gaseous. They may all be reduced to particles of the most 
 unlimited minuteness, and yet each of these minute particles 
 will possess the same qualities as the largest mass of the same 
 substance. 
 
 32. Examples. As this quality of unlimited divisibility in- 
 volves conditions of the most profound interest, as well in the 
 sciences as in the arts, we shall offer here several examples in 
 illustration of it. 
 
 33. The most solid bodies are capable of unlimited comminu- 
 tion, by a variety of mechanical processes, such as cutting, filing, 
 pounding, grinding, &c. If a mass of marble be reduced to a fine 
 powder by the process of grinding, and this powder be then 
 
DIVISIBILITY. 7 
 
 purified by careful washing, its particles, if examined by a power- 
 ful microscope, will be found to consist of blocks having rhomboidal 
 forms, and angles as perfect and as accurate as the finest specimens 
 of calcareous spars. These rhomboids, minute as they are, may 
 be again broken and pulverised, and the particles into which 
 they are divided will still be rhomboids of the same form and 
 possessing the same character. The particles of such powder 
 being submitted to the most powerful microscopic instruments, 
 and the process of pulverization being pushed to the utmost 
 practical extreme, it is still found that the same forms are repro- 
 duced. 
 
 34. The polish of which the surfaces of certain bodies, such as 
 steel, the diamond, and other precious stones, are susceptible, is 
 an evidence at once of the limited sensibility of our organs, and 
 the unlimited divisibility of matter. This polish is produced, as 
 is well known, by the friction of emery powder or diamond dust, 
 and consequently each individual grain of such powder or dust 
 must leave a little trench or trace upon the surface submitted to 
 such friction. It is evident, therefore, that after this process has 
 been completed, the surface which presents to the senses such bril- 
 liant polish, and apparently infinite smoothness, is in reality 
 covered with protuberances and indentations, the height and 
 depth of which cannot be less than the diameter of the particles of 
 powder by which the polish has been produced. 
 
 35. In the detection of matter in a state of extreme comminu- 
 tion, the sense of sight is infinitely more delicate than that of 
 touch. If we rub a piece of gold upon a touchstone, we plainly 
 see the particles of matter which are left upon the surface of the 
 stone. The touch, however, cannot detect them. 
 
 36. In the preceding examples the comminution, however great, 
 cannot be easily submitted to actual measurement. Certain pro- 
 cesses, however, in the arts enable us to obtain exact numerical 
 estimates of a minute divisibility, which without them might ap- 
 pear incredible. If a thin tube of glass, being held before the 
 flame of a blow-pipe until the glass be softened and acquire a 
 white heat, be drawn end from end, a thread of glass may be ob- 
 tained so fine that its diameter will not exceed the two thousandth 
 part of an inch. This filament of glass will have all the fineness 
 and almost all the flexibility of silk, and yet a bore proportional 
 to that which passed through the original tube will still pass 
 through its centre. The presence of this bore may be rendered 
 manifest by passing a fluid through it. 
 
 37. It has been ingeniously conjectured, that if a filament of 
 this degree of fineness could be obtained of a material which would 
 retain sufficient inflexibility, it might be made to penetrate the 
 
 B 4. 
 
8 PROPERTIES OF MATTER. 
 
 flesh without producing either pain or injury, inasmuch as its 
 magnitude would be much less than that of the pores of the in- 
 teguments. 
 
 38. In the application of the telescope to astronomical purposes, 
 the distance between objects which are present at one and the 
 same time within the field of view of the instrument, is measured 
 by fine threads which are extended parallel to each other across 
 the field of view, and which may be moved towards and from each 
 other until they are made to pass through the objects between 
 which we desire to measure the distance. An experiment, then, 
 which determines the distance between these threads measures the 
 distance between the objects. 
 
 But these threads, being placed before the eye-glass of the tele- 
 scope, and therefore necessarily magnified in the same manner as 
 the objects themselves, would, unless such filaments were of an 
 extreme degree of tenuity, appear in the field of view like great 
 broad bands, and would conceal many of the objects which it might 
 be necessary to observe. It was therefore necessary to resort to 
 the use of filaments of extraordinary minuteness for this pur- 
 pose. The threads of spiders' webs are used with more or less 
 success ; but the late Dr. Wollaston invented a beautiful expe- 
 dient by which metallic threads of any degree of fineness might be 
 obtained. 
 
 Let us suppose a piece of platinum wire, one-hundredth of 
 an inch in diameter, a fineness easily obtainable by the process of 
 wire-drawing, to be extended along the axis of a cylindrical mould, 
 one-fifth of an inch in diameter, the wire being thus the twen- 
 tieth part of the diameter of the mould. Let the mould be then 
 filled with silver in a state of fusion. When this is cold we shall 
 have a cylinder of silver, having in its axis a thread of platinum 
 the twentieth part of its diameter. This compound cylinder is 
 then submitted to the common process of wire-drawing, during 
 which the platinum in its centre is drawn with the silver, the pro- 
 portion of their diameters being still maintained. When the wire 
 is drawn to the greatest degree of fineness practicable, a piece of 
 it is plunged in nitric acid, by which the surrounding silver is dis- 
 solved, and the platinum wire remains uncovered. 
 
 39. By this process Dr. Wollaston obtained platinum wire so 
 fine, that thirty thousand pieces, placed side by side in contact, 
 would not cover more than an inch. It would take one hundred 
 and fifty pieces of this wire bound together to form a thread as 
 thick as a filament of raw silk. Although platinum is the heaviest 
 of the known bodies, a mile of this wire would not weigh more 
 than a grain. Seven ounces of this wire would extend from Lon- 
 don to New York. 
 
DIVISIBILITY. 9 
 
 40. The natural filaments of wool, silk, and fur, afford striking 
 examples of the minute divisibility of organized matter. The fol- 
 lowing numbers show how many filaments of each of the annexed 
 substances placed in contact, side by side, would be necessary to 
 cover an inch : 
 
 Coarse wool ...... JQO 
 
 Fine Merino wool ..... 12,50 
 
 Silk Z 5 oo 
 
 The hairs of the finest furs, such as beaver and ermine, hold a 
 place between the filaments of Merino and silk, and the wools in 
 general have a fineness between that of Merino and coarse wool. 
 
 All these objects are sensible to the touch. 
 
 It will be remembered that they are compound textures, having 
 a particular structure, each containing very different elements, 
 which are prepared by the processes of nutrition and secretion. 
 
 41. The optical investigations of Newton disclosed some asto- 
 nishing examples of the minute divisibility of matter. A soap- 
 bubble as it floats in the light of the sun reflects to the eye an 
 endless variety of the most gorgeous tints of colour. Newton 
 showed, that to each of these tints corresponds a certain thickness 
 of the substance forming the bubble ; in fact, he showed in general, 
 that all transparent substances, when reduced to a certain degree 
 of tenuity, would reflect these colours. 
 
 Near the highest point of the bubble, just before it bursts, iu 
 always observed a spot which reflects no colour and appears black. 
 Newton showed that the thickness of the bubble at this black point 
 was the 2,5oooooth part of an inch ! Now, as the bubble at this 
 point possesses the properties of water as essentially as does the 
 Atlantic Ocean, it follows, that the ultimate molecules forming 
 water must have less dimensions than this thickness. 
 
 42. The same optical experiments were extended to the organic 
 world, and it was shown, that the wings of insects which reflect 
 beautiful tints resembling mother-of-pearl owe that quality to 
 their extreme tenuity. Some of these are so thin that 50000 
 placed one upon the other would not form a heap of more than a 
 quarter of an inch in height ! 
 
 43. In the process of gold-beating the metal is reduced to la- 
 minae or leaves of a degree of tenuity which would appear fabulous, 
 if we had not the stubborn evidence of common experience in 
 the arts as its verification. A pile of leaf-gold the height of an 
 inch would contain 282000 distinct leaves of metal! The thick- 
 ness, therefore, of each leaf is in this case the 282OOOth part of an 
 inch. Nevertheless, such a leaf completely conceals the object 
 which it is used to gild ; it moreover protects such object from the 
 action of external agents as effectually as though it were plated 
 with gold an inch thick. 
 
10 PROPERTIES OF MATTER. . 
 
 44. In the manufacture of embroidery, fine threads of silver 
 gilt are used. To produce these, a bar of silver, weighing 180 oz., 
 is gilt with an ounce of gold ; this bar is then wire-drawn until it 
 is reduced to a thread so fine that 3400 feet of it weigh less than 
 an ounce. It is then flattened by being submitted to a severe 
 pressure between rollers, in which process its length is increased 
 to 4000 feet. Each foot of the flattened wire weighs, therefore, 
 the 4<DOOth part of an ounce. But as in the processes of wire- 
 drawing and rolling the proportion of the two metals is maintained, 
 the gold which covers the surface of the fine thread thus produced 
 consists only of the I Both part of its whole weight. Therefore 
 the gold which covers one foot is only the yzooooth part of an 
 ounce, and consequently the gold which covers an inch will be the 
 8,64OOOOth part of an ounce. If this inch be again divided into 
 one hundred equal parts, each part will be distinctly visible with- 
 out the aid of a microscope, and yet the gold which covers such 
 visible part will be only the 864,ooooooth part of an ounce. 
 
 But we need not stop even here. This portion of the wire may 
 be viewed through a microscope which magnifies 500 times; and 
 by these means, therefore, its 5<DOth part will become visible. 
 
 45. In this manner, therefore, an ounce of gold may be divided 
 into 432000,000000 parts, and each part will still possess all the 
 characters and qualities found in the largest mass of the metal. 
 It will have the same solidity, texture, and colour, will resist the 
 same chemical agents, and will enter into combination with the 
 same substances. If this gilt wire be exposed to the action of 
 nitric acid, the silver within the coating will be dissolved, but the 
 hollow tube of gold which surrounds it would still cohere and 
 remain suspended. 
 
 46. The organic world affords most interesting and striking 
 examples of the minute divisibility of matter. None can be selected 
 more remarkable than that presented by the blood of animals, 
 which is not, as it appears to the naked eye, a uniform red liquid, 
 but consists of a transparent colourless fluid called lymph, in which 
 innumerable small red corpuscles of solid matter float. 
 
 47. In different species these red corpuscles differ both in form 
 and size. In the human blood, and in that generally of animals 
 who suckle their young, they are circular discs, their surfaces 
 being slightly concave, like the spectacle glasses used by short- 
 sighted persons. In birds, reptiles, and fishes, they are generally 
 oval, the oval being more or less elongated in different species. 
 The surface of the discs in these species, instead of being concave, 
 are convex, like the spectacle glasses used by weak-sighted per- 
 sons. The thickness of these discs varies from one third to one 
 quarter of their diameter. Their diameter in human blood is the 
 
DIVISIBILITY. i ! 
 
 35OOth part of an inch ; they are smallest in the blood of theNapu 
 musk-deer, where they measure only the 1 2OOOth of an inch. It 
 would require 50000 of these discs as they exist in the human 
 blood to cover the head of a small pin, and 800000 of the disks of 
 the blood of the musk-deer to cover the same surface. 
 
 48. It follows from these dimensions that in a drop of human 
 blood which would remain suspended from the point of a fine 
 needle there must be about 3,000000 of disks, and in a like drop 
 of the blood of the musk-deer there would be about 1 20,000000 , 
 yet these corpuscles are rendered not only distinctly visible to the 
 senses by the aid of the microscope, but their forms and dimen- 
 sions are rendered apparent. Small as they are, they are divisible, 
 and can be resolved into their elements by chemical agents. 
 
 49. But these diminutive globules are exceeded in minuteness 
 by innumerable creatures whose existence the microscope has dis- 
 closed, and whose entire bodies are inferior in magnitude to the 
 globules of blood. 
 
 Microscopic research has disclosed the existence of animals, a 
 million of which do not exceed the bulk of a grain of sand, and 
 yet each of these is composed of members as admirably suited to 
 their mode of life as those of the largest species. Their motions 
 display all the phenomena of vitality, sense, and instinct. In the 
 liquids which they inhabit they are observed to move with the 
 most surprising speed and agility ; nor are their motions and ac- 
 tions blind and fortuitous, but evidently governed by choice and 
 directed to an end. They use food and drink, by which they are 
 nourished, and must, therefore, be supplied with a digestive appa- 
 ratus. They exhibit a muscular power far exceeding in strength 
 and flexibility, relatively speaking, the larger species. They are 
 susceptible of the same appetites, and obnoxious to the same pas- 
 sions, as the superior animals, and, though differing in degree, the 
 satisfaction of these desires is attended with the same results as in 
 our own species. 
 
 Spallanzani observes that certain animalcules devour others so 
 voraciously that they fatten and become indolent and sluggish by 
 over-feeding. After a meal of this kind, if they be confined in 
 distilled water so as to be deprived of all food, their condition be- 
 comes reduced, they regain their spirit and activity, and once more 
 amuse themselves in pursuit of the more minute animals which are 
 supplied to them. These they swallow without depriving them of 
 life, as, by the aid of the microscope, the smaller, thus devoured, 
 have been observed moving within the body of the greater. 
 
 The microscopic researches of Ehrenberg have disclosed most 
 surprising examples of the minuteness of which organized matter 
 is susceptible. He has shown that many species of infusoria exist 
 
12 PROPERTIES OF MATTER. 
 
 which are so small that millions of them collected into one mass 
 would not exceed the bulk of a grain of sand, and a thousand 
 might swim side by side through the eye of a needle. The shells 
 of these creatures are found to exist fossilized in the strata of the 
 earth in quantities so great as almost to exceed the limits of 
 credibility. 
 
 By microscopic measurement it has been ascertained that in the 
 slate found at Bilin, in Bohemia, which consists almost entirely of 
 these shells, a cubic inch contains forty-one thousand millions ; 
 and as a cubic inch weighs two hundred and twenty grains, it 
 follows that one hundred and eighty seven millions of these shells 
 must go to a grain, each of which would consequently weigh the 
 1 87,ooooooth part of a grain. 
 
 All these phenomena lead to the conclusion that these creatures 
 must be supplied with an organization corresponding in beauty 
 with those of the larger species. 
 
 50. If a grain of salt be dissolved in 1000 grains of distilled 
 water, each grain of the water will contain the I oooth part of the 
 grain of the salt ; and if a grain of this water be mixed with I ooo 
 grains of distilled water, the I oooth part of a grain of salt which 
 it holds in solution will be uniformly diffused through the latter, 
 so that each grain of the latter solution will contain the millionth 
 part of a grain of salt. The presence of the salt in this second 
 solution can be detected by certain chemical tests. It is evident 
 that this process may be continued to a still greater extent. 
 
 51. A grain of sulphate of copper, dissolved in a gallon of 
 water, will impart to the whole mass of the liquid a plainly per- 
 ceptible tinge of blue ; and a grain of carmine will give its pecu- 
 liar red to the same quantity of water. It follows, therefore, that 
 a minute drop of such water will contain such a proportion of 
 either of these substances as the drop bears to the gallon. 
 
 52. The sense of smelling, although it does not inform us of the 
 mechanical qualities of minute masses of matter, determines, never- 
 theless, their presence thus, it is known that a grain of musk will 
 impregnate the atmosphere of a room with its odour for a quarter 
 of a century, or more, without suffering any considerable loss in 
 its weight. Every particle of the atmosphere which produces 
 the sense of the odour must contain a certain quantity of the 
 musk. 
 
 53. A thread of a spider's web, measuring four miles, will weigh 
 very little more than a single grain ! Every one is familiar with 
 the fact, that the spider spins a thread, or cord, by which his own 
 weight hangs suspended. It has been ascertained that this thread 
 is composed of about six thousand filaments. 
 
 54. The sense of taste, like that of smelling, may determine the 
 
DIVISIBILITY. 13 
 
 presence of matter, without manifesting, by direct evidence, any- 
 thing concerning its mechanical qualities. 
 
 55. A portion of strychnine, so minute as to be scarcely percep- 
 tible to the sight, if dissolved in a pint of water, will render every 
 drop of the water bitter. Now, it is evident that in this case, the 
 strychnine being uniformly diffused through the water, the minute 
 portion of it above mentioned is subdivided into as many parts as 
 there are drops of water in a pint. 
 
 56. In like manner, a single grain of the salt of silver, called 
 ammoniacal hyposulphite, will impart a flavour of sweetness to a 
 gallon of water. Now, a gallon of water will weigh about 70000 
 grains ; and as the flavour of .the salt is perceptible in each grain 
 of the water, it follows that one grain of this salt is thus divided 
 into 70000 equal parts. 
 
 57. A small lump of sugar, dissolved in a cup of tea measuring 
 half a pint, will sweeten the whole perceptibly. In this half-pint 
 of tea there are 3 1 ooo drops. Each drop, therefore, must contain 
 the 3 1 oooth part of the sugar dissolved, and each such drop is 
 perceptibly sweet. But if the point of a needle be inserted in one 
 of these drops, and withdrawn from it, a film of moisture will 
 remain upon it, and the drop will not be visibly diminished. Yet 
 this film of moisture will still be sweet, and will, therefore, contain 
 a fraction of the 3 1 oooth part of the lump of sugar, too minute 
 to admit of numerical estimation. 
 
 58. It may be asked, whether we are then to conclude, from 
 these various facts, that matter is infinitely divisible, and that there 
 are no original constituent atoms of determinate magnitude and 
 figure, at which all subdivision must cease. Such an inference, 
 however, would be unwarranted, even if we had no other means of 
 deciding the question except those of direct observation, as we 
 should thus impose those limits on the operations of nature which 
 she has imposed upon our powers of observing them. 
 
 59. Although we are unable, by direct observation, to perceive 
 the existence of molecules, or material atoms of determinate figure, 
 yet there are many observable phenomena which render their 
 existence in the highest degree probable, if not positively certain. 
 
 60. Crystallization. The most remarkable of such pheno- 
 mena are observed in the crystallization of salts. When salt is 
 dissolved in distilled water, as in the preceding example, the mix- 
 ture presents the appearance of a transparent liquid like water 
 itself, the salt altogether disappearing from sight and touch. The 
 presence of the salt in the water, however, can be established by 
 weighing the solution, which will be found to exceed the original 
 weight of the water by the exact amount of the weight of the salt 
 dissolved. 
 
H PROPERTIES OF MATTER. 
 
 Now, if this solution be heated to a sufficient temperature, the 
 water will gradually evaporate ; but this process of evaporation 
 not affecting the salt, the remaining water will still contain the 
 same quantity of salt in solution, and it will consequently become, 
 by degrees, a stronger and stronger saline solution, the water bear- 
 ing, consequently, a less and less proportion to the salt. The water 
 will at length be diminished, by evaporation, to that point, that a 
 sufficient quantity does not remain to hold in solution the entire 
 quantity of salt contained in it. When this; has taken place, each 
 particle of water which is evaporated leaving behind it the salt 
 which it held in solution, and this salt not being capable of being 
 dissolved by the water which remains, it will float in such water in 
 its solid and natural state, undissolved, just as particles of dust, or 
 other matter not soluble in the water, would do. But the saline 
 particles which thus remain floating in the liquid undissolved, will 
 not collect in irregular solid pieces, but will exhibit themselves in 
 regular figures, terminated by plane surfaces, always forming re- 
 gular angles, these figures being invariably the same for the same 
 species of salt, but different for different species. There are several 
 circumstances attending the formation of these crystals which 
 merit attention. 
 
 6 1 . If one of these be detached from the others, and the gra- 
 dual progress of its formation be submitted to observation, it will 
 be found to grow larger, always preserving its original figure. 
 Now, since its increase must be produced by the continual ac- 
 cession of saline molecules, disengaged by the water evaporated, 
 it follows that these molecules, or atoms, must have such a shape, 
 that, by attaching themselves successively to the crystal, they will 
 maintain the regularity of its .bounding planes, and preserve the 
 angles which these planes form with each other unvaried. In 
 fact, they must be so shaped, that the structure of the crystal they 
 form may be built up by their regular aggregation into the form 
 which it assumes. If one of these crystals be taken from the 
 liquid during the process of its formation, and be broken, so as to 
 destroy the regularity of its form, and then restored to the liquid, 
 it will be observed soon to recover its regular form, the atoms of 
 salt, successively dismissed by the evaporating water, filling up 
 the irregular cavities produced by the fracture. 
 
 62. Two consequences obviously follow from this phenomenon : 
 First, that the atoms of the salt dismissed by the water evapo- 
 rated have such a form, as enables them, by combination, to give 
 to the crystals the shape which they exhibit ; and, Secondly, that 
 the atoms which are successively attached to the crystals in the 
 process of formation, attach themselves in a particular position, to 
 explain which it is necessary to suppose that corresponding sides 
 
DIVISIBILITY. 1 5 
 
 of the crystals have attractions for each other, so that the atoms 
 of salt not only attach themselves to the sides of the crystals, but 
 place themselves there in a particular position. In a word, we 
 must suppose that the walls of the crystal are built with these 
 atoms in the same manner, and with the same regularity, as the 
 walls of a building are formed with bricks. 
 
 All these, and many similar details of the process of crystalli- 
 zation, are, therefore, very evident indications of a determined 
 figure in the ultimate atoms of the substances which are crystal- 
 lized. 
 
 63. There are certain planes called planes of cleavage, in the 
 direction of which natural crystals are easily divided. In sub- 
 stances of the same kind, these planes have always the same re- 
 lative position ; but they differ in different substances. 
 
 64. We must conclude, therefore, that these planes of cleavage 
 are parallel to the sides of the constituent atoms of the crystals, 
 and their directions therefore form so many conditions for the 
 determination of the shape of these atoms. 
 
 65. It follows, therefore, from these effects, and the reasoning 
 established upon them, that the substances which are susceptible 
 of crystallization consist of ultimate atoms of different figure. 
 Now, all solid bodies whatever are included in this class, for they 
 have severally been found in, or are reducible to a crystallized 
 form. Liquids crystallize in freezing : several of the gases have 
 been already reduced to the liquid and solid forms, and analysis 
 justifies the conclusion that all are capable of being reduced to 
 this form. 
 
 Hence it appears reasonable to presume that all bodies what 
 ever are composed of ultimate atoms, having determinate shape 
 and magnitude ; that the different qualities with which we find 
 different bodies endued, depend upon the shape and magnitude of 
 these atoms ; that these atoms cannot be disturbed or changed so 
 long as the body to which they belong is not decomposed into 
 other elements, as we find the qualities which depend on them 
 unchangeably the same under all the influences to which they have 
 been submitted. 
 
 66. We must conclude also that these atoms are so minute in 
 their magnitudes that they cannot be observed by any means 
 which human art has yet contrived, but nevertheless that such 
 magnitudes still have limits. 
 
 67. Matter indestructible. The extreme division to which 
 bodies are subjected in many natural and artificial processes, and 
 especially when exposed to the application of heat or fire, has 
 naturally suggested to minds not habituated to the rigid process 
 of scientific reasoning, the idea that bodies are destructible. The 
 
1 6 PROPERTIES OF MATTER. 
 
 ancients, instead of the modern practice of inhumation, disposed 
 of the bodies of their dead by burning them, upon the supposition 
 that their component parts were by such operation destroyed. 
 
 The more exact reasoning of modern philosophy, however, 
 teaches us that a power to destroy matter would be as incon- 
 ceivable in a finite agent as a power to create it. It is certain 
 that the quantity of matter which exists upon and in the earth 
 has never been diminished by the annihilation of a single atom. 
 Matter is in fact indestructible by any agency short of divine 
 power. It may be asked, then, what becomes of the matter com- 
 posing a body which, being subjected to the action of fire, gra- 
 dually and completely disappears. The answer is, that in this, as 
 well as in all other cases of the apparent destruction of matter, 
 nothing takes place except its subdivision and the change of its 
 form and position. 
 
 68. When a body is subjected to the action of heat, its elements 
 are decomposed, and its constituent particles separated, many of 
 them combine with other particles of matter, and form new sub- 
 stances possessing other qualities. Thus, when coal or other fuel 
 is burned, the carbon enters into combination with one of the con- 
 stituents of the atmosphere called oxygen, and forms a gaseous 
 substance called carbonic acid, which rises into and mixes with 
 the atmosphere. Another element, hydrogen, combines with the 
 same constituent of the atmosphere and forms vapor, which also 
 disperses in the atmosphere. 
 
 Sulphur, which is also occasionally present in fuel, combines 
 with the same constituent of the air, forming a gas called sul- 
 phureous acid, which also escapes into the atmosphere. Thus the 
 entire matter of the fuel, with the exception of a small portion 
 of incombustible matter which falls into the ash-pit, is dispersed 
 in the air, and no destruction or annihilation takes place. 
 
 That no portion of the matter of the fuel is destroyed or 
 annihilated can be established by the incontrovertible experi- 
 mental proofs of the chem'ist, for by the expedients of his science 
 all the products of the combustion which have been just men- 
 tioned can be preserved, weighed, and decomposed. The oxygen 
 which has entered into combination with each element of the 
 fuel can be reproduced, as well as the constituents of the fuel 
 itself, the latter of which being weighed, as well as the incom- 
 bustible ash, the weight of the whole is found to be precisely 
 equal to the weight of the fuel which was burned and apparently 
 destroyed. 
 
 69. Liquids when subjected to heat are converted into vapor, 
 and this vapor disperses in the atmosphere, so that the liquid 
 seems to be boiled away ; but if the vapor be preserved, as it may 
 
DIVISIBILITY. 17 
 
 be in a separate vessel, and exposed to cold, it will return to the 
 liquid form, and its weight and measure will be found to be pre- 
 cisely the same as that of the liquid evaporated. 
 
 70. Destructive distillation. There is a process in chemistry 
 which is called destructive distillation. The term is objectionable 
 because it implies a destruction where no destruction takes place. 
 If a piece of wood, being previously weighed, be placed in a close 
 retort and submitted to what is called destructive distillation, it 
 will be found that water, a certain acid, and several gases will 
 issue from it, all of which may be preserved, and mere charcoal 
 will remain in the retort at the end of the process. If the 
 water, acid, and gases which thus escape be weighed with the 
 charcoal, the weight of the whole will be found to be precisely 
 equal to that of the wood which was subjected to destructive dis- 
 tillation. 
 
 71. General conclusion. Thus various forms of matter may 
 be fused, evaporated, or submitted to combustion; animals and 
 vegetables may die, organized bodies may be dissolved and de- 
 composed, but in all cases their elementary and constituent parts 
 maintain their existence. The remains of our own bodies after 
 death are deposited in the grave, and enter into innumerable 
 combinations with the materials of the soil, with the vegetation 
 which covers it, and the air which circulates above it. 
 
 Consequently, these parts enter into an infinite series of other 
 combinations, forming parts of other organized bodies, animal 
 and vegetable, and which, after having discharged their functions, 
 are thrown off again, mixing with the soil, the air, or organized 
 matter, and once more running through the round of physical 
 combinations. 
 
 The constituent atoms of matter are thus constantly performing 
 a circle of duties in the economy of nature with infinitely more 
 certainty and regularity than is observed in the most disciplined 
 army or in the best regulated manufactory. 
 
 CHAP, m, 
 
 POROSITY, DENSITY, COMPRESSIBILITY, AND DILATABILITY. 
 
 72. THE volume or magnitude of a body is, as has been already 
 explained, the space which is included within its external sur- 
 faces. The mass of a body, or the quantity of matter of which it 
 consists, is the collection of atoms or molecules which compose it. 
 
 c 
 
18 PROPERTIES OF MATTER. 
 
 If these atoms were in actual contact, the volume would be com- 
 pletely occupied by the mass ; but numerous results of observa- 
 tion prove that this is never the case. There is no body so 
 solid or compact that it cannot, by various processes to be ex- 
 plained hereafter, be forced into less dimensions. Now, if its 
 atoms were in absolute contact, and had no unoccupied spaces 
 between them, this compression could not take place. It follows, 
 therefore, that between the atoms or molecules which form the mass 
 of a body there are vacant spaces in the magnitude or volume, 
 which therefore consists partly of the spaces occupied by the atoms, 
 and partly of the spaces which intervene between these atoms. 
 
 73. Pores. These intervening spaces which separate the con- 
 stituent atoms of a mass of matter are called pores ; and the 
 quality of bodies in virtue of which their constituent atoms are 
 thus separated by vacant spaces is called Porosity. 
 
 74. Density. In bodies of different species, the pores bear a 
 greater or less proportion to the whole volume ; or, in other words, 
 the component atoms of the mass are placed more or less closely 
 together. This circumstance determines what is called the density 
 of bodies. 
 
 75. Porosity and density are therefore correlative terms. The 
 greater the porosity, the less the density ; and the greater the 
 density, the less the porosity. Gold and platinum are bodies of 
 great density ; cork is one of much less density ; and air or the 
 gases still less. When one body is heavier than another under 
 the same bulk, it is concluded that its density is proportionally 
 greater than that of the other, and consequently its porosity pro- 
 portionally less. 
 
 76. When the pores are uniformly diffused throughout the 
 dimensions of a body, the body is said to be uniformly dense. 
 
 77. The porosity of bodies is sometimes illustrated and ex- 
 plained by a sponge, which allows the cavities which pervade it to 
 be filled with water or other fluid ; but such an illustration is not 
 strictly apposite. The cavities of a sponge are not its pores, any 
 more than are the cells of a honeycomb the pores of wax. Cel- 
 lular structures in general present peculiar modifications of matter 
 totally different from what is understood by porosity. 
 
 78. The pores of a body are sometimes filled with another ma- 
 terial substance of a more subtle nature. Thus, if the pores of a 
 body be greater than the atoms of air, such body being surrounded 
 by the atmosphere, the air will pervade its pores. This is found 
 to be the case, for example, with certain sorts of wood, with chalk, 
 sugar, and many other substances. If a piece of such wood, chalk, 
 or sugar be pressed to the bottom of a vessel filled with water, 
 the air wnich fills the pores will b observed to escape in bubbles, 
 
POROSITY DENSITY. ig 
 
 and to rise to the surface, the water pervading the pores, and 
 taking their place. 
 
 79. Examples. The following examples will illustrate the 
 qualities of porosity and density : 
 
 80. If a cylinder of wood be inserted in the bottom of a cup 
 which is attached to the mouth of a glass receiver placed upon the 
 plate of an air-pump, the cup thus placed being filled with mer- 
 cury, on withdrawing the air from the interior of the reservoir 
 the pressure of the external atmosphere will force the mercury 
 through the pores of the wood, arid it will be observed to fall in a 
 shower of silver within the receiver. 
 
 8 1 . Wood in general is lighter, bulk for bulk, than water, and 
 will therefore float in it ; but this comparative lightness is in some 
 cases not a property of the wood, but of the air which fills its 
 pores. To prove this, let a piece of such wood be held beneath 
 the surface of water contained in a vessel placed under the re- 
 ceiver of an air-pump. On exhausting the receiver, the air con- 
 tained in the piece of wood will force its way out by reason of its 
 elasticity, and will rise in bubbles to the surface of the water. If, 
 when the air has been thus expelled from the pores, the pressure 
 of the atmosphere be made to act again upon the surface of the 
 water by opening the cock which admits air to the receiver, the 
 water will be forced into the pores of the wood and will fill the 
 spaces deserted by the air, and the wood will then sink to the 
 bottom. 
 
 82. There is no substance 'so dense as to be divested of pores. 
 The celebrated Florentine experiment, performed at the Academia 
 Del Cimento in 1 66 1, and often repeated since that time with the 
 same result, showed that gold itself has pores sufficiently large to 
 admit the particles of water to pass through them. A globe of 
 gold, being completely filled with water, was closed by a screw, 
 and submitted to a severe pressure. As a globe is the figure 
 which within the same surface contains the greatest possible 
 volume, any change produced in its figure by external pressure 
 must necessarily diminish its volume. When the globe, therefore, 
 thus filled with water, is submitted to a pressure which changes it 
 to a form slightly elliptical, or turnip-shaped, it would necessarily 
 contain less liquid, and either of two effects must ensue, viz., the 
 globe must burst, or a portion of the liquid must force its passage 
 through the pores of the gold. The latter effect ensued ; and as 
 the globe changed its form, the water was seen collecting in a dew 
 on the external surface of the metal. This proved that the par- 
 ticles of water found their way through the pores of the gold 
 without tearing, rupturing, or otherwise doing violence to its 
 general structure. 
 
20 PROPERTIES OF MATTER. 
 
 83. The process of filtration, so extensively used in the arts and 
 sciences, depends on the quality of porosity. The substance 
 through which a liquid is filtrated has pores large enough to allow 
 the particles of the liquid to pass, but too small to permit the pas- 
 sage of the foreign matter suspended in the liquid, and of which 
 it is intended to purify the liquid by the process of filtration. 
 The most ordinary filters are soft stone, paper, and charcoal. 
 
 84. Animal and vegetable petrifactions furnish' striking ex- 
 amples of porosity, since the stony substance which petrifies them 
 must have been infiltrated through their mass, so as to penetrate 
 all their fibres. 
 
 85. Mineral substances are all more or less porous. Opaque 
 stones are in general more porous than transparent ones. Chalk 
 and marble are formed of the same constituents, with different 
 degrees of porosity. If water be poured on chalk, it is instantly 
 observed passing into its innermost pores ; if it be poured on 
 marble, it rests on the surface without penetrating. Stones, how- 
 ever, which resist the admission of water to their pores under 
 ordinary circumstances, will be penetrated by the liquid provided 
 an intense pressure be used for a sufficient length of time. Thus, 
 stones taken from the bottom of the sea, especially if the depth be 
 considerable, are found penetrated by water to their very centre. 
 Among siliceous stones, such as agate and flint, there is one called 
 Hydrophane, whose porosity is attended by a singular phenomenon. 
 A piece of this substance, in its common state, is nearly opaque ; 
 but if it be plunged in water, it is 'found, on withdrawing it from 
 the liquid, to be nearly as transparent as glass. In this case the 
 water penetrates the stone exactly as oil penetrates paper; bubbles 
 of air are disengaged from its pores, which are filled with the water 
 absorbed, the presence of which gives the transparency. 
 
 86. Large mineral masses existing naturally in the strata of the 
 earth present examples of porosity still more striking. Water 
 percolates through the sides and surfaces of caverns and grottos, 
 and, being impregnated with calcareous and other earths, forms 
 stalactites, or pendulous protuberances, presenting curious appear- 
 ances, with which every one is familiar. 
 
 87. Compression. The quality in virtue of which a body allows 
 its volume to be diminished without diminishing its quantity of 
 matter is called compressibility, when the effect is produced by the 
 application of external mechanical force ; and contractibility when 
 produced by change of temperature, or any other agency not me- 
 chanical. When the volume of a body is diminished, whether by 
 compression or contraction, its constituent atoms are brought into 
 closer contiguity, its pores are consequently diminished, and its 
 density proportionally increased. 
 
COMPRESSIBILITY. 2 1 
 
 88. All known bodies, whatever be their nature, are capable of 
 having their dimensions reduced without diminishing their mass ; 
 and this is one of the most conclusive proofs that all bodies are 
 porous. 
 
 89. It is evident in general that the more porous a body is the 
 more easy is its compression. This truth is manifested by innu- 
 merable examples derived from organized bodies, especially those 
 of a fibrous texture. All those whose porosity is such as to allow 
 them to be easily penetrated by fluids can be diminished by the 
 application of pressure ; and in this case, if they have been previ- 
 ously filled with fluids, these fluids are expelled by the pressure 
 exactly as water is squeezed from a piece of sponge. Innumerable 
 processes in the arts supply examples of this. 
 
 90. Wood of even the hardest kind, in its natural state, is so 
 porous as to absorb both air and water in considerable quantities. 
 When such wood is used in the arts, in cases where extreme hard- 
 ness is required, it is previously submitted to severe pressure, by 
 which the fluids absorbed are expelled from the pores, the volume 
 diminished, and the density increased. The wooden wedges used 
 in fastening the rails of railways in their chairs are prepared in 
 this manner. 
 
 9 1 . Even the most solid stone, when loaded with a considerable 
 weight, is found to be compressed. The foundations of buildings, 
 and the columns which sustain incumbent weights in architecture, 
 supply numerous proofs of this. 
 
 92. Malleable metals are compressed by percussion or hammer- 
 ing : they become thus more compact and dense. In the process 
 of coining, medals and pieces of money are struck by a severe 
 pressure, by which they are made to receive the impression and 
 characters upon them more accurately than softened wax would 
 from the pressure of the hand. Under the blow of the press they 
 not only change their form, accommodating themselves to the 
 characters and figures sunk upon the die, but they are at the same 
 time compressed and rendered more dense, so that the coin or 
 medal has a volume sensibly less than the blank piece had before 
 it was struck. 
 
 93. Liquids in general are less easily compressed than solids; 
 so much so, that in practical science they are regarded as incom- 
 pressible. 
 
 They are, however, strictly speaking, capable of a slight com- 
 pression under the operation of considerable mechanical force. 
 
 In the year 1761, the compressibility of water and other liquids 
 was established. It was found that water, submitted to a me- 
 chanical pressure, amounting to fifteen pounds on a square inch, 
 would be diminished in its volume by forty-five parts in a million, 
 
22 PROPERTIES OF MATTER. 
 
 that is to say, a million of cubic inches would be reduced to about 
 forty-five cubic inches less. 
 
 In more recent experiments, a quantity of water was enclosed 
 in a piece of cannon and submitted to a mechanical pressure 
 amounting to fifteen thousand pounds per square inch. Under 
 this pressure it was diminished by one-twentieth of its volume, and 
 the cannon enclosing it was burst. 
 
 94. But if liquids are so little compressible, they are, in a very 
 high degree, susceptible of contraction. 
 
 If a quantity of water coloured with ink or other colouring 
 matter be included in a glass bulb connected with a tube of small 
 bore, it will be found, that when the bulb is exposed to cold, the 
 level of the coloured water in the tube will descend. This is an 
 effect of the contraction which the liquid undergoes in consequence 
 of its diminution of temperature. This , contraction by cold is an 
 universal quality of matter, which will be explained more fully in 
 a subsequent part of this work. 
 
 95. Compression of gases. Of all forms of matter the gases 
 are the most susceptible of compression. This quality has already 
 been briefly noticed. There appears to be no practical limit to the 
 compression of which this form of matter is susceptible, its volume 
 being diminished in the exact proportion of the compressing force 
 applied to it. 
 
 96. Elasticity. This is the quality in virtue of which a body, 
 after having been compressed, recovers its former dimensions, on 
 being relieved from the force which compresses it. Bodies which 
 retain their compressed state after the force ceases to act, and do 
 not resume their original dimensions, are said to be inelastic. 
 
 97. The class of bodies which affords the most striking examples 
 of elasticity are the gases and aeriform bodies. If a quantity of 
 air be included in a syringe under a piston, and be compressed by 
 a force applied to the piston, on the removal of that force the air, 
 by virtue of its elasticity, will force the piston upwards until it 
 resumes the position from which it had been driven by the com- 
 pressing force. 
 
 98. All liquids, when compressed, immediately recover their 
 original dimensions when relieved from the compressing force, and 
 therefore may be said to be perfectly elastic. The play of the 
 compressive and elastic principle, however, in the case of liquids, 
 is so extremely limited, that for all practical purposes this form of 
 body is treated as both incompressible and inelastic. All the 
 theorems of those parts of physical science called Hydrostatics, 
 Hydrodynamics, Hydraulics, &c., are based upon the principle that 
 liquids are incompressible and inelastic ; for although it be true, as 
 has been stated, that within certain very minute limits they are 
 
ELASTICITY 23 
 
 both compressible and elastic, yet these limits are so small as to 
 produce no appreciable effects under ordinary circumstances. 
 
 99. Gaseous bodies are not only compressible and elastic without 
 any practical limit, but also endued with unlimited dilatability. 
 Thus, if a quantity of gas be included in any given volume, and 
 that this volume be augmented in any required proportion, the 
 gas will spontaneously, and without the application of any external 
 agency, dilate itself so as to fill the augmented volume, and this 
 expansion will go on, no matter to what extent the volume be 
 augmented. 
 
 100. The quality of elasticity is manifested in solid bodies, but 
 in a less decided manner than in gases. Caoutchouc, or elastic 
 gum, is perhaps of all bodies that which has most elasticity. This 
 quality, combined with the methods recently discovered of varying 
 the form of this substance, has extended considerably the appli- 
 cation of it to the useful purposes of life. 
 
 101. Examples. The following examples will illustrate the 
 quality of elasticity as found in solid bodies. 
 
 1 02. If a flat and hard surface be smeared with a thin coating 
 of oil, and an ivory ball be allowed to drop upon it, the ball will 
 rebound by reason of its elasticity. On examining that part of 
 the surface of the ball which struck the flat surface from which it 
 rebounded, it will be found that a somewhat extensive circular 
 space will have been stained with the oil. If the ball be brought 
 gently into contact with the flat surface, a minute space only 
 would be stained with the oil. Why, then, it may be asked, did a 
 larger space receive a stain when the ball was allowed to drop 
 with a certain force upon the surface ? The answer to this is, 
 that the force of the impact flattens the surface of the ball to a 
 certain extent ; that, in virtue of its elasticity, the ball recovers its 
 spherical figure ; and that the force with which it recovers this 
 figure causes the rebound. The extent of the surface stained by 
 the oil is a little, but not much greater than the extent of the circle 
 flattened by the impact. 
 
 If the ball be let fall from several different heights, it will be 
 found that the circular space stained by the oil will be greater, the 
 more elevated the point from which the ball is allowed to depart. 
 This effect is only what might have been anticipated ; the greater 
 the height from which the ball falls, the greater will be the force 
 of the impact, and consequently the greater will be the extent 
 over which its surface will be flattened, and the greater, conse- 
 quently, will be the elastic force which produces the rebound. 
 
 103. If such an experiment be made with a ball composed of a 
 substance softer than ivory, and equally elastic, the flattening may 
 be rendered directly perceptible to the senses. This may be made 
 
 04 
 
2>j. PROPERTIES OF MATTER. 
 
 evident by the large caoutchouc balls inflated with air, used in the 
 plays of children. When they strike the ground, they are flattened 
 at the surface over a circle of very considerable magnitude, and 
 which flattening may be exhibited by pressing them on the ground 
 by the force cf the hand. This is only an exaggeration of what 
 would actually take place in the case of a ball of ivory or glass. 
 
 104. Elasticity in bodies is sometimes manifested by their dis- 
 position to recover their form when disturbed by a force which 
 does not affect their volume. For example, a plate of steel when 
 bent would have the same dimensions which it had before the 
 pressure, yet its elasticity will be rendered apparent by its imme- 
 diately recovering its original form after the force which bends it 
 had ceased to act. The play of springs of every form affords 
 examples of this. When a straight bar of steel is bent into a 
 curve, both compression and expansion of its molecules take place. 
 The molecules which compose that side which becomes convex are 
 forcibly drawn asunder, and those which form the surface which 
 becomes concave are forcibly compressed. This is evident, inas- 
 much as the convex side becomes longer, and the concave shorter^ 
 by the change of form. The tendency of the molecules, by virtue 
 of their elasticity, to recover their original position, causes those 
 on the convex surface to contract, and those on the concave sur- 
 face to expand, the combined effects of such contraction and ex- 
 pansion being the restoration of the bar to its original form. 
 
 105. Iiimits of the elastic force. As elasticity results from 
 a derangement of the component molecules of bodies, it will be 
 easily understood that there must be limits, beyond which such 
 derangements cannot be produced without a permanent change in 
 the form of the body ; and there are consequently limits to the 
 play of the elastic principle. These limits will be obviously different 
 in different bodies. 
 
 In the case of the most elastic class of bodies, such for example 
 as caoutchouc, these limits are very extensive. In ivory they are 
 more extensive than glass, -for ivory will recover its figure after a 
 compression which would cause the fracture of glass. These 
 limits are narrow in the case of such metals as lead ; for although 
 considerable compression will not cause the fracture of lead as it 
 would that of glass, yet the derangement which such compression 
 produces amongst the molecules of that metal is greater than their 
 feeble elasticity can resist, and the metal accordingly takes perma- 
 nently any form given to it. 
 
 1 06. Torsion. Elasticity is sometimes manifested by torsion 
 or twisting. Thus, let us suppose a filament of raw silk stretched 
 by a weight attached to it. If this weight be made to revolve 
 several times in the same direction, so as to twist the silk and then 
 
DIL AT ABILITY. 2$ 
 
 be disengaged, the fibre of silk in virtue of its elasticity will un- 
 twine itself, causing the weight to revolve in a contrary direction ; 
 and this process of untwining will continue ufltil the filament re 
 covers its original position ; but the twisting may have been con- 
 tinued to such an extreme as to exceed the limits of the elasticity 
 of the silk ; and in that case a permanent derangement of the 
 molecules of the silk will take place, and it will not recover its 
 original form. 
 
 The same effects would ensue if the weight had been suspended 
 to a fine wire of copper, silver, or any other metal, but the limit 
 at which the twisting would produce a permanent derangement of 
 form, or, in other words, the limit of play of the elastic principle, 
 would be different. 
 
 107. Dilatability. When the extension or augmentation 
 of the volume of a body is produced by any physical agency, 
 such, for example, as heat, not coming under the denomination 
 of mechanical force, it is called dilatation. All bodies whatever, 
 when submitted to the action of heat, are susceptible of having 
 their dimensions enlarged ; and to this augmentation of magnitude 
 or dilatation by increase of temperature there is no practical 
 limit. 
 
 Innumerable examples of the operation of this principle in the 
 arts and sciences may be produced. 
 
 1 08. In the thermometer the dilatation of a liquid is used as the 
 measure of the degree of heat which produces it. This instru- 
 ment consists of a glass bulb attached to a tube of small bore. 
 The bulb and part of the tube are filled with a liquid. As the 
 temperature to which the instrument is exposed is increased or 
 diminished, the liquid affected by it expands or contracts in a 
 much greater degree than does the glass in which the liquid is 
 contained. The consequence of this is, that in order to find room 
 for its increased volume, a portion of the liquid in the bulb is 
 forced into the tube. The column in the tube consequently be- 
 comes longer, and its increase of length, measured by a scale 
 attached to the tube, becomes a measure of the increased tem- 
 perature. 
 
 109. The dilatation and contraction of metal consequent upon 
 change of temperature has been applied some time ago in Paris to 
 restore the walls of a tottering building to their proper position. 
 In the Conservatoire des Arts et Metiers, the walls of a part of the 
 building were forced out of the perpendicular by the weight of the 
 roof, so that each wall was leaning outwards. M. Molard con- 
 ceived the notion of applying the irresistible force with which 
 metals contract in cooling to draw the walls together. Bars of 
 iron were placed in parallel directions across the building, and at 
 
26 PROPERTIES OF MATTER 
 
 right angles to the direction of the walls. Being passed through 
 the walls, nuts were screwed on their ends outside the building. 
 Every alternate bdr was then heated by lamps, and the nuts 
 screwed close to the walls. The bars were then cooled ; and the 
 lengths being diminished by contraction, the nuts on their ex- 
 tremities were drawn together, and with them the walls were 
 drawn through an equal space. The same process was repeated 
 with the intermediate bars, and so on alternately, until the walls 
 were brought into a perpendicular position. 
 
 IIO. General effects of dilatation and contraction. Since 
 there is a continual change of temperature in all bodies on the 
 surface of the globe, it follows that there is also a continual change 
 of magnitude. The substances which surround us are constantly 
 swelling and contracting under the vicissitudes of heat and cold. 
 They grow smaller in winter, and dilate in summer. They swell 
 their bulk on a warm day, and contract it on a cold one. These 
 curious phenomena are not noticed only because our ordinary 
 means of observation are not sufficiently accurate to appreciate 
 them. Nevertheless, in some familiar instances, the effect is very 
 obvious. In warm weather the flesh swells, the vessels appear 
 filled, the hand is plump, and the skin distended. In cold weather, 
 when the body has been exposed to the open air, the flesh appears 
 to contract, the vessels shrink, and the skin shrivels. 
 
 CHAP. IV. 
 
 INERTIA. 
 
 III. All matter inert. The quality of matter which stands 
 foremost in importance in all mechanical inquiries, forming the 
 basis of the whole theory of force and motion, is inactivity or 
 inertia; and important as this quality is, there is perhaps nothing 
 which has given rise to so many erroneous conceptions. 
 
 These errors have chiefly arisen from the adoption of a vicious 
 phraseology on the part of many writers on Natural Philosophy 
 
 112. Inertia. At any given moment of time a body, mechani- 
 cally considered, must be in one or other of two states, rest or 
 motion. Inertia or inactivity is the total absence of all power in 
 the body to change its state. If the body be at rest, it cannot put 
 itself in motion ; if the body be in motion, it can neither change 
 that motion nor reduce itself to rest. Any such change must be 
 produced from some external cause independent of the body. 
 
INERTIA 27 
 
 113. The phrase vis inertia, or force of resistance, used in many 
 treatises on Natural Philosophy, has been a fertile source of error. 
 Such a phrase implies a disposition in matter to resist being put 
 in motion when at rest. Now no such disposition is found to 
 exist ; and if it did exist, it would be as utterly incompatible with 
 the quality of inactivity as is the power to produce spontaneous 
 motion. 
 
 114. Innumerable effects which fall daily under our observation 
 prove to us the inability of mere matter when at rest to put itself 
 in motion, or when in motion to augment its speed ; but, on the 
 other hand, we have not the same direct and manifest evidence of 
 its inability to destroy or diminish any motion which it may have 
 received ; and it happens, therefore, that while few will deny to 
 matter the former effect of inertia, many will at first doubt or fail 
 to comprehend the latter. 
 
 Philosophers themselves, so late as the epoch signalised by the 
 writings of Bacon, held it as a maxim that matter is more inclined 
 to rest than to motion ; and this being so, we cannot be surprised 
 to find those who have not been familiar with physical science still 
 slow to believe that a body once put in motion would continue for 
 ever to move in the same direction and with the same speed, un- 
 less stopped by some external cause. 
 
 But a careful examination of the circumstances which affect 
 the movement of the bodies around us with which we are most 
 familiar will soon convince us, that in every case in which we ob- 
 serve the motion of those bodies gradually diminished, or entirely 
 destroyed, such effects arise, not, as has been erroneously supposed, 
 from any natural disposition of the bodies themselves to be re- 
 tarded or brought to rest, but from the operation of causes of 
 which there is no difficulty in rendering an account. 
 
 In some of the modern popular works on Natural Philosophy, a 
 trifling experiment is mentioned as an example of the effect of 
 inertia, and explained on principles somewhat erroneous. A card 
 being placed on the top of the finger, and a coin placed on the 
 card, a sudden blow being given with the back of the nail to the 
 edge of the card, it will be projected from its place between the 
 coin and the finger, the coin remaining unmoved on the finger. 
 
 This has been explained by stating that the inertia of the coin 
 is comparatively so great, that the friction produced between it 
 and the card is insufficient to move it from its place. 
 
 If by these words it be understood that the coin resists the force 
 exerted upon it by means of the friction, it is erroneous, and would 
 be incompatible with that quality of inertia to which the effect is 
 ascribed. 
 
 The correct explanation of the experiment is as follows. A part 
 
28 , PROPERTIES OF MATTER. 
 
 of the momentum given to the card by the blow is communicated 
 to the coin in consequence of the resistance to the motion of the 
 card produced by the friction which takes place between it and 
 the coin. But the coin contains comparatively so much matter, 
 that this moving force, when distributed among its component 
 particles, which it will necessarily be, will give to the whole coin a 
 velocity in the direction of the motion of the card incomparably 
 smaller than that of the card, and so small that the resultant of the 
 forces produced by the weight of the coin upon the finger and this 
 force scarcely deviates from the direction of the weight of the 
 coin : consequently, although the coin remains on the finger, it 
 does not remain precisely in the same position over the finger 
 which it had when it rested on the card ; its position will be 
 changed in a slight degree in the direction of the motion of the 
 card. 
 
 115. When a stone is rolled along the surface of the ground, 
 the inequalities of its form, as well as those of the ground on which 
 it moves, present impediments which gradually retard its move- 
 ment, and soon bring it to rest. Render the stone round and 
 smooth, and the ground level, and the motion will be considerably 
 prolonged ; a much longer interval will elapse, and a much greater 
 space will be traversed, before it will come to rest. But aspe- 
 rities more or less considerable will still remain on the surface of 
 the stone, and on the surface of the ground. Substitute for it a 
 ball of highly polished metal, moving on a highly polished steel 
 plane truly level, and then the motion will continue for a very 
 long time. 
 
 But, even in this case, asperities will remain on the surface of 
 the moving body, as well as on the surface on which it moves, 
 which will gradually destroy the motion, and ultimately bring it 
 to rest. 
 
 But, independently of the obstructions to the motion of bodies 
 arising from the friction of the surfaces which move in contact 
 with each other, all motions which take place on or near the sur- 
 face of the earth are necessarily made in the fluid medium of the 
 atmosphere. This fluid, however attenuated, still offers consider- 
 able resistance to the motion of bodies through it. An extensive 
 flat surface spread at right angles to the direction of the motion 
 will thus meet a powerful resistance. This resistance arises from 
 the body moved being compelled to push out of its way a volume 
 of air proportional to the extent of the surface which the body 
 presents in the direction of the motion. If on a calm day an open 
 umbrella be carried with its concave surface presented in the 
 direction in which we are moving, a powerful resistance will be 
 encountered, which will increase with every increase of speed. 
 
INERTIA. 29 
 
 1 1 6. Proofs of inertia. As these causes of resistance to the 
 motion of bodies are everywhere present on and near the surface 
 of the earth, we are unable, by direct experiment, to establish the 
 proposition that a body when once put in motion would continue 
 for ever to move in the same direction, and with the same speed, 
 if undisturbed ; but astronomical observations supply an immense 
 mass of evidence to established this principle. In the heavens we 
 find a vast apparatus, every movement of every part of which 
 establishes incontrovertibly the inertia of matter, inasmuch as the 
 reasoning by which all these motions are explained, and by which 
 all these phenomena are predicted, is based upon the fundamental 
 principle of the complete inertia of matter. The celestial bodies, 
 removed from all the casual obstructions and resistances on the 
 surface of the globe which disturb our reasoning, roll on in their 
 appointed paths with unerring regularity, preserving undirninished 
 all that motion which they received at their creation from the 
 hand which launched them into space. These phenomena alone, 
 unsupported by other reasoning, would be sufficient to establish 
 the quality of inertia ; but, viewed in connection with the other 
 circumstances already mentioned, and with the whole superstruc- 
 ture of mechanical science, leading to innumerable truths verified 
 by daily and hourly experience, no doubt can remain that this 
 important principle is a universal law of nature. 
 
 117. Examples. The following examples will illustrate the 
 quality of inertia : 
 
 1 1 8. If a horse or a vehicle of any kind moving with consider- 
 able speed be suddenly stopped by any cause which does not at the 
 same time affect the rider or those who are transported by the 
 vehicle, then the body of the rider, or those who are transported, 
 still retaining the progressive motion of which the horse or vehicle 
 is suddenly deprived, will be projected forwards; and unless some 
 means of resistance be adopted, the rider will be thrown over the 
 head of the horse, and the passengers thrown forwards from the 
 vehicle. 
 
 In the same manner, if a horse or vehicle being at rest be sud- 
 denly started forwards with considerable speed, the rider, or the 
 persons placed upon the vehicle, nofc being as suddenly affected by 
 the same forward motion will be thrown backwards. 
 
 In both these cases the effects are the consequence of the quality 
 of inertia. In the one case they manifest the tendency of the 
 bodies to continue the motion they have already received, and in 
 the other they manifest the disposition of the same bodies to con- 
 tinue at rest. 
 
 119. If a passenger in a carriage which moves with considerable 
 speed leap to the ground, he will fall in the direction in which the 
 
3 o PROPERTIES OF MATTER. 
 
 carriage is moving ; for in descending to the ground his entire 
 body will still retain all the progressive motion which it had in 
 common with the carriage. When his feet touch the ground, they 
 and they alone will be suddenly deprived of this progressive mo- 
 tion, which being retained by the remainder of his body, he will 
 fall as if he were tripped by some object impeding his motion, 
 in the direction of the carriage. 
 
 1 20. The sport of coursing presents many amusing and instruc- 
 tive examples of the force of inertia. From the movements of the 
 hare, one might suppose that it is, indeed, an expert mechanical 
 philosopher. The hound which pursues it being a comparatively 
 heavy body, and moving at the same or a greater speed, cannot 
 suddenly arrest its course, because, in virtue of its inertia, it has a 
 tendency to proceed forward in the same straight line. 
 
 The hare, a comparatively light body, and moreover being pre- 
 pared for the evolution, first gradually retards its motion so as to 
 diminish the effects of inertia, and at the moment when the hound, 
 urged to its extreme speed, is in the act of seizing the game, the 
 hare dexterously turns at an acute angle to its former course, 
 leaving the hound propelled forwards in the direction in which it 
 was previously moving. 
 
 Thus, if the line AB {Jig- 2.) represent the direction 
 in which the hound was pursuing the hare, the hare, 
 having arrived at the point c, suddenly turns in the 
 direction CD; while the hound, unprepared for the 
 trick, and hurried forward by the inertia of its motion, 
 is carried on in the direction AB to the point B, while 
 the hare has passed along the line CD to the point D. 
 The distance now between the hound and the hare is 
 the line BD, the base of the obtuse angle formed by 
 two lines, c B and c D, simultaneously moved over by 
 the hound and the hare. 
 
SPECIFIC PROPERTIES. 31 
 
 CHAP. V. 
 
 SPECIFIC PROPERTIES. 
 
 121. THE qualities of matter which have been illustrated and ex- 
 plained in the preceding chapters, are those which are common, in 
 a greater or less degree, to all bodies, in whatever form or under 
 whatever circumstances they may exist. There remains to be 
 noticed another group of properties which may be denominated 
 for distinction, specific properties, being found in some species of 
 matter and not in others, or at least varying in degree so ex- 
 tremely in different sorts of bodies as to give them specific cha- 
 racters. 
 
 122. Elasticity and hardness. Although the property of 
 elasticity in its general sense may be considered as one which, in 
 various degrees, is common to all bodies, yet it is manifested in so 
 peculiar a manner in bodies of different forms, that it may be not 
 incorrectly considered as giving them a specific character. It is 
 intimately connected with another mechanical quality which may 
 be called Jiardness. This quality consists in a certain degree of 
 coherence, by which the constituent molecules of a body keep their 
 relative position so as to resist any force which tends to change 
 the figure of the body. 
 
 123. Hardness is distinct from density, as we frequently find 
 the most dense bodies possess this quality in a much less degree 
 than lighter substances. Glass, for example, is harder than gold, 
 or even than platinum, which is still harder and denser than gold. 
 A piece of glass will scratch the surface of gold or platinum, an 
 effect which shows that the particles of gold or platinum yield and 
 are displaced more easily than those of glass. 
 
 Again, in comparing different species of metals one with an- 
 other, their hardness is evidently independent of their densities. 
 Gold and platinum, the most dense of metals, are softer than iron 
 or zinc, which are much lighter. Among the hardest of the metals 
 are iron, zinc, copper, manganese, nickel, titanium, and pelagium. 
 The softest of the common metals is lead, but the new metals 
 developed by chemical enquiries, such as potassium and sodium, 
 are so soft as to yield under the finger like putty. 
 
 1 24. Some metals are capable of having their structure modi- 
 fied without the combination of any other substance with them, so 
 as to render them harder or softer within certain limits at plea- 
 sure. Thus steel, when heated, and then suddenly cooled by 
 being plunged in cold water, becomes harder than glass ; but if it 
 be cooled more gradually, then it becomes soft and flexible. 
 
3 ? PROPERTIES OF MATTER. 
 
 125. Elasticity. Elasticity manifests itself in various ways 
 according to the form and character of the body to which it be- 
 longs. The elasticity of a flat and thin bar of steel is manifested 
 by the force with which it will recover its figure when bent by 
 lateral pressure ; the elasticity of an ivory ball is manifested by 
 the force with which it will recover its figure when flattened by 
 impact against some hard surface. In the case of a steel spring, 
 the body yields to a slight pressure, readily changing its form ; in 
 the case of an ivory ball, the body does not yield to mere pres- 
 sure, and requires the force of impact to produce change of form. 
 
 When the force with which a body recovers its form is equal to 
 the force by which its form has been changed, the elasticity is 
 said to be perfect. Thus, if a bent spring recover its position 
 when relieved from the force which bent it with an energy 
 equal to such bending force, then the spring is said to be perfectly 
 elastic ; but when the restoring force is less than the bending 
 force, the elasticity is imperfect. In the same manner, if an ivory 
 ball flattened by a blow recover its form with a force equal to that 
 of the blow which flattens it, the elasticity is perfect, but otherwise 
 imperfect. 
 
 1 26. It is more exact to say that no body whatever is, in an 
 absolute sense, either perfectly elastic or perfectly inelastic. All 
 bodies possess some degree of elasticity, however small, but some 
 bodies, such as the gases, for example, possess elasticity in so high 
 a degree, and others, such as the liquids, in a degree compara- 
 tively so small, that not only, in popular language, is the one 
 considered elastic and the other inelastic, but it has been found 
 convenient to assume hypothetically these two qualities of per- 
 fect elasticity and perfect inelasticity as the bases of those divisions 
 of physical science, in which the laws which regulate the pheno- 
 mena of liquids and gases are developed. 
 
 It has been already shown, however, that liquids themselves 
 admit of some compression, and it may be added that they recover 
 their volume with a force sensibly equal to the compressing force ; 
 and to this extent, and in this sense, they are therefore almost 
 perfectly elastic. 
 
 127. The quality of elasticity is intimately connected with that 
 of hardness ; so much so, that it has sometimes been said that one 
 quality is proportional to the other. This is, however, erroneous. 
 Many of the gums, and eminently that called caoutchouc, are 
 highly elastic, and yet these substances are among the softest of 
 the solids. The elasticity of caoutchouc is nearly perfect, and yet 
 this substance, especially when it is warm, has great softness. 
 On the other hand, glass, flint, marble, ivory, afford examples of 
 solids in which hardness is combined with great elasticity. 
 
SPECIFIC PROPERTIES. 
 
 33 
 
 Putty, wet paste, moist clay, and similar bodies afford examples 
 of substances nearly deprived of elasticity. The figure of any of 
 these may be changed by pressure or by impact, and no tendency 
 to recover the figure so changed is perceptible. 
 
 128. Sound, as will be explained hereafter, is produced by vi- 
 bration imparted to the air by some solid body which is itself in a 
 state of sympathetic vibration. It is obvious, therefore, that the 
 metals best suited for bells, and other forms of matter intended 
 to produce sound, must be those which are most elastic. 
 
 129. The hardness and elasticity of metals are affected in a 
 striking manner by their combination. It often happens that two 
 metals, neither of which is eminently hard or elastic, produce by 
 their combination in certain proportions, one which possesses 
 these qualities in a high degree. Thus, bells formed of pure 
 copper, or of pure tin, will have little sonorous quality ; but if 
 these two metals be united in a certain proportion, their com- 
 bination will give a beautiful musical sound. The compounds of 
 different metals which have this quality are accordingly known as 
 bell metal. 
 
 1 30. Flexibility and brittleness. When a body easily yields, 
 and changes its form in obedience to a force exerted at right 
 angles to its length, as, for example, when a bar being supported 
 at the middle is pressed upon the ends, it is said to be flexible ; 
 but if, upon the action of such a force, instead of yielding and 
 changing its form, it breaks, it is said to be brittle. 
 
 Flexibility and brittleness are specific qualities which bodies 
 possess in an infinite variety of degrees. 
 
 In general, brittleness is connected with hardness, nor is it, as 
 might at first appear, at all inconsistent with certain forms of elas- 
 ticity. Glass, for example, which is highly elastic, is also the 
 most brittle of known substances. Brittleness, like hardness and 
 elasticity, is a quality which the same body may acquire, or be 
 deprived of, according to certain conditions to which it may be 
 subjected. Thus, the metals, iron, steel, brass, and copper, if 
 they be heated and suddenly cooled, by being plunged in cold 
 water, will become brittle ; but if, when heated, they are buried 
 in a hot sand-bath, and allowed to cool very gradually, then they 
 will lose their, brittleness, and acquire the contrary quality of 
 flexibility. 
 
 131. Malleability. This is a quality by which the metals 
 in general are eminently distinguished, but which they possess 
 in extremely different degrees. It is a property in virtue of 
 which a substance admits of being reduced to thin plates or 
 leaves under the blow of a hammer, or the intense pressure of 
 rollers. No process is of more extensive use in the arts. In 
 
 D 
 
34 PEOPERTIES OF MATTER. 
 
 large iron works, great lumps of metal at a white heat, but still 
 solid, are taken from the furnace, stuck upon the end of a long 
 bar of iron, and placed under a sledge hammer of enormous 
 weight, which rapidly strikes them, and reduces them to an elong- 
 ated form approaching to that of an iron bar. The metal, being 
 still red hot, is then passed between rollers, which are formed to 
 the shape of the transverse section of the rails used on our railways. 
 When pressed between and drawn through these rollers, the rail 
 has acquired its proper form, but is still red and soft ; and when 
 received from the rollers is so flexible, that it bends by its own 
 weight like a rod of wax. It is then laid on a flat surface, where 
 it cools and hardens, and assumes the condition of the rails on 
 which we travel. 
 
 The malleability of bodies depends on the combination in them 
 of the qualities of tenacity and softness. Without softness, they 
 could not yield either to the impact of the hammer or the pressure 
 of the roller ; without tenacity, they would be fractured by the 
 severe process of their fabrication. 
 
 The most malleable of the metals are gold, silver, iron, and 
 copper. 
 
 132. The malleability of a metal varies in degree according to 
 its temperature. There are certain temperatures in which this 
 quality exists in the highest degree. Iron is most malleable when 
 it first attains the white heat which follows the red ; zinc becomes 
 malleable at a much lower temperature, possessing this quality in 
 the greatest degree between 300 and 400. Some metals possess 
 the quality of malleability in so slight a degree as to be in this 
 respect specifically different from metals in general : among them 
 may be mentioned antimony, arsenic, bismuth, and cobalt, all of 
 which are brittle. The metals may be rendered brittle as they 
 are rendered hard, by being heated and then suddenly cooled, in 
 which case they lose their malleable quality. This quality, how- 
 ever, may always be restored by again heating them and cooling 
 them gradually, as before described. 
 
 133. Annealing:. This process of gradual cooling, which is of 
 great importance in the arts, is called annealing. 
 
 Metals are also rendered brittle, and deprived of their mal- 
 leability, by constant hammering. Thus, a bar of iron may be 
 hammered until it entirely loses its flexibility. In this case, as 
 before, the malleability may be restored by heating and an- 
 nealing. 
 
 134. Welding-. Metals which are highly malleable admit of 
 being united, piece to piece, by the process called welding. In 
 this process, the two pieces of metal are raised to that heat at 
 which they are most malleable, and the ends being laid one upon 
 
SPECIFIC PROPERTIES. 35 
 
 the other, are rapidly beaten by a welding-hammer. The par- 
 ticles are thus driven into such intimate contact, that they cohere, 
 and form one uniform mass. Different metals may in some cases 
 be thus welded together. 
 
 135. Ductility. The property in virtue of which a metal ad- 
 mits of being drawn into wire, is called ductility. This quality is 
 also eminently specific, being possessed by some sorts of metal in 
 a very high degree, while others are entirely destitute of it. 
 
 136. Ductility is a quality which must not be confounded with 
 malleability; for the same metals are not always ductile and 
 malleable, or, at least, do not possess these qualities in the same 
 degree. Iron possesses ductility in a much greater degree than it 
 possesses malleability, for it admits of being drawn into extremely 
 fine wire, though it cannot be beaten into extremely thin plates. 
 Tin and lead, on the other hand, are highly malleable, being 
 capable of being reduced to extremely attenuated leaves ; but 
 they are not ductile, since they cannot be drawn into small wire. 
 Gold and platinum possess both ductility and malleability in a 
 high degree. Gold has been drawn into wire so fine, that 180 
 yards' length of it did not weigh more than one grain, and an 
 ounce weight would consequently extend over fifty miles. 
 
 137. Tenacity. The property in virtue of which a body resists 
 the separation of its parts, by extension in the direction of its 
 length, is called tenacity. This manifestation of strength must be 
 carefully distinguished from that of which the absence or feeble- 
 ness is expressed by brittleness. The one form of strength may 
 exist in the highest degree in a body in which the other is in the 
 lowest degree. A thin rod of glass, if laid at its middle point on 
 any support, will be broken by the slightest force pressing on its 
 ends ; but the same rod, if suspended by one end in a vertical 
 position, will sustain an immense weight attached to the lower end 
 without being broken. It has at once great brittleness and great 
 tenacity ; while its longitudinal strength is considerable, its lateral 
 strength is almost nothing. 
 
 Different bodies vary extremely in their tenacity. Experiments 
 have been made on an extensive scale for determining the tenacity 
 of those bodies most used in the arts. The tenacity of metals has 
 been tested by suspending a weight from the end of a wire. 
 
 138. In the following table, the greatest weights are given 
 which were found to be supported by wires of the different metals, 
 having a diameter of T^^^jyths of an inch. 
 
 Weight* supported. 
 Gold ... 150-173 IbS. 
 
 Weights supported. 
 
 Iron ... 549-250 Ibs. 
 Copper ... 3oz-278 
 Platinum - - 174-300 
 
 Silver ... 187-137 
 
 Zinc - - - log- 
 
 Tin ... 34 . 
 
 Lead - - - 27621 
 
 09-540 
 34 630 
 
 D 2 
 
3t> PROPERTIES OF MATTER. 
 
 The process of annealing, which improves the malleability and 
 ductility of metals, is found in some cases, as, for example, in iron, 
 copper, and the combinations of zinc and copper, to diminish their 
 tenacity. In organized substances, those which possess a fibrous 
 texture have greater tenacity than those of cellular tissue. Hence, 
 we find that cotton has much less tenacity than thread, rope, or 
 silk. 
 
 139. L'Abbe Labillardiere found that threads of the following 
 substances, having the same diameter, were capable of supporting 
 weights in the proportion of the annexed numbers : 
 
 Silk .... 3400 I Flax (common) - . - 1175 
 
 New Zealand Flax - - 2380 Ditto (Pita) (Agave Americana) 700 
 
 Hemp .... i6jj | 
 
 140. Chemical properties. There is an endless variety of spe- 
 cific properties of bodies, the exposition and investigation of which 
 belong properly to chemistry. It will be sufficient here to notice 
 briefly the distinctions between these qualities and those which 
 form the proper subjects of Natural Philosophy commonly so 
 denominated. 
 
 If two substances, being mixed together, retain respectively 
 their separate qualities, the combination is said to be a mechanical 
 mixture. Thus, if blue and yellow powders be mingled together, 
 the mixture will appear green ; but on examining it with a micro- 
 scope, it will appear to consist of large blocks of matter, of the 
 colours blue and yellow, these being the particles of the separate 
 powders retaining their distinctive qualities, which are mecha- 
 nically mingled. The combination produces upon the eye a green 
 colour, the effect of the separate particles being too minute to be 
 separated by unassisted vision. 
 
 There are two gases, called oxygen and hydrogen, which have 
 the common mechanical properties of atmospheric air. If one 
 ounce weight of hydrogen be mingled with eight ounces of oxygen, 
 the gases will be interfused and mingled ; but the entire mass will 
 retain the same mechanical qualities as before, and the separate 
 particles will remain side by side in the mixture, exactly as did 
 the particles of blue and yellow powder in the preceding example. 
 But if an electric spark be imparted to this mixture, a striking 
 change will take place. The mixture will in an instant be reduced 
 to water, or rather to vapour, which, being cooled, will be soon 
 converted into water. In fact, the oxygen has in this case united 
 with the hydrogen, and the mass has lost the mechanical qualities 
 which it possessed, and has acquired those of the liquid water. 
 This is a chemical phenomenon. 
 
 There is a metal called sodium, and a gas called chlorine, each of 
 which, separately, is poisonous, and destructive of life, if taken into 
 
SPECIFIC PROPERTIES. 
 
 37 
 
 the stomach or lungs. If these two substances be brought together, 
 they immediately explode, and burst into flame. If the substance 
 resulting from this phenomenon be preserved and cooled, it will 
 be found to be common kitchen-salt, one of the most wholesome 
 condiments, and highly antiputrescent. Thus, two ingredients 
 possessing the most noxious properties, when combined chemi- 
 cally, lose those properties, and produce a sulfctance wholly dif- 
 ferent in form and quality. 
 
 The investigation of this and all similar phenomena belongs to 
 the province of chemistry. 
 
BOOK THE SECOND. 
 
 OF FORCE AND MOTION. 
 
 CHAPTER L 
 
 THE COMPOSITION AND RESOLUTION OF FORCES. 
 
 141. ANT agency which, applied to a body, imparts motion to it, 
 or produces pressure upon it, or causes both of these effects to- 
 gether, is called in mechanics a force. To determine a force, 
 therefore, with precision, three things are necessary : 
 
 First) the point of the body to which it is applied, technically 
 called its point of application ; Secondly, its intensity or quantity ; 
 and Thirdly, its direction. 
 
 142. Force expressed by weight. It is in general convenient 
 and customary to express the intensity or quantity of forces by 
 equivalent weights. Weight is the sort of force with which we are 
 most familiar. Every one is acquainted with the effect produced 
 by the pressure of a given weight; and whatever be the force 
 whose intensity or quantity it is required to express, a weight may 
 be named which would produce the same effect. Thus, if a piece 
 of iron, attracted by a magnet, be resisted by any surface, it will 
 press against this surface with a certain force. A weight may in 
 this case be named, which, being placed in the dish of a balance, 
 would press upon the surface of the dish with the same force. The 
 intensity of the attraction of the magnet on the iron would then be 
 expressed by the amount of such an equivalent weight. 
 
 143. Direction. When a force applied to any point of any 
 body causes that point to move, the direction of its motion is the 
 direction of the force. If the force do not produce motion, but 
 mere pressure, then the direction of the force is that in which the 
 pressure is directed, and in which the point would move in obe- 
 dience to the force, if it were free. 
 
 144. Forces in same direction. If two or more forces act 
 upon the same point, and in the same direction, their effect will be 
 equivalent to a single force which is equal to their sum. This is 
 so self-evident that it scarcely needs demonstration. 
 
COMPOSITION OF FORCE. 
 
 39 
 
 If a vehicle be drawn by three horses, one placed before the 
 other, one horse pulling with a force of 80, another with a force of 
 60, and the third with a force of 40 Ibs., then the combined action 
 of the three horses upon the vehicle will be equal to the action of 
 a single horse which should pull with a force of 1 80 Ibs., which is 
 equal to 80 Ibs. -f 60 Ibs. + 40 Ibs. 
 
 145. A single force acting on a body which would thus produce 
 the same motion or pressure as several forces acting together, is 
 called technically the resultant of these forces. Thus, in this pre- 
 ceding example, the force of 1 80 Ibs. acting on the vehicle in the 
 same direction as the three independent forces of 80 Ibs., 60 Ibs., 
 and 40 Ibs., is the resultant of those three. 
 
 146. Opposite forces. If two forces act upon a body in op- 
 posite directions, then the lesser of these forces will neutralize so 
 much of the greater as is equal to its own quantity, and an effective 
 force will remain in the direction of the greater, equal to their 
 difference. This is also self-evident. If, for example, a vehicle 
 be pulled backwards by a weight of I oo Ibs. acting over a pulley, 
 and that it be drawn forwards by a horse acting with a force of 
 150 Ibs., then loolbs. of the horse's force will be neutralized by 
 the weight which draws the vehicle backwards, and an effective 
 force of 50 Ibs. will remain in the direction of the horse's traction. 
 
 This principle is stated generally by saying that the resultant of 
 two forces applied to the same point in opposite directions is equal 
 to their difference and in the direction of the greater. 
 
 If any number of forces act upon the same point, some in one 
 direction, and the others in the direction immediately opposed to 
 it, then the resultant of such a combination of forces will be found 
 by taking the difference between the sum of all the forces which 
 act in the one direction, and the sum of all the forces which act 
 in the other direction, the direction of such resultant being that of 
 the forces whose sum is the greater. 
 
 147. Porces in different directions. When two forces applied 
 to the same point act in the direction of different and diverging 
 
 straight lines, such as A x and A Y 
 (Jig. 3.), then the direction and quan- 
 
 Itity of their resultant is not so evident 
 55 as in the case just mentioned. It is 
 indeed apparent that the combined 
 effect of two such forces on the point A 
 must be in some direction, such as A z, 
 intermediate between A x and AT; but 
 how this direction A z divides the angle 
 formed by the two components is not apparent. 
 
 04. 
 
40 OF FORCE AND MOTION. 
 
 The following example, in which, as usual, weights are used to 
 represent the forces in question, will, however, elucidate this. 
 
 Let two weights A and B (Jig. 4.) be attached to the extremities 
 of a flexible cord which passes over two pulleys, M and N. Let 
 another cord be knotted to this at any inter- 
 mediate point, such as p ; and let a third weight 
 '- c be suspended from it. The weight c will then 
 
 draw the cord which unites A and B into an 
 angle M p N. The system after some oscillations 
 will come to rest, and when it is at rest it will be 
 evident that the point p is solicited by three 
 forces ; 1st, by the weight A acting in the direction 
 of the line P M; zndly, by the weight B acting in the direction 
 r N ; and 3rdly, by the weight c acting in the direction of the 
 line P c. 
 
 Now, it is evident that the weight c acting in the direction p c 
 would equilibrate with an equal force acting in the opposite di- 
 rection P c. Since, then, the weight c would precisely counter- 
 poise an equal weight in the direction of P c, and that it is also in 
 equilibrium with the weights A and B, which act in the directions 
 p M and P N respectively, it follows that the resultant of the forces 
 A and B acting in the directions p M and p N will be a single force 
 equal to c acting in the direction p c. 
 
 It now remains to show in what manner this direction of the 
 resultant of the two diverging forces M and N is connected with 
 their quantities. 
 
 Let us suppose, for example, that the weight A is 6 oz., the 
 weight B 4 oz., and the weight c 6 J- oz. If, then, we take upon the 
 line P o a distance P c of 6^ inches, and if we draw two lines, one 
 c a parallel to p N, and the other c b parallel to p M, so as to form 
 a parallelogram p a c &, we shall find, on measuring the side p a, 
 that it is 6 inches, and on measuring the side p J, that it is 
 4 inches. 
 
 1 48. Composition of forces. Hence it appears, that while the 
 diagonal p c consists of as many inches as there are ounces in the 
 resultant of the two forces, the sides of the parallelogram which 
 are in the direction of these two forces respectively consist of as 
 many inches as there are ounces in these two forces. This result 
 may be enunciated in general terms as follows : 
 
 If two forces acting upon the same point be represented in quan- 
 tity and direction by two lines drawn through that point, then the 
 resultant of such forces will be represented in quantity and direction 
 by the diagonal of the parallelogram of which these lines are the 
 sides. 
 
COMPOSITION OF FORCE. 41 
 
 But it may be objected that the result we have obtained by the 
 use of three particular weights may be accidental, and that it may 
 not always happen that the third weight c, which balances the 
 other two, will throw the cords into such directions as would give 
 the remarkable result here obtained. 
 
 The validity of such an objection may be easily tested by vary- 
 ing the weights at pleasure, and by submitting the position of the 
 string to the same process of measurement as we have given above. 
 It will then be found that in whatever manner the three weights 
 may be varied, the knot which unites the three strings P M, P N, 
 and P c will invariably establish itself in such a position, that while 
 the diagonal P c will measure as many inches as there are ounces 
 in the resultant c, the sides P a and P b will measure as many inches 
 as there are ounces in the components A and B. 
 
 The proposition which we have here established is of the utmost 
 importance in all mechanical investigations, and is known as the 
 principle of the composition of forces. 
 
 1 49. In virtue of this principle, whenever two forces in different 
 directions act upon the same point of a body, a single force deter- 
 mined as above by the diagonal can be substituted for them with- 
 out changing the mechanical state of the body ; or, on the other 
 hand, if a single force act upon any point of a body, two forces 
 acting on the same point may be substituted for them, provided 
 such forces can be represented, in quantity and direction, by the 
 sides of a parallelogram whose diagonal represents in quantity and 
 direction the single force for which they are substituted. 
 
 150. Resultant of any number of forces. If any number of 
 forces whatever act upon the same point of a body, and in any 
 directions whatever, a single force can always be assigned which 
 will be mechanically equal to them, and will therefore be their 
 resultant. After what has been established, nothing is more easy 
 than the solution of this question. Let the several single forces 
 supposed to act upon the point in question be expressed by A, B, 
 c, D, E, &c. I st. Let the resultant of A and B be found by the 
 principle of the parallelogram of forces explained above, and let 
 this resultant be A'. 2nd. Let the resultant of A' and c be found 
 by the same principle, and let this resultant be B'. 3rd. Let the 
 resultant of B' and D be found, and let this resultant be c' ; and so 
 on. In this way we shall finally arrive at the determination of a 
 single force, which will be equivalent to, and will therefore be the 
 resultant of the entire system. 
 
 151. In what precedes, we have supposed the forces whose com- 
 bined effects are to be determined as applied to the same point of 
 the body on which they act. It often happens, however, that the 
 
OF FORCE AND MOTION. 
 
 forces are applied to different points. We shall therefore now 
 proceed to consider this case ; and, first, we shall take the more 
 simple condition under which the force acts in parallel directions. 
 
 152. Parallel forces. As before, we shall consider the forces 
 represented by weights. 
 
 Let P and p' (Jig. 5.) be the points to which the two forces in 
 
 question are applied, 
 and let these two 
 forces be represent- 
 ed in direction by 
 parallel cords p M 
 and P' M' passing 
 over pulleys, and let 
 them be represent- 
 
 ( ed in quantity by 
 
 two weights A and 
 A' suspended from 
 these cords. Now 
 the resultant of 
 these two weights, 
 A and A', or the 
 single force which would be equal to them, may be determined by 
 means precisely similar to those which we have adopted in the case 
 of diverging forces, by ascertaining where a single force may be 
 applied and what will be its amount, so as to balance the two forces 
 A and A'. 
 
 For this purpose, let us suppose a weight, R, to be suspended 
 from a point, o, between r and p'. 
 
 Instead of suspending a determinate weight from the string car- 
 ried over the pulley M, let us suppose the dish of a balance to be 
 suspended there, capable of receiving any heavy matter which 
 may be placed in it. Things being thus arranged, let sand be 
 poured into the dish A, until it is found that the three weights 
 A, A', and R are in equilibrium. Let us suppose, for example, that 
 the weight A' is 6 oz., and the weight R IO oz. If the weight of 
 the sand and of the scale which bears it at A be ascertained, it 
 will be found to be 4 oz. ; and it therefore follows that the sum of 
 the two weights at A and A' being 10 oz., is equal to the weight R. 
 Hence it follows that the resultant of the two parallel forces A and 
 A' is in this case a force equal to their sum. 
 
 Now if the experiment be varied in any manner, the same result 
 will still be obtained. Thus, if the weight A' be 8 oz. and the 
 weight R be 20 oz., then the weight of the sand in the dish A will 
 be found to be 12 oz. ; the sum of A and A', 12 + 8, being still 
 
COMPOSITION OF FORCE. 43 
 
 equal to R, which is 20. In a word, in whatever manner the 
 weights A and B may be varied, so long as B is greater than A, the 
 weight A' will invariably be their difference ; and we therefore con- 
 clude in general that the resultant of two parallel forces acting in the 
 same direction upon two different points of the same body, is a force 
 parallel to their direction, and equal to their sum acting at some in- 
 termediate point. 
 
 Now, it remains to determine what is the intermediate point 
 between p and p' at which this resultant will act. 
 
 After having established an equilibrium by pouring the quantity 
 of sand into the dish A, if we measure the distances p o and p' Q, 
 we shall find that they are invariably in the inverse proportion of 
 the two weights A and A' ; that is to say, if the weight A' be 8 oz- 
 and the weight A 1 2 oz., then the proportion of P o to P' o will be 
 8 to 12, and this will be found to be invariably the case. If the 
 position of the string supporting the weight B be varied, as it may 
 be, it will always be found that the ratio of the two weights A and 
 A', which establish an equilibrium in the system, will be inversely 
 as the distance of the points p and p' from the point o, where the 
 resultant is applied ; while the distance P o represents in quantity 
 the component A', the distance P' o will represent in quantity the 
 component A. 
 
 153. This general principle, which is of great importance in 
 mechanics, may be enunciated as follows : 
 
 The resultant of two forces which act on different points of the 
 same body in parallel lines, and in the same direction, is a single 
 force equal to their sum acting parallel to them, and in the same 
 direction, at an intermediate point which divides the line joining the 
 two points of application of the components in the inverse proportion 
 of the quantities of those components. 
 
 If the forces A' and B be considered as components, the force A 
 may be considered as the opposite of their resultant. It conse- 
 quently follows that the resultant of A' and B is a force equal in 
 quantity to their difference, and applied at p, acting in a line 
 parallel to them, and in the direction of the greater force R. 
 
 154. Parallel forces in opposite directions. This general 
 principle may be enunciated as follows : 
 
 The resultant of two forces which act on different points of the 
 same body in parallel lines in opposite directions, will be a single 
 force equal to their difference, and acting at a point beyond the greater 
 of the two forces, and so situated that the point of application of the 
 greater of the two forces will divide the distance between the lesser 
 and the resultant in the inverse proportion of the quantities of the 
 lesser and of the resultant. 
 
44 
 
 OF FORCE AND MOTION. 
 
 Having the means of determining the resultant of two parallel 
 forces, we can determine the resultant of any number of such 
 forces by taking them respectively in pairs, as we have done in the 
 case of diverging forces. Thus, let any two forces of such a sys- 
 tem be taken, and the resultant found. Then, considering such 
 resultant as a component, let it be combined with a third compo- 
 nent, and their resultant found ; and so on. 
 
 155. There is a case of parallel forces which does not admit of 
 a single resultant, and which is of considerable importance in me- 
 chanical inquiries. This case is that in which two equal forces act 
 upon two points of a body in parallel and opposite directions. The 
 effect of such forces cannot be represented by any single force. 
 In fact, such a combination of forces has no tendency to produce 
 in a body any progressive motion, but has a tendency to cause it 
 to revolve round a point intermediate between the direction of the 
 two forces. Such a system of forces is called a couple. 
 
 156. The mechanical effect of such a system depends, conse- 
 quently, on the intensity of the forces, the perpendicular distance 
 between their lines of direction, and on the direction of the plane 
 which passes through their lines of direction. 
 
 If p and P' (Jig. 6.) be the points of application, and one of 
 the forces act in the direction of p M, while the other acts in the 
 
 Fig. 6. 
 
 direction of p' M', then their effect will depend on their intensity, 
 on the length of the perpendicular distance P p' between their 
 directions, and on the direction of the plane in which the lines P M 
 and P' M' lie. 
 
 Representing such forces by weights, as before, let us suppose 
 strings attached to the points p and p' carried over the pulleys 
 M and M', and supporting the two equal weights, w w'. 
 
 The obvious tendency of these weights is to turn the line P P' 
 
COMPOSITION OF FORCE. 
 
 45 
 
 round in the direction in which the hands of a clock would move. 
 Now this tendency cannot be counteracted by any single force, 
 but it may be resisted by another couple applied to other two points 
 of the body. Let us suppose, for example, that p p' be two other 
 points of the same body, either situate in the same plane as the 
 lines P M and p' M', or in any parallel plane, and that strings be 
 applied to them extended by weights, in the same manner as in the 
 former case, the strings lying in such parallel plane. 
 
 Let p m and p' m', carried over pulleys, support weights w and w', 
 but let them be so applied to the line p p' that they shall have a ten- 
 dency to turn the body round contrary to the motion of the hands 
 of a clock, and therefore contrary to the effect of the former couple. 
 Now let the weights w and w' be so adjusted by trial, that this 
 second couple shall exactly balance the first couple, and keep the 
 body at rest, which may be done by using for the purpose the dish of 
 a balance and sand, as in the former experiment. When the equi- 
 librium is thus established, it will always be found that the weights 
 w and w will bear to each other the inverse proportion of the dis- 
 tance between the parallel cords, that is to say, the weight w will be 
 greater than the weight w in the exact proportion of the distance 
 p p f to the distance P P' ; or, to express this in the usual manner 
 by arithmetical symbols, we should have 
 
 w : w '. : p p' : P P' ; 
 from which it follows that 
 
 w X P P' = w Xpp'. 
 
 157. This conclusion involves the entire mechanical theory of 
 couples, and may be enunciated as follows : 
 
 Two equal and parallel forces acting in contrary directions on a 
 body, have a tendency to make that body revolve round an axis per- 
 pendicular to a plane passing through the direction of such ttvo 
 parallel and opposite forces ; and such tendency is proportional to 
 the product obtained by multiplying the intensity of the forces by the 
 distance between their directions ; and, consequently, all couples in 
 which such products are equal and have their planes parallel are 
 mechanically equivalent, provided that their tendency is to turn the 
 body round in the same direction ; but if two such couples have a ten- 
 dency to turn the body in contrary directions, then two such couples have 
 equal and contrary mechanical effects, and would, if simultaneously 
 applied to the same body, keep it in equilibrium. 
 
 158. Two forces not being parallel in their directions which are 
 applied to different points of the same body, present two different 
 cases, in one of which only they admit of a resultant. 
 
OF FORCE AND MOTION. 
 
 
 Fig. 7- 
 
 1st. If the two forces applied at p and p' (Jlg.J-^ not being 
 
 parallel, are neverthe- 
 less in the same plane, 
 their directions p M 
 and P' M', if prolonged, 
 will necessarily meet at 
 some point such as o. 
 In this case we may 
 imagine the two forces 
 to be applied at o, and 
 their resultant will be 
 represented in quantity 
 and direction by the 
 diagonal o c of a paral- 
 lelogram whose sides represent, in quantity and direction, the 
 two forces, according to the principle already explained. 
 
 2nd. But if the forces applied at the points P and p' (Jig. 8.), 
 
 not being parallel, are at 
 the same time in different 
 planes, then the directions, 
 though indefinitely pro- 
 longed, will never intersect, 
 and they will not have any 
 single resultant; in ether 
 words, their mechanical ef- 
 fect cannot be represented 
 by that of any single force. 
 159. It can be demon- 
 strated, however, that the 
 mechanical effect of such a 
 system of two forces as here 
 described, whose directions 
 lie in different planes, and 
 which, though not parallel, 
 can never intersect, will be 
 mechanically equal to the 
 combined action of a couple 
 such as already described, 
 and a single force ; in other 
 words, such a system will 
 have a double effect on the 
 
 body to which it is applied : 1st, a tendency to produce revolution ; 
 and, zndly, a tendency to produce a progressive motion ; and if it 
 were not held in equilibrio by some equal antagonist forces, the 
 
 M 1 
 
COMPOSITION OF FORCE. 47 
 
 body would at the same time move forward in some determinate 
 direction, and revolve round some determinate axis. 
 
 To render this intelligible, let us imagine the forces to be repre- 
 sented by weights acting on strings, and passing over pulleys. 
 
 In Jig. 8., let P and P' be the two points of application of the two 
 forces ; let the force acting at P be vertical, and be represented by 
 the weight w, suspended by a string at the point p. 
 
 Let the other force applied at P' be horizontal, and in a direction 
 p' M' perpendicular to P' P, and let it be represented by the weight 
 w' suspended by a cord which passes over the pulley M'. Attached 
 to the point P', let another string be carried vertically upwards to 
 M, and then passed over a pulley, and let a weight be suspended to 
 it equal to the weight w. Now take upon the line P' M' a distance 
 p' c, consisting of as many inches as there are ounces in the weight 
 w', or, which is the same, in the force which stretches the cord p' c. 
 Take also upon the vertical line p' M as many inches p' B as there 
 are ounces in the weight w, and draw the line B c. From c draw 
 c A parallel to P' B, and from the point p' draw p' A, parallel to 
 B c. Carry a string from p' along the line P' A, and let it pass 
 over a pulley M", and suspend from it the weight w", consisting of 
 as many ounces as there are inches in the line P' A. 
 
 Now, according to this statement, the weight w will consist of 
 as many ounces as there are inches in p' B, and the weight w" will 
 consist of as many ounces as there are inches in the line P' A. It 
 follows, therefore, from the principle of the composition of forces 
 already established, that the combined effects of these two forces 
 w and w" acting in the line P 7 B, and p' A upon the point p 7 , will 
 be the same as the single action of the weight w' acting in the 
 direction p' c upon the same point p', and that it may be conse- 
 quently substituted for the latter without changing the effects 
 upon the body. Let us then detach the weight w' and relieve the 
 point P' from its action, leaving the weights w and w" acting in its 
 place. The point p' and the body to which it belongs will then be 
 affected in the same manner by the three weights w, w, apd w", 
 as it was by the two original weights \v and w'. It follows, there- 
 fore, that the effect of the two original weights w and w' is mecha- 
 nically equivalent to the effect of the weight w" and the two equal 
 weights w. 
 
 This is equivalent to stating that the two forces acting at the 
 points P and p', in the directions p w and p' c, are equivalent in 
 their effect to three forces, viz., a single force, represented in 
 intensity by the line P' A, and acting along that line, and a couple 
 acting in a vertical plane passing through the line P P'; the dis- 
 tance between the two forces being equal to P P' T and their inten- 
 sities being equal to w. 
 
4.8 OF FORCE AND MOTION. 
 
 The total effect, therefore, would be equivalent to a single force 
 acting in the direction p' A, and a couple producing rotation round 
 an axis perpendicular to a vertical plane through p p'. 
 
 1 60. Suspension bridges. The principles upon which the 
 
 Fig. 9. 
 
 jointed rods and links by which the floors of these bridges are 
 supported, admit of easy explanation upon the elementary theorems 
 of the composition offeree. 
 
 Supposing the floor of the bridge extending between the two 
 quay walls to be represented byABCDEFG (Jig. 9.), and to be 
 sustained by the parallel and equidistant vertical rods M, N, p, Q, R, s, 
 it is necessary that these rods severally should so support the 
 flooring that no one part of it shall have a tendency to rise above 
 or sink below the adjacent part; for it is evident that if any 
 such inequality existed in the sustaining force, one of two conse- 
 quences would ensue, either of which would be incompatible with 
 the necessary conditions of such a structure : if the flooring 
 did not yield to these unequal forces, it would be subject to con- 
 tinual transverse strains, which would impair its strength ; and if 
 it did yield, its surface would not be level, but undulating. The 
 condition, then, being admitted upon which the flooring is sup- 
 ported by the vertical rods in equilibrium in the horizontal position, 
 we may imagine it divided at the middle points B, c, D, E, r of the 
 successive intervals between the rods, as well as at the points A and 
 G, where it meets the quay walls. In this way the several pieces 
 A B, B c, c D, D E, E F, and F G are supported separately and seve- 
 rally by the rods M, N, p, Q, R, s. Although free to move up or down, 
 being quite detached from each other, they will not do so, since 
 such a derangement of their relative positions would be incom- 
 patible with the equilibrium of the forces supporting the flooring 
 which has been supposed. 
 
 It remains now to determine the relative directions of the several 
 links a &, be, &c., which support the vertical rods. For this 
 
COMPOSITION OF FORCE. 49 
 
 purpose we are to consider that the middle link d e is parallel to 
 the flooring at a height above it altogether arbitrary, depending 
 merely on the general height at which it may be found convenient 
 to place the chain above the flooring. The half of the flooring 
 extending from A to D, consisting of the three detached pieces A B, 
 B c, and c D, may be considered as sustained exclusively by the 
 links a b and d e ; for if these last links were cut through, it is 
 evident that the pieces suspended from the rods M, N, p, would fall. 
 This being the case, the entire weight of the three pieces, acting 
 vertically downwards at their centre of gravity N, will necessarily 
 be in equilibrium with the tensions of the links a b and d e. It 
 follows from this, by the principles of the composition of force, 
 that if the line e d be continued to meet the vertical rod N at r, 
 the direction of the link a b will be that of the line drawn from r 
 to a ; this, therefore, determines the direction of the first link a b. 
 The direction of the second link b c is determined in precisely the 
 same manner. We have only to consider the pieces B c and c D as 
 being supported by the links b c and d e ; and it follows, as before, 
 that the weights of these pieces, B c and c D, acting vertically down- 
 wards from their centre of gravity c, will be in equilibrium with the 
 tensions of the links b c and d e. If, therefore, the direction ofde 
 be continued to meet the vertical through c at s, the line drawn 
 from s to b will be the direction of the second link b c, that of the 
 third link being necessarily the line c d. 
 
 The directions of the corresponding links at the other side of the 
 centre will be the same. 
 
 The parts of the chain p q on the other side of the piers will be 
 inclined to the vertical at the same angle with the links a b and p q ; 
 so that, the tensions being the same, the resultant of the two forces 
 in each case will necessarily be vertical, and consequently the 
 weight will be supported by the piers. 
 
 We have here considered only one chain. A bridge, however, 
 must be at least supported by two, one at each side, and may be 
 sustained even by four, two at each side. In such cases the force 
 will be equally distributed among the chains, and the same prin- 
 ciples will be applicable to determine the direction of their several 
 links. 
 
50 OF FORCE AND MOTION. 
 
 CHAP. II. 
 
 COMPOSITION AND RESOLUTION OF MOTION. 
 
 1 6 1. Direction and velocity. Motion has two qualities, direc- 
 tion and velocity. If we would define in a precise and intelligible 
 manner the state of a body which is in motion, we must therefore 
 state, first, the direction in which it moves ; and next, the rate at 
 which it moves, or the speed which it has in such direction. 
 
 If the motion of a body be rectilinear, that is to say, if it move 
 continually in the same straight line, then such straight line is its 
 direction. But it is evident that a body may move in two op- 
 posite directions in the same straight line : thus, if the line of the 
 motion be east and west, the body may move either from east to 
 west, or from west to east, without departing from the line in 
 question. 
 
 A term is wanting in our language for the convenient expres- 
 sion of this condition attending the motion of a body. In French, 
 the word direction expresses the line in which the body moves, 
 and the word sens expresses the direction of its motion in such line. 
 
 162. Direction in a curve. If a body move in a curved path, 
 
 such as AB (Jig' lo.), the di- 
 rection of its motion is con- 
 tinually changed, but at any 
 point of the curve, such as p, 
 Fig . , . it is considered to have the 
 
 direction of a tangent PT at 
 that point. 
 
 163. The velocity of a moving body is expressed by stating the 
 relation between any space through which the body moves, and 
 the time in which such motion is performed. Thus, we say that 
 the speed with which a man walks is four miles an hour, the speed 
 with which a stage-coach travels is ten miles an hour, and the 
 speed of a railway train is thirty miles an hour. 
 
 It is evident that the same speed may be differently expressed, 
 according to the different units of time or distance which may be 
 adopted. We may express the velocity by stating the space 
 moved over in a given time, or the time taken to move over a 
 given space. The given time may be an hour, a minute, or a 
 second ; and the given space a mile, a foot, or an inch. If we say 
 that a railway train moves at thirty miles an hour, or that it moves 
 at eight hundred and eighty yards a minute, or forty four feet 
 
COMPOSITION OF MOTION. 
 
 a second, we express exactly the same speed, the only difference 
 being that different units of time and distance are adopted. 
 
 The selection of the units of time and distance for the expres- 
 sion of velocity i$ of course arbitrary. It is usual, however, to 
 adopt such units that the velocity may be expressed by a number 
 which is neither inconveniently great nor inconveniently small. 
 If the motion of the body be extremely rapid, we express its ve- 
 locity by adopting a large unit of space or a small unit of time ; 
 and if, on the other hand, the motion be very slow, we express 
 the velocity by a small unit of space or a large unit of time. 
 
 As spaces or times which are extremely great or extremely 
 small are more difficult to conceive than those which are of the 
 orders of magnitude that most commonly fall under the observa- 
 tion of the senses, we shall convey a more clear idea of velocity, 
 by selecting for its expression spaces and times of moderate rather 
 than those of extreme length. The truth of this observation will 
 be proved, if we consider how much more clear a notion we have 
 of the velocity of a railway train, when we are told that it moves 
 over fifteen yards per second, or between two beats of a common 
 clock, than when we are told that it moves over thirty miles an hour. 
 We have a vivid and distinct idea of the length of fifteen yards, 
 but a comparatively obscure one of the length of thirty miles ; 
 and, in like manner, we have a much more clear and definite idea 
 of the duration of a second than we have of the duration of an 
 hour. 
 
 1 64. In the following table are collected examples of the velo- 
 cities of various objects, which may be found useful for reference, 
 and which will serve as standards in the memory 'for various 
 classes of motion : 
 
 Table showing the Velocities of certain moving Bodies. 
 
 Objects moving. 
 
 Miles 
 
 Feet 
 
 per 
 Second. 
 
 Objects moving. 
 
 Miles 
 Hour'. 
 
 Feet 
 per 
 
 Second. 
 
 Man walking 
 Horse trotting - 
 Swiftest racehorse 
 
 3 
 
 60 
 
 1 
 
 Air rushing into va- 
 cuum ... 
 Common musket-ball 
 
 8 S 
 850 
 
 1247 
 1247 
 
 Railway train, English 
 Americ. 
 
 ! ! 
 
 
 Rifle-ball - 
 Cannon-ball 
 
 1000 
 
 1091 
 
 3g 
 
 1600 
 
 Belgian 
 
 
 3^1 
 
 Bullet discharged from 
 
 
 
 French 
 
 27 
 
 39! 
 
 air-gun, air being 
 
 
 
 German 
 
 
 7ri 
 
 compressed into a 
 
 
 
 Swift English steam- 
 
 
 * 
 
 hundredth part of its 
 
 
 
 boats navigating the 
 
 
 
 volume - 
 
 477 
 
 700 
 
 channels 
 
 14 
 
 2O- 
 -I 
 
 Sound when atmo- 
 
 
 
 Swift steamers on the 
 
 
 
 sphere is at 32 Fahr. 
 
 743 
 
 1000 
 
 Hudson 
 
 18 
 
 26, 
 
 
 Do. 60 Fahr. 
 
 765 
 
 1 122 
 
 Fast sailing vessels 
 Current of slow rivers 
 
 12 
 
 3 
 
 I? 
 
 4 
 
 
 Earth moving round 
 sun - - - - 
 
 68182 
 
 IOOOOO 
 
 rapid rivers 
 Moderate wind - 
 
 7 
 
 10 
 
 IO 
 
 
 A point on earth's sur- 
 face at equator by 
 
 
 
 Storm ... 
 
 to 
 
 52: 
 
 
 diurnal motion 
 
 1040 
 
 1525 
 
 Hurricane ... 
 
 80 
 
 
 
 
 
 
 E 2 
 
52 OF FORCE AND MOTION. 
 
 165. Uniform velocity. The speed of a body in motion may 
 be uniform or varied. 
 
 The speed is uniform when all equal spaces, great or small, are 
 moved over in equal times. 
 
 It is possible to conceive a body in motion, moving over a mile 
 per minute regularly, and yet the speed not to be uniform ; for 
 though each successive mile may be performed in a minute, the 
 subdivisions of such mile may be performed in unequal times : 
 thus, the first half of each successive mile might occupy forty 
 seconds, and the other half twenty seconds. To constitute a 
 uniform motion, therefore, it is necessary that all equal portions 
 of space, no matter how small they are, shall be passed over in 
 equal times. 
 
 1 66. Composition of motion. As forces which produce pres- 
 sure would, if the bodies on which they act were free to move, 
 produce motion in the direction of such pressure, whose velocity 
 would be proportional to the pressure, all the principles which 
 have been established in the last chapter respecting the com- 
 ponents and resultants of forces will be equally applicable to mo- 
 tion. Hence, without further demonstration, we may consider as 
 established the following principles, called the composition and 
 resolution of motion. 
 
 If a body A (Jig. 1 1 .) receive at the same time two impulses, in 
 virtue of one of which it would 
 
 Zmove in the direction AY, over the 
 space AB, in one second; and in 
 ^ > virtue of the other, it would move 
 
 in the direction A x, over space A c 
 in one second ; then the two im- 
 
 _/_ pulses, acting upon it simultane- 
 
 -A ^^ *- o x ously, will cause it to move over 
 
 Flg< " the diagonal AD of the parallelo- 
 
 gram, whose sides are AB'and AC, in one second. 
 
 Conversely, also, a single motion AD may be considered as equal 
 to the combination of two motions, AB and AC, along the sides of 
 any parallelogram of which AD is a diagonal. 
 
 Now, as an infinite variety of parallelograms may have the 
 
 same diagonal, it follows 
 
 JL^^\^ B i. that any single force may 
 
 be considered as equal to an 
 infinite variety of combined 
 forces or motions. Injfig. 12., 
 the line AD is the diagonal 
 of the parallelograms A B D c, 
 AB'DC', AB"DC", &c. 
 
COMPOSITION OF MOTION. 53 
 
 1 67. The effect of a force which acts upon a body in motion, must 
 be either to change its velocity, or to change its direction, or 
 to produce both these effects together. 
 
 If a body being in uniform motion in a certain straight line, 
 such as AB (Jig- 12.), receive at P the impulse of a force in the 
 
 A 2_> * P B 
 
 Fig. 13. 
 
 direction in which it is moving, the effect of such impulse will 
 be to augment its velocity ; and such increase of velocity will be 
 exactly equal to the velocity which the same force would impart 
 to the body, if the body had been at rest. Thus, if the body had 
 been moving towards B at the rate of ten feet per second, and that 
 it received an impulse at P, in the direction of B, which would 
 have imparted to it, being at rest, a velocity of five feet per second, 
 then the velocity of the body after the impact will be fifteen feet 
 per second. This is evident, because the previous motion which 
 the body is supposed to have had in the direction of B cannot in 
 any way impair the effect of a force tending to make it move in 
 the same direction. 
 
 1 68. But if the impulse which it has received at P had been 
 given in the direction PA, contrary to that of its motion, then 
 such impulse would deprive the body of just so much velocity in 
 the direction PB, as it would have imparted to it in the direction 
 p A had it been at rest. In this case, therefore, the velocity after 
 the impact will be diminished from ten feet per second to five feet 
 per second. 
 
 These consequences are obviously analogous to the correspond- 
 ing conclusions, which were explained in the last chapter, re- 
 specting the combined effects of forces acting upon a body in the 
 same or any parallel directions. 
 
 169. If a body, being in motion from A to B (fig- 14-)? receive 
 
 c / at the point P an impact 
 
 which, had it been at rest, 
 would have caused it to move 
 in the direction p c, then the 
 body commencing from the 
 A ** B point P will have a motion 
 
 Fig> J 4- compounded of the motion 
 
 which it had before' receiving the impact at P, and the motion 
 which that impact would have given to it had it been at rest. 
 The motion, therefore, which it would have on leaving p in this 
 case, will be determined by the parallelogram of forces, according 
 to the principles already established. 
 
 Thus, if we suppose that, by virtue of the motion which the 
 E 3 
 
5* OF FORCE AND MOTION. 
 
 body had along the line AB, it would have moved from p to D in 
 a second, and that the motion which it would, being at rest at P, 
 have received from the impact would have carried it from p to E 
 in one second, then the motion which the body will actually have 
 in leaving p after receiving the impact, will be along the diagonal 
 PF of the parallelogram whose sides are PD and PE. 
 
 If a body moving with a 
 certain determinate velocity 
 in the direction AB {fig. 15.) 
 B encounter at P a fixed object 
 that has a flat surface c D, its 
 motion will not be destroyed, 
 but will be modified by the 
 resistance of this surface. 
 
 The effect of the surface, supposing it, as well as the body, to 
 be perfectly hard and inelastic, will be to destroy so much of the 
 motion of the body as is in a direction perpendicular to it. 
 
 To determine this, let us draw the line P M perpendicular to the 
 surface c D, and, taking p o as the space which the moving body 
 moves over in a second, from o draw o N parallel to P c, and o B 
 parallel to P N, so that P N o B shall be a parallelogram, of which 
 p o is the diagonal. The force, therefore, with which the body 
 will strike the surface at P will be equal to the two forces, one in 
 the direction of B p, and the other in the direction of N P, the sides 
 of this parallelogram, inasmuch as these two forces are, by what 
 has been already explained, equal to the single force represented 
 by the diagonal o P. But the reaction of the surface c D will de- 
 stroy the perpendicular force N P, and therefore the force B p alone 
 will remain in the direction B P D. The body, therefore, after it 
 strikes the surface, will move from p towards D with a velocity, in 
 virtue of which it will describe spaces equal to B P in one second. 
 
 170. Examples. The following examples will illustrate the 
 principles of the composition and resolution of forces which have 
 been explained in the present and the last chapter. 
 
 171. If a swimmer direct his course across a river in which 
 there is a current, his body will be at the same time subject to two 
 motions : first, that which he receives from his action in swimming, 
 which, if there were no current, would carry him directly across 
 the river in a direction perpendicular to its course in a certain 
 time, as, for example, fifteen minutes ; and the other in virtue of 
 the current, by which, if he were to float on the water without 
 swimming, he would be carried down the stream with a velocity 
 equal to that of the stream, at a rate, for example, of five thou- 
 sand feet in fifteen minutes. 
 
 Now, in actually passing over the river, the swimmer is affected 
 
COMPOSITION OF MOTION. 55 
 
 at the same time by both these motions ; and consequently his body 
 will be carried in fifteen minutes along the diagonal of the paral- 
 lelogram of which these motions are sides. 
 
 M M 
 
 Fig. 16. 
 
 If s s' and T T' (fig. 16.) be the banks of the river, the direction 
 of the stream being expressed by the arrow, and P be the point of 
 departure of the swimmer, let P N be the distance down which the 
 stream would carry the swimmer in the time which he would take, 
 if there were no current, to cross the river from P to the opposite 
 point M of the other bank. Then, as he crosses, he will be im- 
 pelled by the two motions. By swimming, he will continually 
 approach the bank T T' ; at which he will arrive as soon as he would 
 do if there were no current, because the current which crosses 
 him laterally will not interfere with his course in swimming, which 
 is perpendicular to it. Therefore, at the time that he would arrive 
 at the middle point, Q, of the stream, if there were no current, he 
 will still have arrived at the middle point, Q', of the stream; but in 
 virtue of the current, that point will lie, not opposite the point 
 from which he started, but lower down on the stream at Q' ; and, 
 in fine, he will arrive at the opposite bank at the point M', just so 
 far below the point M as is equal to the space through which the 
 current moves in the time which the swimmer takes to cross the 
 river. The actual course followed by the swimmer, therefore, by 
 the combined effects of his action in swimming and of the effect of 
 the stream, will be the diagonal P M' of the parallelogram, one side 
 of which, P N, represents the space through which the current 
 moves in the time the swimmer takes to cross the river, and the 
 other, P M, the space through which the swimmer would move in 
 the same time if there were no current. 
 
 By a due attention to the principles of composition of motion, 
 the swimmer may be enabled, notwithstanding the current, to cross 
 the river in a direction perpendicular to its banks. 
 
 T o M T As before, let ?0%M 7.) be the 
 
 point of the bank from which 
 \ he starts, the current of the 
 
 \ river being in the direction of 
 
 S P N S f the arrow. 
 
 Fig. 17. Let P N be the distance through 
 
 which the current would run in the time which the swimmer would 
 
 E4 
 
56 OF FORCE AND MOTION. 
 
 take to cross the river. From N draw the line N M to the point of 
 the bank directly opposite to P, and from p draw the line P o pa- 
 rallel to M N. Now, by the principles of the composition of motion 
 which have been already explained, a force in the direction of P M 
 will be equal to two forces simultaneously acting, one in the direc- 
 tion of P N, and the other in the direction of P o ; consequently, if 
 the swimmer, leaving the point p, direct his course to the point o, 
 and use such action in swimming as would enable him to pass in 
 still water from p to o in the same time which the current requires 
 to run from p to N, then his actual motion, in consequence of the 
 combined effect of his action in swimming and the force of the 
 current, will be in the direction p M, directly across the stream ; as 
 this direction will be the diagonal of the parallelogram, whose 
 sides represent the two forces to which his body is exposed. 
 
 But, under these circumstances, although he will pass directly 
 across the stream from p to M, instead of being carried diagonally 
 down it, as in the last example, along the line p M' (fig. l5-)> h 
 will take a longer time and greater exertion to cross the river. In 
 the former example, as his motion was perpendicular to the stream, 
 the force of the current did not in any wise retard his course in 
 swimming, but merely changed the point of the opposite bank at 
 which he arrived ; and although he passed over a greater space in 
 the water, yet the increased distance over which he moved was due, 
 not to his own exertion, but to the current. In the latter instance, 
 however, a part of his exertion was expended in stemming the 
 current, so as to prevent his descending the stream, and another 
 part in crossing the river. 
 
 172. A vessel impelled at the same time by wind and tide in 
 different directions presents another example of the composition 
 of motion. If we suppose the wind to impel the vessel in the di- 
 rection of the keel, and at the same time the tide to act at right 
 angles to the keel, giving the vessel a lateral motion, the real 
 course of the vessel will be intermediate between the direction of 
 the keel and the direction of the tide at right angles to the keel. 
 The exact direction of this course may be determined by the com- 
 position of motion, provided the force of the tide and the effect of 
 the wind be known. 
 
 Let us suppose, for example, that the force of the tide is four 
 miles an hour, and that the effect of the wind is such that if there 
 were no tide the vessel would be carried in the direction of its keel 
 at the rate of seven miles an hour. The actual course of the vessel 
 will then be the diagonal of a parallelogram, one side of which is 
 in the direction of the vessel, measuring seven miles, and the other 
 at right angles to the keel, and in the direction of the tide, measur- 
 ing four miles. 
 
COMPOSITION OF MOTION. 
 
 57 
 
 173. The action of the oars in impel- 
 ling a boat is an example of the composi- 
 tion of forces. Let A (fig. 1 8.) be the 
 head, and B the stern of the boat. The 
 boatman presents his face towards B, and 
 places the oars so that their blades press 
 against the water in the directions c E 
 D F. The resistance of the water pro- 
 duces forces on the sides of the boat in 
 the directions G i and H , which by the 
 composition of force are equivalent to 
 the diagonal force K L, in the direction 
 of the keel. 
 
 1 74. Similar observations will apply to almost every body im- 
 pelled by instruments projecting from its sides, and acting against 
 a fluid. The motion of fishes, the act of swimming, the flight of 
 birds, are all instances of the same kind. 
 
 175- Numerous other examples maybe derived from naviga- 
 tion, illustrative of the composition and resolution of forces and 
 motion. The action of the wind on the sails of a vessel, and the 
 effects produced by the reaction of the water, is one of these. 
 Let A B (Jig. 19.) represent the position of the sail, and let the 
 wind be supposed to blow in the direction 
 c D oblique to the sail, its force being repre- 
 sented by the line c D. Let c D be considered 
 as the diagonal of a parallelogram whose sides 
 are E D and F D, the former perpendicular to 
 the sail, and the latter in the direction of its 
 surface. We may then consider the actual 
 wind to be represented by two different 
 winds, one of which will blow perpendicular 
 a - to the surface of the sail, and the other will 
 
 i> ig( , g> strike the sail edgewise, and will therefore 
 
 produce no effect. 
 
 The effective part, therefore, of the wind c D will be repre- 
 sented by E D. Now, this force E D, thus acting perpendicular to 
 the sail, will still act obliquely to the keel. Let the force there- 
 fore of the wind upon the sail be represented by D G, the diagonal 
 of a parallelogram, one side of which, D H, is in the direction of the 
 keel, and the other, D i, perpendicular to it. The component of the 
 wind, therefore, expressed by D E, may be imagined to be replaced 
 by two distinct winds, one, D H, in the direction of the keel, and the 
 other, D i, at right angles to the keel. 
 
 The original force of the wind c D will thus be expressed by 
 three distinct winds, one, D r, striking the sail edgewise, and there- 
 
58 OF FORCE AND MOTION. 
 
 fore inefficient ; another, D i, acting at right angles to the vessel, 
 and therefore producing lee way ; and the third, D H, acting in the 
 direction of the keel from stern to stem, and therefore propelling 
 the vessel. 
 
 In this case, it appears that a wind blowing at right angles to 
 the course of the vessel, and having therefore in itself no tendency 
 to propel the vessel, is nevertheless, by the resolution of forces, 
 rendered efficient for propulsion. 
 
 It is easy to show that this principle may be pushed further, and 
 that a wind which may blow in a direction nearly opposite to the 
 course of a vessel may, by the proper application of the same prin- 
 ciple of the resolution of force, be made to propel a vessel. In 
 Jig. 20. the wind is represented as blowing in the direction c D, 
 forming an acute angle with the course of the 
 
 /vessel, and therefore to a proportional extent 
 opposed to it. Let us suppose that the rigging 
 admits of placing the sail so as to form a still 
 more acute angle with the course of the vessel 
 than does the wind. 
 
 The sail will thus lie between the line c D, 
 representing the direction of the wind, and the 
 line D v, representing the direction of the keel. 
 As before, c D, by two successive resolutions of 
 forces, is shown to be equal to three different 
 winds : I st, a wind represented by F D, blowing edgewise on the 
 sail, and therefore inefficient ; 2ndly, a wind, D i, blowing at right 
 angles to the course of the vessel, and therefore producing lee 
 way ; and 3rdly, a wind, D H, in the direction of the keel, from stern 
 to stem, and therefore efficient for propulsion. 
 
 It is evident, however, that the more oblique the wind may be 
 to the course of the vessel, the smaller will be the component of 
 its force represented in the diagram by D H, which is efficient for 
 propulsion; and the greater will be the component r D, acting 
 edgewise on the sail, and therefore inefficient. This will be ap- 
 parent from the mere inspection of the two diagrams. 
 
 The limit of the practical application of this principle in sailing 
 is determined by the play given by the rigging to the sails. There 
 is in each species of rigging a practical limit beyond which it is 
 impossible to place the sails obliquely to the keel. A wind which 
 blows upon the quarter near this limit cannot be rendered useful. 
 It will be apparent, therefore, that different kinds of rigging supply 
 different limits to the application of this principle ; and we ac- 
 cordingly find that one form of vessel is capable of sailing nearer 
 to the wind than another. 
 
 176. But even though the wind should blow in a direction 
 
COMPOSITION OF MOTION, 59 
 
 immediately opposed to the course of the vessel, we are enabled, 
 by another application of the principle of the resolution of force, to 
 press such an adverse wind into the service, and to render it avail- 
 able in carrying the vessel directly against its own force. 
 
 This object is attained by the expedient called tacking, which is 
 nothing but a practical application of the resolution of force. 
 Supposing the course which the vessel has to pursue to be due 
 west, while the wind is blowing due east, then the course of the 
 vessel due west is to be regarded as the diagonal of a parallelogram 
 whose sides are directed alternately to some points north and south 
 of west. 
 
 Let A w (Jig. 2 1 .) be the course of the vessel departing from A ; 
 it sails from A to D, some points north of west ; from D it sails to 
 
 D', some points south of 
 \ V\TV/\ west; from D' it sails to 
 W - < " * D// ara ll e l to A D and 
 
 j j from D" to D'", parallel to 
 
 Fig. zi. D D' ; and so on. Thus, at 
 
 first, instead of sailing di- 
 
 rect from A to B', it reaches B' by sailing over two sides A D and 
 D B' of a parallelogram whose diagonal is A B'. Again, instead of 
 sailing directly from B' to B", it sails from B' to D', and from D' to 
 B", over two sides of a parallelogram whose diagonal is B' B" ; and 
 so on. The vessel is thus conducted over the sides of a series of 
 parallelograms, the succession of whose diagonals forms its right 
 course. 
 
 177. If a ball of lead be taken to the masthead of a vessel which 
 is advancing under sail or steam, and be let drop downwards, it 
 might at first be supposed that this ball would fall at a point ver- 
 tically under that from which it was discharged, in which case it 
 would necessarily strike the deck just so far behind the mast as 
 would be equal to the space through which the vessel had ad- 
 vanced in its course during the time of the fall. 
 
 But we find, on the contrary, that the ball strikes the deck at 
 the foot of the mast precisely in the place where it would have 
 struck it had the vessel been at rest. 
 
 This fact is explained by the effect 
 of the composition of motion. 
 
 Let AB (Jig. 22.) be the course of the 
 ship. Let B T be the position of its mast 
 at the moment the ball is dropped ; and 
 let B'T' be the position of the mast at the 
 moment the ball falls on the deck, the 
 
 F1 M ship having advanced through the space 
 
 B B' during the interval of the fall. The 
 
60 OF FORCE AND MOTION. 
 
 ball by being dropped from the top of the mast has in common 
 with the ship its progressive motion ; and if it had not beeh 
 dropped, it would have moved through the space XT', precisely 
 equal to B B', in the time of the fall. 
 
 This progressive motion during the fall is combined with the 
 vertical motion, and if the vertical descent were made with a uni- 
 form velocity, the ball would, by reason of the combined effect of 
 the two motions, move along the diagonal of the parallelogram, 
 T B B' T'. But the vertical descent of the ball not being uniform, 
 but first moving more slowly, and then moving more rapidly, it 
 will move, not along the diagonal of the parallelogram, but along 
 a curve represented by the dotted line in the figure, which curve 
 will be explained hereafter ; but at the termination of the fall the 
 ball will be found at the foot of the mast. 
 
 178. The same illustration is presented in a still more striking 
 form by the movement of a carriage on a railway. Let us sup- 
 pose that a carriage is moved along a line of railway, at the rate 
 of 60 miles an hour, and a ball is dropped from it at the height of 
 1 6 feet as it moves. If this ball fell vertically it would strike the 
 ground at a point 30 yards behind the point of the carriage from 
 which it was dropped ; for the time of falling 16 feetas one second, 
 and in one second a carriage moving 60 miles an hour would move 
 over 30 yards. The carriage would, therefore, be 30 yards in ad- 
 vance of the point at which the ball was let fall. But it is found 
 that, as in the former case, the ball will meet the ground at a 
 point vertically under the part of the carriage from which it was 
 let fall, which renders it evident that during the fall the ball 
 advances with the same progressive motion as the carriage. 
 
 Let T (fig- 23.) represent the point from which the ball is dis- 
 engaged, and B the point of the rails vertically under it, the height 
 TB being 16 feet. Let BB' be 30 yards. In one second after the 
 ball has been disengaged, the point T will have been carried for- 
 
 T' T 
 
 B' B 
 
 Fig.ij. . 
 
 wards to T' vertically above B'. At the moment the ball was dis- 
 engaged from T, it participated in the progressive motion of the 
 carriage; and consequently, having a motion in the direction TT', 
 which, if it had not been disengaged, would have carried it to x'in 
 one second, during the fall this motion is combined with that of 
 the vertical descent, and the combination of the two motions will 
 
COMPOSITION OF MOTION. 
 
 61 
 
 cause the ball to move over the curve represented by the dotted 
 line, and to arrive at B', the point vertically under T', at the end 
 of a second. 
 
 " These effects may be illustrated in a still more striking manner 
 by supposing a vertical tube 16 feet high carried with the train, 
 like the smoke-funnel of the engine. If a ball were held over the 
 centre of the top of the mouth of this funnel, and let fall, it would, 
 if the train were at rest, strike the bottom of its centre point, fall- 
 ing directly along the centre or axis of the tube. If the tube be 
 carried forward with any velocity, such as 60 miles an hour, the 
 side of the tube will not be carried against the ball by such pVo- 
 gressive motion, as might be imagined, but the ball during its 
 descent will still keep exactly in the centre of the tube, as it would 
 have done had the tube been at rest. 
 
 1 79. The skill of the billiard player depends on his dextrous 
 application of the composition and resolution of force. All the 
 movements of the balls, in obedience to the cue, and whether re- 
 flected from each other or from the cushions, are determined by 
 this principle. Let P (j%. 24.) be a billiard-ball driven in the 
 direction P o by the cue. At o let it strike the cushion M N. The 
 force of the impact may be represented by the dotted line OA, 
 which is the diagonal of a parallelogram, one side of which is o B 
 perpendicular to the cushion, and the other o c in the direction of 
 the cushion. The effect, therefore, is the same as if, at the mo- 
 ment the ball struck the cushion, it were influenced by two inde- 
 pendent forces repre- 
 sented by oc and OB. 
 The force o B being 
 perpendicular to the 
 cushion is destroyed by 
 its reaction; but the 
 ball, being elastic, re- 
 ceives a rebound in the 
 contrary direction o B'. 
 The force o c being in 
 the direction of the 
 cushion is not destroyed, 
 and being combined 
 with that of the re- 
 bound OB' will cause 
 the ball to move along 
 the diagonal OD of the 
 Fi parallelogram, of which 
 
 these two lines o c and 
 o B' are the sides If o, instead of being a point upon the cushion 
 
 2S 
 
 m 
 
 -^\ lj 
 
 M 
 
62 
 
 OF FORCE AND MOTION". 
 
 had been a point upon the surface of another ball, the effects 
 would be the same ; only in this case the line M N would represent, 
 not the cushion, but a tangent plane to the ball at the point of 
 impact. 
 
 In Jjg. 25., the cushion of the 
 table and the ball are shown, 
 AB being the direction in which 
 the ball strikes the cushion ; B c' 
 that in which it is reflected, and 
 BD the force of the blow on the 
 cushion. 
 
 The skill of the billiard player 
 consists in a knowledge of the com- 
 bination of such effects of the composition and resolution of force ; 
 but instead of deriving it from the physical principles, he obtains 
 his knowledge by long- continued experience. He knows the 
 effects, but cannot explain them. 
 
 ~Ln.Jig. 26. a stroke is represented in which a cannon is made, 
 after successive reflections from each of the four cushions, at the 
 
 [Fig. 26. 
 
 points marked o. The ball P is first directed in the line P o, upon 
 the ball P', so that being reflected from it, it strikes the four 
 cushions successively at the points marked o, and is finally re- 
 flected so as to strike the third ball P". At each of the reflections 
 from the ball P' and the four cushions, the same composition and 
 resolution offeree takes place as is represented in Jig. 24. ; and the 
 diagrams showing such composition and resolution are given in 
 fig. 26. 
 
 1 80. The principle of the composition and resolution of force 
 has been ingeniously applied for the purpose of obtaining a direct 
 demonstration of the diurnal motion of the earth. 
 
 If a high tower or steeple be erected on the surface of the earth, 
 it is evident that, in consequence of the revolution of the globe 
 
COMPOSITION OF MOTION. 63 
 
 upon its axis, the top of the tower will be moved in a greater 
 diurnal circle than the base, being more distant from the common 
 centre round which the entire world is moved. The top of the 
 tower, therefore, and anything placed upon it, has a greater velo- 
 city from west to east, which is the direction of the earth's rota- 
 tion, than has the bottom. 
 
 Now if we imagine a heavy ball to be let fall from the top of the 
 tower towards the base, this ball will be affected by two motions : 
 1st, that which it has in common with the top of the tower from 
 west to east, in virtue of the earth's diurnal motion ; and 2ndly, 
 that vertical motion which it has in falling. The course it will 
 follow will therefore depend on the combination of these two mo- 
 tions, and it will strike the ground at a point east of that which it 
 occupied at the commencement of its fall, by a space equal to that 
 through which the top of the tower is carried during the time of 
 the fall. But during this same interval, the base of the tower is 
 also moving eastward, but, as has been explained, through a less 
 space. 
 
 Now as the ball is carried eastward through the space through 
 which the top of the tower is carried, while the base of the tower 
 is carried eastward through a less space, the ball, instead of falling 
 at the base of the tower, which it would do, if there were no diur- 
 nal rotation of the earth, will fall just so much east of the base as 
 is equal to the difference between the motion of the top and the 
 motion of the bottom of the tower. 
 
 As the difference between its two motions must be an extremely 
 minute quantity, it might be supposed that such an experiment, 
 though beautiful in theory, would be impracticable, the quantity 
 which would indicate the effect of rotation being smaller than 
 could be accurately measured. 
 
 The experiment, nevertheless, has been made ; and the result 
 has been, within the limits of error, such as would be produced 
 by a diurnal revolution of the earth in twenty-four hours. 
 
 1 8 1 . Motion is distinguished into absolute and relative. 
 
 182. If a man walk upon the deck of a ship from stem tp stern, 
 he has a motion relative to the deck, measured by the space upon 
 it over which he walks in a given time ; but while he thus walks 
 from stem to stern, the ship and its contents, including himself, are 
 carried in an opposite direction. 
 
 If it should so happen that his own progressive motion from 
 stem to stern is exactly equal to the progressive motion of the 
 ship, then he will be at rest with regard to the -surface of the sea, 
 and will be vertically above the same point of the bottom ; for his 
 own motion on the surface of the deck in one direction is, by this 
 
64 OF FORCE AND MOTION". 
 
 supposition, exactly equal to the motion of the ship in the other 
 direction. 
 
 If he walk upon the deck at a slower rate than the progress of 
 the ship, then he will have a motion relatively to the sea, in the 
 direction of the ship's motion, equal to the difference between the 
 rate of the ship and the rate of his own motion on the deck ; but 
 if he move from stem to stern on the deck at a more rapid rate 
 than the ship advances, then he will have a motion relative to the 
 sea contrary to that of the ship, and equal to the difference be- 
 tween the rate of the ship and the rate of his own motion on the 
 deck. 
 
 But these motions are again compounded with that of the earth, 
 in which the man who walks, the ship which sails, and the sea 
 itself participate ; and if the absolute motion of the man who 
 walks upon deck is required to be determined, it would be neces- 
 sary to combine, by the principles of the composition of motion, 
 I st, the motion of the man upon the deck ; 2ndly, the motion of 
 the ship through the water ; 3rdly, the motion of the earth on its 
 axis ; and 4thly, the motion of the earth in its orbit. 
 
 183. Many of the feats exhibited in gymnastic and equestrian 
 exhibitions are explained by the principles of the composition and 
 resolution of motion. 
 
 As an example, let us take the case in which, the exhibitor 
 standing on the saddle, a table or other elevated object is held 
 before him, above the height of the horse, but below that of the 
 rider, so that the horse may pass under the table, which would ob- 
 struct the progress of the rider. In this case the rider leaps over 
 the table, and returns to the same point of the saddle which he 
 left when the horse had passed under the table, Now, this feat 
 demands from the rider a muscular exertion, extremely different 
 from that which might be expected. 
 
 Let us suppose the course of the horse to be represented by 
 
 A A' (Jig. 27.), the table by 
 TB, and the position of the 
 rider upon the saddle when 
 the horse approaches the 
 table, and when the leap is 
 about to be effected, by 
 ED. At this point the rider 
 leaps, not with a force 
 which would project him 
 over the table, but with 
 one which would project 
 
 him vertically upwards to the position represented 1 by r' d'. This 
 motion, combined with -the progressive motion which the rider 
 
 
COMPOSITION OF MOTION 
 
 fig. 28. 
 
 has in common with the horse, and which is represented by RT, 
 will cause the rider to move, not directly upwards, in immediate 
 obedience to his exertion, but in the diagonal direction R r, so that 
 at the moment the horse comes directly under the table, the rider 
 is directly over it at r d. The upward force of the leap being here 
 expended, the body of the rider begins to fall, and, if not urged by 
 a progressive motion, would fall on the table. But retaining the 
 progressive motion, the descending tendency of the fall is combined 
 with such motion, and the rider accordingly descends in the dia- 
 gonal direction TR', and arrives at the point R' precisely at the 
 moment that the saddle borne by the horse arrives at the same 
 point, so that the rider returns to his position on the saddle at R' 
 which he left at R. 
 
 Strictly speaking, in this example, the motion of the rider does 
 
 not take place in the 
 right lines represented 
 by the diagonals in the 
 figure, but in lines 
 slightly curved (fig. 
 28.). This, however, 
 makes no difference in 
 the principle involved 
 in the case. 
 
 1 84. The flying of a kite presents an example of the principle 
 
 of the composition of 
 forces. Let K (fig. 
 29.) be the point 
 on which the force 
 of the wind may be 
 considered as con- 
 centrated, and let 
 the direction and 
 quantity of this force 
 be represented by 
 the horizontal line 
 w. This line K w 
 may be taken as the 
 diagonal of a paral- 
 lelogram, one side 
 
 Fig . 29 . of which, K L, is in 
 
 the direction of the 
 
 surface of the kite, and the other, K M, perpendicular to it. 
 The former component, K L, being in the direction of the surface 
 of the kite, glides off without effect ; the latter, K M, being perpen- 
 dicular to the kite, is effective. If the , string of the kite be in 
 
 
66 OF FORCE AND MOTION. 
 
 the direction K A, directly opposed to K M, then the tension of 
 the string will balance the force K M, and the kite will remain 
 suspended in the air, without rising higher; but if the direc- 
 tion of the string be KA', making an angle with K M, then the 
 tension of the string acting in the direction K A', and the component 
 of the wind acting in the direction of K M, will produce a resultant 
 action in some intermediate direction such as K N, and this resultant 
 will cause the kite to rise to the point N, describing a circle round 
 the centre A', and the kite will ascend in this circle until the string, 
 K A, takes the direction of the component of the wind to which it is 
 opposed. 
 
 185. That a mass of matter moving in any manner exercises a 
 certain force against any object which may lie in its way, is the 
 physical law which, of all others, is the most early, the most fre- 
 quent, and the most universal result of observation and experience. 
 The child has hardly emerged from the nurse's arms, before it be- 
 comes conscious of the force with which its body would strike the 
 ground if it fell. 
 
 1 8 6. Momentum. The moving mass of a hammer-head will 
 exercise a force upon a nail sufficient to make it penetrate wood ; 
 an effect which no common pressure could produce. The force of 
 the hammer-head of the sledge-hammer will compress and vary 
 the form of a mass of metal under it. By the force of a descending 
 mass of matter, the die impresses upon a piece of metal the image 
 and characters of the coin with more fidelity and effect than the 
 pressure of the hand could produce them, if similarly applied to 
 wax. 
 
 The force of a moving mass will cause a punch to penetrate a 
 thick plate of iron, or a shears to cut the same plate, with as much 
 facility as a needle would penetrate, or a common scissors cut this 
 paper. 
 
 The moving masses of the spear, the javelin, and the arrow, and 
 of the bullet and the cannon-ball, have been used as destructive 
 projectiles in war, ancient and modern. 
 
 187. This quality equally appertains to matter in the liquid 
 form. 
 
 The force of the torrent, when a river overflows its banks, has 
 carried away buildings and levelled towns. The force of the stream 
 is used to turn the mill and discharge various mechanical func- 
 tions. They who have stood at the foot of Niagara, have been 
 conscious of the frightful energy of the mechanical power deve- 
 loped by the motion of the mass of waters forming that cataract. 
 
 1 8 8. The same quality belongs to matter even in the attenu- 
 ated form of air. The force of air in motion carries the ship over 
 the sea, and, acting upon the diverging arms, impels the mill. The 
 
MOMENTUM. 67 
 
 tempest agitates the deep, and flings the largest vessel, with de- 
 structive force, upon the rock. The force of the moving atmo- 
 sphere in the hurricane devastates countries, overturning buildings, 
 and tearing up by the roots the largest trees. 
 
 It will therefore be understood how important it is to investigate 
 the laws which govern a quality of matter which developes such 
 effects. 
 
 189. It is not enough to know that matter in motion will exer- 
 cise this force on any object which it encounters ; we must be able 
 to express, with arithmetical precision, the conditions on which 
 the energy of this force in each case depends ; we must be able to 
 determine, when two different masses of matter are moved under 
 known conditions, what is the proportion between the forces which 
 they would respectively exert on any object which they might 
 strike. 
 
 1 90. If a leaden ball, of a certain magnitude, move with a cer- 
 tain velocity, it will strike an object with a certain determinate 
 force. If another leaden ball, of the same magnitude, moving 
 beside it with the same velocity, strike the same object at the same 
 time, it will evidently strike it with the same force, so that the 
 force exerted by the two balls will be precisely double the force 
 which would be exerted by either of them. 
 
 191. But if we suppose the two balls moulded into one, so as to 
 make a ball of double magnitude, the force of the impact will still 
 be the same. It results, therefore, in general, that if two bodies 
 be moved with the same velocity, one having twice the quantity 
 of matter of the other, the force of the latter will be twice that of 
 the former. 
 
 By the same mode, of reasoning, it may be shown generally that, 
 in whatever proportion the mass of the body in motion be in- 
 creased, the force with which it would strike an object in its way 
 will be increased in the same proportion. This force, exerted by 
 a mass of matter in motion, is called in mechanics by the term 
 momentum, and sometimes by the phrase moving force. 
 
 When the velocity is the same, therefore, the momentum or 
 moving force of bodies is directly proportionate to their mass or 
 quantity of matter. 
 
 It is found that any force which would impel a ball with a given 
 velocity must be doubled, if the ball require to be impelled with 
 double the velocity, and increased in a threefold proportion if the 
 ball be required to be impelled with three times the velocity, and 
 so on. It is evident, therefore, that the moving force of a body 
 will be augmented in the exact proportion in which its velocity is 
 increased, its mass or quantity of matter remaining the same. 
 Thus the force with which a ball weighing an ounce and moving 
 
68 OF FORCE AND MOTION 
 
 at ten feet per second will strike any object, will be exactly ten 
 times the force with which the same ball, moving at one foot per 
 second, would strike such an object. 
 
 These fundamental principles are so obviously consistent with 
 universal experience, that they can scarcely be said to require 
 proof. 
 
 192. Examples. If we project a stone from the hand, we give 
 it but a slight impulse, if our purpose be to impart to it a slow 
 motion ; but if we desire to project it with greater speed, we 
 exercise a greater force of projection. Now, as the stone receives 
 all the force communicated by the hand, it is evident that the 
 increase of force in this case is exhibited by the increase of velo- 
 city of the motion of the projectile, the body projected being the 
 same. 
 
 If we find a certain muscular exertion sufficient to project a 
 ball IO Ibs. weight with a certain velocity, we shall find it neces- 
 sary to use double the force if we desire to project a ball 20 Ibs. 
 weight with the same velocity. 
 
 193. A man's force exerted upon oars can propel a skiff weigh- 
 ing 100 Ibs. through the water at the rate of ten feet per second ; 
 but the same force applied in the same manner to a vessel weighing 
 a hundred tons would not move it faster than a sixteenth of an 
 inch per second ; for since the same force would be then applied 
 to a body more than two thousand times heavier, the speed of the 
 motion would be more than two thousand times less. 
 
 194. To express momentum arithmetically, let M express the 
 mass of the body, and v its velocity ; then M X v will express its 
 moving force : and if M' express the mass of another body, and v' 
 express its velocity, M' X v' will express its moving force. 
 
 Nothing can be more simple or easy than the use of such symbols, 
 provided it be observed that the same units are used in each case 
 for matter, and space, and time. Thus, for example, if in the first 
 case the mass M be expressed by its weight in pounds, and the 
 velocity v by the number of feet moved over per second, then it 
 is necessary, in the second case, that the mass M' should also be 
 expressed in pounds, and the velocity v' also in feet per second. 
 
 For example, if the mass M be I o Ibs., and the velocity v 5 feet 
 per second, then the product M X v= 50; the meaning of which 
 is, that the moving force of the body M having the velocity v, will 
 be the same as the moving force of a body weighing 50 Ibs., moved 
 at one foot per second. 
 
 In like manner, if the mass M' be 5 Ibs. and its velocity v' 10 
 feet per second, then its momentum will be M' X v' = 50, showing 
 that in this ease also the momentum is the same as that of a body 
 weighing 50 Ibs. moving through one foot per second. 
 
COMMUNICATION OF MOMENTUM. 69 
 
 In these two cases the momenta are equal, although the mass of 
 one body be double that of the other ; but it will be observed that 
 the lighter mass obtains as much force by its superior velocity as 
 the other has by its superior quantity. 
 
 195. General condition ot the equality of moving forces. 
 These examples being generalized, we obtain the following theorem, 
 which is of considerable use in mechanics : When the momenta of 
 two bodies are equal, their velocities will be in the inverse proportion 
 of their quantities of matter. The meaning of which is, that the 
 velocity of the lesser body will be just so much greater than the 
 velocity of the greater, as the quantity of matter in the latter is 
 greater than the quantity of matter in the former. 
 
 A ball of cork which strikes upon' a plank of wood, even with 
 the greatest velocity, will scarcely produce an indentation of its 
 surface. A ball of lead striking on the same plank with the same 
 velocity, will penetrate it. The force of the latter will be greater 
 than the former, in the same proportion as lead is heavier than 
 cor. 1 : 
 
 CHAP. III. 
 
 THE COMMUNICATION OF MOMENTUM BETWEEN BODY AND BODY. 
 
 196. Effects of collision. When a body in motion encounters 
 another body, certain changes ensue in the motion and in the 
 moving force of both bodies. These changes are in general of a 
 complicated kind, depending on the degree of elasticity of the 
 bodies, their form, weight, and other physical circumstances. 
 
 197. To simplify the question, however, at present, we shall 
 consider the bodies completely devoid of elasticity, and so consti- 
 tuted that after the one impinges on the other they shall coalesce 
 and move as one body in some determinate direction and with 
 some determinate speed. 
 
 Although these conditions be" not strictly fulfilled in practice, 
 they lead to conclusions of the greatest practical utility, and which 
 approximate more or less to the results of experience, which re- 
 sults, however, are modified by various physical and mechanical 
 conditions, the consideration of which is here omitted. 
 
 198*. If we suppose a mass of matter, M {fig. 30.), moving in a 
 certain direction, A B, with a velocity v, to encounter another mass 
 of matter, M', at rest, and that after the impact the two masses 
 shall coalesce and move together ; let it be required to ascertain 
 
 r 3 
 
70 OF FORCE AND MOTION. 
 
 what will be the direction, velocity, and force of the united mass 
 after such impact. 
 
 The mass M', being supposed to be previously at rest, can have 
 
 Fig. jo 
 
 no motion save what it may receive from the mass M ; and con- 
 sequently, after the impact, it must move in the same direction 
 A B as the mass M moved in before the impact. Whatever moving 
 force M' may obtain by its coalition with M must be lost by the 
 mass M. This is an immediate and very important consequence of 
 the quality of inactivity or inertia, which has been already fully 
 explained. 
 
 Bodies cannot generate force in themselves, nor can they destroy 
 force which they have independent of any external agency. What- 
 ever force, therefore, the mass M' may acquire after the coalition 
 of the two bodies, must be lost by the mass M ; and consequently 
 the total moving force of the uriited masses, after the impact, must 
 be exactly equal to the moving force of the mass M before impact. 
 Now, according to what has been already explained, the moving 
 force of the mass M before impact was M x v. If v be taken to 
 express the velocity of the united mass after such coalition, then 
 since the quantity of matter in the combined mass is M -f- M'', its 
 moving force after coalition will be expressed by (M -f- M') x v. 
 But this moving force cannot be greater or less than the moving 
 force of M before impact ; for to suppose it greater would be to 
 assume that the bodies have a power of creating force in them- 
 selves, with which they were not endued by any external agency ; 
 and to suppose it less would be to suppose them capable of de- 
 stroying a force which they had independently of any external 
 agency. 
 
 Both of these suppositions, however, are equally incompatible 
 with the quality of inactivity or inertia, which implies the absence 
 of all power to create or to destroy any moving force. It follows, 
 therefore, that the momentum of the mass M, before impact, is 
 equal to the momentum of the united mass after impact, which is 
 thus expressed in arithmetical symbols : 
 
 M X v = (M + M') x r. 
 
 That is to say, the mass M, multiplied by its previous velocity, is 
 equal to the united masses multiplied by their subsequent velo- 
 city. 
 
COLLISION. 71 
 
 To find, therefore, the velocity with which the united mass will 
 move, it is only necessary to divide the moving force of the mass M 
 before impact, by the sum of the united masses M and M'. 
 
 199. Examples. As an example of this, let us suppose a boat, 
 connected with a ship by a tow-rope, the latter being slack, to be 
 propelled by rowers, at the rate of ten miles an hour, the boat 
 weighing 5 cwt, and the ship weighing 250 tons, being 1000 
 times the weight of the boat. The moment the rope becomes 
 tight, or, as seamen call it, taut, the moving force of the boat is 
 divided between the mass of itself and of the ship, and the common 
 velocity of the two will be determined by the principles above 
 explained. To find it, therefore, it will be necessary to diminish 
 the previous velocity of the boat in the proportion of the weight 
 of the united masses of the ship and boat to that of the boat, that 
 is, in the ratio of 1001 to I. 
 
 The velocity of the boat having been ten miles an hour, or about 
 1 4 feet per second, the velocity with which the ship would be 
 towed will be, therefore, the -j^+^th part of a foot per second. 
 Thus, the force which was sufficient to propel the towing-boat 
 alone through 145 feet in ten seconds, will only tow the ship 
 through little more than one-sixth of an inch in the same time. 
 
 200. Collision of two bodies moving in same direction. 
 We have here taken the case in which the body M' is at rest. 
 Let us now suppose that it has a motion in the same direction AB, 
 as that of the mass M, but with a velocity v' less than v the 
 velocity of M, so that the body M shall overtake the body M', and 
 that after coalition they shall move together with a common 
 speed ; what will this common speed be ? It follows, as a conse- 
 quence of the quality of inertia, that, . after their coalition, the 
 united mass cannot have a moving force either greater or less than 
 they had before their union ; consequently, the moving force of 
 the coalesced bodies will be found by adding together the two 
 moving forces which they had before their coalition. These two 
 forces, according to what has been explained, being expressed by 
 M X v and M' X v', the moving forces of the coalesced mass will be 
 M X v + M' X v'. But if v express the velocity of the united 
 mass, its moving force must also be expressed by this velocity P, 
 multiplied by the total mass, that is, by M + M'; and, consequently, 
 we have 
 
 M X v + M' X v' = (M -f- M') X v. 
 
 It follows, therefore, from this, that if we divide the sum of the 
 moving forces of the two bodies, before their union, by the sum of 
 their masses, the quotient will be the velocity of the united mass 
 
72 OF FORCE AND MOTION". 
 
 after their union; the moving force of the united mass being equiva- 
 lent to the sum of their moving forces before their union. 
 
 It is evident that this common velocity, which the combined 
 bodies will have after their union, will be less than the speed of 
 the body M, which overtakes the other, and greater than that of the 
 body M', which is overtaken. The one gains and the other loses ve- 
 locity ; and, consequently, it is evident that the one must gain and 
 the other lose momentum. Now, this gain and loss, on the one 
 side and on the other, is always precisely equal. Whatever mo- 
 mentum the body struck receives from the body striking, the latter 
 loses, and neither more nor less. There is, therefore, a precisely 
 equal change of momentum. 
 
 20 1 . Action and reaction. This general principle, which, as 
 we have seen, is a direct consequence of the quality of inertia, and 
 which, if not established, would lead to the conclusion that mere 
 matter has a power in itself to produce or destroy its motion, is 
 usually announced under the form of the following mechanical 
 dogma or maxim : 
 
 ACTION AND REACTION ARE EQUAL AND CONTRARY. 
 
 The term action, applied to the case above explained, is the 
 power which the striking body has to give increased momentum 
 to the body struck ; and reaction expresses the correlative power 
 which the body struck has of depriving the striking body of an 
 exactly equal quantity of momentum. 
 
 This celebrated law of the equality of action and reaction, there- 
 fore, means nothing more than the equal interchange of momentum, 
 in contrary directions, between two bodies which come into col- 
 lision, and, as such, is an immediate consequence of the quality of 
 inertia. 
 
 202. If a boat weighing a hundred-weight, rowed at the rate of 
 fifteen feet per second, be suddenly connected by a towing-line 
 with another boat which is rowed in the same direction at the rate 
 of ten feet per second, and which weighs two hundred-weight, then, 
 the momentum of the first being expressed by 1 5, and that of the 
 second by 20, their combined momenta will be 35, which, being 
 divided by their united weights 3, gives the quotient 1 1, or 1 1 ft. 
 8 in., the space per second through which they will be rowed 
 together. 
 
 203. Finally, let us consider the case in which the two bodies M 
 and M' {fig. 3 1 .) are moving, not in the same, but in contrary 
 directions. Let the body M be supposed to be moving from A 
 towards c, and the body M' from B towards c ; and let c be the 
 point at which they will coalesce, the momentum of M being sup- 
 posed to be greater than that of M'. On their coalition, the mo- 
 
ACTION AND REACTION. 
 
 73 
 
 mentum of M' will destroy just so much of the momentum of M as 
 is equal to its own amount ; for it is evident that equal and con- 
 
 M 
 
 M 
 
 -A 
 
 .in. 
 
 Ot 
 
 Fig. 31. 
 
 trary forces must destroy each other. After the union of the two 
 bodies, the momentum, therefore, that will remain undestroyed, 
 will be the excess of the moving force of M over the moving force 
 of M' ; and this excess will be in the direction c B of the progressive 
 motion of the body M. But, as we have seen, the moving force of 
 the body M is expressed by M X v ; and the moving force of the body 
 M' is expressed by M' X v'. It therefore follows that the moving 
 force which the united mass will have in the direction A B will be 
 M X v M' X v'. But it is also apparent that the moving force of 
 the united mass will be expressed by the quantity of the united 
 mass multiplied by its velocity. If we express, as before, this 
 velocity by 0, then we shall have 
 
 M X v M' X v'= (M + M') X v. 
 
 This statement implies nothing more than that the momentum 
 of the masses, after their coalition in the direction of the greater 
 force, will be equal to the difference between their momenta in 
 contrary directions before coalition. 
 
 It follows, also, from this, that the velocity of the united masses 
 in the direction of that which has the greater force before their 
 coalition, will be found by dividing the difference between their 
 momenta by the sum of their masses. 
 
 204. This case affords another example of the application of 
 the maxim, that action and reaction are equal and contrary. The 
 effect which takes place when the bodies coalesce may be thus 
 explained. 
 
 When the two bodies M and M' meet at c (Jig- 3 1 ) the action 
 of the body M upon the body M' destroys the entire momentuin of 
 the latter in the direction B c, and, in addition to that, imparts to 
 it a certain momentum in the contrary direction o B, which will be 
 expressed by the mass M', multiplied by the velocity which that 
 mass has in common with M, in the- direction c B, after their coa- 
 lition. This velocity is expressed by v; and, consequently, the 
 momentum imparted to M', in the direction c B, after impact by 
 the action of the body M, will be M' X v. The total action, there- 
 fore, of the body M upon the body M' will be made up of the mo- 
 mentum of M', which is destroyed in the direction B c, and which 
 
74 OF FORCE AND MOTION. 
 
 is expressed by M' X v', and the momentum which is imparted to it 
 in the contrary direction c B,. and which is expressed by M' X v. 
 Therefore, the entire action of M upon M' will be expressed by 
 
 M' X v' + M' X v. 
 
 Such, then, being the effect of the action of M upon M', let us now 
 consider what will be the effect of the reaction of M' upon M. Upon 
 the collision at c, when M' loses its entire momentum in the di- 
 rection c A, it will destroy in the body M an exactly equal amount 
 of momentum ; that is to say, the body M will lose, in the direction 
 A c, a momentum equal to M' X v', which is lost by the body M'. 
 But M also imparts, by its action to M', a momentum in the di- 
 rection of c B, which is expressed by M' x v. Now, the reactiori 
 of M' upon M, in receiving this momentum in the direction of c B, 
 must deprive M of exactly the same momentum in that direction ; 
 that is, it must deprive it of a momentum in the direction c B 
 expressed by M' X v. Hence it follows that the total amount of 
 momentum lost by reason of the reaction of M' upon M is ex- 
 pressed by 
 
 M' X v' + M' X v, 
 
 which is precisely equal to the momentum gained by M'. It ap- 
 pears, then, that in this case the law of equal action and reaction 
 is still fulfilled ; the action of M upon M' being precisely equal to the 
 reaction of M' upon M. 
 
 205. When two equal bodies meet, moving with equal velocities 
 in opposite directions, their shock will immediately destroy each 
 other's momentum ; for in this case, the momenta, being equal and 
 contrary, will be mutually destroyed. The force of the shock 
 produced by the two bodies in this case will be equal to the force 
 which either, being at rest, would sustain, if struck by the other 
 moving with double the velocity ; for the action and reaction being 
 equal, each of the two will sustain as much shock from reaction as 
 from action. 
 
 206. Examples. If two railway trains, moving in contrary 
 directions at twenty miles an hour, sustain a collision, the shock 
 will be the same as if one of them, being at rest, were struck by 
 the other moving at forty miles an hour. 
 
 If two steam-boats, of equal weight, approach each other, one 
 moving at twelve miles an hour, and the other fifteen miles an 
 hour, each will suffer a shock from the collision, the same as if it 
 were struck by the other moving at twenty-seven miles an hour. 
 
 207. In the combats of pugilists, the most severe blows are those 
 struck by fist against fist, for the force suffered by each in such 
 case is equal to the sum of the forces exerted by either arm. Skil- 
 
ACTION AND REACTION. 75 
 
 ful pugilists, therefore, avoid such collisions, since both suffer 
 equally and more severely. 
 
 208. In what precedes we have limited our observations to the 
 cases in which the bodies which coalesce are moving either in the 
 same or opposite directions in the same straight line. Let us con- 
 sider now the case in which they move in different straight lines 
 before their coalition. 
 
 Let a body, M (fig. 32.), be supposed to move in the line A B, 
 and from A towards B. At some intermediate point, c, suppose it 
 
 to be struck by another 
 body, M', moving in the 
 line A! B' from A' towards 
 B', and suppose that at the 
 moment M arrives at c, 
 M' also arrives at that 
 point and coalesces with 
 it, and that after their 
 union the bodies move to- 
 gether. The question is, 
 in what direction, with 
 what force, and with what 
 g * ja ' velocity they will be 
 
 moved. This question is easily solved by the application of the 
 principle of the parallelogram of forces already explained. 
 
 Let the velocity with which M moves in the direction A B be 
 expressed by v, and let the velocity with which M' moves in the 
 direction A' B' be expressed by v' ; then M X v will be the moving 
 force of the body M directed from c towards B, and M' X v' will be 
 the moving force of the body M' directed from c towards B'. Let 
 the distance c D represent the force M X v, and let c D' represent 
 the force M' x v', and complete the parallelogram c D E D'. Draw 
 its diagonal c E. This diagonal will then represent the direction 
 and the quantity of the momentum which the combined masses M 
 and M' will have after their coalition. If we would find the ve- 
 locity, it will only be necessary to divide the number expressed by 
 this diagonal c E by the number expressing the sum of the masses 
 M and M' ; the quotient will be the velocity with which the com- 
 bined masses will move from c to E. 
 
 209. Action and reaction modified by elasticity. When a 
 body which strikes a hard surface is elastic, the effects of action 
 and reaction are modified in a manner which it will be neces- 
 sary to explain. 
 
 Let us suppose, for the simplicity of the explanation, that the 
 form of the body is that of a sphere or globe. When it strikes a 
 hard surface with any force, it will be momentarily flattened at the 
 
OF FORCE AND MOTION 
 
 C 
 
 Fig. 34- 
 
 point of impact, and will take an oval form ; the force of the im- 
 pact will compress it in the direction of the blow, and it will be 
 elongated in a direction at right angles to the body. 
 
 Thus, if its spherical form before the blow be represented in 
 jig. 33., and if D c be the diameter which is in 
 the direction of the impact, c being the point at 
 which the impact takes place, and A B be the dia- 
 meter at right angles to D c, then the body at the 
 moment of receiving the force of impact will take 
 an oval form represented in Jig. 34., the diameter 
 A B will be elongated, and D c contracted by the 
 force of the impact. But the body, by force of its elasticity, will 
 make an effort to recover its figure, and the point c will react upon 
 the surface which it strikes ; and by that effort 
 the body will recoil, because the diameter D c, in 
 its effort to recover its original length, will press 
 the matter of the body against the hard surface 
 at c, and this pressure being resisted will cause 
 the body to rebound. 
 2 1 o. Perfect and imperfect elasticity. If the elasticity of 
 the body be perfect, the force with which the spherical figure is re- 
 stored will be equal to that with which it has been compressed into 
 an ellipse, and this force being resisted by the surface, the body will 
 rebound with the same force as that with which it struck it. But if 
 the elasticity of the body be imperfect, then the restoring force will 
 have less intensity than the compressing force, and the body will 
 rebound with less force than that with which it struck the surface. 
 211. If an ivory ball, a substance which possesses elasticity in a 
 high degree, be dropped upon any hard and smooth surface which 
 is level, it will rise very nearly to the height from which it was 
 
 dropped. It would rise ex- 
 actly to that height, but for 
 two causes : first, the want 
 of perfect elasticity; and, 
 secondly, the resistance of 
 the air. 
 
 212. When an elastic body 
 B~ strikes a surface in a direc- 
 tion not perpendicular to 
 that surface, it will be re- 
 flected in another direction, 
 which will depend partly on 
 the direction in which it 
 strikes the surface, and partly 
 on the degree of elasticity of the striking body. Let A B (Jig. 35.) 
 
 Fig, 35- 
 
EFFECTS OF ELASTICITY 
 
 77 
 
 be the surface ; and let the body be supposed to strike it at c, 
 having moved in the direction D c; and let us first suppose that 
 the body is perfectly elastic. The force of the impact at c, being 
 represented by c E, may be resolved into two forces, c F perpen- 
 dicular to the surface AB, and c G parallel to it. We may there- 
 fore suppose the ball at c to be affected by two such forces. 
 Now, since the ball is supposed perfectly elastic, the component 
 c F will cause a rebound in the direction of c F' equal to the 
 force c F. If, therefore, we take c F' equal to c F, the body will, 
 after the impact, be aifected by two forces, c G in the direction of 
 the surface, and c F' perpendicular to it ; and it will accordingly 
 move in the diagonal c E' of the parallelogram of which c F' and 
 c G are sides ; but this parallelogram being in every respect equal 
 to the parallelogram c F E G, the angle formed by c E' with the 
 surface c B will be equal to the angle formed by c E and c D 
 with the same surface. Hence it appears that in this case the 
 body will be reflected from the surface which it strikes at the 
 same angle as that with which it strikes it ; that is to say, the angle 
 D c A will be equal to the angle E' c B. 
 
 213. This principle is usually announced thus: When a per- 
 fectly elastic body strikes a hard surface and rebounds from it, 
 the angle of incidence will be equal to the angle of reflection, and 
 these angles will be in the same plane. 
 
 By the angle of incidence is understood the angle which the 
 direction, DC, of the original motion of the ball forms with the per- 
 pendicular, c i, to the surface struck ; and by the angle of reflection 
 is understood that which the direction, CE', in which the body 
 recoils, forms with the same perpendicular. 
 
 214. But if the body which struck. the hard surface be imper- 
 fectly elastic, then the recoil produced by the component CF 
 
 {Jig. 36.), perpendicular to the 
 surface CB, will be less than the 
 force with which the body strikes 
 the surface at c. The line CF', 
 therefore, which represents this 
 force of recoil, will be less than c F, 
 B and the parallelogram c F' E' G, 
 while it has the side c G in 
 common with the parallelogram 
 CFEG, has the side CF' less than 
 CF. Consequently it is obvious 
 that the angle which c E' makes 
 with the perpendicular c i will be 
 
 greater than the angle which DC makes with the same perpen- 
 
 dicular 
 
 Fig< 
 
/8 OF FORCE AND MOTION. 
 
 This principle is announced thus: When a body imperfectly 
 elastic strikes a hard surface, the angle of incidence will be less 
 than the angle of reflection, and the difference between these 
 angles will be greater, the more imperfect is the elasticity of the 
 body. 
 
 215. The apparatus to illustrate collision experimentally, 
 which is generally found in collections for public instruction, 
 shown in Jigs. 37, 38, 39., consists of balls, inelastic or elastic, 
 
 suspended by di- 
 verging threads, 
 the ends of which 
 are attached to two 
 parallel bars, and 
 which are so ad- 
 justed that the 
 balls, when quies- 
 cent, rest in a ho- 
 rizontal line and 
 in mutual contact. 
 Let us first suppose 
 that two equal in- 
 elastic balls are sus- 
 pended as shown in 
 fig. 37. If one of 
 them, being drawn 
 aside, as in Jig. 38., 
 is allowed to fall 
 against the other, , 
 it will strike it with 
 a force propor- 
 tionate to the dis- 
 tance, B A, from 
 which it fell. After 
 the collision, this 
 force being divided 
 equally between 
 
 Fj the balls, they will 
 
 move together to a 
 
 distance from their position of rest equal to half that from which 
 the ball A fell. Thus, if the distance B A be 12 inches, the two 
 balls, after collision, will move together to the distance of 6 inches, 
 and will then swing together from side to side until they are 
 brought to rest by the resistance of the air. 
 
 If three equal inelastic balls had been thus suspended, one 
 
EFFECTS OF ELASTICITY. 
 
 79 
 
 being, as before, drawn aside to the distance of 1 2 inches, the 
 force, after collision, being equally distributed, all three would 
 move together to the distance of 4 inches. 
 
 In general, if any number be suspended, as inj^g-. 39., the force 
 
 of the single ball will 
 be shared equally 
 among all, after colli- 
 sion, and the whole 
 will move to a dis- 
 tance, which 'will be 
 found by dividing that 
 from which the strik- 
 ing ball fell by the 
 entire number of balls. 
 All these effects will 
 be seen to be in com- 
 plete accordance with 
 the principles ex- 
 plained in the pre- 
 ceding paragraphs. 
 Balls of putty or 
 
 _ moist clay, are suffi- 
 
 * ciently inelastic for 
 
 * lg- 3 8 - such an illustration. 
 
 If the balls be elastic, the effects are different and very re- 
 markable. Let two equal ivory balls, for example, be suspended, 
 as in^g". 37. When one, A, is drawn aside, as in jig. 38., and let 
 fall against the other, B, the immediate effect of the collision, as in 
 the case of inelastic balls, is, that A loses half its force, which it im- 
 parts to B. But B being compressed with this force recovers its 
 figure with equal force, and reacts against A so as to deprive A of 
 the other half of its force and to reduce it to rest. At the same 
 time A being compressed by the reaction of B, recovers its figure 
 with equal force, and reacting upon B imparts to it the other half of 
 its force, so that B moves forward with the full force which A had 
 at the moment of collision, while A remains at rest. The ball B, 
 therefore, will move on to a distance equal to that from which A 
 had descended. Thus, if A fall from 1 2 inches, it will remain at 
 rest after the collision, and B will rise to 1 2 inches, from which it 
 will descend and strike A, which in its turn will rise, B remaining 
 at rest, and so on. 
 
 This alternate motion would continue indefinitely, but for the 
 resistance of the air, by which the range of the vibration is gra- 
 dually diminished. 
 
8o 
 
 OF FORCE AND MOTION. 
 
 If several ivory balls be suspended, as in jig. 39., and one be 
 drawn aside and let fall, the effect of the collision will be trans- 
 mitted through 
 the series to the 
 last ball, which 
 will be affected 
 exactly as if it 
 were immediate- 
 ly acted upon by 
 the first. Each 
 of the interme- 
 diate balls in this 
 case, being equal- 
 ly affected in op- 
 posite directions 
 by the reaction 
 of the contigu- 
 ous balls in re- 
 covering their 
 figures, will re- 
 main at rest, and 
 the extreme balls 
 alone will alter- 
 nately rise and 
 fall until reduced 
 to rest by the re- 
 sistance of the 
 air. 
 
 2l6. The laws 
 of motion. 
 rig. 39. Before the true 
 
 principles of in- 
 ductive science were so well understood and so generally admitted 
 as they are at present, the exposition of the property of inertia, 
 and of its most important consequences, was embodied by Newton 
 in three formularies, called by him the laws of motion, which have 
 attained great celebrity in the history of mechanical science. 
 Although these mechanical maxims have lost much of their im- 
 portance by the more general diffusion of correct principles of 
 physical science, they are, nevertheless, entitled to notice, and 
 ought to be registered in the memory of all students, were it only 
 for their illustrious origin, and to commemorate the difficulty 
 which the true principles of induction had to struggle against, in 
 the extermination of the errors of the old philosophy. These laws 
 of motion are announced as follows : 
 
LAWS OF MOTION. 81 
 
 FIRST LAW. 
 
 Every body must persevere in its state of rest, or of uniform 
 motion in a straight line, unless it be compelled to change that state 
 by a force impressed upon it. 
 
 SECOND LAW. 
 
 Every change of motion must be proportional to the impressed 
 force, and must be in the direction of that straight line in which the 
 force is impressed. 
 
 THIRD LAW. 
 
 Action must always be eaual and contrary to reaction ; or the actions 
 of two bodies upon each other must be equal, and their directions must 
 be opposite. 
 
 217. The first of these propositions is little more than a defi- 
 nition of the quality of inertia. The terms in which the second 
 is expressed require qualification and explanation, without which 
 they might be subject to erroneous interpretation. 
 
 If a body be at rest, it is true that every motion it receives 
 must be proportional to and in the direction of the force impressed, 
 that is, of the force which produces the motion ; but if a body be 
 already in motion in one direction, and receive a force in another 
 direction, then the new direction which the body takes will not be 
 in the direction of the force impressed, nor proportional to it, but 
 will be in the direction of the diagonal of a parallelogram, one 
 side of which represents the force with which the body previously 
 moved, and the other the new force which is impressed upon it. 
 This has been already fully explained. 
 
 In the third law, the word action means the moving force which 
 one body receives when another acts upon it, and the word re- 
 action means the moving force which the latter loses in consequence 
 of communicating force to the former. 
 
 The equality of action and reaction is, therefore, subject to the 
 same qualification as must be given to the terms of the second law 
 of motion. 
 
 Both propositions are literally true only when the two motions 
 in question are directed in the same straight line. When they 
 are directed in different straight lines, then the propositions must 
 be interpreted by the principles of the composition of forces, as 
 already explained. 
 
 218. The consequence of the equality of action and reaction, 
 combined with what has been explained respecting the indestruc- 
 tibility of matter in a former chapter, have led some philosophers 
 to the adoption of the startling physical dogma, that there has 
 been, and must be always, the same quantity of matter and the 
 
 G 
 
82 OF FORCE AND MOTION. 
 
 same quantity of motion in the world ; in other woi ds, that nothing 
 short of divine agency can create or destroy either the smallest 
 portion of matter or the smallest moving force. So far as this 
 applies to matter, it has been already explained ; but a few words 
 may be useful to elucidate the import of this maxim as applied to 
 moving force or momentum, as it may naturally be objected, that 
 since all creatures endowed with animal life have the power of 
 spontaneous motion, how can it with truth be said that there is 
 always the same quantity of moving force in the world ? 
 
 21 9. Must not an animal, who by the act of its will puts its 
 body in progressive motion from one place to another, and after a 
 time, by another act of its will, causes this motion to cease, first 
 create a new motion, and then destroy it? Is there not in such a 
 power manifest contradiction to the maxim which states that 
 there is always the same quantity of moving force in the world ; 
 for while the animal is in motion, is there not the momentum of 
 the mass of its body in existence which did not exist before it 
 began to move ? and is not this momentum destroyed when it 
 ceases to move ? 
 
 220. Let us examine what takes place in such cases. If an 
 animal commence to move its own body on the surface of the earth 
 in any given direction, it obtains its progressive motion by the 
 action of its feet upon the ground. 
 
 Between the mass of the body of the animal and the earth on 
 which it treads, there is therefore an action and reaction, which 
 are equal and opposite. Whatever moving force the body of the 
 animal acquires in one direction, the earth loses in the other ; and 
 therefore the animal may be considered as robbing the earth, so 
 to speak, of the moving force which its body gains. 
 
 But when the animal, after moving to the desired point, brings 
 his body to rest, there is another action upon the .earth. His 
 body is deprived of the momentum which it had acquired by the 
 action of the feet upon the ground, so that the momentum with 
 which the mass of its body moved is now imparted to the earth, 
 gradually, as his motion is retarded, and altogether when he comes 
 to rest. It therefore appears, that the animal takes from the earth 
 his progressive momentum when he begins to move, and returns it 
 to the earth when he comes to rest. 
 
 221. If a railway train, weighing loo tons, be started from a 
 state of rest by a locomotive engine, and attain a velocity of 50 
 miles an hour, it will have acquired a moving force of loo tons, 
 moving at 75 feet per second, which is equivalent to a mass of 
 7500 tons moving at the rate of one foot per second. Now this 
 momentum is obtained by the action of the driving-wheels of the 
 engine on the rails, produced by the force of the steam. 
 
LAWS OF MOTION. 83 
 
 This action is attended with an equal and opposite reaction upon 
 the rail, and through the rail upon the earth. The earth, there- 
 fore, by this reaction, loses as much momentum in the direction in 
 which the train moves as the train gains ; therefore the earth loses, 
 in this case, a momentum equal to 7500 tons moved one foot per 
 second, which momentum the train has acquired. 
 
 Now, when the same train is about to stop, the moving force 
 which it possesses is imparted to the rails, as it must be by the 
 resistance of the rails on the wheels that the train is brought to 
 rest. The rails, therefore, in this case, and with them the earth, 
 receive back, gradually, its moving force, as the train is gradually 
 stopped ; and when the train has been brought to actual rest upon 
 the rails, its entire moving force, equal to 7500 tons moving one 
 foot per second, is restored. 
 
 222. It appears, therefore, that the earth may be regarded as a 
 great reservoir of momentum as it is a great reservoir of matter, 
 and that every moving force, produced upon it by any action 
 whatever, whether mechanical or vital, must be borrowed from, 
 and will be restored to it, when such moving force ceases. The 
 analogy, therefore, of matter and momentum is complete, and 
 the maxim above mentioned must be accepted. As all apparently 
 new bodies must be composed of materials derived from the earth, 
 and as all bodies apparently destroyed are merely decomposed, and 
 their atoms restored to the common stock of matter which consti- 
 tutes the globe, so all momenta must be obtained from the common 
 reservoir of forces in the earth and restored to it. 
 
 223. But perhaps another objection maybe raised in the minds 
 of some to this reasoning. It may be asked whether, if the body 
 of a man or animal, endued with life, could be imagined to be 
 suspended in space, out of contact with the earth, could not such 
 man or animal, by the act of its will, put its body in progressive 
 motion ? We answer at once in the negative. 
 
 Any attempt to move the limbs would produce in the body of 
 such animal an equal action and reaction. If any limb were pro- 
 jected forwards with any given force, a reaction would take place 
 in other parts of the body, which would be projected backwards 
 with the same force, and the general mass of the body would have 
 no progressive motion. 
 
 The memorable declaration of Archimedes, that if he had a 
 point of support he could move the world itself, admits of being 
 converted ; and the philosopher might have said with equal truth, 
 that without a support he could not move forward even his own 
 body. 
 
 o a 
 
84 OF FORCE AND MOTION. 
 
 CHAP. IV. 
 
 TERRESTRIAL GRAVITY. 
 
 224. The plumb-line. A small weight suspended by a light and 
 flexible thread from a fixed point forms a combination called a 
 plumb-line, and so denominated because the weight usually attached 
 to the string is a ball of lead. 
 
 If several plumb-lines be placed near each other, it will be 
 found that the strings, when at rest, will be precisely parallel to 
 each other. 
 
 225. This common direction which the threads of plumb-lines 
 assume is called the vertical direction. 
 
 226. If a quantity of liquid contained in a vessel be at rest, its 
 surface will have a position which is called level. 
 
 If several fluids, near each other, be at rest, their surfaces will 
 be found to be parallel to each other. Thus, the surface of water 
 in a basin, the surface of a pond, lake, or river, are all parallel to 
 each other, and parallel to the surface of the sea itself when calm. 
 
 The surface of the land is unequal and undulating, being formed 
 into hills and vallies, and rising occasionally into mountains of 
 considerable elevation. If this land, however, could be rendered 
 fluid, the mountains and hills would subside, the vallies would rise, 
 and the entire surface would assume one uniform level ; and would, 
 in fact, coincide with the general surface of the sea, and would be 
 parallel to the surface of fluids at rest. 
 
 When it is said, therefore, that the level of a fluid surface at 
 rest is parallel to the surface of the earth, it must be understood 
 that by the surface of the earth is meant the general direction of 
 the surface of the land, or the exact direction which it .would 
 assume, if, being rendered fluid, all the parts were allowed to sub- 
 side to a common level. The direction which is taken by a plumb- 
 line at rest, is found to be exactly perpendicular to such a surface. 
 
 227. These facts, which are the result of universal experience 
 and observation, are explained by the supposition that the earth 
 exercises an attraction upon all bodies placed upon or near it, and 
 that the direction of this attraction is perpendicular to its general 
 surface, that is to say, perpendicular to the surface of the sea, or 
 of a fluid at rest. Thus, the fact that several plumb-lines have 
 their strings parallel is explained by stating, that the weights' sus- 
 pended from the strings, being all attracted in a direction perpen- 
 dicular to the surface of the earth, must be parallel to each other. 
 
 Every particle of a fluid contained in a vessel being equally 
 
TERRESTRIAL GRAVITY. 85 
 
 attracted in the same vertical direction, the surface must become 
 level and perpendicular to that direction ; for if any portion of it 
 were more elevated than another, the weight of such more elevated 
 part would force it downwards, and press upward the lower par- 
 ticles, until all should attain the same level. This supposition, 
 which is adopted to explain these familiar effects, is verified and 
 conclusively established by a vast body of evidence supplied by 
 astronomical researches, from which it appears that all bodies in 
 the universe exercise upon each other attractions depending on 
 their mass and on their mutual distance, in a manner which will 
 be explained hereafter. 
 
 228. The globe which we inhabit participates in this common 
 property. It therefore exercises an attraction upon all bodies 
 placed on or near its surface, which is proportional to their masses, 
 and is directed towards the centre of the earth, and therefore in a 
 direction perpendicular to its surface. 
 
 229. If a body suspended at any height be disengaged, this 
 attraction of the earth will cause it to fall in a vertical direction, 
 that is to say, in the direction of a plumb-line. 
 
 230. And here it may be asked whether such effects are not 
 incompatible with that principle of equality of action and reaction 
 which we so fully developed in the last chapter ? If a heavy body, 
 disengaged at any height, be precipitated to the surface of the 
 earth, does it not exhibit a moving force which did not previously 
 exist, and is not this an action without a reaction ? 
 
 It is, however, established by proofs which will be explained in 
 another part of this work, that the earth not only attracts the 
 bodies around it, but is attracted by them ; and that the moving 
 force which is impressed on them is balanced by a moving force 
 impressed by them upon it. In fact, in the attraction of gravitation, 
 as this physical agent is called, the equality of action and reaction 
 is verified as completely as it is in the mutual impact of bodies. 
 
 For the present, however, it will not be necessary to dwell on 
 this point. What more immediately concerns us is the explanation 
 of those phenomena which are developed in the effects of gravity 
 acting on bodies at or near the surface of the earth. 
 
 231. All bodies fall with the same velocity. Gravity acts 
 equally and independently on all the particles composing a body, 
 and therefore has a tendency to make all these particles move^with 
 equal velocities, and in parallel lines perpendicular to the surface. 
 It is easily conceived that if two leaden balls of equal magnitude 
 be placed side by side at the same height, they will fall together 
 with the same velocity to the surface, and strike the earth at 
 the same moment side by side. Now, if the matte/ of these two 
 balls be moulded into a single ball, the effect will remain the 
 
 G 3 
 
86 OF FORCE AND MOTIOX. 
 
 same, since their form cannot affect the operation of gravity upon 
 them. 
 
 In the same manner, if ten or a hundred leaden balls of equal 
 magnitude be disengaged together, they will fall together ; and if 
 they be moulded into one ball of great magnitude, it will still fall 
 in the same manner. Hence it follows that masses of matter, how- 
 ever they may vary in magnitude and weight, will descend to the 
 surface of the earth with the same velocity, and if they fall from 
 the same height will arrive at the surface of the earth in the same 
 time, provided they be affected by no other force but that of 
 gravity. 
 
 232. There are some circumstances developed in the fall of 
 bodies, and the effects of the resistance of the air upon them, which 
 are apparently incompatible with what has been just stated. If a 
 feather and a leaden ball suspended at the same height be disen- 
 gaged, it is evident that they will not fall with the 
 same velocity. The leaden ball will be propelled 
 with a rapidity much greater than that which affects 
 the feather. But in this case the operation of gravi- 
 tation is modified by the resistance of the air, which 
 is much greater upon the feather than upon the 
 leaden ball. That two such bodies would descend 
 with the same velocity if relieved from the inter- 
 ference of the air, may be shown by the experiment 
 which is familiarly known as that of the guinea and 
 feather. 
 
 233. Let a glass tube (Jig. 40.), of wide bore, as, 
 for example, three or four inches, and of five or 
 six feet in length, be closed at one end, and sup- 
 plied with an air-tight cap and stop-cock at the 
 other end. The cap being unscrewed, let small 
 pieces of metal, cork, paper, and feathers, be put into 
 it, the cap screwed on, and the stop-cock closed. 
 Let the tube be rapidly inverted, so as to let the 
 objects included fall from end to end of it. It 
 will be found that the heavier objects, such as the 
 metal, will fall with greater, and the lighter with less 
 speed, as might be expected. But that this differ- 
 ence of velocity in falling is due, not to any differ- 
 ence in the operation of gravity, but to the resist- 
 ance of the air, is proved in the following manner. 
 Let the stop-cock be screwed upon the plate of an 
 air-pump (^g-. 41.), the cock being open, and let the 
 jjube be exhausted. Let the cock then be closed, 
 *'g-4- an( j unscre wed from the plate. On rapidly invert- 
 
TERRESTRIAL GRAVITY. 
 
 ing the tube, it will then be found that the feathers will be pre- 
 cipitated from end to end as rapidly as the metal, and that, in short, 
 all the objects will fall together with a common velocity. 
 
 234. Since the attraction of the earth acts equally on all the 
 
 component parts of bodies, 
 and since the aggregate 
 forces produced by such 
 attraction constitute what 
 is called the weight of the 
 body, it is clear that the 
 weights of bodies must be 
 in the exact proportion of 
 the number of particles 
 composing them, or of their 
 quantity of matter. 
 
 Hence, in the common 
 affairs of commerce, the 
 quantities of bodies are 
 estimated by their weights. 
 It will appear, hereafter, 
 that, the weight of a body, 
 or the force with which 
 it is attracted to the sur- 
 face, is slightly different in 
 different places upon the 
 earth; but this is a point 
 which need not be insisted 
 on at present. 
 
 At the same place the 
 weights are invariably and 
 exactly proportional to the 
 quantities of matter com- 
 posing the bodies. If one 
 body have double or triple 
 the weight of another, it 
 will have double or triple 
 the quantity of matter in 
 the other. 
 
 235. Motion of a falling: 
 body. It is not enough 
 for the purposes of science 
 to know merely the direc- 
 tion of the motion which 
 Fig. 41. gravity impresses upon bo- 
 
 dies ; we require to know whether the motion be one having a 
 
88 OF FORCE AND MOTION. 
 
 uniform velocity ; or, if not, in what manner does the velocity of 
 the falling body vary ? 
 
 . If a man leap from a chair or table, he will strike the ground 
 without injury. If the same man leap from a house-top, he will 
 probably be destroyed by the fall. These, and innumerable similar 
 effects, indicate that the force with which a body strikes the ground 
 is augmented with the height from which it falls. Now, as this 
 force depends on the velocity of the body at the moment it touches 
 the ground, it follows that the velocity of the fall is augmented 
 with the height. 
 
 In short, when a body is disengaged and allowed to descend in 
 obedience to gravity, its velocity is gradually accelerated as it 
 descends. Meteoric stones which descend from the upper regions 
 of the atmosphere, strike the earth with such force that they are 
 often known to penetrate in it a considerable depth. 
 
 236. It might be naturally enough conjectured that the force 
 with which a body strikes the earth is proportional to the height 
 from which it has fallen, and such an illustration has been accord- 
 ingly used by orators in speaking of the severity of censure pro- 
 ceeding from high quarters ; but, like many other ornaments of 
 eloquence drawn from physical science, this is erro- 
 fr neous. The force of the fall is not, as we shall now 
 
 show, proportional to the height from which the 
 body has descended. A body falling from a double 
 height does not strike the ground with a double 
 force. 
 
 . g 2 37- When a body, such, for example, as a leaden 
 
 ball, is disengaged at any height, and delivered to the 
 action of gravitation, the effect of this force upon it 
 is to impart to it a certain velocity. Now it is evi- 
 dent that the quantity of velocity which the attraction 
 ' ^ of the earth gives to the ball in one second of time 
 must be equal to the force which it would give to it 
 in another second of time. Let us suppose, for ex- 
 ample, that a moveable stage s (jig- 42.) is attached 
 . s" to a wall or pillar, and is so adjusted that the ball 
 disengaged at B shall arrive upon the stage s pre- 
 cisely at the termination of one second. The body 
 will then strike the stage with a certain force. 
 
 Let another stage s' be placed at the same distance 
 from s as s is from B. If the ball, having been brought 
 to rest by the stage s, is again disengaged, it will 
 Fig. ^z. strike the stage s' at the end of another second, and 
 with the same force ; and if the stage s" be fixed at 
 an equal distance below s', the ball, having been brought to rest 
 
TERRESTRIAL GRAVITY. 89 
 
 at s', and then disengaged, will strike the stage s" at the end of 
 the third second, and with equal force. 
 
 In this case we have supposed that while the ball descends, the 
 velocity it has acquired at the end of each successive second is 
 destroyed by the resistance of the stages s, s', and s", &c. But 
 suppose that on arriving at s, at the .end of the first second, 
 the body was not deprived of the velocity it had acquired, but 
 allowed to retain it in its descent, the retention of this velocity 
 would not in the slightest degree prevent the action of gravity 
 in imparting to it an equal quantity of velocity in the second 
 second ; therefore at the end of the second second the body 
 would have the velocity with which it struck the stage s, in ad- 
 dition to the velocity which it had acquired during the second 
 second. In the second second, therefore, the body would descend 
 through a much greater space than s s', and at its termination 
 would have a velocity double that which it had at the end of the 
 first second. In like manner, if the .velocity acquired in the 
 second second were not destroyed by the stage s', the body would 
 at the end of the third second possess this velocity, in addition to 
 the velocity which would be imparted to it by the action of gravity 
 in the third second. 
 
 In fine, it follows that the action of gravitation imparts to a 
 descending body a certain velocity in every successive second of 
 time during its action ; and consequently, the velocity which a 
 falling body has at the end of ten or twenty seconds, is exactly ten 
 or twenty times the velocity it had at the end of one second. 
 
 238. This principle, in virtue of which the velocity imparted 
 by gravity to falling bodies accumulates in them, is expressed as 
 follows : 
 
 The velocity acquired by a body in descending by the force of 
 gravity, increases in proportion to the time of the fall. 
 
 239. A motion, the velocity of which is thus augmented in 
 proportion to the time counted from its commencement, is called 
 uniformly accelerated motion, and the force which produces such a 
 motion is called uniformly accelerating force. 
 
 Gravity, therefore, acting on bodies near the surface of the 
 earth, is an uniformly accelerating force. 
 
 Since a body in falling moves with a velocity gradually and 
 uniformly accelerated, its average or mean velocity will be that 
 which it had precisely at the middle point of the interval which 
 elapses during its fall. Thus, if a body fall during ten seconds, 
 the average speed will be that which it had at the end of the fifth 
 second. This is evident, inasmuch as, the speed imparted in each 
 successive second being the same, the average of all the speeds at 
 the end of each number of seconds, counted from the commence- 
 
90 OF FORCE AND MOTION. 
 
 ment, will necessarily be that which it had at the middle point of 
 the time. 
 
 It follows from this also, that the final speed acquired by a body 
 at the end of any time will be double the average speed counted 
 from the commencement of its fall. This is evident, since, the 
 velocity being proportioned to the time, the final speed is neces- 
 sarily double that which is acquired in half the time, which is, as 
 has been just shown, the average speed. 
 
 240. It follows from this also, that if a body were to move with 
 its final velocity continued uniformly, it would, in a time equal to 
 that of the fall, move over a space equal to double that through 
 which it had fallen ; for the final speed being double the average 
 speed, the space described with the former will be double the space 
 described with the latter in the same time. 
 
 To obtain a more exact estimate of the manner in which the 
 descent of a heavy body is accelerated, it will be useful to investi- 
 gate the spaces through which a body moves in its descent during 
 every successive second of time. 
 
 241. Let us express by H the height through which a body 
 falls from a state of rest in one second. At the end of such 
 second, the body has acquired a velocity, in virtue of which it 
 would, in another second, without the further action of gravity, 
 move through a space 2 H ; but during the next second gravity 
 would cause the body to descend through another space equal to 
 H, supposing it to move from a state of rest. Therefore, during 
 the next second the body is moved through a space equal to three 
 times H ; that is to say, twice H in virtue of the velocity acquired 
 at the end of the first second, and a space nin virtue of the action 
 of gravity upon it during the next second. 
 
 Let us now consider the motion of the body during the third 
 second. At the end of the second second, the body, having fallen 
 through a height expressed by 4 H, has acquired a velocity in 
 virtue of which, without any further action of gravity, it would 
 move through a space equal to 8 times H in two seconds, and 4 
 times H in one second ; but in addition to this, gravity also, in the 
 third second, would move it through a space H ; and from these 
 two effects combined, the body in the third second would descend 
 through a space expressed by 5 H. But we have seen that in the 
 first two seconds it has fallen through a space expressed by 4 H, 
 and therefore at the end of the third second it will have fallen 
 through a height from the state of rest expressed by 9 H. 
 
 Pursuing its course further, we find that it begins its motion 
 during the fourth second with a velocity such as would make it, 
 in three seconds, without the further aid of gravity, move through 
 a space equal to double that which it had fallen through from a 
 
TERRESTRIAL GRAVITY. 
 
 state of rest, that is to say, 1 8 H ; consequently, with this velocity, 
 it would move in the fourth second through a space equal to 6 H ; 
 but, in addition to this, the action of gravity carries it an the fourth 
 second through the space H, and by these combined effects it 
 must move in this second through a space equal to 7 H. In the 
 same manner, it may be shown that the space through which it 
 moves in the fifth second is 9 H, while the space through which it 
 moves in the first five seconds is 2 5 H ; and the space through 
 which it moves in the sixth second is 1 1 H, while the space through 
 which it descends from a state of rest in the first six seconds is 
 36 H; and so on. 
 
 242. In the following table is expressed, in the first column, 
 the number of seconds, or other equivalent intervals of time, 
 counted from the commencement of the fall. In the second column 
 is exhibited the space through which the falling body moves in 
 each successive interval, the unit being understood to express the 
 space through which a body falls in the first second of time. . In 
 the third column is expressed the velocity which the body has 
 acquired at the end of each interval, counted from the commence- 
 ment of the fall, and expressed by the space which, if such velo- 
 city continued uniformly, the body would describe in one second. 
 In the fourth column is expressed the total heights from which the 
 body falls from a state of rest to the end of the time expressed in 
 the first column. 
 
 Tabular Analysis of the Motion of a falling Body. 
 
 Number of Second* 
 in the Fall, counted 
 from a Mate of 
 Rest. 
 
 Spaces fallen 
 through in each 
 successive Second. 
 
 Velocities acquired 
 at the End of 
 Number of Seconds 
 expressed in First 
 Column. 
 
 Total Height fallen 
 through from Rest 
 in the Number of 
 Seconds expres*ed in 
 First Column. 
 
 I 
 
 I 
 
 Z 
 
 j 
 
 Z 
 
 3 
 
 4 
 
 4 
 
 3 
 
 5 
 
 6 
 
 9 
 
 4 
 
 7 
 
 8 
 
 16 
 
 5 
 
 9 
 
 10 
 
 2-S 
 
 6 
 
 ii 
 
 IZ 
 
 ?6 
 
 I 
 
 ij 
 15 
 
 3 
 
 
 
 9 
 
 *7 
 
 18 
 
 81 
 
 10 
 
 19 
 
 2.0 
 
 100 
 
 Although all the circumstances attending the descent of bodies 
 falling freely are included with arithmetical precision in the above 
 table, we may nevertheless render it more easy to obtain a clear 
 conception of these important physical phenomena by the annexed 
 diagram (Jig. 43 ), in which the divided scale represents the ver- 
 tical line along which the body is supposed to fall, o being the 
 point from which it commences its descent. The points which it 
 successively passes at the termination of I, 2, 3, 4, 5; 6, and 7 
 seconds respectively are marked i, n, in, iv, v, vi, vn. The figures 
 
OF FORCE AND MOTION. 
 
 of the scale indicate the total heights through which the body has 
 fallen at the end of each successive second, the unit being the 
 height through which the body falls in the first 
 second. The spaces included between brackets 
 on the right of the diagram are those through 
 which the body falls in each successive second. 
 It will then be apparent, first, that the body 
 is accelerated in its motion, inasmuch as the 
 spaces through which it falls in each successive 
 second are evidently increasing ; secondly, that 
 the space through which it falls in any number 
 of seconds is expressed by the square of this 
 number, the unit being the space fallen through 
 in the first second ; thirdly, that the spaces fallen 
 through in each successive second are expressed 
 by the odd numbers with reference to the same 
 unit. 
 
 A direct experimental verification of the re- 
 sults exhibited in the preceding table and dia- 
 gram, would be attended with several practical 
 difficulties. The heights through which a body 
 
 y ^ r j falls by gravity, acting freely in several seconds, 
 
 are considerable, and a great velocity is soon 
 acquired. The resistance of the air disturbs the 
 result, and some .difficulty would be found in 
 observing, with sufficient precision, the points at 
 which the falling body would be found at each 
 successive second of time. 
 
 243. Atwood's machine. This and other 
 Vi ~.~.. 9 /;J practical difficulties have, however, been sur- 
 mounted by a beautiful and useful experimental 
 apparatus, called from its inventor "Atwood's 
 machine." By this apparatus, the intensity of 
 the force of gravity can be diminished in any 
 desired proportion without divesting it of any 
 of its characters of an uniformly accelerating 
 force. Thus we can make the falling body de- 
 scend at so moderate a rate, that the effect of 
 the atmospheric resistance becomes imperceptible, 
 and the height, and all the circumstances at- 
 tending the fall, can be observed with the 
 greatest precision. 
 Fig. 43- This contrivance consists of two equal cylin- 
 
 drical weights, w, w' (fig. 44.), connected by a fine silken thread, 
 which passes in a groove over a nicely -construe ted wheel K, 
 
 \TT 
 
 -M- 
 
 -U6 
 
TERRESTRIAL GRAVITY. 
 
 93 
 
 turning on a horizontal axis, so as to be subject to an imper- 
 ceptible friction. This wheel, and the stand which supports it, 
 are placed upon a bracket AB, attached to 
 a wall, or supported on a pillar, at a conve- 
 nient height. Adjacent t<? the thread sup- 
 porting one of the weights w, there is a 
 divided vertical scale, by which the circum- 
 stances attending the descent of the weight 
 can be noticed and measured. When the 
 weight w is brought to the highest point 
 of the scale, the weight w' will be near the 
 ground ; but the weight of the thread is so 
 insignificant, that though unequal portions 
 of it hang on each side of the wheel R, the 
 difference of their weights produces no per- 
 ceptible defect, and, accordingly, the two 
 equal weights w and w' rest in equilibrio. 
 
 Now, if a small additional weight w be 
 placed upon the top of the cylindrical 
 weight w, it will cause the weight w to 
 descend, and the weight w' to rise ; and 
 this descent will have all the characters of 
 a uniformly accelerating motion, for the 
 force of gravity impresses on the prepon- 
 derating weight w the same moving force 
 which it would impress upon it if it were 
 free ; but this moving force is, by the very 
 condition of the apparatus, shared by w and 
 1^' the equal weights w and w', so that instead 
 of imparting to w the velocity which such 
 weight would have were it free, the velocity 
 Fig. 44. f tne augmented moving mass, consisting 
 
 of w, w, and w', will be diminished in pre- 
 cisely the same proportion as the mass moved is increased ; there- 
 fore, the weight w bearing upon it w, will fall with a velocity so 
 much less than that with which w would fall, were it free, as the 
 combined weights w', w, and w are greater than w alone. But 
 the other circumstances attending the descending motion will be 
 precisely similar to those which attend the descending motion of 
 any falling body. The machine will, in fact, present a miniature 
 representation of the phenomena of falling bodies ; the effects will 
 be the same as though the attraction of the earth upon a falling 
 body were diminished to such an extent that the velocity of the 
 descent would be reduced to that with which the weight w falls. 
 Now, as we can adopt a preponderating weight w as small as may 
 
94 OF FORCE AND MOTION. 
 
 be desired, it is clear that we may reduce the velocity of the fall in 
 so great a degree, that all the circumstances attending the motion 
 during the descent can be deliberately and accurately observed. 
 
 Let us suppose, for example, that the weights w and w' are each 
 twenty-four ounces, and that the preponderating weight w, placed 
 upon w to produce its descent, is a quarter of an ounce. The 
 total mass moved, therefore, by the action of gravity impressed 
 upon the weight w, will be 193 times the weight w, for the weights 
 w and w', taken together, are forty-eight ounces, that is to say, 
 192 quarters of an ounce ; and the weight MJ, which is one quarter 
 of an ounce, being added to this, will make a total of 193 quarters 
 of an ounce. 
 
 The attraction of gravity, therefore, instead of imparting velo- 
 city to one. quarter of an ounce, has to move 193 quarters of an 
 ounce, and, consequently, the velocity it imparts per second will 
 be 193 times less. 
 
 We have here, with a view to simplify the explanation, avoided 
 all reference to the motion imparted to the wheel over which the 
 strings pass ; but it will be evident that the force impressed by 
 gravity on the preponderating weight w, must be shared with the 
 matter of the wheel, as well as with the weights w and w'. If the 
 matter of the wheel were all collected at its edge, it would then be 
 moved with the same velocity as the weights w and w', and in this 
 case it would be only necessary to consider the weight of the wheel 
 as forming part of the masses w and w', and therefore to diminish 
 the latter so that the total weight of w and w' and the wheel 
 should make up .forty-eight ounces. 
 
 But as the mass of the wheel is not all collected at its edge, it 
 does not all receive the same velocity, but, on the contrary, its 
 different parts are moved with less velocities the nearer they are 
 to its centre. This difference of moving force imparted to dif- 
 ferent parts of the wheel, requires to be allowed for, by calculating 
 how much matter collected at the edge of the wheel would have 
 an equal moving force. Such a calculation, though presenting no 
 difficulty, and subject to no inaccuracy or doubt, would involve 
 mathematical principles and operations which cannot be conve- 
 niently introduced here ; and we may therefore assume that the 
 momentum imparted to the wheel is represented by an equivalent 
 portion of the forty-eight ounces assigned to the weights w and w', 
 and that, in fact, the real weights of these must be a little less 
 than those assigned to them, the difference being represented by 
 the effect of the wheel. 
 
 Being provided with a pendulum beating seconds in an audible 
 manner, and taking the thread which sustains the weight w' be- 
 tween the fingers, let the weight w be elevated until its upper 
 
TERRESTRIAL GRAVITY. 
 
 95 
 
 surface coincides with the 
 zero of the scale. List- 
 ening attentively to the 
 beats of the pendulum, let 
 the thread be disengaged at 
 the moment of any one beat 
 It will be found that, at 
 the moment of the next 
 beat, the weight w will have 
 fallen precisely one inch. 
 During the second beat it 
 will have fallen through 
 precisely three inches more ; 
 during the third beat it will 
 have fallen through five 
 inches ; during the fourth 
 beat it will have fallen 
 through seven inches ; dur- 
 ing the fifth beat it will 
 have fallen through nine 
 inches ; and so on. Now, 
 if these distances be com- 
 pared with those given in 
 the second column of the 
 preceding table, they will 
 be found to correspond ; 
 the spaces through which 
 the weight w descends in 
 successive seconds being, as 
 shown in this table, ex- 
 pressed by the odd num- 
 bers. 
 
 In the same manner, it 
 will appear that the height 
 through which the weight 
 w falls during the first se- 
 cond, being one inch, the 
 height through which it 
 falls during the first two 
 seconds will be four inches, 
 the height through which 
 it falls during the first three seconds will be nine inches, the 
 height through which it falls during the first four seconds will be 
 sixteen inches, the height through which it falls during the first 
 five seconds will be twenty-five inches, and so forth. 
 
 Fig 45- 
 
9 5 
 
 OF FORCE AND MOTION. 
 
 These numbers correspond with and verify those given in the 
 fourth column of the preceding table. 
 
 Fig. 48. 
 
 The principles of Atwood's machine being explained vhfig- 44., 
 its form, as actually constructed, with all its accessories, is shown 
 \nfigs. 45, 46, 47. 
 
TERRESTRIAL GRAVITY. 
 
 97 
 
 244. Morin's apparatus. Another apparatus for the expe- 
 rimental illustration of the laws which regulate the descent of 
 bodies by gravity has recently been invented and constructed by 
 M. Morin, the director of the Conservatoire des Arts et Metiers at 
 Paris. It consists of a cylinder or drum A A, Jig. 48., mounted on 
 a vertical axis, on which it is moved uniformly by a train of 
 clock-work B, impelled by a weight c, rendered uniform by a 
 fly a. 
 
 A small cylindrical weight D, carries a pencil, the point of which 
 is pressed gently against the surface of the cylinder. This weight 
 is guided in its fall between two vertical wires, passing through 
 holes in a plate fixed upon it, which projects from it at either 
 side, so that in its fall it keeps constantly parallel to the cylinder 
 and at the same distance from it. An apparatus for detaching 
 this weight is fixed above the cylinder. By merely pulling a 
 cord, like a bell-pull, shown in the figure beside the weight 
 D, this weight can be let fall. Now, if the cylinder did not 
 revolve, the pencil carried by the weight would evidently trace 
 a vertical line upon its surface. If, on the other hand, the 
 weight were stationary and the cylinder revolved, the pencil would 
 trace a horizontal circle around it. But, if 
 while the cylinder turns uniformly round its 
 axis the weight falls, the pencil will trace 
 around the surface of the cylinder a curved 
 line, such as is shown in fig. 49. To facilitate 
 the experiment, the surface of the cylinder is 
 divided into equal parts vertically by the 
 parallel lines rr, ss, tt, &c. &c.; and since the 
 motion of the cylinder is uniform, the inter- 
 vals between the moments at which these 
 parallels pass under the pencil are equal. 
 The vertical space through which the weight 
 falls in the first interval will be rin ; in the 
 first two intervals p p ; in the first three q'q, 
 and so on. 
 
 It is evident, therefore, that the circum- 
 stances of the descent will be indicated ex- 
 actly by this means. In Atwood's machine 
 it is not, properly speaking, a body falling 
 freely, which is observed, but one, the rate of 
 whose fall is diminished in a known propor- 
 tion. Here, however, it is the actual descent 
 which is exhibited and analyzed. 
 
 245. Itaw of free descent. It is evident 
 that the numbers, which express the heights through which the 
 
 Fig. 49. 
 
98 OF FORCE AND MOTION. 
 
 bodies fall in any number of seconds, counted from the com- 
 mencement of the motion, are the squares of the numbers of 
 seconds ; and hence we have the following general principle : 
 
 When a body is moved by a uniformly accelerating force, such as 
 gravity, the spaces through which it moves, counted from the com- 
 mencement of the motion, will be proportional to the squares of the 
 times, and the spaces through which it moves in equal intervals of 
 time will be as the odd numbers. 
 
 These rules, which are of the highest importance, may be con- 
 veniently reduced to arithmetical symbols. Let us express by ^g 
 the space through which a body, urged by an uniformly accele- 
 rating force from a state of rest, would move in one second, a 
 space which, in the case of gravity, is 1 6 ft. I in., or 193 inches. 
 Thus it is evident, from what has been stated, that we shall find 
 the space which the body would move through in any given number 
 of seconds, counted from the commencement of its motion, by 
 multiplying g by the square of this number of seconds. 
 
 246. Hence, in general, if T express the number of seconds 
 during which the body has been moving from a state of rest, T 2 X ^g 
 will express the entire space through which the body has moved 
 in the number of seconds expressed by T. If this space, then, be 
 expressed by H, we shall have 
 
 In like manner, since it has been established that the velocity 
 which is gained in falling during one second, is such, that in each 
 second the body would with that velocity move through a space 
 equal to twice that through which it had fallen, it follows that the 
 velocity acquired in one second is g ; in other words, it is such, 
 that a body moving with that uniform velocity would move 
 through a space expressed by g in each second. 
 
 But it has also been shown that the velocity augments in pro- 
 portion to the time, and that the velocity in two, three, four, and I 
 five seconds is two, three, four, and five times the velocity in one 
 second. To find, therefore, the velocity acquired in any number I 
 of seconds, we shall' only have to multiply g by that number of 
 seconds. If, then, T express the number of seconds during which 
 the body has been falling, and v the velocity which it has gained 
 in the time T, we shall have 
 
 The two preceding formulae include the whole theory of falling 
 bodies in vacuo. From these may be deduced the following for- 
 
 
 - 
 
 
TERRESTRIAL GRAVITY. 99 
 
 mula, by which the velocity which is acquired in falling through 
 any given height is known : 
 
 V 2 = 2 H X #. 
 
 It remains now to show, that by Atwood's machine the numbers 
 given in the third column of the preceding table may be verified ; 
 that is to say, to demonstrate, by direct experiment, that the 
 velocity imparted to the body in its descent increases in the pro- 
 portion of the time of the fall. 
 
 To accomplish this, the following arrangements are made. The 
 preponderating weight used to produce the descent of w has the 
 form of a long narrow bar D (fig- 44. ), which is laid across the 
 upper surface of the cylindrical weight w. A ring E, large enough 
 to allow the weight w to pass through it, but not large enough to 
 allow the bar resting on this weight to pass, is attached to the 
 scale at the division marked I. If the weight be now brought 
 to such a position that its upper surface shall coincide with the 
 zero of the scale, and if it be let fall at a moment corresponding 
 with one beat of the peadulum, its upper surface will arrive at the 
 ring E at the moment of the next beat, and the ring which allows 
 the weight w to pass freely through will ca'tch the bar, which will 
 rest upon it. After the top of the weight, therefore, has passed 
 the ring, the weight w being relieved from the bar, by whose pre- 
 ponderance its motion was accelerated, will continue to move 
 downwards without further acceleration, with the velocity it had 
 acquired at the end of the first second, such velocity being now 
 continued uniform. If, then, the descent of this weight uniformly 
 downwards be compared with the beats of the pendulum, it will be 
 found to move uniformly at the rate of two inches per second. 
 
 Thus, we infer, first, that the velocity imparted at the end of 
 the first second is such as to make the weight w move uniformly 
 in one second through double the space through which it has fallen, 
 and that such velocity is at the rate of two inches per second- 
 Let the ring be now moved to the fourth division of the scale, 
 and the bar being put upon the weight w, let the experiment be 
 repeated. It wilt be found that at the end of two seconds the bar 
 will strike the ring and the weight will pass below it, moving with 
 a uniform velocity ; and by comparing its motion along the scale 
 with the beats of the pendulum, it will be found that this velocity 
 is at the rate of four inches per second. 
 
 Again, let the position of the ring be fixed at the ninth division 
 of the scale, and, replacing the bar, lei the experiment be once 
 more repeated. It will be found that the bar will strike the ring 
 at the end of the third second, and that the weight w, when dis- 
 engaged from the bar, will continue to descend with the uniform 
 
ioo OF FORCE AND MOTION. 
 
 velocity of six inches per second. The same experiment may be 
 repeated for as many seconds as the height of the scale may admit, 
 and like results will be obtained. We may thus obtain a com- 
 plete verification of the numbers contained in the third column of 
 the table, p. 9 1 . 
 
 247. From these experiments we are enabled to calculate the 
 height through which a body would fall in one second of time by 
 the effect of the force of gravity, and independently of any in- 
 fluence from the resistance of the air. 
 
 It appears from what has been stated, that when the magnitude 
 of the weights w', w, and w was so adjusted that the height of the 
 descent was 193 times less than the height with which w would 
 fall freely, the height through which it fell was one inch. It 
 consequently follows, that if w were submitted to the unimpeded 
 action of gravity, it would fall through 193 inches, or 1 6 ft. I in., 
 in the first second. 
 
 248. The table at page 91. compared with this result, will 
 show all the circumstances attending the descent of bodies falling 
 freely, 1 6 ft. I in. being the unit of the table. Thus, if we desire 
 to ascertain the height from which a body would fa41 in five seconds, 
 we take the number in the fourth column of the table opposite 5 
 seconds, which is 25, and multiply it by 1 6 ft. I in., the product, 
 which is 402 ft. I in., will 'be the height required. 
 
 In the same manner, if it be required to determine what space 
 a falling body would descend through in the fifth second of its 
 motion, we take, in the second column of the table, the number 
 opposite 5 seconds, which is 9 ; we multiply 1 6 ft. I in. by this 
 number, and find the product, which is 1 44 ft. 9 in., which is the 
 space required. 
 
 In like manner, if it b required to determine with what velocity 
 a body would strike the ground after falling'during an interval of 
 five seconds, we take the number in the third column of the table 
 opposite 5 seconds, which we find to be I o, and we multiply 1 6 ft. 
 I in. by this number. The product, which is 1 60 ft. IO in., will 
 be the velocity required ; and we infer that the body thus falling 
 would have, when it strikes the ground, a velocity of 1 60 ft. I o in. 
 per second. 
 
 It will be observed that the numbers in the first column of the 
 table now referred to, and which express the time of the fall, are 
 the square roots of the numbers in the fourth column, which ex- 
 press the height from which the body falls. We have therefore 
 this general principle of uniformly accelerating motion : 
 
 When a body is moved by a uniformly accelerating force, the times 
 required to move through any given space are proportional to the 
 square roots of those spaces. 
 
TERRESTRIAL GRAVITY. V 11M 
 
 By the aid of this rule, and the results alreath? tfotaintfd ' we> aVe! 
 enabled to ascertain the time which a body wetiXTfalio^o full from 
 any given height. Thus, if a body be supposed to fall from a 
 height of loooo feet: Find the number of times which 16 ft. I in. 
 are contained in loooo feet, which is done by dividing 10000 by 
 l6 T V The quotient is 62176. 
 
 This number is then the square of the number of seconds in the 
 time of the fall. The square root of this obtained from a table of 
 square roots being 24-9, we infer that the time a body would take 
 to fall through the height of I oooo feet is 24-9 seconds. 
 
 In the same manner it follows, that since the velocity acquired 
 by a body in its fall is proportional to the time of the fall, and 
 since the time of the fall itself is proportional to the square root of 
 the height, the velocity acquired must also be proportional to the 
 square root of the height. 
 
 If we would, therefore, determine the velocity or force with 
 which a body falling from a given height would strike the ground, 
 independently of the effect of the resistance of the air, we are 
 enabled to do so by these principles. 
 
 Thus, let it be required to determine the force with which a 
 body falling from the height of I oooo feet would strike the 
 ground. It has been just shown that the time of the fall would 
 be 24*9 seconds, and it has been already demonstrated that the 
 velocity acquired by the body would move it uniformly over a 
 space equal to double the height through which it falls, and in the 
 same time. Therefore, the velocity in this case would be such 
 that in 24*9 seconds the body would move through 20000 feet ; 
 and consequently, by dividing 20000 by 24*9, we shall obtain the 
 velocity in feet per second, which appears, therefore, to be 803 feet 
 per second. 
 
 249. It appears, therefore, that the velocity or force with which 
 a falling body strikes the ground increases in a much less pro- 
 portion than the height from which it falls. If the height be 
 augmented in a fourfold proportion, the force of the fall will only 
 be augmented in a twofold proportion ; and if the height be aug- 
 mented in a ninefold proportion, the force of the fall will only be 
 augmented in a threefold proportion ; and so on. 
 
 250. This explains a fact of not unfrequent occurrence, and 
 which sometimes produces surprise. Persons sometimes fall or 
 leap from such heights as would seem to render their destruction 
 inevitable, yet they are frequently fi^ind to escape without con- 
 siderable injury. This is explained by the fact that the momentum, 
 or shock produced by the fall, increases in a proportion so very 
 much less than the height. A man can leap from a height of five 
 feet with perfect impunity ; if, however, he leap from a height of 
 
 H 3 
 
IC2 OF FORCE AND MOTION. 
 
 ten 'feet,, the for'ie with which he will strike the ground, instead of 
 beirfg double;!, will t>e increased in a proportion less than one 
 half; and if he leap from a height of twenty feet, the force with 
 which he strikes the ground will be only doubled. 
 
 251. A further mitigation of the shock produced by a fall arises 
 from the resistance of the air, which further diminishes the velocity 
 acquired. A case occurred some years ago, in which a boy, dressed 
 in a smock frock, accidentally fell down the shaft of a coal-pit 
 having a depth of nearly 100 feet. It was expected that he would 
 have been found dead at the bottom. On searching, however, he 
 was found there almost uninjured. It is probable, that in this 
 case, the frock he wore afforded a resistance to the air, somewhat 
 resembling that of a parachute, which, combined with the principle 
 already explained, that the velocity augments in a very much less 
 proportion than the height, explained his safety. 
 
 252. Retarded motion of bodies projected upwards. All the 
 circumstances attending the accelerating descent of falling bodies, 
 which have been explained in the present chapter, are exhibited 
 in a reversed order when a body is projected upwards. Gravity 
 then acts as a uniformly retarding, instead of uniformly accele- 
 rating force, depriving the body so projected of equal quantities 
 of velocity in equal times ; and further, it is apparent that the 
 velocities which the force of gravity thus destroys in a body pro- 
 jected upwards in any given time are exactly equal to those which 
 it would impart to a body in the same time when falling freely. 
 
 Thus, if a body be projected vertically upwards with the velocity 
 which it would acquire in falling freely during one second, the 
 body so projected will rise exactly to the height from which it 
 would have fallen in one second, and at that point of its ascent it 
 will have the velocity which it would have at the same point if it 
 had descended. 
 
 To determine, therefore, the height to which a body will rise 
 projected upwards with a given velocity, it is only necessary to 
 determine the height from which it would fall to acquire the same 
 velocity. 
 
 In like manner, to determine the time which a body would take 
 to rise to a certain height when projected upwards, it is only 
 necessary to determine the time which it would take to fall freely 
 from the same height. 
 
 253. motion down an inclined plane. A plane and hard 
 surface, which is neither in f the vertical nor horizontal position, is 
 called an inclined plane. In jig. 50., if the line w o be vertical, 
 then w M will represent an inclined plane. 
 
 Bodies which descend upon inclined planes move with a uni- 
 
MOTION DOWN INCLINED SURFACES. 
 
 formly accelerating force similar to that of gravity, omitting, as 
 usual, the consideration of friction, and the resistance of the air. 
 
 Let w be a body placed upon 
 the inclined plane. The force of 
 gravity acts upon it in the vertical 
 direction w o. Let this line w o, 
 so representing the force of gra- 
 vity, be considered as the diago- 
 nal of a parallelogram, of which 
 w N and w M are sides, the side 
 WN being perpendicular to WM. 
 The entire force of gravity, there- 
 fore, represented by w o, and act- 
 ing on the body w, will, by the 
 principle of composition of forces, 
 be equal to the two forces repre- 
 sented by the sides of the paral- 
 lelogram w M and w N. But w , 
 being perpendicular to the plane, 
 is counteracted by it, and exhibits 
 itself merely in pressure upon it. 
 The component w M, however, 
 being in the direction of the plane 
 and downwards, will cause the 
 body to move down the plane. 
 
 The proportion of this accelerating force down the plane to that 
 of gravity acting freely in the vertical direction, will, therefore, be 
 that of the lines w M to w o. If w o be the height through which 
 a body would fall vertically in one second, then w M will be the 
 distance through which the body would fall in the first second on 
 the inclined plane. It is evident, therefore, that by taking w o 
 equal to 193 inches, the distance w M will be actually that down 
 which the body w, independently of friction, &c., would fall in the 
 first second. 
 
 If it be desired to ascertain the force with which the body w 
 presses on the inclined plane, let w o be taken so as to consist of 
 as many inches as there are pounds in the weight w. Then w N 
 will consist of as many inches as there are pounds in the pressure 
 which w exerts on the plane. 
 
 The motion down an inclined plane, therefore, being uniformly 
 accelerated, like gravity, but only mitigated in its intensity 
 m a certain ratio, depending on the inclination of the plane, all the 
 circumstances which have been already explained in reference to 
 the accelerated motion of bodies falling freely, will be similarly 
 exhibited in the motion down an inclined plane. 
 
 H4- 
 
 Fig. 50. 
 
io 4 
 
 OF FORCE AND MOTION. 
 
 Let w M (fig. 51.) be an inclined plane, and w o the vertical 
 line, and let us suppose two bodies dismissed at the same moment 
 from w, one falling down the vertical line 
 w o, and the other down the line w M. 
 Let i, IT, in, iv, v, be the points upon the 
 vertical line o, at which the body is found 
 at the end of one, two, three, four, and 
 five seconds. 
 
 If from this point lines be drawn per- 
 pendicular to w M, the points i', n', in', 
 iv', v', where these perpendiculars will 
 meet the inclined plane, will be those at 
 which the body falling down such inclined 
 plane will be found at the same epochs ; that 
 is to say, at the end of the first second the 
 one body will be found at i and the other 
 at i', at the end of the second second the 
 bodies will be found respectively at 11 and 
 11', at the end of the third second at in 
 and in', and so on. 
 
 The force down the inclined plane is 
 just so much less in intensity than the 
 force of gravity, as the spaces w i', w n', 
 w in', &c. are respectively less than w i, 
 w n, w in, &c. Consequently, it is evi- 
 dent that, these spaces, being in the pro- 
 portion of the forces, will be described in the same time, as, 
 indeed, has been already proved. 
 
 In this manner, therefore, the circumstances of the motion down 
 an inclined plane may always be determined with reference to the 
 circumstances of the motion down a vertical line. 
 
 If it be desired to ascertain the points at which a body falling 
 down an inclined plane will- acquire the same velocities which it 
 acquired in one or more seconds in falling freely in the vertical 
 direction, it is only necessary to consider that the more feeble 
 force down the plane requires a proportionally greater space to 
 produce a given velocity. If, then, w M and w o (fig. 52.) 
 represent, as before, an inclined plane and a vertical line, and if, 
 as before, i, n, in, iv, v represent the points at which the body, 
 falling vertically, would be found at the end of one, two, three, 
 ^ur, and five seconds, then the points on the plane where the 
 same velocity would be attained as the body had at the points, i, 
 n, in, iv, and v, will be determined by drawing lines from the 
 points i, n, in, iv, and v respectively in the horizontal direction 
 because, by these means, the line w v on the plane will be just so 
 
 Fig. 51 
 
MOTION OF PROJECTILES. 
 
 105 
 
 much longer than the line w i as the force of gravity, acting freely, 
 is more intense than the force down 
 the inclined plane ; consequently, the 
 velocity which will be acquired at v 
 on the plane, will be the same as the 
 velocity acquired at i in falling freely. 
 In the same manner, it will appear 
 that the velocities acquired on the 
 plane at the points v', v", v r// , v"" will 
 be the same as the velocities acquired 
 in falling freely at the points u, in, iv, 
 and v. 
 
 254. Projectiles. We have consi- 
 dered the case where a body, acted on 
 freely by the force of gravity, is either 
 allowed to fall vertically downwards, 
 or is projected vertically upwards 
 We shall now consider the other cases, 
 in which a body is projected in any 
 other direction, not vertical, and then 
 abandoned to the action of gravity, a 
 problem which forms the foundation of 
 the doctrine of projectiles. The solu- 
 tion of this problem follows imme- 
 diately from the principles which de- 
 
 Fig. 5*. 
 
 iermine the motion of a body falling freely, as explained in the 
 present chapter, and the composition of motion. 
 
 255. Let us first take the case in which a body w (fig. 53.), as, 
 
 3V 
 
 M 
 
 
 IV 
 
 Fig. 5J- 
 
io6 OF FORCE AND MOTION". 
 
 for instance, a ball shot from a cannon, is projected in the hori- 
 zontal direction w M. If the force of gravity did not act on it, it 
 would move forwards towards M with the velocity of projection 
 continued uniform, and, in virtue of such motion, would pass over 
 equal spaces in equal times. Thus, if, by the velocity of pro- 
 jection, the body would move from w to i' in the first second, it 
 would move from i' to n' in the next second, from n' to m' in the 
 following second, and so on, these successive spaces being equal. 
 
 But if, on the other hand, the body, without being projected at 
 all, were disengaged at w, and left to the action of gravity alone, 
 it would, as has been already explained, descend vertically, and 
 would be found at the points i, n, m, iv, v at the end of the suc- 
 cessive seconds, the distance being, as already explained, repre- 
 sented by the numbers I, 4, 9, &c. 
 
 Now, the body leaving w, being submitted to both these forces 
 simultaneously, will, by the composition of motion, be found at 
 the end of each successive second at the extremities of the dia- 
 gonals of a parallelogram whose sides represent these motions. 
 Thus, at the end of the first second, the body will be found at the 
 point I, being the extremity of the diagonal of a parallelogram 
 whose sides are the space wi', through which the body would 
 move in virtue of the velocity of progression, and wi the space 
 through which it would fall freely in the same time by gravity. 
 If the force of gravity would have made it move over wi with a 
 uniform motion, then the body, in moving from w, would folloAv 
 exactly the diagonal of the parallelogram. But the force of gra- 
 vity imparting to the body not a uniform, but an accelerated mo- 
 tion, first very slow and then more rapid, the body will pass from 
 w to I, not by a strict diagonal course, but by a curved line, as 
 represented in the figure. 
 
 In the same manner, at the end of two seconds, the body will be 
 found at 2. For it is actuated at the same time by two motions ; 
 first, the projectile motion, w-hich, acting alone upon it, would carry 
 it uniformly from w to n', and, secondly, the force of gravity, 
 which, acting alone upon it, would cause it to fall from w to n. 
 At the end of two seconds it will therefore be found at the point 
 2, being the extremity of the diagonal. 
 
 But, as before, the motion from w to n not being uniform but 
 accelerated, first slow but afterwards more rapid, the body will 
 pass from w to 2, not along the diagonal, but over the curved line 
 represented in the figure. 
 
 The same explanation will be applicable to its remaining course, 
 and it will follow that the body will pursue the curved course from 
 w to 5 in five seconds, in consequence of the combination of the 
 projectile velocity imparted to it, and represented by wv', com- 
 
MOTION OF PROJECTILES. 
 
 107 
 
 bined with the descending motion imparted to it by gravity, and 
 
 represented by w v. 
 
 256. In this case we have supposed, for simplicity, the body to 
 
 be projected in the horizontal direction ; but the same principles 
 
 will explain its motion, if projected in an oblique direction, such 
 
 as WM 0%. 54.)- 
 
 As before, let the space which the body would move over in one 
 
 second, in virtue of the pro- 
 'M jectile force alone, gravity 
 being supposed not to act 
 upon it, be w i'. It would 
 move over the equal spaces 
 terminating at i', n", in', iv', 
 v' in the successive seconds. 
 
 On the other hand, suppose 
 the body to be acted on by 
 gravity alone, independently 
 of the projectile force. It 
 would then, as before, moving 
 in the vertical line wo, be 
 found at the end of the suc- 
 cessive seconds at the points 
 
 I, II, III, IV, V. 
 
 Now, by the principle of the 
 composition of motion, the 
 body will actually be found, in 
 consequence of the simulta- 
 neous effects of the two mo- 
 tions imparted to it by gravity, 
 and by the projectile force, at 
 the end of the successive se- 
 conds, at the points i, 2, 3, 4, 
 5, which are the extremities 
 of the diagonals of parallelo- 
 grams, whose sides are respec- 
 tively the spaces which the body would describe in virtue of the 
 projectile force, and of gravity acting separately. The course of 
 the body will.be the curved line represented in the figure, and not 
 the straight diagonal, for the reasons already explained. 
 
 257. Motion in parabolic curves. The path which the pro- 
 jectile follows in this case is a curve, known in geometry as the pa- 
 rabola, the property of which is, that the sides of the parallelogram 
 whose diagonal determines its successive points, are related to each 
 other as the successive whole numbers I, 2, 3, 4? & c> > an( ^ their 
 squares I, 4, 9, 1 6, &c. 
 
 54- 
 
io3 OF FORCE AND MOTION. 
 
 258. Resistance of the air. It must, however, we repeat, be 
 remembered, that these conclusions rest upon the supposition that 
 the body moves in a medium which offers no resistance to it, and 
 which does not deprive it of any of the force imparted to it by pro- 
 jection or by gravity. In the actual case, however, the motion of 
 all projectiles takes place through the atmosphere, which is a re- 
 sisting medium, and, moreover, one of which the resistance varies, 
 increasing in a certain high proportion with the velocity. The 
 real path, therefore, of projectiles differs more or less from the 
 parabola explained above. The deviation is not very considerable 
 when the velocity of the moving body is not great ; but when the 
 projectile is driven with great velocity, as in the practice of gun- 
 nery, then the deviation from the parabolic path is so considerable, 
 that the above theory becomes altogether inapplicable. 
 
 259. Application of projectiles in gunnery. According to 
 what has been explained above, a ball projected from any missile 
 will not follow the direction of the axis of the barrel, but will pro- 
 ceed in a curved line, concave downwards, to which the direction of 
 the barrel is a tangent ; thus, for example, if the barrel be directed 
 horizontally in the line WM,^. 53, the ball proceeding along the 
 dotted curve will fall below the line of aim ; and if that line be 
 directed to the object, the ball will miss it by passing too low ; to 
 hit an object, therefore, at any proposed distance, the gun must be 
 aimed in a direction above that of the object, more or less, accord- 
 ing to the distance of the object and the force of the charge ; thus, 
 for example, if the gun be discharged from w, jig. 54, and the 
 object be at 4, the force of the charge being such as to cause the 
 ball to move in the curve I, 2, 3, 4, 5, it is evident that in order to 
 hit the object at 4, the gun must be aimed in the direction WM. 
 In practical gunnery, the force of the charge of each form and 
 class of gun is known, and expedients are found by which the dis- 
 tance of objects aimed at may be determined with sufficient ap- 
 proximation for practical purposes. With these data the inclina- 
 tion at which aim is to be taken with each sort of weapon, in order 
 to hit the object, is determined by simple rules, which can be, 
 without difficulty, applied in the field. 
 
 Until recently the muskets placed in the hands of soldiers were 
 usually aimed, so that the line of sight was at once parallel to the 
 barrel, and directed to the object, as shown in^g-. 55. But since 
 the improvement which has recently taken place in musketry, and 
 more especially by the introduction of the improved rifle, greater 
 precision of aim has been attained. With an aim directed as in 
 fig. 55., at the object, the ball must necessarily pass below it; so 
 long as the range of a musket was of limited extent, and when 
 great precision was not expected to be attained, this deviation was 
 
MOTION OF PROJECTILES. 109 
 
 disregarded ; but since, by the modern improvements, the range 
 
 Fig- 55- 
 
 has been greatly augmented, the drop of the ball produced by the 
 curvature of the projectile would become so considerable as to de- 
 prive the weapon of the necessary precision. On the modern guns, 
 therefore, a double sight is provided, by which the elevation neces- 
 sary to ensure point blank precision can always be given to the 
 barrel ; one of the sights B,^. 56., is fixed in the usual manner on 
 
 Fig. 56. 
 
 the extremity of the barrel, while the other, A, is one which is gra- 
 duated, and sometimes pro- 
 vided with an adjustment, 
 by which it can be adapted 
 to objects at different dis- 
 tances, so as to hit them 
 point blank. 
 
 260. Effect of a ham- 
 mer. When a nail is dri- 
 ven by the strokes of a 
 hammer, the resistance 
 which the moving force has 
 to overcome, is the friction 
 between the nail and the 
 wood; the momentum of 
 tte hammer is however, 
 imparted to the wood, and 
 if the latter have not suffi- 
 cient resistance to prevent 
 it from unduly yielding to 
 Fig. 57. the force, the nail will not 
 
Fig. 58- 
 
 OF FO&CE AND MOTION. 
 
 penetrate. In such a case, 
 'for example, as that which 
 is represented in jig. 57., 
 where the nail is to be 
 driven into a board, having 
 no support behind it, and 
 not thick enough itself to 
 offer the necessary resist- 
 ance, the blows of the 
 hammer, if strong enough, 
 would break the board, but 
 would not drive in the nail. 
 The object is attained by 
 applying behind the board 
 fig-5%> a block of wood, or, 
 still better, a lump of lead, 
 against which the blows of 
 the hammer will be di- 
 rected. It is not, however, 
 as might be supposed, by 
 any increased resistance 
 thus opposed to the blows 
 that in this case the object is attained. To comprehend the 
 effect produced, it is necessary to consider that in each case 
 the momentum of the hammer is equally imparted to the mass 
 which it strikes; but in the one case (Jig- 57.) this momentum 
 is received, by the board alone, which, having little weight, is 
 driven by it through so great a space as to produce a consider- 
 able flexure, or even fracture ; but in the second case, the same 
 momentum, being shared between the board and the block of wood 
 or metal applied behind it, will produce a flexure of the former, 
 less in the same proportion exactly as the weight of the board and 
 block applied to it together is greater than the weight of the board 
 alone. 
 
 The same principle serves to explain a trick, or tour de force, 
 sometimes shown by public exhibitors. The exhibitor, extending 
 his body horizontally, his legs and shoulders being supported, 
 causes a heavy anvil to be placed upon his chest and abdomen ; 
 men employed for the purpose then give successive blows of heavy 
 sledge-hammers upon the anvil, without injury to the exhibitor : 
 blows which would speedily put an end to his exhibitions if they 
 were received without the interposition of the anvil. 
 
 To explain this feat, it is only necessary to consider that the 
 whole momentum of the sledge, being imparted to the anvil, will 
 give the latter a downward motion, just as much less than the 
 
or 
 
 EFFECTS 
 
 motion of the sledge as the mass of the sledge is less than the mass 
 of the anvil. Thus, if we suppose the weight' of the anvil to be 
 loo times less than that of the sledge, its downward motion upon 
 the" body of the exhibitor will be also loo times less than the 
 motion with which the sledge strikes it, and the body of the ex- 
 hibitor easily yielding to so slight a displacement, he passes through 
 the ordeal with complete impunity. 
 
 Influence of time upon the effect of force. Some extraor- 
 dinary and apparently unaccountable mechanical effects are ex- 
 plained by the fact that the effect of forces are greatly modified by 
 the continuance of their action ; a resistance which will be sufficient 
 to counteract a force whose action continues only for a few seconds, 
 will often yield to it if it act for as many minutes. 
 
 It is a fact familiar to all skaters, that they may pass with im- 
 punity over thin ice, which would break under their weight if they 
 moved over it more slowly, and still more if they stood still upon it. 
 
 The effect of forces acting upon solid bodies is transmitted from 
 molecule to molecule through their dimensions, before it can affect 
 the aggregate of their mass ; and if, from the nature of the force, 
 the entire duration of its action be less than that which is neces- 
 sary to propagate through the mass its effect, that effect can only 
 be produced upon the part of the mass on which it immediately 
 acts. 
 
 Let us suppose, for example, that a musket ball be pressed with 
 the hand against a pane of glass, it will break the glass to pieces 
 the fractures extending to its very edges. If the same ball be 
 flung from the hand with some velocity against the same pane, 
 it will pass through it, making a hole considerably larger than 
 the ball, surrounded by diverging cracks. But if the ball be 
 propelled from a gun against the same pane of glass, it will 
 pierce it with a hole whose diameter is nearly equal to that of the 
 ball, and which shall have clean edges as if the glass were bored 
 through with an augur. Now these effects are easily explained : 
 when the ball is merely pressed against the glass, the force has such 
 continuance of action, that its effects have time to be propagated 
 to the limits of the pane, which is accordingly broken in pieces ; 
 when the ball is projected against the glass with the hand, the ac- 
 tion is more sudden and of shorter continuance, and consequently 
 the fractures round the hole are much less extensive ; but when, 
 in fine, it is discharged from a gun, its action being instantaneous, 
 there is no time for the propagation of the force, and it merely 
 drives before it the portion of the glass which lies in its way. 
 
 If a cannon ball be flung with the hand against the pannel of a 
 door, suspended on hinges, it will cause the door to turn on its 
 hinges, yielding to its force ; but if the same cannon ball be pro- 
 
112 OF FORCE AND MOTION. 
 
 jected against the door by the cannon, it will pass clean through 
 the pannel, making a hole equal in diameter to the ball, and 
 without imparting any motion to the door. 
 
 In the former case, the action is slow enough to allow the force 
 to be propagated to all parts of the door, and therefore to move 
 it ; but in the latter case, being instantaneous, it only affects the 
 parts of the wood which lie immediately in its way. 
 
 CHAP. V. 
 
 CENTRE OP GRAVITY. 
 
 26 1 . Weight. If a body be prevented from moving in obedience 
 to the force of gravity by a fixed axis passing through it, a fixed 
 point from which it is suspended, or a surface placed beneath it, 
 the effect of gravity upon it will be manifested by a pressure 
 produced upon such axis, point of suspension, or surface. This 
 pressure is called the weight of the body. 
 
 As gravity acts separately upon all the component particles of 
 a body, the weight of such body is composed of the aggregate of 
 the weights of all its particles. This, which is manifest from what 
 has been already explained, may be rendered still more clear, from 
 considering that if a body be divided into parts, no matter how 
 minute and numerous, each of these parts will have a certain 
 weight, and the aggregate amount of their several weights will be 
 exactly equal to the weight of the body of which they are the 
 fragments. 
 
 Such a division may be carried to the most extreme practical 
 limit of comminution by pounding, grinding, filing, and other pro- 
 cesses known in the arts, and the weight will still be divided as the 
 matter is divided ; nor is it possible, even in imagination, to con- 
 ceive any degree of comminution so great that the same principle 
 will not prevail; and it may, therefore, be considered as esta- 
 blished that every individual atom which composes a body has 
 weight, and that the weight of the mass is the sum of the weights 
 of all its constituent atoms or molecules. 
 
 262. If the particles composing a body had no mutual coherence 
 or other mechanical connection having a tendency to retain them 
 in juxtaposition, each particle would obey the force of its gravity 
 independently of the others, and they would fall asunder like a 
 
CENTRE OF GRAVITY 
 
 mass of sand. But if they be so connected by their mutual co- 
 hesion, as they are in fact in all solid bodies, this cohesion will 
 resist the tendency of their weights to separate them ; they will 
 maintain their juxtaposition, the body will retain its form, and the 
 several forces with which gravity affects them will become com- 
 pounded, so as to produce a single force or pressure, which is the 
 resultant of all the separate forces impressed upon the particles. 
 
 263. As this resultant enters as a condition into every mecha- 
 nical question affecting bodies, it is of the greatest importance to 
 investigate the conditions by which in every case its intensity and 
 the line of its direction may be determined. 
 
 It has been already shown that the weights of all the particles 
 composing a body act in directions parallel to a plumb-line, or 
 perpendicular to a level surface. But it has been also demon- 
 strated (153.) that when any number of forces act in the same 
 direction in parallel lines, their resultant is a force acting in a line 
 parallel to them, and in the same direction in this line, and that its 
 intensity or quantity is equal to the sum of these forces. The 
 resultant, therefore, of the forces of gravity affecting all the par- 
 ticles of any mass of matter is a single force acting vertically 
 downwards, which is equal to the sum of all the forces affecting 
 the particles severally, and therefore equal to the weight of the 
 mass. 
 
 If, for example, A B, fig. 59., represent a mass of matter, and 
 the small arrows pointing vertically downwards 
 represent the direction of the gravitating forces of 
 the particles composing such mass, then it follows, 
 from what has been explained, that the resultant 
 of all these forces, or a single force equal to them, 
 will have a direction parallel to them, such as 
 D E, and will, in its intensity, be equal to their 
 
 But this is not yet sufficient to indicate this 
 resultant in a definite manner. We as yet only 
 know that its direction is parallel to the common 
 direction of the gravity of the particles ; but in- 
 numerable lines may be imagined passing through 
 the body vertically downwards, and the question 
 still remains to be determined which of these lines is the direction 
 of the resultant. 
 
 When the body in question has a determinate form and a uni- 
 form density, or even a density varying according to some known 
 conditions, the principles of mathematical science supply methods 
 by which the line of direction of the resultant may be determined , 
 
U4 OF FORCE AND MOTIOX. 
 
 but we shall here adopt a more simple and generally intelligible 
 method of explanation. 
 
 If we suppose the line represented by the great arrow i> E 
 (.fig- 59-) * be that of the resultant, then it is evident that if any 
 point such as c in that line be supported, the body will remain at 
 rest, because the resultant of all the forces acting upon the body 
 having the direction D E will be expended in pressure on the 
 fixed point c. The effect, therefore, will be that the whole weight 
 of the body will press upon c, and the body will remain at rest. 
 
 The same would be true for any point whatever in the direction 
 of the great arrow. If, for example, D were a pin from which a 
 thread was suspended, and that this thread were attached to the 
 body at any point in the line D c, then the body would still remain 
 at rest, the whole weight being expended in pressure upon the pin 
 at D ; for, as before, the resultant of all the forces of gravity acting 
 upon the component particles of the body would have the direction 
 D E, and would therefore be supported by the fixed pin at D. 
 
 But if a point of support be selected which is not in the direction 
 of the resultant i> E, such as p, and a string be carried from p to 
 any point of the body, such as c, then the body, although it will 
 not be permitted to descend vertically, in obedience to gravity, 
 will not nevertheless remain at rest. 
 
 If we suppose the weight of the body to be expressed by the 
 line c F, let this line be taken as the diagonal of a parallelogram 
 whose sides are c H and c i, one in the direction of the cord, and 
 the other at right angles with it, that portion of the weight 
 which is represented by c H, and which is in the direction of the 
 string, will act upon the fixed point P, and produce pressure upon 
 it. The portion of the weight which acts in the direction c I will 
 move the body towards the vertical line p G, which passes directly 
 downwards from the point of suspension. The body will there- 
 fore begin to move towards that vertical line. If the body had 
 been on the other side of the vertical line P G, it would still have 
 moved towards it, and therefore in a direction contrary to its 
 present motion. 
 
 It follows, therefore, that if a body be supported by a fixed 
 point, it cannot remain at rest, unless the resultant D E of all the 
 parallel forces which gravity impresses upon its particles pass 
 through that point. 
 
 264. We are thus supplied with a practical means of ascer- 
 taining the direction of the resultant of the weights of all the com- 
 ponent parts of a body with reference to any given point taken 
 upon it, since we have only to suspend the body by a string attached 
 to the given point, and allow it to settle itself to rest. When thus 
 at rest, the resultant of the weights of all its particles will be in the 
 direction of the string by which it is suspended. 
 
CENTRE OF GRAVITY. 1 1 5 
 
 If the same body be suspended by different points upon it, the 
 parallel directions of the gravitating forces of its particles will 
 differ in reference to the body, although they are the same in 
 reference to the direction of the suspending string, being always 
 parallel to it. Thus, for example, if an egg be suspended with its 
 length vertical, the parallel forces which gravity impresses on its 
 particles will be parallel to its length ; but if it be suspended with 
 its length horizontal, then the parallel directions of the gravity of 
 its particles will be perpendicular to its length. 
 
 265. Since in each case the resultant of these parallel forces 
 will coincide with the direction of the string, it must in the one 
 case pass through the egg in the direction of its length, and in 
 the other in a direction at right angles to its length. In like 
 manner, the body being supported by any point whatever taken 
 upon it, the direction of the string will be different for each such 
 point ; and consequently, there will be an infinite variety of re- 
 sultants of the gravitating forces of the particles of the body, 
 according to the different points by which it may be suspended. 
 
 Now, a question arises, whether there is any relation between 
 this infinite variety of resultants ; for if such, be not the case, the 
 determination of tBe resultant of the gravitating forces of a body, 
 would be a problem which would present itself under an infinite 
 diversity of forms and conditions for every individual body. 
 
 266. This question may be solved by a very simple experiment ; 
 and its solution is attended with a remarkable and important 
 result. 
 
 Take a solid body of any form, regular or irregular, and com- 
 posed of a material which is easily perforated without diminishing 
 its mass, or considerably deranging its structure. Take, for ex- 
 ample, a mass of putty of any form. Let this mass be suspended 
 by a thread attached to a fixed point, which it may easjjy be if 
 previously surrounded by a thread forming a loop. When at rest, 
 the resultant of the forces of gravity, acting upon all its particles, 
 will be a vertical line penetrating its dimensions in the direction 
 of the suspending thread. Take a needle, and pierce the putty in 
 this direction. The hole which is thus made through it will 
 represent the direction of the resultant of the gravitation of its 
 particles. 
 
 Let the mass be now detached from the thread of suspension, 
 and let it be again suspended, but in a different position, which 
 may be easily accomplished by the loops of thread surrounding it. 
 The mass will again settle itself into a position of rest, and, as 
 before, the direction of the resultant of all its gravitating particles 
 wiil be a vertical line in the exact direction of the suspending 
 
 i ^ 
 
1 1 6 OF FORCE AND MOTION. 
 
 thread. Let the putty, as before, be thoroughly pierced in this 
 direction with a needle. 
 
 Let the same experiment be repeated in three or four other 
 different positions of the mass, so that we shall obtain several holes 
 pierced through the body by the needle, representing the direc- 
 tion of the resultant of the gravitating forces, in the several posi- 
 tions in which the body was suspended. 
 
 Now a curious relation will be found to exist between the several 
 directions in which the needle has pierced through the putty. It 
 will be found, in fact, that all these lines of perforation intersect, 
 at a common point, within the dimensions of the body. This fact 
 may be easily established. 
 
 Let a needle be inserted in any one of the perforations, and it 
 will be found that another needle cannot pass through any of 
 them, for its progress will be stopped by the needle already in- 
 serted. All the perforations, therefore, must intersect each other 
 at a common point within the putty. 
 
 It appears from this experiment, that there is a certain point, 
 within the dimensions of the body, through which the resultant of 
 all the gravitating forces of the particles of the mass must pass, no 
 matter in what position the body may be placed. 
 
 267. Centre of gravity. This result, which is of high import- 
 ance, may be further illustrated and verified in the following 
 manner : 
 
 Let a flat thin plate of metal, or a piece of card, of any form, 
 however irregular, be pierced with small holes at several points, 
 so that it may be suspended upon a horizontal pin, the plate itself 
 being vertical. When so suspended, it can only remain at rest, 
 provided the resultant of the gravitating forces of its particles pass 
 through the pin ; for otherwise, as has been already explained, the 
 body \ipuld move, in, one direction or other, round the pin on 
 which it is suspended. 
 
 If a plumb-line be suspended from this pin, it is evident that 
 when the plate is at rest, the direction of the resultant of the gra- 
 vitating forces must coincide with the direction of the plumb-line. 
 Let a line then be traced upon the plate coinciding with the 
 direction of the plumb-line. Let the body be then detached from 
 the pin, and let the pin be inserted in another hole. The body 
 will now hang in another position, the resultant of the gravitating 
 forces of its particles again coinciding with the plumb-line. Let 
 the direction of the plumb-line be traced upon the plate as before. 
 In fine, let this experiment be repeated, with all the holes pierced 
 in the plate, and it will be found that the lines traced upon the 
 plate, indicating the various directions of the resultant of the 
 gravitating forces of its particles, will intersect each other at one 
 common point. 
 
CENTRE OF GRAVITY. n 7 
 
 zf>8. This common point, through which the resultants of. the 
 gravity of the particles of bodies pass, is called its centre of gravity. 
 A line drawn in the vertical direction through the centre of gra- 
 vity of a body, is called the line of direction of the centre of gravicy* 
 
 269. If the centre of gravity of a body be supported on a point, 
 or axis, and the body is free to turn round such axis, the body 
 will, in that case, remain at rest in any position in which it may 
 be placed ; for, according to what has been already stated, the 
 resultant of the gravitating forces of all its particles must be in 
 the direction of a vertical line passing through the centre of gra- 
 vity, and the whole weight of the body may be considered as 
 acting in that line. But, if the centre of gravity be suspended by 
 a pivot, or an axis, then the whole weight of the body will press 
 upon such pivot or axis, no matter what be the position in which 
 the body is placed. This may be easily verified by experiment. 
 
 Let the centre of gravity of any solid body be determined, by 
 suspending it from different points, in the manner explained above, 
 and let the body be placed upon a pivot or axis, passing through 
 this point. It will be found to rest indifferently on such axis or 
 pivot, in any position in which it may be placed. This experiment 
 may be easily performed with a piece of card or pasteboard. The 
 centre of gravity being determined, let a pin be passed through it, 
 and it will be found that the card will rest in any position upon 
 the pin. 
 
 270. Case of regular figures. If a body, being of uniform 
 density, have any regular figure, its centre of gravity will coincide 
 with its centre of magnitude ; for the matter composing the body 
 will, in such case, be symmetrically arranged round that point, so 
 that it is self-evident, that if this point be supported, the body will 
 have no tendency to turn in any direction round it. 
 
 271. Thus, for example, it is evident, without experiment, that 
 a ball or sphere of uniform density, such as a billiard-ball, has its 
 centre of gravity at the centre of its magnitude. In like manner, 
 a cube has its centre of gravity at the point where straight lines 
 joining its opposite corners would intersect each other, that is to 
 say, at its centre of magnitude. 
 
 272. If the figure of a body be such, that the matter composing 
 it is uniformly distributed round any line passing through it, its 
 centre of gravity must lie in that line, because, if it be suspended 
 by a string in the direction of that line, it will remain at rest ; 
 since the gravity of its particles, acting equally on every side of 
 such line, will have no tendency to move it, it will equilibrate. 
 Thus, it is evident that the centre of gravity of a cone, being of 
 uniform density, must be situate in its axis, that is, in a straight 
 line drawn from the point of the cone to the middle of its base. 
 
 i 3 
 
Ji8 OF FORCE AND MOTION. 
 
 In the same manner it may be shown, that the centre of gravity 
 of solids of an oval figure will be in the axis of the oval ; the 
 centre of gravity of a cylinder will be at the middle point of its 
 axis ; the centre of gravity of a straight rod of uniform thickness 
 will be at the middle point of its length, and at the centre of its 
 thickness. 
 
 It will be easy, in this and all similar cases, to verify the con- 
 clusions, by suspending the body in the manner already described. 
 
 273. Imaginary centre of gravity. The centre of gravity of 
 a body is not always placed within its dimensions. Thus, for example, 
 the centre of gravity of a hoop is at its centre, an imaginary point, 
 Avhich does not constitute any part of the body in question. In like 
 manner, in all hollow bodies the centre of gravity is an imaginary 
 point. Thus it is in the centre of a hollow sphere. The centre of 
 gravity of an empty box or cask is within it, at an imaginary point. 
 
 If a piece of wire, which when straight has its centre of gravity 
 at its middle point, be bent into a curved form, its centre of gra- 
 vity will be an imaginary point within the concave part of the 
 curve. In like manner, if the wire be bent into the form of a v, 
 the centre of gravity will be an imaginary point within the angle 
 of the v. 
 
 These conclusions may be verified, and the centre of gravity in 
 all such cases found, by suspending the body in different positions 
 in the manner already explained. 
 
 274. Although the centre of gravity in such cases be not a 
 material point, and not included within the dimensions of the body, 
 it nevertheless still possesses those properties which it would pos- 
 sess were it actually included within the mass of the body. 
 
 To verify this by experiment, let us suppose a bar of metal, A B 
 (Jig. 60.). bent into a curved form. Let its centre of gravity be 
 determined by suspension. When supported 
 D //^~~ ^^\\ r by the point A, let AC be the direction of 
 
 // " '"--..O. .- \\ the plumb-line, and when supported by the 
 
 (/.-- -\J point B, let BD be the direction of the 
 
 A B plumb-line. It follows, therefore, that the 
 
 Fig. 60. point o within the concavity of the circle 
 
 where these two lines intersect, will be the centre of gravity. Let 
 a light silk cord be attached to the points A and c, and stretched 
 tight between them, and let another silk cord be stretched between 
 the points B and D in the same manner. Now the point o, where 
 these two cords cross each other, will be the centre of gravity. 
 
 Let a cord be tied to the junction of the strings at o, and let 
 the upper extremity of this cord be attached to a fixed point, so 
 that the wire may be thus suspended. It will be found that in 
 this case the hoop of wire will rest in equilibrium in any position 
 
CENTRE OF GRAVITY. 
 
 in which it may be placed. In this case, the weight of the silk 
 string, being insignificant in comparison with the weight of the 
 wire, does not disturb the position of the centre of gravity, which 
 still remains at o. 
 
 275. If a body, without being absolutely fixed in its position so 
 as to be immovable, be nevertheless partially restrained, so as to be 
 capable of moving only under certain conditions or within certain 
 limits, then the centre of gravity will have always a tendency to 
 move into the lowest position which the conditions under which the 
 body is placed will admit of; and in all cases it can never remain 
 at rest unless its line of direction that is to say, a vertical line 
 passing through it should pass through a point of support. 
 
 276. It may therefore be assumed as a principle of the highest 
 generality, that in all cases in which a body is at rest, a vertical 
 line passing through its centre of gravity must also pass through 
 a point of support. If the point of support, therefore, through 
 which this line passes be placed above the centre of gravity, the 
 body is said to be suspended ; if it be placed below, it is said to 
 be supported. 
 
 277. If a body be suspended from a fixed point by a string, it 
 will remain at rest, as has been already explained, provided its 
 centre of gravity be placed in a vertical line under the point of 
 support. But if the body be drawn out of that position, so that the 
 centre of gravity will be on either side of such vertical line, then 
 
 the body when disengaged 
 
 P 
 
 A 
 
 will fall from such position 
 to the vertical line, and in 
 consequence of its inertia 
 -will continue its motion 
 beyond the vertical line 
 until it comes to rest; it 
 will then return to the ver- 
 tical line, and thus oscillate 
 from side to side. 
 
 278. Pendulum. Such 
 a body constitutes what is 
 called the pendulum. 
 
 Let P, fig. 6 1 ., be the point 
 II of suspension. Let PA re- 
 present the string, and c 
 the centre of gravity of the 
 body. Let the weight of 
 Fig ' 6l . the body be represented by 
 
 the vertical line CD. Let 
 
 this be taken as the diagonal of a parallelogram, one of whose sides 
 
 14- 
 
 J 
 
 'II 
 
 .1, 
 
 \ 
 \ 
 D 
 
J20 OF FORCE AND MOTION. 
 
 CH is in the direction of the string, and the other ci at right 
 angles to it. The weight represented by the diagonal will thus, 
 by the resolution of forces, be equal to two forces, one represented 
 by c H and the other by c i. That which is represented by c H 
 expends itself in pressure on the point of suspension ; the other, 
 represented by ci, will cause the body to move towards the ver- 
 tical line P v, and in so moving the centre of gravity will describe 
 the circular arc c G. When the centre of gravity arrives at G, it will 
 be in the vertical line P v, passing through the point of suspension ; 
 and if the body were at rest it would remain there ; but on arriv- 
 ing at G, the body has a certain velocity and moving force, which 
 it will retain in virtue of its inertia, until deprived of it by some 
 external agency. It will therefore continue to move to the left of 
 G, and the centre of gravity will describe the circular arc GC'. 
 In ascending this circular arc, the force of gravity has a tendency 
 to destroy its velocity. 
 
 Let the weight of the body, as before, be represented by the 
 vertical li'ie G' D' : it will be equivalent to the two forces repre- 
 sented by c' H' and c' i'. The force c' H' is expended in pressure 
 upon the point of suspension, P ; the other, c' i', has a tendency to 
 carry the centre of gravity, c', back to the point G, along the cir- 
 cular arc c' G. This component of gravity, while the body moves 
 from G to c, gradually deprives it of its momentum, and if the 
 momentum be entirely destroyed at the point c, then this same 
 component of gravity, c' i', will cause the body to return along 
 the circular arc to the point G. In this manner the body would 
 oscillate continually from side to side of the vertical line P v, the 
 centre of gravity describing alternately equal arcs, G c and G c'. 
 But the resistance of the air and other impediments have a ten- 
 dency continually to dimmish the length of the arcs, by which it 
 departs from the vertical line, until at length the body loses its 
 vibration and settles itself in such a position that the centre of 
 gravity c will be quiescent in the vertical line P v. 
 
 279. Stability. The stability of a body resting in any position 
 is estimated by the magnitude of the force required to disturb and 
 overturn it, and therefore will depend on the position of its centre 
 of gravity with respect to the base. If its position can be dis- 
 turbed or deranged without raising its centre of gravity, then the 
 slightest force will be sufficient to move it ; but if its position 
 cannot be changed without causing its centre of gravity to rise to 
 a higher position, then a force will be necessary which would be 
 sufficient to raise the entire weight of the body through the height 
 to which its centre of gravity must be elevated ; for, according to 
 what has been already explained, the whole weight of the body 
 may be considered concentrated at its centre of gravity. 
 
CENTRE OF GRAVITY. 121 
 
 280. Let B A c, jig. 62., represent a pyramid, the centre of 
 
 A 
 
 J 
 
 t! 
 
 B C B C 
 
 Fig. 62. Fig. 63. Fig. 64. 
 
 gravity of which is G. To turn this over the edge B, the centre of 
 gravity must be carried over the arc G E, and must therefore be 
 raised through the height H E. If, however, the pyramid were 
 taller relative to its base, as in fig. 63., the height H E, through 
 which the centre of gravity would have to be elevated, would be 
 proportionally less ; and if the base were still smaller in reference 
 to the height, as in Jig. 64., the height H E would be still less, and 
 so small, that a very slight force would throw the pyramid over 
 the edge B. It is evident, from examining these diagrams, that 
 the principle may be generalized, and that it may be stated that 
 the stability of any body depends, other things being the same, 
 upon the distance of the line of direction of its centre of gravity 
 from the edges of its base. The nearer this direction is to one 
 edge of the base, the more easily will the body be turned over this 
 edge. 
 
 281. Line of direction. If the line of direction of the centre 
 
 of gravity fall outside the edge, as in Jig. 65., then 
 the weight of the body concentrated at G will be 
 unsupported, and the body will fall over its edge. 
 
 This will always take place, if the body be not 
 attached to the ground at its base ; but it happens, 
 in some cases, that the body is so rooted to the ground 
 at its base, that it will resist the tendency of its 
 weight to make it fall, even though the line of direction of its 
 centre of gravity should fall a little outside its base. Thus, we see 
 trees not unfrequently leaning in such a position, that their centre 
 of gravity obviously falls outside the limits of their trunk. Yet 
 the trees nevertheless remain standing, the tenacity of the roots 
 and their hold upon the soil being sufficient to resist the effect of 
 their weight acting at the centre of gravity. 
 
 282. In the case of the celebrated leaning towers of Pisa and 
 Bologna, although they are inclined considerably from the perpen- 
 dicular, the lines of direction of their centres of gravity still fall 
 within their bases. The tower of Pisa is 315 feet high; and it 
 is inclined so that if a plumb-line hang from the side towards 
 
122 
 
 OF FORCE AND MOTION. 
 
 which the inclination takes place, it will meet the ground at 1 2 ft. 
 4 in. from the base. The tower of Bologna is 1 34 feet high, and 
 a plumb-line similarly suspended would fall at 9 ft. 2 in. from the 
 base. Nevertheless, these structures have stood, and will probably 
 stand, as permanently as if they were erected in the true perpen- 
 dicular; for the line of direction of their centre of gravity falls 
 sufficiently within the base to render their overthrow impossible by 
 any common force to which they will be exposed. 
 
 283. If the line of direction of the centre of gravity, however, 
 fall upon the edge, as in fig. 66., then the body 
 will still stand ; but it will be in a condition in 
 which the slightest possible force will turn it over, 
 as in this case it can be overturned without causing 
 the centre of gravity to rise. 
 
 In figs. 67. 68. and 69. the line of direction falls 
 within the base ; and the position is stable. 
 
 Fig. 67. F^ 68. Fig. 69. 
 
 In fig. 70. it falls within, and in fig. 71. outside the base. The 
 
 Fig. 70. 
 
 Fig. 71. 
 
 position is stable in the former case ; and in the latter the body 
 falls. 
 
 284. Hence appears the principle upon which the stability of 
 loaded carriages or waggons depends. When the load is placed at 
 a considerable elevation above the wheels, the centre of gravity is 
 elevated, and the carriage becomes proportionally unstable. In 
 coaches for the conveyance of passengers, the luggage is therefore 
 very unsafely placed when collected on the roof, as is generally 
 
CENTRE OF GRAVITY. 123 
 
 done. It would be more secure to pack the heavier luggage in 
 the lower parts of the coach, placing light parcels on the top ; for 
 in such case the centre of gravity of the loaded vehicle would be 
 in a lower position. 
 
 285. Drays for the carriage of heavy loads are often constructed 
 in such a manner, that the load would be placed below the axle 
 of the wheels. If a waggon or cart, loaded in such a manner that 
 its centre of gravity shall be in an elevated position, pass over an 
 inclined road, so that the line of direction of the centre of gravity 
 would fall outside the wheels, it would cause the vehicle to be 
 overturned. 
 
 The same waggon will have a greater stability when loaded 
 with a heavy substance which occupies a small space, such as metal, 
 than when it carries the same weight of a lighter substance, such 
 as hay, because the centre of gravity in the latter will be much 
 more elevated. 
 
 286. If a large table be placed upon a single leg in its centre, 
 it will be impracticable to make it stand firm ; but if the pillar on 
 which it rests terminate in a tripod, it will have the same stability 
 as if it had three legs attached to the points directly over the 
 places where the feet of the tripod rest. 
 
 It will be still more stable if it be supported on three legs placed 
 as in fig. 72. 
 
 287. When a solid body is supported 
 by more points than one, it is not necessary 
 for its stability that the line of direction 
 should fall on one of these points If 
 there be only two points of support, the 
 line of direction must fall between them. 
 The body is in this case supported as 
 effectually as if it rested on an edge coin- 
 ciding with a straight line drawn from one point of support to the 
 other. If there be three points of support, which are not ranged 
 in the same straight line, the body will be supported in the same 
 manner as it would be by a base coinciding with the triangle formed 
 by straight lines joining the three points of support. In the same 
 manner, whatever be the number of points on which the body 
 rests, its virtual base will be found by supposing straight lines 
 drawn, joining the several points of support. When the line of 
 direction falls within this base, the body will always stand firm ; 
 and otherwise not. The degree of stability is determined in the 
 same manner as if the base were a continued surface. 
 
 If there be four or more legs, the table may be unstable, even 
 though the centre of gravity fall within the base, and will be so if 
 
OF FORCE AND MOTION. 
 
 the ends (A, B, c, D, Jig. 73.) of the legs be not in the same plane ; 
 for in that case they cannot all rest together on a level floor. 
 
 287. All the attitudes, 
 gestures, and movements of 
 animals are governed with 
 reference to the centre of 
 gravity of their bodies. 
 When a man stands, the 
 line of direction of his 
 weight must fall within the 
 base formed by his feet. If 
 A B, c D (fig. 74.), be the 
 
 Fi _- feet, this base is the space 
 
 A B D c. It is evident that 
 
 the more his toes are turned outwards, the more contracted the 
 ^, b ase W *U be in the direction E r, and the 
 more liable he will be to fall backwards or 
 CL..U j. l.JLa forwards. Also, the closer his feet are toge- 
 ther, the more contracted the base will be to 
 the direction GH, and the more liable he will 
 be to fall towards either side. 
 288. When a man walks, the legs are alter- 
 nately lifted from the ground, and the centre of gravity is either 
 unsupported or thrown from the one side or the other. The body 
 is also thrown a little forward, in order that the tendency of the 
 centre of gravity to fall in the direction of the toes may assist the 
 muscular action in the propelling the body. This forward incli- 
 nation of the body increases with the speed of the motion. 
 
 But for the flexibility of the knee-joints, the labour of walking 
 would be much greater than it is, for the centre of gravity would 
 be more elevated by each step. The line of motion of the centre 
 of gravity in walking is represented by 7?-. 75., and deviates but 
 
 Fig. 74. 
 
 Fig. 75- 
 
 Fig. 76. 
 
 little from a regular horizontal line, so that the elevation of the 
 centre of gravity is subject to very slight variation. 
 
 289. But if there were no knee-joint, as when a man has wooden 
 legs, the centre of gravity would move as in fig. 76., so that at 
 each step the weight of the body would be lifted through a more 
 considerable height, and therefore the labour of walking would 
 
CENTRE OF GRAVITY. 
 
 125 
 
 be much increased. If a man stand on one leg, the line of direc- 
 tion of his weight must fall within the space on which his foot 
 treads. The smallness of this space, compared with the height of 
 the centre of gravity, accounts for the difficulty of this feat. 
 
 290. The position of the centre of gravity of the body changes 
 with the posture and position of the limbs. If the arm be ex- 
 tended from one side, the centre of gravity is brought nearer to 
 that side than it was when the arm hung perpendicularly. When 
 dancers, standing on one leg, extend the other at right angles to 
 it, they must incline the body in the direction opposite to that in 
 which the leg is extended, in order to bring the centre of gravity 
 over the foot which supports them. 
 
 291. When a porter carries a load, his position must be regu- 
 
 lated by the centre of gravity of his body and 
 the load taken together. If he bore the load 
 on his back, the line of direction would pass 
 beyond his heels, and he would fall back- 
 wards. To bring the centre of gravity over 
 his feet, he accordingly leans forward (Jig. 
 77.). If a. nurse carry a child in her arms, 
 Flg- 77 ' she leans back for a like reason. When a 
 
 load is carried on the head, the bearer stands upright, that the 
 
 centre of gravity may be over his feet. 
 
 292. In ascending a hill we appear to incline forward, and in 
 
 descending to lean backward; but, 
 in truth, we are standing upright 
 with respect to a level plane. This 
 is necessary to keep the line of di- 
 rection between the feet, as is evi- 
 dent from. Jig. 78. 
 
 293. A person sitting on a chair 
 cannot rise from it without either 
 stooping forward to bring the centre 
 of gravity over the fyet, or drawing 
 back the feet to bring them under the centre of gravity. 
 
 If a person stand with his side close against a wall, his feet 
 being close together, he will find it impracticable to raise the out- 
 side foot ; for if he did, the line of direction of the centre of 
 gravity of his body would fall outside the inner foot, and he would 
 be unsupported. 
 
 294. When a quadruped stands with his four feet on the 
 ground, the centre of gravity of his body is over a point found by 
 drawing the two diagonals of the quadrilateral formed by his feet ; 
 that is to say, if a line be drawn joining his right fore foot with his 
 left hind foot, and another joining his left fore foot with his right 
 
 Fig. 78. 
 
126 OF FORCE AND MOTION. 
 
 hind foot, then the centre of gravity of his body will be very 
 nearly over the point where these lines cross each other. Strictly 
 speaking, it will generally be a little nearer to his fore feet than 
 this point. It will, however, still be very nearly on the centre of 
 the quadrilateral base formed by his four feet, and, therefore, in a 
 position to give complete stability to the animal. 
 
 When a quadruped walks, he raises his right fore and left hind 
 foot (the former leaving the ground a little before the latter), the 
 diagonal line joining his left fore foot, and right hind foot sup- 
 porting his weight. The centre of gravity of his body is a little 
 in advance of this line, and his gravity, therefore, assists his for- 
 ward motion. The left fore foot is raised a moment before the 
 left hind foot is brought to the ground, and, in like manner, the 
 right hind foot is raised immediately after the right fore foot 
 comes to the ground. The effect of these motions is, that the 
 weight of the animal is thrown alternately upon the two diagonal 
 lines joining the right fore and left hind foot, and the left fore and 
 right hind foot. . 
 
 When a quadruped trots, he also raises his legs from the ground, 
 alternately, by pairs, placed diagonally ; but in this case the two 
 feet leave the ground and return to it precisely together, and each 
 pair springs from the ground a moment before the other pair 
 returns to it ; so that there are short intervals between the suc- 
 cessive returns of the feet, by pairs, to the ground, during which 
 the entire body is unsupported. The weight is, in these intervals, 
 projected upwards by the spring of the legs, so that the centre of 
 gravity of the body describes a succession of arcs concave towards 
 the ground. It is this motion of the body which produces the 
 action sustained by the rider of a horse in trotting. 
 
 When a quadruped gallops, he raises simultaneously his two 
 fore legs, and by the muscular action of his hind legs he projects 
 his weight forwards. During the spring the centre of gravity is 
 unsupported, but is thrown forward, describing a circular arc con- 
 cave towards the ground. When this arc has been completed, the 
 fore legs reach the ground, and immediately afterwards the hind 
 legs ; and the centre of gravity is again momentarily supported, 
 and the animal is in an attitude to repeat the same action. 
 
 295. If a cylindrical body of uniform density be placed upon a 
 horizontal plane, A &,jig. 79., its centre 
 of gravity being its centre of magni- 
 tude, c, the line of direction c p will 
 necessarily meet the plane at the point 
 where the cylinder touches it, and the 
 body will consequently remain at rest. 
 If the cylinder be rolled upon the plane, 
 
CENTRE. .OF GRAVITY. 127 
 
 the centre of gravity will be carried in a horizontal line, parallel 
 to the plane represented by the dotted line in the figure. Since, 
 therefore, in this motion the centre of gravity does not rise, any 
 force applied to the body, however slight, will cause it to move, 
 since no elevation of its weight is required. But, on the other 
 hand, the body will have of itself no tendency to change its po- 
 sition, because the centre of gravity is not only supported, but 
 because by no change of the body can it assume a lower position. 
 296. If a board of uniform density be cut into the form of an 
 ellipse, A B, fig. 80., and be placed upon a 
 level surface, D E, with the longer axis A B of 
 the ellipse parallel to the surface D E, the 
 centre of gravity, c, will then be vertically 
 over the point P, at which the board touches 
 Fig. 80. the surface, and the body will be supported 
 
 at rest. If the body be disturbed slightly 
 from this position, the end A being depressed, and the end B ele- 
 vated, then the centre of gravity, c, will be elevated towards the 
 point o ; and if, on the other hand, the end B be depressed, and the 
 end A elevated, then the centre of gravity will be raised towards 
 the point o'. In either case, this elevation of the centre of gravity 
 will require the application of such a force as would be sufficient 
 to raise the body through that height, whatever it be, through 
 which the centre of gravity has been elevated ; and if, after such 
 elevation, the body be disengaged, and left to the free action of 
 gravity, the centre of gravity will descend to the lowest pos- 
 sible position, that is to say, to the position represented in the 
 figure, and will oscillate from side to side of this position until the 
 vibrating motion be destroyed by the resistance of the air and by 
 friction. The centre of gravity will then rest in the position 
 represented in the figure. 
 
 Let us now suppose that the same board is placed on the hori- 
 zontal plane D E, with its longer axis vertical, as 
 represented in fig. 8 1 . The line of direction, c P, 
 of the centre of gravity will now pass through the 
 point of support of the body, and consequently the 
 board will be supported. But if in this case the 
 body be slightly turned from its position to the right 
 or to the left, the centre of gravity will descend 
 towards o or towards o', and cannot resume the 
 original position at c until a force be applied to 
 it which would be sufficient to raise the weight of the body througli 
 the height to which the centre of gravity has fallen. It is evident, 
 therefore, that in this case, if the position of equilibrium, c P, be 
 disturbed in the slightest degree by inclining the body a little 
 either to the right or the left, the centre of gravity will move 
 
128 OF FORCE AND MOTION. 
 
 downwards, and the body will fall until it take the position repre- 
 sented in fig. 80., with its long axis horizontal. 
 
 297. Stable, unstable, and neutral equilibrium. Now, it 
 will be observed, that in each of the three cases represented in 
 Jigs. 79., 80., and 8 1. these is equilibrium; but this equilibrium 
 
 is characterized in each case by particular conditions. 
 
 298. In the case represented in Jig. 80., the equilibrium is 
 called stable, because, if it be deranged either to the right or to the 
 left, the body will of itself return to it, and settle definitively into 
 it, after some oscillation, the centre of gravity resuming its position 
 over the point P. This position of stable equilibrium is deter- 
 mined by the condition that no change of position can take place 
 in the body without causing an elevation in the centre of gravity ; 
 or, what is the same, it is that position in which the centre of 
 gravity is at the lowest point it is capable of assuming consistently 
 with the conditions under which the body is placed. 
 
 The state of equilibrium represented in Jig. 8 1 . is called unstable, 
 or tottering equilibrium. It is such a state of equilibrium, that if 
 the slightest derangement takes place in the position of the centre 
 of gravity, it will not return to the same point, but the body will 
 assume another position, in which the centre of gravity will be in 
 a state of stable equilibrium, as represented in Jig. 80. 
 
 299. Unstable equilibrium, then, is characterized by the quality 
 that the centre of gravity is at 'the highest point which it can 
 assume compatibly with the conditions in which the body is 
 placed ; and although it is vertically over the point of support, it 
 is nevertheless in such a condition that the slightest derangement 
 will cause it to descend, and the body to be overturned. 
 
 300. The case represented in Jig. 79. is an intermediate con- 
 dition between these two extremes, and is called the state of 
 neutral equilibrium. It is neither stable nor unstable. It is not 
 stable, because the slightest force applied to the body will perma- 
 nently change its position. It is not unstable, because the body 
 will not be overturned by the action of its own weight, however 
 its position may be changed. 
 
 Another case of neutral equilibrium is shown in Jig. 82. 
 
 301. Examples. The effects of 
 a variety of children's toys are ex- 
 plained by this principle. The centre 
 of gravity is, by loading one ex- 
 tremity in a manner not perceptible 
 to the eye, moved to a considerable 
 distance from the centre of magni- 
 Fig ga a tude. The object, therefore, will 
 
 only stand when the point which is 
 the real centre of gravity is in the lowest position. 
 
CENTRE OF GRAVITY. 
 
 129 
 
 Fig. 8j. 
 
 Thus, a figure (fig. 83.) made of some light substance, such as elder 
 pith or cork, has a piece of lead at- 
 tached to one of its extremities. If 
 it be placed on the other extremity, 
 the end being rounded, it will appa- 
 rently, by a spontaneous movement 
 invert its position, and a sort of 
 tumbler will be formed. 
 
 In fig. 84. a toy is represented, the 
 principle of which will be easily un- 
 derstood from what has been explained. The figure rests steadily 
 on the toe, because, by the weights at the 
 lower ends of the rods, the centre of gra- 
 vity is brought below the point of sup- 
 port. 
 
 302. Many of the feats exhibited by 
 sleight-of-hand performers are explained 
 by the principles of stable and unstable 
 equilibrium. If any object, such as a 
 sword, be supported on its point, it will 
 be in unstable equilibrium so long as its 
 centre of gravity is directly over its point ; 
 but as it cannot be maintained precisely 
 so, the finger or other support of the point 
 is moved slightly in one direction or 
 another, so as to keep nearly under the 
 centre of gravity, and to check the ten- 
 dency of the sword to fall on the one side 
 or on the other. 
 
 But these and similar feats are greatly 
 facilitated, if the object thus balanced 
 
 is made to spin upon its point; for in that case the centre of 
 gravity, though not vertically over the point of support, is 
 continually revolving round a vertical line passing through the 
 point of support, and the tendency which it has at one moment to 
 make the body fall on one side is instantly checked by a contrary 
 tendency when it passes to the opposite side. 
 
 303. It is in this manner that the common effect of a spinning-top 
 is explained. It would be stable, if placed as in fig. 85.; but it 
 would be quite impracticable to make it stand on its point, as in 
 fig. 86., if it did not revolve, or if it revolved very slowly ; but if it 
 have a very rapid motion of gyration, then it will stand steadily on 
 its point. 
 
 It. may be asked how the rapidity of the gyration affects the 
 
 Fig. 84. 
 
130 OF FORCE AND MOTION. 
 
 question. This is easily explained. If the centre of gravity re- 
 volve round the line so slowly that the time taken in half a revo- 
 lution is so considerable as to allow it to fall to any considerable 
 
 Fig. 85. Fig. 86. 
 
 depth, then it cannot recover itself when it passes to the other 
 side ; but if the revolution be so rapid that half the time of one 
 gyration is so small that the centre of gravity cannot fall through 
 any sensible height, the top will maintain its position. 
 
 304. Public exhibitors place a circular plute on the point of a 
 sword, the point being placed as near the centre of the plate as 
 possible. But, however near the centre it may be placed, it is not 
 always possible to ensure its coincidence with the centre of gravity 
 of the plate. If in this case the plate were at rest on the point of 
 the sword, it would not be balanced, but would incline and fall on 
 that side at which the centre of gravity would lie. The exhibitor, 
 therefore, prevents this effect, by giving to the plate a rapid 
 rotation on the point of the sword. The centre of gravity of the 
 plate rapidly moves in a small circle round the point of support, 
 and its tendency at one moment to fall down at one side is checked 
 the next moment by a contrary tendency to fall down at the other 
 side, and the plate accordingly remains balanced on the point. 
 
 305. In some cases the centre of gravity of a body apparently 
 
 ascends; but this is always deceptive, and its 
 real motion is invariably a descending one. Let 
 a cylinder of wood, A &,jig. 87., be pierced by a 
 hole, o, near its surface, B, and let a cylinder of 
 lead be inserted in this hole. The centre of 
 gravity of the mass will then be, not at its centre 
 of magnitude, but between that point and the 
 centre of the cylinder of lead which fills the hole 
 o, and will not be far removed from the centre 
 Fig. 87. of the lead, in consequence of the great coin-* 
 
 parative weight of that substance. 
 
 If such a cylinder as this be placed upon an inclined plane, M N, 
 Jig. 88., in such a position that the lijie of direction, o B, of the 
 centre of gravity shall fall above the point of contact, P, of the 
 cylinder with the plane, the cylinder will roll up the plane, because 
 
CENTRE OF GRAVITY 
 
 131 
 
 its weight, concentrated at the centre of gravity, o, acting down- 
 wards in the line o B, 
 will have a tendency 
 to descend towards 
 N the plane, and will so 
 descend, causing tftie 
 body to roll towards 
 N; and it will con- 
 tinue to descend un- 
 til the cylinder take 
 such a position that 
 the line of direction 
 of the centre of gra- 
 vity, o B, shall pass through the point of contact of the cylinder 
 with the plane. 
 
 In these and similar cases, although the general mass of the Lody 
 rises, the particular part occupied by its centre of gravity falls ; 
 that point is, in effect, simultaneously affected by two motions, one 
 produced by the progressive motion of the cylinder up the plane 
 from M to N, and the other by the motion of revolution of the 
 cylinder round its centre, c. In virtue of the former, the centre 
 of gravity would rise ; and in virtue of the latter, it would fall. 
 The effect of the latter predominates until the centre of gravity 
 comes into the vertical line, passing through the point of contact 
 of the cylinder with the plane. 
 
 306. A case of the ascent of the centre of gravity is sometimes 
 produced by public exhibitors, the explanation of which may here 
 be found instructive. The exhibitor places a sphere of wood 
 upon an inclined plane, and standing upon it, he places his feet on 
 that side of the centre which is towards the elevation of the plane. 
 Immediately the globe begins to roll up the plane, and the ex- 
 hibitor, by moving his feet so as to keep them near the highest 
 point of the globe, but still on the side next the elevation of the 
 plane, the globe continues to .roll up the plane, the exhibitor 
 dexterously maintaining his position as here described. 
 
 In this case there is a real ascent of the common centre of 
 gravity of the globe and the body of the exhibitor. Now the 
 question is, What force in this case produces this ascent ? for it is 
 evident that in the time during which it rolls to the top of the 
 plane, the entire weight of the globe and the body of the exhibitor 
 has been elevated through a perpendicular space equal to the 
 height of the plane. 
 
 The force which accomplishes this is the muscular action of the 
 feet of the exhibitor upon the surface of the globe. As the globe 
 rolls up the plane, if the feet of the exhibitor pressed upon the 
 
132 OF FORCE AND MOTION. 
 
 same point of its surface, they would descend ; and in that case 
 the common centre of gravity of the globe and the body of the 
 exhibitor, instead of ascending, would in fact descend, until the 
 feet of the exhibitor would sink down to the surface of the plane ; 
 but this is prevented by the feet of the exhibitor continually 
 stepping backwards upon the surface of the globe, so as to stand 
 near the top; and thus, by moving his feet on the globe back- 
 wards continually towards the top of it, the exhibitor elevates the 
 centre of gravity of his body, while the action of his feet upon the 
 globe causing it to roll up the plane at the same time, raises the 
 centre of gravity of the globe. 
 
 307. Magic clock. Time-pieces, with a transparent glass dial 
 
 and without any apparent works, thus called, 
 often excite the wonder of the curious. The 
 principle of their motion depends on the 
 tendency of the centre of gravity to take the 
 lowest possible position. In the centre of the 
 dial is a small circular gilt plate of metal, 
 which conceals a very short arm, A,of the hand 
 A c B, Jig. 89. This short arm carries a 
 hollow ring, in the interior of which a small 
 weight is moved by clockwork, through 
 the intervention of an arbor which passes 
 Flg< 89< through the centre of the dial, the works 
 
 which turn the rod being at a distance behind the dial, so as not 
 to be visible through it. This weight A, is thus made to move 
 uniformly round the ring, and to change continually the position 
 of the centre of gravity of the hand, so as to make it describe 
 uniformly round the centre of the dial the small dotted circle c G 
 in twelve hours, so that the successive hours are indicated by its 
 extremity B. The ring and weight A being concealed, the hand 
 seems to move without any apparent cause of impulsion ; and what 
 adds to the illusion is, that the hand can be freely turned by apply- 
 ing the finger to it, but when left to itself it always returns to the 
 point from which it was turned, so as still to indicate the time, and 
 to be incapable of being set to a wrong hour. 
 
 308. Centre of gravity of fluids. In all that we have stated 
 respecting the centre of gravity, we have supposed the body to be 
 solid ; but this quality also plays an important part in the pheno- 
 mena of fluids. The centre of gravity of a fluid mass is deter- 
 mined by the same conditions as if it were solid. It is that point 
 which would have the properties already defined, if the fluid mass 
 were supposed to be congealed. 
 
 Thus the centre of gravity of the water forming a lake is that 
 point which would have the properties already explained, if the 
 
CENTRIFUGAL FORCE. 133 
 
 water of the lake were converted into a mass of ice. It will ap- 
 pear hereafter, however, to possess, in reference to fluid bodies, 
 many important characters. 
 
 309. The centre of gravity of two separate and independent 
 bodies is that point between them which would possess the charac- 
 ters already denned, if the two bodies were united by a straight 
 rod which is itself devoid of weight. This point may be deter- 
 mined by a vary simple mathematical process. Let the centres of 
 gravity of the two bodies in question be conceived to be connected 
 by a straight line, and let a point be found upon this straight line 
 which shall divide it into two parts, which shall be in the inverse 
 proportion of the weights of the two bodies. Then this point will 
 be their common centre of gravity. 
 
 Thus let A and K,jig. 90., be the two bodies, and let a, b be 
 their centres of gravity. Draw the line a &, and take upon it 
 
 a point c, so that b c shall 
 bear to a c the same propor- 
 fj\ tion as the weight of A bears 
 
 c to the weight of B. In this 
 
 case, c will be the centre of 
 gravity of the two bodies. 
 
 Xow if the line a b were a rigid rod, devoid of gravity, the point c 
 would have all the properties which have been already explained 
 as belonging to the centre of gravity; thus, the bodies would 
 balance themselves on c in any position. 
 
 CHAP. VI. 
 
 CENTRIFUGAL FORCE. 
 
 310. IF a ball of metal or other heavy substance, placed upon a 
 smooth and level surface, be attached to the extremity of a string, 
 the other extremity of which is fastened to a fixed point upon the 
 surface, and then whirled round in a circle, it is known, by uni- 
 versal and constant experience, that the string will be stretched 
 with a certain force, which will be augmented as the velocity of 
 the whirling motion is increased, or as the string is lengthened. 
 That such force is not produced by gravity is evident, inasmuch 
 as the level surface upon which the body moves supports its 
 weight. 
 
 311. This force, which always attends matter that is moved 
 K 3 
 
'34 
 
 OF FORCE AND MOTION. 
 
 round a centre, in what manner and under what form soever the 
 motion be produced, is called centrifugal force, because it is mani- 
 fested by a tendency of the matter which revolves to recede from 
 the centre of revolution; this tendency in the case just mentioned 
 being manifested by the force with which the string connecting the 
 body with the fixed point is stretched. This tension resists the 
 tendency of the ball to fly from the centre, and is therefore the 
 measure of its centrifugal force. 
 
 312. That centrifugal force is a mere effect of the inertia of 
 matter, may be easily shown. It has been already explained, that 
 in virtue of its inertia, a body, if in motion, can only move uni- 
 formly in a straight line. If, therefore, it be deflected from one 
 straight line into another straight line, it must be by the action of 
 some force impressed upon it at the moment of deflection ; and if 
 a body be continually deflected from a straight direction, which it 
 must be if it move in a curve, then such body must be under the 
 operation of a force continually acting upon it, producing such 
 incessant change of direction. 
 
 Let P, ./?#.' 9 1., be the fixed point to which the string is at- 
 tached. Let A be the ball, and let A c r be the circle in which the 
 ball is whirled round. Let A c be a 
 small arc of this circle moved over in a 
 given interval of time. Starting from 
 A, the motion of the ball has the direc- 
 tion of the tangent A D to the circle, 
 and it would move from A to D in the 
 given interval of time, if it were not 
 deflected from the rectilinear course ; 
 but it is deflected into the diagonal 
 A c, and this diagonal, by the composi- 
 tion of forces, is equivalent to two 
 forces represented by the sides AD, 
 A B. But the motion A D is that which 
 the body would have in virtue of its inertia ; and therefore the 
 force A B, directed towards the fixed point P, is that which is im- 
 pressed upon it by the tension of the string, and which, combined 
 with the motion A D, causes it to move in the diagonal A c. 
 
 The tension of the string, therefore, is in fact a force directed 
 to the centre, P, which continually deflects the body from the 
 tangent to the circle in which it has a constant tendency to move 
 in virtue of its inertia. 
 
 313. It follows from the elementary principles of geometry, that 
 the space A B, which is that which represents the force of the string 
 upon the ball, or the centrifugal force, and which is in fact the 
 space through which the body is moved in a given small interval 
 
 d A ^VwwCCvv r [/$ (/1[/J 
 
 -j ~ An. .' .. / 
 
CENTRIFUGAL FORCE. ,35 
 
 of time by the tension of the string which measures the centrifugal 
 force, is found by dividing the square of the number representing 
 A c by the number representing the diameter A F of the circle, or 
 twice the length of the string A p. But A c, being the space 
 described in a given time by the revolving body, is its velocity. 
 
 If we would then compare the centrifugal force of the body with 
 its weight, we have only to compare the space which the body 
 would be moved through by the centrifugal force acting alone upon 
 it, with the space which gravity would move the same body 
 through acting equally alone upon it. 
 
 Let us then express the physical quantities involved in this 
 question as follows : 
 
 w = the weight of the revolving body. 
 c = its centrifugal force. 
 v = its velocity in feet per second. 
 R = length of string in feet. 
 ig = i(5 T i_ f ee t, being the height through which w would fall 
 
 freely in one second. 
 We shall have then the following proportion : 
 
 2R 
 
 and therefore we have 
 
 v s 
 
 
 This formula, expressed in ordinary language, is as follows : 
 
 314. The centrifugal force of a body revolving in a circle is found 
 by multiplying its weight by the square of the number of feet which 
 it moves through in a second, and dividing the product by the number 
 of feet in the radius of the circle it described, multiplied by 3 2^. 
 
 But it is more convenient in practice to express the centrifugal 
 force of a revolving body by reference to the number of revolu- 
 tions it performs in a given time. Let us therefore express by N 
 the number of revolutions, or fraction of a revolution, performed 
 by the body in one second. The circumference of the circle 
 which it describes, the length of the string being R, will be 
 6-2832 R. 
 
 If, then, this be multiplied by N, we shall obtain the space 
 through which the body moves in one second, or its velocity ; and 
 since the square of 6*2832 is 39-4786, we shall have 
 
 V 2 = 39*4786 R 2 XN 2 ; 
 and therefore we shall have the centrifugal force expressed by 
 
 C J "2 27 3 ^^ X R X N 
 K 4 
 
136 OF FORCE AND MOTION. 
 
 In this formula it must be understood, however, that the length 
 of the string must be expressed in feet or fractions of a foot, and 
 that N must express the number of revolutions, or fraction of a 
 revolution, made by the body in one second. 
 
 This formula, expressed in ordinary language, is as follows : 
 
 315. To find the centrifugal force of a revolving body, multiply 
 its weight by the number 1*2273. Multiply this product by the number 
 of feet in its distance from the centre round which it turns, and finally 
 multiply this product by the square of the number of revolutions, or 
 fraction of a revolution, which it makes round that centre in one 
 second of time. 
 
 Example. Let it be required to find with what force a body 
 weighing 2lbs. would stretch a string 3 feet long, revolving four 
 times per second. Multiply 2lbs. by 1*2273, an< ^ we obtain 
 2*4546 Ibs. ; multiplying this by 3, the number of feet in the 
 length of the string, we obtain 7*3638. In fine, multiply this by 
 1 6, which is the square of 4, the number of revolutions per second, 
 and we have 1 17*8208. So that the centrifugal force with which 
 the string is stretched would be 1 1 7 T % Ibs. very nearly. 
 
 From the preceding conclusions it follows, that if two bodies of 
 eaual weights be whirled round their centres by strings or rods 
 01 the same length, their centrifugal forces will be in proportion to 
 the squares of the number of revolutions which they perform in a 
 given time. Thus, if one of them make three revolutions while 
 the other makes two, the centrifugal force of the former will be to 
 that of the latter as 9 to 4. 
 
 Again, if two bodies of equal weight are attached to centres by 
 strings of different lengths, but perforn the same number of revo- 
 lutions in a given time, their centrifugal forces will be in propor- 
 tion to the lengths of the strings. Thus, if one be attached by a 
 string of two feet, and the other by a string of three feet, the cen- 
 trifugal force of the former will be to the centrifugal force of the 
 latter as 2 to 3. 
 
 In general, if two bodies of equal weight be at different dis- 
 tances from the centres round which they revolve, and also make 
 a different number of revolutions in the same time, then their 
 centrifugal forces will be as the products found by multiplying 
 their distances from the centre by the squares of the number of 
 revolutions which they make in the same time. 
 
 316. Whirling-table. These conclusions may be experi- 
 mentally verified by an apparatus called a whirling-table, usually 
 
 found in collections of philosophical apparatus. 
 
 A part of this instrument is represented in i /g-. 92., where c is a 
 metallic bar, having two upright pieces f f at its ends, in which a 
 polished metal rod is fixed parallel to c.. On this rod a ball g 
 
CENTRIFUGAL FORCE. 1 37 
 
 slides, being perforated by a hole 
 corresponding to the rod. At 
 the centre c is a vertical frame- 
 ' work, which contains a number 
 of thin circular weights h placed 
 FigToz. one above the other, and sup- 
 
 ported on a sliding stage, which 
 
 is capable of rising and falling. At the centre of the top of this 
 stage, carrying the weights, is a hook, to which two strings are 
 attached, which are carried over grooves in the pulley A, and 
 then pass over corresponding grooves in the lower pulley k', from 
 which they are carried to the ball g to which they are attached. 
 Now if the ball g be drawn along the rodff towards f with suffi- 
 cient force, the weights h will be raised, and the force necessary to 
 effect this may be increased or diminished by varying the number 
 of weights at h. This apparatus has a square hole at the centre of 
 its lower part c, by which it can be attached to a spindle, by which 
 a regulated revolution can be imparted to it. The apparatus is 
 provided with two such spindles, so that two instruments like that 
 represented in fig. 92. can be fixed upon the table and put in rota- 
 tion with any required velocities, the number of revolutions which 
 they make in a given time having any desired ratio to each other. 
 By this apparatus, the centrifugal force with which the ball g is 
 affected can always be estimated by the weight which such centri- 
 fugal force is capable of raising at h. A rotatory motion is given 
 to c, such that the centrifugal force of g is just sufficient to lift the 
 weights A, but not to carry them to the top of the frame. When 
 this takes place, the centrifugal force of g will be equal to the 
 weight. 
 
 A variety of conditions affecting revolving bodies can be ex- 
 amined and determined by this apparatus. By varying the dis- 
 tance of the weights from the centres round which they revolve, 
 the centrifugal forces in circles with different radii can be deter- 
 mined ; and by varying the velocities of rotation, the effects of 
 different angular motions, or of a different number of revolutions 
 in a given time, can be ascertained. By this apparatus, therefore, 
 the general principles which have been already established respect- 
 ing centrifugal force can be verified. 
 
 Thus we can show : 
 
 I st. That when equal weights are at equal distances from the 
 centres of revolution, their centrifugal forces will be proportional 
 to the squares of the numbers of revolutions which they make in a 
 given time. 
 
 2nd. When they revolve in the same time, then the centrifugal 
 
138 OF FORCE AND MOTION. 
 
 forces will be in the direct ratio of their distances from the 
 centre. 
 
 3rd. When they are at different distances from the centre, and 
 revolve in different times, then their centrifugal forces will be in 
 the ratio of the products found by multiplying their distances from 
 the centre by the squares of the numbers of revolutions which they 
 make in a given time. 
 
 4th. But if the weights be unequal, then let the proportion of 
 the centrifugal forces which they would have if they were equal 
 be first found, and let the numbers expressing them be multiplied 
 by the numbers expressing the weights; the product will then 
 express the proportion of the centrifugal forces. 
 
 These propositions involve the whole theory of centrifugal 
 force. 
 
 317. If two bodies revolve round a common centre in the same 
 time, but at different distances from it, their centrifugal forces 
 would be in the proportion of these distances if they were equal, 
 the body at the greater distance having a greater proportional 
 centrifugal force. But if the body at the lesser distance be in- 
 creased in weight, so as to exceed the other in the same ratio as 
 the distance of the other from the centre of rotation is greater, then 
 the centrifugal forces will be equal ; for what the centrifugal force 
 of the lesser gains by distance, that of the greater gains by weight. 
 Thus, if one of the bodies weigh three ounces, and the other five 
 ounces, and if, further, the latter be at three inches from the 
 centre, the other being at five inches from it, they will have the 
 same centrifugal force, provided they revolve in the same time. 
 This, which is an important proposition, may be experimentally 
 proved by the whirling-table. 
 
 Let A and B, fig. 93., be two bodies connected by a wire, and 
 
 let a point, c, be taken upon this 
 wire, in such a position that 
 B c shall bear to A c the same 
 proportion as the weight of A 
 bears to the weight of B ; then 
 let the wire be attached to the spindle of the whirling-table at c, 
 so that the balls shall be made to whirl round c. It will be found 
 that the wire will maintain its position, although free to move in 
 the direction of its own length ; showing that the centrifugal force 
 exerted by the greater ball, A, upon the wire in the direction c A, 
 is equal to the centrifugal force exerted by the lesser ball, B, upon 
 the wire in the direction c B. The lesser ball, B, therefore, gains 
 as much centrifugal force by its greater radius, B c, as the greater, 
 A, gains by its superior weight. 
 
 The point c, which divides the distance between the balls in the 
 
CENTRIFUGAL FORCE. 1 39 
 
 inverse ratio of their weights, is, as has already been shown, their 
 common centre of gravity ; and it therefore follows that if two 
 bodies revolve round their common centre of gravity in the same 
 time, they will exert equal centrifugal forces upon it. 
 
 The same important principle may be illustrated experimentally 
 by the apparatus shown in Jig. 94., which will be readily under- 
 
 4 
 
 *' 94- 
 
 stood on inspection. The ball D is fixed, while E, sliding on the 
 wire, re-acts on the spiral spring, c E. The centrifugal force 
 causes E to compress the spring until it retires to such a dis- 
 tance from D that the centre of gravity coincides with that of 
 rotation. 
 
 318. Examples. Examples are presented of the effects of 
 centrifugal force in almost all the motions which fall within our 
 daily observation. 
 
 3 1 9. A horseman, or a pedestrian passing round a corner, moves 
 in a curve, and consequently suffers a centrifugal force directed 
 from the centre of the curve, which increases with his velocity, 
 and which impresses on his body a force directed from the corner. 
 He resists this force by inclining his body towards the corner. 
 An animal made to move in a ring, as is customary in training 
 horses, inclines his body towards the centre of the ring. 
 
 320. In all the equestrian feats exhibited in the circus, it will 
 be observed, that not only the horse, but the rider, inclines his 
 body towards the centre, fg. 95., and according as the speed of 
 the horse round the ring is increased, this inclination becomes 
 more considerable. When the horse walks slowly round a large 
 ring, the inclination of his body is 'imperceptible ; if he trot, there 
 
OF FORCE AND MOTION. 
 
 is a visible inclination inwards ; and if he gallop, he inclines still 
 more ; and when urged to full speed, almost lies down upon his 
 side, his feet acting against the partition which separates the circus 
 from the adjacent parts of the theatre. 
 
 Fig- 95- 
 
 In all these cases, the facts are explained by considering that 
 the centrifugal force and the weight of the horse are compounded 
 together, and form a resultant which is directed upon the ground, 
 and is represented by the pressure of the horse's foot. 
 
 321. The actual amount of the centrifugal force, and the pro- 
 portion which it bears to the weight of the animal or other body 
 which is moved in the circle, can be determined by the principles 
 already explained, if the radius of the circle and the velocity of 
 the body moving in it are known ; and from these may be calcu- 
 lated the inclination which the body of the animal must assume in 
 order to be whirled round the circle without falling outwards by 
 the effect of the centrifugal force. 
 
 Let c (fig> 96.) be the centre of the ring round which the 
 
 animal moves. Let c r be 
 its radius, F being therefore 
 the point at which the feet of 
 the animal would act. Take 
 the line F A, perpendicular to 
 p c, and consisting of as many 
 inches as there are pounds 
 weight in the animal. Take 
 A B parallel to F c, consisting 
 of as many inches as there 
 are pounds weight in the cen* 
 
CENTRIFUGAL FORCE. 141 
 
 trifugal force. Then r B will represent the inclination which the 
 animal must assume in order to prevent it from falling either out- 
 wards or inwards. A less inclination than this would cause him 
 to fall outwards, and a greater inwards. 
 
 To demonstrate this, let the weight be conceived as acting at B, 
 and to be represented by B D, which is equal to A F ; the centri- 
 fugal force is represented by B A, and these two forces combined 
 will produce a resultant represented by the diagonal B F. If the 
 body of the animal be inclined according to this line B F, then the 
 resultant will press upon its feet ; but if it be inclined at a less 
 angle, the resultant will cause it to fall outwards ; and if at a 
 greater angle, it would cause it to fall inwards. 
 
 If the centrifugal force be increased, as will be the case if the 
 animal moves with increased speed, then it would be represented 
 by A B'; and the resultant of it, and of the weight, would be repre- 
 sented by B' F. Again, if the centrifugal force be further in- 
 creased, and represented by A B", the weight being represented 
 by B" D", the resultant of these will be represented by B'' F, which 
 line must then be the inclination of the body of the animal. It is 
 clear, then, that an increase of the centrifugal force, which arises 
 from increased speed, will cause the body of the animal to incline 
 more and more towards the centre of the circle. 
 
 For example, if a horse move in a ring of 60 feet diameter, with 
 a speed of 1 5 feet per second, the ratio of the centrifugal force to 
 his weight will be that of the square of 15, or 225, to 60 X 32^-, 
 which is equal to 1930; the ratio, therefore, of the centrifugal 
 force to the weight is I to 8^ very nearly. We shall therefore 
 find the inclination corresponding to this, by taking A F (Jig. 96.) 
 equal to 8^ inches, and A B equal to I. inch : the line F B would 
 represent the inclination of the .horse. 
 
 322. A carriage not having voluntary motion cannot make this 
 compensation for the disturbing force which is 
 called into existence by the gradual change of di- 
 rection of the motion in turning a corner ; conse- 
 quently it will, under certain circumstances, be 
 overturned, falling, of course, outwards, or from 
 the corner. If A B be the carriage, and c (Jig. 97.) 
 the place at which the weight is principally col- 
 lected, this point c will be under the influence of 
 two forces ; the weight, which may be represented 
 by the perpendicular c D ; and the centrifugal force, 
 which will be represented by a line c F, which 
 shall have the same proportion to c D as the centrifugal force has 
 to the weight. Now, the combined effect of these two forces 
 will be the same as the effect of a single force represented by 
 
142 
 
 OF FORCE AND MOTION. 
 
 C G. 
 
 unresisted. 
 
 Thus, the pressure of the carriage on the road is brought 
 nearer to the outer wheel B. If the centrifugal 
 force bear the same proportion to the weight as 
 c r (or D B), jig. 98., bears to c D, the whole 
 pressure is thrown upon the wheel B. 
 
 If the centrifugal force have to the weight a 
 greater proportion than D B has to c D, then the 
 line c r, which represents it (Jig. 99.), will be 
 greater than D B. The diagonal c G, which re- 
 presents the combined effects of the weight and 
 centrifugal force, will, in this case, pass outside 
 the wheel B, and therefore this resultant will be 
 To perceive how far it will tend to overthrow the 
 carriage, let the force c G be resolved into 
 two ; one in the direction of c B, and the 
 other, c K, perpendicular to c B. The former, 
 c B, will be resisted by the road ; but the 
 latter, c K, will tend to lift the carriage over 
 the external wheel. If the velocity and the 
 curvature of the course be continued for a 
 sufficient time to enable this force c K to ele- 
 vate the weights so that the line of direction 
 shall fall on B, the carriage will be over- 
 thrown. 
 
 It is evident, from what has been now 
 
 stated, that the chances of overthrow, under these circumstances, 
 depend on the proportion of B D to c D, or, what is to the same 
 purport, of half the distance between the wheels to the height of the 
 principal seat of the load. It was shown in the last chapter, that 
 there is a certain point, called the centre of gravity, at which the 
 entire weight of the vehicle and its load may be conceived to 
 be concentrated. This is the point which, in the present inves- 
 tigation, we have marked c.. The security of the carriage, there- 
 fore, depends on the greatness of the distance between the wheels 
 and the smallness of the elevation of the centre of gravity above 
 the road ; for either or both of these circumstances will increase 
 the proportion of B D to c D. 
 
 323. If a stone or other weight be placed in a sling which is 
 whirled round by the hand in a direction perpendicular to the 
 ground, (Jig- 100.) the stone will not fall out of the sling, even 
 when it is at the top of its circuit, and consequently has no support 
 beneath it. The centrifugal force in this case acting from the 
 hand, which is the centre of rotation, is greater than the weight of 
 the body, and therefore prevents its fall. 
 
 324. In like manner, a bucket of water may be whirled so rapidly, 
 
CENTRIFUGAL FORCE. 
 
 that even when the mouth is presented downwards, the water 
 till be retained in it by the centrifugal force (fig. 1 o I .) . 
 
 Fig. ioo. 
 
 Fig. 101. 
 
 The tendency of the water to recede from the centre of rotation 
 may be also shown by the apparatus represented mjig. 102. 
 
 325. If a glass of water be fixed upon the whirling- table, and 
 be made to revolve rapidly round its central line A B (fig. 103.)? as 
 an axis, the water will rise upon its sides and sink upon its centre, 
 the surface assuming a concave form. This effect is due to the cen- 
 trifugal force, which affects every particle of the water, and gives 
 it a tendency to fly from the centre of revolution ; it would be 
 possible to impart to the glass a revolution so rapid as to cause the 
 water to flow over the sides. 
 
144 OF FORCE AND MOTION. 
 
 326. Centrifugal drying-machine for laundries. The 
 
 agency of the centrifugal force has been applied with great in- 
 genuity in a machine for drying linen in some of the larger class 
 
 Fig. toz. 
 
 of laundries in Fi ance. This machine, which is represented in 
 Jig. 104., consists of a large hollow drum, within which there is a 
 central space left empty, through which the axle passes. The 
 
 Fig. 105. 
 
 space A A, shown in section, surrounding this vacant central space, 
 in enclosed by two cylindrical surfaces, an exterior and an interior. 
 This circular cylindrical chamber is closed by covers, by opening 
 which the linen to be dried can be introduced. The bottom of 
 this chamber is pierced with holes like a sieve, through which the 
 water expressed from the linen can flow off; a rapid rotation 
 
CENTRIFUGAL FORCE. 145 
 
 being given to this cylinder, the linen, by the effect of the centri- 
 fugal force, is urged against the exterior surface of the cylinder, 
 and is there squeezed with a force which increases with the 
 
 Fig. 104. 
 
 rapidity of the rotation, by the effect of which the water is pressed 
 out of it, and escapes through the holes in the bottom. A rotation 
 so rapid as to produce 25 turns per second, or 1500 per minute, 
 is given to these drying cylinders, by. which the linen, however 
 moist it may be, is rendered so nearly dry that a few minutes' ex- 
 posure in the air renders it perfectly so. 
 
 The mechanism by which this rapid rotation is produced con- 
 sists of a horizontal shaft c, connected with the vertical shaft B by 
 a pair of bevelled wheels ; the shaft c receives its motion from the 
 system of toothed wheels F E, F' E', F" E", which are themselves 
 driven by a system of hoops D, D', D". These hoops receive their 
 motion from a shaft driven by any moving power with which they 
 are connected by an endless band. 
 
 As the very rapid rotation necessary to produce the desired 
 effect could not be instantaneously or even very rapidly produced 
 without injury to the mechanism, an ingenious expedient is 
 adopted by which it is gradually imparted without any injurious 
 shocks. The endless band, which imparts the motion, and which 
 is shifted horizontally by the fork H, moved by the screw and 
 winch K, is first placed upon the hoop D ; this hoop is fixed upon 
 
 L 
 
146 OF FORCE AND MOTION. 
 
 the axle of the toothed wheel E, which drives the wheel P, the axle 
 c, and the cylinder A A, with a moderate velocity ; the wheels F' F", 
 also fixed upon the axle c, are moved at the same rate, and they 
 impart motion to the wheels E' and E'', which are respectively fixed 
 upon hollow axles, which pass one within the other, and which 
 carry respectively the hoops D' and D", 
 
 When the apparatus is thus put in motion, with a certain mode- 
 rate velocity, the endless band is transferred by the apparatus 
 from the hoop D to the hoop D', and the motion is then conveyed 
 to the shaft c, through the wheels E' and F', instead of E and F ; 
 and the proportion of these wheels is so arranged, that the same 
 velocity of the hoop D' will impart an increased velocity to the 
 shaft c and the cylinder A A. 
 
 In the same manner exactly, by transferring the band from 
 D' to -D", the velocity is still further increased, and the full effect 
 produced. 
 
 When the motion of the machine is suspended, to remove the 
 dried linen and recharge the cylinder, the band is transferred to 
 the hoop G, which turns on the axle without being attached to it. 
 
 327. If a body B (Jig. 105.) move down a curved surface G F, 
 
 whose centre is at c, it may acquire such a 
 velocity that the centrifugal force will cause 
 it to leave the surface, and to be projected 
 forwards to the ground. The conditions 
 under which this would take place are easily 
 explained. 
 
 Let the weight of the body be expressed 
 by B A, and its centrifugal force by B E, and 
 let the parallelogram be completed. Then 
 the diagonal B D will be the resultant of these 
 two forces, and will be the line in which the 
 
 body has a tendency to move. So long as this diagonal forms an 
 acute angle with B c, the b. ody will remain on the curved .surface ; 
 but so soon as it forms a right angle with B c, then it will become 
 a tangent to the surface, and the body will fall off. 
 
 328. Body revolving: on a fixed axis. The most important 
 classes of problems in mechanical science in which the principles 
 determining the centrifugal force are practically applied, are those 
 which relate to solid bodies revolving on an axis. If a solid body 
 be pierced by a straight and round hole in which a cylindrical rod 
 is inserted, and the body be made to turn rapidly round this rod 
 in an axis, each particle of matter composing the body will revolve 
 in a circle round such axis ; all these circles will be described in 
 the same time, and consequently the centrifugal forces of the par- 
 ticles exerted upon the axis will be in proportion to their distances 
 
CENTRIFUGAL FORCE. 
 
 '47 
 
 from the axis, such distances being the radii of the circles which 
 they describe respectively round it. 
 
 All the particles of the body which are at the same distance 
 from the axis will therefore exert equal centrifugal forces upon it, 
 and those which are at greater distances will exert centrifugal 
 forces proportionally greater than those at less distances. The 
 particles distributed round the axis will produce forces directed 
 from the axis in the direction of perpendiculars connecting them 
 with the axes respectively. 
 
 Now it is evident that as many different forces will thus be 
 exerted upon the axis as there are different particles of matter 
 composing the mass of the revolving body. 
 
 Three cases are presented which may arise in the combination of 
 these forces. 
 
 329. They may be in equilibrium; that is to say, the centrifugal 
 forces exerted by all the particles composing the body on the axis 
 may neutralise each other. In this case the axis would suffer no 
 strain in consequence of the centrifugal forces, and the body would 
 spin round it without producing any effect upon it. 
 
 In such a case, if the axis were withdrawn from the hole in 
 which it is inserted, the body would still continue to spin as before ; 
 because, since the axis suffered no pressure or strain from the 
 revolving matter, its presence or absence can make no difference in 
 the motion. 
 
 330. The centrifugal forces produced by the particles of the 
 revolving mass may be such as to Ue represented by a single force 
 applied at some point of the axis, and at right angles to it. In 
 this case the axis will suffer a corresponding pressure or strain at 
 this point. If this point could be fixed, the remainder of the axis 
 might in that case be withdrawn ; because, since no other point of 
 it suffers any strain from the effect of the motion, its presence or 
 absence can produce no difference in the motion. 
 
 331. The centrifugal forces may be such that their combined 
 effect cannot be represented by a single force, but may be repre- 
 sented by two equal and parallel forces acting on two different 
 points of the axis, and in contrary directions. This combination 
 is what has been already called a couple, and its effect is to twist 
 the axis round some point intermediate between the two contrary 
 forces. In this case the axis could not be withdrawn unless two 
 fixed points were provided, representing the points to which the 
 opposite forces of the couple are applied. 
 
 332. The combination of forces produced by the revolving 
 matter may be such as to be incapable of being represented either 
 by a single force or by a couple. In this case it can be proved that 
 their conibiued effects will be represented by a single force and a 
 
148 OF FORCE AXD MOTIOX. 
 
 couple taken together. The effect, in such a case, is a pressure 
 upon the axis at right angles to its length at the point where the 
 single force is placed, and a tension or twist produced at the same 
 time by the couple. 
 
 These are all the possible cases which can be presented by a solid 
 body revolving on a fixed axis. 
 
 333. Examples. Their complete analysis and demonstration 
 would require the use of the principles and formulae of the higher 
 parts of mathematical science, the introduction of which would 
 not be suitable to the purpose of the present treatise. We shall 
 therefore limit ourselves to some examples which will convey a 
 sufficiently clear notion of the general effects produced by the 
 rotation of solid bodies on fixed axes. 
 
 334. If a series of particles of matter placed in the circum- 
 ference of a circle are made to revolve by a common motion round 
 an axis, passing through such circle, and perpendicular to its plane, 
 their centrifugal forces will be evidently in equilibrium, and no 
 
 pressure on the axis will be produced. A 
 circular series of such particles is repre- 
 sented in fig. 1 06. ; the radii represent 
 the direction of the centrifugal forces, 
 which are all equal, because the particles 
 are equal and the distances from the 
 centre are equal. 
 
 It is evident on inspection that these 
 forces equilibrate round the centre, and 
 that the central point, therefore, would 
 suffer no pressure in one direction rather than in another. 
 
 335. A flat circular plate of uniform thickness and density may 
 be considered as consisting of a series of concentrical rings of 
 such particles. If such a plate revolve round an axis passing 
 through its centre, and perpendicular to its plane, the centrifugal 
 forces of the particles will be in equilibrium, and no pressure will 
 be produced on the axis. 
 
 336. A cylinder may be considered as composed of a number ot 
 such circular plates placed one upon the other, and the axis of the 
 cylinder will be the line formed by the centres of these plates. 
 If such a cylinder, therefore, revolve round its axis, the centrifugal 
 force of its mass may be in equilibrium, because each separate plate 
 being in equilibrium, the entire pile would necessarily alsb be in 
 equilibrium. 
 
 337. There is an extensive and important class of solid bodies 
 having a geometrical form, which gives to their axes this property. 
 They are called solids of revolution. 
 
 Let A B D (Jig. 107.) be a triangle with a right angle at A. If 
 
CENTRIFUGAL FORCE. 
 
 149 
 
 this be supposed to revolve round the side A B as an axis, it will 
 by its revolution generate a solid called a cone ; that is to say, the 
 space through which it would pass as it revolves, and which it 
 would include within its sides in revolving if filled by any solid 
 matter, would form a cone. 
 
 If a right-angled parallelogram (Jig. 1 08.) revolve round its side 
 A B, it would generate in the same way a cylinder. 
 
 If a triangle A B D (Jig. 109.) revolve round a side A B, not in- 
 cluding a right angle, it would generate the double cone. 
 
 D 
 
 Fig. 108. 
 
 Fig. 109. 
 
 If a semicircle {fig. 1 1 o.) revolve round its diameter A B, il 
 would generate a sphere or globe. 
 
 If a semi-ellipse (fig. in.) revolve round its shorter axis A B, 
 it would generate an oblate spheroid, being a figure resembling an 
 orange or a turnip. 
 
 If a semi-ellipse (fig. 112.) revolve round its longer axis, it 
 would generate a prolate spheroid, being a figure resembling an 
 egg, only that its ends are similar. 
 
 Dl 
 
 Fig. in. 
 
 Now it is evident, that in these and all solid bodies whose figures 
 are determined by the same principle, all sections made by planes 
 perpendicular to their axes of revolution are circles through the 
 
150 OP FORCE AND MOTION. 
 
 centre of which such axes pass. Since the centrifugal forces of the 
 particles of matter composing each of such sections are in equi- 
 librium, the centrifugal force of the entire mass of the body is 
 necessarily also in equilibrium, such mass being composed of these 
 several sections. The body, in short, may be considered to be 
 made up of a number of circular plates laid one upon another, 
 varying in their diameter according to the form of the body. 
 
 If a solid of revolution, therefore, be made to revolve upon its 
 geometrical axis, the centrifugal forces of its mass will be in equi- 
 librium, and no pressure or strain whatever will take place upon its 
 axis. 
 
 If a globe or sphere be composed of any materials capable of a 
 change of figure, it will be converted into an elliptical flattened 
 spheroid by the centrifugal force produced by its rotation. This 
 fact is shown experimentally by an apparatus composed of elastic 
 hoops put in rotation by the whirling-table, as shown in fig. 113. 
 
 Fig. 113. 
 
 338. The same reasoning will be applicable to all solids which 
 have an axis round which the particles of such section made by a 
 perpendicular plane are so arranged, that every particle of matter 
 at one side has a corresponding particle at an equal distance at the 
 other side ; for in this case every pair of such equidistant par- 
 ticles will exert equal and opposite centrifugal forces on the axis. 
 
 A great number of solid bodies, including the class of solids of 
 revolution described above, fulfil this condition. 
 
 If the sections of a solid be all equal, and have any regular 
 geometrical figure having a point within it forming its geometrical 
 centre, and through which all lines drawn are bisected, then such 
 a solid participates in the above property, and the centrifugal 
 
CENTRIFUGAL FORCE. 
 
 forces of its mass, when revolving round such an axis, will neu- 
 tralize each other. 
 
 339. A column formed by laying one upon another equal flat 
 plates of the same figure has this property. Such a column is 
 called, in geometry, a rectangular prism. 
 
 340. A pyramid formed by laying upon each other similar plates 
 not equal, but diminishing gradually in magnitude, will have the 
 same property. 
 
 It follows, therefore, that the axes of the rectangular prisms 
 and pyramids whose bases have a centre of magnitude enjoy the 
 above property. 
 
 341. It will be observed that the axis round which the centri- 
 fugal force equilibrates in the examples given above will pass 
 through the centre of gravity of the bodies in question. It may 
 therefore be asked, whether such property belongs to all lines 
 whatever passing through the centre of gravity of a body ; that is 
 to say, whether, if a body is made to revolve upon an axis passing 
 through its centre of gravity, the centrifugal force will be in 
 equilibrium, and whether such axis will be free from strain or 
 pressure. It is easy to show that this will not be the case in 
 general, and that it is only certain lines passing through the centre 
 of gravity which have the property above mentioned. 
 
 If a rectangular plate having unequal sides, A B and B c {fig. 
 1 14.), be made to revolve round a line M , passing through the 
 
 middle points of the oppo- 
 site sides A B and D c, the 
 centrifugal forces will be in 
 equilibrium; because every 
 point on the one side of M N 
 has a corresponding point 
 at an equal distance on the 
 other side. 
 
 In like manner, the line 
 m n, passing through the 
 middle points of the sides 
 A D and B c, would enjoy a 
 like property. But if any 
 other line, such as P Q, be 
 drawn through the centre 
 of gravity, o, and the plate 
 
 Fig. 114. 
 
 be made to revolve round such line, then it will be evident that 
 the centrifugal force will not be in equilibrium. For through a 
 point such as E, let a line r G be drawn perpendicular to the axis 
 p Q. This line r G will be divided into unequal parts at E, and the 
 centrifugal force produced by the particles between E and F will 
 
152 OF FORCE AXD MOTION. 
 
 be greater than the centrifugal forces produced by the particles 
 between E and G. The same will be true of all lines drawn per- 
 pendicular to o P ; and the combined effect of all the centrifugal 
 forces acting in that part of the axis between o and P will be to 
 produce a greater strain on the side of the angle A than on the 
 side of the angle D, and the centrifugal force will have a resultant 
 directed towards the side A. By the same reasoning it may be 
 shown that the centrifugal forces of that part of the plate which is 
 below K L will have a resultant directed to the side of the angle c, 
 and this resultant will be equal to the resultant of the centrifugal 
 forces above K L, directed to the side of the angle A. 
 
 It follows, therefore, that the axis P Q round which the plate 
 revolves will be affected by forces having two resultants parallel 
 to each other and in opposite directions, one perpendicular to o P, 
 and directed towards the side A, and the other perpendicular to 
 o Q, and directed towards the side c. These two forces form a 
 couple, and have a tendency to turn the axis of revolution towards 
 the position M N, as represented by the arrows in the diagram. 
 
 342. This property is general, although it cannot be demon- 
 strated in all its universality without the aid of the language and 
 principles of the higher analysis. It may be stated thus : In all 
 bodies whatever, there are three lines passing through the centre 
 of gravity which are at right angles to each other, each of which is 
 so placed, in reference to the mass of the body, that the centri- 
 fugal forces produced by the revolution of the body round them 
 respectively will be in equilibrium ; and such lines, when the body 
 revolves round them, will suffer no strain or pressure. 
 
 But if the body be made to revolve round any other line passing 
 through the centre of gravity, the centrifugal force will produce a 
 strain, which will be represented by two equal opposite and 
 parallel forces acting upon the axis at opposite sides of the centre 
 of gravity, and having a tendency to turn the axis in the position 
 of one or other of the three axes of equilibrium here mentioned. 
 
 343. Principal axes. An axis round which the centrifugal 
 force equilibrates is called a principal axis; and from what has 
 been explained, it appears that there are three principal axes 
 through the centre of gravity at right angles to each other. 
 
 344. There are some particular cases in which every line 
 passing through the centre of gravity is a principal axis ; such, for 
 example, is the case with a sphere or globe of uniform density. 
 Such a solid, whatever diameter it may revolve round, will be a 
 solid of revolution, and the sections perpendicular to such diameter 
 will be circles. The same principle is true of all the regular 
 solids. All lines, for example, passing through the centre of a 
 cube are principal axes. 
 
CENTRIFUGAL FORCE. 153 
 
 345. If a solid be made to revolve round an axis which does- not 
 pass through the centre of gravity, but which is parallel to one or 
 other of the principal axes passing through that point, the centri- 
 fugal forces will not equilibrate round such axis, but they will be 
 represented by a single force perpendicular to it, and passing 
 through the centre of gravity. 
 
 346. Such an axis is called a principal axis, and in general 
 there are three such axes corresponding with such points taken in 
 the body, which are parallel respectively to the principal axes 
 passing through the centre of gravity. 
 
 347. It may therefore be stated, in general, that if any point in 
 a body be taken different from the centre of gravity, there are 
 three lines passing through it at right angles to each other, round 
 each of which, if the body is made to revolve, an effect will be pro- 
 duced by the centrifugal forces which can be represented by a 
 single force, perpendicular to the circumference, and passing 
 through the centre of gravity. 
 
 348. If the body be made to revolve round any line passing 
 through a given point in it which is not a principal axis in the 
 sense just referred to, then the centrifugal forces produced by 
 such revolution cannot be represented either by a single force or 
 by a pair of equal and opposite parallel forces, but will be repre- 
 sented by both of these together. This, therefore, is the character 
 of all axes, round which a body would move, which do not pass 
 through the centre of gravity, and are not parallel to either of the 
 principal axes passing through that point. 
 
 349. Experimental illustrations. Many of these important 
 properties admit of being experimentally illustrated in a very 
 striking manner by a simple apparatus. . 
 
 Let a body be suspended by a thread, or, better still, by a skein 
 composed of several threads, from a fixed point. Let this fixed 
 point be so arranged, that a rapid rotatory motion may be given 
 to it. This rotatory motion will soon be imparted to the body 
 suspended to the threads by the twisting of the skein, and, after a 
 time, the rotation of the body will become extremely rapid. It 
 will first take place round the line, passing vertically through it in 
 the direction of the skein, when it hangs quiescent. If this -line 
 happen to be one of the principal axes passing through its centre 
 of gravity, it will continue to revolve round it ; but if it be not 
 one of these principal axes, the centrifugal force, as has been 
 already explained, will produce an effect upon it, represented by 
 two equal, opposite, and parallel forces tending to twist it. This 
 effect will be soon rendered manifest in a remarkable manner : the 
 body, when it revolves rapidly, will not continue in the same posi- 
 tion which it had when it was quiescent. The line which was in 
 
154 
 
 OF FORCE AND MOTION. 
 
 the direction of the string will begin to take another direction, the 
 point where the string is attached to it will not remain in its first 
 position, and the body will throw itself into a position at a greater 
 or lesser angle to the string, and, in a word, will at length, after 
 the revolving motion has become sufficiently rapid and continued, 
 assume such a position, that a principal axis, passing through its 
 centre of gravity, will become vertical, and the body will spin 
 round it. 
 
 It is evident, therefore, that this effect takes place in spite of 
 the opposing influence of the gravity of the body ; for in the new 
 position which the body assumes, the line of direction of its centre 
 of gravity ceases to be represented by the skein. This experiment 
 may be varied in a great variety of ways, exhibiting most instruc- 
 tive and amusing effects. 
 
 If a metallic ring be suspended by the skein by being attached 
 to a point in its side, the ring, when quiescent, will hang with its 
 plane vertical ; but when the rotation becomes rapid, the ring will 
 throw itself into a horizontal position, and will spin round a ver- 
 tical axis through its centre, and perpendicular to its plane. 
 
 If the experiment be made with an oblate spheroid suspended 
 by a point P (fig. 115.) in its equator PQ, its lesser axis AB 
 
 Fig. 115. 
 
 Fig. 116. 
 
 B 
 
 Fig. 117. 
 
 Fig. 118. 
 
 being horizontal, it will, when it acquires a rapid motion, take the 
 position represented in fig. 1 1 6., in which the shorter axis A B is 
 
MOLECULAR FORCES. 155 
 
 vertical, and the equator PQ horizontal, and it will spin in this 
 position round the principal axis AB, although the centre of 
 gravity is unsupported by the string. 
 
 If a cylindrical rod A B (Jig. 1 1 7.) be suspended by a point at 
 the centre of one of its ends, and a rapid revolution be imparted 
 to it, it will not continue in this position, but will assume the posi- 
 tion represented in fig- 1 1 8., in which its length will be horizontal, 
 and it will revolve round an axis passing through its centre, and 
 at right angles to the length. 
 
 If a metallic chain, the ends of which are united, as in a neck- 
 lace or bracelet, be suspended from any point, and put in rapid 
 revolution, the chain will at first make some irregular gyrations, 
 but after a time it will gradually open, and settle itself into the 
 form of a precise circle, the plane of which will be horizontal, and 
 the string will then be attached to a point in this circle. In short, 
 the chain will assume the form of a solid ring in a horizontal 
 plane. 
 
 CHAP. VH. 
 
 MOLECULAR FORCES. 
 
 350. Pores the region of molecular forces. It has been 
 demonstrated in a former chapter, that the space included within 
 the external surface of a body is not all occupied by the matter 
 composing that body. We have seen that bodies, however dense 
 or ponderous, are capable of being diminished in their volume by 
 compression, or by diminution of temperature. It follows, there- 
 fore, that the component particles forming the mass of a body of a 
 uniform density are uniformly distributed throughout its volume, 
 each particle being separated from those around it by a space of 
 greater or less extent unoccupied by matter. 
 
 These interstitial spaces are the regions which form the theatre 
 of action of those important physical agents called molecular 
 forces. A multitude of phenomena, familiar to all observers, show 
 that between the particles which compose the mass of a body, 
 there exist attractive or repulsive forces, the sphere of whose action 
 is in general limited to distances imperceptible to the senses, and 
 which only admit of being proved by indirect means. 
 
 351. Cohesion. The qualities of solidity and hardness, and, 
 in general, those properties by which a body resists fracture or 
 flexure, or any other derangement of its form, arise from the 
 
156 OF FORCE AND MOTION. 
 
 energy with which its component particles attract 2ach other and 
 resist any force which tends to separate them. 
 
 Molecular attraction manifested in this manner is called the 
 attraction of cohesion. 
 
 352. Adhesion. If the surfaces of two bodies be brought into 
 very close contact, it is found that they cannot be separated with- 
 out the exertion of some force of greater or less intensity, accord- 
 ing to the circumstances of the contact. 
 
 Molecular force manifested in this manner is called the attrac- 
 tion of adhesion. 
 
 353. Capillary attraction. Certain bodies being placed in 
 contact with a fluid, the fluid will enter their dimensions and 
 occupy their pores ; as, for example, when a sponge or a lump of 
 sugar is brought in contact with water. The fluid in these cases 
 rises in opposition to its gravity, and fills all the interstices of the 
 sponge or the sugar. Molecular force manifested in this manner 
 is called capillary attraction* The effects of this force will be 
 explained in a subsequent part of this work. 
 
 354. Chemical affinity. When two bodies of different kinds 
 are mixed together, their constituent particles will in certain cases 
 unite, and form by their combination the constituent particles of a 
 compound, differing in its sensible qualities from either of the 
 components. For example, if two gases called oxygen and hy- 
 drogen be mixed together in a certain proportion, and a light be 
 applied to the mixture, an explosion will take place ; the atoms of 
 the two gases will unite one with another, and the entire mass 
 will be converted into water, the weight of the water being exactly 
 equal to the sum of the weights of the two gases. In this case, 
 each atom of the oxygen is attracted by an atom of hydrogen, and 
 their combination forms an atom of water. 
 
 Molecular attraction manifested in this manner is called che- 
 mical attraction, or chemical affinity. 
 
 355. The atomic attraction and repulsion. It has been 
 shown that all bodies submitted to the action of mechanical forces 
 of sufficient energy are capable of being compressed and dimi- 
 nished in their volume. By such means, therefore, their component 
 particles are forced into closer proximity. 
 
 But all of these resist such compression with a certain force, 
 and most of them have a tendency to recover the volume which 
 they had before compression. This general fact indicates the ex- 
 istence of another force, contrary in its direction to the attraction 
 of cohesion, the sphere of whose action is within that of the latter 
 
 * So called from capilla, a hair ; the magnitude of the pores being in this 
 case estimated at about the thickness of a hair. 
 
MOLECULAR FORCES. 
 
 157 
 
 attraction. To explain this phenomenon we are compelled to sup- 
 pose that each atom composing a body is surrounded with a sphere 
 of repulsion within which adjacent atoms cannot enter unless 
 urged by a certain force. But outside this sphere of repulsion 
 there exists the sphere of attraction, by which such atoms attract 
 all the surrounding atoms, which gives the cha"racter of solidity 
 and hardness to the mass. 
 
 356. The attraction of cohesion is manifested in solid bodies by 
 the force which is necessary to derange their form by fracture or 
 flexure, or by any other change of figure. The same force is mani- 
 fested in liquids in their tendency to form into spherical drops, a 
 globe being the greatest volume which can be contained within a 
 given surface. 
 
 357. Thus particles of water falling in the atmosphere attract 
 each other, and collect in spherules forming rain. If such 
 spherules after their formation be exposed to cold, they harden and 
 form hailstones. If a little mercury be let fall on a sheet of paper, 
 it will collect in small silvery globules, notwithstanding the ten- 
 dency of the gravity of its particles to make it spread over the 
 paper in fine dust. Innumerable examples present themselves of 
 this class of phenomena. The tear as it falls from the eye collects 
 in a spherule upon the cheek ; the dew forms a translucent globule 
 on the leaves of plants. 
 
 358. The manufacture of shot presents one of the most striking 
 examples of this phenomenon in the arts. The lead, in a state 
 of fusion, is poured into a sieve, the meshes of which determine 
 the magnitude of the shot, at the height of about two hundred 
 feet from the ground. The shower of liquid metal, after pass- 
 ing through the sieve, forms, like rain in the atmosphere, sphe- 
 rules, which, before they reach the ground, are cooled and 
 solidified. 
 
 These spherules form the common shot used in sporting, and the 
 precision of their spherical form shows how regularly the liquid 
 obeys the geometrical law, that a sphere contains the greatest 
 volume within a given surface. 
 
 359. This disposition of fluids to affect the spherical form may 
 be further elucidated in considering that any other figure which a 
 body could take would necessarily place different parts of its sur- 
 face at different distances from its central point, a circumstance 
 which would be incompatible with the combined qualities of attrac- 
 tion of cohesion and fluidity. By fluidity, all the particles forming 
 the mass are free to move amongst each other, and by the attrac- 
 tion of cohesion they are drawn round their common centre with 
 the same force. To suppose that they could rest at different dis- 
 tances from their common centre, would necessarily involve the 
 
158 OF FORCE AND MOTION. 
 
 supposition either that the attraction by which they are affected 
 was unequal, or that the mass had not perfect fluidity. 
 
 This principle, which is so evident, may be inverted, and we may 
 assume that in all cases where natural bodies are found in the 
 spherical form, even though they be solid, they must have been, at 
 the epoch of their formation, in a fluid state. 
 
 360. Hence it is inferred that the earth and the other bodies of 
 the solar system were once fluid, and that our globe existed for- 
 merly in a liquid state. 
 
 361. The mutual repulsion of the atoms of gases is manifested 
 by their indefinite expansibility and compressibility. Air included 
 in a cylinder under an air-tight piston will expand so as to fill the 
 increased volume as the piston is drawn up, and to this expansion 
 there is no practical limit. This is explained by supposing that 
 uriund each molecule of the air or gas there is a sphere of repul- 
 sion, so that each particle repels those around it. When the piston 
 is raised to twice its former height; the air beneath it will expand 
 into double its former volume. 
 
 In this case it must be concluded that the vacant spaces between 
 the particles of air are twice as great as they were before the piston 
 was raised. If the piston be again raised to double its present, 
 height, the same effect will take place. The air will again expand 
 in virtue of the repulsive force prevailing among its particles, and 
 the interstitial spaces separating the particles will be proportionally 
 augmented. 
 
 There is no known limit to this expansive quality, and it con- 
 sequently follows that the region through which the repulsive 
 forces of gases act has a corresponding extent. 
 
 362. Methods will be explained in a succeeding part of this 
 work by which several of the gases have been reduced to the liquid 
 state, and analogy justifies the conclusion that all gaseous bodies 
 are capable of this change. In the liquid state, the attraction of 
 cohesion is rendered manifest, as has been already shown. But we 
 have a still further evidence of the attraction of cohesion amongst 
 the particles of gases, inasmuch as some of them have been reduced 
 to the solid state ; and by analogy we may c6nclude that all are 
 capable of this change. It has been already shown that the solid 
 state is only a consequence of the attraction of cohesion. 
 
 363. These and other phenomena lead to the conclusion that in 
 this case of the gaseous bodies, there is beyond the sphere of co- 
 hesion a sphere of repulsion. When the particles, either by the 
 application of cold or compression, or both of these agencies, are 
 brought into such close contact as to be within the space of co- 
 hesion, then they become a liquid or solid, as the case may be. 
 
 364. The mutual repulsion found to prevail among the con- 
 
MOLECULAR FORCES. 159 
 
 stituent particles of bodies is by some attributed to the agency of 
 heat, and it is certain that the energy of this repulsion is increased 
 or diminished, according as heat is imparted to or subtracted from 
 bodies. In a solid body, such as a mass of gold, in its ordinary 
 state, the attraction of cohesion between its particles greatly pre- 
 dominates over the influence of the repulsion already mentioned ; 
 but if heat be applied to this mass, the energy of the repulsion is 
 gradually increased, until at length it becomes so nearly equal to 
 that of cohesion, that the gravity of the particles overcomes that 
 part of the cohesion not balanced by the repulsion, and the con- 
 stituent parts of the mass no longer holding together in the solid 
 form, the metal is converted into a liquid. If heat be still applied 
 to this liquid, the temperature will rise and the liquid will expand ; 
 but after a certain quantity of heat has been imparted to it, the 
 repulsive force between the particles themselves is so great, that, in 
 spite of their gravity and of the attraction of cohesion, they sepa- 
 rate and disperse into a vapour which possesses the qualities of gas, 
 being capable of expanding without limit. 
 
 365. Thus it appears that the same body may exist in the solid, 
 liquid, and gaseous forms, according to the conditions under which 
 it is placed in reference to heat. 
 
 366. Adhesion of solids. If the surfaces of two pieces of 
 metal, being rendered perfectly smooth, are brought into close 
 contact by a strong pressure, they will adhere together with con- 
 siderable force. That this adhesion is not due to atmospheric 
 pressure can be demonstrated by showing that the adhesion will 
 continue in a vacuum. Jn this case the superficial molecules of 
 the two bodies are brought into contact so close as to be within the 
 sphere of each other's attraction. 
 
 Innumerable examples of the adhesion of solid bodies are fami- 
 liar to daily experience. We may write with chalk, or with a 
 pencil, or charcoal on a wall or on a ceiling, although the effect of 
 gravity would be to cause the particles abraded from the chalk, 
 the lead, or the charcoal to fall from the wall or the ceiling. Dust 
 floating in the air sticks to the wall or ceiling, in spite of the 
 tendency of its gravity to fall from them. 
 
 The force of adhesion of solid surfaces one to another may be 
 ascertained by placing the adhering surfaces in a horizontal posi- 
 tion, the lower one being attached to a fixed point, and the upper 
 one connected with the arm of a balance. The weight necessary 
 to separate them is the measure of the adhesion. If we desire to 
 ascertain the amount of adhesion per square inch of surface, it is 
 only necessary to divide such weight by the magnitude of the 
 adhering surface expressed in square inches. 
 
 367. It is on the adhesion between metallic surfaces when pressed 
 
160 OF FORCE AND MOTION 
 
 strongly together that the efficacy of a locomotive engine depends. 
 The driving-wheels press with a great weight upon the rails, and 
 are made to revolve round their own centres by the force of the 
 engine. If there were no adhesion, or even insufficient adhesion 
 between the tire of the wheel and the rail on which it is pressed, 
 the wheel would turn without advancing ; and this actually does 
 happen in cases where the rails are greasy, and very frequently 
 when they are covered with a hoar frost, the contact being then 
 interrupted, and the matter between the wheel and the rail not 
 offering the necessary adhesion. 
 
 368. Effect of lubricants. On the other hand, when the 
 force applied to break the adhesion is directed perpendicularly to 
 the adhering surfaces, a fluid or unctuous matter smeared upon the 
 surface often increases the adhesive force. 
 
 Thus two metallic plates will adhere with greater force together 
 if they are smeared with oil than when they are clean. This may 
 partly arise, however, from the fact that the film of oil which covers 
 them excludes air more effectually than could be accomplished in 
 the case of surfaces so considerable by mere pressure. 
 
 369. The bite. The effect known amongst workers in metal 
 as the bite is the adhesion of two metallic surfaces brought into ex- 
 tremely close contact. It may be doubted whether this adhesion 
 would not be diminished if some fluid were introduced between the 
 surfaces. 
 
 370. Glues, solders, and like adherents. The adhesion of 
 the surface of solids may be rendered more intense than even the 
 cohesion of the particles of the solids themselves by interposing 
 between them some substance in a liquefied form, which hardens 
 by cold, and which when hard has a strength equal to or greater 
 than that of the solids which it unites. Glues, cements, and solders 
 supply remarkable examples of this. Two pieces of wood glued 
 together will break anywhere rather than at their joint. The pro- 
 cesses of gilding and plating also supply examples of the adhesion 
 of metals to each other. 
 
 371. The process of silvering mirrors is an example of the ad- 
 hesion of metal to glass ; and that of mortar in building is an 
 example of the adhesion of earthy matters to each other. 
 
 Two pieces of caoutchouc, if pressed together upon freshly cut 
 surfaces, will be found to unite as completely as if they composed 
 one independent piece. 
 
BOOK THE THIRD. 
 
 THEOKY OF MACHINERY. 
 
 CHAPTER I. 
 
 GENERAL PRINCIPLES. 
 
 372. What constitutes a machine. A machine is an instru- 
 ment or apparatus by which a force applied at a certain point, and 
 having a certain determinate intensity and direction, is made to 
 exert a force at another point, more or less distant from the 
 former, and generally different in intensity and direction. Thus, 
 for example, a horse moving on a horizontal road in a circle is 
 made to raise a weight vertically in the shaft of a mine, or water 
 from the shaft of a well. 
 
 Men pulling at a rope in some direction more or less oblique 
 are enabled to raise a mass of heavy matter from the hold of a 
 ship, and to transfer it to an adjacent wharf. 
 
 373. The force which is applied to and transmitted by a machine 
 is technically called the power ; the point at which it is applied is 
 called the point of application ; its direction is the line in which 
 the force has a tendency to make the point of application move ; 
 and its intensity is usually expressed by 'a weight which, acting at 
 the same point of application, would produce a like effect upon it. 
 
 The moving powers applied to arid transmitted by machinery 
 are infinitely various. In the capstan of a ship, the moving power 
 is human force applied to it ; in a common pump, the same moving 
 force is used ; a horse is the moving power applied to vehicles of 
 transport on common roads, and a steam-engine on railways ; the 
 wind is the moving power applied to a sailing-vessel, and to a 
 windmill ; the momentum of water acting against the float-boards 
 of a wheel, or its weight acting in the buckets, is the moving 
 power of a water-wheel ; the elastic force of steam acting on the 
 piston in the cylinder is the moving power of the steam-engine. 
 
 374. That part of a machine which is immediately applied to 
 the resistance to be overcome is called the working point. 
 
 375. The resistance, whatever be its nature, to which the 
 
 M 
 
1 62 THEORY OF MACHINERY. 
 
 working point is applied, is technically called the weight or load. 
 In many cases weight is the actual resistance which machines are 
 applied to overcome ; as, for example, in raising water from a 
 well, or from mines ; also in raising ore. In some cases the re- 
 sistance to be overcome is friction, used for the purpose of frac- 
 turing and pulverising material substances. This is the case in 
 flour-mills. 
 
 In some cases tne resistance to be overcome is the friction of 
 surfaces, and the resistance of the air. This is the case when 
 carriages are moved on level roads or railways. 
 
 Whatever be the nature of the resistance, a weight which would 
 produce an equivalent force acting against the moving power may 
 be assigned. Thus, for instance, if the traces of a carriage drawn 
 by horses, or the chain connecting a locomotive with a railway 
 train, be stretched by the resistance of the carriage or the train, a 
 weight may be substituted, which, being suspended vertically from 
 the traces or the chain, would produce the same tension. The 
 resistance in such case is expressed by stating the amount of this 
 weight. 
 
 376. In the exposition of the theory of machinery, it is expe- 
 dient to omit, in the first instance, the consideration of many 
 circumstances, of which, however, a strict account must be subse- 
 quently taken before any practically useful application of them 
 can be made. A machine, such as we must for the present con- 
 template it, is a thing which can have no real or practical exist- 
 ence. Its various parts are considered to be free from friction. 
 Thus, the surfaces composing it, which move in contact with each 
 other, are assumed to be infinitely smooth and polished ; the solid 
 parts are all considered to be absolutely rigid and inflexible. 
 
 The weight and inertia of the matter composing the machine 
 itself are wholly neglected, and we reason upon it as if it were 
 divested of these qualities. Cords, ropes, and chains are supposed 
 to have neither stiffness, thickness, nor weight ; they are regarded 
 as mathematical lines, infinitely flexible and infinitely strong. 
 The machine, when it moves, is assumed to encounter no resistance 
 from the air, and to be in all respects circumstanced as if it were 
 in vacuo. 
 
 These suppositions being all false, it follows that none of the 
 consequences immediately deduced from them can be true. Never- 
 theless, as it is the business of art to bring machines as near to 
 this state of ideal perfection as possible, the conclusions which are 
 thus obtained, though false in a strict sense, yet deviate from the 
 truth in no considerable degree. 
 
 These conclusions may, in fact, be regarded as a first approxi- 
 mation to truth. 
 
GENERAL PRINCIPLES. 16; 
 
 377. The various effects which have been previously neglected 
 are afterwards taken into account. The roughness of surfaces, 
 the imperfect rigidity of the solid parts, the imperfect flexibility of 
 cords and chains, the resistance of the air and other fluids, and the 
 effects of the weight and inertia of the machine itself, are after- 
 wards severally examined, their properties explained, and the 
 manner in which they modify the transmission of the power to the 
 weight developed. These modifications and corrections being 
 applied to the conclusions obtained, a second approximation to the 
 truth is made, but still only an approximation ; for, in investigating 
 the laws which govern the several effects last mentioned, we are 
 compelled to proceed upon a new group of false suppositions. 
 
 To determine the laws which regulate the friction of surfaces, 
 it is necessary to assume that the surfaces are uniformly rough, 
 and subject to uniform pressure ; that the solid parts which are 
 imperfectly rigid, and the cords and chains which are imperfectly 
 flexible, are constituted throughout their entire dimensions of a 
 uniform material, so that the imperfections do not prevail more in 
 one part than in another. Thus, all irregularity is left out of 
 account, and a general average of the effects taken. It is obvious, 
 therefore, that, even in this second system of reasoning, we have 
 still failed in obtaining a result exactly conformable to the real 
 .state of things. But it is equally obvious that we have obtained 
 one much more conformable to that state than had been previously 
 accomplished; and, in fine, it is found that the conclusions thus 
 obtained are sufficiently near the truth for practical purposes. 
 
 378. The imperfections in our process of investigation, mani- 
 fested in this laborious system of successive approximation of the 
 truth, is not peculiar to Natural Philosophy. It pervades all de- 
 partments of natural science. In Astronomy, the motions of the 
 celestial bodies, and their various changes and appearances as 
 developed by theory, assisted by observation and experience, con- 
 sist of a like series of approximations to the real motions and 
 appearances which take place in nature. 
 
 It is the same in Art. The first labours of the artist produce 
 from the rude block of marble a rough and rude resemblance of 
 the human form. The next attempts remove the greater ine- 
 qualities and protuberances, and reduce the form to a closer re- 
 semblance to the original. It is not, however, until after a long 
 succession of operations, in which smaller and smaller portions of 
 the stone are detached, that the last labour of the chisel of the 
 master completes the resemblance. 
 
 379. We shall therefore for the present consider the machine, 
 by which the effect of the power is transmitted to the working 
 point, as divested of weight and inertia ; we shall consider all the 
 
164 THEORY OF MACHINERY. 
 
 pivots, axles, and surfaces which move in contact absolutely de- 
 void of friction ; we shall consider all cords, ropes, and chains to 
 be absolutely and perfectly flexible, and to be moved in contact 
 with the grooves and wheels without friction ; and, in fine, we shall 
 consider the machine itself, as well as the agent exerting the power, 
 and the matter composing the weight, to move without resistance 
 from the air or any other fluid. 
 
 380. The exposition' of the effects of machinery is often invested 
 with the appearance of paradox. Astonishment is excited at what 
 seems incompatible with the results of common experience, rather 
 than admiration of the genius and skill by which simple and ob- 
 vious principles are so applied as to produce unexpected results. 
 
 Thus it is stated that, by means of a machine, a power of com- 
 paratively insignificant amount is capable of supporting or raising 
 a vast weight ; as, for example, it is affirmed that the fingers of an 
 infant pulling a thread of fine silk, which a pound weight could 
 snap asunder, are capable by this or that machine of supporting or 
 raising several hundred weight. 
 
 Statements like these, if literally understood, are fallacious ; if 
 rightly explained, they involve nothing which is not consistent with 
 our habitual experience. 
 
 381. In every machine there are some fixed points or props, 
 and the arrangement of the parts is always such that all that 
 portion of the weight not directly acting against the power is dis- 
 tributed among these props. If the weight, for example, amount 
 to 20 cwt., it is possible so to arrange it that any proportion of it, 
 however great, may be thrown upon the fixed points or props of 
 the machine : the remaining part only can properly be said to be 
 supported by the power, and this part so supported can never be 
 greater than the power. Considering the effect of a machine in 
 this manner, it appears that the power supports just so much of 
 the weight, and no more, as is equal to its own force, and that all 
 the remaining part of the weight is sustained by the machine. 
 
 The force of this observation will become more and more appa- 
 rent when the conditions are explained, under which a power and 
 weight can maintain each other in equilibrium, through the inter- 
 vention of a machine, whether simple or complex. 
 
 382. But if the power, instead of merely supporting the weight 
 at rest, be employed to raise this weight a given height, it may be 
 asked how it can be explained that a power indefinitely small can 
 lift a weight indefinitely great. The paradoxical character of this 
 statement arises, as is the case generally in such propositions, from 
 the .omission of an important condition. It is quite true that a 
 feeble power is capable of raising a great weight, but it is necessary 
 to add that in doing this, the feeble power must act through a 
 
GENERAL PRINCIPLES. 165 
 
 space just so much greater than that through which the weight is 
 raised, as the weight itself is greater than the power. Thus, if a 
 weight of I ooo Ibs. be raised one foot by a power whose force is 
 only equal to lib., then such power in raising the weight must 
 move through I ooo feet. 
 
 Now when this condition is stated, the proposition is stripped 
 altogether of its paradoxical character. There is nothing at all 
 astonishing in the fact, that one thousand successive exertions of a 
 force of one pound, each exertion being made through the space of 
 one foot, should raise a I ooo Ibs. weight through the height of one 
 foot. There is nothing more surprising in such a fact than if the 
 I ooo Ibs. weight, being divided into I ooo equal parts, were raised 
 by a thousand successive efforts of the power without the inter- 
 vention of any machinery. 
 
 383. It will be necessary to consider the effect produced by 
 means of a machine under three distinct relations between the 
 power and weight, viz., 
 
 I. When the power equilibrates with the weight. 
 II. When the power is greater than that which equilibrates 
 
 with the weight. 
 
 III. When the power is less than that which equilibrates with 
 the weight. 
 
 384. The power and weight are said to be in equilibrium when 
 they are so related to each other that when placed at rest they will 
 remain so. It is a great, but very common error, to suppose that 
 equilibrium, as applied to a machine, necessarily implies rest or 
 the absence of motion. It is easy to show that if the power and 
 weight, being in equilibrium, are put in uniform motion, they will 
 continue that uniform motion exactly as a mass of matter would 
 do in virtue of its inertia, if moving independently of any machine ; 
 for if we were to suppose that such motion would cease either 
 suddenly or gradually, we must necessarily also suppose a definite 
 force applied to the machine to stop its motion. Since the power 
 and weight are in equilibrium, they cannot of themselves stop or 
 retard the motion. It is true that the motion will in practical ap- 
 plications be gradually retarded, but that will be the effect of fric- 
 tion and atmospheric resistance, both of which are at present 
 excluded from consideration. 
 
 We cannot, on the other hand, suppose the uniform motion im- 
 parted to the power and weight to be accelerated, without sup- 
 posing the application of some adequate force to produce such 
 acceleration, the power and weight being excluded by the very 
 condition of their equilibrium. 
 
 Let us suppose the power and weight to be connected with the 
 machine by cords, by which they are suspended from their respec- 
 
166 THEORY OF MACHINERY. 
 
 tive points of application, both being, as usual, represented by 
 equivalent weights. Now the cords by which they are suspended 
 will be stretched with the same force whether the power and weight 
 be at rest or in uniform rectilinear motion ; and consequently the 
 relation between them in both cases must be the same. 
 
 385. The most common state of machines which are under the 
 operation of equilibrating forces, is that of uniform motion, and 
 not that of rest, as commonly stated. If a wind or water-mill be 
 in regular operation, its driving-wheel moving with a uniform 
 speed, then the power of the wind or water will be in equilibrium 
 with the resistance, whatever that may be. If a steam-engine be 
 in regular operation, its piston will move at a uniform rate, and 
 the force of the steam upon it will be in equilibrium with the re- 
 sistance which it is applied to overcome. If a locomotive engine 
 draw a railway train at a uniform speed, then the power exerted 
 by the engine will be in equilibrium with the resistance opposed 
 by the train. 
 
 386. Let us now consider the case in which the power is 
 greater than that which equilibrates with the weight or resistance. 
 In this case the motion imparted to the object moved will be acce- 
 lerated ; for so much of the power as would equilibrate with the 
 weight or resistance would impart, as has been already shown, a 
 uniform motion to the object moved. The surplus power above 
 this amount, therefore, must be employed in accelerating the 
 motion. 
 
 387. For example, if a locomotive engine exert a greater power 
 than is equivalent to the resistance opposed by the train which it 
 moves, then such surplusage of power can only act upon the 
 inertia of the train, and will impart to it an equivalent amount of 
 moving force. So long as this surplus power, therefore, acts, the 
 mass of the train will receive from it a corresponding augmentation 
 of its momentum, and consequently will receive a proportionate 
 increase of speed. If the resistance, however, opposed by the 
 train to the moving power augments with the speed, then it may 
 at length become equal to the amount of the moving power ; and 
 when it does, their equilibrium is established, and the train is 
 moved by the power at a uniform speed. 
 
 These conditions are by no means imaginary. They are realized 
 in every case in which a train is started from a state of rest, and 
 in general when any machine whatever is first put in motion. 
 
 388. The power in commencing its action must necessarily be 
 greater than the resistance opposed by the load ; for if it were not, 
 it would only equilibrate with the resistance, and no motion would 
 ensue. The surplus power is absorbed by the. momentum acquired 
 by the moving mass ; and as the velocity augments, more and 
 
GENERAL PRINCIPLES. 167 
 
 more momentum is imparted. The velocity will at length become 
 uniform, either because the energy of the power will be diminished, 
 so as to become equal to the resistance, or because the resistance 
 will be augmented, so as to become equal to the power ; or, in fine, 
 as most generally happens in practice, both of these effects are 
 combined, the resistance increasing and the power diminishing. 
 This is always the case, therefore, when a machine is impelled by a 
 surplus power, and when, on the other hand, there is a less than 
 ordinary resistance on the side of the machinery and of the load. 
 When first starting, the velocity being inconsiderable, the resist- 
 ance of the air and other agencies depending upon speed is less. 
 As the velocity increases, these resistances augment. This aug- 
 mentation of resistance, however, as the speed increases, is gene- 
 rally much less than the diminution of the moving power; in 
 short, a considerable surplus power is generally necessary, at 
 starting, to impart to the load, and to the moving parts of the 
 machinery the necessary momentum. But after this momentum 
 has once been imparted, then nothing remains for the power but 
 to balance the resistance of the load, properly so called. 
 
 This excess above the equilibrating power and the accelerated 
 motion are reciprocal consequences. Such excess necessarily 
 infers the accelerated motion of the load, and the accelerated 
 motion of the load indicates such excess. 
 
 389. Effects directly the reverse of these are developed when 
 the power applied is inferior to that which would equilibrate with 
 the weight. Let us suppose, in this case, the machine to have been 
 in uniform motion, and therefore the power and weight to have 
 been in equilibrium. Let the power then be diminished by any 
 amount, however small: the moment this diminution takes place 
 equilibrium is destroyed, the power becomes inferior to the resist- 
 ance, and ther.e is an action in a direction contrary to that of the 
 power, and therefore contrary to that of the motion which the 
 load had already acquired, equivalent in amount to the difference 
 between the resistance and the power. This force will act against 
 the momentum of the load, and will continually diminish it, until, 
 at length, it brings the load to rest. From the moment, therefore, 
 that the power becomes less than the resistance, the motion of the 
 load will be gradually retarded. The inferiority of the power to 
 the resistance, and a gradually retarded motion, are, therefore, re- 
 ciprocal consequences of each other. 
 
 It may be useful to illustrate still further these effects, which are 
 of considerable importance in practice. Let us suppose the resist- 
 ance which a machine is employed to overcome to be represented 
 by the weight A, ;%. 119., and let the power which acts against 
 such resistance throuh the intervention of the cord ABC be re 
 
1 68 
 
 THEORY OF MACHINERY. 
 
 presented by the force of an animal H. When the animal is at 
 rest before starting, the cord A B c H 
 is stretched with a force exactly 
 equal to that of the weight. When 
 the power begins to move, a mo- 
 mentum is imparted to the weight 
 through the intervention of the cord. 
 The cord is therefore stretched with 
 an additional force proportional to this 
 momentum. The speed of the power 
 gradually increases from the moment 
 its motion commences until it at- 
 tains that speed which is continued 
 uniform. During this increase of 
 
 Fig. 119. 
 
 the speed of the power H, a corresponding and continual increase 
 of momentum is imparted to the weight A, and consequently, 
 during this interval, the tension of the cord i constantly greater 
 than the weight. When, however, the speed of the power H be- 
 comes uniform, then no further momentum will be imparted to 
 the weight, and the force exerted by the power will diminish so 
 as to become exactly equal to the weight. During this uniform 
 motion the tension of the cord will be the same as it would be if 
 the power and weight were at rest. 
 
 When the weight approaches its point of destination, and is 
 about to be brought to rest, the power slackens its exertion, and, 
 at the moment that it becomes less than the weight, a moving force 
 takes effect equal in intensity to the difference between the power 
 and weight directed from B to A. But against this there is the 
 momentum of the weight directed from A to B in virtue of the 
 uniform velocity with which it had been moving. The moving 
 force, therefore, from B to A, represented by the difference between 
 the power H and the weight A, will act against this momentum, and 
 will gradually diminish it. 
 
 Although the upward motion of the weight, therefore, will con- 
 tinue after the diminution, of the power, it will be gradually 
 retarded, and after a certain interval will be altogether exhausted, 
 and the weight will come to rest. 
 
 These effects take place in all machines whatever when they are 
 started and stopped, and the circumstances and mechanical laws 
 which govern them are precisely the same as in the illustration here 
 given. 
 
 390. Proper functions of a machine. The use of a machine 
 is to adapt the power to the resistance. If the intensity, direction, 
 and velocity of the power were identical with the intensity and 
 direction of the resistance, and the velocity required to be imparted 
 
GENERAL PRINCIPLES. 169 
 
 to it, then there would be no need of a machine ; the power might 
 be applied immediately to the resistance. But if a power of feeble 
 intensity is required to act against a great resistance, then a ma- 
 chine must be interposed which will augment the intensity of the 
 power. Or if a power moving in one direction be required to 
 impart motion to a resistance in another direction, then a machine 
 must be interposed which will transmit the effect of the power to a 
 new direction. Or if a power having a certain velocity be required 
 to impart a greater or less velocity to the resistance, then a machine 
 must be interposed which will modify the velocity in the required 
 proportion. 
 
 But even these, though the principal, are only a few of the 
 infinite varieties of change and modification which machines are 
 required to effect in the transmission of the power to the resist- 
 ance. Independently of the directions, intensities, and velocities 
 of the moving power and resistance, the character of the respective 
 motions may differ in an infinite variety of ways : thus the moving 
 power may be one which acts with a reciprocating motion between 
 two points ; as, for example, that of the piston of a steam-engine ; 
 and this moving power may be required to produce a continuous 
 motion in a straight line, like the motion of a train along a railway. 
 
 The machine which connects such a power with such a weight 
 must therefore be so constructed as to convert the reciprocating 
 motion between two points into a continuous motion in a straight 
 line. 
 
 The moving power may act in a straight line, while the resist- 
 ance requires a circular motion. Thus, the wind which acts upon 
 the arms of a windmill is a continuous rectilinear force. The mill- 
 stones to which that force is transmitted, revolve by a continual 
 circular motion round vertical axes. The machinery of the wind- 
 mill must therefore be adapted to convert the rectilinear force of 
 the wind into the circular motion of the stones. 
 
 In every class of machines, and in every individual machine of 
 each class, the relation between the velocities and directions of the 
 power and weight, and the change produced on the character of 
 the motion of the power when transmitted to the weight, depends 
 solely on the structure of the machinery. 
 
 No variation in the magnitude of the power and weight can 
 alter this relation. Thus the ratio of the velocities of the power 
 and weight on a lever or an inclined plane, so long as their form and 
 proportions remain the same, will be unaltered, and, whatever 
 power or weight be applied to them, they will have this particular 
 ratio. 
 
 391. Wo machine can really add to the mechanical 
 energy of the power. A machine being composed of inert 
 
1 70 THEORY OF MACHINERY 
 
 matter cannot generate force, and consequently the working point 
 cannot exert more force than is transmitted to it from the point of 
 application of the power. 
 
 It will, in fact, exert less, because friction and other sources of 
 resistance must intercept a portion of the action of the power in 
 its transmission from its point of application to its working point ; 
 but as, for the present, the consideration of this species of resist- 
 ance is neglected, and machines are considered as exempt from 
 them, we shall assume that the influence of the power is transmitted 
 undiminished to the working point. But it is important on the 
 other hand to remember, that no more moving force can be so 
 transmitted. 
 
 392. Now the energy or momentum of the power is determined 
 by multiplying the weight which is equivalent to it, by the space 
 through which it is moved ; and, on the other hand, the moving 
 force imparted to the resistance is also estimated by multiplying 
 the weight which is equivalent to this resistance by the space 
 through which it is moved. 
 
 The moving force of the power is determined in the same man- 
 ner as the moving force of a weight equivalent to it, and moving 
 with the same velocity, would be determined. Thus, if we mul- 
 tiply the power, or its equivalent weight, by the space through 
 which it moves in a given time, that is to say, by its velocity, 
 we shall obtain a product which expresses its moving force or 
 mechanical effect. 
 
 393. This product is called the moment of the power. Thus if r 
 express the power and p the space through which it moves in one 
 second, then p X p will be its moment. In like manner, the moving- 
 force imparted to the resistance at the working point will be ex- 
 pressed by multiplying the resistance, or the weight equivalent to 
 it, by the space through which it is moved in a given time. Thus 
 if w be the weight, and w be the space through which it is moved 
 in one second, then w x w will be the moving force of the weight, 
 and this product is called the moment of the weight. 
 
 394. The relation between the moments of the power and weight 
 determines their mechanical state. 
 
 Three cases are here presented : 
 
 I. When the moment of the power is equal to the moment 
 of the weight ; that is, when 
 
 p x p = w X w. 
 
 II. When the moment of the power is greater than the 
 moment of the weight ; that is, when 
 
 p X p is greater than w x K\ 
 
GENERAL PRINCIPLES. 171 
 
 III. When the moment of the power is less than the moment 
 of the weight ; that is, when 
 
 p X p is less than w x w. 
 
 In the first case, it is manifest that the power and weight will be 
 in equilibrium, and that they will be either at rest or in uniform 
 motion. For, since the moment of the power is the expression of 
 its moving force, and since this moving force is transmitted without 
 increase or diminution by the machinery to the weight, and since, 
 by the supposition we have made, it is equal to the moving force 
 of the weight, these two forces must balance each other, and there- 
 fore be in equilibrium. 
 
 395. The condition therefore of equilibrium is, that the moment 
 of the power is equal to the moment of the weight, or 
 
 p x p = w X w. 
 
 396. If the moment of the power be greater than the moment 
 of the weight, then the moving force of the power, exceeding that 
 of the weight, and being transmitted to the working point undimi- 
 nished, will prevail over it, and the power and weight must either 
 have an accelerated motion in the direction of the power, or a 
 retarded motion in the direction of the weight. 
 
 397. If the moment of the power be less than the moment of the 
 weight, then the moving force of the power being transmitted to 
 the weight and being less than the moving force of the latter, the 
 latter will prevail, and therefore the power and weight must have 
 either a retarded motion in the direction of the power, or an 
 accelerated motion in the direction of the weight. 
 
 398. If the moments of the power and weight be equal we may 
 infer that the power will bear to the weight the same ratio as the 
 velocity of the weight bears to the velocity of the power, or 
 
 or, as it is sometimes expressed, the power and weight will be to 
 each other inversely as their velocities. This is another mode, 
 then, of expressing the conditions under which the power and 
 weight will be in equilibrium. 
 
 399. Power always grained at the expense of time. It is 
 this inverse proportion which is intended to be expressed, when it 
 is said that power is never gained save at the expense of time ; the 
 meaning of which is, that if a small power work against a great 
 resistance, the rate at which it moves the resistance will be just so 
 much slower than that at which the power itself moves, as the 
 resistance is greater than the power. 
 
 400. This condition of equilibrium, when rightly understood, 
 
172 THEORY OF MACHINERY. 
 
 removes all paradox from the statement of the effects of machinery 
 A small power working through a large space, raising a great 
 weight through a small space, is merely an expedient by which a 
 feeble power is enabled to accomplish its task, by a long succession 
 of efforts, without dividing the weight. To raise the weight of a 
 ton by a single effort one foot, would require a force equivalent to 
 the weight of a ton. But if, by the intervention of a machine, a 
 power is enabled to accomplish this object by 2240 distinct efforts, 
 each effort working through one foot, then such power need not 
 be more than one pound, or 2240 efforts made through the space 
 of one foot, each effort exerting the force of one pound will be 
 mechanically equivalent to 2240 Ibs., or one ton raised through 
 one foot, and the effect produced will be the same as if the weight 
 were actually divided into 2240 equal parts, and the power applied 
 successively to raise each of these parts one foot. 
 
 401. A very inadequate estimate would, however, be formed 
 of the objects and the utility of machinery, if we were to suppose 
 them only directed to this particular class of problems. Cases 
 innumerable occur, on the contrary, where small resistances are 
 moved by great powers. For example, in a locomotive engine, 
 while the piston in the cylinder moves once backwards and for- 
 wards, the train, which is the resistance overcome, is moved 
 through a space equal to the circumference of the driving-wheel. 
 Now, - if we suppose the length of the cylinder, as frequently 
 happens, to be one foot, and the circumference of the driving- 
 wheel to be fifteen feet, then the velocity of the piston, or the 
 power, will be to the velocity of the train, or the resistance, as 2 
 to 1 5 ; and consequently, the power which acts upon the piston 
 must be greater than the resistance of the train, which is moved 
 in the proportion of 1 5 to 2, omitting, as usual, the consideration 
 of friction, &c. In like manner, in a watch or clock, the resistance 
 of the object moved is merely that which is opposed to the motion 
 of the hands on the dial-plate, while the moving power is the 
 energy of the main-spring, or of a descending werght. In both 
 these cases it is obvious that the power is vastly greater than the 
 weight. 
 
 The machinery, therefore, may be stated generally as being the 
 means by which the force and motion of the power are modified, 
 so as to adapt them to the force or motion which is required 
 to be imparted to the object moved. 
 
 402. In all that precedes, it has been assumed that the point of 
 application of the power moves in the direction in which the power 
 acts, and that the motion imparted to the working point is in a 
 direction immediately opposed to the action of the weight or re- 
 sistance. This, in fact, is what generally takes place in the prac- 
 
GENERAL PRINCIPLES. 
 
 173 
 
 tical construction and operation of machinery ; for it is evident 
 that if the point of application of the power were not free to move 
 in the direction in which the power acts, a part of the power 
 would necessarily be lost ; and that if the working point did not 
 move in a direction immediately opposed to the weight or resist- 
 ance, a part of the force transmitted to the working point would 
 be inefficient. But as, in certain cases, these conditions would not 
 be fulfilled, it will be useful to state how the principles which 
 have been established in the present chapter must be modified in 
 such cases. 
 
 If by the construction of the machinery the point of application 
 of the power moves in a line different from that in which the power 
 acts, then the effective part of the power will be found by the 
 parallelogram of forces. 
 
 Let A (Jig. 1 20.) be the point of application of the power, and 
 
 m 
 
 Fig. izo. 
 
 let B be the working point. Let A P represent the direction of 
 the power, and B w the direction of the resistance or weight. Let 
 A jo be the direction in wKch the point of application is free to 
 move, and let the working point B be free to move in the direction 
 opposite to B w. Let right-angled parallelograms be formed, 
 having for their diagonals A P and B w, representing the power 
 and resistance, and having their sides in the directions A p and 
 B w, in which the point of application and the working point are 
 respectively free to move. 
 
 The power will then be equivalent to two forces represented 
 by A m and A p ; the latter, being in the direction in which alone 
 the point of application can move, is alone effective ; that part of 
 the power represented by A m will necessarily be expended in 
 pressure and strain upon the fixed points of the machine. In like 
 manner the weight or resistance represented by B w is equivalent 
 to two forces, B w and B n ; the force B w, being in the direction 
 against which alone the working point can act, is that portion of 
 the weight or resistance which the working point will act against : 
 the remainder of the weight will produce strain or pressure on the 
 fixed point. 
 
 In the application of the principles determining the relation 
 of the power and weight in cases of equilibrium which have been 
 
174 THEORY OF MACHINERY. 
 
 established in the present chapter, the effective portion only of the 
 power and weight must be to be taken into account. Thus, the power 
 is to be considered as represented by A p, and the weight by B w, 
 These principles will be rendered more clearly intelligible when 
 they have been illustrated in their application -to the simple 
 machines. 
 
 CHAP. II. 
 
 SIMPLE MACHINES. 
 
 403. Machines simple and complex. Machines which are 
 composed of two or more parts acting one upon another are called 
 complex machines. Machines which consist only of one part are 
 called simple machines. The several parts composing a complex 
 machine are themselves simple machines. In a complex machine 
 the effect of the power is transmitted successively through each of 
 the parts composing it until it reaches the working point. The 
 effect of complex machines is determined by combining together 
 the separate effects of the simple machines of which they are com- 
 posed. 
 
 To estimate the effects of machinery, therefore, it will be neces- 
 sary, in the first instance, to explain the principles of simple 
 machines. 
 
 404. Classification of simple machines. Simple machines 
 have been differently enumerated by different writers. If the 
 object be to group in the smallest possible number of distinct 
 classes those machines whose efficacy depends on the same prin- 
 ciple, the simple machines may be comprised under the following 
 three denominations : 
 
 I. A solid body turning on an axis. 
 II. A flexible cord. 
 
 III. A hard and smooth inclined surface. 
 
 Notwithstanding the infinite variety of machinery, and of the 
 parts composing it, it will be found that those parts may invariably 
 be brought under one or other of the above classes. 
 
 405. In a machine composed of a solid body turning on an axis, 
 all the parts are carried round such axis as a common centre, and 
 describe circles round it in the same time. It is evident that the 
 magnitude of these circles, and consequently the velocities of the 
 different parts, will be proportional to their respective distances 
 from the axis in which their common centre lies. 
 
SIMPLE MACHINES. 
 
 175 
 
 But since it has been already shown that when the power and 
 weight are in equilibrium they must be inversely as the velocities 
 of the points to which they are applied, it follows that any power 
 and weight applied to such a machine will be in the inverse pro- 
 portion of their distances from the axis when they are in equi- 
 librium. 
 
 It must be understood, in the application of this important 
 principle, that the power and weight are supposed to act in the 
 direction of the motion of the parts to which they are respectively 
 applied. If they do not act in this direction, then they must be 
 resolved by the principle of the composition of force into two 
 forces, one acting in the direction of the motion of the point of 
 application, and the other in a direction passing through a point 
 upon the axis. This has been already explained (402.). 
 
 406. The second class of simple machines includes all those in 
 which a force is transmitted by means of flexible threads, ropes, 
 or chains. The principle by which the effects of these machines is 
 estimated is, that the tension throughout the whole length of the 
 same cord, provided it be flexible and free from the effects of 
 friction, must be the same. Thus, if a force acting at one end be 
 balanced by a force acting at the other end, however the cord may 
 be bent, or whatever course it may be compelled to take, by any 
 cause which may affect it between its ends, these forces must be 
 equal, provided the cord be free to move over any obstacles 
 which may deflect it. This class includes all the various forms 
 of pulleys. 
 
 407. The third class includes all those cases in which the weight 
 or resistance is supported or moved upon a hard surface inclined 
 to the direction in which the weight or resistance itself acts. The 
 effects of such machines may be estimated by the principles already 
 explained. The force of the weight or resistance being resolved 
 into two other forces by the principle of the composition of force, 
 one of these two forces will be perpendicular to the surface, and 
 thus supported by its reaction ; the other will be parallel to it, and 
 will act against the power. 
 
 408. Mechanic powers. The first class of simple machines 
 above mentioned, consisting of a solid body revolving on an axis, 
 is usually subdivided into two. 
 
 1st. The lever, which consists of a solid bar, straight or bent, 
 resting upon a prop, pivot, or axis. 
 
 2nd. A cylinder connected with a wheel of much greater dia- 
 meter moving round a centre or axis. This combination is called 
 the wheel and axle. 
 
 The second class includes the pulley. The third class includes 
 the simple machines, commonly known as the inclined plane, the 
 
176 THEORY OF MACHINERY. 
 
 wedge, and the screw ; the last being, as will appear hereafter, 
 nothing more than an inclined plane rolled round a cylinder. 
 
 The classes, therefore, of the simple machines, as they are gene- 
 rally received, and which are known as the mechanical powers, are 
 the six following : 
 I. The lever. 
 II. The wheel and axle. 
 
 III. The pulley. 
 
 IV. The inclined plane. 
 V. The wedge. 
 
 VI. The screw. 
 
 We shall accordingly explain these, and show the most important 
 varieties and combinations of which they are susceptible. 
 
 THE LEVEE. 
 
 409. A straight and solid bar turning on an axis is called a lever. 
 The arms of the lever are those parts of the bar extending on each 
 side of the axis. The axis is called the fulcrum or prop. 
 
 Levers are commonly divided into three kinds, according to the 
 position which the fulcrum has in relation to the power and 
 weight. 
 
 If the fulcrum be between the power and weight, as in fig. 1 2 1., 
 the lever is of the first kind. 
 
 Fig. in. 
 
 If the weight be between the fulcrum and power, as in Jig. 122., 
 the lever is of the second kind. 
 
 w 
 
 
 
 . Fig. iz2. 
 
 If the power be between the fulcrum and weight, as in fig. 123., 
 the lever is of the third kind. 
 
 410. Of whatever kind the lever may be, the conditions of 
 equilibrium of the power and weight will be such that they arc 
 
 
THE LEVER. 
 
 7 
 
 inversely as their distances from the fulcrum, this being the general 
 condition of equilibrium for all machines which turn round a fixed 
 axis (405.)- It follows, therefore, that in figs. 121, 122, and 
 123., we shall have 
 
 p : w :: F A : F B; 
 
 or, if p express the distance of the power from the fulcrum, and w 
 the distance of the weight from the fulcrum, we shall have 
 
 p : w :: w.p; 
 or, what is the same, 
 
 p X p = w X iv. 
 
 This statement, as will be perceived, is nothing more than a 
 repetition of the general principle affecting machines which turn 
 on an axis, in virtue of which forces upon them are in equilibrium 
 when their moments round the axis are equal. The moment of 
 the power is P X p, and the moment of the weight is w x w. The 
 tendency of the power to turn the lever round its fulcrum in the 
 direction of the power is expressed by the moment P X j, and the 
 tendency of the weight to turn the lever in the contrary direction 
 is expressed by w X w. 
 
 411. It follows, therefore, that the tendency of the power to 
 turn the lever would be augmented either by increasing the amount 
 of the power p, or by increasing its distance p from the fulcrum. 
 In either case the effect will be increased in a corresponding pro- 
 portion. Thus, if we remove the power to double its distance from 
 the fulcrum, we shall double its effect ; and if we remove it tojialf 
 its distance, we shall diminish its effect one half. The distance of a 
 force, whether power or weight, from the fulcrum, is called its lever~ 
 age ; and it is evident from what has been stated, that the effect 
 of any force applied to a lever will be proportional to its leverage. 
 
 412. If the forces applied to a lever do not act perpendicular to 
 it, their effect will be iound by drawing from the fulcrum a per- 
 pendicular on their directions. This perpendicular will be their 
 
 leverage. Thus in jig. 124, 
 
 j^ / if the power act in the direc- 
 
 /"\ tion B P, draw FN perpendi- 
 
 "--... cular to the direction PEN; 
 
 B / '^ , the power -will have the same 
 
 I effect in turning the lever, as 
 W if it acted at N upon the lever 
 Fig N F. The moment of the 
 
 power, therefore, in this case, 
 
 will be found by multiplying it by F N, the perpendicular distance 
 of its direction from the fulcrum. In general, therefore, the lever- 
 
 N 
 
 I 
 
 , 
 
THEORY OF MACHINERY. 
 
 age of any force applied to such a machine is estimated by the 
 perpendicular distance of the direction of such force from the 
 fulcrum. 
 
 413. In a lever of the first kind, the power and weight may be 
 equal, and will be so when their leverages are equal. The weight 
 may be less than the power, and it will be so when it is at a greater 
 distance from the fulcrum than the power. 
 
 In a lever of the second kind, the weight, being between the ful- 
 crum and the power, must be at a less distance from the fulcrum than 
 the power, and must consequently be always greater than the power. 
 
 In a lever of the third kind, the power being between the ful- 
 crum and the weight, will be at a less distance from the fulcrum 
 than the weight, and consequently in this, case the power must 
 always be greater than the weight. 
 
 414. The balance. Numerous examples of levers of the first 
 kind may be given. A balance is a lever of this kind with equal 
 arms, in which the power and weight are necessarily equal. The 
 dishes are suspended by chains or cords from points precisely at 
 equal distances from the fulcrum, and being themselves adjusted 
 so as to have precisely equal weights, the balance will rest in 
 equilibrium when the dishes are empty. To maintain this equili- 
 brium, it is evident that equal weights must be put into the two 
 dishes ; the slightest inequality would give a preponderance to one 
 or the other dish. 
 
 To obtain all the necessary precision, the centre of the beam in 
 
 well-constructed balances 
 (fig- 125.) has angular- 
 shaped steel prisms at- 
 tached to each side of it, 
 with their edges presented 
 downwards. These, which 
 are called knife edges, rest 
 upon a hard surface, such as 
 steel or agate. The hooks 
 to which the chains sup- 
 
 j ; i \ porting the dishes are 
 
 I \ !yV | { i attached are similarly sup- 
 
 |l Mi ported. The instrument 
 
 / \ | I I is so constructed that the 
 
 \ 1 \ centre of gravity of the 
 
 ^LL '-"* '*^^* beam and its appendages 
 
 is a little below the knife 
 edge on which it is sus- 
 pended when the beam is 
 
 horizontal, the index attached to the centre of the beam then 
 
 Fig. IZ5- 
 
BALANCES. 179 
 
 pointing vertically upwards. If the beam be inclined downwards 
 on either side, the centre of gravity will deviate to the other side, 
 and, by its tendency to take the lowest position, it will bring the 
 beam back to the horizontal position when the force which dis- 
 turbed it is removed. 
 
 By the conditions of equilibrium of the lever, as explained 
 above, the beam can only rest in the horizontal position, with the 
 index vertical, "when the weights in the dishes are equal. The 
 article to be weighed being placed in one dish, known weights are 
 placed in the other, until the index points to the zero on the 
 graduated scale behind it, which is so arranged that the index is 
 then vertical. The weight of the article is then equal to the 
 known weights in the other dish. 
 
 To verify the exactitude of a balance, let it be first observed 
 whether the index is vertical, and the beam horizontal when the 
 dishes are empty. The article to be weighed being then placed in 
 one dish, and balanced by weights in the other, let the article and 
 the weights be transposed, the former being placed in the dish 
 containing the latter, and vice versa. If the index is still vertical , 
 and the beam horizontal, the balance is exact ; if not, the arms are 
 unequal, and the indications erroneous. Such a balance should 
 be rejected. 
 
 415. Sensibility of a balance. This term is used to express 
 the facility with which the index is turned from its position by a pre- 
 ponderating weight placed in either dish, and it is greater or less, 
 according as the distance of the centre of gravity below the point of 
 support is less or greater. Let A B (Jig. 126.) be the beam, c the 
 
 point of support, and G the 
 centre of gravity. If the beam 
 be 'slightly turned from the 
 horizontal position to the po- 
 sition A' B', the centre of gra- 
 vity will take the position G' 
 tig. 126. and would be balanced by a 
 
 weight upon A', which would 
 
 be less than that of the beam in the proportion of the distance 
 of o'-from G c to that of A' from G c. The nearer G is to c, the 
 less will be this proportion, and therefore the less will be the 
 weight which would suffice to turn the beam in a given degree 
 from the horizontal position, and it is evident that the less this 
 weight is, the more sensitive will be the balance. 
 
 In very sensitive balances the beam does not immediately come 
 to rest when weights, whether equal or unequal, are put in the 
 dishes, but oscillates for a long time slowly on the one side and 
 the other. If the zero of the graduated arc on which the index 
 
I So THEORY OF MACHINERY. 
 
 plays be exactly in the middle of the arc of oscillation, the weights 
 in this case will be equal ; but if the middle of the arc of oscillation 
 be on either side of the zero, the preponderating weight will be on 
 the same side. 
 
 However inaccurately a balance may be constructed, the exact 
 weight of a body may be ascertained by it. For this purpose let 
 the body to be weighed be placed in one dish, and let it be equi- 
 poised by sand placed in the other. Leaving the sand undis- 
 turbed, let the body be removed, and replaced by weights of 
 known value which will equipoise the sand. The amount of these 
 weights will then be the weight of the body. 
 
 4 1 6. A steelyard is a lever with unequal arms ; the power 
 being represented by a sliding weight, is adjusted so that its lever- 
 age may be changed at pleasure. These and similar instruments 
 are used for the purpose of weighing in commerce. 
 
 This instrument, sometimes called the Roman balance, is con- 
 structed in various forms, one of the most common of which is 
 represented in fig. 1 27. The weight Q slides upon a graduated 
 
 Fig. 1*7. 
 
 arm, and in some ascertain able position balances the article p 
 which is to be weighed. The weight of P is just as many times 
 that of Q as D c is greater than A c. 
 
 It may happen that even when Q is moved to the last division of 
 the arm B c, the article P will still preponderate. In that case the 
 steelyard is held by the ring nearer to A, which hangs down in the 
 figure. By this means the distance of p from the fulcrum being 
 less, the weight Q, at a given distance from the fulcrum, will 
 balance a proportionally greater weight. 
 
BALANCES. 
 
 181 
 
 417. letter balance. One of the most simple forms of these 
 useful instruments, which gives exact indications, is shown in 
 
 fig. 128. It is a bent lever of the 
 first kind. The dish E is suspended 
 from a short arm A c, and is balanced 
 by the arm c B, having a heavy knot 
 at G, and a point at B, which moves 
 upon a graduated arc. When A is 
 depressed, c B rises, and its centre of 
 gravity G moves to a greater distance 
 from the vertical line through the 
 point of support c, and takes a posi- 
 tion in which it balances the weight 
 placed in the dish E. The indications 
 
 expressed on the arc show the weights in E, which correspond to 
 
 the various positions of the index. 
 
 418. Spring: balances. Besides the instruments constructed 
 upon the principle of the lever, various forms of weighing instru- 
 ments are made, which depend on the elastic tension of springs. 
 
 Fig 128. 
 
 Fig. 109. 
 
 Fig. 130. 
 
 One of these is shown in fig. 129. and another in 130., which will 
 be easily understood upon inspection, without further explanation. 
 
 419. A crowbar is a lever of the first kind. In this instru- 
 ment, when used, for example, to raise a block of stone, the ful- 
 crum, fig. 1 3 1 ., is another stone F, placed near that which is to 
 be raised, and the power of the hand H is placed at the other end 
 of the bar. A poker applied to raise fuel is a lever of the first 
 kind, the fulcrum being the bar of the grate. 
 
 The force exerted by the hand at B (fig. 132.) will in this case 
 overcome a resistance greater than itself in the proportion of B c to 
 
 3 
 
THEORY OF MACHINERY. 
 
 Fig.iji. 
 
 A. c. The instrument may be rendered still more elficacious by 
 removing the point of support c to the line r 5, or on the other 
 hand a greater range of motion with less power may be obtained 
 by removing it to the line m n. 
 
 Scissors, shears, nippers, pincers, and other similar instruments 
 are composed of two levers of the first kind, the fulcrum being 
 the joint or pivot, and the weight the resistance of the substance 
 to be cut or seized, the power being the fingers applied at the other 
 end of the levers. 
 
 The brake of a pump is a lever of the first kind, the pump-rods 
 and piston being the weight to be raised. 
 
 420. Examples of levers of the second kind, though not so 
 frequent, are not uncommon. 
 
 An oar is a lever of the second kind. The reaction of the water 
 against the blade is the fulcrum. The boat is the weight, and the 
 hand of the boatman the power. 
 
 The rudder of a ship or boat is an example of this kind of lever, 
 and explained in a similar, way. 
 
 The chipping-knife, {.fig- I33-) ^ s a lever of the second kind. 
 The end r attached to the bench is the ful- 
 crum, and the weight the resistance of the 
 substance R to be cut. 
 
 A door moved upon its hinges is another 
 example. 
 iJ3- Nutcrackers are two levers of the second 
 
LEVERS. !8 3 
 
 kind, the hinge which unites them being the fulcrum, the resistance 
 of the shell placed between them being the weight, and the hand 
 applied to the extremity being the power. 
 
 A wheelbarrow is a lever of the second kind, the fulcrum being 
 
 the point at which the 
 wheel presses on the 
 ground, and the weight 
 being that of the barrow 
 and its load collected at 
 their centre of gravity M 
 
 (fig- I34-). 
 
 The same observation 
 may be applied to all two- 
 wheeled carriages which 
 are partly sustained by 
 Fig. 134. the animal which draws 
 
 them. 
 
 42 1 . Levers of the third kind, acting, as has been explained, to 
 mechanical disadvantage, the power being less than the weight, are 
 of less frequent use. They are adopted only where rapidity and 
 dispatch are required more than power. 
 
 The most striking examples of levers of the third kind are found 
 in the animal economy. The limbs of animals are generally levers 
 of this description. The socket of the bone is the fulcrum, a 
 strong muscle attached to the bone near the socket is the power, 
 and the weight of the limb, together with whatever resistance is 
 opposed to its motion, is the weight. A slight contraction of the 
 muscle in this case gives a considerable motion to the limb : this 
 effect is particularly conspicuous in the motion of the arms and 
 legs in the human body ; a very inconsiderable contraction of the 
 muscles at the shoulders and hips gives the sweep to the limbs, 
 from which the body derives so much activity. 
 
 The treadle of the turning-lathe is a lever of the third kind. 
 The hinge which attaches it to the floor is the fulcrum ; the foot 
 applied to it near the hinge is the power ; and the crank upon the 
 axis of the fly-wheel, with which its extremity is connected, is the 
 weight. 
 
 - Tongs are levers of this kind, as also the shears used in shear- 
 ing sheep. In these cases the power is the hand, placed imme- 
 diately below the fulcrum or point where the two levers are 
 connected. 
 
 422. The pressure on the fulcrum of a lever, when the power 
 and weight are in equilibrium, is determined by the principle of 
 the composition of forces. In a lever of the first kind, the re- 
 sultant of the power and weight is a single force passing through 
 
1 84 THEORY OF MACHINERY. 
 
 the fulcrum equal to their sum; consequently, the pressure on 
 such point will be equal to the sum of the power and weight. 
 
 In a lever of the second and third kind, the power and weight, 
 acting in contrary directions, will have a resultant equal to their 
 difference passing through the fulcrum. This resultant will there- 
 fore express the pressure on the fulcrum. 
 
 423. In the rectangular lever, the arms are perpendicular to 
 
 each other, and the fulcrum, r (jig- I35-)> is at 
 R A the right angle. The moment of the power in 
 
 this case is P multiplied by A F, and that of the 
 P A weight w multiplied by B r. When the instru- 
 
 ment is in equilibrium, these moments must be 
 equal. 
 
 i, _ When the hammer is used for drawing a nail, 
 
 Jw it is a lever of this kind ; the claw of the hammer 
 ffl is the shorter arm, the resistance of the nail is 
 Fig- 135- the weight, and the hand applied to the handle 
 
 is the power. 
 
 424. When a beam rests on two props, A B (fig- 136.), and 
 
 supports at some intermediate place, c, a 
 weight w, this weight is distributed between 
 the props in a manner which may be deter- 
 mined by the principles already explained. 
 
 If the pressure on the prop B be considered 
 as a power sustaining the weight w by means 
 of the lever of the second kind B A, then this power multiplied by 
 B A must be equal to the weight multiplied by c A. Hence the 
 pressure on B will be the same fraction of the weight as the part 
 AC is of A B. In the same manner it may be proved that the 
 pressure on A is the same fraction of the weight as B c is of B A. 
 Thus, if A c be one third, and therefore B c two thirds of B A, the 
 pressure on B will be one third of the weight, and the pressure on 
 A two thirds of the weight. - 
 
 It follows from this reasoning, that if the weight be in the 
 middle, equally distant from B and A, each prop will sustain half 
 the weight. The effect of the weight of the beam itself may be 
 determined by considering it to be collected at its centre of 
 gravity. If this point, therefore, be equally distant from the 
 props, the weight of the beam will be equally distributed between 
 them. 
 
 According to these principles, the manner in which a load borne 
 on poles is distributed between the bearers may be ascertained. 
 As the efforts of the bearers and the direction of the weight are 
 always parallel, the position of the poles relatively to the horizon 
 makes no difference in the distribution of the weights between 
 
COMPOUND LEVERS. 185 
 
 them. Whether they ascend or descend, or move on a level 
 plane, the weight will be similarly shared between them. 
 
 If the beam extend beyond the prop, as in fig. 137., and the 
 weight be suspended at a point not placed 
 between them, the props must be applied at 
 different sides of the beam. The pressure 
 which they sustain may be calculated in the 
 Kig IJ7i same manner as in the former case. 
 
 The pressure of the prop B may be con- 
 sidered as a power sustaining the weight w by means of the lever 
 B c. Hence, the pressure of B multiplied by B A must be equal to 
 the weight w multiplied by A c. Therefore, the pressure on B 
 bears the same proportion to the weight as A c does to A B. In 
 the same manner, considering B as a fulcrum, and the pressure of 
 the prop A as the power, it may be proved that the pressure of A 
 bears the same proportion to the weight as the line B c does to A B. 
 It therefore appears that the pressure on the prop A is greater that 
 the weight. 
 
 425. Compound lever. A combination consisting of several 
 
 levers acting one upon 
 
 F . * *"' another, as represented 
 
 Pi i~F F ' P" ^ fw in fig. 138., is called a 
 
 compound lever. 
 
 Fig. 138. The manner in which 
 
 the effect of the power 
 
 is transmitted to the weight may be investigated by considering 
 the effect of each lever successively. The power at P produces an 
 upward force at P', which bears to P the same proportion as P r to 
 p' F. Therefore, the effect at P' is as many times the power as the 
 line P F is of p' F. Thus, if p F be ten times p' F, the upward force 
 at P' is ten times the power. The arm p' F' of the second lever is 
 pressed upwards by a force equal to ten times the power at P. In 
 the same manner this may be shown to produce an effect at p" as 
 many times greater than that at P' as p' F' is greater than p" F'. 
 
 Thus, if P' F' be twelve times p" F', the effect at P" will be 
 twelve times that at p'. But this last was ten times the power, 
 and therefore the effect at P" will be one hundred and twenty 
 times the power. In the same manner it may be shown that the 
 weight w is as many times greater than the effect at P", as P" F" is 
 greater than w F". If p" F" be five times w F", the weight will 
 be five times the effect at P". But this effect is one hundred and 
 twenty times the power, and therefore the weight would be six 
 hundred times the power. 
 
 In the same manner, the effect of any compound system of levers 
 may be ascertained by taking the proportion of the weight to the 
 
1 86 
 
 THEORY OF MACHINERY. 
 
 power in each lever separately, and multiplying these numbers 
 together. 
 
 In the example given, these proportions are 10, 12, and 5, 
 which, multiplied together, give 600. In^o-. 138. the levers com- 
 posing the system are of the first kind ; but the principles of the 
 calculation will not be altered, if they be of the second or third 
 kind, or some of one kind and some of another. 
 
 426. Weighing-machines. These are usually compound 
 levers, and are constructed in very various forms, but all depend - 
 
 Fig. 139. 
 
 ing nearfy on the same mechanical principles; one of these varie- 
 ties is shown \nfgs. 139. and 140. 
 
 -M 
 
 Fig. 140. 
 
 The platform A B, upon which the object to be weighed is placed, is sup- 
 ported by the short arm M L, of a lever L N, through the intervention of a 
 compound lever, KHDCBEFGL, the longer arm of which, MN, sup- 
 ports a dish in which the counterpoising weight is placed. 
 
 The platform A B, is supported at two points, i by the lever E o L at E, 
 
WEIGHING MACHINES. 
 
 187 
 
 and z by the arm L M at K, through the intervention of the rod K H, the 
 diagonal piece D c, and the upright piece c B. The proportions of the levers 
 are so regulated that the pressure at L is less than that at K, in the same 
 proportion as K M is less than L si, and these two pressures together are 
 balanced by the weight p. 
 
 Before commencing the operation the lever L M N is rendered horizontal 
 by small weights or sand put into the dish a, and its horizontal position is 
 ascertained by the coincidence of the two points b and c, the former of which 
 is fixed, and the latter attached to the arm M N of the lever. 
 
 It is easy to understand that the proportion of the arms of the lever may 
 be so arranged that each hundredweight of the article Q, may be balanced 
 by a pound or any lesser denomination cf weight placed in p. 
 
 427. The knee lever. A form of compound lever, known as 
 the knee lever, is much used in the arts. This combination con- 
 sists of a metal rod A B, Jig. 
 141., having a fixed point of 
 support A, on which it works. 
 Another bar G c is jointed to 
 it at c, a point intermediate 
 between A and B. This bar 
 c G is jointed at G to a plate, 
 such as R, or any other object 
 to which it is desired to trans- 
 mit an intense force acting 
 through a very limited space, 
 as, for example, in the case of 
 the printing press, where the 
 paper is pressed upon the type 
 by a plate which is driven 
 upon it by a sudden and 
 severe force. The handle B 
 of the lever being pressed in 
 the direction of the arrow, exerts a corresponding pressure on the 
 point c, which is driven in the direction c D, perpendicular to A B. 
 This motion c D is resolved into two by the parallelogram of forces, 
 one in the direction c B, and the other in the direction c r ; the 
 latter exerts pressure on the fixed point A, and the other acts upon 
 the plate B, by means of the joint G forcing it downwards. As the 
 joint c advances, the angle A c G becomes more and more obtuse, 
 and the component c E of the force acting at B bears a rapidly 
 increasing proportion to the force itself, so that when the levers 
 A c and c G come nearly into a right line, the pressure exerted at 
 B is augmented at G in an almost infinite proportion. 
 
 428. Key of Erard's pianoforte. In this instrument, the 
 object is to convey from the point where the finger acts upon the 
 key, to that at which the hammer acts upon the string, all the 
 
 Fig. 141. 
 
i88 
 
 THEORY OF MACHINERY. 
 
 fig- 
 
 delicacy of action of the fin 
 ger, so that the piano may 
 participate, to a certain extent, 
 in that sensibility of touch 
 which is observable in the harp, 
 and which is the consequence 
 of the finger acting immediately 
 on the string in that instru- 
 ment, without the intervention, 
 of any other mechanism. 
 
 The combination of levers, 
 by which the action of the 
 finger is transmitted to the 
 string, in Erard's pianoforte, is 
 represented in fig. 142. 
 
 The key is represented at b a c, 
 the centre or pivot on which it 
 plays being a, and the ivory table 
 upon which the finger acts being 
 at b. The point to which its mo- 
 tion is communicated is at c. 
 
 The motion is transmitted by a 
 double-jointed piece d to an inter- 
 mediate lever ef, the pivot of 
 which is at e. At the joint / is a 
 rod g called the sticker, which car- 
 ries up the hammer to the string. 
 The hammer is supported by the 
 head of the sticker g, and at the 
 same time rests upon the oblique 
 lever i, vrhich latter is acted upon 
 by the spring h. 
 
 When the key is pressed down 
 by the finger at b, the piece d is 
 raised, and by it the lever ef and 
 the sticker g. This lever raises the 
 lever t, and acts upon the rod of 
 the hammer at a point near the 
 joint, making the hammer rise 
 along the dotted curve so as to 
 strike the string. 
 
 The proportions of this com- 
 bination of levers are such, 
 that, after the blow of the ham- 
 mer on the string, the check k 
 comes forward and receives the 
 hammer in its fall, at about 
 one third of its original dis- 
 
WHEEL WORK. 189 
 
 tance from the string ; so that while the finger continues to keep 
 down the key, the hammer remains at a distance from the string, 
 equal to one third of its distance when the key is not depressed. 
 
 In the meantime, the spring h has given way under the weight 
 of the hammer, and, under these circumstances, the key b being 
 allowed to rise by the finger through one third of its play, and 
 then again being depressed, another stroke of the hammer on the 
 string will be produced; for in this case the hammer will be 
 brought back to the level of the head of the sticker g, by 
 which means it will be driven upwards upon the depression of 
 the key. 
 
 In the combination of levers used in other pianofortes, the note 
 cannot be repeated without allowing the key to rise to the position 
 it has before it is depressed ; consequently in this case, a repetition 
 of the note is produced with one third of the motion of the finger 
 which is necessary in other pianofortes. 
 
 429. Power of a machine. That number which expresses 
 the proportion of the weight to the equilibrating power in any 
 machine, we shall call the power of the machine. Thus if, in a 
 lever, a power of I Ib. support a weight of 10 Ibs., the power of 
 the machine is 10. If a power of 2 Ibs. support a weight of 1 1 Ibs., 
 the power of the machine is 5^. 
 
 430. Equivalent lever. As the distances of the power and 
 weight from the fulcrum of a lever may be varied at pleasure, 
 and any assigned proportion given to them, a lever may always be 
 conceived having a power equal to that of any given machine. 
 Such a lever may be called, in relation to that machine, the equiva- 
 lent lever. 
 
 43 1 . As every complex machine consists of a number of simple 
 machines acting one upon another, and as each simple machine 
 may be represented by an equivalent lever, the complex machine 
 will be represented by a compound system of equivalent levers. 
 From what has been proved, it therefore follows that the power 
 of a complex machine may be calculated by multiplying toge- 
 ther the power of the several simple machines of which it is 
 composed. 
 
 WHEEL-WOBK. 
 
 432. The wheel and axle. The form of simple machine de- 
 nominated the wheel and axle, consists of a cylinder which rests in 
 pivots at its extremities, or is supported in gudgeons, and is capable 
 of revolving between those pivots, or in those gudgeons. At- 
 tached to this cylinder, and supported on the same pivots or 
 gudgeons, a wheel is fixed, so that the two revolve together 
 with a common motion. The wheel is often supplied with a rope 
 
190 
 
 THEORY OF MACHINERY. 
 
 or chain, which winds round the axle, and the power with another 
 
 rope or chain which winds round the wheel. 
 
 Such an arrangement is represented in jig. 143., where w is 
 the weight, A and B the pivots or gudgeons, 
 c the wheel, and p the power. 
 
 433. The condition of equilibrium is, 
 according to what has been already proved, 
 the inverse proportion of the power and 
 weight to the diameters of the wheel and 
 axle, that is to say, the power is to the 
 weight as the diameter of the axle is to the 
 diameter of the wheel. The weight is ge- 
 nerally applied, as represented in the figure, 
 by means of a rope coiled upon the axle. 
 
 434. The manner of applying the power is very various. Some- 
 times the circumference of the wheel is furnished with projecting 
 points, as represented in fig. 143., to which the hand is applied 
 when human force is the power. Examples of this are numerous 
 a familiar one is presented in the steering-wheel of a ship. 
 
 435. In the common windlass the power is applied by means of 
 a winch D c, as represented in Jig. 144. The arm B c of the 
 winch represents the radius of the wheel, and the power is applied 
 to D c at right angles to B c. In some cases no wheel is attached 
 to the axle, but it is pierced with holes, directed towards its centre, 
 in which long levers are incessantly inserted, and a continuous 
 action produced by several men working at the same time, so that 
 whilst some are transferring the levers from hole to hole, others are 
 working the other levers, fig. 145. 
 
 D 
 
 B 
 
 Fig. 144. 
 
 Fig. 145. 
 
 436. The axle is sometimes placed in a vertical position, the 
 wheel or levers being moved horizontally. The capstan is an example 
 of this. A vertical axis is fixed in the deck of the ship, the cir- 
 cumference being pierced with holes presented towards its centre. 
 
WINDLASS CAPSTAN TREADMILL. 
 
 191 
 
 These holes receive long levers, as represented in jig. 146. The 
 men who work the capstan walk continually round the axle, press- 
 ing forward the levers near their extremities. 
 
 More generally the head of the capstan is circular, and pierced 
 with many holes, in each of which a lever can be inserted, so that 
 many men can work together when great power is required, as in 
 weighing anchor, fig. 14-7- 
 
 Fig. 147. 
 
 Fig. 146 
 
 437. The tread-mill, <Scc. In some cases the wheel is turned 
 by the weight of animals placed at its circumference, who move 
 forward as fast as the wheel descends, so as to maintain their posi- 
 tion continually at the extremity of the horizontal diameter. The 
 tread-mill, fig. 148., and certain cranes, such as fig. 149., are 
 example? of this. 
 
 Fig. 148. 
 
 Fig. 149- 
 
 438. French quarry wheels. In the neighbourhood of 
 Paris, and other parts of France, stone for building is obtained in 
 subterranean quarries, from which it is raised through deep ver- 
 tical shafts, by means of large wheels, worked by the weight of 
 men, on the same principle as that of the tread-mill. One of these 
 quarry wheels, and the manner of, working it, is shown \nfig. 150. 
 
 From what has been already explained, it is evident that the 
 effect of the power upon the weight would be augmented by 
 
19* THEORY OF MACHINERY, 
 
 diminishing the thickness of the axle, and diminished by increasing 
 that thickness. 
 
 Fig. 150. 
 
 439. It sometimes happens that an invariable power has to act 
 against a variable resistance, or a variable power against a con- 
 stant resistance. In such a case, the effect of the wheel and axle 
 as just described would vary: an augmentation of the power or 
 diminution of the resistance would throw the power and weight 
 out of equilibrium. 
 
 If, however, the axle were made to increase in thickness in the 
 same proportion as the ratio of the power to the weight is aug- 
 mented, then the change of such ratio would be compensated by a 
 corresponding change in the leverage, and an equilibrium would 
 
WHEEL AND AXLE. 193 
 
 be maintained between the power and weight, notwithstanding 
 their variation. Numerous instances of this are presented in the 
 arts, some of which will be noticed hereafter. 
 
 440. When a weight or resistance of comparatively great 
 amount is to be raised by a very small power by means of the 
 simple wheel and axle, either of two inconveniences would ensue ; 
 either the diameter of the axle would become too small to support 
 the weight, or the diameter of the wheel would become so great 
 as to be unwieldy in its operation. This has been remedied, with- 
 out having recourse to a complex machine, by a simple expedient 
 represented in^g-. 151. The axle of the windlass here consists of 
 two parts, one thicker than the other, and the rope by which the 
 
 Fig. 151. 
 
 weight is raised rolls on the thicker while it rolls off the thinner. 
 In each revolution, therefore, the part which is rolled on exceeds 
 that which is rolled off by the difference between the circumfer- 
 ence of the two parts of the axle. The effect, accordingly, is the 
 same as if an axle had been used, whose diameter is equal to the 
 difference between the diameters of the thicker and thinner part. 
 Since, then, without diminishing the thickness of the axle, we may 
 dimmish without limit the difference between the thicker and 
 thinner parts, the ratio of the weight to the power may be aug- 
 mented indefinitely without diminishing the strength of the axle. 
 
 441. When great power is required, wheels and axles may be 
 combined in a manner analogous to the compound lever already 
 explained. The power being supposed to act on the circumference 
 
194 THEORY OF MACHINERY. 
 
 of the first wheel, its effect is transmitted to the circumference of 
 the first axle ; this circumference acts on the circumference of the 
 second wheel, and transmits motion thereby to the circumference 
 of the second axle, which, in its turn, acts on the circumference of 
 the second wheel, transmitting motion to the circumference of the 
 third axle, and so on. 
 
 There is nothing different in the mechanical effect of such a 
 combination from that of a system of compound levers, except that 
 it admits more conveniently of a continuous action, and produces 
 continued and regular motion. The relation between the power 
 and the weight of resistance, when in equilibrium, is determined in 
 exactly the same manner as in the case of the compound lever. 
 
 'If the diameters of all the wheels be multiplied together, and 
 the diameters of all the axles be also multiplied together, then the 
 power will be to the weight as the product of the diameters of all 
 the axles to the product of the diameters of all the wheels. Thus, 
 if the diameters of all the axles be expressed by the numbers 2, 
 3, 4, and the diameters of all the wheels be expressed by the 
 numbers 20, 25, and 30, then the ratio of the power to the weight 
 will be as 2 X 3 X 4=24 to 20 X 25 X 30 = 1 5000. 
 
 442. The manner in which the wheels and axles act one upon 
 another is very various. Sometimes a strap or cord is placed in a 
 groove in the circumference of the axle, and carried round a 
 similar groove in the circumference of the wheel. This, which 
 is called an endless band, is represented mjigs. 152. and 153. 
 
 OP 
 
 Fig. 153. 
 
 443. In the case represented in jig. 152. the wheels are driven 
 in the same direction ; in that represented in fig. 153., they are 
 driven in opposite directions. Examples of this method of trans- 
 mitting the motion from wheel to wheel are presented in every 
 department of the arts and manufactures. In the turning-lathe 
 and the grinding-wheel a cat-gut cord carried round the treadle 
 wheel imparts motion to the maundrell or the grindstone. In the 
 great factories shafts, as shown in Jig. 154., are carried along the 
 ceilings of the room, round which, at certain points, endless straps 
 are carried which are conducted round the wheels, thus giving 
 motion to the lathes or other machines. One of the chief advan- 
 
WHEEL WORK, 
 
 195 
 
 tages of this method of transmitting motion by wheels and axles is, 
 that the bands by which the motion is conveyed may be placed at 
 
 Fig. 154- 
 
 any distance from each other, and even in any position with respect 
 to each other, and may, by a slight adjustment, receive motion in 
 either one direction or the other. 
 
 444. When the circumference of the axle acts immediately on 
 that of the wheel, which it moves without the intervention of a 
 strap or cord, means must be adopted to prevent them from 
 moving in contact, (fig- 155.) without transmitting motion, which 
 they would do if both surfaces were perfectly smooth and free 
 from friction. 
 
 This is accomplished by different expe- 
 dients. In cases where great power is not 
 required, motion is communicated through 
 a series of wheels and axles by rendering 
 their surfaces rough, either by facing them 
 with rough leather, or making them of wood 
 cut across the grain. This method is used 
 in spinning machinery, where a large buffed 
 wheel, placed in a horizontal position, is sur- 
 rounded by a series of small buffed rollers 
 pressed close against it, each roller communicating motion to a 
 
 o z 
 
 Fig. 155- 
 
1 96 THEORY OF MACHINERY. 
 
 spindle. As the wheel revolves, revolution Is imparted to the 
 rollers, the velocity of which exceeds that of the wheel in the same 
 proportion as the diameter of the wheel exceeds that of the roller. 
 This method is very convenient in cases where the motion of the 
 rollers requires to be occasionally suspended, each roller being 
 provided with a means by which it can be thrown out of contact 
 with the wheel, and thus stopped. 
 
 445. The most frequent method of transmitting motion through 
 a train of wheel-work is by the construction of teeth upon their 
 circumference, so that the teeth of each falling between those of the 
 other, the one wheel necessarily pushes forward the other. When 
 teeth are used, the axles are usually called pinions, and the teeth 
 raised upon them are called leaves. 
 
 446. In the formation of the teeth of wheels and pinions, expe- 
 dients are adopted to prevent them from rubbing one upon 
 another, when they move in contact with each other. A particular 
 form is adopted for the teeth, in virtue of which the surfaces are 
 applied one to the other with a rolling motion like that of a carriage 
 wheel upon the road. By this expedient the rapid wear of the 
 teeth, which would be produced by constant friction accompanied 
 by pressure, is prevented. 
 
 447. In computing the mechanical effects of toothed wheels and 
 pinions, the number of teeth may be substituted for their circum- 
 ferences and diameters. 
 
 The condition of equilibrium will therefore be obtained by 
 multiplying together the number of teeth in all the wheels, and 
 the number of teeth in all the pinions, the power being to the 
 weight, when in equilibrium, as the latter product to the former. 
 
 448. Spur wheels. Teeth are formed upon the edges of wheels 
 
 and pinions in different positions. 
 When they are formed in the plane of 
 the wheel so as to diverge from the axis 
 as a centre, the wheels are called spur 
 wheels ; and a train of such wheels and 
 pinions, such asjig. 156., is called spur 
 
 tr- -m gearing. By such gearing the several 
 
 axes round which motion is produced 
 are parallel one to another, as appears 
 
 Fig. 156. b 7 the figure- 
 
 It is evident in this case that the 
 
 wheel and pinion, which are upon the same axis, need not be in 
 immediate juxtaposition, but may be separated one from the other 
 by any length of the shaft. In this way, if the shaft be carried 
 along the factory, a pinion at one end of a room may give motion 
 to or receive it from a wheel upon the same shaft at the other end. 
 
WHEEL WORK. 
 
 '97 
 
 449- Crown wheels. When the teeth 
 are formed in the surface of a hoop or 
 cylinder, so as to be directed parallel to 
 li the axis, as shown in fig. 157., the wheel 
 is called, from its form, a crown wheel. If 
 the teeth of such a wheel be engaged in 
 those of a spur wheel, as in the figure, the 
 axis of the latter, will be at right angles to 
 that of the former. 
 
 Sometimes the crown wheel works in a 
 Fig. 157. sort of cylindrical basket on a shaft at right 
 
 angles to its axis, as shown in fig. 158. 
 
 450. Bevelled wheels are such as have their teeth inclined to 
 the shaft, as shown in fig. 159. By this expedient motion may be 
 imparted from one shaft to another inclined to it at any angle. 
 
 Fig. 158. 
 
 Fig. 159. 
 
 451. Rack and pinion. When it is desired to impart a 
 rectilinear motion of limited range by 
 means of the rotation of an axle, the 
 object is attained by the combination of 
 a spur wheel or pinion with a straight bar 
 having teeth of corresponding form and 
 magnitude formed upon it. Such a toothed 
 bar is called a rack. The mode of action 
 of such a combination is shown in fig. 
 160. 
 
 The efficacy of the power is generally increased by the combi- 
 nation of two or more wheels and pinions, as shown in fig. 1 6 1 ., 
 where the handle drives the pinion D which works in the wheel B, 
 
 o 3 
 
 Fig. ,60. 
 
198 
 
 THEORY OF MACHINERY. 
 
 Fig. i6z. 
 
 Fig. 161. 
 
 on the axle of which is the pinion 
 which drives the rack A. 
 
 452. Ratchet wheel When 
 
 it is desired to allow an axle or 
 shaft to revolve in one direction, 
 but not in the other, the object is 
 accomplished by fixing upon it a 
 wheel having teeth, n (Jig. 162.), 
 inclined in the direction contrary 
 to that in which it is intended the 
 shaft should be free to move. A 
 catch, o ?tt, falls between these 
 teeth, and stops all motion of the 
 wheel directed against it ; but when 
 the wheel turns in the other direc- 
 tion, the catches fall from tooth to 
 tooth, producing a clicking noise. 
 
 PULLEYS. 
 
 453. Although the term pulley implies the combination of a rope 
 and a wheel on which it runs, the practical effects of the simple 
 machine so denominated depend altogether on the rope ; the wheel 
 being introduced for the mere purpose of diminishing the effects 
 of friction and imperfect flexibility, the consideration of both of 
 which are omitted in theory. If a rope were perfectly flexible, and 
 were capable of being bent over a sharp edge, and of moving upon 
 it without friction, we should be enabled by its means to make a 
 force in any one direction overcome a resistance or communicate 
 a motion in any other direction. 
 
 Thus, if a perfectly flexible rope, r s {fig. 163.), pass over a 
 sharp edge P, and be connected with a weight R 
 vertically, a force acting obliquely in the direction 
 p F will raise the weight vertically in the direction 
 R Q ; but as no materials of which ropes can be 
 made can render them perfectly flexible, and as 
 in proportion to the strength by which they are 
 enabled to transmit force their rigidity increases, 
 it is necessary in practice to adopt means to 
 remedy or mitigate those effects which attend the 
 absence of perfect flexibility, and which would 
 otherwise render cords practically inapplicable as machines. 
 
 But, beside the want of perfect flexibility, the surface of the 
 rope is always rough, and often considerably so. This surface, in 
 passing over an edge, would produce a degree of friction which 
 would altogether stop its movement. If a rope were used in the 
 
 Fig. 163. 
 
PULLEYS. 199 
 
 manner represented in the figure, to transmit a force in one 
 direction to a resistance in another, some force would be necessary 
 to bend it over the angle p, which the two directions form one 
 with the other : and, if the angle were sharp, the effect of such a 
 force might be the rupture of the rope. 
 
 454. But if, instead of bending the rope at one point over a 
 single acute angle, the change of direction were produced by 
 successively deflecting it over several angles, each of which would 
 be less sharp, the force necessary for the deflection and the liability 
 of breaking the cord would be diminished. But such object will 
 be still more effectually attained if the cord be deflected over the 
 surface of a curve. 
 
 If the rope were applied merely to sustain a weight without 
 moving it, a curved surface would therefore be sufficient to remove 
 the inconvenience arising from imperfect flexibility; but when 
 motion is required, the rope, in passing over such a surface, would 
 be subject to great friction and rapid wear. This inconvenience 
 is removed by causing the surface on which the rope runs to move 
 with it, so that no more friction is produced than would arise from 
 the curved surface itself rolling upon the rope. These objects are 
 attained by the common pulley, which consists of 
 a wheel called a sheave, fixed in a block turning 
 on a pivot. A groove is formed in the edge of 
 the wheel, in which the rope runs, the wheel 
 revolving with it. 
 
 This apparatus is represented in jig. 164. 
 [_J Notwithstanding, however, that this expedient 
 removes the effects of friction and rigidity to so 
 Fig 164. great a degree as to render the use of the cord 
 
 practically available,- it must not be supposed that 
 these effects are altogether "Overcome ; they still produce some 
 impediment to the full efficiency of the power, as will be ex- 
 plained more fully hereafter. For the 
 present, however, we shall consider the 
 rope as rendered by this expedient flexible 
 and free from friction. 
 
 455. By means of a single sheave, a 
 power acting in any one direction may be 
 made to transmit its effects to a resist- 
 ance in any other direction, provided the 
 two lines of direction be in the same plane, 
 and not parallel. Thus, let A B (fig. 165.) 
 be the direction in which the power acts, 
 Fig. 165. and lefc c D be the direction in which the 
 
 weight or resistance acts. 
 04 
 
200 THEORY OF MACHINERY. 
 
 To transmit in this case the power to the weight, let the two 
 directions B A and D c be prolonged until they meet, which they 
 will do at o. In the angle o, formed by the two directions, let a 
 sheave be placed, and let the power be connected with a rope in 
 the direction B A. This rope, being carried from A over the 
 sheave at o, must be brought down in the direction CD, and 
 connected with the resistance. 
 
 456. But if the direction of the power and resistance be 
 --*7v parallel, as in Jig: 166., then the effect of the 
 vSj power might be transmitted to the weight by 
 means of a cord and two fixed pulleys, one 
 placed over the direction of the force, and the 
 other over the direction of the weight, as re- 
 presented in the figure. 
 
 In fine, if the direction of the power and 
 the weight be not placed in the same plane, 
 then the effect of the power may still be 
 Fig. 166. " transmitted to the weight by means of two 
 pulleys. Let us suppose, for example, that a 
 power acting in a given horizontal line is required to be trans- 
 mitted to a weight acting at some distance from it, in a certain 
 vertical line, which is not in the same plane with the direction of 
 the power. 
 
 Let two fixed pulleys be placed at two points in any convenient 
 positions on the lines of direction of the power and weight, and let 
 a line be supposed to join these points. Let the axis of one of the 
 pulleys be placed at right angles to the plane formed by the line of 
 direction of the power and the line joining the two pulleys, and let 
 the other be placed with the axis at right angles to the plane pass- 
 ing through the line of direction of the weight and the line joining 
 the two pulleys. By this arrangement, the cord being passed suc- 
 cessively over both pulleys, the effect of the power will be trans- 
 mitted to the weight. 
 
 457. In all these cases, the same coi*d by which the weight is 
 suspended being directly connected with the power, and its tension 
 throughout its entire length being the same, the weight and the 
 power must be equal when they are in equilibrium. This condition 
 is also rendered manifest by the fact, that from the motion of the 
 mechanism connecting it, the weight and power will move with 
 the same velocity. It appears, therefore, that no mechanical ad- 
 vantage is gained by a single rope acting over one or more fixed 
 pulleys ; nevertheless, there is scarcely any engine, simple or com- 
 plex, which is attended with more convenience. 
 
 458. In the applications of power, whether of man or animals, 
 or arising from other natural forces, there are always some direc- 
 
PULLEYS. 
 
 201 
 
 tions in which it may be exerted to greater convenience and 
 advantage than others, and in many cases the power is capable of 
 acting only in one particular direction. Any expedient, therefore, 
 which can give the most advantageous direction to the moving 
 power, whatever be the direction of the resistance opposed to it, 
 contributes as much practical convenience as one which enables 
 a small power to balance or overcome a great weight. 
 
 459. By means of the fixed pulley a man may raise himself to a 
 considerable height, or descend to any proposed depth. If he be 
 placed in a chair or a basket attached to one end of a rope which 
 is carried over a fixed pulley, by laying hold of this rope on the 
 other side, he may, at will, descend to a depth equal to half of the 
 entire length of the rope, by continually yielding rope on the one 
 side, and depressing the basket or chair by his weight on the other. 
 Fire escapes have been constructed on this principle, the fixed 
 pulley being attached to some part of the building. 
 
 4.60. A single movable pulley is represented in jig. 167.; a 
 cord is carried from a fixed point r, and passing through a block B 
 attached to a weight w, passes over a fixed pulley c, 
 the power being applied at P. We shall first suppose 
 the parts of the cord on each side the wheel B to be 
 parallel : in this case the whole weight w being 
 sustained by the parts of the cords B c and B r, and 
 these parts being equally stretched, each must sustain 
 half the weight, which is therefore the tension of the 
 cord. This tension is resisted by the power at P, 
 which must therefore be equal to half the weight. 
 In this machine, therefore, the weight is twice the power. 
 46 1 . If the parts of the cord B c and B r be not parallel, as 
 in jig. 1 68., a greater power than half the weight is therefore 
 necessary to sustain it. To determine the power necessary 
 
 ___^__ to support a given weight in this 
 
 case, take the line B A in the vertical 
 direction, consisting of as many inches 
 as the weight consists of ounces ; from 
 A draw A D parallel to B c, and A E 
 parallel to B F : the force of the weight 
 represented by A B will be equivalent 
 to two forces represented by B D and 
 B E. The number of inches in these 
 lines respectively will represent the 
 number of ounces which are equivalent to the tensions of the parts 
 B F and B c of the cord. But as these tensions are equal, B D and 
 B E must be equal, and each will express the amount of the power 
 p, which stretches the cord at P c. 
 
 Fig. 167. 
 
 Fig. 168. 
 
202 THEORY OF MACHINERY. 
 
 it is evident that the four lines A E, E B, B i>, and D A are equal, 
 and as each of them represents the power, the weight which is 
 represented by A B must be less than twice the power which is 
 represented by A E and E B taken together. It fol- 
 lows, therefore, that as the parts of the rope which 
 support the weight depart from parallelism, the ma- 
 chine becomes less and less efficacious, and there are 
 certain obliquities at which the equilibrating power 
 would be much greater than the weight. 
 
 462. If several sheaves be constructed in the same 
 movable block, the mechanical advantage may be 
 proportionally augmented. In fig. 169. a system is 
 represented in which three sheaves are inserted in the 
 movable block bearing the weight, the same number 
 being inserted in the fixed block.' The cord is carried 
 from the power first over the fixed sheave A, then 
 over the movable sheave B, then over the fixed 
 Fig. 169. sheave c, the movable sheave D, the fixed sheave E, 
 and the movable sheave r, and finally attached to 
 the block at G. 
 
 Now, since the cord throughout its whole length is stretched 
 with the same force, and since it is evident that at the part where 
 the power is applied, this force of tension must be equal to the 
 power, it follows that the six parts of the cord which support the 
 weight will each be stretched by a force equal to the power, and 
 that consequently the weight, when in equilibrium, must be equal 
 to six times the power. 
 
 This condition may also be inferred from the fact, that if the 
 power move the cord, its velocity will be six times that of the 
 weight ; for if six feet of the rope be drawn over the fixed pulley, 
 these six feet must be equally distributed between the six parts of 
 the rope which sustain the weight, and consequently each part 
 must be raised through one foot, which is therefore the height 
 through which the weight would be raised for every six feet 
 through which the power passes. 
 
 463. In general it may therefore be inferred, that, in a pulley 
 which consists of a single movable block containing one or more 
 sheaves, the weight, when in equilibrium, will be just as many 
 times the power as is represented by the number of cords, or the 
 number of parts of the cord, which sustain the weight. 
 
 464. In the form of movable block represented in fig. 170., 
 the cord, after passing successively over the sheaves, is finally 
 attached to the lower block. This, by increasing the parts of the 
 cords supporting the lower block by one, augments the efficiency 
 of the instrument without increasing the number of sheaves. 
 
PULLEYS. 
 
 203 
 
 465. Two of the most powerful forms of pulley, consisting of a 
 single movable block, are represented in figs. 171. and 172. 
 The combination represented in fig. 1 7 1 . is called Smeaton's pulley, 
 
 Fig. 170. 
 
 Fig. 171 
 
 Fig. 172. 
 
 having been invented by that celebrated engineer. The fixed and 
 movable block contain each ten sheaves, and the order in which 
 the rope is carried over them is represented in the figure by the 
 numbers I, 2, 3, 4, &c. The total number of parts of the cord 
 supporting the lower block is in this case twenty, and consequently 
 the power is to the weight as 20 to I. 
 
 The form of pulley represented in fig. 172. is called White's 
 pulley. 
 
 466. If instead of one movable block and a single rope, two or 
 more movable blocks with independent ropes be used, the power 
 of the pulley may be augmented on the same principle as in the 
 case of compound levers or compound wheel-work. 
 
 Different combinations of this kind are represented \nfigs. 173, 
 174. 175, 176. 
 
 The figures which are annexed to the ropes in each case repre- 
 sent the power they respectively exert. 
 
 The first rope in fig. 173. is stretched by the force of the power 
 only. The first movable block being supported by two parts of 
 this first rope will exert a force equal to double the power on the 
 second rope ; and the second movable block being supported by 
 two parts of this rope will exert a force upon the third rope equal 
 to four times the power; and in the same way it follows that the 
 third block will exert a force in supporting the weight equal to 
 
204 
 
 THEORY OF MACHINERY. 
 
 eight times the power. In such a system, the addition of each 
 movable block doubles the mechanical effect. 
 
 Fig. 173. 
 
 Fig. 174. 
 
 Fig. 175. 
 
 Fig. 176. 
 
 But without augmenting the number of movable blocks, but 
 only adding fixed blocks, the effects may be augmented in a three- 
 fold instead of a two-fold proportion, as represented in jig. 174., 
 where each successivemovable block is supported by three parts 
 of the same cord. In Jig. 175. the ends of the cords, instead of 
 being attached to fixed points, are attached to the weight. In this 
 case, the weight is supported by each of the several cords, these 
 cords being stretched by different forces. The first is stretched 
 with a force equal to the power, the second with a force equal to 
 double the power, and the third with a force equal to four times the 
 power, and so on. In such a system, the mechanical effect for the 
 same number of blocks is greater than in that represented in fi%. 
 173., where three blocks only support a weight four times the 
 power ; whereas in the system represented in fig. 175., three blocks 
 support seven times the power. The effect of this system may be 
 still further increased by attaching blocks to the weight, as repre- 
 sented in fig. 176., and carrying the ropes to the pulleys above. 
 The effect produced is indicated by the numbers fixed to the 
 cords. 
 
 467. From its portable form, cheapness of construction, and the 
 facility with which it may be applied in almost every situation, the 
 pulley is one of the most useful of the simple machines. The 
 mechanical advantage, however, which it appears in theory to pos- 
 sess, is considerably diminished in practice, owing .to the stiffness of 
 the cordage and the friction of the wheels and blocks. By these 
 means it is computed that in most cases so great a proportion as 
 two thirds of the power is lost. The pulley is much used in build- 
 ing when weights are to be elevated to great heights ; but its most 
 
INCLINED PLANE. 
 
 205 
 
 extensive application is found in the rigging of ships, where almost 
 every motion' is accomplished by its means. 
 
 In all these examples of pulleys, we have supposed the parts of 
 the rope sustaining the weight, and each of the movable pulleys, 
 to be parallel to each other. If they be subject to considerable 
 obliquity, the relative tensions of the different ropes must be esti- 
 mated according to the principle applied in 46 1 . 
 
 INCLINED PLANE. WEDGE AND SCREW. 
 
 468. Effect of an inclined surface. A hard surface pressed 
 against a weight or resistance in a direction at right angles to it, 
 would support it, and the whole amount of such weight or resist- 
 ance, would in this case press upon the surface. But if, instead of 
 being at right angles to it, it were placed in an oblique direction, 
 then the weight or resistance would be resolved by the parallelo- 
 gram of forces into two, one of which would act perpendicularly 
 to the plane and produce pressure upon it, and the other would be 
 parallel to the plane, and be free to produce motion. 
 
 Let A B )t /g% 177., be such a surface, and let w be a body pro- 
 ducing some resistance, or having a tendency to move in the direc- 
 tion B' A' oblique to B A. Let 
 the whole force with which w 
 would move if not supported by 
 the plane be expressed by B' A'. 
 Then, taking B' A? as the diagonal 
 of a parallelogram one of whose 
 sides is B' c' at right angles to 
 B A, and the other B D in the 
 direction of the plane, the force 
 B' A' will be equivalent to two 
 forces, one represented by B' c', 
 and the other by B' D. The for- 
 mer, being perpendicular to the 
 plane, will be resisted by its re- 
 action, and the other only will' 
 take effect. To support the 
 weight, therefore, in this case, 
 would require a force acting parallel to the plane, and opposite to 
 the force represented by B' D. If we take on the plane the length 
 B. A equal to B' A', and draw A c parallel to B' A', and B c perpen- 
 dicular to it, then the triangle ABC will be in all respects equal 
 and similar to A' B' c' ; in fact, it may be considered as the same 
 triangle, but in a different position. Since, therefore, A' B', B' c', 
 and A' c' represent respectively the whole force of the body w, its 
 pressure on the plane, and its tendency to move in the direction ot 
 
 Fig. 177. 
 
206 THEORY OF MACHINERY. 
 
 the plane, these three forces will be exactly represented in their 
 effects by the lines A B, u c, and A c. 
 
 469. We have here taken the general case, and supposed the 
 body w to exercise a force in any direction whatever ; but if we 
 apply the principle to the case of a heavy body resting upon a 
 plane inclined to the vertical direction, then the machine becomes 
 what is commonly called the inclined plane. A B is called the 
 length of the plane, A c its height, and B c its base. 
 
 From what has been just proved, then, it follows, that if a 
 weight be placed upon an inclined plane, the weight consisting of 
 as many pounds as there are inches in the length of the plane, the 
 pressure on the plane will consist of as many pounds as there 
 are inches in the base, agid the tendency to move down the plane 
 would be balanced by as many pounds as there are inches in the 
 height. 
 
 470. The apparatus represented in jig. 1 78. is intended to repre- 
 
 sent this experimentally. The weight 
 placed upon the plane is a roller so 
 formed as to move freely upon it. A 
 string is attached to it, which being 
 carried parallel to the plane is con- 
 ducted over a fixed pulley, and sup- 
 Fig. 178. ports a dish bearing a weight. On 
 comparing this weight, including the 
 
 weight of the dish, with the weight of the roller upon the plane, 
 and by varying the angle of elevation of the plane, we find that in 
 every case the weight necessary to produce equilibrium will be 
 expressed by the height of the plane, the entire weight of the 
 roller being expressed by its length. 
 
 It is evident from what has been just explained that the less the 
 elevation of the plane is, the less will be the power requisite to 
 sustain a given weight upon it, and the greater will be the pres- 
 sure upon it ; for the less the elevation of the plane is, the less will 
 be its height and the greater will be its base. 
 
 471. Inclined roads. Roads which are not level may be con- 
 sidered as inclined planes, and loads drawn upon them in carriages, 
 regarded in reference to the powers which impel them, are sub- 
 ject to all the conditions which have been established for inclined 
 planes. 
 
 The inclination of the road is estimated by the height cor 
 responding to some proposed length : thus we say, a road rises one 
 foot in twenty-five, or one foot in thirty ; meaning that if twenty- 
 five or thirty feet of the road be taken as the length of an inclined 
 plane, the corresponding height of such plane would be one foot ; 
 and if twenty* five or thirty feet be measured upon the road, the 
 
INCLINED PLANE. 
 
 207 
 
 difference of the level of the two extremities will be one foot. 
 According to this method of estimating the inclination of roads, 
 the power required to sustain a load upon them, friction apart, is 
 always proportional to this rate of elevation. If a road rise one 
 foot in twenty, then a power of one ton will be sufficient to sustain 
 twenty tons ; and so on. 
 
 472, Inclined planes on railways. When a power is em- 
 ployed in moving a load upon a road thus inclined, the action of the 
 power may also be regarded in another point of view. Let us sup- 
 pose a railway train weighing 200 tons moving up an inclined plane, 
 which rises at the rate of one in two hundred ; what mechanical 
 effect does the moving power produce in moving up 200 feet of 
 such a plane ? First, it acts against the friction, the atmospheric 
 and other resistances to which it would be exposed if the plane 
 had been level. Secondly, it is employed in raising the entire 
 weight of the train through the elevation which corresponds to 
 200 feet in length, that is, through one perpendicular foot. 
 
 The mechanical effect, therefore, is precisely the same as if the 
 load of 200 tons had been first moved along a level plane 200 feet 
 long, and then elevated up a step one foot high ; but instead of 
 being called upon to make this great exertion of raising 200 tons 
 directly through one perpendicular foot, the moving power is 
 enabled gradually to accomplish the same object by a longer con- 
 tinuance of a more feeble exertion of force, such exertion being 
 spread over 200 feet instead of being condensed into a single 
 foot. 
 
 473. In all that precedes, we have assumed that the power acts 
 parallel to the plane; in some cases, however, it acts obliquely 
 to it. 
 
 Let w p, fig. 1 79., be the direction of the power. Taking that 
 
 Fig. 179. 
 
 Fig. 180. 
 
 as the diagonal of a parallelogram, it will be equivalent to w D and 
 w E, the former perpendicular, and the latter parallel to the plane, 
 w D will have the effect of diminishing the pressure on the plane, 
 and w E will be efficient in drawing the weight up the plane. In 
 
208 
 
 THEORY OF MACHINERY. 
 
 some cases the direction of the power is below the plane, as in 
 fig. 1 80. In this case, as before, the power w p is resolved into 
 two forces, WE parallel to the plane, and w D perpendicular to it. 
 The latter augments the pressure of the weight on the plane, and 
 the former is efficient in drawing it up the plane. 
 
 474. The method of drawing up or letting down barrels from or 
 into a cellar, represented in fig. 181., and that of drawing them 
 
 Fig. 181. 
 
 upon a dray, shown in fig. 182., are cases of the inclined plane 
 which will be easily understood. 
 
 Fig. i8z. 
 
 475. Double inclined plane. It sometimes happens that a 
 weight upon one inclined plane is raised or supported by another 
 weight upon another inclined plane. Thus, if A B and A B', 
 fig. 183., be two inclined planes, forming an angle at A, and ww 
 
INCLINED PLANES. 
 
 209 
 
 Fig. 18}. 
 
 be two weights placed upon these 
 planes, and connected by a cord 
 passing over a pulley at A, the one 
 weight will either sustain the other, 
 or one will descend, drawing the 
 other up. To determine the cir- 
 cumstances under which these effects 
 will ensue, draw the lines w D and 
 w' D' in the vertical direction, and take upon them as many inches 
 as there are ounces in the weights respectively, w D and w' D' 
 being the lengths thus taken, and therefore representing the 
 weights, the lines w E and w' E' will represent the effects of these 
 weights respectively down the planes. If w E and w' E' be equal, 
 the weights will sustain each other without motion ; but if w E be 
 greater than w' E', the weight w will descend, drawing the weight 
 w' up ; and if w' E' be greater than w E, the weight w' will descend, 
 drawing the weight w up. In every case w F and w' F' will repre- 
 sent the pressures upon the planes respectively. 
 
 476. Self-acting: planes. It is not necessary, for the effect 
 just described, that the inclined planes should, as represented in 
 the figure, form an angle with each other. They may be parallel, 
 or in any other position, the rope being carried over a sufficient 
 number of wheels, placed so as to give it the necessary deflection. 
 This method of moving loads is frequently applied in great public 
 works where railroads are used. Loaded waggons descend one 
 inclined plane, while other waggons, either empty or loaded, so as 
 to permit the descent of those with which they are connected, are 
 drawn up the other. 
 
 477. Tne wedge. When the weight is not moved upon the 
 plane, but is stationary, the plane being itself moved under the 
 weight, the machine is called a wedge. 
 
 Let D E, fig. 1 84., be a heavy beam, secured in a vertical posi- 
 
 Fig. 184. 
 
 Fig. 185. 
 
 tion between guides, F G and H i, so that it is free to move upwards 
 
 p 
 
210 THEORY OF MACHINERY. 
 
 and downwards, but not laterally. Let A B c be an inclined plane, 
 the extremity of which is placed beneath the end of the beam. A 
 force applied to the back of this plane A c, in the direction c B, 
 will urge the plane under the beam so as to raise the beam to the 
 position represented in fig. 185. Thus, while the inclined plane 
 is moved through the distance c B, the beam is raised through the 
 height c A. 
 
 It follows, therefore, that in the case of the wedge the velocity 
 of the resistance is to the velocity of the weight as 
 the base of the inclined plane, which forms the 
 wedge, is to its height. 
 
 478. Wedges, however, are more generally formed 
 of two inclined planes, connected base to base, as 
 represented in fig. 1 86. In this case, the back of 
 the wedge is the sum of the heights of the two 
 inclined planes, and the length of the wedge is 
 their common base. The force, therefore, which 
 drives the wedge is to the resistance with which it 
 Fig. 186. equilibrates, as half the back of the wedge is to 
 its length. 
 
 479. This theory of the wedge is not applicable in practice 
 with any degree of accuracy. This is owing chiefly to the enor- 
 mous disproportion which friction in these machines bears to the 
 power ; but independently of this there is another difficulty in the 
 theory of this machine. 
 
 480. The power commonly used in the case of a wedge is not 
 pressure, but percussion. The force of a blow is of a nature so 
 different from continued force, such as the pressure of weights, 
 that it admits of no numerical comparison with the resistance 
 offered by cohesion, to overcome which it is generally applied. 
 We cannot properly state the proportion which a blow bears 
 to a weight. The wedge is almost invariably urged by per- 
 cussion, while the resistances which it has to overcome are as 
 constantly forces of another kind. Although, however, no exact 
 numerical computation can be made, yet it may be stated in 
 general that the wedge is more and more powerful as the angle is 
 more acute. 
 
 48 1 . The cases in which wedges are most generally used in the 
 arts and manufactures, are those in which an intense force is re- 
 quired to be exerted through a very small space. This instru- 
 ment is therefore used for splitting masses of timber or stone, for 
 raising vessels in docks, when they are about to be launched, by 
 being driven under their keels, in presses where the juice of seeds, 
 fruits, or other substances are required to be extracted, as, for 
 example, in the oil mill, in which the seeds from which the oil is 
 
WEDGE. , 211 
 
 extracted are introduced into hair bags, which being placed be- 
 tween planes of hard wood are pressed by wedges. The pressure 
 exerted by the wedges is so intense that the dry seeds are con- 
 verted into solid masses as hard and compact as the most dense 
 woods. Wedges have been used occasionally to restore to the 
 perpendicular, edifices which have inclined owing to the sinking of 
 their foundations. 
 
 482. Practical examples. All cutting and piercing instru- 
 ments, such as knives, razors, shears, scissors, chisels, nails, pins, 
 needles, &c., are wedges. The angle of the wedge in all these cases 
 is more or less acute, according to the purpose to which it is ap- 
 plied. Chisels intended to cut wood have their edge at an angle 
 of about 30 ; for cutting iron from 50 to 60, and for brass about 
 80 to 90. In general, tools which are urged by pressure admit 
 of being sharper than those which are driven by percussion. The 
 softer or more yielding the substance to be olivided is, the more 
 acute the wedge may be constructed. 
 
 483. Utility of friction. In many cases the efficiency of the 
 wedge depends on that which is entirely omitted in its theory, 
 viz., the friction which arises between its surface and the substance 
 which it divides. This is the case when pins, bolts, or nails are 
 used for binding the parts of structures together, in which case, 
 were it not for the friction, they would recoil from their places and 
 fail to produce the desired effect. Even when the wedge is used 
 as a mechanical engine the presence of friction is absolutely indis- 
 pensable to its practical utility. 
 
 The power, as has already been stated, generally acts by succes- 
 sive blows, and is therefore subject to constant intermission, and 
 but for the friction the wedge would recoil between the intervals 
 of the blows with as much force as it had been driven forward. 
 Thus the object of the labour would be continually frustrated. 
 The friction in this case is of the same use as a ratchet-wheel, but 
 is much more necessary, as the power applied to the wedge is 
 much more liable to intermission than in the cases where ratchet- 
 wheels are generally used. 
 
 484. The screw. In ascending a steep hill it has been the 
 practice of road engineers, instead of making an inclined plane 
 directly from the base to the summit, to carry the road round the 
 hill, gradually rising as it proceeds. If we desire to ascend with 
 ease to the top of a high column, we could do so if a path or ledge 
 were formed on the outer surface, gradually winding round and 
 round the column from the bottom to the top. Such a path would 
 be, in fact, an inclined plane carried round the column. But it 
 will be evident that such an arrangement would constitute a screw. 
 
 r 2. 
 
212 
 
 THEORY OF MACHINERY. 
 
 This will be rendered still more apparent by the following con- 
 trivance. 
 
 Fig. 187. 
 
 Let AB, jig. 187., be a cylindrical roller, and let CDE be an 
 inclined plane cut in paper, the height of which c D is equal to the 
 length of the roller. Let the edge c D be pasted on the roller, and 
 then -let the roller be turned so that the paper shall be wrapped 
 round it. When it makes one revolution of the roller, the portion 
 of the edge c G will have made one spiral coil ; the next revolution 
 will make an equal spiral coil, and so on until all the paper has 
 been rolled upon the roller, when the edge of the paper so coiled 
 will show a regular spiral line round the roller, as represented in 
 fig. 1 88. Taking c H G,fig. 187., as the inclined plane thus rolled 
 round the roller, it is evident that c H is its height, and H G its 
 base. But c H is the distance between two successive coils of the 
 spiral, and H G is the circumference of the roller. The coils of the 
 spiral are called the threads of the screw, and the distance c H 
 between the successive coils is called the distance between the 
 threads. 
 
 485^111 the application of the screw, the weight of resistance is 
 not, as '4n, the inclined plane and wedge, placed upon the surface 
 of the plane or thread. The power is usually trans- 
 mitted by causing the screw to move in a concave 
 cylinder, on the interior surface of which a spiral 
 cavity is cut, corresponding exactly to the thread of 
 the screw, and ia which the thread will move by 
 turning round the screw 'continually in the same 
 .direction. This hollow cylinder is usually called the 
 nut or concave screw. The screw surrounded by its 
 spiral thread is represented in fig. 1 89. 
 
 4.86. There are several ways in which the power is 
 transmitted to the resistance by means of a screw ; 
 but by whatever means it may be so transmitted, it is evident that 
 the screw will move the resistance in a single revolution through 
 a space equal to the distance between two contiguous threads. 
 The comparative velocities, therefore, of the power and weight 
 will always be found in this class of simple machines by com- 
 paring this space described by the power, in imparting one revo- 
 
 Fig.-i8 9 . 
 
SCREW. 2 1 3 
 
 lution to the screw with the distance between two contiguous 
 
 threads. 
 
 487. The most common manner of urging the screw is by a 
 
 lever attached to its head, as represented in Jig. 190. at E r. Sup- 
 
 posing the power to be applied 
 at F, it will in producing one re- 
 volution of the screw, and there- 
 fore in moving the resistance 
 through a space equal to the dis- 
 tance between two contiguous 
 threads, make one complete revo- 
 lution in a circle whose radius is 
 the length of the lever on which 
 
 Fig . 190 . it acts. The velocity, therefore, 
 
 of the power will be to the velo- 
 
 city of the weight as the circumference of the circle described by 
 the power is to the distance between two contiguous threads ; and 
 consequently, the condition of equilibrium between the power and 
 weight will be this, that the power is to the weight as the distance 
 between the contiguous threads is to the circumference described 
 by the power. 
 
 488. The great mechanical force exerted by the screw will 
 hence be easily understood. There is no limit to the smallness of 
 the distance between the threads, except the strength which is 
 necessary to be given to them, and there is no limit to the magni- 
 tude of the circumference to be described by the power, except 
 the necessary facility of moving in it. We can conceive the 
 power acting by a lever, and therefore moving through a great 
 circumference while the screw moves through a comparatively 
 minute space ; and consequently, in such case, the power will be 
 so much less than the resistance, as the distance between the 
 threads is less than the circumference described by the power. 
 
 489. The manner of acting upon the resistance by means of the 
 screw is very various. Sometimes the nut is fixed and the screw 
 movable; sometimes the screw is fixed and the nut movable; 
 sometimes the nut, though incapable of revolving, can be moved 
 progressively ; and sometimes the screw is incapable of revolving, 
 but is moved progressively. These conditions admit of various 
 combinations which are severally adopted in practice. In Jig. 190., 
 the nut A B being supposed to be fixed, if the lever F be turned, 
 the end D of the screw will descend or ascend, according to the 
 direction in which E F is turned, and will act upon the resistance 
 accordingly. If the screw be fixed, the nut may be moved upon 
 it, either by turning the nut or the screw. In either case the nut 
 will ascend or descend, according to the direction of the motion. 
 
 p 3 
 
2I 4 
 
 THEORY OF MACHINERY. 
 
 In each revolution it will move through a space equal to the dis- 
 tance between two contiguous threads. 
 
 If we suppose the nut A B, fig. 1 90., to be incapable of ascending 
 or descending, but to be capable of revolving, then, by turning it 
 round, the screw which plays in it will ascend or descend through 
 a space equal to the distance between two contiguous threads for 
 every revolution made by the nut. 
 
 On the other hand, the apparatus may be so arranged that the 
 screw, though capable of revolving, is incapable of a progressive 
 motion, and the nut, though capable of a progressive motion, is 
 incapable of revolving. In this case, when the screw is made to 
 revolve, the nut in which it plays will be moved upwards or 
 downwards, through a space equal to the distance between two 
 threads, by the revolution of the screw. 
 
 The screw is generally used in cases where severe pressure is to 
 be exercised through small "spaces; it is, therefore, the agent in 
 most presses. 
 
 InJ%-. 191. the nut is fixed, and by turning the lever which 
 passes through the head of the screw a pressure is exercised upon 
 any substance placed upon the plate immediately under the end of 
 the screw. In Jig. 192. the screw is incapable of revolving, but 
 
 Fig. 191. 
 
 Fig. I9Z. 
 
 is capable of advancing in the direction of its length. On the 
 other hand, the nut is capable of revolving, but does not advance 
 in the direction of the screw. When the nut is turned by means 
 of the lever inserted in it, the screw advances in the direction of 
 its length, and urges the board which is attached to it downwards, 
 so as to press any substance placed between it and the fixed board 
 below. 
 
 490. Examples. In cases where liquids or juices are to be 
 
SCREW. 
 
 215 
 
 expressed from solid bodies, the screw is the agent generally em- 
 ployed. It is also used in coining, where the impression of a die is 
 to be made upon a piece of metal, and in the same way in pro- 
 ducing the impression of a seal upon wax or other substance 
 adapted to receive it. When soft and light materials, such as 
 cotton, are to be reduced to a convenient bulk for transportation, 
 the screw is used to compress them, and they are thus reduced 
 into hard dense masses. In printing, the paper is sometimes 
 urged by a severe and sudden pressure upon the types by means 
 of a screw. 
 
 49 1 . A screw may be cut upon a cylinder by placing the cylinder 
 in a turning-lathe, and giving it a rotatory motion upon its axis. 
 The cutting point is then presented to the cylinder and moved in 
 the direction of its length at such a rate as to be carried through 
 the distance between the intended threads while the cylinder re- 
 volves once. The relative motions of the cutting point and the 
 cylinder being preserved with perfect uniformity, the thread will 
 be cut from one end to the other. The shape of the threads may 
 be either square, as \nfig. 189., or triangular, as in Jig. 191. 
 
 492. If the lever by which the power acts on the screw were 
 capable of indefinite increase, or the thread of indefinite fineness, 
 there would be no limit to the mechanical effect of the instrument ; 
 but to both of them there are practical limits. The lever cannot 
 be increased so as to render the operation of the power unwieldy 
 and impracticable, and the thread cannot be diminished beyond 
 that limit which will give it sufficient strength ; and the cases in 
 which the greatest mechanical efficacy is needed, are precisely 
 those in which the thread requir es to be strongest. 
 
 493. Hunter's screw. To obtain an indefinite augmen- 
 
 tation of the power of the screw, 
 without diminishing the strength of 
 the thread, Mr. Hunter proposed an 
 arrangement which is known by his 
 name, as the Hunterian screw. This 
 (represented in Jig. 193.) consists in 
 the use of two screws, the threads of 
 which may have any strength and mag- 
 nitude, but which have a very small 
 difference of breadth. 
 
 While the working point is urged 
 forward by that which has the greater 
 thread, it is drawn back by that which 
 has the less ; so that during each revo- 
 lution, the screw, instead of being ad- 
 vanced through a space equal to the magnitude of either of the 
 
 Fig. 193. 
 
216 
 
 THEORY OF MACHINERY. 
 
 threads, moves through a space equal to their difference. The 
 mechanical power of such a machine will be the same as that of a 
 single screw having a thread whose magnitude is equal to the dif- 
 ference of the magnitudes of the two threads just mentioned. 
 
 Thus, without inconveniently increasing the sweep of the power 
 on the one hand, or on the other diminishing the thread until the 
 necessary strength is lost, the machine will acquire an efficacy 
 limited by nothing but the smallness of the difference between the 
 two threads. 
 
 494. Micrometer screw. The slow motion which may 
 be imparted to the end of a fine screw by a considerable 
 motion of the power, renders it an instrument peculiarly well 
 adapted to the measurement of very minute motions and spaces, 
 the magnitude of which could scarcely be ascertained by any other 
 means. To explain the manner in which it is applied : suppose a 
 screw to be so cut as to have fifty threads in an inch, each revolu- 
 tion of the screw will advance its point through the fiftieth part of 
 an inch. Now, suppose the head of the screw to be a circle whose 
 diameter is an inch, the circumference of the head will be some- 
 thing more than three inches : this may be easily divided into a 
 hundred equal parts, distinctly visible. If a fixed index be pre- 
 sented to this graduated circumference, 
 the hundredth part of a revolution of the 
 screw may be observed by noting the pas- 
 sage of one division of the head under the 
 index. Since one entire revolution of the 
 head moves the point through the fiftieth 
 of an inch, one division will correspond to 
 the five-thousandth of an inch. In order 
 to observe the motion of the point of the 
 screw in this case, a fine wire is attached to 
 it, which is carried across the field of view 
 of a powerful microscope, by which the 
 motion "is so magnified as to be distinctly 
 perceptible. 
 
 495. Endless screw. When the 
 thread of a screw acts in the teeth of a 
 wheel, the screw being fixed so that it can 
 revolve without advancing, it is called an 
 endless screw. It has the effect of giving 
 motion to the wheel in the same manner 
 as does a pinion, one tooth of the wheel 
 being driven forward by each revolution 
 of the screw. Such an arrangement is 
 applied, for example, to tighten the strings of double bases,^. 194. 
 
 Fig. 194. 
 
PENDULUM. 
 
 217 
 
 CHAP. III. 
 
 THE PENDULUM, THE BALANCE WHEEL, AND THE FLY WHEEL. 
 
 496. Or the class of mechanical contrivances whose purpose is to 
 render motion uniform, the principal are the pendulum, the 
 balance wheel, and the fly. 
 
 497. Simple pendulum. The pendulum consists of a heavy mass 
 attached to a rod, the upper extremity of which rests upon a point of 
 support in such a manner as to have as little friction as possible. 
 Such an instrument will remain at rest when its centre of gravity is 
 in the vertical line immediately under the point of suspension or sup- 
 port. But if the centre of gravity be drawn from this position on 
 either side, and then disengaged, the instrument will swing horizon- 
 tally from the one side to the other of the position in which it would 
 remain at rest, the centre of gravity describing alternately a cir- 
 cular arc on the one side or the other of its position of rest. If 
 there were neither friction nor atmospheric resistance, this motion 
 of vibration or oscillation on either side of the position of equi- 
 librium would continue for ever; but, in consequence of the 
 combined effects of these resistances, the distances to which the 
 pendulum swings on the one side and on the other are continually 
 diminished, until, after the lapse of an interval more or less pro- 
 tracted, it comes to rest. 
 
 Suppose a small ball of lead suspended by a fine silken string, 
 the .length of which is incomparably greater than the diameter 
 of the leaden ball. Such an arrangement is called the simple 
 pendulum. 
 
 Let s,^. 195., be the point of suspension ; let SB be the fine 
 silken thread by which the ball B is suspended, and the weight of 
 which, in the present case, is neglected. 
 Let B be the position of the ball when 
 in the vertical under the point of sus- 
 pension s. In that position the ball 
 would remain at rest ; but if we suppose 
 the ball drawn aside to the position -A, it 
 will, if disengaged, fall down the arc A B, 
 of which the centre is s, and the radius 
 the length of the string. Arriving at B, 
 it will have acquired a certain velocity, 
 which, in virtue of its inertia, it will 
 have a tendency to retain, and with this 
 velocity it will commence to move 
 through the arc B A'. Supposing neither 
 
2 1 8 THEORY OF MACHINERY. 
 
 the resistance of the atmosphere nor friction to act, the ball will 
 rise through an arc B A' equal to B A ; but it will lose the velocity 
 which it had acquired at B, for it is evident that it will take the 
 same space and the same time to destroy the velocity which has 
 been acquired, as to produce it. Thus, the velocity at B, being 
 acquired in falling through the arc A B, will be destroyed in rising 
 through the equal arc B A?. 
 
 Having arrived at A', the ball, being brought to rest, will again 
 fall from A' to B, and at B will have again acquired the same velo- 
 city which it had obtained in falling from A to B, but in the 
 contrary direction ; and in the same manner it may be explained 
 that this velocity will carry it from B to A. Having arrived at A, 
 the ball, being again brought to rest, will fall once more from A 
 to B, and so the motion will be continued alternately between 
 A and A'. 
 
 The motion of the pendulum from A to A 7 , or from A' to A, is 
 called an oscillation, and its motion between either of those points 
 and B is called a semi-oscillation, the motion from B to A or from 
 B to A' being called the ascending semi-oscillation, and the motion 
 from A or A? to B, the descending semi-oscillation. 
 
 The time which elapses during the motion of the ball between 
 A and A! is called the time of one oscillation. 
 
 It is evident, from what has been stated, that the time of moving 
 from either of the extremities A A' of the arc of oscillation to the 
 point B, is half the time of an oscillation. If, instead of falling 
 from the point A, the ball had fallen from the point c, intermediate 
 between A and B, it would have then oscillated between c and c'; 
 two points equally distant from B, and the arc of oscillation would 
 have been c c', more limited than A A'. But in commencing its 
 motion from c, the declivity of the arc down which it falls towards 
 B would be evidently less than the declivity at A ; consequently, 
 the force which would accelerate it, commencing its motion at c, 
 would be less than that which would accelerate it, commencing its 
 motion at A. The ball, therefore, commencing its motion at A, 
 would be more rapidly accelerated than when it commences its 
 motion at c. 
 
 The result of this is, that, although the arc A B may be twice 
 as long as the arc c B, the time which the ball takes to fall from 
 A to B will not be sensibly different from the time it takes to 
 fall from c to B, provided that the arc of oscillation A B A' is not 
 considerable. 
 
 It was at first supposed, as we have just stated, that, whether 
 the oscillations were longer or shorter, the times would be abso- 
 lutely the same. Accurately speaking, however, this is not the 
 case ; but if the total extent of the oscillation A A' do uot exceed 
 
PENDULUM. 219 
 
 5 or 6, the time of oscillation in it may be considered, prac- 
 tically, the same as in the lesser arcs. 
 
 498. This important principle may be easily experimentally 
 verified. Let two small leaden balls be suspended from the same 
 point of support, but one being in advance of the other, so that 
 in oscillating the two balls shall not strike each other. This 
 being done, let one of the balls be drawn from its point of re^t 
 through an angle less than 3, and let it be disengaged. It will 
 oscillate as described above. Let the other ball be now drawn 
 from its point of rest through a much less angle, and let it be so 
 disengaged that it shall commence its oscillation at the same 
 moment with the commencement of one of the oscillations of 
 the other ball. 
 
 Let it, in short, be so managed, that when the one ball is at A, 
 the other shall be at c ; and that both shall commence their de- 
 scending motion towards B at the same moment. It will be then 
 found that their oscillations will be synchronous for a considerable 
 length of time, that is to say, the balls will arrive at A' and c', 
 respectively, at the same instant ; and returning, will simultane- 
 ously arrive at A and c respectively. 
 
 If, in this case, the oscillation of the ball A were made through 
 an arc, even as great as 10, that is to say, 5 on each side of the 
 vertical, the oscillation of the ball c being made through an arc of 
 2, it would be found that 10001 oscillations of the latter would be 
 equal to I oooo oscillations of the former, so that the actual differ- 
 ence between their times of oscillation would not exceed the ten 
 thousandth part of such time. 
 
 499. It might be expected that the time of oscillation of dif- 
 ferent pendulums would depend, more or less, upon the weight of 
 the matter composing them, and that a heavy body would oscillate 
 more rapidly than a lighter one. Both theory and experience, 
 however, prove the result tr be otherwise. The force of gravity 
 which causes the pendulum to oscillate acts separately on all the 
 particles composing its mass ; and if the mass be doubled, the effect 
 of this force upon it is also doubled ; and, in short, in whatever 
 proportion the mass of the pendulum be increased or diminished, 
 the action of the force of gravity upon it will be increased or 
 diminished in exactly the same proportion, and consequently the 
 velocity imparted by gravity to the pendulous mass at each instant 
 will be the same. 
 
 It is easy to verify this by experiment. Let different balls of 
 small magnitude, of metal, ivory, and other materials, be suspended 
 by light silken strings of the same length, and made to oscillate ; 
 ~1heir oscillations will be found to be equal. 
 
 500. If pendulums of different lengths have similar arcs of oscilla- 
 
22O 
 
 THEORY OF MACHINERY. 
 
 tion, the times of oscillation 
 of those which are shorter 
 will be less than the times 
 of oscillation of those which 
 are longer. Let a, &, c, c?, 
 and e, fig. 196., be five 
 small leaden balls, sus- 
 pended by light silken 
 strings to the point of sus- 
 pension s, and let all of 
 those be supposed to form 
 pendulums, having the 
 same angle of oscillation. 
 The arc of oscillation of 
 the ball a will be a a"", 
 that of b will be W", thai 
 of c, c c"", and so on. In 
 commencing to fall from 
 the points a, 5, c, e?, e, to- 
 wards the vertical line, these five balls are accelerated equally, 
 inasmuch as the circular arcs down which they fall are all equally 
 inclined at this point to the vertical line. The same will be true 
 if we take them at any corresponding points, such as a', #', c', e '. 
 It may therefore be concluded that throughout the entire range 
 of oscillation of each of these five pendulums, they will be impelled 
 by equal accelerating forces. 
 
 Now it has been shown that when bodies are impelled by the 
 same or equal accelerating forces, the spaces through which they 
 move are proportional to the squares of the 
 times of their motion; therefore it follows 
 that the lengths of these arcs of oscillation are 
 proportional to the squares of the times. But 
 the lengths of these arcs are evidently in the 
 same proportion as the lengths of the pen- 
 dulums ; that is to say, the arc a a "" is to 
 b b"" as s a is to s Z>, and the arc b b'" is to 
 c c"" as s b is to s c, and so on. 
 
 It follows, therefore, that the squares of the 
 times of oscillation of pendulums are as their 
 lengths, or, what is the same, the times of os- 
 cillation are as the square roots of their lengths. 
 This principle is easily verified experimentally. 
 501. Experimental illustration. Let 
 three small leaden balls be suspended vertically under each other 
 by means of loops of silken threads, as represented in fig. 197., 
 
PENDULUM. 221 
 
 and in such a manner that they can all oscillate in the same 
 plane at right angles to the plane of the diagram, the suspending 
 loops not interfering with each other. 
 
 Let the loops be so adjusted that the distance of the ball I 
 below the line M N shall be I foot, the distance of the ball 4, 
 4 feet, and the distance of the ball 9, 9 feet. 
 
 Let the ball 9 be put in a state of oscillation through small 
 arcs, and let the ball 4 be then drawn from its vertical position, 
 and disengaged so as to commence one of its oscillations with 
 an oscillation of the ball 9 ; and in the same manner let the 
 ball I be started simultaneously with one of the oscillations of the 
 ball 9. 
 
 It will be found that two oscillations of the one-foot pendulum 
 are made in exactly the same time as a single oscillation of .the 
 four-foot pendulum; consequently, the time of each oscillation of 
 the latter will be double that of the former, while its length is 
 fourfold that of the former. 
 
 In the same manner, while the one-foot pendulum makes three 
 oscillations, the nine-foot pendulum will make one, and, conse- 
 quently, the time of oscillation of the latter will be three times 
 that of the former, while its length is nine times that of the 
 former. 
 
 502. By this principle, the length of a pendulum which would 
 oscillate in any proposed time, or the time of oscillation of a 
 pendulum of any proposed length, can be ascertained, provided 
 we know the length of a pendulum which oscillates in any given 
 time. Thus, suppose L to be the length of a pendulum which 
 oscillates in the time T. Let it be required to determine the 
 length of a pendulum i/, which would oscillate in any -other time 
 T'. We shall have the following proportion : 
 
 L : i/ :: T 2 : T ' 3 . 
 
 From this proportion, if L and T be both given, we can find the 
 time T' of oscillation of the other pendulum if i/ be given ; or we 
 can find the length i/ if T' be given. 
 
 In the first case we have 
 
 in the second we have 
 
 L X 
 
 -- . 
 
 3 
 
 503. Since the force which produces the oscillation of a pen- 
 dulum is the accelerating force of gravity urging the pendulous 
 body alternately from the extremities of the arc of oscillation to 
 
222 THEORY OF MACHINERY. 
 
 the middle point of that arc, it is evident that if this force were 
 increased in its intensity, the velocity with which the pendulous 
 body would be precipitated to its lowest position would be 
 increased, and consequently the time of oscillation diminished ; 
 and if, on the other hand, the impelling force of gravity were 
 diminished, the force urging the pendulous body being enfeebled, 
 it would be moved with a diminished velocity, and consequently 
 the time of oscillation would be increased. It follows, therefore, 
 that the same pendulum will oscillate more slowly or more rapidly, 
 according as the force of gravity which acts upon it is diminished 
 or increased. 
 
 504. But it is not enough to state that a variation in the force 
 of gravity will change the time of oscillation of the pendulum. It 
 is required to ascertain in what proportion it will produce this 
 change ; that is to say, if the force of gravity acting on the pen- 
 dulum be augmented in any given ratio, in what corresponding 
 ratio will the time of oscillation of such pendulum be diminished. 
 It is proved in the theory of accelerating forces that under such 
 circumstances, the squares of the times of oscillation will vary in 
 the inverse proportion of the force ; that is to say, in whatever 
 ratio the force of gravity be augmented, the squares of the times 
 of oscillation of the pendulums will be diminished. 
 
 505. Pendulum indicates variation of gravity in different 
 latitudes. But as the squares of the times of oscillation are 
 proportional to the lengths of the pendulums, it follows from this 
 that the lengths of the pendulums which oscillate in the same time 
 under the influence of different accelerating forces will be propor- 
 tional to these forces ; and that consequently, if in any two places 
 it be found that the pendulums which oscillate in the same time 
 have different lengths, it must be inferred that the forces of 
 gravity in these two places are in the exact proportion of these 
 lengths. 
 
 It is in virtue of this principle that the pendulum supplies means 
 of determining the variation of the forces of gravity upon different 
 parts of the earth's surface. 
 
 506. Compound pendulum. We have hitherto supposed that 
 the pendulous body is a heavy mass of indefinitely small magnitude, 
 suspended by a wire or string having no weight. These are condi- 
 tions which cannot be fulfilled in practice. Every real pendulous 
 body has a definite magnitude, its component parts being at differ- 
 ent distances from the point of suspension ; the rod which sustains 
 it has considerable weight, and all the points of this rod, as well as 
 those of the pendulous mass itself, are at different distances from 
 the point of suspension. In estimating, therefore, the effect of 
 pendulums, it is necessary to take into account this circumstance. 
 
PENDULUM. 
 
 223 
 
 Let us suppose a, J, c, d, e,y, g {fig- 198.) to be as many small 
 heavy balls connected by independent strings, the weight of which 
 may be neglected, with a point of suspension s, and let these seven 
 balls be supposed to vibrate between the positions s M and s M'. 
 Now, if these balls were totally independent of each other, and 
 
 Fig. 198. 
 
 connected with the point of suspension by independent strings, 
 they would all vibrate in different times, those which are nearer 
 the point s vibrating more rapidly than those which are more 
 distant from it. If, therefore, they be all disengaged at the same 
 moment from the line s M, those which are nearest to s will get the 
 start of those which are more distant, and at any intermediate 
 position between the extremes of their vibration they will assume 
 the positions a', V, c', d', e', /', g'. That which is nearest to the 
 point s, and which is the shortest pendulum, will be foremost, 
 since it has the most rapid vibration. The next in length, &, will 
 follow it, and so on ; the most remote from s, being the longest 
 pendulum, will be the last in order. 
 Now if, instead of supposing these seven balls to be suspended 
 
22 ^ THEORY OF MACHINERY. 
 
 by independent strings, we imagine them to be fixed upon the 
 same wire, so as to be rendered incapable of having any inde- 
 pendent motion, and compelled to keep in the same straight line, 
 then it is evident that while the whole series vibrates with a 
 common motion, those which are nearest to the point of suspension 
 will have a tendency to accelerate the motion of those which are 
 more distant, while those which are more distant will have a ten- 
 dency to retard the motion of those which are nearer 
 
 These effects will produce a mutual compensation ; b and c will 
 vibrate slower than they would if they were moving freely, while 
 e and f will evidently move more rapidly than if they were moving 
 freely. Among the series, there will be found a certain point 
 which will separate those which are moving slower than their 
 natural rate from those which are moving faster than their natural 
 rate ; and a ball placed at this point would vibrate exactly as it 
 would do if no other balls were placed either above or below it. 
 Such a ball would, as it were, be the centre which would divide 
 those which are accelerated from those which are retarded. 
 
 507. Centre of oscillation. Such a point has been denomi- 
 nated the centre of oscillation. It is evident, that a pendulous mass, 
 of magnitude more or less considerable, will vibrate in the same time 
 as it would do if the entire mass were concentrated at its centre 
 of oscillation, and formed there a material point of insensible 
 magnitude. By the length of a pendulum, no matter what be its 
 form, is always to be understood the distance of its centre of 
 oscillation from its point of suspension. 
 
 508. The centre of oscillation has the following remarkable and 
 important quality, which is established by the higher mathematics, 
 and verified by experiment : If a pendulum be inverted and 
 suspended by its centre of oscillation, its former point of suspen- 
 sion will become its new centre of oscillation, and the time of 
 vibration will remain the same as before. This property is usually 
 expressed by stating that the " centres of suspension and oscillation 
 are interchangeable." 
 
 This property can be verified by experiment. If the centre 
 of oscillation of anjj pendulous body, ascertained by mathema- 
 tical calculation, be taken as a point of suspension, it will be 
 found that the time of oscillation of the pendulum will be the 
 same as it was with the first point of suspension. Since the length 
 of a pendulum, and, therefore, the time of its oscillation in a given 
 place and subject to a given intensity of the force of gravity, 
 depends upon the distance between its point of suspension and its 
 centre of oscillation, it is evident that any variation which may 
 take place in this distance will cause a corresponding change in 
 the time of ostillation ; and if the pendulum be applied to a chro- 
 
PENDULUM. 
 
 22C 
 
 nometer, it will cause a variation in the rate of that instrument, 
 and a corresponding error in its indications. 
 
 509. Now, it is found in practice that all pendulums are subject 
 to a change, more or less, in their form and magnitude, in conse- 
 quence of the change of temperature of the atmosphere to which 
 they are exposed. With this change they expand and contract, 
 and with every expansion and contraction the distance between 
 their centres of oscillation and suspension will be varied, unless 
 expedients be adopted to counteract such an effect. Although the 
 variations produced by these causes are not sufficiently great to 
 render it necessary to provide a correction for them in common 
 time-pieces, yet, in cases where extreme accuracy is required, 
 expedients have been adopted to prevent the consequent error. 
 
 510. These expedients are called compensation pendulums. 
 The principle upon which all these depend is the combination 
 
 of two substances in the structure of the pendulum, which expand 
 in unequal degrees for the same change of temperature ; and they 
 are so arranged, that while the expansion of the one increases the 
 distance of the centre of oscillation from the point of suspension, 
 the expansion of the other has the contrary effect, and the dimen- 
 sions of these two substances are so adjusted that the increase of 
 distance produced by the one shall be exactly equal to the dimi- 
 nution of distance produced by the other ; so that the result is that 
 the centre of oscillation remains at the same distance from the 
 point of suspension, and therefore the time of oscillation of the 
 pendulum remains unaltered. 
 
 511. The first, the most important, and the most universal use 
 of the pendulum, is as a measure of time. The uniformity of the 
 rate of its vibration is the property which renders it so eminently 
 qualified for this purpose. A pendulum vibrating alone, inde- 
 pendently of any mechanism, would measure the time which 
 elapses during its motion. It would be only necessary for an 
 observer to sit by it and count the number of its oscillations. If 
 the time of one oscillation were previously known, then the number 
 of oscillations performed in any interval would at once give the 
 length of such interval. But, in order to supersede the attention 
 and vigilance of such an observer, a train of wheel-work is placed 
 in connection with the pendulum, the movement of which it regu- 
 lates ; and in connection with this train of wheel-work are fixed 
 the dial-plate and the hands of the clock, by which the number of 
 oscillations of the pendulum which take place in a day, or in any 
 part of a day, are indicated and registered. 
 
 512. When the same pendulum is transported to different parts 
 of the earth's surface, it is found that the rate of its vibration 
 varies, and this variation is proved to take place even after pre- 
 
 o. 
 
2-sb THEORY OF MACHINERY. 
 
 cautions have been taken to keep the centres of oscillation and 
 suspension at the same distance from each other. Now this change 
 in the rate of vibration under such circumstances can only be 
 explained by a change in the intensity of the force of gravity by 
 which the pendulum is moved. It is found that when the pen- 
 dulum is carried towards the terrestrial equator, the time of its 
 vibrations is longer ; and that when it is carried towards the pole,' 
 the time of its vibrations is shorter : the inference deduced from 
 which is, that the force of gravity diminishes as we approach the 
 equator, and increases as we approach the pole. 
 
 If the earth had the form of an exact sphere, did not revolve on 
 its axis, and was of uniform density, the force of gravity of all 
 parts of its surface would be the same, and no such variation in 
 the rate of a pendulum could take place when transported from 
 one point of the surface of the earth to another. But if the earth 
 be an exact sphere, revolving upon its polar axis in 23 h. 56 min., 
 then the effect of such motion of rotation would be to produce a 
 certain small diminution of the intensity of the force of gravity in 
 approaching the equator, and an increase in such intensity in 
 approaching the pole. The amount of such diminution or increase 
 produced by such rotation is capable of calculation, and, being 
 computed and compared with such change of intensity of the force 
 of gravity indicated by the variations of a pendulum, is found not 
 to correspond exactly with it. This absence of complete corre- 
 spondence indicates another cause affecting the force of gravity 
 besides the rotation of the earth. 
 
 If the earth be not an exact sphere, but have a form of which a 
 turnip and an orange are exaggerated representations, called in 
 geometry an oblate spheroid, such a form, combined with the 
 rotation of the earth, would produce a further effect in varying the 
 force of gravity in proceeding towards the equator or towards the 
 pole. Now it is found, by calculation, that a certain degree of 
 this form, combined with the diurnal rotation of the earth, would 
 produce exactly that variation in the force of gravity going 
 towards the equator and going towards the pole which is indicated 
 by the variation in the time of vibration of the same pendulum. 
 
 513. Hence it appears that the pendulum becomes an instru- 
 ment by which not only the doctrine of the diurnal rotation of the 
 earth is verified and corroborated, but by which the departure of 
 the earth from an exact globular form is also established. From 
 what has been stated, it will appear that the length of a pendulum 
 which vibrates seconds in different parts of the earth will be dif- 
 ferent ; the force of gravity in lower latitudes being less than in 
 higher, the length of the pendulum which vibrates seconds will be 
 proportionally less. 
 
PENDULUM. 227 
 
 514. Pendulum measures the velocity of falling: bodies. 
 
 As the pendulum thus supplies a measure of the intensity of the 
 force of gravity, it necessarily also affords the means of calculating 
 the height from which a body falling freely would descend in a 
 second if it moved in vacuo. The method of determining this by 
 the pendulum is susceptible of much greater accuracy than that 
 which has been already indicated by Atwood's machine. To find 
 the space through which a body will fall at any place in a second 
 of time, let the length of a pendulum which vibrates seconds in 
 that place, expressed in inches, be multiplied by 4-9348, and the 
 product will express in inches the height through which a body 
 would fall in a second in that place, independently of the resist- 
 ance of the air. 
 
 515. Curious property of a swing. It is well known to all 
 persons who have amused themselves with exercising in this species 
 of pendulum, that by humouring the motion by certain changes 
 of attitude, the oscillation to and fro, which, if they remain station- 
 ary, would be gradually diminished and would ultimately cease, by 
 i eason of the resistance of the air and friction, can be not only sus- 
 tained, but increased in its range. This is explicable by the well 
 established principles which govern the motion of pendulous bodies. 
 
 Let A (Jig- 199-) be a small weight suspended upon a fine thread 
 from the centre B, and let it be 
 supposed to fall from the position A 
 to the vertical line B c ; when it 
 arrives at c, it will have exactly 
 the same -velocity in the horizontal 
 direction as a body would have 
 after having fallen vertically from 
 F to c. Now let us suppose that at 
 the moment of arriving at c, the 
 body A is suddenly raised to some 
 point such as D, nearer to B, re- 
 taining ne\ertheless the horizontal 
 
 velocity it had acquired. With this velocity it will begin to move 
 up the arc D N, and it will continue to rise in that arc, until it 
 arrives at a point H whose height D K, above D, is equal to c F. 
 
 Now it is evident that, in this case, the angle H B D, through 
 which it will swing from the vertical B D, will be much greater 
 than the angle ABC. This will be still more evident if it be con- 
 sidered that if B E be drawn, making the angle E B D equal to A B c, 
 the height D G will be less than c E, and therefore less than D K in 
 the same proportion as D B is less than c B ; and since D G is less 
 than D K, the angle D B E, and therefore the angle A B c, is less than 
 the angle I^B H. 
 
27.S 
 
 THEORY OF MACHINERY. 
 
 It follows, therefore, that if by any expedient the centre of 
 gravity of a pendulous body be lowered in each descending seini- 
 oscillation, and raised in each ascending one, the range of the 
 oscillation will be constantly augmented ; but this, as will be appa- 
 
 Fig. zoo. 
 
 rent, is precisely what the swinger does when, by humouring his 
 attitude with the changes of oscillation, he increases the range of 
 
 Fig. act. 
 
 the swing. If he stand upon the swing board he lowers his body, 
 as shown in Jig. zoo., in each descending motion of the swing. 
 
BALANCE WHEEL. 229 
 
 and raises it, as in fig. 201., in each ascending motion; in the 
 one case he lowers the centre of gravity, and in the other he 
 raises it. 
 
 If he sit upon the swing board, he lies back with his body at 
 right angles to the cord and his legs downwards in descending, and 
 with his body upwards and his legs at right angle to the cord in 
 ascending. By these changes of attitude, the centre of gravity, as 
 before, is lower in descending than in ascending. 
 
 In both cases the same effect is produced as we have described 
 above, in supposing the body A (fig. 199.) to be lower upon the 
 string while it descends than while it ascends. 
 
 516. Tlie actual length of a pendulum which vibrates 
 seconds is found by accurate experiments to be 39 139 inches at 
 London, 39^021 inches at the Line, and 39^215 at Spitzbergen, 
 lat. 79 49' 58"N. 
 
 517. The balance wheel applied to regulate watch work is a 
 
 light steel wheel ABC, {fig. 
 202.) nicely balanced upon its 
 axis. Under this wheel a highly 
 elastic spiral spring, made of 
 very fine steel wire, is placed, 
 the inner end of which is at- 
 tached to the centre of the 
 Fjg> 20Z> wheel, and the outer end e is 
 
 attached to the fixed plate of 
 
 the watch under the wheel. The wheel naturally assumes a certain 
 position of equilibrium, from which if it be disturbed in either 
 direction, by turning it so as to coil or uncoil the spring, it will, 
 when disengaged, return to its position of equilibrium; but at 
 the moment of arriving at that position it will have acquired 
 such a velocity as will cause it to go on in virtue of the inertia of 
 the wheel, and it will thus swing to the other side of the position 
 of equilibrium. In this manner the wheel will swing alternately 
 on the one side and the other of its position of equilibrium, with 
 a motion altogether analogous to the motion of a pendulum ; and 
 this analogy is rendered still more close by the fact that the spiral 
 spring has a property which renders the oscillations of the wheel, 
 whatever be their range, great or small, isochronous ; that is to say, 
 they will be performed in exactly equal times. It is therefore by 
 this property that the balance wheel enjoys the same regulating 
 power as the pendulum. 
 
 518. The fly is a regulator, consisting of a wheel having four 
 arms, upon each of which a thin oblong vane of metal is so fixed 
 that it can be turned with its plane at any desired inclination to 
 the axis ; the wheel is accurately balanced so as to be in equili- 
 
 <i 3 
 
2 3 o THEORY OF MACHINERY. 
 
 brium upon its axis, whatever be the position of the vane. When a 
 motion of revolution is imparted to such a wheel the four vanes 
 acting against the atmosphere will be resisted by it with a force 
 which will be greater as the velocity of revolution is augmented, 
 and it is demonstrable that the rate of increase of this resistance 
 is proportional to the square of the velocity of the wheel : thus, 
 with a double velocity, there will be a fourfold resistance ; with a 
 triple velocity, a ninefold resistance, and so on. An apparatus for 
 illustrating experimentally the effect of such a wheel is represented 
 in jig. 203., where two such wheels, A and B, are mounted on 
 
 horizontal axles so as to be capable of turning with very little 
 friction. Upon each of these axles is fixed a small pinion, engaged 
 in the teeth of a rack c ; to the lower ends of these racks is at- 
 tached a heavy weight. When this weight is pushed up to the 
 pinions, the lowest teeth of the racks will be engaged in those of 
 the latter, and if the weight be then let go, it will fall, drawing the 
 racks with it, and imparting revolution to the two pinions, so that 
 
FLY. 231 
 
 when the weight shall have dropped upon the bottom of the stand, 
 the two pinions, and consequently the two flies, will have received 
 precisely the same velocity of rotation ; but the racks having ter- 
 minated just before the arrival of the weight upon the stage, the 
 pinions are no longer engaged in their teeth, and will therefore be 
 free to revolve, as well as the flies, with the velocity acquired. As 
 represented in the figure, the vanes of the fly B are presented with 
 their surfaces at right angles to the direction of the motion, while 
 the vanes of A are presented with their edges to the motion ; it is 
 evident, therefore, that one must encounter more resistance than 
 the other from the air, and this is proved by the fact that while B 
 is very soon brought to rest, A will continue to move for a consi- 
 derable time. The experiment may be varied by inclining the vanes 
 at various angles, when it will be found that the more near their 
 position is to that represented at B, the sooner the wheel will be 
 brought to rest by the resistance ; and the nearer it is to the posi- 
 tion shown in A, the longer will be the continuance of the motion. 
 The velocity imparted to the wheels may also be varied by varying 
 the weight attached to the racks. 
 
 The practical application of the fly as a regulator depends 
 upon the two properties here explained : first, that with a given 
 position of the vanes, the resistance of the air will increase inde- 
 finitely with the velocity ; and, secondly, that this resistance will 
 vary with the inclination of the vanes to the direction of the 
 motion. 
 
 Let us suppose, for example, that a fly be applied to equalise the 
 accelerating force produced by a descending weight ; the fly is then 
 mounted on an axis which receives motion, in common with the 
 other parts of the mechanism to which it is applied, from the weight. 
 At the commencement, the motion will be accelerated because the 
 weight will be greater than the resistance, and consequently a con- 
 tinually increasing velocity will be imparted to the fly ; according 
 to what has been explained, the fly will therefore encounter a re- 
 sistance which, increasing as the square of the velocity, will soon 
 become equal to the force of the descending weight, and then all 
 acceleration must cease, and the motion must become absolutely 
 uniform, since any, even the least, increase of velocity would pro- 
 duce a resistance greater than the moving power ; the uniform 
 velocity thus attained will continue so long as the moving power 
 continues to act with the same force upon the mechanism. 
 
 If it be desired to increase the uniform velocity which a given 
 moving force will thus impart, it is only necessary to turn the 
 vanes of the fly so as to present them more obliquely to the direc- 
 tion of the motion. 
 
 The fly possesses a property as a regulator which renders it 
 Q4 
 
232 THEORY OF MACHINERY. 
 
 available in many cases in which neither the pendulum nor the 
 balance wheel would be applicable : these two regulators, as we 
 shall more fully explain hereafter, produce, not an uniform, but an 
 intermitting motion, as will be readily comprehended by ob- 
 serving the motion of the seconds-hand of a time-piece, whether it 
 be regulated by a pendulum or a balance wheel ; such a hand, as 
 every one knows, moves not continuously, but by starts ; and the 
 same is true of the minute and hour hands, only that in these the 
 intermissions are made througa spaces so minute that they are not 
 observable. Now, in many cases, such an intermitting motion is 
 altogether inadmissible, as, for example, in the apparatus for illus- 
 trating the motion of falling bodies represented in jig. 48., where 
 the fly placed at the top regulates the rotation of the cylinder, 
 which must be continuous and uniform. 
 
 Musical boxes, self-acting pianofortes, barrel organs, played by 
 mechanism, are all examples where the regulator must be a fly, 
 and where neither pendulum nor balance wheel would be ad- 
 missible. 
 
 CHAP. IV. 
 
 REGULATION AND MODIFICATION OF FORCE AND MOTION. 
 
 519. REGULATION and uniformity are two of the conditions most 
 universally indispensable in the operation of machinery ; sudden 
 changes of speed are always a source of loss of power, often inju- 
 rious and sometimes destructive to the apparatus, and never fail to 
 produce imperfection in the articles fabricated. 
 
 Much attention, therefore, has been directed to, and much me- 
 chanical ingenuity expended on, contrivances for insuring these 
 conditions of regularity and uniformity in the movement of 
 machinery, by removing those causes of inequality which can be 
 avoided, and by compensating those which cannot. 
 
 520. Irregularity in the motion of machinery will result in any 
 one of the following causes : 
 
 1. When a varying power is opposed to a uniform resistance. 
 
 2. When a uniform power is opposed to a varying resistance. 
 
 3. When the power and resistance both vary, but not propor- 
 tionally to each other. 
 
 4. When the power is not transmitted with uniform effect 
 to the working point in the successive positions assumed by the 
 machine. 
 
REGULATION OF MOTION. 233 
 
 521. Varying power opposed to uniform resistance. The 
 
 force of the prime mover is seldom regular. The force of water 
 varies with the copiousness of the stream ; the force of wind is 
 proverbially capricious ; the power of steam varies with the in- 
 tensity of combustion in the furnace ; and the force of animal 
 power, depending on the temper and health, is difficult of control, 
 human labour being the most unmanageable of all. No machine 
 works so irregularly as one that is manipulated. 
 
 In some cases the prime mover is subject, by the very con- 
 ditions of its existence, to constant variation ; as, for example, 
 where it is a main spring, which gradually loses its energy as it is 
 relaxed. In some cases, the prime mover is liable to intermission, 
 and is totally suspended during certain intervals. An example of 
 this is presented in the single-acting steam-engine, where the force 
 of the steam acts only during the descent of the piston, but is 
 suspended during its ascent. 
 
 522. Uniform power opposed to varying resistance. In 
 almost all the applications of machinery, the load or resistance is 
 subject to continual fluctuation. In mills, a multiplicity of parts 
 are liable to be occasionally and irregularly disengaged, and to 
 have their operations suspended. In large factories for spinning, 
 printing, dyeing, &c., a great number of separate spinning- 
 machines, looms, presses, and other engines, are usually driven by 
 a common power, such as a water-wheel or a steam-engine. In 
 such cases, the number of machines worked at the same time must 
 necessarily vary according to the employment supplied to the 
 factory, and to the fluctuating demand for the articles produced. 
 Under such circumstances, the velocity with which the machinery 
 is moved would suffer corresponding changes, increasing with 
 each diminution, and being retarded with each increase of the 
 resistance. 
 
 523. In many cases, the variation of the power and the varia- 
 tion of the resistance are both from their conditions inevitable, and 
 yet a uniform effect is indispensable. It is evident that this can 
 only be insured by a class of contrivances which have for their 
 object to proportion the power to the resistance, by either causing 
 a diminution or increase of resistance to diminish or augment the 
 supply of power ; or, on the other hand, by causing the variation 
 of the power to act in a corresponding manner upon the resistance 
 or load. In a word, uniformity of action in machinery can only 
 be insured by providing means by which the power and the resist- 
 ance, no matter what be their respective variations, shall always be 
 proportional to each other. 
 
 Whenever the power is less than that which is in equilibrium 
 with the resistance, the motion will be retarded, and if this con- 
 
2 34 
 
 THEORY OF MACHINERY. 
 
 dition continue, it will ultimately stop; and whenever the power 
 is greater than that determined by the condition of equilibrium, 
 the motion will be accelerated ; and if this condition should con- 
 tinue, the acceleration would continue until the machine would be 
 destroyed by its own momentum. 
 
 5 24. There is scarcely any machine in which the energy of the 
 power is transmitted uniformly to the resistance in all the phases 
 of the mechanism. In all machines the moving parts assume in 
 succession a variety of positions, in each of which their effect to 
 transmit the power to the resistance is different; and thus the 
 effective energy of the machine in acting against the resistance is 
 subject to continual fluctuation. It is not easy to convey, without 
 numerous examples, a general idea of those causes of inequality 
 to those who are not familiar with machinery. It will, however, 
 be more clearly understood when we come, to explain the methods 
 of equalising the action of the power and the resistance. 
 
 525. Regulators. The class of contrivances which have for 
 their object to render the power and resistance proportionate to 
 each other are called regulators. They generally act upon that 
 point of the machine which commands the supply of the power by 
 means of some mechanical contrivances, which check the quantity 
 of the moving principle conveyed to the machine whenever the 
 motion becomes accelerated, and increase the supply whenever it 
 becomes retarded. In a water-wheel, for example, this is accom- 
 plished by acting upon the shuttle, in a wind-mill by the adjust- 
 ment of the sails, and in a steam-engine by acting on a valve called 
 the throttle valve placed in the main pipe, through which steam 
 flows from the boiler to the cylinder. 
 
 526. Tlie governor. One of the most interesting and instruc- 
 
 tive examples of this class 
 of contrivances is called 
 the governor. This expe- 
 dient, which was long used 
 to regulate mill work and 
 other machinery, owes its 
 beautiful adaption to the 
 steam-engine to the inge- 
 nuity of Watt. It consists 
 of two heavy balls, B B, Jig. 
 204., attached to the ex- 
 tremities of rods B F jointed at E, and passing through a mortice 
 in the vertical stem D D'. 
 
 When the balls B are driven from the axis by the centrifugal 
 force arising from their rotation, their upper arms E F are caused 
 to increase their divergence in the same manner as the blades 
 
 Fig. 404. 
 
REGULATORS. 
 
 2 35 
 
 of scissors are opened by separating the handles. These acting 
 upon the ring H, by means of the short rods F H, draw it down 
 the vertical axis from D towards E. A contrary effect is produced 
 when the balls B are brought closer to the axis, and the divergence 
 of the rods B E diminished. A horizontal wheel w is attached to 
 the vertical axis D D', having a groove to receive a rope or strap 
 upon its rim. This strap passes round the wheel or axis, by which 
 motion is transmitted to the machinery to be regulated so that the 
 spindle or shaft D D' will be always made to revolve with a speed 
 proportionate to that of the machinery. 
 
 As the shaft DD' revolves, the balls B are carried round it with 
 a circular motion, and consequently acquire a centrifugal force 
 which causes them to recede from the axle, and therefore to de- 
 press the ring H. On the edge or rim of this ring is formed a 
 groove, which is embraced by the prongs of a fork i at the ex- 
 tremity of one arm of a lever whose fulcrum is at G. The ex- 
 tremity K of the other arm is connected by some means with the 
 part of the machine which supplies the power. In the present 
 instance we shall suppose it a steam-engine, in which case the rod 
 K i communicates with a flat circular valve v placed in the prin- 
 cipal steam pipe, and so arranged that when K is elevated as far as 
 by their divergence the balls B have power over it, the passage of 
 the pipe will be closed by the valve v, and the passage of steam 
 entirely stopped ; and, on the other hand, when the balls subside 
 to their lowest position, the valve will be presented with its edge 
 in the direction of the tube, so as to interrupt no part of the 
 steam. 
 
 The property which renders this instrument so well adapted to 
 its purpose is, that there is but one velocity at which the balls can 
 remain in equilibrium. 
 
 527. Fusee. The effect of a power of variable energy may 
 be rendered uniform by transmitting it to the working point, 
 through the agency of a leverage of corresponding variation so 
 regulated, that, in the same proportion as the power diminishes in 
 energy, the leverage shall increase, and vice versa. A well-known 
 example of this occurs in the construction of certain watches, 
 where the moving power, being a main-spring inclosed in a barrel, 
 has a gradually diminished energy as the spring is relaxed. The 
 chain as it is discharged from the barrel is coiled upon a conical 
 spiral, called a fusee, represented in fg. 205. The leverage by 
 which the force of the spring is transmitted being the semi- dia- 
 meter of the fusee ; and the motion commencing from the top, or 
 narrowest end, it follows that when the energy of the spring is 
 greatest the leverage is least ; and as the chain coils upon the barrel 
 containing the spring, and is discharged from the fusee, the radius 
 
2 3 6 
 
 THEORY OF MACHINERY. 
 
 of each part of the fusee which discharges the chain gradually 
 increases. The form of the fusee is such that this increase of 
 
 Fig. 205. 
 
 leverage is in the exact proportion of the diminished force of the 
 spring. 
 
 528. The several parts of a machine have certain periods of 
 motion, in which they pass through a variety of positions, return- 
 ing constantly to similar positions after equal intervals. 
 
 In the different positions assumed by the moving parts during 
 these periods, the effect of the power transmitted to the working 
 point is different; and cases even occur in which for a moment this 
 effect is altogether interrupted, and the machinery is then in a pre- 
 dicament in which the power loses all effect upon the resistance. 
 The consequence of this would be, that, supposing the power and 
 resistance to be both uniform, the action of the former upon the 
 latter would be subject to periodical variation, being at one time 
 more and at another time less than what would be necessary to 
 keep the whole in equilibrium. 
 
 Under these circumstances, it is possible to suppose that the 
 movement of the machine may continue, and even that its average 
 rate may be uniform ; but its motion would be subject to periodical 
 variations, being alternately accelerated and retarded. This would 
 be attended not only with an injurious effect upon the work pro- 
 duced by the machine, but would be also detrimental to the machine 
 itself, whose moving parts would be subject to continual starts and 
 strains arising from the alternate reception and destruction of 
 momentum. 
 
 529. Crank. To render these general observations more 
 clearly intelligible, we shall take, as an example, the action 
 of a common crank, used in steam-engines and many other ma- 
 chines. 
 
 A crank is nothing more than a double winch. It is represented 
 
REGULATORS. 
 
 237 
 
 a 
 
 Fig. 206. 
 
 complete with both its arms in fig. 206. 
 Attached to the middle of CD, by a joint, 
 is a rod, which is the means of imparting 
 the effect of the power to the crank. This 
 rod is driven by an alternate motion like 
 the brake of a pump. The bar c D is 
 carried with a circular motion round the 
 axis A F. 
 
 Let the machine viewed in the direction 
 A B E F of the axis be conceived to be represented in fig. 207., 
 where A represents the centre round which the motion is to be 
 produced, and G the point where the connecting rod G H is attached 
 to the arm of the crank. The circle through which G is to be 
 urged by the rod is represented by the dotted line. In the position 
 represented in Jig. 207., the rod acting in the direction HG has its 
 full power to turn the crank G A round the centre A. As the 
 crank comes into the position represented in fig. 208., this power 
 is diminished ; and when the point G comes immediately below A, 
 as in Jig. 209., the force in the direction H G has no effect in 
 
 Fig. 207. 
 
 Fig. 108. 
 
 Fig. 109. 
 
 turning the crank round A, but, on the contrary, is entirely 
 expended in pulling the crank in the direction G A, and therefore 
 only acts on the pivots or gudgeons which support the axle. 
 
 At this crisis of the motion, therefore, the whole effective energy 
 of the power is annihilated. 
 
 After the crank has passed to the position represented in fig. 
 2 1 o., the direction of the force which acts upon the connecting rod 
 is changed, and now the crank is drawn upward in the direction 
 G H. In this position the moving force has some efficacy to produce 
 rotation round A, which efficacy continually increases until the 
 crank attains the position shown in fig. 2 1 1 ., when its power is 
 greatest. Passing from this position, its efficacy is continually 
 diminished until the point G comes immediately above the axis 
 A (fig. 212.). Here again the power loses all its efficacy to turn 
 
238 THEORY OF MACHINERY. 
 
 the axle. The force in the direction G H or H G can obviously 
 produce no other effect than a strain upon the pivots or gudgeons. 
 
 Fig.zio. Fig. an. Fig. ziz. 
 
 It will be evident from this that the action of the power trans- 
 mitted to the working point G is very variable. At the dead 
 points represented mfigs. 209. and 2 1 2., the machine, if depending 
 solely upon the moving power, must come to rest, for at both 
 points the whole effect of the power would be exerted in producing 
 pressure on the axle and gudgeons of the crank. Through a 
 small space at either side of those dead points, the effect trans- 
 mitted to G, though not absolutely nothing, is almost evanescent, 
 so that it may be considered that through a small arc at either side 
 of each of the dead points the machine is still inert. 
 
 It must, however, be considered that, in virtue of its inertia, the 
 motion which the machinery had previously to its arrival at its 
 dead points has a tendency to continue ; and if the resistance of 
 the load and the effects of friction be not too great, this disposition 
 to preserve its state of motion will extricate the machinery from 
 the mechanical dilemma in which it is involved in these cases by 
 the particular disposition of its parts. Although, however, the 
 motion will not therefore be actually suspended, on the arrival of 
 the crank at the dead points, it will be greatly retarded ; and, on 
 the other hand, when the power acquires its greatest activity, as 
 it does in the position represented mfigs. 207. and 21 1., it will be 
 unduly accelerated. 
 
 530. Ply wheel. These irregularities are equalised by fixing 
 upon the axis of the crank, or at any other convenient part of the 
 machine, a fly wheel, which is a massive ring of metal, connected 
 with a central box or nave by comparatively light spokes, and 
 turning on an axis with but little friction. If any force be applied 
 to it, with that force, making some slight deduction for friction, 
 it will move and will continue to move until some obstacle retard 
 it, which obstacle will receive from it as much force as the fly 
 wheel loses. 
 
 The effect of such a wheel applied to the parts moved by the 
 
REGULATORS. 239 
 
 prank will equalise the inequality which has just been described. 
 When the crank assumes the position represented mjigs. 207. unii 
 
 Fig. 
 
 211., where the power has full play upon it, the effect of the power 
 is partly transmitted to the machine, and partly received by the 
 movable rim of the fly wheel, to which it imparts increased mo- 
 mentum. There is here, it is true, an acceleration of the motion, 
 but one which is comparatively small, inasmuch as the great mass 
 of the fly wheel receives the momentum without sensible increase 
 of speed. When the crank gets into the predicament represented 
 at the dead points (Jigs. 209. and 212.), the momentum of tL, 
 fly wheel, received when the crank acted with the most advantage, 
 immediately conveys its force to the working-point G, extricates 
 the machine, and carrying the crank out of the neighbourhood of 
 the dead point, brings the power again to bear upon it. 
 
 531. It happens frequently in the practical application of 
 machinery, that the moving power is much too intense, or much 
 too feeble, for the resistance ; and in the one case contrivances are 
 required by which it may be greatly attenuated, and in the other 
 by which it may be greatly augmented. 
 
 Main-spring. In the case of watch work, the resistance to be 
 overcome is nothing more than that presented by the hands which 
 move upon the dial-plate. In this case the moving power is the 
 force of the fingers, by which once in twenty- four hours the main- 
 spring is wound up. The main -spring itself must be regarded, in 
 this case, as a mere depository for the power exerted in winding 
 
24.0 THEORY OF MACHINERY. 
 
 up the watch, and not as a prime mover. The force which is thus 
 deposited once in twenty-four hours in the main-spring is delivered 
 gradually and regularly, by such spring, to the fusee, and trans- 
 mitted, through the system of wheels, to the hands. 
 
 In the case of clocks moved by main-springs instead of a weight, 
 this attenuation of the moving power is still more extensive. 
 These, being wound up, will frequently go for fifteen days. In 
 this case, therefore, the mechanical force exerted by the hand in 
 winding them up, and which is developed in less than a minute, is 
 spread over fifteen times twenty-four hours by the mechanism of 
 the clock. 
 
 532 . Clockwork moved by a weight. In the case of clocks 
 which move by a descending weight, the original moving force is 
 also the application of the human hand in winding up the weight. 
 The weight, being lifted a height of three or four feet, descends 
 slowly through the same height, imparting . its descending force 
 gradually and regularly to the clockwork. In this case, therefore, 
 the descending force of a weight through a small height is so 
 attenuated as to impart a motion to the hands which will continue 
 sometimes for a month or longer. 
 
 533. It is frequently required, on the contrary, to impart to the 
 resistance a force vastly greater in intensity than the moving power. 
 Numerous examples of this have been already given in illustrating 
 the simple machines. In all cases where the leverage of the power 
 is greater than the leverage of the resistance, there will be an 
 augmented intensity of mechanical action in the same proportion ; 
 and this intensity, by combining levers or other simple machines, 
 may be augmented without any practical limit. 
 
 534. Hammers, sledges, &.c. But in some cases a force is 
 required more intense than can be obtained even by these means. 
 In such cases, it becomes necessary to convert the continued agency 
 of the moving power into one which acts instantly and by inter- 
 mission. If, for example, it be required to cause a nail to penetrate 
 a beam of wood, we should attempt in vain to accomplish this by 
 producing any pressure, however great, on the head of the nail. 
 A few blows of a hammer, nevertheless, easily effect this. In this 
 case, the moving power is the hand, or other force which raises the 
 hammer. The mass of the hammer, in falling on the head of the 
 nail, imparts instantly to the nail the entire force which was exerted 
 in lifting it, but with this difference, that such force, in raising the 
 hammer, was developed in a certain definite time, whereas it is 
 discharged upon the head of the. nail in an instant. 
 
 The same observations apply to all cases in which percussion is 
 used. In all these cases the force is developed in a definite time, 
 but is discharged upon the resistance in an instant. 
 
ACCUMULATION OF FORCE. 241 
 
 535. Inertia supplies means of accumulating; force. In 
 
 some cases where a severe instantaneous action is required, the 
 moving power is accumulated by means of the inertia of matter. 
 A mass of matter retains, by virtue of its inertia, the whole 
 amount of any force which may be given to it, except that part of 
 which friction and the atmospheric resistance deprives it. To 
 render this method of accumulating force intelligible, let us first 
 imagine a polished level plane, on which a heavy globe of metal, 
 also polished, is placed. It is evident that the globe will remain at 
 rest on any part of the plane without a tendency to move. Sup- 
 pose, then, a slight impulse be given to it, which will cause it to 
 move with any given velocity, for example, three feet per second. 
 It will then continue to move with this velocity, for any length of 
 time, except so far as it may be impeded by the resistance already 
 mentioned. 
 
 Let us then imagine a second impulse given to it equal in force 
 lo the former : this will increase its velocity to six feet per second ; 
 a third impulse will augment it to nine feet per second, and so on. 
 Now there is no limit to the number of impulses which may be 
 successively given to the moving body, provided only space were 
 given for its motion. Thus, ten thousand repetitions of the im- 
 pulse would make the body move at the rate of thirty thousand 
 feet a second. If the body to which these impulses were trans- 
 mitted were a cannon-ball, it might, by the constant application of 
 a feebly impelling force, be made to move at length with as much 
 force as if it were impelled from a powerful piece of ordnance. 
 The force with which such a ball would strike a buttress might be 
 sufficient to reduce it to ruins ; and yet such force may be nothing 
 more than the accumulation of a number of feeble forces, not 
 beyond the power of a child to exert, which are stored up and 
 preserved in the moving mass, and then brought to bear at the 
 same moment on the resistance against which the force is directed. 
 It is the same for any number of actions exerted successively and 
 during a long interval, brought into operation at one and the same 
 moment. 
 
 But the case here supposed cannot actually occur, because we 
 have not in general practical means of moving a body for a con- 
 siderable time in the same direction, without much friction, and 
 without encountering other obstacles which would impede its pro- 
 gress. If, however, a leaden ball be attached to the end of a string 
 and whirled rapidly round, a great force would be given to it, and 
 it will strike a board with such intensity as to penetrate it. 
 
 536. A weapon called a life-preserver consists of a piece of lead 
 sometimes attached to the end of a piece of cane or whalebone, 
 with which a blow may be given with great force. Innumerable 
 
 1 
 
242 
 
 THEORY OF MACHINERY. 
 
 examples of the application of this principle will present themselves 
 to every mind. Flails used in threshing, clubs, canes, whips, and 
 all instruments used for striking, axes, hatchets, cleavers, and all 
 instruments which act by a blow, present examples of this 
 principle. 
 
 537. Where very intense force is required, as, for example, in 
 
 certain presses, two heavy balls 
 are attached to the ends of a 
 horizontal lever A B, with equal 
 arms, jig. 2 1 4. This lever works 
 a screw, at the lower end of which 
 is the working point. A rapid 
 motion is imparted to the balls 
 by the hand, and the working 
 point is driven against the re- 
 sistance by the accumulated mo- 
 mentum acquired by the balls, 
 
 augmented by the leverage of the arms to which they are attached, 
 
 and the mechanical force of the screw. 
 
 538. The surprising effects produced by the accumulation of 
 force are apt to lead to erroneous suppositions, that instruments 
 thus acting by inertia have the effect of actually augmenting the 
 amount of moving power. When the quality of inertia, however, 
 is rightly understood, such an error cannot occur. The instru- 
 ments by which force is thus accumulated, so far from augmenting 
 the effect of the moving power, must to some extent diminish it ; 
 inasmuch as they are liable to friction and atmospheric resistance, 
 by which more or less force is intercepted. An accumulator of 
 force, whatever be its form, can never have more force than has 
 been applied to put it in motion. Whether it be a falling weight, 
 a revolving mass, a string which is coiled up, or air which is con- 
 densed, it cannot develop a greater amount of force than that 
 which is imparted in raising it if it be a weight, in putting it in 
 motion if it be a moving mass, in winding it up if it be a spring, or 
 in compressing it if it be air. The only difference between the 
 power which is imparted to these agents, and the effects which they 
 produce respectively upon the resistance, is in the time during 
 which the effects are developed. The power is in general imparted 
 slowly, while the effects are produced instantaneously. 
 
 539. Mills for rolling metals, or for punching boiler-plates, 
 supply striking examples of this. The water-wheel, or steam- 
 engine, or whatever other power be used, is allowed for some time 
 to act upon the fly wheel alone, no load being placed upon the 
 machine. When a sufficient momentum has been imparted to the 
 mass of metal forming the fly wheel, the metal to be rolled or 
 
FLY WHEEL. 243 
 
 pierced is submitted to the machine, and is immediately flattened 
 or perforated by it, depriving at the same time the fly wheel of a 
 corresponding quantity of its momentum. 
 
 In the same manner, a force may be obtained by the arms of 
 men acting on a fly for a few seconds, sufficient to impress an 
 image on a piece of metal by an instantaneous stroke. The fly is 
 therefore the principal agent in coining-presses. 
 
 Some presses used in coining have flies with arms four feet long, 
 bearing a hundredweight at each of their extremities. If such a 
 velocity be imparted to such an arm that it shall make one revolu- 
 tion per second, the die will be driven against the metal with the 
 same force as that with which 3f tons would fall from the height 
 of 1 6 feet, which is an enormous power if the simplicity and com- 
 pactness of the machine be considered. 
 
 540. Position of tlie fly wheel. The place to be assigned to 
 a fly wheel relatively to the other parts of the machinery is deter- 
 mined by the purpose for which it. is used. If it be intended to 
 equalise the action, it should be near the working point. Thus, 
 in a steam-engine, it is placed near the crank which turns the axle, 
 by which the power of the engine is transmitted to the object it is 
 finally designed to affect. On the contrary, in hand-mills, such as 
 those commonly used for grinding coffee, &c., it is placed upon the 
 axis of the winch by which the machine is worked. 
 
 541. The open work of fenders, fire grates, and similar orna- 
 mental articles constructed in metal, is produced by the action of 
 a fly in the manner already described. 
 
 The cutting tool, shaped according to the pattern to be executed, 
 is attached to the end of the screw, and the metal being held in a 
 proper position beneath it, the fly is made to urge the tool down- 
 wards with such force as to stamp out pieces of the required 
 figure. When the pattern is complicated, and it is necessary to 
 preserve with exactness the relative situation of its different parts, 
 a number of punches are impelled together, so as to strike the en- 
 tire piece of metal at the same instant, and in this manner the most 
 elaborate open work is executed by a single stroke of the hand. 
 
 542. Modification of motion. Although the simple ma- 
 chines, classed under the general denomination of mechanic powers, 
 include in principle all the solid parts which can enter into the 
 composition of any piece of mechanism, there are still many expe- 
 dients which, on account of their simplicity of construction and 
 extensive utility in practice, merit more particular notice. 
 
 In the arts and manufactures, the kind of motion produced by 
 a given force is often of much greater importance than either the 
 velocity imparted to the working-point, or the amount of the 
 useful effect obtained from the moving power : the latter may 
 
244- THEORY OF MACHINERY. 
 
 affect the quantity of work done in a given time; but the former, 
 in particular cases, is essential to the performance of the work, in 
 any quantity whatever. 
 
 In the practical application of machinery, the object aimed at 
 is often to communicate to the working-point some peculiar sort 
 of motion adapted to the uses to which the machine is applied, and 
 it rarely, indeed almost never, happens that the moving power is 
 capable of this sort of motion. Expedients must, therefore, be 
 discovered by means of which the motions which the moving power 
 is capable of directly producing can be converted into those which 
 are necessary for the purposes to which the machine is applied. 
 
 The varieties of motion which most commonly present them- 
 selves in the practical application of mechanics may be divided 
 into rectilinear and rotatory. In rectilinear motion the several 
 parts of the moving body proceed in parallel straight lines with 
 the same speed. In rotatory motion the several points revolve 
 round an axis, each performing a complete circle, or similar parts 
 of a circle, in the same time. 
 
 Each of these may again be resolved into continued and recipro- 
 cating. In a continued motion, whether rectilinear or rotatory, 
 the parts move constantly in the same direction, whether that be 
 in parallel straight lines, or in rotation on an axis. In recipro- 
 cating motion the several parts move alternately in opposite 
 directions, tracing the same spaces from end to end continually. 
 Thus, there are four principal species of motion which more fre- 
 quently than any others act upon, or are required to be trans- 
 mitted by, machines : 
 
 1 . Continued rectilinear motion. 
 
 2. Reciprocating rectilinear motion. 
 
 3. Continued circular motion. 
 
 4. Reciprocating circular motion. 
 
 These will be more clearly understood by examples of each kind. 
 
 Continued rectilinear motion is observed in the flowing of a 
 river, in a fall of water, in the blowing of the wind, in the motion 
 of an animal upon a straight road, in the perpendicular fall of a 
 heavy body, in the motion of a body down an inclined plane. 
 
 Reciprocating rectilinear motion is seen in the piston of a common 
 syringe, in the rod of a common pump, in the hammer of a pavier, 
 the piston of a steam-engine, the stampers of a fulling-mill. 
 
 Continued circular motion is exhibited in all kinds of wheel- 
 work, and is so common, that to particularise it is needless. 
 
 Reciprocating circular motion is seen in the pendulum of a 
 clock, and in the balance wheel of a watch. 
 
 We shall now explain some of the contrivances by which a 
 power having one of these motions may be made to communicate 
 
MODIFICATION OF MOTION. 245 
 
 either the same species of motion changed in its velocity or direc- 
 tion, or any of the other three kinds of motion. 
 
 543. Continued rectilinear motion. The most obvious 
 expedient for producing this motion is by means of one or more 
 pulleys. It has been already shown that, by a single fixed pulley, 
 a moving power acting in the direction of any line may impart 
 continued rectilinear motion to a resistance in any other line, 
 provided only that the two directions can be connected by a fixed 
 pulley. This, however, can only be done when lines drawn in the 
 two directions intersect each other. In that case a fixed pulley, 
 being placed in the angle formed by the two lines, with its axis 
 perpendicular to them, will serve to guide a rope, one part of 
 which will be directed to the resistance, and the other to the 
 moving power. 
 
 If, however, the direction of the power do not intersect that of 
 the resistance, they must either be parallel or in different planes. 
 
 If they be parallel, as, for example, when the power acts verti- 
 cally downwards, while the weight is moved vertically upwards, 
 the object will be attained by two fixed pulleys having their axes 
 horizontal, one placed directly above the power, and the other 
 directly above the resistance. 
 
 If the direction of the power and that of the weight be in 
 different planes, the object may also be attained by two fixed 
 pulleys, placed at any two points arbitrarily taken. The axis of the 
 
 pulley o (Jig. 215.), which 
 guides that part of the rope 
 directed to the power p, 
 must in that case be tit 
 right angles to the direc- 
 tion of the power and that 
 of the line joining the 
 pulleys ; and the axis of 
 the pulley o', which guides 
 that part of the rope di- 
 Fi g 115 rected to the resistance w, 
 
 must be at right angles to 
 
 the direction of the line joining the pulleys and that of the resist- 
 ance. 
 
 If it be necessary to change the velocity, any of the systems of 
 pulleys described in 453., etseq., may be used in addition to the 
 fixed pulleys. \ , - 
 
 By the wheel and axle any one continued rectilinear motion 
 may be made to produce another in any other direction, and with 
 any other velocity. It has been already explained that the pro- 
 portion of the velocity of the power to that of the weight is as the 
 
 n 3 
 
246 
 
 THEORY OF MACHINERY. 
 
 diameter of the wheel to the diameter of the axle. The thickness 
 of the axle being therefore regulated in relation to the size of the 
 wheel, so that their diameters shall have that proportion which 
 subsists between the proposed velocities, one condition of the 
 problem will be fulfilled. The rope coiled upon the axle may be 
 carried, by means of one or more fixed pulleys, into the direction 
 of one of the proposed motions, while that which surrounds the 
 wheel is carried into the direction of the other by similar means. 
 
 By the wheel and axle a continued rectilinear motion may be 
 made to produce a continued rotatory motion, or vice versa. If 
 the power be applied by a rope coiled upon the wheel, the con- 
 tinued motion of the power in a straight line will cause the machine 
 to have a rotatory motion. Again, if the weight be applied by a 
 rope coiled upon the axle, a power having a rotatory motion 
 applied to the wheel will cause the continued ascent of the weight 
 in a straight line. 
 
 Continued rectilinear and rotatory motions may be made to pro- 
 duce each other, by causing a 
 toothed wheel to work in a rack 
 Such an apparatus is represented 
 in fg. 216. In some cases the 
 teeth of the wheel work in the 
 links of a chain. The wheel is 
 then called a rag-wheel (Jig. 217.) 
 Straps, bands, or ropes, may com- 
 municate rotation to a wheel, by 
 their friction in a groove upon its 
 edge. 
 
 A continued rectilinear motion 
 is produced by a continued circu- 
 lar motion in the case of a screw. 
 The lever which turns the screw has a continued circular motion, 
 while the screw itself advances with a continued rectilinear 
 motion. 
 
 The continued rectilinear motion of a stream of water acting 
 upon a wheel produces continued circular motion. In like manner 
 the continued rectilinear motion of the wind produces a continued 
 circular motion in the arms of a windmill. 
 
 Cranes for raising and lowering heavy weights convert a circu- 
 lar motion of the power into a continued rectilinear motion of the 
 weight. 
 
 544. Reciprocating: rectilinear motion. Continued circu- 
 lar motion may produce reciprocating rectilinear motion, by a 
 great variety of ingenious contrivances. 
 
 Reciprocating rectilinear motion is used when heavy stampers 
 
 Fig. 216. 
 
 Fig. 117. 
 
MODIFICATION OF MOTION. 247 
 
 are to be raised to a certain height, and allowed to fall upon some 
 object placed beneath them. This may be accomplished by a 
 wheel bearing on its edge curved teeth, called wipers. The stam- 
 per is furnished with a projecting arm or peg, beneath which the 
 wipers are successively brought by the revolution of the wheel. 
 As the wheel revolves the wiper raises the stamper, until its 
 extremity passes the extremity of the projecting arm of the 
 stamper, when the latter immediately falls by its own weight. 
 . It is then taken up by the next 
 
 L wiper, and so the process is con- 
 
 tinued. A similar effect is pro- 
 
 {; rrV duced if the wheel be partially 
 
 - furnished with teeth, and the 
 
 stamper carry a rack in which 
 these teeth work. Such an appa- 
 ratus is represented in^g-. 218. 
 
 It is sometimes necessary that 
 the reciprocating rectilinear mo- 
 tion shall be performed at a certain 
 varying rate in both directions. 
 This may be accomplished by the ar- 
 rangement represented in fig. 210. 
 Fig. 218. Fig.zig AII ,> , , 
 
 A wheel turns uniformly in the 
 
 direction A B D E. A rod m n moves in guides, which only permit 
 it to ascend and descend perpendicularly. Its extremity m rests 
 upon a path or groove raised from the face of the wheel, and 
 shaped into such a curve that as the wheel revolves the rod m n shall 
 be moved alternately in opposite directions through the guides, 
 with the required velocity. The manner in which the velocity 
 varies will depend on the form given to the groove or channel 
 raised upon the face of the wheel, and this may be shaped so as to 
 give any variation to the motion of the rod m n which may be 
 required for the purpose to which it is to be applied. 
 
 The rose-engine in the turning-lathe is constructed on this prin- 
 ciple. It is also used in spinning machinery. 
 
 It is often necessary that the rod to which the reciprocating 
 motion is communicated shall be urged by the same force in both 
 directions. A wheel partially furnished with teeth, acting on two 
 racks placed on different sides of it, and both connected with the 
 bar or rod to which, the reciprocating motion is to be communi- 
 cated, will accomplish this. Such an apparatus is represented in 
 fig. 220., and needs no further explanation. 
 
 Another contrivance for the same purpose is shown in Jig. 221., 
 where A is a wheel turned by a winch H, and connected with a 
 rod or beam moving in guides by the joint a b. As the wheel A is 
 
248 THEORY OF MACHINERY. 
 
 turned by the winch H, the beam is moved between the guides 
 alternately in opposite directions, the extent of its range being 
 
 Fig. 220. 
 
 Fig. aza. 
 
 governed by the length of the diameter of the wheel. Such an 
 apparatus is used for grinding and polishing plane surfaces, and 
 also occurs in silk machinery. 
 
 An apparatus applied by M. Zureda in a machine for pricking 
 holes in leather is represented in jig. 222. The wheel A B has its 
 circumference formed into teeth, the shape of which may be varied 
 according to the circumstances under which it is to be applied. 
 One extremity of the rod a b rests upon the teeth of the wheel upon 
 which it is pressed by a spring at the other extremity. When the 
 wheel revolves, it communicates to this rod a reciprocating rectili- 
 near motion. 
 
 Leupold applied this mechanism to move the pistons of pumps. 
 Upon the vertical axis of a horizontal hydraulic wheel is fixed 
 another horizontal wheel, which is furnished with seven teeth in 
 the manner of a crown wheel. -These teeth are shaped like inclined 
 planes, the intervals between them being equal to the length of the 
 planes. Projecting arms attached to the piston rods rest upon the 
 crown of this wheel ; and, as it revolves, the inclined surfaces of 
 the teeth, being forced under the arm, raise the rod upon the 
 principle of the wedge. To diminish the obstruction arising from 
 friction, the projecting arms of the piston rods are provided with 
 rollers, which run upon the teeth of the wheel. In one revolution 
 of the wheel each piston makes as many ascents and descents as 
 there are teeth. 
 
 545. Continued circular motion. Wheelwork furnishes 
 numerous examples of continued circular motion round one axis, 
 
MODIFICATION OF MOTIOX. 
 
 249 
 
 producing continued circular motion round another. If the axles 
 be in parallel directions, and not too distant, rotation may be 
 transmitted from one to the other by two spur wheels (448.) ; and 
 the relative velocities may be determined by giving a correspond- 
 ing proportion to the diameter of the wheels. 
 
 If a rotatory motion is to be communicated from one axis to 
 another parallel to it, and at any considerable distance, it cannot in 
 practice be accomplished by wheels alone, for their diameters would, 
 in that case, be too large. In this case, the motion is communi- 
 cated from wheel to wheel, by endless bands. If the two wheels 
 are desired to turn in the same direction, the bands are placed as in 
 jig. 223. If in contrary directions, they are crossed, as in fig. 224. 
 
 w^ 
 
 Fig. zzj. Fig. 134. 
 
 In this case, as in toothed wheels, the relative velocities are de- 
 termined by the proportion of the diameters of the wheels. 
 
 In some cases it happens that the strain upon the wheels is too 
 great to allow of the motion being transmitted by a band. In 
 
 such cases, the motion may 
 be transmitted from one 
 shaft to the other, by means 
 of bevelled wheels. In fig. 
 225. an arrangement is 
 shown in which a revolv- 
 lg ' M5 ' ing shaft imparts motion to 
 
 a series of other shafts, perpendicular to it. Each of the latter 
 might impart motion to other shafts, at any convenient distance, 
 by like means. 
 
 In figs. 226. and 227. another arrangement is shown, by which 
 
 Fig. za6. Fig. ^^^. 
 
 a shaft w imparts motion to another shaft w', by the intervention 
 
2 5 
 
 THEORY OF MACHINERY. 
 
 Fig. 128. 
 
 of a third shaft w w f on which bevelled wheels are fixed. The 
 velocities may in this case be varied in any desired proportion by 
 establishing a corresponding relation between the magnitudes of 
 the wheels. 
 
 The methods of transmitting rotation fix an an axis or shaft to 
 another at right angles to it, by spur and crown wheels, and by an 
 endless screw and a spur wheel, have been already explained in 
 450. 
 
 546. Universal joint. The axis to which rotation is to be 
 given, or from which it is to be taken, is sometimes variable in its 
 position. In such cases, an ingenious con- 
 trivance, called a universal joint, invented 
 by the celebrated Dr. Hook, may be used. 
 The two shafts or axles A B, Jig. 228., be- 
 tween which the motion is to be commu- 
 nicated, terminate in semicircles, the diame- 
 ters of which, c D and E r, are fixed in the 
 form of a cross, their extremities moving 
 freely in bushes placed in the extremities 
 of the semicircles. Thus, while the central 
 cross remains unmoved, the shaft A and its 
 semicircular end may revolve round c D as 
 an axis ; and the shaft B and its semicircular end may revolve 
 round E r as an axis. If the shaft A be made to revolve without 
 changing its direction, the points c D will move in a circle whose 
 centre is at the middle of the cross. The motion 
 thus given to the cross will cause the points E F 
 to move in another circle round the same cen- 
 tre, and hence the shaft B will be made to re- 
 volve. 
 
 This instrument will not transmit the motion 
 if the angle under the directions of the shafts 
 be less than 140. In this case a double joint 
 as represented in Jig. 229. will answer the pur- 
 pose. This consists of four semicircles united 
 by two crosses, and its principle and operation 
 is the same as in the last case. 
 
 Universal joints are of great use in adjusting 
 the position of large telescopes, where, while the 
 Fig. 229. observer continues to look through the tube, it 
 
 is necessary to turn endless screws or wheels whose axes are not 
 in an accessible position. 
 
 The cross is not indispensably necessary in the universal joint. 
 A hoop, with four pins projecting from it at fcur points equally 
 distant from each other, or dividing the circle of the hoop into 
 
MODIFICATION OF MOTION. 
 
 251 
 
 four equal arches, will answer the purpose. These pins play in the 
 bushes of the semicircles in the same manner as those of the cross. 
 The universal joint is much used in cotton-mills, where shafts 
 are carried to a considerable distance from the prime mover, and 
 great advantage is gained by dividing them into convenient 
 lengths, connected by a joint of this kind. 
 
 547. Alternate circular motion. In the practical appli- 
 cation of machinery, it is often necessary to connect a part having 
 a continued circular motion with another which has a recipro- 
 cating or alternate motion, so that either may move the other. 
 There are many contrivances by which this may be effected. 
 
 One of the most remarkable examples of it is presented in the 
 escapements of watches and clocks. 
 
 A beam vibrating on an axis, and driven by the piston of a 
 steam-engine, or any other power, may communicate rotatory 
 motion to an axis by a connecter and a crank. 
 
 A wheel A (Jig. 230.) armed with wipers, acting upon a sledge- 
 hammer B, fixed upon a centre 
 or axle c, will, by a continued 
 rotatory motion, give the ham- 
 mer the reciprocating motion 
 necessary for the purposes to 
 which it is applied. The man- 
 ner in which this acts must 
 be evident on inspecting the 
 figure. 
 
 The large shears used in 
 factories for cutting plates of 
 
 metal are worked upon a similar principle. The lower edge of the 
 shears (Jig. 231.) is fixed, and the upper movable upon a joint. 
 
 Fig. 230. 
 
 Fig. 1JI. 
 
 Under the extremity of the movable arm is placed a wheel, the 
 contour of which has the form shown in the figure. In the 
 
252 THEORY OF MACHINERY. 
 
 position there shown, the arm is in its lowest position, and there- 
 fore the blades of the shears diverge as much as possible. When 
 the wheel turns, so as to raise the movable arm, the shears will be 
 closed. In this way, by the continual revolution of the wheel, the 
 shears will be alternately closed and opened. 
 
 The treddle of the lathe furnishes an obvious example of a 
 vibrating circular motion producing a continued circular one. 
 The treddle acts upon a crank, which gives motion to the principal 
 wheel, in the same manner as already described in reference to the 
 working beam and crank in the steam-engine. 
 
 By the following ingenious mechanism an alternate or vibrating 
 force may be made to communicate a circular motion continually 
 in the same direction : Let A B (j%. 232.) be an axis receiving 
 
 an alternate motion from some 
 force applied to it, such as a 
 swinging weight. Two ratchet 
 wheels, m and w, are fixed on this 
 axle, their teeth being inclined in 
 opposite directions. Two toothed 
 wheels c and D are likewise placed 
 upon it, but so arranged that they 
 turn upon the axle with a little 
 friction. These wheels carry two 
 catches o p, which fall into the teeth of the ratchet wheels m w, 
 but fall on opposite sides conformably to the inclination of the 
 teeth already mentioned. The effect of these catches is, that if 
 the axis be made to revolve in one direction, one of the two 
 toothed wheels is always compelled (by the catch against which 
 the motion is directed) to revolve with it, while the other is permit- 
 ted to remain stationary in obedience to any force sufficiently great 
 to overcome its friction with the axle on which it is placed. The 
 wheels c and D are both engaged by bevelled teeth with the wheel E. 
 According to this arrangement, in whichever direction the axis 
 A B is made to revolve, the wheel E will continually turn in the 
 same direction ; and therefore, if the axle A B be made to turn 
 alternately in the one direction and the other, the wheel E will not 
 change the direction of its motion. Let us suppose the axle A B is 
 turned against the catch p. The wheel c will then be made to 
 turn with the axle. This will drive the wheel E in the same 
 direction. The teeth on the opposite side of the wheel E being 
 engaged with those of the wheel D, the latter will be turned upon 
 the axle ; the friction, which alone resists its motion in that direc- 
 tion, being overcome. Let the motion of the axle A B be now 
 reversed. Since the teeth of the ratchet wheel n are moved 
 against the catch o, the wheel D will be compelled to revolve with 
 
MODIFICATION OF MOTION. 253 
 
 the axle. The wheel E will be driven in the same direction ag 
 before, and the wheel c will be moved on the axle A B, and in a 
 contrary direction to the motion of the axle, the friction being 
 overcome by the force of the wheel E. Thus, while the axle A B 
 is turned alternately in the one direction and the other, the wheel 
 E is constantly moved in the same direction. 
 
 It is evident that the direction in which the wheel E moves may 
 be reversed by changing the position of the ratchet wheels and 
 catches. 
 
 It is often necessary to communicate an alternate circular 
 motion, like that of a pendulum, by means of an alternate motion 
 in a straight line. A remarkable instance of this occurs in the 
 steam-engine. The moving force in this machine i^ the pressure 
 of steam, which impels a piston from end to end alternately in a 
 cylinder. The force of this piston is communicated to the working 
 beam by a strong rod, which passes through a collar in one end of 
 the piston. Since it is necessary that the steam included in the 
 cylinder should not escape between the piston rod and the collar 
 through which it moves, and yet that it should move as freely and 
 be subject to as little resistance as possible, the rod is turned so as 
 to be truly cylindrical, and is well polished. It is evident that, 
 under these circumstances, it must not be subject to any lateral or 
 cross strain, which would bend it towards one side or the other of 
 the cylinder. But the end of the beam to which it communicates 
 motion, if connected immediately with the rod by a joint, would 
 draw it alternately to the one side and the other, since it moves in 
 the arc of a circle, the centre of which is at the centre of the beam. 
 It is necessary, therefore, to contrive some method of connecting the 
 rod and the end of the beam, so that while the one shall ascend and 
 descend in a straight line, the other may move in the circular arc. 
 The method which first suggests itself to accomplish this is, to 
 construct an arch-head upon the end of the 
 beam, as in t /g\ 233. Let c be the centre on 
 which the beam works, and let B D be an 
 arch attached to the end of the beam, being 
 a part of a circle having c for its centre. To 
 the highest point B of the arch a chain is 
 attached, which is carried upon the face of 
 the arch B A, and the other end of which is 
 attached to the piston rod. Under these 
 circumstances it is evident that when the 
 force of the steam impels the piston down- 
 wards, the chain p A B will draw the end of 
 the beam down, and will, therefore, elevate 
 the other end. 
 
254 THEORY OF MACHINERY. 
 
 When the steam-engine is used for certain purposes, such as 
 pumping, this arrangement is sufficient. The piston in that case 
 is not forced upwards by the pressure of steam. During its ascent 
 it is not subject to the action of any force of steam, and the other 
 end of the beam falls by the weight of the pump-rods drawing the 
 piston, at the opposite end A, to the top of the cylinder. Thus the 
 machine is, in fact, passive during the ascent of the piston, and 
 exerts its power only during the descent. 
 
 If the machine, however, be applied to purposes in which a con- 
 stant action of the moving force is necessary, as is always the case 
 in manufactures, the force of the piston must drive the beam in its 
 ascent as well as in its descent. The arrangement just described 
 cannot effect this ; for although a chain is capable of transmitting 
 any force, by which its extremities are drawn in opposite direc- 
 tions, yet it is, from its flexibility, incapable of communicating a 
 force which drives one extremity of it towards the other. In the 
 one case the piston first pulls down the beam, and then the beam 
 pulls up the piston. The chain, because it is inextensible, is per- 
 fectly capable of both these actions ; and, being flexible, it applies 
 itself to the arch-head of the beam, so as to maintain the direction 
 of its force upon the piston continually in the same straight line. 
 But when the piston acts upon the beam in both ways, in pulling 
 it down and pushing it up, the chain becomes inefficient, being, 
 from its flexibility, incapable of the latter action. 
 
 The problem might be solved by extending the length of the 
 piston rod, so that its extremity shall be above the beam, and 
 using two chains ; one connecting the highest point of the rod 
 with the lowest point of the arch-head, and the other connecting 
 the highest point of the arch-head with a point on the rod below 
 the point which meets the arch-head when the piston is at the 
 top of the cylinder,^. 234. 
 
 The connection required may also be made by arming the arch- 
 
 Fig. 234. Fig. 
 

 MODIFICATION OF MOTION. 255 
 
 head with teeth, -fig. 235., and causing the piston rod to terminate 
 in a rack. In cases where, as in the steam-engine, smoothness of 
 motion is essential, this method is objectionable ; and under any 
 circumstances such an apparatus is liable to rapid wear. 
 
 The method contrived by Watt, for connecting the motion of the 
 piston with that of the beam, is one of the most ingenious and elegant 
 solutions ever proposed for a mechanical problem. He conceived 
 the notion of two straight rods, A B, 
 c~D,fig. 236., moving on centres or 
 pivots A and c, so that the extre- 
 mities B and D would move in the 
 arcs of circles having their centres 
 at A and c. The extremities B and 
 D of these rods he conceived to be 
 connected with a third rod B D 
 united with them by pivots on 
 which jt could turn freely. To the 
 system of rods thus connected let 
 
 an alternate motion on the centres A and c be communicated : the 
 points B and D will move upwards and downwards in the arcs ex- 
 pressed by the dotted lines, but the middle point P of the connect- 
 ing rod B D will move upwards and downwards without any sensible 
 deviation from a straight line. 
 
 To prove this demonstratively would require some abstruse 
 mathematical investigation. It may, however, be rendered in some 
 degree apparent by reasoning of a looser and more popular nature. 
 As the point B is raised to E, it is also drawn aside towards the 
 right. At the same thne the other extremity D of the rod B D is 
 raised to E', and is drawn aside towards the left. The ends of the 
 rod B D being thus at the same time drawn equally towards oppo- 
 site sides, its middle point P will suffer no lateral derangement, 
 and will move directly upwards. On the other hand, if B be 
 moved downwards to r, it will be drawn laterally to the right ; 
 while D being moved to r' will be drawn to the left. Hence, as 
 before, the middle point P sustains no lateral derangement, but 
 merely descends. Thus, as the extremities B and D move upwards 
 and downwards in circles, the middle point P moves upwards and 
 downwards in a straight line. 
 
 An elegant contrivance, by which a continued motion of rotation 
 can be made to produce a reciprocating rectilinear motion, is 
 shown in jig. 237., where a wheel A B is surrounded on the inside 
 with teeth, and a wheel c of half the diameter revolves within it ; 
 the wheel A B being fixed. While the centre of c moves in a 
 circle round the centre of A B, any point taken on the circum- 
 ference of c will move alternately across A B, in the direction of 
 
THEORY OF MACHINERY. 
 
 
 
 Fig.zjy- 
 
 one of its diameters. Thus, 
 that point of c which, in the 
 figure, is at the highest point 
 of A B, will move along the ver- 
 tical diameter of AB, to the 
 lowest point, while c rolls over 
 half the circumference of A B ; 
 and the same point will again 
 traverse the same vertical dia- 
 meter of A B upwards, while the 
 wheel c rolls over the other 
 semicircle of A B. 
 
 Another method of produc- 
 ing a reciprocating rectilinear 
 motion by a reciprocating cir- 
 cular motion is shown in fig. 238. The beam AB vibrates on 
 w w the centre c, and a cord 
 
 applies itself to a groove in 
 the semicircular arch D E F. 
 This cord is extended over 
 the grooves of two fixed 
 rollers M N ; it is evident 
 that as the beam AB vi- 
 brates, any point L, upon 
 ' N the cord, between M N, will 
 move alternately right and 
 left in a straight line. 
 
 By the arrangement 
 
 shown in fig. 239., the alternate circular motion of the unequal 
 toothed sectors will impart recipro- 
 cating rectilinear motions to two racks. 
 These motions will be unequal, and 
 will be in the same ratio as the radii 
 of the two circles. 
 
 By a mechanical contrivance repre- 
 sented in fig. 240., called the lever of 
 La Garousse, a reciprocating circular 
 motion is made to produce an inter- 
 mitting rectilinear motion. The lever 
 A B vibrates on the centre c, the arm 
 which ascends drawing up one of the 
 hooks E, while the arm which descends 
 passes the other hook E' from one tooth to another of the vertical 
 beam r G. By the continuance of this operation, the beam F G 
 is elevated by an intermitting motion. 
 
 Fig. 238. 
 
JOINTS-GIMBALS. 
 
 25- 
 
 Fig. 140. Fig. 241. 
 
 In Jig. 241. an expedient is represented by which a similar 
 effect is produced upon a wheel M N. , A lever A B vibrates on a 
 centre c, while two arms D and E extend from it, and act upon 
 pins projecting from the wheel M N. The arms D and E are thus 
 alternately pushed forwards and drawn backwards : that which is 
 pushed forwards drives before it one of the pins, and that which is 
 drawn backwards falls from one pin to the other. 
 
 548. Joints. It frequently happens in mechanical combina- 
 tions, that it is required to make temporary connections between 
 separate parts, or so to unite them together that they shall admit 
 of having their relative position changed at pleasure. In some 
 cases one, and in others both, of the parts thus connected are 
 susceptible of receiving motions more or less limited in their di- 
 rection. In some cases the motion of one part necessarily produces 
 some corresponding motion in the other ; and in others, the motion 
 of one is quite independent of the other, which may be fixed. The 
 class of contrivances which fulfil these purposes are called joints. 
 
 The universal joint already explained belongs to this class. One 
 of the arms thus connected may be inclined to the other in almost 
 any direction whatever, when the joint is constructed as in Jig. 229. 
 But in thus changing its direction, no corresponding motion is im- 
 parted to the other arm. If, however, a twisting or rotatory mo- 
 tion round its own longitudinal axis be given to either arm, a 
 corresponding rotatory motion round its longitudinal axis will be 
 imparted to the other. In this manner, for example, a screw may 
 be turned by a twisting motion given to an arm having a direction 
 inclined at any angle to the longitudinal axis of the screw. 
 
 549. Gimbals. This is a mechanical contrivance, which bears 
 a close relation to the universal joint. An example of it, familiar 
 to every one, is presented by the apparatus for suspending a ship's 
 
258 THEORY OF MACHINERY. 
 
 compass (fig. 242.). The object is to keep the suspended body 
 vertical, whatever be the derangements to which the points of sus- 
 pension are liable. 
 
 A brass hoop is supported by two pins, projecting from points 
 of its external surface which are diametrically opposed to each 
 other. Another brass hoop is supported within the former, also by 
 two pins projecting from points of its external surface, diametri- 
 cally opposed one to the other, which play in two holes made in 
 the former hoop at the extremities of that diameter, which is at 
 right angles to the diameter at the extremities of which the first 
 points of support are placed. 
 
 Fig. 14*. 
 
 A similar mode of suspension is applied to the marine barometer. 
 550. Ball and socket. The joint thus called has some, but 
 not all, of the qualities of the universal joint. The extremity of one 
 of the arms is attached to a solid ball of metal, and that of the other 
 to a hollow ball, the internal diameter of which is equal to the 
 diameter of the solid ball. If one of 
 these balls were contained within the 
 other, the arm of the solid ball, pass- 
 ing through a hole made to fit it in 
 that of the hollow ball, the two arms 
 would be incapable of any change 
 whatever, in their relative position, 
 and the joint so formed would form a 
 rigid and invariable connection. But 
 if the hole in the hollow ball, through 
 which the arm of the solid ball passes, 
 be much larger than the arm, as shown 
 in Jig. 243., then the arm will have a 
 certain play, the solid ball moving 
 within the hollow ball, and the extent 
 Fi *- * of such play would obviously depend 
 
BALL AND SOCKET CRADLE. 
 
 259 
 
 on the magnitude of the opening. A joint thus constructed is that 
 which is called the ball and socket joint. 
 
 This joint is represented in^. 244. 
 
 It is evident that the inclination which 
 can be given to the arms will be limited 
 by half the circumference of a section of 
 the socket. 
 
 The greatest play which such a joint 
 can have will be produced when the 
 socket exceeds an exact hemisphere by 
 just enough to enable it to retain the 
 ball within it ; and it follows, therefore, 
 that the angle under the arms thus con- 
 nected, when they are most inclined, will 
 be a little greater than 90. 
 
 If either arm be fixed, the play of the 
 other will be limited, by the directions of 
 the sides of a cone, whose vertex is at the 
 centre of the ball, and whose base is 
 limited by the part of the ball left un- 
 covered by the socket. 
 
 The arms connected by this joint are 
 moved independently of each other. 
 Either may be turned round its longi- 
 tudinal axis, without imparting a corre- 
 sponding motion to the other. 
 
 The socket is usually made of such a magnitude, that the ball 
 moves in it with a certain degree of friction sufficient to retain the 
 arms in whatever position may be given to them. 
 
 551. Cradle joint. By this joint, either arm connected is 
 allowed to move in a given plane round 
 the other arm. The joint is usually 
 formed by a pin passing through the 
 centre of circular discs of metal formed 
 at the end of the arms, having equal 
 diameters, and placed concentrically face 
 to face ; sometimes, one such disc is at- 
 tached to one arm passing between two, 
 which are attached to the other : such a 
 joint is represented in Jig. 245. 
 
 It is by a joint of this kind that the 
 legs of compasses are connected. 
 
 552. Hinge. This is, in fact, a va- 
 riety of the cradle joint. Its construction 
 is rendered so familiar by the lids of 
 
 fig. Z44. 
 
260 THEORY OF MACHINERY. 
 
 boxes, and the suspension of doors, that it will not require further 
 explanation. 
 
 553. Trunnions. The method of supporting heavy bodies, 
 which require to be freely movable in a vertical plane, thus deno- 
 minated, consists of two strong cylindrical pins, projecting from 
 its sides, which rest in semi-cylindrical grooves formed at equal 
 heights, in two pillars, so that a line, connecting the trunnions, shall 
 be horizontal. A body thus supported will be free to assume 
 every possible direction, in a vertical plane at right angles to the 
 line joining the supports of the trunnions. 
 
 554. Axles. These are, in general, the centres round which 
 wheels revolve ; but the wheels may either turn upon or with the 
 axles. 
 
 The wheels of ordinary carriages turn upon their axles, the 
 axles being strong iron bars, extending horizontally and trans- 
 versely under the body of the vehicle. Their extremities, which 
 pass through the centre of the wheels, have a cylindrical or slightly 
 conical form. The centre part of the wheel in which the spokes 
 are inserted, is called the nave, and the central part within it, into 
 which the extremity of the axle passes, is called the box. The 
 inner surface of the box has a form corresponding to that of the 
 axle, but a little larger, so as to admit the play of some lubricating 
 fluid between them, by the interposition of which the actual con- 
 tact of the metallic surfaces of the axle and the box is prevented, 
 no matter what be the pressure by which they are urged one 
 against the other. 
 
 In railway carriages, the wheels and axle form one solid piece ; 
 the cylindrical or conical extremities of the axle project outside 
 the wheels, and certain parts called bearings rest upon them. In 
 this case, therefore, the extremities of the axle revolve with the 
 wheels under the bearings. Reservoirs of lubricating matter, called 
 the grease boxes, are placed immediately over the axles, by which 
 the grease is let continually down upon the axles. 
 
 555. Telescope joint. -This mode of connection, which is of 
 very extensive use in mechanics, is rendered familiar to every one, 
 by the manner in which the tubes of telescopes and opera-glasses 
 are adjusted. It is used, in general, where a rod or pillar requires 
 to have its length varied occasionally, according to the circum- 
 stances in which it is used. 
 
 556. Bayonet joint. The joint thus called takes its name 
 from the simple and well known mechanical contrivance by which 
 the handle of a bayonet is fastened on the end of a musket barrel. 
 Its simplicity and efficiency have rendered it of extensive use in 
 practical mechanics. As in the telescope joint, one tube passes 
 ^within another ; but in this case they are held in a fixed position 
 
BAYONET JOINT- CLAMPS. 
 
 261 
 
 by means of a pin which projects from the inner tube, and passes 
 through a rectangular opening in the outer tube. 
 
 fig. 246. represents the position of the pin when the inner 
 tube is pressed into the outer, until the pin comes against the edge 
 of the opening and stops its further progress. In this position, 
 
 Fig. 146. 
 
 Fig. 247. 
 
 however, the inner tube might be detached from the outer by any 
 force tending to draw it out. To prevent this, it is turned round 
 its axis, until the pin enters the rectangular opening in the outer 
 tube, as shown in Jig. 247. It cannot be detached then without 
 two successive motions being given to it ; one round, and the other 
 parallel to the axis of the tube. 
 
 557. Clamps and adjusting: screws. "When parts of mecha- 
 
 nism usually separated require to be tem- 
 porarily connected, the object is attained by 
 clamps, which are made in an infinite variety 
 of forms according to the circumstances in 
 which they are applied. An example of this 
 class of contrivances is shown in Jig. 248. The 
 ends of the parts to be united being placed 
 one upon the other, are introduced into the 
 rectangular opening shown in the figure, and 
 the screw is then turned until it urge them by a pressure suffi- 
 ciently strong to hold them together. 
 
 558. Couplings. In the practical operation of machinery, it 
 is frequently necessary to be enabled to suspend at pleasure, or to 
 recommence, the motion of a wheel or wheels. This is accomplished 
 by a class of contrivances called couplings. 
 
 When the motion of a wheel is imparted to another by an endless 
 band of leather stretched tightly round them, the motion can be 
 suspended or resumed, at pleasure, by making the band loose, and 
 giving it the requisite tightness by pressing upon it a roller at any 
 intermediate point between the wheels ; so long as the pressure of 
 the roller is maintained, the rotation of one wheel will be imparted 
 
 s 3 
 
 Fig. 248. 
 
262 THEORY OF MACHINERY. 
 
 to the other ; but the moment the roller is removed, the communi- 
 cation of the motion will be suspended. 
 
 In some cases, the communication of the motion is discontinued 
 or altered by removing the band from one or other of the wheels 
 by lateral pressure. By this expedient, the velocity of the motion 
 imparted may be modified. Thus, if several wheels be fixed side 
 by side, on the same axis, having different diameters, the strap may 
 be shifted from one to another, the velocity of rotation imparted 
 being increased in the same proportion as the diameter of the wheel 
 receiving the motion is diminished. 
 
 Couplings are sometimes constructed by providing two wheels 
 upon the same shaft, one turning upon and the other with the shaft. 
 Let us suppose that the moving power keeps the wheel which 
 turns upon the shaft constantly in revolution, but does not act 
 upon that which turns with the shaft. In that case, it is evident 
 thaf no rotation would be imparted to the shaft. Now let us sup- 
 pose that the wheel w w (Jig- 249.) is that which turns upon the 
 
 Fig. 249. 
 
 shaft, that attached to it is a collar c embraced by a fork r at 
 the end of a lever L, that the wheel with the collar is capable of 
 sliding longitudinally on the shafts, and that its surface towards s' 
 is cut into a sort of angular teeth corresponding with similar ones 
 formed in the face of the wheel w' w', which turns with the axle ; 
 When, by means of the lever L and the fork F, the wheel w w is 
 moved towards the wheel w' w', so that the teeth are inserted the 
 cne within the other, the two wheels thus forming a single one, the 
 wheel w w will impart its motion to w' w', and therefore to the 
 shaft s s'. 
 
 On the contrary, when it is desired to suspend the motion of the 
 shaft, the wheel w w is drawn by the lever L r towards s, and 
 disengaged from w' w'. 
 
 A coupling by which the rotation of a shaft can be suspended or 
 
COUPLINGS. 
 
 263 
 
 reversed at pleasure is shown in fig. 250., where w and w' are 
 two bevelled wheels, both driven by a third bevelled wheel R, and 
 therefore, made to revolve in contrary directions. Let us suppose 
 that these wheels w and w' turn upon the shaft s s', and carry at 
 their centres two small wheels N N', the faces of which are cut into 
 
 angular teeth; let M be two 
 similar wheels similarly cut, pre- 
 senting their opposite faces to 
 wards those of N N' ; let these 
 wheels M revolve with the shaft, 
 but so as to be capable of slid- 
 ing upon it ; let the fork r, of 
 the lever L F, embrace between 
 its prongs the collar M. By 
 pressing this lever in the one 
 direction or the other, the wheels 
 M may be coupled, either with 
 N or N', so as to revolve with 
 them, and thus impart to the 
 shaft s s' a motion in the one 
 F . direction or the other. 
 
 If the lever L F be kept in the 
 
 vertical position, the wheels M not being connected with either N or 
 N', the motion will be suspended. 
 
 Another method of coupling is represented in fig. 251., where 
 
 Fig. 151. 
 
 the wheel w w has two or more holes in it, corresponding with 
 pins projecting from w r w' ; the former revolves with and the 
 latter upon the axle ; the wheel w w slides upon the axle, so that 
 when pressed towards .w' w', the pins enter the holes and the 
 wheel w' w' revolves with w w 
 
264 THEORY OF MACHINERY. 
 
 CHAP. V. 
 
 RESISTING FORCES. 
 
 559. A physical agent capable of imparting motion to a quiescent 
 body is called a force. It is evident that such an agent would also 
 be capable of increasing the velocity of a body already in motion 
 if it were applied in the direction of the motion, or diminishing the 
 velocity, or even altogether destroying the motion and bringing 
 the body to rest if it were applied in an opposite direction. 
 
 This principle is not, however, convertible : although it follows 
 that an agent capable of imparting motion is also capable of dimi- 
 nishing or destroying it, it does not follow that an agent capable 
 of diminishing or destroying motion is also capable of imparting or 
 increasing it. 
 
 Forces of the class to which we now refer are capable of dimi- 
 nishing the velocity of a body in motion, and of bringing it to rest, 
 but they are incapable of imparting motion to a body at rest, or 
 of augmenting any motion it may have. The former class of 
 forces, which are capable of producing or increasing motion, may 
 be described for distinction active forces, and the latter passive 
 forces. 
 
 The force of gravity, for example, comes under the former class. 
 A body freely suspended, being disengaged and submitted to the 
 action of gravity, is put in motion, and its motion is continually 
 accelerated as it moves downwards, until it encounters some 
 obstacle which brings it to rest. 
 
 If the same body be projected upwards with the velocity with 
 which it strikes the ground, the force of gravity will then gradually 
 diminish its motion until it rises to the height from which it fell, 
 where its motion will altogether be destroyed. 
 
 Of the passive or resisting forces, the most important are fric- 
 tion and the resistance of fluid media, such as air or water. 
 All bodies moving at or near the surface of the eai th are subject 
 to some, or all, of these forces, and, consequently, all terrestrial 
 motions whatever are liable to constant retardation ; and, to be 
 maintained, require the constant agency of some impelling force to 
 repair the loss produced by the resisting forces to which they are 
 exposed. 
 
 The smallest attention to the phenomena which form the subject 
 of mechanical inquiries, will render manifest the great importance 
 of investigating and comprehending the effects of resisting forces. 
 
 In the preceding part of this volume, the construction and pro- 
 
FKICTIOtf. 265 
 
 perties of machinery have been explained, on the supposition that 
 the moving force of the power is transmitted to the working point 
 with undiminished effect. In order to disembarrass the questions 
 of their complexity, and present them in the most simple and 
 intelligible form, machines have been considered as absolutely free 
 from the effects of all resisting forces ; surfaces moving in contact 
 have been considered to be perfectly free from friction; axles 
 were regarded as mathematical lines; pivots as mathematical 
 points ; ropes as perfectly flexible ; and, in a word, the effect of 
 the moving power has been considered as absolutely undiminished 
 by any resistance whatever, in its transmission through the ma- 
 chinery to the working point. 
 
 It is scarcely needful to observe, that none of these suppositions 
 are perfectly true. The surfaces of the machinery which move in 
 contact are never perfectly smooth ; axles have always definite 
 thickness, and move in ^ sockets never perfectly polished; ropes 
 have considerable rigidity, and this rigidity is necessarily greater 
 in proportion to their strength. Much has been accomplished, it 
 is true, by a variety of expedients, to diminish these resistances ; 
 highly polished surfaces and effective lubricants have been applied 
 to obtain additional smoothness ; but, still the surfaces in contact 
 continue to be studded with small asperities, which, coming con- 
 stantly in opposition to each other in their motion, produce consi- 
 derable resistance and, robbing the moving power of a great part 
 of its efficacy, transmit it with proportionally diminished intensity 
 to the working point. 
 
 To estimate therefore, correctly, the practical effects of any 
 machinery, it is essential that we should calculate the effect of this 
 resistance, and subduct it from that effect of the power which has 
 been computed on the theoretical principles established in the 
 preceding chapters ; the overplus of effect after this deduction is 
 all that part of the power which can be regarded as practically 
 available. 
 
 560. Friction. The effect of friction on a power supporting 
 a weight or resistance at rest is different from its effect when the 
 weight or resistance is moved. In the one case, friction assists ; 
 in the other, it opposes the power. 
 
 Let us suppose, for example, a power p supporting a weight w, 
 by means of a single movable pulley. From what has been already 
 proved, it is evident that if the power P be half the weight of w, 
 they would be in equilibrium, and the power would keep the 
 weight at rest if the pulley were subject to no friction ; and in 
 that case the slightest diminution of the power would cause the 
 weight to descend, and draw the power upwards. But if the 
 pulley be subject, as it always is in practice, to friction, then a 
 
266 THEORY OF MACHINERY. 
 
 small diminution of the power will be resisted by this friction, 
 and the weight will not descend and overcome the power until the 
 diminution of the power shall become so great as to enable the 
 weight to overcome the friction. 
 
 561. It follows, therefore, that when a pulley or any other 
 machine is subject to friction, a less power is sufficient to support 
 a weight at rest than would be necessary if there were no friction ; 
 and the greater the friction is, the less will be the power necessary 
 to support the weight. It is in this sense that friction is said to 
 aid the power, when the weight or resistance is supported at rest. 
 But if the power be required, not merely to support the weight, 
 but to raise it, then we shall find that the friction, instead of 
 aiding, opposes the power. 
 
 Let us suppose, for example, the power p acting on the weight 
 through the intervention of a single movable pulley, the power 
 being half the weight. In the absence of friction, the slightest 
 addition to the power will cause it to descend, and to raise the 
 weight ; but when the machine is subject to friction, then the 
 power will not descend until it shall receive such an addition as 
 will be sufficient to overcome the friction. 
 
 562. This circumstance modifies materially the conditions of 
 equilibrium. Representing by p that amount of the power applied 
 to any machinery whatever which would keep the weight w in 
 equilibrium in the absence of friction, let f express the addition 
 which must be made to P, in order to enable it to overcome the 
 friction and put 'the weight in motion; then / will also express 
 the amount by which the power must be diminished, in order to 
 enable the weight to prevail over it and to descend. It is evident, 
 therefore, that any power which is less than p -f /, and greater 
 than P /, would keep the weight in equilibrium and at rest. 
 
 563. It may therefore be inferred, generally, that when a 
 machine of any kind is used simply to sustain a weight or to 
 balance a resistance, the friction, acting in common with the 
 power, becomes a mechanical advantage. In many instances this 
 resisting force constitutes the entire efficiency of the instrument. 
 Thus, when screws, nails, or pegs are used to bind together the 
 parts of any structure, their friction with the surface with which 
 they are in contact prevents their recoil, and gives them their 
 entire binding power. In the ordinary use of the wedge itself we 
 have another striking example of the mechanical advantage of 
 friction. When the wedge is used for any purpose, as, for example, 
 to split timber, it is urged forward by percussion, the action of the 
 moving power being only instantaneous, and being totally sus- 
 pended between each successive blow. 
 
FRICTION. 267 
 
 But for the resisting force of the friction which takes place 
 between the surface of the wedge and the surface of the timber, 
 the wedge would react after each blow, and render abortive the 
 action of the moving power. The friction, therefore, in this case 
 plays the part of a ratchet wheel, preventing the reaction of the 
 wedge, and making good the action of the power. 
 
 564. Notwithstanding the disadvantages which attend the 
 presence of friction in machines, it is an agent eminently useful in 
 giving stability to structures, and in giving efficiency to the move- 
 ments of almost all bodies, natural and artificial. Without friction, 
 most structures, natural and artificial, would fall to pieces. The 
 stones and bricks used in building owe to the mutual friction of 
 their surfaces a great part of their solidity. Manual exertion 
 would become impracticable if no friction existed between the 
 limbs and the objects upon which they act. The friction between 
 the foot and the ground gives a purchase to the muscular force, 
 so as to enable it to produce a progressive motion. Without 
 friction, every effort of the foot to propel the body forward would 
 be attended with a backward action, so that no progressive motion 
 would ensue. The difficulty of moving upon ice, or upon ground 
 covered with greasy or unctuous matter, illustrates this. Without 
 friction we could not hold any body in the hand. The difficulty 
 of holding a lump of ice is an example of this. Without friction, 
 a locomotive engine could not propel its load ; for if the rail and 
 the tire of the driving-wheels were both absolutely smooth, one 
 would slip upon the other, without affording the necessary pur- 
 chase to the steam power. 
 
 565. Sliding: and rolling:. Friction is manifested in different 
 ways, according to the kind of motion which one surface has upon 
 the other. When one surface slides upon the other in the manner 
 of a sledge, the friction is called sliding or rubbing friction. When 
 a, body rolls upon another, so that different points of such body 
 come into successive contact with each other, it is called rolling 
 friction. 
 
 566. The laws which regulate friction are derived exclusively 
 from experiments, independently of theory. There are no simple 
 or general principles from which they can be deduced by mathe- 
 matical reasoning. It is a matter of regret that, even amongst the 
 best conducted experiments that have been made, considerable 
 discrepancies are observable, and that differences of opinion pre- 
 vail between the most respectable authorities respecting many 
 particulars connected with the properties and laws of these resisting 
 forces. 
 
 Although these laws, so far as they are known, depend thus 
 
z68 
 
 THEORY OF MACHINERY. 
 
 Fig. 2 5 Z. 
 
 wholly on experiment, yet the general principles of science, as 
 applied to them, are far from being useless. They serve as a 
 guide in the selection of the experiments which are best adapted 
 to develop those laws which are the subject of inquiry, as well as 
 to show the inconclusiveness of some experiments on which reliance 
 might otherwise be placed, and thus enable us further to deduce 
 from the results of experimental inquiries numerous useful prao 
 tical results. 
 
 567. There are two methods by which the quantity of friction 
 produced when two surfaces are moved one upon another can be 
 ascertained. 
 
 1st. The surfaces being rendered perfectly flat, let one be fixed 
 in a horizontal position on a table 
 T T (fig. 252.), and let the other be 
 attached to the bottom of a box B E, 
 adapted to receive weights so as to 
 vary the pressure. 
 
 Let a flexible cord be attached to 
 this box, and being carried parallel 
 to the table, let it pass over a fixed 
 pulley at P, and have a dish sus- 
 pended to it at D. 
 If no friction existed between the surfaces, the smallest weight 
 suspended from D would cause the box B E to move with a uni- 
 formly accelerated motion along the table towards the point ; but 
 the resistance of friction renders it necessary, before motion can 
 take place or be maintained, that the weight D shall be equal to 
 the amount of this friction. If the weight D and the friction be 
 equal, then, the power and the resistance being in equilibrium, the 
 box B E, if put in motion, will move towards the point with any 
 velocity, continued uniform, which may be imparted to it. If the 
 weight D be greater than the friction, then the motion of the box 
 towards P will be accelerated ; and if the weight be less than the 
 friction, then any motion which may be given to the box will be 
 retarded, and will soon cease altogether. 
 
 The determination, therefore, of the weight acting at D, which 
 represents the exact amount of the 
 friction, will depend upon the velocity 
 given to the box in the direction B P, 
 being maintained uniform. 
 
 2ndly. Let one of the surfaces be 
 attached, as before, to a flat plane 
 A B (fig' 2 53-) but instead of being 
 horizontal, let it be inclined, and so arranged that the incli- 
 
LUBRICANTS. 269 
 
 nation may be varied at pleasure. The box w being constructed 
 as before, and placed upon the plane, let such an elevation be 
 given to the plane that the box shall be capable of moving down 
 it with an uniform velocity, without acceleration or retardation. 
 If the elevation be greater than this, the motion of the box down 
 the plane will be accelerated, and the gravity of the plane will be 
 greater than the friction ; if it be less, the motion of the box will 
 be retarded, and the gravity will be less than the friction. 
 
 568. That particular inclination of the plane corresponding to 
 the friction of any given surface, which renders the gravity of the 
 plane equal to the friction, is called the angle of repose. According 
 to the principles already explained, it follows that in these cases, 
 if the length of the plane A B represent the total weight w, the 
 gravity down the plane, which is equal to friction, will be repre- 
 sented by the height A E, and the pressure upon the plane will be 
 represented by the base B E, and consequently the ratio of the 
 friction to the pressure will be that of the height A E to the base 
 B E. Experiments conducted according to both these methods 
 have given nearly the same results, which may be summarily stated 
 to be as follows. 
 
 569. The proportion of the friction to the pressure, when the 
 quality of the surface is given, is always the same, no matter how 
 the weight or the magnitude of the surface may be varied, except 
 in extreme cases, when the proportion of the pressure to the 
 surface is very great or very small. 
 
 570. If the surfaces in contact be placed with their grains in 
 the same direction, the friction will be greater than if their grains 
 cross each other. Smearing the surfaces with unctuous matter 
 diminishes the friction, probably by filling the cavities between 
 those minute projections which produce the friction. 
 
 571. The pivots of pendulums or balances are usually made of 
 steel, and rest upon hard polished stones, different surfaces being 
 used for the purpose of diminishing the amount of friction. Brass 
 sockets are generally used for iron axles on the same principle. 
 
 572. lubricants. In the selection of lubricants, those of a 
 viscous nature are selected, in the case of the rough surfaces of 
 softer bodies, and those which are more fluid are applied to the 
 smoother surfaces of harder bodies. Thus, when metal moves 
 upon wood, tallow, tar, or some solid grease is generally used ; 
 but when metal moves upon metal, oil is preferred ; and the harder 
 and the smoother the metal, the finer the oil. Finely pulverised 
 plumbago is found to be a very efficient agent in diminishing 
 friction, especially as applied to the axles of carriages and the 
 shafts of machinery. 
 
270 THEORY OF MACHINERY. 
 
 573. Rolling: friction. The friction which attends a rolling 
 motion is very much less than that which would attend a sliding 
 or rubbing motion with the same surfaces and under the same 
 pressure. Hence it is that rollers are used with so much success 
 as an expedient for diminishing friction. A roughly chiselled 
 block of stone, weighing 1 080 Ibs., was drawn from the quarry on 
 the surface of the rock by a force of 758 Ibs. It was then laid 
 upon a wooden sledge, ;ind drawn upon a wooden floor, the trac- 
 tive force being 606 Ibs. When the wooden surfaces moving 
 upon one another were smeared with tallow, the tractive force was 
 reduced to 182 Ibs. ; but when the load was, in fine, placed upon 
 wooden rollers three feet in diameter, the tractive force was 
 reduced to 28 Ibs. 
 
 574. Sledges. Although it is therefore obvious, on general 
 principles, that the friction of sliding considerably exceeds that of 
 rolling, vehicles supported on straight and parallel edges, sliding 
 over the surface of the ground, are often found more convenient 
 than wheel carriages. In climates where snow and frost prevail 
 during the winter, sledges supersede all other carriages. In New 
 
 Fig. 154. 
 
 York and other cities of North America, public vehicles, such as 
 omnibuses, hackney coaches, and all other conveyances, are taken 
 off the wheels, and placed upon sledges, on which they are drawn 
 along the streets and roads. 
 
 At all seasons and in all climates sledges are, for certain 
 purposes, more convenient than wheel carriages; thus, dray- 
 men are provided with small ones to take barrels into narrow 
 streets, where the approach of the dray would be difficult or im- 
 practicable. 
 
SLEDGES ROLLERS. 
 
 Fig. 155. 
 
 Use of rollers. 
 
 Fig. z 5 6. 
 
 When heavy weights, such as large 
 blocks of stone, are 
 required to be moved 
 through short dis- 
 tances, the applica- 
 tion of rollers is at- 
 tended with great ad- 
 vantages ; but when 
 loads are to be trans- 
 ported to consider- 
 able distances, the 
 process is inconvenient and slow, owing to the necessity of continu- 
 ally replacing the rollers 
 in front of the load, as 
 they are left behind 
 by each progressive ad- 
 vancement (fg. 256.). 
 
 The wheels of car- 
 riages may be regarded 
 
 Fig 257 as "rollers which are con- 
 
 tinually carried forward 
 
 with the load. In addition to the friction of the rolling motion on 
 the road, they have, it is true, the friction of the axle in the nave ; 
 but, on the other hand, they are free from the friction of the 
 rollers with the under surface of the load or the carriage in which 
 the load is transported. The advantage of wheel carriages in 
 diminishing the effects of friction is sometimes attributed to the 
 slowness with which the axle A (fig. 257.) moves within the box, 
 compared with the rate at which the wheel moves over the road ; 
 tut this is erroneous. The quantity of friction does not in any 
 case vary considerably with the velocity of the motion, but least 
 of all does it in that particular kind of motion here considered. 
 
272 
 
 THEORY OF MACHINERY. 
 
 Castors (figs. 258, 259.) placed on the feet of tables and other 
 articles of furniture facilitate their movement from one place to 
 another by substituting rolling for sliding friction. 
 
 Fig. 259. 
 
 Fig. 260. 
 
 576. Friction rollers (s s, fig. 260.) are sometimes inter- 
 posed between an axle and its bearings, to diminish the friction 
 attending their motion one upon the other. 
 
 577. Friction wheels. In certain cases where it is of great 
 
 importance to remove the effects of 
 friction, a contrivance called fric- 
 tion wheels or friction rollers is 
 used. The axle of a friction wheel 
 A (fig. 261.), instead of revolving 
 within a hollow cylinder which is 
 fixed, rests upon the edges of wheels 
 B B, which revolve with it : the 
 species of motion thus becomes that 
 in which the friction is of least 
 amount. 
 
 578. In carriages, the roughness 
 of the road is more easily overcome 
 by large wheels than by small ones ; 
 hence we see wheels of very great 
 magnitude used for carrying beams 
 of timber of extraordinary weight. 
 The animals drawing these, notwithstanding their weight, do not 
 manifest any considerable exertion. The cause of this arises, 
 partly from the carriage- wheels bridging over the cavities in the 
 road, instead of sinking into them, and partly becaiise, in surmount- 
 ing obstacles, the load is elevated less abruptly. 
 
 Fig.i6i. 
 
LINE OF DRAUGHT. 273 
 
 579. Xiine of draught. If a carriage were capable of moving 
 on a road absolutely free from friction, the most advantageous 
 direction in which the tractive force could be applied, would be 
 parallel to the road ; but when the motion is impeded by friction, 
 as in practice it always is, it is better that the line of draught 
 should be inclined to the road, so that the drawing force may be 
 exerted partly in lessening the pressure on the road, by in some 
 degree elevating the carriage, and partly in advancing the 
 load. 
 
 It can be established by mathematical reasoning, that the best 
 line of draught, in all cases, is determined by the angle of repose, 
 that is to say, the traces should form an angle with the road equal 
 to the elevation of a plane which would exactly overcome the fric- 
 tion. Hence it appears that the smoother the road, and the 
 more perfect the carriage, and consequently the less the fric- 
 tion, the more nearly parallel to the road the line of draught 
 should be. 
 
 In wheel carriages, there exist two sources of friction : one 
 which prevails between the tires of the wheel and the road on 
 which they run, the amount of which depends on the quality of 
 the road ; and the other which prevails between the axle and the 
 nave of the wheel in which it turns. This latter is sliding friction ; 
 but the rubbing surface is small, being the line of contact of the 
 axle with the nave or socket. 
 
 580. From the structure of the axle and the nave, this source 
 of friction admits of being almost indefinitely diminished, by the 
 application of lubricants and other expedients. The other resist- 
 ance, depending on the action of the tires of the wheels, amounts, 
 on well-paved roads, to about ^th of the load. On gravelled 
 roads it is but -g^th ; and when a fresh layer of gravel has been 
 laid, it is increased to the -^th of the load. 
 
 It is found, however, on a well macadamized road, when in good 
 order, that the resistance does not exceed the ^th or ? \yth of the 
 load. 
 
 581. Railway. The most perfect modern road is the iron 
 railway, by which the resistance due to friction is reduced to an 
 extremely small amount. 
 
 The rolling portion of the wheels is in this case diminished by 
 substituting for the surface iron bars called rails, supported upon 
 cross beams of timber (fig. 262.), at distances apart corresponding 
 to that of the wheels, which are formed with ledges or flanges 
 projecting from their tires. These falling within the rails (fig. 263.) 
 confine the vehicle to the rails, the even surfaces of which in con- 
 tact with the tires produce very little resistance. 
 
 Various experiments have been made, with a view to determine 
 
274 THEORY OF MACHINERY 
 
 Fig. 261. 
 
 this resistance ; but much difficulty arises, owing to the effects of 
 atmospheric resistance being combined with those of friction. An 
 extensive series of experiments was made by the author of this 
 
 volume, in the year 1 838*, with a view to determine the amount 
 of resistance to railway trains ; the result of which showed that 
 this resistance is much more considerable than it had been pre- 
 viously supposed to be ; but that it depends in a great degree upon 
 the velocity, and probably arises more from the resistance of the 
 air, than from friction, properly so called. 
 
 582. Brakes. Friction is generally resorted to as the most 
 convenient method of retarding the motion of bodies, and bringing 
 
 * The details of these experiments will be found in the published reports 
 of the meetings of the British Association in 1838 and 1841. 
 
BRAKES. 
 
 275 
 
 them to rest. Expedients of various forms, called brakes^ have 
 been contrived for this purpose. 
 
 Fig. 164. 
 
 The form of brake called a shoe, used in travelling carriages, is 
 shown \nfig> 264.. 
 
 Fig. ^65. - 
 
 The brake used in diligences and other heavy vehicles on the 
 continental roads, which is similar in principle to those used in 
 
 railway carriages, is shown in 
 jig. 265. Surfaces of wood 
 are pressed against the tires 
 of the wheels by a combination 
 of levers pressed by the hand 
 of the guard or conductor. 
 
 A brake used in machinery 
 is shown in^. 266., consisting 
 of a strap tightly drawn upon 
 the tire of the wheel intended 
 
 Fig. 166. 
 
 to be retarded or stopped. 
 
 583. Friction is sometimes used as a point of resistance, as 
 
2 7 6 
 
 THEORY OF MACHINERY. 
 
 where a cable is coiled round a post to arrest the progress of a 
 vessel (fig. 267.). 
 
 Fig. 267. 
 
 584. Imperfect flexibility of ropes. When ropes or cords 
 form a part of machinery, the effects of their imperfect flexibility 
 are in a certain degree counteracted by bending them over the 
 grooves of wheels. But although this so far diminishes these 
 effects as to render ropes practically useful, yet still, in calculating 
 the power of machinery, it is necessary to take into account some 
 consequences of the rigidity of cordage, which even by these means 
 are not quite removed. 
 
 To explain the way in which the stiffness of a rope modifies the 
 operation of a machine, we shall suppose it bent over a wheel, and 
 stretched by weights A B, fig. 268., at its extremities. The weights 
 A and B, being equal, and acting at c and D in opposite ways, 
 balance the wheel. If the weight A receive an addition, it will 
 overcome the resistance of B, and turn the wheel in the direction 
 
 Fig. 169. 
 
 Fig. 470. 
 
 DEC. Now, for the present, let us suppose that the rope is per- 
 fectly inflexible, the wheel and weights will be turned into the 
 position represented in fig. 269. The leverage by which A acts 
 will be diminished, and will become o r, having been before o c ; 
 
RIGIDITY OF ROPES. 277 
 
 and the leverage by which B acts will be increased to o G, having 
 been before o D. 
 
 But the rope, not being inflexible, will yield to the effect of the 
 weights A and B, and the parts A c and B D will be bent into the 
 forms represented in fig. 270. The preponderating weight A still 
 has a less leverage than the weight B, and consequently, a propor- 
 tionate part of the effect of the moving power is lost. 
 
 The extent to which the rigidity of cordage affects the motion 
 of machinery has been ascertained by experiment in a still more 
 imperfect manner than the results of friction. Many incidental 
 circumstances vary the conditions, so as to throw great difficulties 
 in the way of such an investigation. Different ropes, and the same 
 ropes at different times, produce extremely different effects, in- 
 fluenced by the circumstances of their dryness or humidity, the 
 quality of their material, the mode in which they are prepared and 
 twisted, &c. These circumstances, it is evident, do not admit of 
 being estimated or expressed with any degree of accuracy. It 
 may, however, be stated generally that the resistance produced by 
 the rigidity of a rope is directly proportional to the weight that 
 acts upon it, and to its thickness. Other things being the same, it 
 is also in the inverse proportion of the diameter of the wheel or 
 axle upon which the rope is coiled ; the greater the weights, there- 
 fore, which are moved, and the stronger the ropes, the greater will 
 be the resistance proceeding from rigidity ; and, on the other hand, 
 the greater the diameter of the wheel or axle on which the rope 
 runs, the less in proportion will be the force necessary to overcome 
 the rigidity. 
 
 The resistance from new ropes is greater than from those of the 
 same quality which have been some time in use. Ropes saturated 
 with moisture offer increased resistance on that account. 
 
 585. Resistance of fluids. Since all the motions which com- 
 monly take place on the surface of the earth are made in the 
 atmosphere or in water, it is of great practical importance to 
 ascertain the laws which govern the resistance offered by these 
 fluids to the motion of bodies passing through them. A body 
 moving through a fluid must displace as it proceeds as much of 
 that fluid as fills the space which it occupies ; and in thus impart- 
 ing motion to the fluid, it loses by reaction an equivalent quantity 
 of its momentum. 
 
 If a body thus moving were not impelled by a motive force in 
 constant action, it would be gradually deprived of its momentum, 
 and at length brought to rest. Hence it is, that all motions which 
 take place on the surface of the earth, and which are not sustained 
 by the constant action of an impelling power, are observed 
 gradually to diminish, and ultimately cease. 
 
278 THEORY OF MACHINERY. 
 
 Since the resistance produced by a fluid to the motion of a solid 
 through it is equivalent to the momentum imparted by the solid to 
 the fluid, which it thrusts out of its way in its motion, it follows 
 evidently that, other things being the same, this resistance will be 
 proportional to the density or weight of the fluid. Thus the re- 
 sistance produced by air is less than that produced by water, in 
 the proportion of the weight of air to the weight of an equal bulk 
 of water. 
 
 The resistances are proportioned to the quantity of the fluid 
 which the moving body thrusts from the path ; and this again de- 
 pends upon the form and magnitude of the body, and more especially 
 on its frontage. 
 
 586. The resistance which a body encounters in moving through 
 a fluid is greater therefore, with a broad end foremost, than with 
 a narrow end foremost. A ship would evidently encounter a much 
 greater resistance if it were driven sideways, than if it move in the 
 direction of the keel. It would also encounter a greater resistance 
 if it moved stern foremost than in the usual direction. 
 
 The blade of a sword would be wielded with difficulty if moved 
 with its flat side against the air, whereas it is easily flourished when 
 moved edge foremost. Bodies whose foremost; ends have the form 
 of a wedge or a point, move through a fluid with less resistance 
 than if the pointed ends were cut off, and they presented a flat 
 surface to the fluid medium, because in their motion they act upon 
 the principle of the wedge, and more easily cleave the fluid. Na- 
 ture has formed birds and fishes in this manner to facilitate their 
 passage through the air and through the water. 
 
 587. The resistance which a body moving through a fluid en- 
 counters, increases in a high ratio with its velocity. If the bodv 
 move with the velocity of one foot per second, it will act in each 
 second upon a column of the fluid, whose base is equal to its own 
 transverse section, and whose length is one foot, and it will impel 
 such a column with a velocity of one foot per second ; but if the 
 velocity of the moving body be doubled, it will not only drive 
 before it in one second a column of fluid two feet long that is, 
 double the former length, but it will impel this column with double 
 the former speed. 
 
 The resistance, therefore, will be doubled on account of the 
 double quantity of the fluid, and again doubled on account of this 
 quantity receiving double the velocity. The moving force, there- 
 fore, imparted by the body to the fluid when the velocity is 
 doubled, will be increased in a fourfold proportion. 
 
 In the same manner it may be shown that if the velocity of the 
 body be increased in a threefold proportion, it will drive from its 
 path three times as much of the fluid per second, and impart to it 
 three times as great a velocity ; consequently, the moving force 
 
RESISTANCE OF FLUIDS. 279 
 
 which it will impart to the fluid will be nine times that which it 
 imparted moving at the rate of one foot per second. 
 
 In general, therefore, it follows that the moving force which the 
 body imparts to the fluid in moving through it, will be increased in 
 proportion to the square of the velocity ; but as the resistance 
 which the body suffers must be equal to the momentum which it 
 imparts to the fluid, it follows that the resistance to a body moving 
 through a fluid will be proportional to the square of its velocity. 
 
 588. Ponderous missiles. Hence it follows that missiles lose 
 a less proportion of their moving force in passing through the air, 
 as their weight is increased ; for, according to what has been stated, 
 the resistance which they suffer at a given velocity will be propor- 
 tional to their transverse section, which in this case is in the ratio 
 of the squares of their diameters ; but as their weight increases as 
 the cubes of their diameters, and is proportional to their moving 
 force, it follows that in increasing their magnitude their moving 
 force is increased in a higher ratio than the resistance they encoun- 
 ter. For example, if two cannon-balls have diameters in the propor- 
 tion of 2 to 3, the resistances which they will encounter at the same 
 velocity of projection will be in the ratio of 4 to 9, while their weights 
 willbein the ratio of 8 to 27. The resistance, therefore, of the smaller 
 ball will bear to its weight a greater ratio than that of the larger. 
 
 589. Resistance of the air to the motion of falling: bodies. 
 It has been shown that a body obedient to the action of gravity 
 would descend in a vertical line with a uniformly accelerated 
 motion. Its velocity would increase in proportion to the time of 
 its fall, so that in ten seconds it would acquire ten times the velo- 
 city which it acquired in one second ; but these conclusions have 
 been obtained on the supposition that no mechanical agent acts 
 upon the body save gravity itself.- If, however, the body fall 
 through the atmosphere, which in practice it must always do, it 
 encounters a resistance which augments with the square of the ve- 
 locity. Now, as the accelerating force of gravity does not increase, 
 while the resistance continually increases, this resistance, if the 
 motion be continued, must at length become equal to the gravita- 
 tion of the falling body, and, when it does, the velocity of the 
 falling body will cease to increase. It follows, therefore, that when 
 a body falls through the atmosphere, its rate of acceleration is 
 continually diminished ; and there is a limit beyond which the 
 velocity of its fall cannot increase, this limit being determined by 
 that velocity at which the resisting force of the air will become 
 equivalent to the gravity of the body. 
 
 As the resisting force of the air, other things being the same, 
 increases with the magnitude of the surface presented in the direc- 
 tion of the motion, it is evidently possible so to adapt the shape of the 
 
 T 4 
 
z8o THEORY OF MACHINERY. 
 
 falling body that any required limit may be impressed upon the 
 velocity of its descent. It is upon this principle that parachutes 
 have been constructed. 
 
 When a body attached to a parachute is disengaged from a bal- 
 loon, its descent is at first accelerated, but very soon becomes 
 uniform, and as it approaches the earth, the air becoming more 
 and more dense, the resistance on that account increases, and the 
 fall becomes still more retarded. 
 
 The theory of projectiles, which is founded upon the supposition 
 of bodies moving in vacuo, is rendered almost inapplicable in prac- 
 tice, in consequence of the great effect produced by atmospheric 
 resistance to bodies moving with such velocities as those which are 
 generally imparted to missiles. According to experiments and 
 calculations, it has been found that a 4~lb. cannon-ball, the range 
 of which in vacuo would be 23226 feet, was reduced to 6437 feet 
 by the resistance of the air. Hutton showed that a 6-lb. ball, pro- 
 jected with a velocity of 2000 feet per second, encountered a re- 
 sistance a hundred times greater than its weight. 
 
 CHAP. VI. 
 
 STRENGTH OF MATERIALS. 
 
 590. THE solid materials of which structures, natural and artificial, 
 are composed, are endued with certain powers, in virtue of which 
 they are capable of resisting forces applied to bend or break them. 
 These powers constitute an important class of resisting forces, and 
 are technically called in mechanics the strength of materials. 
 
 Experimental inquiries into the conditions which determine the 
 strength of solid bodies, and their power to resist forces tending to 
 tear, break, or bend them, are obstructed by practical difficulties, 
 the nature and extent of which have deterred many from encoun- 
 tering them. 
 
 591. These difficulties arise partly from the great force which 
 must be employed in such experiments, but more from the pecu- 
 liar nature of the bodies upon which such experiments are made. 
 
 The object of such an inquiry must necessarily be the establish- 
 ment of a general law, or such a rule as would be strictly observed 
 if the materials were perfectly uniform in their texture, and sub- 
 ject to no casual inequalities. In proportion, however, as such 
 inequalities are frequent, experiments must be multiplied, so that 
 
STRENGTH OF MATERIALS. 281 
 
 they shall include cases varying in both extremes, so that the 
 peculiar effects of each may be effaced from the general average 
 result which shall be obtained. These inequalities of texture, 
 however, are so great, that even when a general law has been 
 established by a sufficiently extensive series of experiments, it can 
 only be regarded as a mean result from which individual examples 
 will be found to depart in so great a degree, that the greatest cau- 
 tion must be observed in its practical observation. 
 
 Although the details of this subject belong more properly to 
 engineering than to an elementary treatise like the present, it may 
 nevertheless be useful to give a general view of the most important 
 principles which have been established. 
 
 592. A mass of solid matter may be submitted to the action of 
 a force tending to separate its parts in several ways, of which the 
 principal are 
 
 I. A direct pull ; as when a weight is suspended to a wire or a 
 
 rope, or when a tie-beam resists the separation of the walls 
 of a structure. 
 
 II. A direct pressure or thrust ; as when a weight rests upon a 
 
 pillar, or a roof upon walls. 
 
 III. When a force is applied to twist or wrench a body asun- 
 der by turning a part of it round a point within it. 
 
 IV. A transverse strain ; as when a beam, being supported at 
 its centre, weights are suspended from its ends, or, being 
 supported at its ends, a weight is suspended from its centre. 
 
 593. When a rod, rope, or wire is extended between forces 
 applied to its ends, and tending directly to stretch it, its strength 
 to resist such force is, other things being the same, in proportion 
 to the magnitude of its section. 
 
 Thus, suppose an iron wire stretched by a weight which it is 
 just able to support without breaking. It is evident that a wire 
 having twice the quantity of iron of the same quality in its thick- 
 ness would support double the weight, because such wire would be 
 in effect equivalent to two wires like the former combined. In 
 the same manner, a wire having three times the quantity of iron 
 of the same quality in its thickness, being equivalent to three wires 
 like the first, would support three times the weight, and so on. 
 Thus the power of bodies to resist a direct pull will be in general 
 in proportion to the area of their transverse section. In practice 
 it is found that when the length is much increased, the strength to 
 resist a direct pull is diminished. 
 
 This departure, however, from the general law is explained by 
 the increased probability of casual defects of structure in the in- 
 creased length; and, subject to such qualification, it may be stated 
 generally, that the strength of a body to resist tension, or a direct 
 
282 
 
 THEORY OF MACHINERY. 
 
 pull drawing from end to end, is in the direct ratio of the area of 
 its section made at right angles to the direction in which it is 
 stretched. 
 
 594. The strength of bodies to resist a direct pull is experi- 
 mentally estimated by attaching the upper extremity securely to a 
 point of support, and suspending weights to the lower extremity, 
 which are increased gradually until the body under experiment is 
 broken ; the weight which breaks it is taken as the expression of its 
 strength. The bodies which have been subjected to experiments 
 of this kind are chiefly metals, woods, and ropes, these being most 
 generally used in structures, in which their capacity for resisting 
 tension is of great importance, as in suspension bridges, iron roofs, 
 cables, cordage, &c. 
 
 595. In the following table are collected the results of the most 
 recent and' extensive experiments on this subject. The bodies 
 subjected to experiment are supposed to be in the form of long 
 rods, the cross section of which measures a square inch. In the 
 second column is given the amount of the breaking weights, which 
 are the measure of their strength. This table is selected from 
 various recent works. 
 
 Table showing the strength with which prisms of the undermentioned 
 
 substances resist a direct pull, expressed in Ibs. per square inch of 
 
 the area of their transverse section. 
 
 Name. Ibs. Ibs. 
 
 Name. Ita, Ibs. 
 
 ist. Metals: 
 
 2nd. Woods: 
 
 Steel, untempered - from 110690 10127094 
 ^_ tempered - 114791153741 
 
 Teak 
 Sycamore 
 
 - from 12915 to 15405 
 - 9030 
 
 cast - - 134256 
 
 Beech 
 
 12225 
 
 Iron, bar - - 53182 84611 
 
 Elm 
 
 9720 15040 
 
 plate, rolled - 53920 
 wire - - 58730 112905 
 
 Memel fir 
 Christiana deal 
 
 - 9549 
 - ., 12346 
 
 Swedish malle- 
 
 Larch 
 
 12240 
 
 able - 72064 
 
 Oak 
 
 - 10367 25851 
 
 English do. - 55872 
 
 Alder 
 
 11453 21730 
 
 cast - - 16243 19464 
 
 Lime 
 
 - 6991 20796 
 
 Silver, cast - 40997 
 
 Box 
 
 14210 24043 
 
 Copper, do. - - 20320 37380 
 
 Pinus fylv 
 
 - 17056 20395 
 
 hammered - ., 37770 39968 
 
 Ash 
 
 - 13480 23455 
 
 Brass, cast - 17947 . T 9472 
 
 Pine 
 
 10038 149*5 
 
 wire - - 47"4 58931 
 
 Fir 
 
 - 6991 12876 
 
 plate - 52240 
 
 
 Gold 20490 65237 
 
 3rd. Cords : 
 
 Tin - 3228 6666 
 
 Hemp twisted Ibs. 
 
 cast - - 4736 
 Bismuth, cast - 3137 
 
 to I inch thick . - 8746 
 
 Zinc - - 2820 
 Antimony, cast - 1062 
 
 I to 3 - - 6800 
 3 to 5 . - - 5315 
 
 Lead, molten - 887 1824 
 
 5 to 7 - - - 4860 
 
 wire - - 2543 3823 
 
 
 596. From this table it appears that the strongest of the bodies 
 for resisting tension is iron, and that the strongest condition of 
 iron is that of tempered steel. In general, metals when cast are 
 less strong than when hammered Thus, cast iron has not one 
 third of the tenacity of wrought iron. It is also found that metals 
 
STRENGTH OF MATERIALS. 283 
 
 which are composed of two or more alloyed together are often 
 stronger than any of their components. Thus, brass wire, which 
 is composed of zinc and copper, has greater tenacity than copper 
 wire, although the tenacity of zinc, as appears by the table, is ex- 
 tremely small. 
 
 597. It is also found that the strength of metals is affected by 
 their temperature, being diminished in general as their tempera- 
 ture is raised. Sudden, frequent, and extreme changes of tem- 
 perature impair tenacity. 
 
 598. The woods are subject to extreme variations, produced 
 in general by the great inequalities which are incidental to them. 
 Thus the strength of oak varies, as appears by the table, between 
 the limits of loooo and 25000 Ibs. It is found that trees which 
 grow in mountainous places have greater strength than those which 
 grow on plains, and also that different parts of the same tree, such 
 as the root, trunk, and branches, vary in strength within wide 
 limits. 
 
 599. It is found friat cords of equal thickness are strong in pro- 
 portion to the fineness of their strands, and also to the fineness of 
 the fibres of which these strands are composed. It is found also 
 that their strength is diminished by being overtwisted. Ropes 
 which are damp are stronger than ropes which are dry, those 
 which are tarred than the untarred, the twisted than the spun, and 
 the unbleached than the bleached. Other things being the same, 
 silk ropes are three times stronger than those composed of flax. 
 
 The strength of many substances is increased by compressing 
 them ; this is the case with leather and paper, for example. 
 
 600. Animal and vegetable substances which, being originally 
 liquid, are rendered solid by evaporation, change of temperature, 
 or exposure, often possess extraordinary strength. Examples of 
 this are presented in the gums, glue, varnish, &c. Count Rumford 
 found that a copper plate having the thickness of I-2Oth of an 
 inch, rolled into the form of a cylinder, had its strength doubled 
 when coated with well-sized paper of double its own thickness ; 
 and that a cylinder composed of sheets of paper glued together, 
 having a sectional area of one square inch, was capable of support- 
 ing a weight of 1 5 tons for every square inch in its sectional area; 
 and, in fine, that a cylinder composed of hempen fibres glued 
 together had a strength greater than that of the best iron, being 
 capable of supporting 46 tons per square inch of sectional area. 
 
 60 1. According to the theory of Euler, the strength of a column 
 composed of any material and of any prismatic form to resist the 
 crushing force of a weight placed upon it, increases as the square 
 of the number expressing its sectional area divided by the square 
 of the number expressing its height. This law, which is of great 
 
284 THEORY OF MACHINERY. 
 
 practical importance on account of its application to architectural 
 purposes, is found to be in very near accordance, within practical 
 limits, with the experiments of Musschenbrock and others ; and 
 more recently with the still more extensive experiments of Mr. 
 Eaton Hodgskinson made on pillars of wrought iron and timber. 
 
 It is necessary to observe here, that when the height is reduced 
 to a very small magnitude, the pillar is more easily crushed than 
 would be indicated by this law. Exceptions are also presented in the 
 case of pillars formed of particular materials; such, for example, as 
 cast iron, which Mr. Hodgskinson found to vary in rather a higher 
 proportion in reference both to the sectional area and to the height. 
 
 According to Eytelwein's experiments, the strength of rectan- 
 gular columns to resist compression is directly as the product of 
 the larger side of their section multiplied by the cube of its shorter 
 side, and inversely as the square of their height. This will coin- 
 cide with that of Euler when the pillar is square. 
 
 602. Eytelwein gives the following table of the weights neces- 
 sary to crush pillars composed of the materials expressed in the first 
 column, the numbers expressed in the second column being the total 
 crushing weights in Ibs. per square inch : 
 
 from 3860 to 5147 
 1928 
 1606 
 1184 
 
 Sandstone, Rothenburg 2556 
 1092 
 
 Name. 
 
 Ibs. Ibs. 
 
 Name. 
 
 I. Metals : 
 
 
 2. Woods: 
 
 Cast iron 
 Brass, fine 
 
 from 115813 to 177776 
 164864 
 
 Oak 
 Pine 
 
 Copper, molten 
 
 117088 
 
 Pinus sylv. 
 
 hammer* 
 
 :d 103040 
 
 Elm 
 
 Tin, molten 
 Lead, molten - 
 
 ., 15456 
 
 77*8 
 
 3. Stones: 
 Gneiss 
 
 .->anusione, ixoineuuui 
 Brick, well baked - 
 
 603. Mr. Hodgskinson gives the following results of his experi- 
 ments as to the strength of timber pillars : 
 
 
 Strength per Square 
 Inch in Lbs. 
 
 
 Strength per Square 
 Inch in Lbs. 
 
 Description of Wood. 
 
 Moderately 
 dry. 
 
 -After sub- 
 jected to 
 drying 
 Process. 
 
 Description of Wood. 
 
 Moderately 
 dry. 
 
 After sub- 
 jected to 
 drying 
 Process. 
 
 Alder 
 
 6831 
 
 6960 
 
 Mahosrany 
 
 8198 
 
 8198 
 
 Ash 
 
 8683 
 
 9363 
 
 Oak, Quebec 
 
 42.31 
 
 5982 
 
 Bay 
 Beech 
 
 7518 
 
 7733 
 
 7518 
 9363 
 
 _ English 
 Pine, Pitch 
 
 6484 
 6790 
 
 10058 
 6790 
 
 English Birch - 
 
 3Z97 
 
 6402 
 
 Red - 
 
 5395 
 
 7S I8 
 
 Cedar 
 
 5674 
 
 5863 
 
 Poplar 
 
 3107 
 
 5124 
 
 Red Deal 
 
 ?748 
 
 6586 
 
 Plum, dry 
 
 8241 
 
 10493 
 
 White Deal 
 
 0781 
 
 7*93 
 
 Teak 
 
 
 I2IOI 
 
 Elder 
 
 7451 
 
 9973 
 
 Walnut - 
 
 6063 
 
 7227 
 
 Elm 
 
 
 10331 
 
 Willow - 
 
 2898 
 
 6128 
 
 Fir (spruce) 
 
 6499 
 
 6819 
 
 
 
 
 The numbers in the first column are the number of Ibs. per 
 
STRENGTH OF MATERIALS. 285 
 
 square inch, deduced from experiments made on cylinders one inch 
 in diameter and two inches in height, with flat sides, the wood be- 
 ing moderately dry. The second column gives the strength of simi- 
 lar pillars of the same woods, which had been subjected to a dry- 
 ing process in a warm place for two months and upwards. The 
 great difference in the strength of the two cases, shows in a strik- 
 ing manner the effect of dryness upon the strength of timber. 
 
 604. Neither theory nor experiment has thrown much light on 
 the laws which govern this mode of resistance to the separation of 
 the constituent parts of bodies. The following results, showing 
 the comparative strength of various metals, were obtained from a 
 series of experiments made by Mr. Rennie. 
 
 Table showing the comparative Strength of various Metals to resist 
 Torsion. 
 
 Lead 
 
 Tin 
 
 Copper 
 
 Brass 
 
 Gun metal 
 
 Swedish iron 
 
 1000 | English iron - - 101*5 
 
 1438 
 
 5000 
 9500 
 
 Cast-iron - - 10600 
 
 Blister-steel - - 16688 
 
 Shear-steel - 17063 
 
 Cast-steel - - 19562 
 
 It is stated by Mr. Bankes, that a square bar of cast-iron which 
 would measure an inch, is wrenched asunder by an average force 
 of 631 Ibs. applied at the extremity of a lever two feet long. 
 
 605. Bodies submitted to a transverse strain are usually con- 
 sidered as having the form of prismatic beams, at right angles to 
 which the strain is applied. A prismatic beam is one, all whose 
 transverse sections are equal and similar figures, and the charac- 
 ter of the beam is determined by the figure of this section. Thus, 
 a cylindrical beam is one whose transverse section is a circle ; a 
 rectangular, one whose transverse section is a right-angled paral- 
 lelogram ; a square, whose transverse section is a square ; and so 
 on. The force producing the strain may be resisted by one or by 
 two points of support, in the latter case the weight being placed 
 between these two points. 
 
 In the first case, let B c,jig. 271., be a prismatic beam, fixed 
 in a wall, or other vertical means of support, at B A, and let a 
 weight w be supported from its end c. We shall, for the present, 
 omit the consideration of the weight of the beam itself. 
 
 The effect of the weight w produced at B A will be to turn the 
 beam round the point A, as if it were a hinge, as represented in 
 fig. 272., and thus to tear asunder the fibres which unite the parts 
 of the body forming its transverse section at B A. Let f be the 
 force corresponding to the unit of surface of this section, let A be 
 the area of the section ; then/" x A will express the total force of 
 the body over the whole surface of the section. But if the beam 
 
286 
 
 THEORY OF MACHINERY. 
 
 tends to turn round the point A as a fulcrum, this force acts by a 
 leverage at each point, corresponding to the distance of such point 
 from A. The total force expressed by / X A may therefore be 
 
 Fig. vj\. 
 
 Fig. ayz. 
 
 considered as having an average leverage, determined by the vari- 
 ous distances of all the parts in the section from a horizontal line 
 passing through A. 
 
 Xow the point determined by these conditions is the centre of 
 gravity of the section of the beam at A B. Let the distance of this 
 centre of gravity from the horizontal line passing through the point 
 A be c ; we shall then have the effect of the forces of cohesion by 
 which the body is united in the surface of the section expressed 
 by/" x A X c. Against this the weight w acts with the leverage 
 c B, that is to say, the length of the beam. 
 
 Let this length be expressed by L, and we shall have, according 
 to the properties of the lever, 
 
 and therefore 
 
 XL=/XAXC, 
 
 It appears from this, therefore, that the weight necessary to frac- 
 ture a beam placed in this manner, will be in the direct ratio of 
 the area of the transverse section multiplied by the height of the 
 centre of gravity above the lowest point of this section, and divided 
 by the length of the beam. 
 
 For beams of the same length, their strength is proportional to 
 the area of their section, multiplied by the distance of the centre 
 of gravity above the lowest point ; and for beams of the same sec- 
 tion, their strength is inversely as their length. 
 
 By the length of the beam in this case is to be understood the 
 distance of the part of the beam at which the weight is suspended 
 from the point of support. 
 
STRENGTH OF MATERIALS. 
 
 287 
 
 606. Let us now consider the case in which the body is sup- 
 ported at two points, the breaking weight being placed at some 
 
 _j> A M 5^ intermediate point. 
 
 Let such a beam be 
 represented in fig. 
 273., supported at the 
 points D and c, the 
 weight being placed at 
 an intermediate point 
 A. In this case, the 
 tendency of the weight 
 is to rupture the fibres 
 in the vertical section 
 
 of the beam passing through A, and in doing so the beam would 
 
 break as if a hinge 
 were placed at A, on 
 the upper surface of 
 
 M v LA the section supporting 
 
 the weight, as repre- 
 sented in Jig. 274. 
 
 The fracture may in 
 this case be considered 
 to be produced by the 
 reaction of the points 
 of support D and c. 
 
 To determine the 
 effects of this reaction, 
 we are to consider that, by the properties of the lever, the part of 
 the weight which acts it c as expressed by 
 
 Fig. Z74. 
 
 \v X 
 
 DA 
 
 and the part of the weight which acts at D is expressed by 
 
 c A 
 w X . 
 
 But the former of these acts upon the section at A with the 
 leverage c A, and the latter with the leverage D A. If each be 
 therefore multiplied by its respective leverage, we find that the 
 effect of each of these actions in producing the fracture at A will 
 be expressed by 
 
 DA X C A 
 
 w X 
 
 C D 
 
 the total effect therefore will be 
 
288 THEORY OF MACHINERY. 
 
 D A X C A 
 2 W X - . 
 C D 
 
 Against this force the strength of the beam acts, and the strength 
 of the beam at the section A is determined and expressed in ex- 
 actly the same manner as in the former case, except that, in the 
 present case, the average leverage by which the strength of the 
 fibres resists rupture, is the distance of the centre of gravity below 
 the highest point A of the section. Expressing this distance as 
 before by c, we shall have the condition of equilibrium at the mo- 
 ment of fracture expressed by 
 
 D A X C A 
 
 2 W X -- = / X A X C. 
 CD 
 
 This may be simplified in the following manner : 
 Let M,J%-. 273., be the middle point of the beam between the 
 two points of support, and let a express D M, half its length ; let 
 x express M A, the distance of the point where the breaking weight 
 is placed from the middle point. We shall then have, by well- 
 known principles of geometry, 
 
 D A X c A = a 2 x z . 
 Hence we shall have 
 
 ~~ 
 
 2 w x 
 
 2 a 
 and consequently we shall have 
 
 _ 
 
 _/X A X c X a 
 
 -- 
 
 Hence it follows that if the weight be placed at the middle point 
 between the two points of support, we shall have x = o ; and con- 
 sequently, 
 
 It appears from this, that the weight placed at the centre of a 
 beam, between two points of support necessary to break it, is 
 double that which would be sufficient to break the same beaux, if 
 it were supported only at one point, the weight being placed at 
 the other point. 
 
 Such are the general practical principles for determining the 
 transverse strength of beams as established by Galileo ; and al- 
 though other practical formulae have been proposed by later writers, 
 the above have been found in such near accordance with the aver- 
 age results of experiments made upon a large scale, that they have 
 
STRENGTH OF MATERIALS 289 
 
 been generally adopted by mechanical writers as the basis of their 
 investigations upon the strength of materials. 
 
 Let us now see how the preceding formulae are modified for 
 beams of particular forms. 
 
 607. If the beam be rectangular, let its breadth be b, its 
 depth e?, and its length 2 a. The distance of the centre of gravity 
 of its section, therefore, from its upper or its lower surface will 
 be d. Now the area of its section will be b X d : hence we 
 have 
 
 A = b X d, 
 C~f4 
 L, = 2 a ; 
 
 and consequently we have, if the beam be supported at one end only, 
 
 In like manner, if the beam be supported at both ends, the 
 weight being placed at the middle, we shall have 
 
 2 a 
 
 If the beam be square, its breadth will be equal to its depth, 
 and the breaking weight, where there is but one point of support, 
 will be 
 
 and when there are two points of support, 
 
 608. From the general principles here established it is evident 
 that, so long as the quantity of matter composing the beam, and 
 therefore its sectional area, remains the same, its strength will be 
 augmented by any modification of form which will carry its centre 
 of gravity to a greater distance from its lower surface if the beam 
 have but one, and from its upper surface if it have two points of 
 support. Some curious and instructive consequences ensue from 
 this. 
 
 Thus, the strength of a rectangular beam, when its narrow side 
 is horizontal, is greater than when its broad side is horizontal, in 
 the same proportion as the width of its broad side is greater than 
 the width of its narrow side. Hence, in, all parts of structures 
 
 u 
 
290 THEORY OF MACHINERY. 
 
 where beams are subject to transverse strain, as in the rafters of 
 floors, roofs, &c., they are placed with their narrow sides hori- 
 zontal, and their broad sides vertical. 
 
 If a beam, supported at two points, bear a strain at any inter- 
 mediate point, a given quantity of matter composing it will have 
 greater strength if it be so formed that its area shall be less in the 
 upper part than in the lower. 
 
 Thus, a beam of triangular section, such as A B c (Jig. 27 5.), 
 will be stronger than a rectangular beam 
 of equal section, such as A B c D (f,g. 
 276.), because the centre of gravity of the 
 triangle with its vertex upwards will be 
 further from B than that of the parallelo- 
 gram from B D. 
 
 But if the beam be supported at one 
 point only, then the position of the tri- 
 Fig. a7S . Fig.z 7 6. angle must be inverted. 
 
 609. If a solid and a hollow cylinder of 
 
 equal lengths have the same quantity of matter, so that their 
 sectional areas shall be equal, then their strength will be propor- 
 tional to the distances of their centres of gravity from their external 
 surface. But their centres of gravity being at their geometrical 
 centres, it follows that the strength of the solid cylinder will be 
 less than the strength of the hollow cylinder, in the ratio of the 
 diameter of the solid cylinder to the diameter of the external sur- 
 face of the hollow cylinder. 
 
 It appears, therefore, that the strength of a tube is always 
 greater than the strength of the same quantity of matter made into 
 a solid rod ; the practical limitation of the application of this prin- 
 ciple being, that the thinness of the tube should not be so great as 
 to cause a local derangement of its form by the application of a 
 strong force. 
 
 610. Innumerable striking and beautiful examples of this prin- 
 ciple occur in the organised world. The bones of animals of every 
 species are hollow cylinders, thereby combining strength with 
 lightness. The stalks of numerous species of .vegetables which 
 have to bear a weight at their ,upper end are also tubes, -whose 
 lightness is remarkable when their strength is considered. 
 
 These are intended to resist not only the crushing weight of the 
 ear which they bear at their summit, but also the lateral strain 
 produced by the movement of the air. 
 
 The quills and the plumage of birds, and especially of their 
 wings, present a still more striking example of the application of 
 this principle in the animal structure. The surface of the extended 
 wing acting on the air produces a strong transverse strain upon 
 
STRENGTH OF MATERIALS. 291 
 
 the quill, which has a single point of support near the joint of the 
 wing. 
 
 The number expressed by f in the preceding formulae is the 
 strength of a beam, whose sectional area is the square unit, which 
 in this case is generally taken as the square inch. This number is 
 always the same for the same species of material, but different for 
 different materials. In tabulating the strength of materials 
 obtained from experiment, this is accordingly the number which 
 is taken to designate the strength of each species of substance. 
 When this number /is known, the strength of a beam of any pro- 
 posed dimensions can be immediately calculated by the formulas ; 
 for it is only necessary to multiply this number / by the number 
 of square inches in the sectional area, and the number of inches in 
 the distance of the centre of gravity of this area from the upper or 
 lower surface, as the case may be, and to divide the product by 
 the length of the beam, and the result of this arithmetical process 
 will give the breaking weight. 
 
 It must be understood that this weight, and the force expressed 
 by the number/, are to be expressed in the same unit. 
 
 6 1 1 . In ascertaining by experiment the average strength of 
 each material, the number which it is important to determine is 
 therefore that which is here expressed by /, and -which may be 
 considered the unit of strength ; because, this being once deter- 
 mined, the strength of a beam of the proposed materials of any 
 proposed denomination can be immediately computed. 
 
 This quantity / can be determined by direct experiment. If 
 the amount of the breaking weight upon a beam of given dimen- 
 sions be ascertained, we shall know the numbers severally expressed 
 by w, &, d and a in the preceding formulas ; and in this case we 
 ^all have 
 
 In practice it will be found that the values obtained for the 
 number / will be subject to considerable variations in different 
 experiments, owing to casual inequalities and defects which affect 
 each particular beam submitted to experiment. 
 
 The value of/ must therefore, for each material, be obtained by 
 taking an average of the results of numerous experiments, the 
 casual inequalities being 'therein made to disappear from the result 
 in proportion to the number and variety of trials. 
 
 612. In the following table is given the results of a series of 
 experiments made by Tredgold upon the transverse strength of 
 prismatic beams. The numbers in the second column represent 
 the values of /in the preceding formulae, while the numbers in the 
 
292 THEORY OF MACHINERY. 
 
 third column express the number of Ibs. weight in a cubic foot of 
 the material under experiment. 
 
 Table. 
 
 
 
 Ltis. 
 
 
 
 LI*. 
 
 Name. 
 
 /* 
 
 Weight 
 in Cubic 
 
 Name. 
 
 / 
 
 
 
 
 Foot. 
 
 
 
 Foot. 
 
 i. Metals : 
 
 
 
 2. Woods: 
 
 
 
 Malleable iron - 
 
 17800 
 
 475 
 
 Oak ... 
 
 3960 
 
 51 
 
 Hammered iron 
 Cast iron - 
 Brass 
 Zinc 
 Tin 
 
 15300 
 6700 
 5700 
 2880 
 
 45 
 586.25 
 4392.5 
 455-7 
 
 Fir (red or yellow) . 
 Pine (American yel 
 low) - 
 Fir (white) 
 Ash 
 
 4290 
 
 3900 
 3630 
 354 
 
 34.8 
 
 26.75 
 29.3 
 47-5 
 
 Lead 
 
 1500 
 
 709.5 
 
 Elm 
 
 2.340 
 
 34 
 
 613. Professor Barlow, of Woolwich, gives a table of the 
 strength of beams to resist transverse strain, from which the 
 following is extracted. The numbers in the second column in 
 this case represent the same number f as those in the second 
 column of the preceding table. 
 
 
 Talk. 
 
 
 Name. 
 
 f. Name. 
 
 /. 
 
 Teak - 
 
 - 9848 
 
 Elm - - - 
 
 4052 
 
 Poon ... 
 
 - 8884 
 
 Pitch pine 
 
 28 
 
 English oak, ist specimen 
 
 - 472.4 
 
 New England fir 
 
 4^08 
 
 2nd specimen 
 Canadian oak - 
 
 - 6688 
 - 7064 
 
 Riga fir 
 Mar Forest fir - 
 
 T ---u 
 
 4432 
 5048 
 
 Beech ... 
 
 6224 
 
 Norway spar 
 
 :.'# 
 
 614. If the weight producing the strain upon a beam, instead of 
 being concentrated at a single point, as supposed in the cases here 
 investigated, be equally distributed over the whole beam, the 
 power of suspension becomes twice as great as if it were applied at 
 the middle point of the beam. 
 
 It has been also shown that each point of the beam has a greater 
 supporting power the nearer it is to either point of support. It is 
 evidently, therefore, advantageous in all structures, in the distri- 
 bution of the weight upon them, to throw a less quantity at the 
 centre, and to increase the quantity towards the points of support. 
 In this manner of loading, a far greater weight may be placed near 
 the walls than at the centre. In all cases, however, a concen- 
 tration of weight at a single point is to be avoided. A sheet of ice 
 
 * The numbers given by Tredgold are the values of 
 
 4BX O 
 
 We have obtained the numbers given above by multiplying this by 4. 
 
STRENGTH OF MATERIALS. 
 
 2 93 
 
 which would break under the weight of a skater would sustain the 
 same skater if his weight were equally distributed over its surface. 
 In allowing for the weight of the beam itself, this weight may be 
 considered as uniformly spread over its surface. 
 
 615. According to Peschel, the transverse strength of a beam 
 of timber may be greatly increased by sawing down from one third 
 to one half of its depth, and driving in a wedge of metal or hard 
 wood until the beam is forced at the middle out of the horizontal 
 line, so as to form an angle presented upwards. It was found by 
 such an experiment that the transverse strength of a beam thus 
 cut to one third of its depth was increased one nineteenth ; when 
 cut to one half of its depth, it was increased one twenty-ninth ; 
 and when cut to three fourths of its depth, it was increased one 
 eighty-seventh. 
 
 6 1 6. It follows from the principles which have been explained, 
 that if any structure be increased in magnitude, the proportion of 
 its dimensions being preserved, the strength will be augmented as 
 the squares of the ratio in which it is increased. Thus, if its 
 dimensions be increased in a two-fold proportion, its strength will 
 be increased in a four-fold proportion ; if they be increased in a 
 three-fold proportion, its strength will be increased in a nine-fold 
 proportion, and so on. But it is to be considered that, by 
 increasing its strength in a two-fold proportion, its volume, and 
 consequently its weight, will be increased in an eight-fold pro- 
 portion ; and by increasing its dimensions in a three-fold propor- 
 tion, its volume and weight will be increased twenty- seven times, 
 and so on. Thus it is apparent that the weight increases in a 
 vastly more rapid proportion than the strength, and that conse- 
 quently, in such increase of dimensions, a limit would speedily be 
 attained at which the weight would become equal to the strength, 
 and beyond this limit the structure would be crushed under its 
 own weight. On the other hand, the more below this limit the 
 dimensions of the structure are kept, the greater will be the pro- 
 portion by which the strength will exceed the weight. 
 
 617. The strength of a structure of any kind is therefore not to 
 be determined by its model, which will always be much stronger 
 relatively to its size. All works, natural and artificial, have limits 
 of magnitude which, while their materials remain the same, cannot 
 be exceeded. Small animals are stronger in proportion than 
 larger ones. We find insects and Animaleulas capable of bodily 
 activity, exceeding almost in an infinite degree the agility and 
 muscular exertion manifested by the larger class. The young 
 plant has more available strength in proportion than the forest 
 tree. 
 
 u 3 
 
294 THEORY OF MACHINERY. 
 
 An admirable instance of beneficence in the consequences of this 
 principle is, that children, who are so much more exposed to acci- 
 dents, are less liable to injury from them than grown persons. 
 
 CHAP. VII. 
 
 MOVING POWERS. 
 
 6 1 8. Mechanical agents. The natural forces used for the 
 production of mechanical effects are very various. Those applied 
 to move machines, sometimes called prime movers, are animal 
 power, water, wind, and steam. Besides these, however, there are 
 several others which have been applied on a more limited scale to 
 particular purposes, or under special circumstances. Some of 
 these would be altogether inapplicable as movers of machinery, 
 because frheir development cannot be rendered sufficiently constant 
 and uniform. Others would be applicable ; but their use has 
 been hitherto excluded because of the expense which attends their 
 production. 
 
 619. The dynamical unit. Since all moving powers are 
 applied to overcome resistances, and all resistances can be repre- 
 sented by equivalent weights, while the spaces through which they 
 are urged can be represented by corresponding heights, the effect 
 produced by a moving power is always expressed by a certain 
 weight raised a certain height. Thus, for example, if a horse 
 draw a loaded waggon with a force by which the traces are 
 stretched with the same force as if 200 Ibs. were suspended verti- 
 cally from them, and if the horse thus acting draw the waggon 
 over the space of IOO feet, the mechanical effect produced is said 
 to be 200 Ibs. raised I oo feet ; or, what is the same, 20000 Ibs. 
 raised I foot. 
 
 In the expression, therefore, of all mechanical effects, it must be 
 understood that the dynamical unit adopted is I Ib. raised I foot, 
 and that the effect produced by any moving power or mechanical 
 agent is expressed by an equivalent number of such units. 
 
 The great convenience attending this conventional mode of 
 expression is, that the mechanical effects produced in different 
 times, places, and circumstances by different mechanical agents 
 can be immediately compared one with another, and their nu- 
 merical ratio ascertained. 
 
 620. Machines and tools. It happens rarely that a natural 
 mechanical agent is applied immediately to the object on which it 
 
MOVING POWERS. 
 
 295 
 
 is intended to act. In most cases some instrument or expedient 
 is interposed which, when its form is simple and its magnitude 
 small, is called a tool, and when greater and more complex, a 
 machine. The line of distinction, however, between tools and 
 machines, is not clearly defined ; for we sometimes find large and 
 complicated engines, such, for example, as those used for planing, 
 punching, and boring metal, called tools. 
 
 The expedient by which an agent acts upon an object, not 
 immediately, but by the intervention of a tool, is a mark of intelli- 
 gence which has been considered by some as the most conspicuous 
 distinction between man and the lower animals. Many of these 
 display unmistakeable evidence of skill, sagacity, and reasoning 
 under circumstances which will not allow of the supposition of 
 mere instinct; but we believe that no example can be found 
 among the lower animals of the production of any effect upon 
 objects external to them otherwise than by the immediate use of 
 those members or organs with which nature has furnished them, 
 but never by the intervention of anything analogous to a tool. 
 
 621. Animal power. The strength of animals may be 
 employed as a moving power in various ways : the animal 
 may remain without change of position, working by the action 
 of its members, as when a man works at a windlass ; or it may 
 advance progressively, transferring its own body, and carrying, 
 drawing, or pushing a load. The mechanical effect produced by 
 such a power is subject to extreme variation, according to the 
 various conditions under which it is exerted ; and in an economical 
 point of view it is therefore of extreme importance to determine, 
 with respect to each animal, the circumstances and conditions 
 under which the greatest amount of useful effect can be obtained. 
 
 In considering this question,, two extreme cases present them- 
 selves in which the useful effect altogether vanishes, and between 
 which it continually varies, increasing gradually from nothing to a 
 maximum quantity, and then decreasing again to nothing. 
 
 622. There is a certain amount of load or resistance whi:h 
 requires the entire strength of the animal to sustain, no force 
 remaining for motion. Thus, we can conceive a load which a 
 man or horse is just able to support, but unable to move ; in this 
 case, therefore, the useful effect is nothing. 
 
 We can conceive, on the other hand, a speed of motion so great 
 that with it the animal is unable to carry or draw any load, how- 
 ever small. At all intermediate speeds, however, some load more 
 or less can be carried or drawn, the amount of such load being 
 obviously greater as the speed is less. A horse can draw a greater 
 load at three miles an hour than at ten miles an hour. The useful 
 effect of the agent being, however, measured by multiplying the 
 
 174. 
 
295 THEORY OF MACHINERY. 
 
 load by the speed, it is evident that if either the load or speed be 
 unduly . diminished, the useful effect will consequently be di- 
 minished, and there is a certain relation or proportion between the 
 load and speed which will give a maximum effect. This relation 
 is not only different with different animals, but varies even with 
 the same animal, according to the different ways in which it acts. 
 
 623. If the force of a man be applied, for example, to the mere 
 elevation of a weight, without any reference to the continuance of 
 its action, it is found that the most advantageous mode of exerting 
 the force is when he endeavours to raise a weight placed between 
 his legs. The greatest weight which can be raised in this case 
 varies from 400 Ibs. to 600 Ibs., according to the strength of the 
 individual ; its average amount, however, is much less, not ex- 
 ceeding 250 Ibs. 
 
 The force which can be exerted by human strength varies, as 
 has been already observed, extremely, according to the way in 
 which it is applied. Thus, if men work at a pile- engine, such as 
 represented in Jig. 288., it is found that each can pull with a force 
 sufficient to raise about 144 Ibs. weight of the ram I foot high, 
 and that working thus they can make about 20 strokes per minute, 
 which, being continued for 4 or 5 minutes, they require to reci uit 
 their strength by an interval of repose. 
 
 624. When men work at a capstan, it is found that each can 
 exercise a force of about 28 Ibs. on the extremity of the lever 
 which he pushes, and that with this force they can move at the rate 
 of about I o feet per minute. 
 
 When a man works at a winch, of which the arm is from 1 2 to 
 1 4 inches from the centre on which it revolves, it is found that he 
 can exercise a force of about 1 6 Ibs. on the handle, and can make 
 from 20 to 25 turns per minute. 
 
 In general, however, human labour produces, in the long run, 
 the greatest effect when it is exercised with frequent intervals of 
 rest; and accordingly the greatest effect is produced when an 
 animal ascends, raising nothing but its own weight, and produces 
 the mechanical effect by that weight itself in descending, so that 
 the animal actually reposes in the intervals during which the 
 mechanical effect is produced. 
 
 625. Thus, for example, if two baskets be connected by a rope 
 which passes over a pulley at the level to which heavy matter is to 
 be raised, the length of the rope being such that when one of the 
 baskets is brought up to that level the other shall have descended 
 to the level from which the heavy matter is to be elevated, the 
 lower basket being charged, let one or more men get into the 
 upper basket, so as to give it, by their weight, a sufficient prepon- 
 derance to make it descend, drawing up the lower basket with its 
 
MOVING POWERS. 
 
 297 
 
 load. When the latter has , 
 been brought up and dis- 
 charged at the upper level, 
 the men leave the lower 
 basket and charge it, when 
 men at the upper level get 
 into the upper basket, and 
 again bring up the lower 
 as before. Meanwhile, the 
 men who had previously 
 gone down again ascend to 
 the upper level by a ladder 
 or stair-case. In this man- 
 ner the work may be con- 
 tinued without any inter- 
 mission, the labour of the 
 men consisting exclusively 
 in elevating the weights of 
 their own bodies while as- 
 cending the ladder or stair- 
 case. 
 
 It is found by experi- 
 ments, conducted upon a 
 large scale, that men work- 
 ing in this manner for eight 
 hours a day can produce in 
 each day an effect equiva- 
 lent to 2,000000 Ibs. raised 
 I foot. 
 
 The great advantage ob- 
 tained by this mode of ap- 
 plying animal force will be 
 apparent, when it is stated 
 that the man who can thus 
 produce an effect of two 
 millions of pounds could 
 not produce, in working at 
 a windlass, a greater effect 
 than a million and a quarter ; 
 while, if he acted upon a 
 pile-engine, such as that re- 
 presented in Jig. 288., he would produce an effect of only three 
 quarters of a million. 
 
 626. An apparatus such as that here described is represented 
 in fig. 277 It was employed in making the earth- works at Vin- 
 
 Fig. z 77 . 
 
298 THEORY OF MACHINERY. 
 
 , cennes, near Paris, and was found to be attended with great 
 economy. 
 
 Analogous, though not quite identical with this mode of action, 
 are the tread mill and the ladder wheel which are represented in 
 figs. 148. and 150. In these cases the labour is expended upon 
 the continual elevation of the body, which, however, descends as 
 fast as it is elevated. The daily effect produced by men oi 
 the average strength working in this way for eight hours is found 
 to be equivalent to 1,875000 Ibs. raised I foot. Thus it appears 
 that this method of employing human force is a little less advan- 
 tageous than the method of ascent and descent, described above, 
 which produces a daily effect of 2,000000. 
 
 The comparative disadvantage, slight as it is, is explained by 
 the fact that in the one case the labour is intermitting, frequent 
 intervals of repose alternating with those of labour, the most 
 advantageous conditions under which animal force can be em- 
 ployed ; while in the other case the labour is continuous. 
 
 627. Spade labour is one of the most disadvantageous forms in 
 
 which human force can be applied, The spade or shovel is a lever 
 of the first or third kind, according as the one hand or the other 
 applied to its handle is regarded as the fulcrum, the other being 
 the power, and the earth taken up upon it being the weight. And 
 since that part of the handle included between the two hands is 
 always less than that between the lower hand and the earth, the 
 force exerted is always greater than the weight of the earth 
 raised. 
 
 In all extensive works, such, for example, as the formation of 
 
MOVING POWERS. 299 
 
 embankments on railways, the labour is executed by contract, the 
 quantity of work being determined by the number of cubic yards 
 of earth excavated and raised to a given height, or transported to 
 a given distance. The number of cubic yards is computed by 
 multiplying the superficial extent of the excavation, measured 
 horizontally, by its average depth. 
 
 628. In the formation of railways, the surveying engineer 
 endeavours, as far as possible, to render the quantity of earth- 
 work in the cuttings, by which the road is carried through 
 elevations, equal to the quantity necessary to form the embank- 
 ments by which it is carried across the adjacent and intermediate 
 valleys. In this case the quantity of labour is computed by taking 
 the average distance between all the points of the cutting and all 
 the points of the embankment, and assuming that the entire quan- 
 tity of earth excavated in the cutting must be transported over 
 this distance. 
 
 It rarely happens, however, that this exact equilibrium is attain- 
 able between the cuttings and embankments. If the earth of the 
 cutting be in excess, the surplus is deposited in the nearest con- 
 venient place, forming what is called a spoil-bank ; and if the earth 
 of the cutting fall short of the quantity necessary for the embank- 
 ment, the deficiency is made up by excavations made in the ground 
 beside the embankment, called side cuttings. 
 
 629. Horse-power. Before the invention and improvement 
 of the steam-engine, the force of horses was very extensively used 
 as a moving power ; and although its application to machinery is 
 now much less frequent, it is still resorted to, especially in places 
 where fuel is expensive. Much of what has been said of the 
 relative advantage of so applying the force of men is also applicable 
 to that of other animals ; and accordingly one of the most advan- 
 tageous methods of applying such force is that represented in 
 fig. 149., which is upon the principle of the tread mill, and in 
 which the animal uses its strength in continually endeavouring to 
 ascend, while, by the nature of the machine, those efforts are 
 continually frustrated by its descent. 
 
 It is found, however, in practice, that the most convenient 
 method of applying horse-power to machinery is by means of a 
 large toothed wheel fixed on a strong vertical axis, and therefore 
 turning horizontally ; to the arms of this wheel two or more 
 horses, called machiners, are yoked in such a manner as, by 
 travelling constantly in a circle of which the axle of the wheel is 
 the centre, and thus pushing against their collars, to make the 
 wheel revolve. The wheel in this case may have the form called a 
 crown wheel, as represented in fig. 280., in which case it gives 
 motion to a horizontal shaft by means of a spur-pinion or basket 
 
300 
 
 THEORY OF MACHINERY. 
 
 Fig. 279. 
 
 fixed on that shaft on which it acts ; or the same object may be 
 attained, perhaps preferably, by a bevelled wheel acting on a 
 bevelled pinion, as explained in 450. 
 
 Fig. 280. 
 
 The greatest average maximum force which a horse can exert 
 in drawing is about 900 Ibs. ; but when he works continuously, 
 the force exerted must be much less than this. A good draught- 
 horse working 6 days a week at 1 6 miles per day, and travelling 
 5 miles an hour, will bear a tractive force of about 1 1 o Ibs., and 
 his daily labour will be equivalent to about 10,000000 Ibs. raised 
 I foot. Ahorse driving a machine, as represented in jig. 280. 
 
MOVING POWERS. 
 
 301 
 
 produces a somewhat less effect than a draught-horse. To prevent 
 the animal from needless fatigue by travelling in too small a 
 circle, the arm to which he is yoked should not have less than 
 20 feet in length. 
 
 630. The effect produced by a horse of average strength so 
 working is found to be equal to that of 7 average men working at 
 a windlass. 
 
 The average effect produced per second of time by a machiner 
 is equivalent to about 300 Ibs. raised I foot. 
 
 An ox harnessed to a vehicle can exercise a tractive force equal 
 to that of a horse, but he produces less than half the effect, owing 
 to his natural slowness ; but as a machiner, where speed is not 
 required, the efficiency of an ox is nearly equal to that of a horse. 
 
 An ass of average force acting as a machiner has about one 
 fourth of the efficiency of a horse. 
 
 631. When it is considered how very variable animal power is, 
 whether we compare one animal of the same species with another, 
 or even the same animal with itself, at different times and under 
 different circumstances, it will be easily understood, that the esti- 
 mates of different observers of the average force exercised by each 
 species of animal differ considerably ; it will, therefore, be useful 
 here to give a few of the most generally received of these estimates. 
 
 In the following table is shown the estimates made by the ob- 
 servers named in the first column, of the average speed with which 
 a man can walk without a load, for any continuance. 
 
 Observer. 
 
 Feet per Second. 
 
 Miles per Hour. 
 
 Scliulze ... 
 
 . 
 
 
 J.66 
 
 Bernoulli 
 
 _ m 
 
 6.<6 
 
 4.47 
 
 Guenyveau 
 
 -I 
 
 from 6.56 
 to 9.84 
 
 i 
 
 The average powers exerted by a man employed in working a 
 pump, turning a winch, ringing a bell, and rowing a boat are, ac- 
 cording to Mr. Buchanan, in the proportion of the numbers 100, 
 167, 227, and 248. It follows from this that the act of rowing is 
 that in which the force is exerted with most advantage. 
 
 632. According to Mr. Peron, the following is the average rela- 
 tive power exerted by men employed in continuous labour in the 
 countries indicated : 
 
 England - 
 
 France 
 
 Timor 
 
 Van Diemen's Land 
 
 New Holland 
 
 51.8 
 50.6 
 
 633. According to Tredgold, the average speed with which a 
 
THEORY OF MACHINERY. 
 
 horse can travel without a load will vary according to the number 
 of hours per day he works, as shown in the following table : 
 
 Time of March in 
 
 Hours. 
 
 Greatest Velocity per 
 Hour in Miles, 
 
 Time of March in 
 
 Greatest Velocity per ! 
 Hour in .Miles. 
 
 I 
 
 14-7 
 
 6 
 
 6.0 
 
 Z 
 
 10.4 
 
 7 
 
 5.5 
 
 3 
 
 8.5 
 
 8 
 
 5.z 
 
 4 
 5 
 
 n 
 
 9 
 
 10 
 
 $ 
 
 According to observations made upon the performance of draught 
 horses in Scotland, it appears that the average load, which a single 
 horse can draw at the rate of 22 miles per day in a cart weighing 
 7 cwt. is a ton. 
 
 634. The following estimates ,of the relative forces of a man 
 and other animals have been given by the authorities whose names 
 are indicated, Coulomb's estimate of the labour of a man being, 
 in each case, taken as the unit : 
 
 Carrying loi 
 road : 
 
 loads on the back on a level 
 
 Horse, according to Brunacci - J..8 
 
 Do., according to Wesermann - o.i 
 
 Mule, according to Brunacci - 7.6 
 
 In drawing loads on a level road with a 
 
 wheeled vehicle : 
 Man with wheelbarrow, according 
 
 to Coulomb - 10.0 
 
 In drawing loads on a level road with a 
 
 wheeled vehicle: 
 Horse in four-wheeled waggon, 
 
 according to Wesermann - 175.0 
 Do. in two-wheeled cart, according 
 
 to Brunacci - - - 143.0 
 
 Mule in do. do. do. - 233-0 
 
 Ox in do. do. do. - izz.o 
 
 Hassenfratz gives the following comparative estimates : 
 
 In carrying loads on a level road : 
 
 In drawing a load on a level road : 
 
 Man - 1.0 
 
 Man - 
 
 I.O 
 
 Horse 
 
 8.0 
 
 Horse 
 
 7.0 
 
 Mule 
 
 8.0 
 
 Mule - 
 
 7.0 
 
 Ass - 
 
 40 
 
 Ass - 
 
 z.o 
 
 Camel 
 
 
 Ox - 
 
 4107.0 
 
 Dromedary 
 Elephant 
 
 Z5.o 
 - 147.0 
 
 Dog - 
 Rem,deer 
 
 - 0.6 
 
 O.Z 
 
 Dog - - i.o 
 Rein-deer - - 3.0 
 
 
 
 635. Steam-horse. Every one is familiar with the term 
 horse-power as applied generally to steam-engines, as well as to 
 water mills and other machines, but few have a definite notion of 
 its import ; horse-power thus applied, is a term merely conven- 
 tional, having no reference whatever to the actual work of the 
 animal from which its name is taken. A steam-engine which is 
 capable of a mechanical effect per minute equivalent to 33000 Ibs. 
 raised I foot, is a steam-engine of I horse power ; and in general, 
 if the effect produced per minute by any machine, whatever be the 
 power which impels it, be expressed as usual by an equivalent 
 number of pounds' weight, the number of horse-power which 
 characterises it will be found by dividing the latter number by 
 33000. 
 
MOVING POWERS. 303 
 
 636. Water-power. This power is rendered available for 
 the motion of machinery by intercepting its fall or progress by 
 wheels having buckets or leaves at their circumference, upon which 
 the water acts either by its impact or weight so as to keep them in 
 revolution with a corresponding force ; the form and effect of these 
 wheels, however, will be explained more fully under the head of 
 hydraulics. 
 
 637. Wind-power. The force of the atmosphere in motion 
 is rendered available as a mover for machinery by means of arms, 
 called sails, which are driven round in a plane nearly vertical, 
 giving revolution to a horizontal axis. These machines will be 
 explained more fully under the head of pneumatics. 
 
 638. Steam-power. When heat is applied in sufficient quan- 
 tity to water, the liquid is converted into vapour which has all the 
 mechanical qualities of air, being elastic, expansive, and com- 
 pressible. Its expansive force cannot be said to have any practical 
 limit, but the space through which it acts will decrease in nearly 
 the same proportion as that ill which its intensity increases ; thus, 
 if it be confined by a resistance equivalent to a pressure of 1 5 Ibs. 
 per square inch, the steam will move the resistance until it has 
 swelled to about 1 800 times the bulk it occupied as water. If 
 the resistance is 30 Ibs. per square inch, it will move it, until it 
 has obtained a space equal to about 900 times its bulk as water, 
 and so on. 
 
 It follows from this, that when water contained in a vessel such 
 as a cylinder, confined by a movable piston pressing on it with a 
 force of 1 5 Ibs. per square inch, is converted into steam, the piston 
 will be raised through about 1 800 inches for every inch of depth 
 of the water evaporated. A moving force would, therefore, be 
 developed equivalent to a weight of 1 5 Ibs. raised 1 800 inches, or 
 150 feet ; this would be in effect the same as 1 5 times 150 Ibs., 
 or 2250 Ibs. raised I foot. It may, therefore, be stated in round 
 numbers, that when a cubic inch of water is evaporated a mecha- 
 nical force is produced equivalent to a ton weight raised I foot 
 high. It does not matter under what pressure the evaporation is 
 produced, since any increase of pressure will be attended with a 
 proportionate decrease of the space through which the resistance 
 is moved. 
 
 The water, however, must in this case be regarded merely as a 
 medium, by which the mechanical effects of heat are evolved. The 
 real moving power is therefore, not the water, but the combustible 
 by which heat is produced, and that combustible being usually pit 
 coal, it becomes a question of the highest economical importance 
 to determine the average quantity of coal which is consumed in 
 
304 THEORY OF MACHINERY. 
 
 the evaporation of a given quantity of water, and in the consequent 
 evolution of a given amount of moving force. 
 
 639. The consumption of coal in the evaporation of water in 
 steam boilers, varies considerably according to the circumstances 
 under which the fuel is applied. In Cornwall, however, where the 
 conditions most favourable to economy are strictly observed, it has 
 been found that in experiments conducted under circumstances of 
 the greatest precision, a bushel of coal, that is, 84 Ibs. weight, of 
 good quality, has produced a mechanical effect equivalent to 
 1 20,000000 Ibs. raised I foot. This must, however, be regarded 
 as an extreme experimental result. We may take, perhaps, 
 100,000000 pounds as the maximum mechanical effect attainable 
 in regular work by the combustion of a bushel of coals. 
 
 Since it has been shown that the average maximum daily labour 
 of a man, working under the most favourable circumstances, is 
 2,000000, and that of a horse 1 0,000000, it follows that I bushel 
 of coals consumed daily can perform the work of 50 men or 10 
 horses. 
 
 640. It has been computed, on data such as these, that the mate- 
 rials of which the great pyramid of Egypt is formed, could have been 
 raised from the ground to their actual position by the combustion 
 of something less than 700 tons of coal. Herodotus states that 
 I ooooo men were employed for 20 years in raising this structure. 
 
 The Menai bridge consists of about 2000 tons of iron, placed 
 at 1 20 perpendicular feet above the water level ; its entire weight 
 could have been raised to that height by the combustion of 400 
 Ibs. weight of coals. 
 
 A train of coaches weighing 80 tons, and conveying 240 pas- 
 sengers, is drawn from Liverpool to Birmingham and back by the 
 combustion of 4 tons of coke, the cost of which is 5/. To carry 
 the same number of passengers daily, in stage coaches, on a com- 
 mon road, would require an establishment of 20 coaches and 3800 
 horses. 
 
 The circumference of the earth measures about 25000 miles; 
 if it were begirt with an iron railway, such a train would be drawn 
 round it in five weeks, by the combustion of about 500 tons of 
 coke. 
 
 641. Springs and weights. In a certain qualified sense, 
 springs and weights may be considered as moving powers ; they 
 are, in fact, the movers in all watch and clock work. It must be 
 observed, however, that the title of movers cannot be given to 
 them in the same absolute sense as that in which it is applied to the 
 several powers which we have described above. It has been 
 already explained that when a watch or clock is wound up, the 
 force which is exerted by the hand is expended in coiling up the 
 
MOVING POWERS. 
 
 305 
 
 main spring or elevating the weight, and that the motion of the 
 watch or clock from that moment until it ceases is merely pro- 
 duced by the spring or weight giving back by slow degrees and 
 during a comparatively long interval the force which was exerted 
 by the hand in winding them up. Strictly speaking, therefore, 
 the moving power is in this case the force of the hand, and the 
 spring or weight is merely the depository in which that force is 
 collected, and from which it is given out until it is completely 
 exhausted, and the clock ceases to move. 
 
 642. Forces produced by beat. It is a physical law of high 
 generality that when heat is imparted to or withdrawn from a 
 body, expansion or contraction is produced. The fdrce with 
 which such expansion or contraction takes place has been some- 
 times used as a moving power or mechanical agent. 
 
 Air being eminently susceptible of such expansion and contrac- 
 tion by variation of temperature, attempts have been made, with 
 more or less success, to apply the force of such expansion and con- 
 traction as a moving power. One of the most recent and ingenious 
 of these projects is the caloric-engine invented by Captain Ericcson 
 as a substitute for the steam-engine. This engine has been con- 
 structed on a large scale, and tried experimentally at New York 
 in the propulsion of a vessel. The piston is urged, as in the steam- 
 engine, on one side by the expansive force of heated air, the resist- 
 ance of the air on the other side being diminished by a great reduc- 
 tion of temperature. 
 
 Hitherto this experiment has not been successful ; the engine is 
 reported to have operated with a certain efficiency, but under 
 conditions not favourable to economy. 
 
 The expansion and contraction of metals has been used with 
 great success in many places as a mechanical agent where great 
 force has been required to be exerted through small spaces ; ex- 
 amples of this have already been given. 
 
 643. Mechanical foree of congelation. Water at a certain 
 point of the thermal scale exhibits a striking exception to the 
 general law of contraction by cold. Just before congelation com- 
 mences, the reduction of its temperature is attended, not with con- 
 traction, but with expansion ; and this expansion takes place with 
 irresistible force. When water, having percolated into the fissures 
 and clefts of rocks, is frozen there, the force with which it expands 
 in the process of congelation is such that the rocks are split 
 asunder, and fragments of enormous weight and magnitude 
 detached from them. 
 
 This force has been sometimes applied artificially for the fracture 
 of large masses of stone. 
 
 644. Electro-magnetic force. By expedients which will be 
 
 x 
 
3 o6 THEORY OF MACHINERY. 
 
 fully explained in another part of this Hand-book, a mass of soft iron 
 can be rendered instantaneously magnetic, and as instantaneously 
 deprived of its magnetism ; and these alternations in its magnetic 
 condition can be made to succeed each other at any desired rate, 
 however rapid. The magnetism imparted to it may also have any 
 desired intensity. It is, therefore, easy to imagine that such a 
 piece of iron will attract another situate at a limited distance from 
 it, with a certain definite force while it is magnetic, and that the 
 moment it loses its magnetism, the iron which it had thus attracted 
 may be made to recoil from it either by its weight, by the reaction 
 of a spring attached to it, or by any other convenient means. A 
 force, acting alternately in one direction and another, of definite 
 intensity, will thus be produced, which may be applied as a moving 
 power in the same manner exactly as that in which the alternate 
 motion of the piston of a steam-engine is applied. 
 
 Various attempts have been made to render this available as a 
 general moving power for machinery ; although the desired effect 
 has been in many cases attained, the expense has hitherto so much 
 exceeded that of steam power, that no successful practical result 
 has been attained, except in the case of the workshop of M. Froment, 
 the well known philosophical instrument maker of Paris, who 
 has for some years used electro-magnetism as a moving power for 
 the mechanism by which the divisions are engraved upon the limbs 
 of circles, quadrants, and other similar instruments of precision. 
 We shall describe this machinery more fully in another volume of 
 this Hand-book. 
 
 645. Chemical agency. A great variety of powerful me- 
 chanical effects are produced by the chemical combination or 
 decomposition of bodies. These phenomena, however, are gene- 
 rally developed under circumstances and conditions which render 
 their application as mechanical agents often difficult, and some- 
 times impracticable. The circumstances under which the mecha- 
 nical force is thus developed are generally those in which solid or 
 liquid bodies of comparatively small dimensions are suddenly con- 
 verted into gaseous bodies of great volume, the change being 
 usually attended with a large and intense development of heat. 
 The gases thus evolved at a high temperature, expanding with 
 prodigious force, drive before them whatever resistances may be 
 opposed to their dilatation. 
 
 One of the most familiar examples of this class of phenomena, 
 and that which has been most extensively applied, is gunpowder. 
 This substance is a mixture of nitre, charcoal, and sulphur, .all in 
 a very pure state, and in certain definite proportions ; thus, in 
 100 Ibs. weight of gunpowder there are generally 75 Ibs. of nitre, 
 1 5 Ibs. of charcoal, and IO Ibs. of sulphur These proportions are 
 
MOVING POWERS. 307 
 
 subject to a small variation in different qualities of powder, which 
 need not be noticed more particularly here. 
 
 When a spark is applied, the charcoal and the sulphur heated 
 by it, attract the oxygen, which is one of the constituents of the 
 nitre ; and, combining with it, form gases, called carbonic oxide, 
 carbonic acid, and sulphurous gas ; while the constituent of the 
 nitre, disengaged from the oxygen, is the gas called nitrogen, or 
 azote. The result, therefore, of the phenomenon is the instanta- 
 neous evolution of a mixture of the four gases above named ; this 
 evolution being attended with such intense heat, that the gases are 
 incandescent, or luminous. It is found that the gunpowder, in 
 this process of explosion, swells into 2000 times its volume. Count 
 Rumford found that 28 grains of gunpowder, screwed into a cylin- 
 drical space within a piece of iron, tore it asunder with a force of 
 400000 Ibs. 
 
 646. Gun cotton. If a piece of common raw cotton, usually 
 called cotton wool, be steeped in a mixture composed of equal 
 measures of sulphuric and nitric acids, and be then pr. essed and 
 dried, it will, to all external appearance, be the same as before ; 
 but if it be weighed it will be found to be nearly one half heavier. 
 The change which it has undergone is this : the cotton has lost a 
 quantity of water which was combined with it, equal to about one 
 third of its weight, but it has entered into combination with such 
 a quantity of nitric acid as to give the whole a weight 50 per 
 cent, greater than that of the original weight of the cotton. 
 
 The cotton thus prepared is highly explosive, and the effects of 
 its explosion are explained on the same principle as those of gun- 
 powder. It is considered by chemists that, weight for weight, the 
 force of this substance is greater than that of gunpowder. It ex- 
 plodes also at a lower temperature, and, when of good quality, 
 leaves no perceptible residuum. It is, therefore, probable that 
 when the process of making it has undergone those improvements 
 which it is likely to receive, it will replace gunpowder in fire-arms, 
 as it has already done to a certain extent in the blasting of rocks. 
 
 647. Capillary attraction. The mechanical agency supplied 
 by this principle in the arts, and its use in the vegetable economy, 
 will be explained in another volume of the Hand-book. 
 
 648. Perpetual motion. There is no mechanical problem on 
 which a greater amount of intellectual ingenuity has been wasted 
 than that which has for its object the discovery of the perpetual 
 motion. Since this term, however, is not always rightly under- 
 stood, it will be useful here to explain what the perpetual motion 
 is not, as well as what it is. 
 
 The perpetual motion, then, which has been the subject of such 
 anxious and laborious research, is not a mere motion which ig 
 
308 THEORY OF MACHINERY. 
 
 continued indefinitely. If it were, the diurnal and annual motion 
 of the earth, and the corresponding motions of the other planets 
 and satellites of the solar system, as well as the rotation of the sun 
 upon its axis, would be all perpetual motions. 
 
 To understand the object of this celebrated problem, it is 
 necessary to remember that, in considering the construction and 
 performance of a machine, there are three things involved : 1st, 
 the object to which the machine gives motion ; 2ndly, the struc- 
 ture of the mechanism ; and 3rdly, the moving power, the effect of 
 which is transmitted by the machine to the object to be moved. 
 In consequence of the inertia of matter, the machine cannot trans- 
 mit to the object more force than it receives from the moving 
 power ; strictly speaking, indeed, it must transmit less force, since 
 more or less of the moving force must be intercepted by friction 
 and atmospheric resistance. If, therefore, it were proposed to 
 invent a machine which would transmit to the object to be moved 
 the whole amount of force imparted by the moving power, such a 
 problem would be at once pronounced impossible of solution, inas- 
 much as it would involve two impracticable conditions ; first, the 
 absence of atmospheric resistance, which would oblige the ma- 
 chine to be worked in a vacuum ; and secondly, the absence of all 
 friction between those parts of the machine which would move in 
 contact with one another. 
 
 But suppose that it were proposed to invent a machine which 
 would transmit to the object to be moved a greater amount of 
 force than that imparted by the moving power, the impossibility of 
 the problem would in this case be still more glaring, for even 
 though the machine were to work in a vacuum, and all friction 
 were removed, it could do no more than convey to the object the 
 force it receives. To suppose that it could convey more force, it 
 would be necessary to admit that the surplus must be produced by 
 the machine itself, and that, consequently, the matter composing 
 it would not be endowed with, the quality of inertia. Such a sup- 
 position would be equivalent to ascribing to the machine the quali- 
 ties of an animated being. 
 
 But the absurdity would be still greater, if possible, if the 
 problem were to invent a machine which would impart a certain 
 motion to an object without receiving any force whatever from a 
 moving power ; yet such is precisely the celebrated problem of the 
 perpetual motion. 
 
 In short, a perpetual motion would be, for example, a watch or 
 clock which would go so long as its mechanism would endure 
 without being wound up ; it would be a mill which would grind 
 corn or work machinery without the action upon it of water, wind, 
 steam, animal power, or any other moving force external to it. 
 
MOVING TOWERS. 3 c y 
 
 It is not only true that such a machine never has been invented, 
 but it is demonstrable that so long as the laws of nature remain 
 unaltered, and so long as matter continues to possess that quality 
 of inertia which is proved to be inseparable from it, not only in all 
 places and under all circumstances on the earth, but throughout 
 the vast regions of space to which the observations of astronomers 
 have extended, the invention of such a machine is an impossibility 
 the most absolute. 
 
BOOK THE FOURTH. 
 
 ILLUSTRATIONS OF THE APPLICATION OF MECHANICAL 
 PRINCIPLES IN THE INDUSTRIAL ARTS. 
 
 CHAPTER I. 
 
 CRANES. 
 
 649. Cranes are pieces of mechanism usually consisting of com- 
 binations of toothed wheels and pulleys, by means of which mate- 
 rials for building are raised to 
 the stage where the builders are 
 engaged goods elevated from 
 vessels to the quays or wharves, 
 and loaded on the waggons by 
 which they are to be trans- 
 ported, or, on the contrary, trans- 
 ferred from the quays or wharves 
 into the vessels and for all similar 
 purposes. 
 
 650. movable cranes. One 
 of the most simple forms of mov- 
 able cranes commonly used for 
 building in France is shown in 
 fig- 281. 
 
 It consists of a sort of strong trian- 
 gular ladder, at the top of which is a 
 fixed sheave c, over which the rope 
 attached to the object to be elevated 
 passing is carried down to the cylin- 
 drical axle T, upon which it is rolled 
 by means of bars inserted ia holes, as 
 shown infg. 145. 
 
 This ladder is inclined more or less 
 from the vertical by means of a rope 
 c D, which is carried to some fixed 
 object at a distance, to which it ia 
 Fi S- ** attached. 
 
CRANES. 311 
 
 651. Another form of movable crane, also used for building 
 purposes, is shown in fig. 282., which consists of a shaft driven 
 
 Fig. 181. 
 
 by a combination of toothed wheels and a pair of winches. Upon 
 the shaft a rope is coiled, which, being carried up, is passed over 
 three sheaves, and then, passing downwards, is attached to the 
 object to be raised. 
 
 x 4 
 
3 I2 
 
 TKACTICAL ILLUSTKATIOXS. 
 
 652. Movable cranes for manufactories and railway 
 stations. Another form of movable crane, much used in large 
 
 engineering establishments and railway stations, is shown in 
 fig. 283. The figure is drawn upon a scale of I inch to 12 feet. 
 
CRANES. 
 
 313 
 
 The stand upon which the apparatus is fixed is supported on four small 
 wheels, which, in some cases, are made to run upon rails like those of a 
 railway waggon. It is evident, from what has been explained in Chap. V. 
 Book II., that the apparatus must be so constructed that the centre of gra- 
 vity of it and its load must be in a vertical line which shall fall within the 
 quadrangle formed by the lines which join the four wheels. 
 
 A strong pillar of wood, A A, is secured upon the base of the apparatus by 
 diagonal ties. In this a circular vertical .hole is perforated, in which the 
 round vertical beam B which supports the crane is inserted, so that it can 
 revolve horizontally without much resistance from friction. The horizontal 
 beams c c and D D are attached to collars, which also turn freely round A, 
 while they are supported by it. The diagonal pieces E E are attached 
 to these horizontal pieces, and also tied by iron rods to the central axis, both 
 in horizontal and diagonal directions. Two winders with systems of wheel- 
 work attached are fixed at c and c, by which the axles upon which the 
 ropes are coiled are turned. In the position represented in the figure, the 
 right-hand winch alone is worked. The rope F coiled upon its axle passes 
 over a sheave at the top of the right-hand diagonal beam E, from which it 
 is carried horizontally over a sheave at the top of the vertical pillar B, and 
 thence to a sheave in the top of the left-hand diagonal beam E, from whence 
 it passes downwards and under the movable pulley which supports the 
 weight to be elevated, and is finally attached to a hook or ring at the top of 
 the left-hand diagonal beam E. 
 
 The winch c and its apparatus of wheelwork acts in a similar manner 
 over another series of sheaves in juxtaposition with the former upon the 
 movable pulley suspended from the right-hand diagonal beam E. 
 
 653. Fixed cranes. When cranes are stationary, as is usually 
 the case upon wharves and landing-places, they are constructed in 
 a more efficient manner, giving greater 
 efficacy to the power by the interposition 
 of a greater number of wheels, axles, and 
 pulleys. They are usually made to re- 
 volve upon a centre, so that the goods 
 raised from the vessel may, by lowering 
 the crane, be brought over the waggon on 
 which they are to be deposited. This is ac- 
 complished by inserting a round piece of 
 metal, P Q (fig. 284.), at the lower part of 
 the apparatus, into a hole of corresponding 
 magnitude made in the wharf. The level 
 of the wharf being R Y, the crane, when it 
 turns in the hole, will carry the object 
 elevated in a circle, of which Q x is the radius, round P Q, the 
 centre or axis. 
 
 A crane thus constructed is represented, with all its accessories, mfig. 285., 
 upon a scale of i to 100, or an inch to 8 ft. 4 in. The wheelwork is shown 
 upon a larger scale and more evidently in fig. 286. 
 
 The weight is attached to a single movable pulley suspended from the 
 upper end of the crane. The rope which is carried over the fixed sheave, 
 
3H 
 
 PRACTICAL ILLUSTRATIONS. 
 
 and along the inclined arm of the crane to the axle A (fig. 285.), is stretched 
 by a force equal to half the weight raised. Upon the axle A is fixed a 
 toothed wheel B, which is driven by a pinion c (fig. 286.). Upon the axle 
 of this pinion is fixed another large toothed wheel D, which is driven by a 
 
 i-ig.286. 
 
 Fig. 185. 
 
 pinion E. Upon the axle of this pinion E is fixed another large toothed 
 wheel F. The two axes of the wheels D and F being upon the same level, 
 as shown in the figure, one conceals the other. 
 
 Two winches G and H work an axle upon which two pinions L and K are 
 fixed, which are capable of being moved through a small space right and 
 left horizontally by a lever M. These pinions are formed to engage in the 
 wheels D and E, and can be thrown into or out of connection with their 
 wheels by the lever M. As shown in the figure, both are out of connection 
 with D and F, and the crane is inoperative. If we suppose them both in 
 connection with the wheels, the pinion L, driven by the winch H, will drive F, 
 and consequently E, which will drive D. At the same time the pinion K, 
 
CRANES AND DROPS. 315 
 
 driven by the other winch o, will drive D. Thus, the wheel D will receive 
 at once the effect of the two powers applied to the winches G and H. This 
 Avheel D will drive the pinion c fixed upon its axle, which will drive B, and 
 consequently will cause the axle A on which the rope or chain is coiled to 
 revolve, and thus to raise the object to be elevated. 
 
 After what has been explained respecting the relation of the power and 
 weight, it will be evident that the efficiency of such an apparatus will alto- 
 gether depend on the relative dimensions of the wheels and pinions. But in 
 all cases it is apparent that the weight must bear a very large ratio to the 
 power. A crane constructed upon this principle by M. Cave, the well-known 
 engine maker at Paris, has been erected in the government dockyard at 
 Brest, in which the wheels have the following numbers of teeth : 
 
 The wheel B. 66 teeth. 
 The pinion c, n teeth. 
 The wheel D, 54 teeth. 
 
 The wheel F, 54 teeth. 
 The pinion K, 9 teeth. 
 The pinion L, 9 teeth. 
 
 The pinion B, 9 teeth. 
 
 By this crane a power of 20 Ibs. applied to each of the winders G and H will 
 be sufficient to raise 25 tons. 
 
 Either of the pinions K or L may be put into connection with the wheels 
 without the other, so that, when the weights to be elevated are not too great, 
 the crane may be worked by one winch only. 
 
 654. Drops for loading 1 and unloading 1 vessels at quays 
 
 and wharves. To facilitate the process of loading and unload- 
 ing vessels, an ingenious contrivance, which is very nearly self- 
 acting, has been adopted, which is represented in jig* 287. 
 
 An iron railway, properly supported on the level of the quay or wharf, is 
 continued a few feet beyond its edge, so as to overhang the water. To the 
 end of a long lever which turns on a centre attached to the quay wall, nearly 
 level with the deck of the vessel, is suspended a platform, upon which the 
 waggon containing a portion of the matter with which the vessel is to be 
 freighted stands. The end of this lever is connected by a rope G with the 
 axle of a windlass c, established on the wharf. Coiled upon the same axle, 
 but in the contrary direction, is another rope F, attached to the end of 
 another lever E, to which a counter-weight D is suspended. The waggon B 
 loaded preponderates a little over the weight D, and by this preponderance 
 it descends upon the deck of the vessel, suitably placed to receive it. When 
 the contents of the waggon are discharged in the vessel, the weight D pre- 
 ponderates, and draws the waggon up. 
 
 In this manner the weight itself of the goods to be loaded becomes the 
 moving power by which the process is executed. 
 
 If, on the contrary, it be required to unload a vessel, the windlass c must 
 be worked by some power adequate to raise the weight deposited in the 
 waggon to the level of the wharf. No advantage would be gained in that 
 case by increasing the counter-weight D, so as to give it a preponderance 
 over the loaded waggon, since a power should in that case be applied to let 
 the empty waggon down. But if it could be so managed that while one 
 cargo was being unloaded another could be loaded, the work would be exe- 
 cuted with a very small power, by adjusting the counter-weight D so as to 
 balance, as nearly as possible, the weight of the loaded waggon both in 
 ascending and descending. In that case power need only be applied sufficient 
 to overcome the friction and other resistances of the machinery, and any 
 
PRACTICAL ILLUSTRATIONS. 
 
 small difference that might exist between the effect of the loaded waggon 
 and the counterpoise. 
 
PILE-EXGIXE. 
 
 317 
 
 CHAP. II. 
 
 ENGINES WORKING BY IMPACT. 
 
 655. Pile-engines. This class of machine is an example of 
 the application of the momentum acquired by a heavy mass falling 
 from a certain height as a mechanical agent. 
 
 Heavy weights are raised by the moving power to a greater or 
 
31 8 PRACTICAL ILLUSTRATIONS. 
 
 less height, from which they are let fall on the object upon which 
 they are intended to act. The forms and magnitudes of such 
 weights, and the mechanism by which they are elevated and de- 
 tached, vary according to the purposes to which the machines are 
 applied. 
 
 The form of machine properly called a Pile-engine is applied to 
 drive piles into the ground, upon which bridges or other struc- 
 tures are intended to be erected. 
 
 One of the most simple and inartificial forms of such a machine is repre- 
 sented in fig. 288., where D is the pile to be driven into the earth, the top 
 of which is bound by a ring of iron to prevent it from being split or flattened 
 by the blows; A is the weight let fall upon it, which is usually a large 
 block of cast iron called a ram, having two ears cast upon its sides, which 
 play in vertical grooves formed in the two uprights c c, by which the ram 
 is guided in its fall, and prevented from deviating from the vertical line. 
 In this case the ram is elevated by human labour, the extremity of the rope 
 being connected with several independent cords each of which is pulled by a 
 man ; the rope passes at the top of the framing over a fixed sheave B, from 
 which it descends, and is tied to a ring at the top of the ram. 
 
 When the ram is elevated to the top of the framing, the men let go the 
 ropes simultaneously by a signal. It is very important that the ropes should 
 be all let go at the same instant, since if some men held on after the others 
 had let go they would be dragged up, and grave accidents would ensue. 
 Where such machines are used, the men are accustomed to accompany their 
 work by a sort of chaunt which regulates the moment of dismissal. 
 
 A less inartificial pile-driver is shown in fig. 289., where the 
 rope which raises the ram, instead of being directly drawn by 
 human power, is worked by a windlass having two winches, A A, 
 by means of which two or four men can raise a ram of enormous 
 weight. 
 
 The axle B, turned by the winches, has upon it a pinion E which works in 
 the teeth of a large wheel attached to the drum upon which the rope is 
 coiled. When the ram is raised by the windlass to the top of the framing, 
 the pinion E is disengaged from the teeth of the large wheel by means of a 
 lever c D E, which turns horizontally on a centre D, and is terminated in a 
 fork at E, which, laying hold of the axle B, moves it horizontally so as to 
 make it slide in its bearings, and thus to withdraw the teeth of the pinion E 
 from those of the great wheel. The wheel being thus detached, the ram 
 would be free to fall upon the pile, but in that case it would draw the rope 
 after it with a force and velocity which would soon wear it out ; this is pre- 
 vented by an expedient which is represented in fig. 290. 
 
 The rope, instead of being immediately attached to the head of the ram o, 
 is attached to apiece of mechanism consisting of a sort of pincers H o K, the 
 two arms of which play upon centres and are pressed outwards by the springs 
 1 1, which keep the points of the hooks K in close contact. The lower sur- 
 faces of these hooks are curved, so that, when they descend upon the ram, the 
 ring forces itself between them, the springs 1 1 yielding to the pressure. When 
 the upper part of the ring has passed above the edges of the hooks, the 
 latter are forced into the hole in the centre of the ring by the reaction of 
 
PILE-ENGINE. 
 
 
 Fig. 489. 
 
 the springs, the ring then resting upon the upper surface of the hooks. The 
 apparatus is thus firmly connected with the ram G, so that the whote can be 
 raised by the windlass. When it is elevated to the top of the framing, the 
 
PRACTICAL ILLUSTRATIONS. 
 
 tops of the curved handles H H pass into an open- 
 ing with oblique sides, which, pressing upon them, 
 force the handles together, the springs 1 1 yield- 
 ing to the pressure; the hooks K K are thus 
 separated, and the ring attached to the ram 
 being no longer supported by them, the latter 
 falls upon the pile. Meanwhile, the block to 
 which the apparatus H o K is attached, slightly 
 preponderating over the resistance of the rope, 
 'slowly descends, and the hooks K K again 
 engage themselves in the ring so that the ram 
 can be again elevated by the windlass. 
 
 656. Stamping-engine. One of the 
 most extensively useful forms which en- 
 gines constructed upon this principle as- 
 sume, is that which is applied to the 
 pounding of ore raised from mines. The 
 raw ore consists, as is well known, of 
 parts which are more or less metallic, me- 
 chanically mixed with earthy, stony, and 
 non-metallic matter, from which it is se- 
 parated by reducing it to dust by pound- 
 ing. The pieces of mechanism by which 
 this is accomplished, called " stamping 
 mills," consist of a combination of pile- 
 engines, identical in principle with those described above, but 
 
 worked in somewhat a different 
 manner. 
 
 A front view of a set of four 
 stampers, with the apparatus which 
 drives them, is shown in fig. 29 1 ., 
 and a side view of part of a stamp- 
 ing mill is shown in Jig. 292. 
 
 The stampers are mounted in sets of 
 four or more in immediate juxtaposition, 
 and set in framing, as shown in fig. 291, 
 To each stamper c c (^.292.) a project- 
 ing piece B is attached, upon which four 
 curved projecting pieces A A A A, called 
 cams, attached to an axis, made to re- 
 volve by the moving power, act. When 
 one of these cams A comes against B it 
 throws up the stamper c; and after it 
 passes from under B, the stamper c falls, 
 and the pounder, at its lower end, strikes 
 upon the metal placed under it. The 
 moment that this takes place the next 
 cam A comes under the piece B, and again 
 
 Fig. 190. 
 
 Kg. 291. 
 
STAMPING-ENGINE. 3 2 i 
 
 raises and lets it fall. In this manner each stamper is raised and let fall 
 four times in each revolution of the axis. There is a distinct series of foul 
 cams for each stamper, and each series of four cams holds a different position 
 on the axis, so that no two of them will act at the same moment upon the 
 stamper. It may be easily conceived that the 16 cams which thus work a 
 system of four stampers, let fall those stampers at 16 equal intervals in the 
 time of a single revolution of the axis. Thus the ore will receive 16 blows 
 in each revolution. A stream of water flowing through the pipe D, passes 
 through a spout, along E, to the ore, which it washes, the earthy and stony 
 being separated from the metallic matter by the process of straining. 
 
 A fly wheel is represented in the figure as attached to the ap- 
 paratus, to equalise the action of the moving power ; but such an 
 appendage is not necessary when the cams divide a revolution of 
 
 Y 
 
3-22 
 
 PRACTICAL ILLUSTRATIONS. 
 
 the axis into so many parts as is here supposed, the resistance be- 
 in^ in that case nearly continuous. 
 
 657. The sledge hammer is another example of accumulated 
 force obtained by the descent of a heavy mass. This apparatus, 
 
 ui its largest and most efficient form, is represented in jig. 293. 
 upon a scale of I in. to 7 teet. 
 
SLEDGE HAMMER. 323 
 
 The head of the hammer A, is a mass of cast iron, weighing from ij to 
 i tons, pierced with a large opening, in which the head of the handle B is 
 inserted and secured by a wedge. This handle turns upon a centre c, and 
 is acted upon at its extremity by cams D D, projecting from an axle, driven 
 bv an arm E, which receives its motion from some moving power, such as 
 steam or water. When the cams descend, and strike the end of the handle 
 H, the hammer A is raised, and as the motion is rapid, it is jerked upwards 
 against a horizontal beam of wood above it, by the elasticity of which it is 
 repelled, and descending falls upon the mass of red hot iron to be forged, 
 which is previously laid upon the anvil L. Since the axle driven by E 
 encounters no resistance, except while the cams act upon the handle of the 
 hammer, the machine would be unduly accelerated in these intervals of sus- 
 pension of the resistance, were it not that a heavy fly wheel F F is mounted 
 on the axle; which, receiving the force of the moving power, becomes a 
 depository thereof, and combines with the moving power in acting upon the 
 hammer when the next cam encounters the handle. 
 
 A workman holds a long iron rod, to the end of which the mass of red hot 
 iron to be forged is attached, and turns it between stroke and stroke of the 
 hammer, from one position to another, so that it can be beaten in any de- 
 sired manner. 
 
 A rod of iron, G, is governed by a lever H o, by which it can be brought at 
 will under the handle of the hammer, so as to prevent it from falling on the 
 'anvil when it is desired to remove one piece of iron already forged and to 
 replace it by another. During this process the axis continues to be driven, 
 and the cams still act upon the hammer, but the hammer only plays between 
 the rod G and the beam above it. When it is desired to recommence the 
 forging on another piece of iron, the rod G is withdrawn and the hammer 
 again allowed to fall. 
 
 The horizontal beam, which reacts by its elasticity on the heel of the ham- 
 mer, adds nothing to its force, for if it were not present, the hammer would 
 rise to a greater height, and would acquire an increase of force in its fall 
 exactly equal to that which it receives from the reaction of the beam, or, 
 more strictly speaking, a little more, the beam not being perfectly elastic. 
 The advantage of the beam is that it makes the hammer return to the anvil 
 in shorter intervals, and consequently to give a greater number of blows to 
 the iron in a given time. 
 
 658. The coining: press. The first process in the fabrication 
 of coin consists in the production of the proper composition of 
 metal of which the coin is to be made, for the silver and gold used 
 for coining, when absolutely pure, do not possess the requisite hard- 
 ness, a quality which is, howerer, easily imparted to them, by 
 alloying them with a small proportion of baser metal. When thus 
 properly alloyed, and formed into ingots or bars, they are sub- 
 mitted to the process of rolling, so as to be reduced to the form of 
 plates of a regulated thickness, an object which is attained with 
 great precision by forcing them between rollers of tempered steel, 
 placed at a distance one from the other, equal to the thickness of 
 the intended plate of precious metal. 
 
 The plates thus produced are then submitted to the action of 
 
 T 2 
 
324 PRACTICAL ILLUSTRATIONS. 
 
 circular punches of hard steel, by which circular discs are punched 
 from them of the exact size of the coins to be made. These discs, 
 which are called blanks, are then submitted to the coining press. 
 
 This press has been constructed in various forms at different 
 mints, but is generally constructed upon the principle of the screw. 
 
 The characters and figures to be raised upon the coin are first 
 engraved with the greatest precision in sunk characters, or " inta- 
 glio" upon circular surfaces of hardened steel. These are called 
 dies, and two of them are of course required for each coin. These 
 dies are placed in the press, one at its base, with its engraved face 
 upwards, and the other attached to the end of a rod driven by the 
 screw, with its engraved face downwards. The blank is then placed 
 upon the lower die, and the upper die is driven down upon it by 
 the screw with a force sufficient to drive the metal of the blank 
 into all the sunk characters of the two dies, in the same manner 
 exactly as wax is forced into the sunk characters of a seal by the 
 pressure of the hand. 
 
 Such being the general principle of the coining press, the details 
 of its mechanism, which vary much in different places, may be illus- 
 trated by those of the press represented in elevation in fig. 294., 
 the parts being shown on a larger scale in Jigs. 295. and 296. 
 
 Fig. 294. 
 
 The lever by which the screw is driven is a long metal bar c c, having 
 heavy knobs at the ends, and strong bunches of ribbon or cord, which the 
 men who drive the press lay hold of, to impart the necessary motion to the 
 screw. The masses c c, receiving thus from the hand a considerable velocity, 
 acquire a great momentum, the effect of which, transmitted to the upper die 
 attached to the screw, is augmented in the proportion of the circumference 
 
COINING PRESS. 32 j 
 
 described by the knobs c c, to the intervals between the threads ; a propor- 
 tion which, it is obvious, is always very great. 
 The base plate of the press, with the circular cavity to receive the lower 
 
 Fig. z 95 . 
 
 die, is shown in jig. 296. Upon this plate is attached a piece H, which is 
 moved horizontally by the mechanism M N L (fig. 295.)* so that when the 
 
3 26 
 
 PRACTICAL ILLUSTRATIONS. 
 
 Rcrew rises after making the impression upon the blank, the piece H turns 
 upon a joint at its extremity, moves towards the blank Avith a rapid jerk, 
 throwing out the coined piece, and dropping into its place another blank to 
 be coined by the next stroke of the press. 
 
 There are various other practical details in this mechanism, which 
 it will not be necessary here to discuss, the general principle being 
 rendered sufficiently clear from what has been stated above. 
 
 In some still more recent coining presses the screw action ha? 
 been replaced by a mechanism constructed upon the principle of 
 the knee-lever explained in 427. One of these presses, of the 
 
 Fig. 297. 
 
 most recent construction and improved form, is represented iny?y. 
 297., being that which was constructed by M. Thonnelier for the 
 mint at Paris, where it is now worked. 
 
BORING TOOLS. 327 
 
 The working shaft which imparts motion to the machine carries a large 
 fly wheel z to equalise its motion. Upon this shaft a short crank G is 
 connected with a lever H by a rod F. The lever H is made to vibrate on the 
 centre a bv the motion of the crank, and a corresponding oscillatory motion 
 is imparted to the short arm a b, which is at right angles to the long arm n. 
 This short arm is connected with a piece o, which carries the upper die, by 
 the rod i, and the knee-joint at b transmits to the upper die an intense 
 pressure, upon the principle already explained in 427. 
 
 This action differs essentially by substituting pressure for a blow, and is 
 consequently more favourable to the preservation and durability of the 
 mechanism. 
 
 The rod i acts upon the piece o by a ball resting in a corresponding 
 socket, and this piece o is pressed upwards against the ball by heavy counter- 
 weights N acting upon the lever o J c through the intervention of tlie 
 lever M and the vertical rod L. The lower die is placed in a cavity in the 
 upper part of the fixed block Q, and the blanks are placed upon it and 
 removed, after being coined, by mechanism which does not differ much in 
 principle from that already described, and which is worked by the levers s' 
 and s, the latter of which receives an oscillating motion by a piny, which is 
 moved along a sinuous groove formed in a central plate fixed upon the 
 main shaft, and turning with the fly wheel. This groove is indicated in the 
 figure by a dotted line. 
 
 CHAP. III. 
 
 BORING, PLANING, SAWING, AND SCREW AND TOOTH CUTTING 
 ENGINES. 
 
 659. Boring: tools. All instruments of this class, great and 
 small, present examples of continued rectilinear, produced by con- 
 tinued circular motion. The point of the boring tool is driven 
 continually in one direction by means of a circular motion im- 
 parted to a disc or lever attached to it. 
 
 The common bow-drill (fig. 298.) presents a familiar example 
 of this. 
 
 Fig. * 9 8. 
 
 The bit and brace (fig. 299.) presents another example. The 
 handle is turned rapidly, while the bit is pressed with its point 
 upon the object to be bored. 
 
3 z8 
 
 PRACTICAL ILLUSTRATIONS. 
 
 The form of the point of the bit varies with the material to be 
 bored, and the magnitude and shape of the hole. Different forms 
 of bits are shown in Jigs. 300 306. 
 
 I 
 
 Fig. 199. 
 
 Fig. 300. 
 
 Fig. joi. 
 
 Fig. 302. 
 
 Fig. 303. 
 
 Fig. 304. 
 
 Fig. 305. 
 
 Fig. 306. 
 
 For heavier work , drilling and boring machines, with self-acting 
 mechanism, and driven by steam or water power, are constructed 
 on a large scale. Machines of this kind are used for drilling the 
 holes in the end plates of large tubular boilers, in the heads and 
 flanges of steam cylinders, and other heavy work. The principle 
 is the same as that of the ordinary hand- drill, the details of the 
 mechanism for imparting the motion, for changing the position of 
 the drill or of the work, or both, and the scale of the machine, only 
 being different. 
 
PLANING MACHINE. 
 
 329 
 
 660. Planing machine. This machine presents an example 
 of the production of a continuous rectilinear by a continuous 
 circular motion. The purpose to which it is applied is to render 
 the surfaces of pieces of metal perfectly plane. 
 
 Fig. 307. 
 
 The block or plate of metal to be planed is firmly attached to a bed B 
 (fid- 37')> which is connected with a screw s s', which extends longi- 
 tudinally from end to end of the machine, and which is turned by a winch 
 A H, or other similar contrivance. The screw plays in nuts attached to the 
 bottom of the bed B ; and as it turns in sockets in the ends s s' of the 
 frame, and has therefore no progressive motion, it must impart a progressive 
 motion to the frame B, and consequently to the plate of metal to be planed. 
 By each revolution of the screw the metal to be planed is thus advanced 
 through a space equal to the interval between the threads. This motion 
 may be rendered as slow as is necessary by regulating the motion of the 
 winch AH. A frame D is fixed horizontally in grooves in the upright 
 pieces E ff. In this frame is inserted a screw F', worked by a winch G G'. 
 This screw carries a piece I, in which is inserted a cutter c, formed of 
 hardened steel, presenting its point in a direction contrary to that in which 
 the metal to be planed is moved. This cutter is raised and lowered at 
 pleasure by a winch L. The piece i which carries the cutter is moved trans- 
 versely, by means of the winch G G' turning the screw F'. 
 
 When, by means of the winches G' and L, the position of the cutter c is 
 properly adjusted, the surface of the metal is moved slowly against it, by 
 turning the winch H. The cutter thus acting detaches from the metal a 
 narrow shaving, and forms in it a very narrow and shallow groove. 
 
 When the entire length of the metal has thus been drawn under the 
 cutter, the action of the machine is suspended ; the cutter is raised by the 
 
330 PRACTICAL ILLUSTRATIONS. 
 
 winch L,, and the metal advanced towards the end s' of the frame by turning 
 the winch H in the proper direction. 
 
 The cutter c is now advanced transversely through a very small space 
 towards E, by turning the winch G'. It will be understood that this motion 
 is susceptible of any required degree of minuteness, since an entire revolution 
 of the winch G' would advance the cutter through no greater space than the 
 interval between the threads of the screw F', the threads of which may be 
 made fine, and the winch G' may be turned only through a small fraction of 
 an entire revolution. The machine is now prepared to recommence its 
 operation ; and, by turning H continuously, another groove will be cut in the 
 surface of the metal parallel to the former, and equal to it in breadth and 
 depth. 
 
 In fine, by continuing this process, the surface of the metal will 
 be rendered not plane but grooved ; and by making the grooves 
 finer and finer, they will at length, for all practical purposes, dis- 
 appear, and the surface will be plane. 
 
 All planing engines, whatever be their form or magnitude, are 
 constructed upon the general principle which has been illustrated 
 in that here described.- They differ, however, in their details. It 
 will be observed that in the form of machine here described the 
 operation is intermitting, being suspended while the bed support- 
 ing the object to be planed is carried back from s' to s. It 
 is, therefore, desirable that this returning motion should be 
 more rapid than the slow motion which takes p lace from s' 
 to s. In the best engines a provision is accordingly made for 
 a quick return motion ; but in some a contrivance is intro- 
 duced for reversing the cutting tool, so that the operation can 
 be executed while the bed B is moved backwards as well as 
 forwards.. 
 
 In some machines a circular motion can be given to the bed, so 
 that the cutter, instead of making longitudinal grooves, makes 
 concentric circular or spiral grooves. 
 
 By thus varying the adjustments and motions, as well of the bed 
 or table supporting the object operated on, as of the cutting tool, 
 these machines can be applied to the formation of all sorts of work, 
 such as shaping and planing levers, cranks, straps, and crossheads, 
 and, in short, all the parts of the larger class of machines. They 
 are so constructed as to be self-acting, being driven by steam or 
 water power, and are sometimes constructed on a scale of great 
 magnitude. 
 
 66 1. Saw-mills. The mechanical action of a saw, whatever 
 be the substance it is applied to divide, is so regular and uniform, 
 that it is obvious that machinery may easily be adapted to move it 
 instead of the hand. It was, therefore, one of the earliest forms 
 of labour, in which the human arm was relieved by inanimate power. 
 Saws driven mechanically by wind mills, have been worked from 
 
SAW MILLS. 
 
 33 
 
 time immemorial in Holland. The application of water power to 
 this industry is very general in mountainous and woody countries, 
 where the frequent falls of water supply means of working the 
 machinery, by which the timber fc divided and adapted for trans- 
 port at almost a nominal cost 
 
 Fig. 308. 
 
 The forms of saws thus driven is either circular or straight. 
 A circular saw is, as the name implies, a thin round disc of 
 steel, having the teeth of a saw formed upon its edge, which is 
 fixed upon a shaft to which a very rapid motion of revolution 
 is imparted by the moving power. The saw in this case re- 
 maining stationary in its position, the object to be divided by it 
 is urged against its teeth, with a regulated pressure, and con- 
 stantly advanced, as the incision is made. When it is applied to 
 divide wood, lor example, the blade of the saw, being vertical, 
 rises through a slit in a level table, at which the operator stands, 
 and upon which he places the block of wood to be cut, pressing 
 it gently against the teeth of the saw, and advancing it continually 
 until it is completely divided. 
 
332 
 
 PRACTICAL ILLUSTRATIONS. 
 
 Such saws are used for squaring the ends of the iron rails used 
 upon railways. The saws adapted for this purpose are usually 
 about 40 inches in diameter and i-ioth of an inch thick, turning 
 at the rate of about 900 revolu^ns per minute, or 1 5 per second. 
 The iron rail is then applied to them red hot, and the lower part 
 of the saw is immersed in a trough of water to keep it cool. 
 
 Straight saws worked by machinery consist of an oblong rect- 
 angular blade, like that of a common knife mounted upon a rect- 
 angular frame, with screw adjustments, for the purpose of giving 
 the necessary tension to the blade. When these are applied to divide 
 timber into boards or planks, they are placed vertically, and are 
 moved with an up and down motion, similar exactly to that given 
 to them when worked by hand in a saw pit. Instead, however, of 
 advancing against the timber, as in the latter case, the timber 
 being stationary, the timber is advanced against the saw, the saw 
 being stationary. Saws worked in this way make from 1 20 to 1 50 
 strokes per minute, and cut at the rate of from one tenth to 
 
SAW MILLS. 
 
 333 
 
 Fig. 310. 
 
 half an inch per stroke, according to the softness or hardness of 
 the wood. 
 
 To render intelligible this im- 
 portant application of the saw, 
 we shall here describe one of the 
 most improved forms of the saw 
 mill adapted to cut veneering 
 from mahogany and other pre- 
 cious ornamental woods. 
 
 The moving power, as is customary 
 in mills for all purposes, imparts revo- 
 lution to a shaft or shafts, which are 
 carried along the rooms under the ceil- 
 ings in which the machines are esta- 
 blished. Upon these shafts over each 
 machine a drum A (fig. 308.) is fixed, 
 which revolves with the shaft. This 
 drum is connected by an endless band 
 with the axle B of a fly wheel E E, and 
 is kept tight by a bar D c fixed upon 
 the centre D, and pressing against it 
 with its weight by a roller c. When 
 
 it is desired to suspend the motion of the machine, it is only necessary to 
 raise the bar D c, when the band becoming slack, the motion of A will not 
 be imparted to B. 
 
 A connecting rod G F is attached by a pin F to one of the arms of the fly 
 wheel E E, at a distance from the centre equal to half the stroke intended to 
 be given to the saw. The other end o is attached in like manner by a pin 
 to the frame of the saw, which is movable in horizontal grooves, so that by 
 each revolution of the fly wheel the pin G, and with it the saw, is moved 
 back and forward through the space m n, which is equal to twice the distance 
 of F from the centre of the fly wheel. 
 
 A view of the sawing machine, driven by the rod F G, is given in fig. 309., 
 where the end G of that rod appears broken off. One side of the horizontal 
 frame which carries the saw is shown at H H, the blade of the saw being 
 behind this, and therefore not. visible in the figure. A rod of strong wire is 
 seen in front of H H, which passes through blocks at each end of the frame, 
 and is secured by nuts screwed upon it, by means of which the rod is 
 shortened so as to increase the tension of the blade of the saw. This blade 
 is kept exactly vertical, the teeth being downwards, by a plate of iron, K, the 
 lower edge of which is bevelled towards the saw. 
 
 According to what has been explained, the saw receives an alternate 
 motion right and left through a space equal to m n (fig. 308.). If, therefore, 
 the wood to be cut be pressed upwards against its teeth in a suitable manner, 
 the desired separation will be effected. We shall now explain how this is 
 accomplished. 
 
 In fig. 309. there is a vertical frame u u fixed behind the saw, to which a 
 rack T T is attached. In the teeth of this rack a pinion s, fixed upon the 
 axle of a ratchet wheel Q Q, works. In the teeth of this ratchet wheel are 
 engaged two catches : one R, playing on a centre attached to the frame, and 
 urged against the teeth by a spring ; and the other fixed upon the lower end 
 
33 + PRACTICAL ILLUSTRATIONS. 
 
 of a bar p, which turns on the centre N, and which is urged against the teeth 
 by a spring which appears in the figure. A lever L works upon a centre 
 attached to the sliding frame of the saw, and which, therefore, moves right 
 and left with it. This lever imparts a corresponding motion to M, and the 
 latter to N. By this means the short lever N moves p alternately upwards 
 and downwards with each stroke of the saw. When p is raised, it draws the 
 hook at its lower end upwards from the tooth of the ratchet wheel in which 
 it was engaged ; and when it has risen to the limit of its play, the spring 
 presses it over the next tooth. When p descends, the hook presses the 
 tooth of the wheel down ; and, causing the wheel to move through a small 
 space, the other catch it passes to the next tooth. In this manner with 
 each stroke of the saw the ratchet wheel is advanced one tooth ; and 
 the pinion s receiving a corresponding motion, the rack T x, and the 
 frame u u to which it is attached, are raised through a corresponding small 
 space. 
 
 Now, between this frame u u and the blade of the saw a space sufficiently 
 wide is left for the introduction of the block of wood from which the sheets 
 of veneering are to be cut. That block, therefore, being so introduced, is 
 glued to the frame u u, its upper edge at the commencement of the operation 
 being a little below the teeth of the saw. 
 
 The framing which carries the saw is traversed by two screws v v, con- 
 nected by an endless chain, both of which receive a common motion from one 
 handle which appears in the figure, attached to the right-hand screw. By 
 means of these screws the frame u u, with a block of wood attached to it, 
 can be moved through any desired space to or from the saw. In the com- 
 mencement of the operation the upper edge of the block is by this means 
 brought under the teeth of the saw, in such a position that a sheet of 
 veneering of the requisite thickness shall be cut from it. 
 
 These arrangements being made, and the machine being put in motion by 
 throwing the bar D c (Jig, 308.) on the band, the work proceeds, and the saw 
 detaches a sheet of veneering. The manner in which the saw thus operates 
 is shown more evidently in fig. 310., where x x is the block of wood, and y y 
 some bars, by means of which it is fastened to the frame u u ( fig. 309.). It 
 will be remembered that by each stroke of the saw, the frame u u, and with 
 it the block x x, is raised through a small height by the action of the levers 
 L M N P. The mechanism is so adjusted that this height shall be equal to 
 the depth of the wood through which the saw can make an incision at each 
 stroke. The bevelled bar K K separates the sheet of veneering from the 
 block, so as to prevent the saw from being locked between them ; and as this 
 sheet is too thin to support its own weight, it is sustained by a loop of wire, 
 a little below the top of the block, as shown in the figure. 
 
 In the practical operation of the machine, the drum A revolves at the rate 
 of about one revolution per second, and if its diameter be 5 times that of 
 the axle B, the latter will revolve 5 times per second, and the same number 
 of strokes will be made per second by the saw. If the wood from which the 
 veneering is cut be mahogany, the depth cut by each stroke will be about the 
 fiftieth of an inch ; and such a saw will be capable of detaching about 400 
 square feet of veneering per day. 
 
 After each sheet of 'veneering is cut off, the action of the machine is sus- 
 pended by raising the bar D c, and the frame u u is then lowered by disen- 
 gaging the two catches from the ratchet wheel, until the upper edge of the 
 block x x is brought below the teeth of the saw, and the screws v v are then 
 turned until the upper edge of the block is brought sufficiently within the 
 
SCREW-CUTTING ENGINE. 
 
 335 
 
 teeth of the saw to detach another sheet of veneering of the necessary thick 
 ness. 
 
 It is found that the thickness of wood converted into saw dust 
 by this process is about the fiftieth of an inch, and if the thickness 
 of the sheet of veneering be also the fiftieth of an inch, which it 
 may be. and generally is, it is evident that 50 per cent, of the 
 wood will be wasted in the process. 
 
 It is obvious that in the case of highly valuable woods, which are 
 precisely those that are cut in the thinnest sheets, it is highly de- 
 sirable to avoid a waste so extensive, and various attempts have 
 been accordingly made from time to time to substitute the plane 
 for the saw in the cutting of veneering, so that the wood would be 
 detached in thin shavings without any waste at all. I am not 
 aware, however, that as yet these attempts have been attended 
 with any practical success. 
 
 662. Screw-cutting engine. This engine affords another 
 example of the reciprocal production of circular and rectilinear 
 motion. It is represented in^. 311. 
 
 Fig jn. 
 
 The whee 1 A imparts rotation to the cylindrical rod s s', upon which it is 
 desired to cut a thread. A toothed wheel B is fixed upon the rod s S', and 
 
33* 
 
 PRACTICAL ILLUSTRATIONS. 
 
 revolves with it. This wheel imparts revolution to a smaller toothed wheel 
 c, which is fixed upon the end of a fine screw by which a progressive motion 
 is imparted to the piece H. in which a cutter D is inserted. It is evident that 
 the proportion between the progressive motion imparted to the cutter D, and 
 the rotatory motion of the rod s s>, is regulated by the proportion of the 
 wheels B and c, and the thread of the screw driven by c. 
 
 According to this combination, while the cutter D moves slowly from s to 
 s', the rod s s> revolves, and it is evident that the incision made by the cutter 
 in the rod, will be of a spiral or helical form, and that the interval between 
 the threads of the screw thus produced will be equal to the space through 
 which the cutter D advances during one revolution of the rod s S'. 
 
 The depth of the incision made by the cutter is regulated by a screw E, 
 which advances or withdraws the cutter at pleasure. 
 
 663. Engine for cutting- the teeth of wheels. This engine 
 furnishes an example of the combination of a continuous with an 
 intermitting circular motion. It is represented in Jig. 312. 
 
 Fig. jiz. 
 
 The disc A, upon which the teeth are to be cut, is fixed upon a rod c, which 
 is turned by a disc B, the circumference of which is divided into equal inter- 
 vals by holes, into which a point projecting from the end of a lever D, can 
 be inserted so as to render the disc B, and consequently A, stationarj-. The 
 teeth are cut in A, by means of a circular cutter E, driven by a band r, 
 receiving its motion from any moving power. The piece carrying the cut- 
 ter is moved longitudinally by a screw driven by a winch G, and it is moved 
 to and from the wheel to be cut by a screw driven by the winch H. 
 
FLOUR MILLS. 337 
 
 The form of the teeth is determined by the shape of the cutter E, and their 
 depth by the extent to which the incisions are continued by acting on the 
 winch H. 
 
 The wheel A is placed upon the shaft and removed from it by means of 
 the lever i, by which the shaft c can be removed from the frame at pleasure. 
 
 CHAP. IV. 
 
 FLOUR MILLS. 
 
 664. THE process of converting grain into flour is executed by 
 passing the grain between two cylindrical stones, having a common 
 vertical axis, smooth horizontal surfaces, and mounted so that the 
 distance between these surfaces is less than the thickness of the 
 grain. The lower stone is stationary, and the upper kept in revo- 
 lution round its axis, with a regulated velocity, by some moving 
 power, such as wind, water, or steam. The grain is let fall in 
 small and regulated quantities through an opening which surrounds 
 the axis in the upper stone ; it descends thus into the space be- 
 tween the two stones, and while it is broken between their surfaces, 
 its particles are carried round by the rotation of the upper stone, 
 and being slightly affected by the centrifugal force produced by 
 the rotatory motion, it recedes gradually from the centre as it re- 
 volves, each particle describing a sort of spiral. The particles, at 
 length arriving at the edge, fall into a groove made to receive them, 
 from which they are discharged through openings provided for the 
 purpose into a proper receptacle. 
 
 Such being the general principle upon whicn flour mills are con- 
 structed, their details and mode of operation will be better under- 
 stood by reference to figs. 313. and 314. 
 
 The horizontal shaft A (fig. 314.) receives a motion of rotation from the 
 moving power, whatever it may be, outside the mill ; it imparts by the 
 bevelled wheels a motion of rotation 1 to the vertical shaft B, upon which a 
 large toothed wheel c is fixed ; this wheel works into small toothed wheels 
 D and E, fixed upon two vertical shafts N and r. These wheels D and E are 
 capable of moving through a limited space with a sliding motion of the shafts 
 N and F, so as to be thrown into or out of connection with the great wheel c 
 at pleasure. The moving power may thus be made to impart revolution to 
 either or both of the shafts N and F. These shafts pass respectively through 
 the centres of two pairs of mill-stones, that through which F passes being 
 shown in section in the figure, and that through which x passes being 
 enveloped in its octagonal wooden casing, on the top of which is placed the 
 apparatus by which grain is supplied to the stones. 
 
333 
 
 PRACTICAL ILLUSTRATIONS. 
 
 The shafts pass through holes in the centres of the lower stones, in which 
 they turn freely, without moving these stones, and the space around them is 
 
 Fig. 314. 
 
 stopped up with leather so as to prevent the grain from falling between. 
 The upper stono is fixed upon the top of the shaft and turns with it ; a space, 
 
FLOUR MILLS. 
 
 339 
 
 however, being left open between the shaft and the stone to let in the grain. 
 The grain is supplied from a funnel-shaped box i, with an opening in the 
 bottom, of regulated magnitude, through which it falls into the inclined 
 spout L, which receives a slight oscillating motion right and left, from pieces 
 which project from the rod K. By this motion the grain is shaken out, and 
 falling little by little through the central opening in the upper stone, passes 
 between the stones, where it is ground. 
 
 Mill-stones have been generally formed of simple blocks of stone, 
 but these having been found to be often defective, a better class 
 of stones have been produced, by cementing together, by means of 
 plaster, chosen pieces of stone of good quality, the whole being 
 bound together by hoops of iron. The diameters of the mill-stones, 
 which were formerly from 5 to 6 feet are now generally reduced 
 to 3^ feet. 
 
 Each pair of stones is capable of grinding about forty-two im- 
 perial bushels in twenty-four hours, and it is found that the best 
 velocity for the stones is about seventy revolutions per minute. 
 
 The mere grinding of the grain is not, however, all that is ac- 
 complished in the flour mill. The grain, as is well known, consists of 
 the material which constitutes the flour, properly so called, and the 
 husky envelope, which forms the bran, so that the immediate pro- 
 duct of the mill stones is a mixture of flour and bran, generally 
 called " whole meal" To produce fine flour from this, it is necessary 
 to separate the bran from the flour, and this process, which may 
 be executed with a greater or less degree of perfection, according 
 to the fineness of the flour intended to be produced, is accomplished 
 in the mill by machinery driven by the same power which moves 
 the mill-stones. For this purpose, proper shafts are connected 
 with the main shaft driven by the power, and their motion is con- 
 veyed to those parts of the mill in which is an apparatus consisting 
 of wire sieves of more or less fineness. The whole meal, as it comes 
 from the stones, is passed over a series of these sieves, through the 
 meshes of which the flour falls, the bran being retained. 
 
 CHAP. V. 
 
 CLOCK AND WATCH WOEK. 
 
 665, Time measurer a necessity of civilised life. After the 
 supply of the absolute necessaries of physical existence food, 
 clothing, and lodging one of the first wants of a society emerg 
 
 z ^ 
 
340 PRACTICAL ILLUSTRATIONS. 
 
 ing from barbarism, is the means of measuring and registering 
 time. In civilised society, all contracts for labour, and for all 
 kinds of service, are based upon time. Even in the cases of the 
 highest public functionaries, and where the service rendered is 
 purely social and intellectual, still it is regulated, limited, and com- 
 pensated with relation to time. Time measurers or chronometers 
 were therefore among the earliest mechanical and physical inven- 
 tions. 
 
 Although nature has supplied visible signs to measure and to 
 mark the larger chronometric units, such as days, months, and 
 years, she has not furnished any corresponding measures of the 
 lesser units of hours, minutes, and seconds. There are no visible 
 marks on the firmament by passing from one to another of which 
 the sun can note the hours, still less are there any signs for minutes 
 or seconds. These subdivisions are therefore merely artificial and 
 conventional, and, to measure and mark them, artificial motions 
 must be contrived. 
 
 666. Sun-dials. Rough approximations were first made to 
 the chief divisions of the day, by observing the apparent motion of 
 the sun from rising to setting. Thus the direction of the meridian, 
 or of the south, being once known, and marked by some fixed and 
 visible object, the time of noon was known by observing when 
 the sun had this direction. The hours before and after noon were 
 roughly estimated by the position of the sun between noon and the 
 times of its rising and setting. Greater precision was given to 
 this method, by erecting a wand or gnomon, the shadow of which 
 would fall upon a level surface, in a direction always opposite to 
 that of the sun. Thus, .after sunrise, the shadow would be inclined 
 towards the west, the sun being then towards the east. From 
 the moment of sunrise until noon, the shadow would move con- 
 tinually nearer and nearer to the direction of the north, and at 
 noon it would have exactly that direction. From noon to 
 sunset the shadow would be more and more inclined towards 
 the east. 
 
 It is evident, however, that such a dial would not afford uniform 
 indications at all seasons of the year, so that the hour-lines of the 
 shadow determined in spring, for example, would not show the 
 same hours in winter as in summer. Without much astronomical 
 knowledge, it is easy to be convinced of this. At the equinoxes, 
 the sun rises and sets at six o'clock, and at the east and west points 
 precisely ; and, therefore, at these seasons, the six o'clock hour- 
 lines of such a dial would be for the morning due west, and for 
 the evening due east. But on the first day of summer (2 1st 
 June), the sun rises and sets at points of the horizon very much 
 north of the east and west points, and at six o'clock in the fore- 
 

 SUN-DIALS. 
 
 noon and afternoon its bearing is north of the east and west 
 points. 
 
 A dial so constructed at any given place would be useless as a 
 time indicator. To render it useful, it would be necessary that 
 the shadow of the style should fall in the same directions at the 
 same hours at all seasons of the year. Now, to attain this object, 
 the style must be not vertical, but must be directed to the celes- 
 tial pole. It is easy to comprehend that in that case a plane pass- 
 ing through the style and the sun would always be carried round 
 the style with an uniform motion by the diurnal motion of the sun, 
 and that at all seasons this plane would at the same hours have the 
 same position. 
 
 It is for this reason that the gnomon of sun-dials is placed at 
 such an inclination with the plate of the dial, that when the dial 
 is properly set the gnomon will be directed to the north pole of the 
 heavens, and being so placed, its shadow will fall upon the same 
 lines of the dial at the same hours, whatever be the season of 
 the year. 
 
 667. Position of gnomon. It is evident, therefore, that dials 
 must be differently constructed for places which have different 
 latitudes. It is shown in astronomy that the elevation of the celes- 
 tial pole is equal to the latitude of the place, and consequently the 
 inclination of the gnomon of a sun-dial must be also equal to the 
 latitude of the place where the dial is intended to be set. It fol- 
 lows, therefore, that a dial constructed for London would not be 
 suitable for York, Newcastle, or Edinburgh. 
 
 The position of the plate of the dial upon which the shadow of 
 the gnomon is projected is quite unimportant. All that is really 
 important, is the direction of the gnomon, which must always be 
 1 that of the celestial pole, whatever be the position of the plate of 
 the dial. Thus the plate of the dial may be either horizontal, ver- 
 tical, or oblique. Its position will depend upon the place where 
 it is to be erected. If it be in an open space, as in a garden or 
 field, having a clear exposure on all sides, it will be generally most 
 convenient to make it horizontal ; and, hence, :oi such cases, it is 
 usual to fix it upon the top of a column of three or four feet high, 
 so that it may be easily observed by a person of ordinary height 
 standing near it. Sometimes it is convenient to place it upon the 
 wall of a building, such as a church. A wall with a southern ex- 
 posure is in that case the most convenient ; but to indicate the 
 hours of the early morning in the spring and summer, an eastern 
 exposure would be required, and to indicate those of the late 
 evening, a western exposure would be necessary. 
 
 Where these vertical dials are erected, it is therefore frequently 
 z 3 
 
342 PRACTICAL ILLUSTRATIONS. 
 
 the practice to establish them at the same time on different walls 
 of the same building. 
 
 Whatever be the position of the plate of the dial, the position 
 of the hour-lines upon it is a matter of mere technical calculation, 
 for which the formulae and principles of spherical trigonometry are 
 necessary, but which is not attended with any difficulty. 
 
 It must, however, be observed, that generally the hour-lines are 
 inclined to each other at unequal angles, as may be seen by inspect- 
 ing any ordinary sun-dial. There is one, and one only, position 
 which could" be assigned to the plate of the dial, such that the hour- 
 lines would make equal angles with each other. That position 
 would be at right angles to the gnomon, and a dial so constructed 
 would be suitable to any place, whatever be its latitude. All that 
 would be necessary would be to set it so that the gnomon would 
 be directed to the celestial pole. The sun, however, would shine 
 upon the upper or north side of it during the spring and summer, 
 and on the lower or south side during the autumn and winter. 
 It would, therefore, be necessary that it should be marked on both 
 sides with hour-lines, and that a gnomon should be fixed on both 
 sides. 
 
 The name dial is derived from the Latin word dies, a day, and 
 the invention and use of the instrument as a time indicator is very 
 ancient. According to Herodotus, the invention came to Greece 
 from Chaldsea. The first dial recorded in history is the hemisphere 
 of Berosus, who is supposed to have lived 540 B.C. 
 
 668. Clepsydra. The first attempts to measure time by mo- 
 tions artificially produced, consisted in arrangements by which a 
 fluid was let fall in a continuous stream through a small aperture 
 in the pipe of a funnel, the time being measured by the quantity 
 of the fluid discharged. The CL.EPSYDRA, or water-clock, of the 
 ancients, was constructed upon this principle. This and the sun- 
 dial were the only instruments contrived or used by the ancients 
 for the measurement of time. 
 
 Clepsydras were contrived by the Egyptians, and were in com- 
 mon use under the reign of the Ptolemys. In Rome, sun-dials 
 were used in summer and clepsydras in winter. These instruments, 
 though subject to very obvious defects, were, nevertheless, when 
 skilfully used, susceptible of considerable accuracy, as may be 
 easily conceived when it is stated that, before the invention of 
 clocks and watches, they were the only chronometric instruments 
 used by astronomers. The chief sources of their irregularities 
 were the unequal celerity with which the fluid is discharged, 
 owing to its varying depth in the funnel and its change of tem- 
 perature. 
 
 669. Sand-glass. The common hour-glass (Jig. 315.) come** 
 
HOUR-GLASS. 343 
 
 under this class of chronometric 
 instruments, but is the most im- 
 perfect of them. Nevertheless, for 
 certain purposes, it is even now, 
 advanced as we are in the applica- 
 tion of science to the arts, still 
 found the most convenient chro- 
 nometer. The process of ascer- 
 taining a ship's rate of sailing or 
 steaming by means of the log affords 
 an example of 'its use. One man 
 holds the reel from which the line 
 
 runs off, while another holds the sand-glass, and gives the signal 
 when the sand has run out. The number of knots run off from the 
 reel is then the number of miles per hour in the rate of the vessel. 
 The intervals between the knots, the quantity of sand in the glass, 
 and the aperture through which it falls, are so adapted to each 
 other as to give this result. 
 
 670. Mercurial time measurer. Notwithstanding the great 
 perfection to which the art of constructing chronometers has 
 attained, an apparatus was not long since proposed by the late 
 Captain Kater for the measurement of very small intervals of time, 
 fractions of a second for example, which is a modification of the 
 clepsydra. A quantity of pure and clean mercury is poured into 
 a funnel with a small aperture at its apex, so that a stream of the 
 quicksilver shall fall through it. The flow is rendered uniform by 
 keeping the mercury in the funnel at a constant level. The appa- 
 ratus is intended in scientific researches to note the exact duration 
 of phenomena, and it is so managed, that the stream issuing from 
 the funnel is turned over a small receiver at the instant the phe- 
 nomenon to be observed commences, and is turned away from it 
 the instant the phenomenon ceases. The mercury discharged into 
 the receiver is then accurately weighed, and the number of grains 
 and parts of a grain it contains being divided by the number of 
 grains which would be discharged in a second, the number of 
 seconds and the parts of a second which elapsed during the con- 
 tinuance of the phenomenon is found. 
 
 671. Clocks and watches. For the purposes of civil life, as 
 well as for the more precise objects of scientific research, all these 
 contrivances have been superseded by clocks and watches, which 
 are now so universal as to constitute a necessary article of furni- 
 ture in the most humble dwellings, and a necessary appendage of 
 the person in all civilised countries. 
 
 All varieties of this most useful mechanical contrivance include 
 five essential parts. 
 
344 PRACTICAL ILLUSTRATIONS. 
 
 I. A moving power. 
 
 II. An indicator, by whose uniform motion time is measured. 
 III. An accurately divided scale, upon which the indicator 
 moves and by which its motion is measured. 
 
 IV. Mechanism, by which the motion proceeding from the mov- 
 ing power is imparted to the indicator. 
 
 V. A regulator, which renders the motion imparted to the indi- 
 cator uniform, and which fixes its celerity at the required rate. 
 
 Thus, for example, in a common clock, the moving power is the 
 weight suspended by cords over a pulley fixed upon the axle of a 
 wheel, to which the weight in descending imparts a motion of 
 rotation. The indicator is the hand. The scale is the dial-plate 
 upon which the hours, minutes, and sometimes the seconds, are 
 marked by equal divisions, over which the point of the hand 
 moves. The mechanism is a train of wheelwork, so constructed that 
 the rate of rotation of the last wheel, upon the axle of which the 
 hand is fixed, shall have a certain proportion to the rate of rotation 
 of the first wheel, upon the axle of which the weight is suspended. 
 And if, as is generally the case, there be two or three hands, then 
 the wheelwork is so constructed that while one of the hands 
 makes one revolution, another shall make twelve revolutions, and 
 the third shall make sixty revolutions during a single revolution of 
 the latter, and therefore seven hundred and twenty during a single 
 revolution of the former. 
 
 672. Pendulum. If no other appendage were provided, the 
 weight would, in such an apparatus, descend with a continually 
 increasing velocity, and would therefore impart to the hands a 
 motion of rotation more and more rapid, which would not con- 
 sequently serve as a measure of time. This defect is removed by 
 the addition of a pendulum, combined with a wheel upon which it 
 acts called the escapement. It is the property of the pendulum 
 that its oscillations are necessarily made always in equal times, 
 and its connection with the escapement wheel is such, that one 
 tooth of that wheel, and no more, is allowed to pass the upper part 
 of the pendulum during each oscillation right and left. But this 
 escapement wheel itself forms part of the train of wheelwork by 
 which the first wheel, moved by the descending weight, is con- 
 nected with the wheels which move the hands, and consequently, 
 by regulating and rendering uniform the motion of this escape- 
 ment wheel, the pendulum necessarily regulates and renders uni- 
 form the motion of the entire apparatus. 
 
 The instrument thus arranged, therefore, imparts an uniform 
 motion of rotation to each of the hands, but this is not enough to 
 render it a convenient time measurer. It is necessary that the 
 motion of the hands should have some definite and simple relation 
 
CLOCK AND WATCH WORK. 345 
 
 to the natural and conventional division of time into days, hours, 
 minutes, and seconds. For this purpose it is required not only 
 that the hands should move uniformly, but that the first, or slowest 
 of them, should make two complete revolutions in a day, or a single 
 revolution in twelve hours; and, as a necessary consequence of 
 this, that the second should make a single revolution in an hour, 
 and the third in a minute. 
 
 From what has been stated, it will be apparent that the actual 
 rate of motion imparted to the hands will be determined by the 
 rate of oscillation of the pendulum. It has been shown that for 
 each oscillation, right and left, of the pendulum, one tooth of the 
 escapement wheel passes, and if the escapement wheel have thirty 
 teeth, and if the pendulum take one second to make a single swing, 
 it will allow the escapement wheel to make a complete revolution 
 while it makes thirty swings from right to left, and thirty from 
 left to right, that is, in sixty seconds, or one minute ; so that, if 
 the axis of the third hand were in this case fixed upon the axle of 
 the escapement wheel, that hand would make one complete revo- 
 lution in a minute, and consequently the second would make one 
 complete revolution in one hour, and the third in twelve hours. 
 The required conditions would therefore be in this case fulfilled. 
 
 To render this explanation of the regulating property of the 
 pendulum complete it will be sufficient to show 1st, that the time 
 of vibration must be always rigorously the same with the same 
 pendulum ; 2nd, that this time can be made shorter or longer by 
 varying the length of the pendulum, so that a pendulum can 
 always be constructed which will vibrate in one second, or in half 
 a second, or, in short, in any desired time ; and 3rd, that the 
 connection of the pendulum with the escapement wheel can be so 
 constructed, that the motion of the latter shall be governed by the 
 vibrations of the former, in the manner already described. 
 
 The isochronism of the pendulum, and the manner in which its 
 rate of oscillation may be made to vary by varying its length 
 having been already explained, it only remains to show how it is 
 connected with the escapement wheel, so as to regulate the motion 
 of the hands. 
 
 The pendulum acts upon the escapement wheel by means of a 
 fork G (fig- 316.) between the prongs of which its rod passes, 
 so that as the pendulum swings alternately from side to side, it 
 carries the fork with it. This fork G projects from a vertical rod 
 F, which is connected with a horizontal shaft D. This shaft is 
 connected with the escapement, to the anchor of which it gives 
 an alternate motion that will be explained hereafter. 
 
 673. Compensation pendulums. It has been explained 
 that the centre of oscillation of a pendulum will vary with every 
 
PRACTICAL ILLUSTRATIONS, 
 
 change in the relative positions of its point of suspension and 
 its centre of gravity. Now, since all bodies are 
 susceptible of dilatation by increase, and contrac- 
 tion by decrease of temperature, pendulums will 
 necessarily be subject to a change in the position 
 of their centres of oscillation, and therefore of the 
 rate of vibration, with every change of their tem- 
 perature. To ensure the uniformity of the rate of 
 vibration, a class of contrivances has been invented 
 by which every variation of temperature is made 
 to produce two effects on different parts of the 
 pendulum, which counteract each other, and keep 
 the centre of oscillation at an invariable distance 
 from the point of suspension. 
 
 One of the means of accomplishing this, is, by 
 connecting the bob of the pendulum with the point 
 of suspension by rods composed of materials ex- 
 pansible in different degrees, so arranged that 
 the dilatation of one shall augment the distance of 
 the centre of oscillation from the point of sus- 
 pension, while the expansion of the other di- 
 minishes it. 
 
 Let s (./%. 317-) be the point of suspension, and 
 o the centre of oscillation, and let s be supposed 
 to be connected with o by means of two rods of 
 metal, s A and A o, which are united at A, but 
 independent of each other at every other point. 
 If such a pendulum be affected by an increase of temperature, 
 the rod s A will suffer an increment of length, by which the point 
 A and the rod A o attached to it will be lowered ; but, 
 at the same time the rod A o, being subject to the same 
 increase of temperature, will receive an increment of 
 length, in consequence of which the point o will be 
 raised to an increased distance above the point A, at 
 which the rods are united. If the increment of the 
 length of the rod A o be in this case equal to the incre- 
 ment of the rod s A, then the point o will be raised as 
 much by the increase of the length of A o as it is 
 lowered by the increase of the length of s A, and, con- 
 sequently, its distance from the point s will remain the 
 same as before the change of temperature takes place. 
 
 To fulfil these conditions, it is only necessary that 
 the length of the rod A o shall be less than the length 
 of the rod s A in exactly the same proportion as the 
 expansibility of the metal composing A o exceeds the 
 
 Fig. 316. 
 
 A 
 
 Fig. 317. 
 
CLOCK AND WATCH WORK. 
 
 347 
 
 expansibility of the metal composing s A. If 
 the lengths of s A and A .o were equal, their 
 increments of length would be proportional to 
 their dilatations ; but the length of the more 
 dilatable rod AO, being less than that of the less 
 dilatable s A, in the same proportion as the dilata- 
 bility of the former is greater than that of the lat- 
 ter, the absolute increments of their length wiL 
 necessarily be equal, the greater dilatability of 
 A o being compensated by its lesser length. 
 
 674. Harrison's gridiron pendulum. 
 This principle is variously applied in different 
 pendulums ; that which is best known is Har- 
 rison's gridiron pendulum, represented in fig. 
 318. A heavy disc of metal L, is suspended 
 from a cross piece of brass a a, fixed to the 
 lower extremities of two iron rods b b. These 
 rods themselves are attached at the upper 
 extremities to another brass bar c, which is 
 supported upon the upper ends of two zinc 
 rods dd. These zinc rods are supported below 
 by a third cross bar e e, which is fixed to the 
 lower end of a brass tube/, into which the iron 
 rod g passes, and is held by a linch-pin h. 
 This rod g is continued upwards to the point 
 of suspension. When the temperature rises, the 
 rod g and tube /, being elongated by expan- 
 sion, the cross piece e e is removed farther 
 from the point of suspension. If the rods d d 
 suffered no change of dimension, the cross 
 piece c c would be lowered just as much as 
 e e, and would be removed farther from the 
 point of suspension. The dilatation which the 
 iron rods b b undergo causes the cross piece 
 a a to be removed farther from c c, and in con- 
 sequence of these several dilatations of the 
 rod g, the tube /, and the rods b &, the lens L 
 would descend to a lower position than that 
 which it previously had, but the zinc rods dd, 
 which we have here supposed to remain un- 
 changed, not only dilate like the other rods, 
 but dilate in a much greater degree in proportion to their length, 
 and, by dilating, cause the cross pieces a a and c c and the rods b b 
 which connect them, and consequently the lens L, to rise and there- 
 fore to come nearer to the point of suspension. 
 
 Fig. 318. 
 
PRACTICAL ILLUSTRATIONS. 
 
 It appears, therefore, that while the dilatation of g, f, and b b 
 cause the lense L to descend, that of d d causes it to ascend ; and if 
 the length of these be properly proportioned, the ascent produced 
 by the latter dilatation will be exactly equal to the descent pro- 
 duced by the former, and consequently, the centre of oscillation 
 and the rate of the pendulum will remain unaltered. 
 
 675. Mercurial pendulum. This pendulum, which depends 
 
 on the same principle, consists of two cylindrical 
 glass vessels b J, (fg. 319.) containing mercury, 
 which supply the place of the bob or lens, the rod 
 of the pendulum a passing between them and 
 supporting a brass plate upon which they rest. 
 An increase of temperature, causing an expansion 
 of the rod a and its appendages, would cause the 
 vessels of mercury to descend ; but the same 
 increase of temperature producing a still greater 
 expansion in the mercury, causes it to rise in the 
 glass cylinders, and consequently to produce a 
 corresponding elevation in the centre of gravity 
 of the mercury. These two dilatations, like the 
 former, may be so arranged that they shall pre- 
 cisely counteract each other, and so keep the 
 centre of oscillation, and consequently the rate 
 of the pendulum, unvaried. 
 
 676. Another compensation pendulum. If two straight 
 bars of differently dilatable metals be soldered together, every 
 change of temperature will bend the combined bar into the form 
 of a curve, the more dilatable metal being on the convex side of 
 the curve when the temperature is raised, and on the concave side 
 of it when it is lowered. 
 
 Let the more dilatable metal be called A, and the less dilatable B. 
 Now, if the temperature be raised, A will become longer than B, 
 and, as they cannot separate, they must assume such a form, being 
 still in contact, as is consistent with the inequality of their lengths. 
 This is a condition which will be satisfied by a curve in which the 
 bar A is on the convex and the bar B on the concave side. 
 
 If the temperature, on the other hand, be lowered, the more 
 dilatable metal being also the more contractible, the bar A will be 
 more diminished in length than the bar B, and being, therefore, the 
 shorter, will necessarily be on the concave side of the curve. 
 
 This principle has been ingeniously applied as a compensator. 
 
 Such a compound bar as we have just described is placed at right 
 angles to the rod of the pendulum, and has, at its extremities, 
 two bobs. When the temperature rises, and the centre of os- 
 cillation is, by expansion of the pendulum, removed to a greater 
 distance from the point of suspension, this compensating bar is 
 
 Fig. 319. 
 
CLOCK AKD WATCH WORK. 
 
 349 
 
 bent into the form of a curve concave towards the point of sus- 
 pension, as represented in /#-. 320. ; and the bobs which it 
 carries at its extremities, being brought closer to the point of 
 suspension, compensate for the increased distance of the bob of 
 the pendulum from that point. 
 
 If, on the other hand, the temperature falls, and the rod of the 
 pendulum contracting brings the bob and the centre of oscillation 
 nearer to the point of suspension, the compensating bar is bent 
 into a curve, which is concave downwards, as represented in 
 fig. 321. ; and the bobs which it carries being removed to an in- 
 creased distance from the point of suspension, compensate for the 
 diminished distance of the bob of the pendulum. 
 
 M( 
 
 )M 
 
 
 
 Fig. 320. Fig. 321. 
 
 677. Connection between pendulum and escapement. 
 
 Supposing, then, the pendulum to be so adjusted that it shall 
 make its vibrations at any required rate, one per second for ex- 
 ample, let us see how the motion of the indicating hands is 
 governed by such vibrations. 
 
 Upon the axis on which the pendulum oscillates is fixed a 
 piece of metal in the form of an anchor, such as D B A c 
 (.fig- 3 22 -)> so that this piece shall swing alternately right and left 
 with the pendulum. Two short pieces, m and ro', called pallets, 
 project inwards at right angles to it from its extremities A and c. 
 
 The form and dimensions of the anchor ABC are accommodated 
 to those of the escapement wheel, w w', which is part of the 
 clockwork, and which, in common with the other wheels forming 
 the train, is moved in the direction indicated by the arrow by the 
 weight or mainspring. When the anchor swings to the right the 
 pallet m enters between two teeth of the wheel, the lower of which 
 coming against it, the motion of the wheel is for the moment 
 arrested. When it swings to the left, the pallet m is withdrawn 
 from between the teeth, and the wheel is allowed to move, but 
 only for a moment, for the other pallet m' enters between two teeth 
 at the other side, the upper of which coming against it the motion 
 of the wheel is again arrested. 
 
 The wheel, therefore, is thus made to revolve on its axis E, not with 
 a continuous motion, as would be the case if it were impelled by the 
 
350 
 
 PRACTICAL ILLUSTRATIONS 
 
 weight or mainspring, without the interference of any obstacle, but 
 with an intermitting motion. It moves by starts, being stopped 
 alternately by one pallet or the other coming in the way of its teeth. 
 When the pendulum, and therefore the anchor, is at the extreme 
 right of its play, the pallet w, having entered between two teeth, 
 a tooth rests against its lower side, the wheel is arrested, and the 
 
 pallet m', is quite dis- 
 engaged from, and 
 clear of, the teeth of 
 the wheel. When in 
 swinging to the left 
 the arm D B becomes 
 vertical, the tooth of 
 the wheel on the left 
 has just escaped from 
 the pallet ?>*, and the 
 wheel, being liberated, 
 has just commenced to 
 be moved by the force 
 of the weight or main- 
 spring. But at the 
 same moment the pal- 
 let m', enters between 
 the teeth of the wheel 
 on the right, and when 
 the anchor has arrived 
 at the extreme left of 
 its play, the tooth of 
 the wheel which is 
 above the pallet m\ 
 will have fallen upon 
 ' it, so that the motion 
 will again be arrested. 
 Thus it appears that 
 during the first half 
 of the swing from right to left, the motion of the wheel is arrested 
 by the pallet m, and during the remaining half of the swing the 
 wheel moves, but is arrested the moment the swing is completed. 
 
 In like manner it may be shown that during the first half of 
 the swing from left to right, the motion of the wheel is arrested by 
 the pallet m', that it is liberated and moves during the latter half- 
 swing, and is again arrested when the swing is completed. 
 
 678. Motion of band intermitting-. The motion which is 
 imparted to the hands upon the dial necessarily corresponds with 
 this intermitting motion of the escapement wheel. If the clock be 
 
 JZ2> 
 
CLOCK AND WATCH WORK. 351 
 
 provided with a seconds-hand, the circumference of the dial being 
 divided into sixty equal parts by dots, the point of the seconds- 
 hand moves from dot to dot during the second half of each swing of 
 the pendulum, having rested upon the dot during the first half swing. 
 The whole train of wheelwork being affected with the same in- 
 termitting motion, the minute and hour hands must move, like the 
 seconds-hand, by intervals, being alternately moved and stopped 
 for half a second. This intermission, however, is not so observable 
 in them as in the seconds-hand, owing to their comparatively slow 
 motion. Thus, the minute-hand, moving sixty times slower than 
 the seconds-hand, moves during each half swing of the pendulum 
 through only the sixtieth part of the space between the dots, and 
 the hour-hand, moving twelve times slower than the minute-hand, 
 moves in each half swing of the pendulum through the 36oth part 
 of the space between the dots. It is easy, therefore, to comprehend 
 how changes of position so minute are not perceptible. 
 
 679. How the motion of the pendulum is sustained. If 
 the pendulum vibrated upon its axis of suspension unconnected 
 with the clockwork, the range of its oscillation would be gradually 
 diminished by the combined effects of the friction upon its axis 
 and the resistance of the air, and this range thus becoming less and 
 less, the oscillation would at length cease altogether, and the pen- 
 dulum would come to rest. Now this not being the case when the 
 pendulum is in connection with the wheelwork, but on the contrary, 
 its oscillations having always the same range, it is evident that it 
 must receive from the escapement wheel some force of lateral 
 impulsion, by which the loss of force caused by friction and the 
 resistance of the air is repaired. 
 
 It is easy to show how the effect is produced. It has been 
 shown that during the first half of each swing, a tooth of the 
 escapement wheel rests upon one or other pallet of the anchor. 
 The pallet reacts upon it with a certain force, arresting the motion 
 of the wheelwork, and receives from it a corresponding pressure. 
 This pressure has a tendency to accelerate the motion of the pen- 
 dulum, and this continues until the tooth slips off, and is liberated 
 from the pallet. It is this force which repairs the loss of motion 
 sustained by the pendulum by friction and atmospheric resistance. 
 
 Thus we see that while on the one hand the pendulum regulates 
 and equalises the motion imparted to the wheelwork by the weight 
 or mainspring, its own range is equalised by the reaction of the 
 weight or mainspring upon it. 
 
 680. Its action upon the escapement. If the action of the 
 anchor of the pendulum upon the escapement wheel be attentively 
 considered, it will be perceived that one tooth only of the escape- 
 ment passes the anchor for each double vibration made by the pen- 
 
352 PRACTICAL ILLUSTRATIONS. 
 
 dul urn. Thus, if we suppose that when the pendulum is at the 
 extreme left of its range, the right-hand pallet is between the 
 teeth m and n\ the tooth n' will escape from the pallet c when the 
 pendulum, swinging from left to right, comes to the vertical posi- 
 tion, which is the middle of its swing. While it rises to the ex- 
 treme right of its range, the tooth n' advances to the place which 
 mf previously occupied, and at the same time the tooth m advances 
 to the place which n previously occupied ; but, at the same time, 
 the pallet A, carried to the right, enters between m and the suc- 
 ceeding tooth, and arrests the further progress of the wheel. 
 When the pendulum then swings to the left, the wheel continues 
 to be arrested until it arrives at the middle of its swing, when the 
 tooth below m escapes from the pallet A, but at the same moment 
 the pallet c enters below the tooth which is above n', and, receiving 
 it at the end of the swing, stops the motion of the wheel. Thus it 
 appears that tooth after tooth, in regular succession, falls upon 
 the pallet c upon the arrival of the pendulum at the extreme left 
 cf its play after each double oscillation. 
 
 If the pendulum be so constructed that it shall vibrate in a 
 second, and that it be desired that the escapement wheel shall 
 make a complete revolution in a minute, that is, during sixty 
 vibrations of the pendulum, the wheel must have thirty teeth. 
 In that case, one tooth passing the anchor during each double 
 oscillation from right to left, and back from left to right, thirty 
 teeth, that is the whole circumference of the wheel, will pass the 
 anchor in thirty double oscillations, or in sixty single swings of the 
 pendulum, the time of each swing being one second. 
 
 68 1. IVIotion of bands, how proportioned. The manner in 
 which different rates of revolution can be imparted to the different 
 hands of a clock or watch, by tooth and pinion work, is easily ren- 
 dered intelligible. 
 
 The wheels commonly used in watch and clock work are formed 
 from thin sheets of metal, usually brass, which are cut into circular 
 plates of suitable magnitude, upon the edges of which the teeth are 
 formed. The edges of the wheels thus serrated are brought together, 
 the teeth of each being inserted between those of the other, so that 
 if one be made to revolve upon its axle, its teeth, pressing upon 
 those of the other, will impart a motion of revolution to the other. 
 
 When a large wheel works in the teeth of a much smaller one, 
 which is a very frequent case in all species of wheelwork, the 
 smaller wheel is called for distinction a pinion, and its teeth are 
 called leaves. 
 
 682. Method of cutting wheels and pinions. The method 
 of manufacturing the pinions and smaller wheels used in watch 
 and clock work is very ingenious. A rod of wire, the diameter of 
 
 
CLOCK AND WATCH WORK. 353 
 
 which a little exceeds that of the wheel or pinion to be made, is 
 drawn through an aperture cut in a steel plate, having the exact 
 form and magnitude of the wheel or pinion to be formed. After 
 being forced through this aperture by the ordinary process of wire- 
 drawing, it is converted into a. fluted wire, the ridges of the fluting 
 corresponding exactly in form and magnitude to the edge of the 
 aperture, and therefore to the teeth or leaves of the pinion or 
 wheel. 
 
 This fluted wire, called pinion wire, is then cut by a cutter, 
 adapted to the purpose, into thin slices, at right angles to its length. 
 Each slice is a perfect wheel, or pinion ; and it is evident that all 
 of them must be absolutely identical in form and magnitude. 
 
 Such a wire-drawing plate, with apertures of different forms and 
 sizes, is represented inj%-. 323. 
 
 * * * 
 
 Fig. 323. 
 
 683. Moving- power. Weight. It has been already stated 
 that the moving power applied to clock or watch work is either a 
 weight or a mainspring. 
 
 If a weight be the moving power, it is suspended to a cord which 
 is coiled upon a drum fixed upon an horizontal axis, the first 
 wheel of the train which gives motion to the hands being fixed on 
 the same axis, so that it shall turn when the drum turns. 
 
 Such an arrangement is represented in jig. 324., where A B is 
 the drum, c D the wheel attached to and moved by it, w the weight 
 which is the moving power suspended to the cord E, which 
 is coiled upon the drum A B. The end, r, of the axis of the drum 
 projecting beyond it, is made square, so as to receive a key 
 made to fit it, by which it is turned, so as to coil the cord 
 upon the axis, when it has been uncoiled by the descending motion 
 of the weight. 
 
 The direction in which the wheel c D is turned by the force of 
 the descending weight is indicated by the arrow, and in that di- 
 rection it will continue to turn so long as the weight acts upon the 
 coil of the cord upon the drum. But as soon as the cord, by the 
 continued descent of the weight, shall have been discharged from 
 the drum, the rotation imparted to c D must cease. It is then 
 that the key must be applied to the square end, r, of the axis of 
 the drum, and turned continually in the direction contrary to that 
 in which the weight would turn the drum in descending . 
 
 A A 
 
354 
 
 PRACTICAL ILLUSTRATIONS. 
 
 It will be perceived 
 that, in this case, the 
 hands of the clock would 
 be always turned back- 
 wards while the clock is 
 being wound up, unless 
 some special provision 
 
 ^ were made against such 
 
 l||B|Sjjjj^^ik an effect ; for it is evident 
 * =3 HSr that if the wheel c D, when 
 
 turned by the descent 
 of the weight w, in the 
 direction of the arrow, 
 give a progressive mo- 
 tion to the hands, the 
 motion imparted to c D, 
 by the ascent of the 
 weight w, while the clock 
 is being wound up, must 
 necessarily impart to the 
 hands a motion in the 
 contrary direction, that 
 is, a backward motion. 
 
 In all clocks this is 
 prevented by an expe- 
 dient called a ratchet 
 wheel and catch, the one 
 being attached to the 
 barrel A B, and the other 
 to the face of the wheel 
 c D, the effect of which is to allow the barrel A B to be turned 
 while the clock is being wound up in the direction contrary to that 
 indicated by the arrow without turning the wheel c D ; but when 
 the barrel A B is turned "by the descent of the weight w, in the 
 direction of the arrow, the catch acting in the teeth of the 
 ratchet wheel, the motion of A B is imparted by the action of 
 the catch on the ratchet wheel to the wheel c D, and by it to the 
 hands. 
 
 The form and mode of application of the ratchet wheel will be 
 presently more clearly explained. 
 
 684. mainspring:. The moving power of a weight can only 
 be applied to time-pieces where the space necessary for the play of 
 the weight in its descent and ascent can be conveniently obtained. 
 This condition is obviously incompatible with the circumstances 
 attending pocket-watches, all portable and movable time-pieces, 
 

 CLOCK AND WATCH WORK. 
 
 355 
 
 chimney, table, and console clocks, and in general all time-pieces 
 constructed on a small scale. 
 
 The moving power applied to these universally is a mainspring, 
 which is a ribbon of highly tempered steel bent into a spiral form, 
 as represented in fig. 325. At one end, A, an eye is provided, by 
 
 which that extremity may be attached either to a fixed point or to 
 the side of the barrel to which the spring is intended to impart 
 motion. In the centre of the spiral an arbor, or axle, is intro- 
 duced, to which the inner extremity of tlie spring is attached. 
 Supposing the extremity A to be attached to a fixed point, let 
 the arbor B {fig. 326.), be turned in the direction indicated by 
 
 Fig. 3*6 
 
 the arrow. The spring will then be coiled closer and closer round 
 the arbor A B, while its exterior coils will be separated one from 
 another by wider and wider spaces. 
 
 After the spring has been thus coiled up by turning the arbor, 
 it, will have a tendency to uncoil itself and recover its former 
 state, and if the arbor B be abandoned to its action and be free to 
 revolve, it will receive from the reaction of the spring a motion 
 
 A A 2 
 
35& PRACTICAL ILLUSTRATIONS. 
 
 of revolution contrary in direction to that which was given to the 
 arbor in coiling up the spring, and such motion would be imparted 
 to a wheel fixed upon the axle, and might from it be transmitted 
 to the hands in the same manner as if the arbor wheel received its 
 motion from the power of a weight. 
 
 But between such a moving force and that of a weight there is 
 an obvious difference. The tension of the cord by which the 
 weight is suspended, and consequently its effect in giving revolution 
 to the barrel upon which the cord is coiled, is always the same 
 until the clock altogether goes down. The moving force of the 
 spring, on the contrary, is subject to a continual decrease of in- 
 tensity. At first, when completely coiled up, its intensity is 
 greatest, but as it turns the arbor B it becomes gradually relaxed, 
 and its intensity is continually less and less. It exerts, therefore, 
 a continually decreasing power upon the wheel fixed upon the 
 arbor, and therefore upon the hands to which the motion is trans- 
 mitted. 
 
 685. Fusee. As a varying power would be incompatible with 
 that uniformity and regularity which are the most essential and 
 characteristic conditions of all forms of clockwork, such a spring 
 would be quite unsuitable if some expedient were not found by 
 which its variation could be equalised. 
 
 This has been accordingly accomplished, by a very beautiful 
 mechanical contrivance, consisting of the combination of a flexible 
 chain and a conical barrel arranged to receive its coils, called a 
 fusee. 
 
 This arrangement is represented in^. 327. The mainspring is 
 
 Fig. ja 7 . 
 
 attached by its inner extremity to the fixed arbor A, and by its 
 outside end at E, to a barrel B, which is capable of being turned 
 round the fixed axis A. A jointed chain is attached by one ex- 
 tremity to the barrel at E, and, being coiled several times round it, 
 
CLOCK AND WATCH WORK. 357 
 
 is extended in the direction c, to the lowest groove of the fusee E, 
 to which its other end, e, is attached. This fusee is a conical- 
 shaped barrel, upon which a spiral groove is formed, continued 
 from the base to the summit, to receive the chain. The base is a 
 toothed wheel, by which the motion imparted by the mainspring 
 and chain to the fusee is transmitted to the hands through the 
 wheelwork. The fusee is fixed upon an arbor, D G, the lower end 
 of which, projecting outside the case containing the works, is 
 formed square, to receive a key made to fit it, by which the clock 
 or watch is wound up. 
 
 The action of the spring transmitted by the barrel to the chain, 
 and by the chain to the fusee, has a tendency to impart to the fusee 
 a motion of rotation in the direction of the arrows. The fusee is 
 connected with the wheel, w, by means of a ratchet wheel and 
 catch similar to that described in the case in which the moving 
 power is a weight, by means of which the fusee F imparts rotation 
 to the wheel when it turns in the direction of the arrows, but does 
 not move it when turned in the opposite direction. 
 
 These arrangements being understood, let us suppose the key 
 applied upon the square end G of the arbor D G of the fusee, and 
 let it be turned round in the direction contrary to that indicated 
 by the arrows. The fusee will then be turned, but will not carry 
 the wheel w with it ; the chain c will give to the barrel B a motion 
 of revolution contrary to the direction of the arrows, the chain 
 will be gradually uncoiled from the barrel B, and will be coiled 
 upon the spiral groove of the fusee, winding itself from groove to 
 groove, ascending on the spiral until the entire length of the chain 
 has been uncoiled from the barrel B, and coiled upon the fusee r, 
 as represented in jig. 328. 
 
 During this process, the external extremity of the mainspring, 
 attached to the barrel at E, is carried round with the barrel, while 
 
 AA 3 
 
35& PRACTICAL ILLUSTRATIONS. 
 
 the internal extremity is fixed to the arbor A, which does not 
 turn with the barrel. By this means the spring is more and more 
 closely coiled round the arbor A, until the entire chain has been 
 discharged from the barrel to the fusee, when the spring will be 
 coiled into 4 the form represented in jig. 328., and in this state 
 the intensity of its force of recoil, and the consequent tension 
 of the chain c, extended from the barrel B to the fusee F, is 
 greatest. 
 
 The clock being thus wound up and left to the action of the 
 spring, the tension of the chain c, directed from the fusee to the 
 spiral, will make the fusee revolve in the direction indicated by 
 the arrows. This tension at the commencement acts upon the 
 highest and smallest groove of the fusee. As the chain is gradually 
 discharged from, the fusee to the barrel, the tension is gradually 
 decreased by reason of the relaxation of the spring, and at the 
 same time the chain acts upon a larger and larger groove of the 
 fusee. In this way the tension of the chain is continually de- 
 creased, and the radius of the groove on which it acts is continu- 
 ally increased, until the entire action has passed from the fusee to 
 the barrel, and the clock goes down. 
 
 Now the power of the chain to impart a motion of revolution to 
 the fusee depends on two conditions ; first, the force of its tension, 
 and secondly, the leverage by which this tension acts upon the 
 fusee. This leverage is the semi-diameter of the groove, upon 
 which the chain is coiled at the point where it passes from the 
 fusee to the barrel. It will be easy to perceive that, it re- 
 quires less force to turn a wheel or barrel if the force be applied 
 at a great distance from the axle than if it be applied at a small 
 distance from it. Upon this principle generalised, it follows that 
 the power of the tension of the chain to impart revolution to the 
 fusee is augmented in exactly the same proportion as the magni- 
 tude of the groove on which it acts is increased. 
 
 The form given to the fusee is such that as the chain is gradually 
 discharged from it, the diameter of the groove on which it acts 
 increases in exactly the same proportion as that in which the ten- 
 sion of the chain decreases. It follows, therefore, that the power 
 of the chain upon the fusee gains exactly as much by the increase 
 of its leverage as it loses by the decrease of its tension, and con- 
 sequently it remains invariable. 
 
 Complete compensation is therefore obtained by this beautiful 
 and simple expedient, and a variable force is thus made to produce 
 an invariable effect. It may be useful to state that this is only a 
 particular application of a mechanical principle of great generality. 
 In all cases whatever, the varying energy of a moving power may 
 be equalised by interposing between it and the object to be moved 
 
 
CLOCK AND WATCH WORK. 359 
 
 some mechanism by which the leverage, whether simple or com- 
 plex, through which its force is transmitted, shall vary in the exact 
 inverse proportion of the variation of the power, increasing as 
 the intensity of the power is decreased, and decreasing as the 
 intensity of the power is increased. 
 
 686. Balance wheel. Whatever be the moving power, 
 whether it be a weight or mainspring, it would, if not controlled 
 and regulated, impart to the hands a motion more or less accele- 
 rated, and therefore unsuitable to the measurement of time, which 
 requires a motion rigorously uniform. It is on that account that 
 the moving power must be controlled and governed by some ex- 
 pedient by which it shall be rendered uniform. 
 
 How the combination of a pendulum and escapement wheel 
 accomplishes this has been already explained. But this expedient 
 requires that the time-piece to which it is applied shall be stationary ; 
 the slightest disturbance of its position would derange the mutual 
 action of the pendulum and the escapement wheel, and would 
 either stop the movement or permanently derange the mechanism. 
 It is evident that a pendulum is not only inapplicable to all forms 
 of pocket time-piece, but that it cannot even be used for marine 
 purposes, the disturbances incidental to which would be quite 
 incompatible with the regularity of its action. 
 
 The expedient which has been substituted for it with complete 
 success in all such cases is the balance wheel. 
 
 This is a wheel, like a small .fly wheel, having a heavy rim con- 
 nected with the centre by three or more light arms, as shown at 
 ABC, in Jig. 329. Under, and parallel to it, is placed a spring 
 resembling in form the mainspring, but much finer and lighter, 
 
 and having much less force. 
 This spring is formed of ex- 
 tremely fine and highly tem- 
 pered steel wire, so fine that 
 it is sometimes called a hair- 
 spring. One extremity of this 
 spring is attached to the axis 
 
 Fig " J2g of the balance wheel, and the 
 
 other to any convenient fixed 
 point in the watch. The spring is so constructed that when at 
 rest it has a certain spiral form, to which it has a tendency to 
 return when drawn from it on the one side or the other. If we 
 suppose it, therefore, to be drawn aside from this position of rest 
 and disengaged, it will return to it, but on arriving at it, having 
 acquired by the elasticity a certain velocity, it will swing past it 
 to the other side, to a distance nearly as far from its position of 
 rest as that to which it had been originally drawn on the other side. 
 A A 4 
 
360 PRACTICAL ILLUSTRATIONS. 
 
 It will then swing back, and will thus oscillate on the one side and 
 the other of the position of rest, in the same manner exactly as that 
 in which a pendulum swings on the one side or the other of the 
 vertical line which is its position of rest. 
 
 687. Isochronism. The balance wheel thus connected with a 
 spiral spring, like the pendulum, is isochronous ; that is, it performs 
 all vibrations long and short in the same time. It will be 
 recollected that this property of the pendulum depends on the fact 
 that the wider is the range of its vibrations the more intense is the 
 force with which it descends to the vertical direction, and con- 
 sequently wide vibrations are performed in as short a time as more 
 contracted ones Now the vibrations of the balance wheel are 
 subject to like conditions. The wider the range of its vibrations, 
 the more intense is the force with which the recoil of this spring 
 carries it back to its position of rest, and consequently it swings 
 through these wide vibrations in the same time as through more 
 contracted ones, in which the force of the spring is proportionally 
 less intense. 
 
 688. Compensation balances. The balance wheel, like the 
 pendulum, being subject to expansion and contraction by variation 
 
 of temperature, will suffer a correspond- 
 ing change in its diameter, and a con- 
 sequent change in the time of its oscil- 
 lation, and the rate of the time -piece 
 which it regulates. 
 
 This irregularity has been compen- 
 sated by attaching to the rim of the 
 wheel a compound metallic arc, such as 
 that already described". When the tem- 
 perature rises, and the diameter of the 
 wheel is augmented, this arc, (B c, 
 Fj fig. 330.) with its concavity towards the 
 
 centre of the wheel, becomes more con- 
 cave, and a weight D which it carries is brought nearer to the 
 centre of the wheel, and this compensates for the increased mag- 
 nitude of the diameter A A. If, on the other hand, the tempe- 
 rature is lowered, and the diameter A A of the wheel diminished 
 by contraction, this arc B c becomes less concave, and the 
 weight which it carries is removed to a greater distance from 
 the centre, and this compensates for the diminished diameter of 
 the wheel. 
 
 689. Movement imparted to the hands of a watch. The 
 oscillation of the balance wheel regulates the motion of watchwork 
 in the same manner by means of an escapement wheel, as that in 
 which the pendulum regulates the motion of clockwork. The 
 

 CLOCK AND WATCH WORK. 361 
 
 pallets and the escapement wheel are, however, very variously 
 formed in different watches. 
 
 Having thus explained generally the powers by which clocks 
 and watches are moved and regulated, it now remains to show 
 how the necessary motions are conveyed to the hands by suitable 
 combinations of wheels and pinions. 
 
 In fig. 331. are represented the works of a common watch, moved by a 
 mainspring A, and regulated by a balance wheel H ; the wheels and pinions, 
 
 Fig. 331. 
 
 however, being changed in their relative positions, and the fusee being 
 omitted, so as to show more visibly the connections and mutual dependency 
 of the many parts. The external extremity of the mainspring is attached 
 to the base, o, of a column of the frame. Its internal extremity is attached 
 to the lower end of an axle, of which the square end, T, at the top enters a 
 hole in the dial-plate into which the key is inserted when the watch is to be 
 wound up. The ratchet wheel B is fixed upon this axis so as to turn with it, 
 but the other wheel c under the ratchet wheel is not fixed upon it, the axis 
 being free to turn in the hole in the centre of c, through which it passes. A 
 catch n o is attached by a pin on which it plays to the face of the wheel c, 
 and its point o is pressed against the teeth of the ratchet wheel B, by a 
 spring provided for that purpose. When the key is applied upon the end 
 T, and turned in the direction in which the hands move, the ratchet wheel is 
 turned with it, and the point o of the catch pressed constantly against the 
 teeth while it turns falls from tooth to tooth with an audible click, and 
 thus produces the peculiar sound, with which every ear is familiar, While the 
 watch is being wound up. During this process the wheel c does not turn 
 with the axle, which only passes through the hole in its centre without being 
 fixed upon it, but the mainspring A, being attached to the axle, is coiled more 
 and more closely round it, and reacts against the fixed point o with greater 
 and greater force. 
 
3 62 PRACTICAL ILLUSTRATIONS. 
 
 If the fusee, which is omitted in this figure, were introduced, it would 
 occupy the place of the spring, and would be turned by the axle imparting a 
 like revolution to the axis of the spring by means of the chain. 
 
 When the watch is wound up, the reaction of the spring, rendered uniform 
 in its force by the fusee, imparts a motion of revolution to the ratchet wheel 
 B, in the direction of the arrow. By this motion the tooth of the ratchet 
 wheel in which the point o of the catch is engaged, presses against the catch 
 so as to carry it round with it in the direction of the arrow ; but the catch 
 being attached to the face of the wheel c, at n, this wheel is carried round 
 also in the same direction, and with a common motion. 
 
 The teeth of the wheel c act in those of the pinion d, which is fixed upon 
 d D. Upon the same axle is fixed the wheel D, so that the wheel D and the 
 pinion d receive a common motion of revolution from the wheel c. 
 
 The wheel D, in precisely the same manner, imparts a common motion of 
 revolution to the pinion e, and the wheel K ; and the wheel E imparts a com- 
 mon motion of revolution to the pinion / and the wheel F. 
 
 This last wheel F is of the form called a crown wheel, and acts upon the 
 pinion g, imparting to it, and to the escapement wheel o, a common motion 
 of revolution. This escapement wheel is acted upon and controlled by the 
 pallets or other contrivances attached to the axis of the balance wheel H, so 
 as to regulate its motion by the oscillations of that wheel in the same man- 
 ner as the escapement wheel of a clock is regulated by the anchor of the 
 pendulum. 
 
 It may be asked why so long a series of wheels and pinions are interposed 
 between the mainspring and the balance wheel, and why the first pinion d 
 may not act directlv upon the escapement wheel. The object attained by 
 the multiplication of the wheels and pinions is to cause the mainspring, by 
 acting through a small space, to produce a considerable number of revolu- 
 tions of the escapement wheel, for without that the spring would be speedily 
 relaxed, and the watch would require more frequent winding up. Thus by 
 the arrangement here shown, while the mainspring causes the wheel c to re- 
 volve once, it causes the pinion d and the wheel D to revolve as many times 
 as the number of teeth in c is greater than the number in d. Thus, if there 
 are ten times as many teeth in c as in d, one revolution of c will produce ten 
 of d and D. In like manner if D have ten times as many teeth as e, one 
 revolution of D will produce ten of e and E, and so on. In this way it is 
 evident that one revolution of the first wheel c, which is on the. axis of the 
 fusee, can be made, by the mutual adaptation of the intermediate wheels and 
 pinions, to impart as many revolutions as may be desired to the escapement 
 wheel G. 
 
 The wheels which govern the motion of the hands are those which appear 
 in the figure between the watch face and the frame x Y. The relative power 
 of the mainspring and balance wheel must be so regulated that the wheel n 
 shall make one revolution in an hour. The axis upon which this wheel is 
 fixed, passing through the centre of the dial, carries the minute-hand, which 
 therefore revolves with it, making one complete revolution on the dial in an 
 hour. 
 
 Upon this axle of the minute-hand is fixed a pinion ft, which drives the 
 
 wheel /, on the axle of which is fixed the pinion m, which drives a wheel 77, 
 
 through the centre of which the axle of the minute-hand passes without being 
 
 fixed upon it. Upon the axle of the minute-hand a small tube is placed, 
 
 . within which it can turn. Upon this tube the hour-hand, as well as the 
 
CLOCK AND WATCH WORK. 363 
 
 wheel/?, is fixed. The pinion k, therefore, fixed upon the axis of the minute- 
 hand, imparts motion to the hour-hand by the intervention of the wheel /, 
 the pinion m, and the wheel p. Since the hour-hand must make one revolu- 
 tion while the minute-hand makes twelve, it is necessary that the relative 
 numbers of the teeth of these intermediate wheels shall be such as to produce 
 that relation between the motions of the hands. An unlimited variety of 
 combinations would accomplish this, one of the most usual being the fol- 
 lowing : 
 
 Pinion k ..... 8 teeth. 
 
 Wheel/ ...... 24 
 
 Pinion m ..... 8 
 
 By this arrangement p will make eight revolutions, while m and I make 
 thirty-two ; or, what is the same, p will make one revolution, while m and / 
 make four. In like manner, / will make eight revolutions, while k, and there- 
 fore the minute-hand, makes twenty-four ; or, what is the same, / will make 
 four revolutions, while k and the minute-hand make twelve. It follows, 
 therefore, that p, and therefore the hour-hand, makes one revolution, while 
 k, and therefore the minute-hand, makes twelve, which is the necessary pro- 
 portion. 
 
 In this case there is no seconds-hand : but, if there were, its motion would 
 be regulated in like manner by additional wheels and pinions. 
 
 690. Movement of the hands of a clock. The manner in 
 which the moving power of a weight, and the regulating power of 
 a pendulum are applied in a clock to produce the motion of the 
 hands, does not differ in any important respect from the arrange- 
 ment explained above. Nevertheless, it may be satisfactory to 
 show the details of the mechanism. The train of wheels connect- 
 ing the weight with the anchor of the pendulum is shown in 
 
 fig-W- 
 
 A side view of the mechanism, showing the wheels which more 
 immediately govern the motion of the hands, and also the pen- 
 dulum, with its appendages, is given in Jig. 333. 
 
 The weight w acts by a cord on a barrel, as already explained. This bar- 
 rel and the ratchet wheel, with its catch, are mounted upon the axis of the 
 great wheel c, and are behind it, as represented in fig. 332., their form and 
 position being shown by the white lines. The catch is attached to the wheel 
 c by the screw n, and its point o acts on the teeth of the ratchet wheel, 
 which is attached to the barrel on which the rope is coiled. The spring 
 which presses the catch against the teeth of the ratchet is also shown. When 
 the clock is wound up by the key applied to the square end T (fig. 333-) of 
 the axis of the barrel, the barrel is turned in the direction opposite to that 
 indicated by the arrows, and the catch falls from tooth to tooth of the 
 ratchet wheel, making the clicking noise which attends the process of wind- 
 ing up. When the clock has been wound up, the weight acting on the 
 barrel presses the tooth of the ratchet wheel against the catch, and there- 
 by carries round with it the wheel c. This wheel transmits the motion 
 to the escapement wheel G,fig. 332., through the series of wheels and pinions, 
 d, D, e, E,/, F, and g, in the same manner exactly as has been already de- 
 
3 6 4 
 
 PRACTICAL ILLUSTRATIONS. 
 
 scribed ; and the pendulum, by means of the anchor N, regulates the 
 motion. 
 
 The wheels which more immediately govern and regulate the motion of 
 the hands are those which appear in fig. 333. in front of the plate x y. 
 
 The pendulum consists of a heavy disc of metal, seen edgeways at v in fig. 
 333., attached to the end of a metal rod, R R, represented broken, to bring it 
 
 W 
 
 Fig. 
 
 within the limits of the figure. This rod is suspended by various means, 
 but often, as in the figure by two elastic ribbons of steel, s s, which permit 
 its swing right and left. It passes between the prongs of a fork u, by which 
 a rod r r is terminated, so that this rod swings right and left with the pendu- 
 lum. Upon the axis of this rod, and over the escapement wheel G, is fixed 
 the anchor N of the escapement. 
 
CLOCK AND WATCH WORK. 365 
 
 **g. 333 
 
306 PKACTICAL ILLUSTRATIONS. 
 
 691. To regulate the rate. Whether the movement be 
 regulated by a pendulum or a balance wheel, it is necessary to 
 provide some means of adjustment by which the rate ot vibration 
 may be increased or diminished at pleasure within certain limits, 
 for although in its original construction the regulator may be 
 made so as to oscillate nearly at the required rate, it cannot be 
 made to do so exactly. Besides, even though it should vibrate 
 exactly at the required rate, it will be subject, from time to time, 
 to lose that degree of precision, and to vibrate too fast or too 
 slowly, from the operation of various disturbing causes. 
 
 It has been already shown that the rate of vibration of the 
 pendulum is rendered more or less rapid by transferring the centre 
 of gravity nearer to, or farther from, the point of suspension. 
 Upon this principle, therefore, the adjustment of the rate of vibra- 
 tion depends. The heavy disc v, Jig. 333., is made to slide upon 
 the rod B E, and can be moved upon it, upwards or downwards, by 
 a fine screw attached to it, which works in a thread cut in the rod. 
 In this manner the centre of gravity of the disc v may be trans- 
 ferred nearer to the suspension s s, so as to shorten the time of its 
 vibrations, or removed farther from s s, so as to lengthen the time. 
 If the clock is found to lose or go too slowly, it is screwed up, 
 and if it gain or go too fast, it is screwed down. 
 
 In chimney and table time-pieces, the pendulum is regulated in 
 a different manner. It is usually suspended upon a loop of silken 
 thread, which can be drawn up or let down through a certain 
 limited space, by means of a rod, upon which one end of the thread 
 which forms the loop is coiled. This rod, passing through the 
 dial-plate, has a square end, upon which a key can be applied, by 
 turning which in the one direction or the other, the loop is drawn 
 up or let down. 
 
 The rate of oscillation of the balance wheel cannot in the same 
 manner be so easily regulated by modifying its form ; but, on the 
 other hand, while the force which moves the pendulum, being that 
 of gravity, is beyond our control, that which moves the balance 
 wheel, being the force of the spiral spring, is at our absolute dis- 
 position. It is accordingly by modifying this spring that we are 
 enabled to regulate the time of oscillation of this regulator. 
 
 One of the most common expedients by which this is accom- 
 plished is represented in^/?g-. 334. 
 
 Near the fixed point G, at the external extremity of the spiral, 
 is placed a small bar E, near the end r, of which is a notch, or 
 hole, through which the wire of the spiral passes. This arrests 
 the action of the spiral, so that the only part of it which oscil- 
 lates is that which is included between r and its internal ex- 
 tremity. In a word the point r is the virtual external extremity 
 
CLOCK AND WATCH WORK 367 
 
 of the spiral. Now this 
 point F can be moved in 
 the one direction or the 
 other, so as to increase 
 or diminish the virtual 
 length of the spiral at 
 pleasure, by means of the 
 toothed arc AB and the 
 pinion c, which latter is 
 turned by the index D. 
 If the index D be turned 
 to the left, the bar E, 
 and the point F, is moved 
 Fig JJ4 towards G, and the length 
 
 of the spiral is increased. 
 
 If it be turned to the right, the point F is moved from G, and the 
 length of the spiral is diminished. 
 
 In this manner the rate of vibration of the balance wheel may 
 be adjusted by varying at will the vertical length of the spiral 
 spring. 
 
 692. Recoil escapement. The precision of the movement of 
 all forms of time- pieces depends in a great degree on the mechanism 
 of the escapement, and accordingly much mechanical skill and 
 ingenuity have been directed to its improvement, and several 
 varieties of form have been adopted and applied. 
 
 The recoil escapement, represented in Jig. 331., consists of two 
 pallets, which project from the axis of the balance wheel at right 
 angles with each other, one of which acts at the top, and the other 
 at the bottom of the escapement wheel G, the axis of which is 
 horizontal and the wheel vertical. These pallets, as the balance 
 wheel oscillates, engage alternately in the teeth of the escapement 
 wheel, exactly in the same manner as do the pallets of the anchor 
 of the escapement attached to the pendulum already described. 
 This form of escapement was long the only one used, and is still 
 continued in the more ordinary sorts of watches. 
 
 It has, however, been superseded, in watches and chronometers 
 where greater precision is required, by others of more improved 
 construction. 
 
 Cylindrical escapement. In pocket watches, where great 
 flatness is required, the cylindrical escapement is used, in which 
 the axis of the balance wheel, instead of having pallets attached 
 to it, is formed into a semicylinder, having a sort of notch in it, 
 ao represented on an enlarged scale in jig. 335. The semi- 
 cylinder a & is cut away at c, through about half its height. The 
 axis AB,y?-. 336., of the escapement wheel is vertical, and the 
 
368 
 
 PRACTICAL ILLUSTRATIONS 
 
 Fig- 335 
 
 teeth, raised at right angles to its plane, and therefore parallel to 
 its axis, have the peculiar form represented in the figure. As the 
 balance wheel oscillates on the one side and the other, the semi- 
 cylinder upon its axis interposes itself alternately between the 
 teeth of the escapement wheel, stopping them and letting them 
 escape in the usual way. The manner in which the action takes 
 
 Fig. 337- 
 
 place will be more clearly understood by ^5.337. and 338., in 
 which a view in plan upon an enlarged scale is given of the 
 

 CLOCK AND WATCH WORK. 369 
 
 position of the semicylinder and the teeth of the escapement wheel 
 after each successive oscillation. 
 
 In^g-. 337., the balance wheel swinging from right to left, throws 
 the convex side A D of the semicylinder before the tooth c of the 
 escapement wheel, and thus for the moment arrests it, while the 
 side A E of the semicylinder has turned out of the way of the 
 preceding tooth, and has let it pass. The balance wheel then 
 swings from left to right, and the convex side A of the semi- 
 cylinder slides against the point of the tooth c. When the edge D 
 of the semicylinder passes the point of the tooth, the latter, in 
 slipping over it, gives to it a slight impulse, which' restores to the 
 balance wheel the small quantity of force which it lost by the pre- 
 vious reaction of the tooth upon its convex surface. 
 
 The side A E of the semicylinder is now thrown before the tooth 
 c. the point of which having advanced through a space equal to 
 
 Fig. 338. 
 
 the diameter of the semicylinder, is thrown against the concave 
 surface of AE, as shown \njig. 338. 
 
 The balance wheel now swinging again from right to left, the 
 point of the tooth c slides upon the concave surface of the semi- 
 cylinder A E, until the edge E comes to it. The tooth then slips 
 over the inclined face of E, and in doing so gives the semicylinder, 
 and consequently the balance wheel, another slight impulse, re- 
 storing to it the force of which it deprived it while previously 
 sliding upon its concave side. 
 
 The explanation here given of the action of this form of escape- 
 ment is well calculated to render the conditions which all escape- 
 ments should fulfil intelligible. These arrangements are primarily 
 directed to the regulation of the movement of the wheelwork, so 
 as to secure its uniformity. This will obviously be accomplished 
 provided that the escapement, whatever be its form, lets a tooth 
 
370 PRACTICAL ILLUSTRATIONS. 
 
 of the wheel pass for each oscillation of the balance wheel. But 
 owing to the friction of the axis of the balance wheel, and of the 
 pallets on the teeth of the escapement wheel, and the resistance of 
 the air, the range of its oscillations would be gradually diminished, 
 so that at last it would not be sufficient to allow the successive 
 passage of the teeth of the escapement wheel, and the watch would 
 stop unless some adequate means be provided by which the 
 balance wheel shall receive from the mainspring through the 
 escapement wheel as much force as it thus loses. All escapements 
 accomplish this by the peculiar forms given to the edges of the 
 pallets and the teeth of the escapement wheel. In the present 
 case, the object is attained by making the edge D round and the 
 edge E inclined, and by giving to the teeth the form shown in the 
 figure. 
 
 This form of escapement supplies a sufficiently good regulating 
 power for the best sorts of pocket watches, and is attended with 
 the advantages of allowing the works to be compressed within a 
 very small thickness. It is the form most commonly used in the 
 French and Swiss watches. 
 
 693. Duplex escapement. The form of escapement used in 
 the best English-made watches consists of an escapement wheel, 
 which partakes at once of the double characters of a spur and 
 crown wheel, and is hence called the duplex escapement. 
 
 The spur teeth, ABC, &c. (fig. 339-)? are similar in their form 
 
 339- 
 
 and arrangement to those of the cylindrical escapement described 
 above. The crown teeth, a b c, &c., project from the face of the 
 wheel, and have a position intermediate between the spur teeth. 
 Upon the axle of the balance wheel just above the plane of the 
 escapement wheel is fixed a claw pallet, called the impulse pallet, 
 which, by the combination of the oscillations of the balance and 
 the progressive motion of the escapement wheel, fall successively 
 
CLOCK AND WATCH WORK. 371 
 
 between the crown teeth of the latter, receiving from their reaction, 
 as they escape from it, the restoring action which maintains the 
 range of the oscillations of the balance wheel. 
 
 Under the pallet and in the plane of the spur teeth is placed a 
 small roller usually formed of ruby or other hard stone, having a 
 notch on one side of it, into which the spur teeth successively fall. 
 After any crown tooth, a, for example, passes the pallet, the corre- 
 sponding spur tooth A falls into the notch of the roller, and this 
 alternate action continues so long as the watch goes. It will be 
 perceived therefore that the pallet and roller in the duplex 
 escapement play the same part as the two edges of the semicylinder 
 in the cylindrical escapement, and as the two pallets in the com- 
 mon recoil escapements. 
 
 The chief advantage claimed for this system is that there is but 
 
 one pallet, and that the action 
 does not require as perfect 
 execution of the teeth of the 
 escapement wheel as other ar- 
 rangements. 
 
 lever escapement. This 
 is much used for English pocket 
 watches. A lever A B (Jig 
 340.), with a notch at one end, 
 is attached to the anchor c. 
 A pin at a, on a disc D, on the 
 verge or arbor of the balance, 
 Fig ' 34 ' enters this notch at each vi- 
 
 bration, and first moves the dead part of the pallet off the tooth of 
 the scape wheel E, and then receives an impulse, which restores the 
 force it has lost, leaving another tooth engaged in the opposite 
 pallet. As the lever is detached from the balance, except for an 
 instant at the middle of each vibration, the amount of friction is 
 very small. Another advantage of this movement is, that it is but 
 little liable to derangement, and when it is injured, is easily and 
 cheaply repaired, while the duplex and cylindrical escapements are, 
 expensive to make, and can only be mended by such skilful work- 
 men as are not often found, except in the metropolis or large 
 towns. 
 
 694. Detached escapement. In the class of portable time- 
 pieces used for the purposes of navigation, where the greatest 
 attainable regularity of motion is required, an arrangement is 
 adopted called the detached escapement. This system is repre- 
 sented in Jig, 341. 
 
 Upon the arbor of the balance wheel is attached a disc, in which 
 there is a notch i A smaller disc G, is also attached to it, from 
 
372 PRACTICAL ILLUSTRATIONS. 
 
 which a small pin a projects. By the oscillations of the balance 
 wheel the notch i and the pin oscillate alternately right and left. 
 A fine flexible spring A, attached to a fixed block B, carries upon 
 it a projecting piece c. To the block D is attached another fine 
 
 Fig. 341. 
 
 spring E, which extends to the edge of the small disc G. The pro- 
 jecting piece c, is so placed that when the spring A is not raised, 
 it encounters a tooth of the scape wheel, but when slightly raised it 
 allows the tooth to pass. The spring E rests in a small fork behind 
 the extremity of A, presented downwards. 
 
 Now, let us suppose the balance wheel to swing from left to right. 
 The pin a, projecting from the small disc G, coming against the end 
 of the spring E, raises it ; and this spring, acting in the fork behind 
 it, raises the spring A, and therefore lifts the piece c, and liberates 
 the tooth which that piece previously obstructed. The scape wheel 
 therefore advances, but before the next tooth comes to the place 
 occupied by the former one, the balance swings back from right to 
 left. The spring E, no longer supported by the fork at the end of 
 A, readily lets the pin pass, "and the piece c, returning to its former 
 position, comes in the way of the succeeding tooth and stops it. 
 
 At the moment that the balance is about to commence its swing: 
 from left to right, and when the piece c is about to liberate the 
 tooth which rests against it, another tooth behind it rests against 
 the side of the notch i in the disc G, and when the escapement 
 wheel is liberated, and the swing of the balance from left to right 
 is commencing, this tooth, pressing on the side of the notch i, gives 
 it and the balance wheel an impulse which is sufficient to restore 
 to it all the force which it lost in the previous oscillation. Except 
 at this moment the balance wheel in this form of escapement is 
 entirely free from all action of the mainspring. 
 
CLOCK AND WATCH WORK. 
 
 373 
 
 695. Maintaining: power In clocks. While a clock or watch 
 is being wound up, the weight or main- 
 spring no longer presses upon the catch 
 nor upon the ratchet wheel, through which 
 the motion is imparted to the wheelwork. 
 The motion of the hands is therefore sus- 
 pended during the time occupied in wind- 
 ing up ; consequently, if the watch or clock 
 keep regular time while it goes, it must 
 lose just so much time as may be em- 
 ployed in the process of winding it up. Al- 
 though this, in common clocks and watches, 
 does not produce any sensible effect, the 
 errors incidental to their rates of going 
 generally exceeding it, yet in clocks used in 
 observatories and in chronometers used for 
 the purposes of navigation, where the 
 greatest degree of regularity is required, 
 provisions are made by which the motion 
 of the clock or watch is continued not- 
 withstanding the process of winding up. 
 
 Such expedients are called the -main- 
 taining power. 
 
 One of the most simple arrangements by 
 which this is accomplished in clocks moved 
 by a weight is shown in Jig. 342. 
 
 The weight p, which is the moving power, is 
 connected with another much smaller weight p, by 
 
 means of an endless cord, which passes over the grooves of a series of pulleys, 
 A, B, c, and D, of which A and B are movable, and c and n fixed. The 
 force with which p descends is the excess of its weight above that of p. 
 
 The pulley c, being prevented from revolving by the catch E during the 
 descent of the weight p, and the cord being prevented from sliding upon its 
 groove by the effects of its friction and adhesion, the parts b a and c d, 
 which descend from the pulley c to the pulleys A and B, may be regarded 
 as being virtually attached to fixed points at b and c, so that they cannot 
 descend. This being the case, the weight p, descending by its prepon- 
 derance over p, and consequently the weight/) being drawn up, the part of 
 the cord c d must pass over the pulley B, the part e f over the pulley D, and 
 the part g h over the pulley A. In this way the parts b a and g h will be 
 gradually lengthened as the weight p descends at the expense of the parts 
 c d and fe, which will be shortened to an equal extent, so that the weight 
 p will be raised through the same space as that through which the weight 
 v is lowered. During this process the wheel D is kept in constant revolu- 
 tion; and the first wheel of the train of clockwork being fixed on its axle, 
 a motion is imparted by it through that train to the hands. 
 
 When it is desired to wind up the clock, the hand is applied to the cord 
 c d, which is drawn downwards, so that the fixed pulley revolves, the catch E 
 
 BBS 
 
374 PRACTICAL ILLUSTRATIONS. 
 
 dropping from tooth to tooth until the weight r has been raised to the 
 highest, and the weight/) has fallen to the lowest point. The parts g h and 
 fe of the cord not being at all affected by this, the pulley D continues to turn 
 as before by the effect of the preponderance of p, which acts as powerfully 
 while it ascends as it did when it descended. 
 
 It appears, therefore, that by this arrangement the motion of the works 
 and of the hands is not suspended during the process of winding up. 
 
 696. Maintaining power in watches. If the works of a 
 watch be impelled by the force of a mainspring without a fusee, in 
 the manner represented in fig. 331., it is evident that the move- 
 ment must be suspended during the process of winding up; 
 because the ratchet wheel B, by which the force of the spring is 
 transmitted to the works, is then relieved from the action of the 
 catch n o. This defect may, however, in such case be removed by 
 a very simple expedient. Instead of connecting the external: 
 extremity of the mainspring with a fixed point, let it be attached 
 to the inside of a barrel surrounding it, and let the wheel c be 
 attached to this barrel. In that case, when the axle of the ratchet 
 wheel is turned in winding up, and the ratchet wheel, therefore, 
 relieved from the action of the catch, the wheel c will be acted 
 upon by the barrel, which will itself be impelled by the reaction of 
 the external extremity of the spring which is attached to it, the 
 winding up being effected only by the internal extremity. 
 
 This is the arrangement generally adopted in chimney and table 
 time-pieces, as constructed in France and Switzerland, and also in 
 flat watches, in all of which the adoption of the cylindrical escape- 
 ment {fig. 336.) enables the constructor to dispense with the 
 fusee. 
 
 It will, of course, be understood that in such arrangements, 
 while the wheel c is attached to the barrel, and by it to the 
 external extremity of the mainspring, the ratchet wheel B is 
 attached to the axle T B (fig. 331.)) an d by it to the internal 
 extremity of the mainspring. 
 
 When a fusee is used, the ratchet wheel being fixed upon its 
 axis, and not on that of the barrel containing the mainspring, this 
 method of obtaining a maintaining power is not applicable. In 
 such cases the object is attained by two ratchet wheels upon the 
 axle of the fusee having their teeth and catches turned in opposite 
 directions, one of them being impelled by a provisional spring, 
 which only comes into play when the action of the mainspring is 
 suspended during the process of winding up. 
 
 The fusee, with its appendages, as commonly constructed, with- 
 out a maintaining power, is drawn in^. 343., the grooved cone 
 with the ratchet wheel attached to its base being raised from the 
 cavity in the toothed wheel c D, in which it is deposited, to 
 
CLOCK AND WATCH WORK. 
 
 375 
 
 A 
 
 B 
 
 Fig. 34? 
 
 show the arrangement more 
 clearly, and in the edge of 
 which the catch n is placed, 
 so that it shall fall into the 
 teeth of the ratchet wheel. 
 When the watch is being 
 wound up, the chain passing 
 from the barrel to the 
 grooves of the fusee, the 
 teeth of the ratchet wheel 
 A B pass freely round the 
 cavity, the catch n falling 
 from tooth to tooth, and 
 producing the clicking noise 
 already noticed. But when 
 the watch is going, the tension of the chain draws the fusee and 
 the ratchet wheel attached to it round in the contrary direction, 
 and, pressing the teeth of the ratchet wheel against the catch n, 
 carries round the wheel c D, which gives motion to all the other 
 wheels, and through them to the hands. Now, it will be evident 
 that when the watch is being wound up, and the catch n relieved 
 from the pressure of the teeth of the ratchet wheel, no motion will 
 be imparted to c D, and consequently the movement of the entire 
 works will be suspended. 
 
 The modification by which a maintaining power is obtained by the com- 
 bination of two contrary ratchet wheels is shown in fig. 344., where c D is the 
 first wheel of the train from which the watch receives its motion. The 
 ratchet wheel A is fixed upon the base of the fusee, so as to move with it. 
 The catch m, which is pressed by a spring into the teeth of this ratchet wheel, 
 
 is attached to the second ratchet 
 wheel B, so that when A is carried 
 round by the chain acting on the 
 fusee, it must carry the wheel B 
 round with it in the direction of 
 the arrow f. The wheel B is 
 connected with the wheel o D by 
 a semicircular spring a be, which 
 is attached to the wheel c D at c, 
 and to the wheel B at a. The 
 catch n, which falls into the teeth 
 of the ratchet wheel B, is at- 
 tached to a fixed point on the 
 bed of the watch. 
 
 While the watch is going, the 
 wheel B, driven by the fusee, 
 draws after it the spring a 6 c, 
 
 bending it round to a certain extent ; and this spring, acting at c on 
 the wheel c D, draws it round in the direction of the arrow/. Now, 
 
 B B 4 
 
 Fig- 344- 
 
3?o PRACTICAL ILLUSTRATIONS. 
 
 let us suppose that the watch is being wound up. The ratchet wheel A, 
 being turned by the key in the direction of the arrow r, the catch m falls 
 from tooth to tooth, and the force it before received from the ratchet wheel 
 A is suspended. But the spring a b c has been drawn from its form of equi- 
 librium, to a certain small extent, in drawing round after it the wheel c D, 
 as already stated, and it has still a tendency to draw that wheel after it, so 
 as to recover its form of equilibrium. In doing this if will continue to act 
 upon the wheel c D, and to carry it round while the action of the fusee 
 upon it is suspended during the process of being wound up. The spring 
 a b c is so constructed as to act thus for an interval more than is necessary 
 to wind up the watch. 
 
 697. Cases in which, weights and springs are used. 
 
 From what has been explained, it will be observed that time- 
 pieces, in general, are constructed with one or other of two moving 
 powers a descending weight or a mainspring ; and with one or 
 other of two regulators a pendulum or a balance wheel. These 
 expedients are variously adopted and variously combined, according 
 to the position and circumstances in which the time-piece is used, 
 and the purpose to which it is appropriated. 
 
 A descending weight as a moving power, combined with a pen- 
 dulum as a regulator, supply the best chronometrical conditions. 
 But the weight can only be used where a sufficient vertical space 
 can be commanded for its ascent and descent, and neither it nor 
 the pendulum is applicable, except to time-pieces which rest in a 
 fixed and stable position. 
 
 In the case of time-pieces whose position is fixed, but where the 
 vertical space for the play of the weight cannot be conveniently 
 obtained, the mainspring is applied as a moving power, combined 
 with the pendulum as a regulator. Chimney and table clocks 
 present examples of this arrangement. The height being limited, 
 it is necessary also in these cases to apply short pendulums. The 
 length of a pendulum which vibrates seconds being about 39 
 inches, such pendulums can only be used where considerable height 
 can be commanded. 
 
 It has been shown that -the lengths of pendulums are in the 
 proportions of the squares of the times of their vibration. It 
 follows, therefore, that the length of the pendulum which would 
 vibrate in half a second, will be one fourth the length of one which 
 vibrates in a second, and since the latter is 39 inches, the former 
 must be 9^ inches. Such a pendulum can, therefore, be con- 
 veniently enough applied to a chimney or a table-clock high enough 
 to leave about ten inches for its play. 
 
 The pendulum is so good a regulator, and the anchor escapement 
 renders it so independent of the variation of the moving power, 
 that in time-pieces where it is combined with a mainspring, a fusee 
 is found to be unnecessary. In such cases, therefore, the axis of 
 
 
CLOCK AND WATCH WORK. 
 
 377 
 
 the first wheel is placed in the centre of the mainspring, as repre- 
 sented in fig. 331. 
 
 698. Watches and chronometers. The cylindrical escape- 
 ment shown in fig. 336., is nearly as independent of the variation 
 of the moving power as the pendulum, and therefore in common 
 watches, where this escapement is used, the fusee is dispensed 
 with. 
 
 In the class of time-pieces called chronometers, used for the 
 purposes of navigation, and in general for all purposes where the 
 greatest attainable perfection is required in a portable time-piece, 
 all the expedients to insure regularity are united, and accordingly 
 the detached escapement is combined with fusee and mainspring. 
 
 Besides the expedients above mentioned, for insuring uniformity 
 of rate, provisions are made in the most perfect chronometers to 
 prevent the variations of rate which would arise from the expan- 
 sion and contraction of the metal composing the balance wheel by 
 the variation of temperature. 
 
 Marine chronometers. These are usually suspended in a 
 box on gimbals, like those which support the ship's compass. The 
 balance wheel usually vibrates in half seconds, being a much 
 slower rate than that of common watches. They are of immense 
 utility in navigation, and especially in long voyages. 
 
 699. Observatory clocks. In observatories where stationary 
 time-pieces can be used, the clock moved by weights and regulated 
 by a pendulum is invariably adopted. The pendulum, in such 
 cases, is always so constructed that its rate of vibration shall not 
 be affected by variations of temperature. 
 
 700. Striking: apparatus. In clocks adapted for domestic 
 and public use, it is found desirable that they should give notice 
 of the time, not only to the eye, but to the ear ; and for that pur- 
 pose a bell is attached to them, which tolls at given intervals, the 
 number of strokes being equal to that of the units in the number 
 expressing the hour. This appendage is called the striking 
 train. 
 
 The striking train, though connected with that which moves the 
 hands, is quite independent of it, having its own moving and 
 regulating power, and its own system of wheels by which the effect 
 of the moving power is submitted to the regulator, and transmitted 
 to the tongue of the bell. 
 
 Unlike the train which moves the hands, the striking train is 
 not in continual motion. Its motion is always suspended, except 
 at the particular moment at which the clock strikes. The mecha- 
 nism partakes of the character of an alarum, being stopped by a 
 certain catch until the hands point to some certain hour, and then 
 being set free by the withdrawal of the catch. It remains free, 
 
378 PRACTICAL ILLUSTRATIONS. 
 
 however, only so long as is necessary for the tongue or hammer to 
 make the necessary number of strokes upon the bell, after which 
 'the catch again engages itself in the striking mechanism, and 
 stops it. 
 
 Some clocks only strike the hour. Others mark the half hours, 
 and others the quarters, by a single stroke. 
 
 The general principle of the striking mechanism will be rendered 
 intelligible by Jig. 345., which represents it in the case of a common 
 clock moved by a weight. 
 
 The weight suspended from the cord E moves tiie train in the same 
 manner as in the case of the train -which moves the hands. The motion of 
 the first wheel c is transmitted to the last wheel i, which corresponds to the 
 escapement wheel, by the intermediate wheels and pinions f t F, g, G, h, H, 
 and i. The wheel i drives the pinion k, upon the axle of which is fixed the 
 regulator K. This regulator is a fly, a side view of which, upon a larger 
 scale, is shown in fig. 346. 
 
 The pinion, which is made to revolve by the wheel w, gives a motion of 
 rotation to the fly A A' B' B, which consists of a thin rectangular plate of 
 metal, along the centre of which the prolongation M i, of the axle of the 
 pinion is attached. The flat surfaces of the fly, revolving more or less 
 rapidly, strike against the air, which resists them with a force which in- 
 creases in the proportion of the square of the velocity of the rotation. Thus, 
 if the velocity of rotation be increased in a two-fold ratio, the resistance 
 to A A' B' B is increased in a four-fold ratio ; if the rotation be increased 
 in a three-fold ratio, the resistance is increased in a nine-fold ratio, and so 
 on. It is evident, therefore, that by this very rapid increase the resistance 
 to the motion of the train must soon become equal to the descending force 
 of the weight, and then the motion will become uniform ; for if it were to 
 increase, the resistance would exceed the force of the weight, and would 
 slacken the rate of motion ; and if it were to decrease, the resistance being 
 less than the force of the weight, the latter would accelerate the motion. 
 In either case the motion would immediately be rendered uniform. 
 
 Projecting from the face of the wheel H (fig. 345.) there is a small pin 
 which rests upon the end m of a lever m n, which turns upon the centre n. 
 The lever m n, when in this position, stops the motion of the striking ti;ain. 
 Behind the same lever m n, and projecting from it, there is another piece, 
 which in the position represented in the figure rests in a notch of the wheel 
 op, lying behind the striking train, and indicated in the figure by dotted 
 lines. Around the edge of this wheel there is a series of similar notches 
 at unequal distances, determined in the manner which we shall presently 
 explain. 
 
 Upon the face of the wheel o, at equal distances one from another sur- 
 rounding it, a series of pins project, which, as the wheel turns, successively 
 encounter a lever 6, which plays upon a centre a'. Upon the same centre 
 a' is fixed the handle a a' of the hammer A by which the bell v is struck. A 
 spring fixed upon the same centre a' causes the lever b to rest in the 
 position represented in the figure, and to return to that position if raised 
 from it. The hammer handle a a' is made either elastic itself, or is provided 
 with a like spring. 
 
 When the wheel G is made to revolve at a uniform rate by the weight E, 
 
CLOCK AND WATCH WORK. 
 
 Fig 
 
 regulated by the ny K, the pins pro- 
 jecting from the face of the wheel o 
 encounter successively the lever i, and 
 raising it, throw back the handle a a' 
 of the hammer which .is in connection 
 with the lever b. After the pin has 
 passed the lever b, the latter is brought 
 back with a jerk by the action of the 
 spring ; and the hammer A, receiving 
 the same jerk, strikes upon the bell 
 v, and instantly, recoils from it ; and 
 if the wheel G continues to revolve, 
 one pin after another upon it will 
 encounter the l/>ver b, and the ham 
 
380 PRACTICAL ILLUSTRATIONS. 
 
 mer A will make a stroke upon the bell each time that a pin passes the 
 iever. 
 
 The wheel H is so constructed that it will make one revolution in the 
 interval between two successive strokes of the bell ; or, what is the same, in 
 the interval between the moments at which two successive pins pass the 
 lever b. 
 
 In the train which moves the hands, an expedient is provided by which 
 each time that the minute-hand passes twelve upon the dial, the lever m n is 
 thrown back from the position which it has in fig. 345. ; and the top m being 
 withdrawn from under the pin upon the wheel H, that wheel and the entire 
 striking train is liberated and set in motion. At the same time the piece is 
 taken out of the notch in which it rested on the wheel op, and that wheel, 
 in common with the other parts of the striking train, is put in motion. 
 
 For every complete revolution that the wheel H makes, the hammer A 
 makes a stroke upon the bell, and the motion of H and of the entire striking 
 train will continue until the end m of the lever m n again comes under the 
 pin projecting from the face of H. During the motion the lever m n is kept 
 back by the edge of the wheel o p acting against the piece projecting from 
 the lever m n ; but when the wheel o p has turned so as to bring the next 
 notch to the projecting piece, it will be thrown into the notch ; and the end 
 m of the lever m n coming under the pin projecting from the wheel H, the 
 motion of the train will be suspended. 
 
 Now it will be evident, from what has-been stated, that when the lever 
 in n is thrown back by the works, it is kept back by the edge of the wheel 
 op acting against the projecting piece; and so long as it is thus kept back, 
 the striking train continues to move, and the hammer continues to strike the 
 bell. But the duration of this motion will depend on the space between the 
 notches on the wheel o p, since it is while that space upon the edge passes 
 under the projecting piece on m n that the end m of the lever m n is kept 
 back, so as to be out of the way of the pin projecting from the wheel H. 
 These spaces between the notches are therefore so proportioned one to 
 another that the lever m n at each hour is held back a sufficient time for the 
 hammer to make upon the bell the number of strokes denoting the hour, and 
 no more. 
 
 The arrangement for striking half hours and quarters is based upon similar 
 principles. 
 
 CHAP. VI. 
 
 THE PRINTING PRESS. 
 
 WHETHER it be regarded as a mere piece of mechanism or viewed 
 in relation to the vast influence it has exercised on the progress of 
 the human race in knowledge and civilisation, the printing press 
 must ever be an object of profound interest. 
 
 To render more easily intelligible the beautiful mechanical contrt 
 vances of which it consists, it will be necessary, in the first instance, 
 to explain the succession of operations they are intended to execute 
 
PRINTING PRESS. 381 
 
 701. Setting: the types. The types are in the first instance put 
 together or composed, as it is technically called, by persons there- 
 fore called compositors, who, standing with the manuscript before 
 them, collect the letters one by one from an inclined desk, consist- 
 ing of as many small compartments as there are letters, in which 
 the types are assorted. The types thus put together are ranged 
 in lines, pages, or columns, according to the sheet to be printed; 
 and all the pages of type to be impressed upon the same side of 
 a sheet are placed in juxtaposition, in a strong iron frame, corre- 
 sponding in size and shape to the sheet, proper intervals being left 
 to represent the margins between page and page, or column and 
 column. The frame thus including the composed types to be 
 impressed on one side of a sheet of paper is called a form. 
 
 Types are similarly composed and arranged, corresponding to 
 the matter to be printed on the opposite side of the same sheet. 
 
 702. Old method of printing:. There are two classes of 
 printing machines, single and double. By the former, the sheets 
 are only printed on one side, and by the latter they are printed on 
 both sides in the same operation. 
 
 The form to be printed being laid upon a horizontal table, with 
 the faces of the types uppermost, the following operations are to 
 be executed: 1st, printing ink is to be applied to the faces of 
 the type, so evenly that there shall be no blotting or inequalities 
 in the printing ; 2nd, the sheet of paper to be printed must be laid 
 upon the form so as to receive the impression of the type in its 
 proper position, and in the centre of it ; 3rd, this paper must be 
 urged upon the type by a sufficient pressure to enable it to receive 
 the printed characters, such pressure, however, not being so great 
 as to cause the type to penetrate or deface the paper : 4th, the 
 paper is, in fine, when thus printed, to be withdrawn from the type 
 and laid upon a table, where the printed sheets are collected. 
 
 In the old process these operations were performed by two men, 
 one of whom was employed to ink the types, and the other to print. 
 The former was armed with two bulky inking balls, consisting of 
 a soft black substance of leathery appearance, spherical form, and 
 about 1 2 inches in diameter. He flourished these with dexterity, 
 dabbed them upon a plate smeared with ink, and then with both 
 hands applied them to the faces of the types until the latter were 
 completely charged with ink. This accomplished, the other func- 
 tionary the pressman having prepared the sheet, of paper while 
 the type was being inked, turned it down upon the type, drew it 
 under the press, and with a severe pull of the lever gave the neces- 
 sary pressure by which the paper took the impression of the type. 
 A contrary motion of the apparatus withdrew the type from under 
 the press, and the pressman, removing the paper, now nrinted de- 
 
382 PRACTICAL ILLUSTRATIONS. 
 
 posited it upon a table placed near him to receive it. The same 
 series of operations was then repeated for producing the impression 
 of another sheet, and so on. In this manner two men in ordinary 
 book work usually printed at the rate of 250 sheets per hour on 
 one side. 
 
 703. Inking rollers. One of the first improvements which 
 
 F'g- 347- 
 
 took place upon this apparatus consisted in the substitution t of a 
 cylindrical roller for the inking balls. This roller was mounted 
 with handles, so that the man employed to ink the type first rolled 
 it upon a flat surface smeared with ink (Jig- 347-)i an d having 
 thus charged it, applied it to the type form, upon which he rolled 
 it in a similar manner, thus transferring the ink from the roller to 
 the faces of the type. The substitution of these inking rollers for 
 the inking balls constituted one of the most important steps in the 
 modern improvement of the art of printing. The rollers were 
 composed of a combination of treacle and glue, and closely resem- 
 bled caoutchouc in their appearance and qualities. 
 
 704. Modern printing: presses. The printing presses 
 which served the purposes of publication for some hundred years, 
 during which they received no other improvements than such &s 
 
PRINTING PRESS. 383 
 
 might be regarded merely as modifications in the detail of their 
 mechanism, have been entirely superseded by engines of vastly 
 increased power and improved principles of construction. Although 
 these admirable machines differ one from another in the details of 
 their mechanism, according to the circumstances under which 
 they are applied, and the power they are expected to exert, they 
 are nevertheless characterised by certain common features. . 
 
 705. The form to be printed is laid in the usual manner upon a 
 perfectly horizontal table, with the. faces of the types uppermost; 
 and upon the same table, in juxtaposition with the form, and level 
 with the faces of the type, or nearly so, is placed a slab upon 
 which a thin and perfectly regular stratum of printing ink is dif- 
 fused ; the table thus carrying the form and inking slab is moved 
 by proper machinery right and left horizontally, with a reciprocat- 
 ing rectilinear motion through a space a little greater than the 
 length of the form. 
 
 Above the form and slab are mounted, also in juxtaposition, a 
 large cylinder or drum which carries upon it the sheet of paper to 
 be printed, and three or four inking rollers similar to that already 
 described. There are also three or four other rollers in juxtapo- 
 sition with the latter, one of which supplies ink to the others, which 
 severally spread it in a uniform stratum upon the slab. The paper 
 cylinder and the inking and diffusing rollers are so mounted 
 that, when the horizontal table, carrying the form and inking slab, 
 moves alternately backwards and forwards under them, they roll 
 upon it. 
 
 In this way, when the table is moved towards the rollers, the 
 form, passing under the inking rollers right and left, receives from 
 them the ink upon the faces of the type ; and at the same time the 
 slab, moving backwards and forwards under the diffusing rollers, 
 receives from them, upon its surface, the proper stratum of ink to 
 supply the place of that which was taken from it by the inking 
 rollers. 
 
 706. Single printing- machines. When the table is moved 
 alternately towards the other side, the form, with the types already 
 inked, passes under the cylinder carrying the paper, the motion of 
 which is so regulated as to correspond exactly with the rectilinear 
 motion of the table carrying the form. The cylinder is urged upon 
 tjie type with a regulated force, sufficient and not more than suffi- 
 cient, to impress the type upon the paper. 
 
 The sheets of paper are supplied in succession to the cylinder, and 
 held evenly upon it by bands of tape while they pass in contact with 
 the type. After receiving the impression of the type, the tapes 
 which bound them are separated, and the printed sheets discharged. 
 
 Such is the general principle of single printing machines. 
 
384 PRACTICAL ILLUSTRATIONS. 
 
 707. Double printing: machines. In these the table which 
 is moved alternately right and left carries two forms, one corre- 
 sponding to the pages to be printed on one side of the sheet, and 
 the other to those to be printed on the other side. There are also 
 two inking slabs, one to the left of the left hand form, and the 
 other to the right of the right hand form. There are also two 
 paper cylinders, and two sets of inking and diffusing rollers. Each 
 sheet of paper to be printed, being held between tapes, as already 
 described, is carried successively round the two cylinders, being so 
 conducted, in passing through one to the other, that one side of it 
 passes in contact with, and is printed by, one form, and the oppo- 
 site side by the other form. The proportion of the motions is so 
 nicely regulated, that the impression of each page or column made 
 on one side of the paper, corresponds exactly with that of the cor- 
 responding page or column on the other side. 
 
 This general description will be more clearly understood by 
 reference to the following illustrative diagrams : 
 
 Fig. 348. 
 
 Fig. 348. illustrates the operation of a single printing machine. The form 
 A and the inking slab B, are placed on a horizontal table; above them is the. 
 paper cylinder D, the inking rollers i i i, the diffusing rollers c c, and the 
 rollers c, which supply the ink fo the diffusing rollers. The first of these, o, 
 is called the ductor roller. When the table x Y is moved towards the left 
 from Y to x, the form A passes under the inking rollers i i i, and receives ink 
 from them on the faces of the type ; at the same time the slab B passes under 
 the diffusing rollers c r, and receives from them a supply of ink to replace 
 what it has just given to the inking rollers. 
 
 When the table is moved in the contrary direction from x towards Y, the 
 form once more passed under the inking rollers, and afterwards under the 
 paper cylinder, which, being pressed upon it while it moves in exact accord- 
 ance with it, the types discharge upon the paper the ink they have just 
 received from the rollers ; and the printirlg of the paper being thus effected 
 on one side, the sheet is discharged from the tapes. The table is then again 
 moved to the left, and the types are again inked, and the same effects ensue as 
 have already been described. 
 
PRINTING PRESS. 385 
 
 In this manner sheet after sheet is printed. 
 
 The inking and diffusing rollers rest upon the slab and the types 
 by their weight, the axes projecting from their ends being in- 
 serted in slits formed in upright supports attached to opposite 
 sides of the frame which supports the moving table. The two 
 upright pieces in which the axes of each roller are inserted, are 
 not placed in exact opposition to each other ; the consequence of 
 which is, that the rollers are placed with their axes slightly inclined 
 to the sides of the table. This arrangement is attended with a very 
 important effect : for, in consequence of the friction or adhesion 
 of the rollers with the slab, they are moved alternately in contrary 
 directions with the longitudinal motion across the table. This 
 motion, combined with their rolling motion upon the slab, aids 
 materially in diffusing the ink in a perfectly uniform stratum. 
 
 An illustrative diagram of a double printing machine is shown in jig. 349. 
 where D and D' are the two paper cylinders ; A and A', the two forms ; i t t 
 
 Fig. 349. 
 
 and t' V i 1 , the inking rollers ; c c and d c', the diffusing rollers ; and c and 
 c', the ductor rollers. The pile of paper placed on the table E, is supplied 
 sheet by sheet to the tapes between which it is held ; and being passed over 
 the roller R, and under the cylinder D', it receives the impression of the types 
 of the form A' ; it then passes successively over the roller T', under T, and 
 round the cylinder D, at the lower point of which its unprinted side comes 
 into contact with the types of the form A, by which it is printed ; after which, 
 the tapes opening, it is thrown out by the centrifugal force upon the receiv- 
 ing table F. 
 
 It will be apparent by the figure, that while the sheet is printed on one 
 side by the form A', the form A is passing between the inking rollers 1 1 1 
 and the slab B ; and on the contrary while the paper is printed on the other 
 side by the form A, the form A' is passing between the inking rollers ' i 1 i 1 , 
 and the slabR'. 
 
 In this manner, by each alternate motion from right to left and from left 
 to right, a sheet is printed on both sides. ^ 
 
 708. A perspective view of a double-acting printing machine, as 
 constructed by Messrs. Applegath and Cowper, is shown \njig. 350. 
 
 A boy, called the layer on, E, standing at an elevated desk, pushes the 
 paper, sheet by sheet, towards the tapes, which, closing upon it, carry it over 
 
 C C 
 
386 
 
 PRACTICAL ILLUSTRATIONS. 
 
PRINTING PRESS. 387 
 
 a roller R, passing under which it is carried to the right of the cylinder i> 
 under which it passes, and being carried up to the left of it, passes succes- 
 sively over the roller T, under the roller T', over the cylinder u, and drawn 
 along its left side, after which it passes under it, and is flung into the hands 
 of bov, F, called the taker off, seated before a table, placed between the two 
 cylinders D and i>', upon which he disposes the sheets as he receives them. 
 
 In this manner, the layer on feeds the machine in constant succession with 
 blank sheets, which, being carried under the cylinder i>, are printed on one 
 side, and afterwards under the cylinder i>', are printed on the other, when they 
 are received by the taker off. 
 
 The ductor rollers, c and c',-are kept in revolution by endless bands, car- 
 ried over rollers at the lower parts of the frame, and then over grooved 
 wheels fixed on the axes of the cylinders. The table carrying the forms 
 and slabs is moved alternately right and left by means of a pair of bevelled 
 wheels, w, under the frame, and a double rack and pinion above ; one of 
 the bevelled wheels, having a horizontal axis, receives motion from a 
 steam-engine or other moving power; it imparts motion to the other 
 bevelled wheel, having a vertical axis. This latter axis has a pinion fixed 
 at its upper end, which works in a double rack attached to the movable 
 type table, which is so constructed that the continuous rotation of the pinion 
 imparts an alternate rectilinear motion right and left to the rack and to the 
 table attached to it. 
 
 The manner in which the tapes lay hold of and conduct the paper succes- 
 sively round the cylinders, and finally discharge it upon the table of the taker 
 off, will be easily understood by reference to fig. 351. 
 
 Fig. 351- 
 
 c and D are two grooved rollers, surrounded by an endless band which 
 pushes the paper from the table of the layer on towards the tapes. The two 
 endless tapes, between which the paper is held, are represented in the dia- 
 gram by the continuous and dotted lines, and the direction of their motion 
 round the rollers and cylinders is represented by the arrows. It will be per- 
 ceived that opposite the table of the layer on, the tapes converge, from two 
 small rollers d and h, and come into contact at the top of the roller E. The 
 edges of the sheets of paper, being advanced from the table of the layer on, 
 are caught between the tapes immediately above the roller E. 
 c C a 
 
388 PRACTICAL ILLUSTRATIONS 
 
 It must be understood that there are two or three pairs of tapes parallel to 
 each other, which correspond to the margins of the pages or columns ; but 
 only one pair is shown in the figure. 
 
 The paper, being thus seized between the tapes above the roller E, is car- 
 ried successively, as shown in the figure, still held between the tapes, under 
 and over F H I and G, until it arrives at i, where the tapes separate, that 
 which is indicated bv the continuous line being carried to the roller a, and 
 that by the dotted line, over the roller i, to the roller k. The tapes being 
 thus separated the printed sheet is discharged at i, upon the table of the 
 taker off; meanwhile, the tape indicated by tha continuous line is carried 
 successively over the roller a, under b, under c, outside d, and is finally 
 returned to the roller E. 
 
 In the same manner the tape indicated by the dotted line is carried suc- 
 cessively under the rollers ft and m, outside n, over v and /t t from which it 
 returns to E, where it again joins the other tape proceeding from d. 
 
 709. The first machines constructed upon this improved princi- 
 ple for printing newspapers, were erected at the printing office of 
 The Times newspaper, and it was announced in that journal, on 
 the 28th of November, 1814, that the sheet which was then placed 
 in the hands of the reader, was the first printed by steam-impelled 
 machinery. 
 
 By this, with some improvements which the apparatus received 
 soon afterwards, the effective power of the printing press was aug- 
 mented in a very high proportion. With the hand presses previously 
 in use, not more, as we have seen, than 250 sheets could be printed 
 on one side in an hour. Each of the two machines erected at The 
 Times office produced 1 800 impressions per hour. 
 
 710. Further improvement. The power of the printing 
 machine constructed upon this principle was soon after augmented, 
 by increasing the number of printing cylinders to four, the princi- 
 ple of the machine, however, remaining the same. 
 
 The manner in which this was accomplished will be easily understood by 
 the aid of the illustrative diagram (fa. 35*-)' where I, 2, 3, and 4, are the 
 printing cylinders; PP p'r', are the tables of the four layers on, and o o o'o', 
 lead to those of the four takers off. The course followed by the sheets of 
 paper, in passing to and from the cylinders, are indicated by the arrows. 
 Inking rollers are in this case placed at R, between the printing cylinders ; the 
 two type forms are inked twice, while they move from right to left, and twice 
 again, while they move from left to right. The printing cylinders are alter- 
 nately let down upon the type and raised from them in pairs ; while the typo 
 table moves from left to right, the cylinders i and 3 are in contact with the 
 table, the cylinders 2 and 4 being raised from it, and, on the contrary, when 
 it moves from right to left, the cylinders 2 and 4 are in contact with it, I 
 and 3 being raised from it. 
 
 By this improvement, which was adopted in The Times office in 
 '1 827, the proprietors of that journal obtained the power, then un- 
 precedented, of printing from 4000 to 5000 sheets per hour on one 
 
PRINTING TRESS. 
 
 389 
 
 cc 
 
390 PRACTICAL ILLUSTRATIONS. 
 
 side of the paper. By this means they were enabled to satisfy the 
 demands of a circulation amounting to 28000. 
 
 In reference to newspaper printing, it must be here observed 
 that the great object to be attained, is to increase the celerity 
 by which printing on one side only of the sheet can be augmented. 
 It is found convenient so to arrange the letter-press that the 
 matter appropriated to one side of the sheet shall be ready for 
 press at an early hour, and may be printed before the contents of 
 the other side, in which the most recent intelligence is included, 
 can be prepared. Hence the advantage of using machines adapted 
 to print one side only with the most extreme celerity for news- 
 papers. 
 
 711. " Times " printing: machine. This machine continued to 
 serve the purposes of The Times newspaper until a later epoch, 
 when again the exigencies of the press exceeded even its immense 
 powers, and another appeal was made to the inventive genius of 
 Mr. Applegath. It was, in short, necessary to provide a machine 
 by which at least 10000 sheets an hour could be worked off from a 
 single form ! 
 
 In considering the means of solving this problem, it is necessary 
 to observe, that whatever expedient may be used, the sheets of 
 paper to be printed must be delivered one by one to the machine 
 by an attendant. After they once enter the machine they are car- 
 ried through it and printed by self-acting machinery. But in the 
 case of sheets so large as those of the newspapers, it is found that 
 they cannot be delivered with the necessary precision by manipu- 
 lation at a more rapid rate than two in five seconds, or twenty-five 
 per minute, being at the rate of 1 500 sheets per hour. Now, in 
 this manner, to print at the rate of loooo per hour, would require 
 seven cylinders, to place which so as to be acted upon by a type 
 form moving alternately in a horizontal frame, in the manner 
 already described, would present insurmountable difficulties. 
 
 In the face of these difficulties, Mr. Applegath, to whom the 
 world is indebted for the invention of The Times printing machine, 
 decided on abandoning the reciprocating motion of the type form, 
 arranging the apparatus so as to render the motion continuous. 
 This necessarily involved circular motion, and accordingly he 
 resolved upon attaching the columns of type to the sides of a large 
 drum or cylinder, placed with its axis vertical, instead of the hori- 
 zontal frame which had been hitherto used. A large central drum 
 is erected, capable of being turned round its axis. Upon the 
 sides of this drum are placed vertically the columns of type. These 
 columns, strictly speaking, form the sides of a polygon, the centre 
 of which coincides with the axis of the drum, but the breadth of 
 the columns is so small compared with the diameter of the drum, 
 
PRINTING PRESS. 391 
 
 that their surfaces depart very little from the regular cylindrical 
 form. On another part of this drum is fixed the inking table. 
 The circumference of this drum in The Times printing machine 
 measures 200 inches, and it is consequently 64 inches in diameter. 
 
 The general form and arrangement of the machine are repre- 
 sented in fig- 353-, where D is the great central drum which car- 
 ries the type and inking tables. 
 
 This drum in The Times machine is surrounded by eight 
 cylinders, R, R, &c., also placed with their axes vertical, upon 
 which the paper is carried by tapes in the usual manner. Each 
 of these cylinders is connected with the drum by toothed wheels, in 
 such a manner that their surfaces respectively must necessarily 
 move at exactly the same velocity as the surface of the drum. 
 And if we imagine the drum, thus in contact with these eight 
 cylinders, to be put in motion, and to make a complete revolution, 
 the type form will be pressed successively against, each of the eight 
 cylinders, and if the type were previously inked, and each of the 
 eight cylinders supplied with paper, eight sheets of paper would 
 be printed in one revolution of the drum. 
 
 It remains, therefore, to explain, first, how the type is eight times 
 inked in each revolution ; and, secondly, how each of the eight 
 cylinders is supplied with paper to receive their impression. 
 
 Beside the eight paper cylinders are placed eight sets of inking 
 rollers ; near these are placed two ductor rollers. These ductor 
 rollers receive a coating of ink from reservoirs placed above them. 
 As the inking table attached to the revolving drum passes each of 
 these ductor rollers, it receives from them a coating of ink. It 
 next encounters the inking rollers, to which it delivers over this 
 coating. The types next, by the continued revolution of the drum, 
 encounter these inking rollers, and receive from them a coating of 
 ink, after which they meet the paper cylinders, upon which they 
 are impressed, and the printing is completed. 
 
 Thus in a single revolution of the great central drum the inking 
 table receives a supply eight times successively from the ductor 
 rollers, and delivers over that supply eight times successively to 
 the inking rollers, which, in their turn, deliver it eight times suc- 
 cessively to the faces of the type, from which it is conveyed finally 
 to the eight sheets of paper held upon the eight cylinders by the 
 tapes. 
 
 Let us now explain how the eight cylinders are supplied with 
 paper. Over each of them is erected a sloping desk, A, A, &c., 
 upon which a stock of unprinted paper is deposited. Beside this 
 desk stands the " layer on," who pushes forward the paper, sheet 
 by sheet, towards the tapes. 
 
 These tapes, seizing upon it, first draw it down in a vertical 
 c c 4 
 
392 
 
 PRACTICAL ILLUSTRATIONS. 
 
 direction between tapes in the eight vertical frames, until its 
 edges correspond with the position of the form of type on the 
 printing cylinder. Arrived at this position, its downward motion 
 is stopped by a self-acting apparatus provided in the machine, and 
 
PRINTING PRESS. 
 
 393 
 
 it is then impelled by vertical rollers towards the printing cylin- 
 der, these rollers having upon them marginal tapes which carry 
 the paper round the cylinder, from which it receives the impression 
 of the types. After this the central and lower marginal tapes dismiss 
 the sheet of paper, which the upper ones only become charged 
 with, and carry it to its proper place, where the tapes are stopped 
 with the paper suspended between them, until the "taker off" 
 draws the sheets down, ranging them upon his table. These move- 
 ments are continually repeated ; the moment that one sheet passes 
 from the hands of the " layer on," he supplies another, and in this 
 manner he delivers to the machine at the average rate of two sheets 
 every five seconds, and the same delivery taking place at each of the 
 eight cylinders, there are 1 6 sheets delivered and printed every five 
 seconds. 
 
 It is found that by this machine in ordinary work between 
 I oooo and 1 1 ooo per hour can be printed ; but with very expert 
 men to deliver the sheets, a still greater speed can be attained. 
 Indeed, the velocity is limited, not by any conditions affecting the 
 machine, but by the power of the men to deliver the sheets 
 to it. 
 
 In case of any misdelivery a sheet is spoilt, and, consequently, 
 the effective performance of the machine is impaired. If, however, 
 a still greater speed of printing were required, the same description 
 of machine, without changing its principle, would be sufficient for 
 the exigency ; it would be necessary that the types should be 
 surrounded with a greater number of printing cylinders. 
 
 It may be right to observe, that the cylinders and rollers are 
 not uniformly distributed round the great central drum; they 
 are so arranged as to leave on one side of that drum an open space 
 equal to the width of the type form. This is necessary in order to 
 give access to the type form so as to adjust it. 
 
 One of the practical difficulties which Mr. Applegath had to en- 
 counter in the solution of the problem, which he has so success- 
 fully effected, arose from the shock produced to the machinery by 
 reversing the motion of the horizontal frame which, in the old ma- 
 chine, carried the type form and inking table, a moving mass which 
 weighed 25 cwt. This frame had a. motion of 88 inches in each 
 direction, and it was found that such a weight could not be driven 
 through such a space with safety at a greater rate than about 45 
 strokes per minute, which limited its maximum producing power 
 to 5000 sheets per hour. 
 
 Another difficulty in the construction of this vast piece of ma- 
 chinery was so to regulate the self-acting mechanism that the im- 
 pression of the type form should always be made in the centre of 
 the page, and so that the space upon the paper occupied by the 
 
394 PRACTICAL ILLUSTRATIONS. 
 
 printed matter on one side may coincide exactly with that occu- 
 pied by the printed matter on the other side. 
 
 The type form fixed on the central drum moves at the rate of 
 about 80 inches per second, and the paper is moved in contact 
 with it of course at exactly the same rate. Now, if by any error 
 in the delivery or motion of a sheet of paper, it arrive at the print- 
 ing cylinder i-8oth part of a second too soon or too late, the rela- 
 tive position of the columns will vary by I -Both part of 80 inches 
 that is to say, by one inch. In that case the edge of the printed 
 matter on one side would be an inch nearer to the edge of the 
 paper than on the other side. 
 
 This is an incident which rarely happens, but when it does, a 
 sheet, of course, is spoilt. In fact, the waste from that cause is 
 considerably less in the present vertical machine than in the former 
 less powerful horizontal one. 
 
 The vertical position of the inking rollers, in which the type is 
 only touched on its extreme surface, is more conducive to the 
 goodness of the work than the horizontal machine, where the 
 inking rollers act by gravity ; also any dust shaken out of the 
 paper, which formerly was deposited upon the inking rollers, 
 now falls upon the floor. 
 
 With this machine 50000 impressions hare been taken without 
 stopping to brush the form or table. 
 
 712. IWarinoni's newspaper printing: press. Messrs. 
 Marinoni and Co., of Paris, have within the last few years con- 
 structed improved printing presses for newspapers of large circu- 
 lation, several of which have been erected and brought into opera- 
 tion in the printing office of the Paris journal La Presse. This 
 printing machine, which is capable of working off 6000 copies per 
 hour, printed on both sides of the paper, is represented \nfig. 354. 
 It will be perceived that eight men are employed in the process, 
 four layers on and four takers off. The machine is double, the 
 parts at each side of a vertical line drawn through the axis of the 
 fly wheel being perfectly similar. The manner in which the sheets 
 pass to and from the printing rollers will be more readily under- 
 stood by^g-. 355., where A is the upper and A' the lower deliver- 
 ing board, and B the upper and B' the lower receiving board on 
 the right, the two delivering and receiving boards on the left being 
 similarly placed. The motion of the sheets, as they are conducted 
 to and from the rollers by the tapes, is indicated by arrows, and 
 the course followed by each sheet from the moment it leaves the 
 delivering board until it arrives at the receiving board, is indicated 
 by the numbers I, 2, 3, 4, &c. Thus, the sheet delivered from 
 the board A is taken by the tapes which pass round the roller M, 
 and carried from I to 2. Arriving at the lower roller, it passes 
 
 
PRINTING PRESS. 
 
396 PRACTICAL ILLUSTRATIONS. 
 
PRINTING PRESS. 397 
 
 as shown by the arrow between the rollers, 3, and is carried from 
 4 under the printing roller i, where it is printed on one side, after 
 which it is carried up between the tapes to 5, from whence it is 
 discharged between the tapes of 6, and carried up over the roller 
 B at 7, from which it is carried down between the tapes 8, and 
 thrown, as shown by the arrow, to the tapes 9, by which it is again 
 carried under the roller and printed on the other side ; after which, 
 it is carried up successively between the tapes 10 and II to 12, 
 and finally discharged from 13 at 14 upon the receiving board B. 
 
 The sheet delivered from the lower receiving board A', follows 
 a course precisely similar, entering at I ' and passing round the 
 printing roller at 2' 3', from which it passes between the tapes 4' 
 round the roller R' at 5' 6', and thence from the tapes 7' 8' round 
 the printing roller i' at 9', by which it is printed on the other side ; 
 after which it is carried by I o', 1 1 ', 1 2' to the lower receiving 
 board B' at 13'. 
 
 The inking rollers are shown at E p D and D', and are arranged 
 in the usual manner, at T, T', T", T"', and T"", to spread the ink on 
 the types. 
 
 It will be perceived that the power of this press is equal to that 
 of The Times, the difference being that The Times prints 1 2000 
 sheets on one side only, while this prints 6000 on both sides. The 
 Times machine requires 8 layers on and 8 takers off, being double 
 the number required by Marinoni's press. It must, however, be 
 observed that, in the practical management of newspaper printing, 
 as conducted in The Times office, the power of Marinoni's press, 
 though in a certain sense equal to that of The Times, would be 
 altogether insufficient ; for it is indispensably necessary for that 
 journal to print 60000 copies on one side of the paper during the 
 last 5 hours of the morning. The matter allotted to the other 
 side of the paper is so selected that it can be composed and printed 
 in the earlier part of the night, or even of the previous day ; the 
 pressure falling exclusively on the matter which occupies the other 
 side of the paper, consisting chiefly of the latest intelligence and 
 Parliamentary reports. 
 
 It may be asked, therefore, how the journal of La Presse, of 
 which the circulation, though inferior to that of The Times, is still 
 very large, being understood to be at this time (June, 1855) 
 40000 copies, can be printed with the necessary celerity ? The 
 answer is, that La Press* does not contain as much as the tenth 
 part of the letter-press of a copy oi The Times, and that, therefore, 
 it is found practicable to compose the matter in type twice or 
 oftener, so as to produce two or more distinct forms, as they are 
 called, which are put to work at as many different presses. The 
 expense of composition is further economised at the printing office 
 
3Q8 
 
 PRACTICAL ILLUSTRATIONS. 
 
PRINTING PRESS. 
 
 399 
 
4 oo 
 
 PRACTICAL ILLUSTRATIONS. 
 
 of La Presse by stereotyping the matter, which is composed at a 
 sufficiently early hour to admit of that process, the stereotype 
 plates being melted down the next day. By this expedient double 
 or triple composition is only necessary for the intelligence which 
 comes too late to allow of being stereotyped. 
 
 713. Marinoni's book printing: machine. A convenient 
 form of printing engine for books, constructed by the same engi- 
 neers, is shown in fig. 356., by which, however, the sheets are 
 printed on one side only. The layer on delivers the sheets upon the 
 board M (./%" 357-)' fr m which they pass round the printing 
 roller i, and are discharged as indicated by the arrow upon the 
 receiving board B. The rollers for delivering and spreading the 
 ink on the types are arranged in the usual way. 
 
INDEX. 
 
 NOTE. This Index refers to the numbers of the paragraphs, and not to the pages. 
 
 Actfon and reaction, 201. 
 
 Adhesion, 552 ; of wheels, 367. 
 
 Adjusting screw, 556. 
 
 Affinity, chemical, 354. 
 
 Air, resistance of, 589. 
 
 Alloys, strength of, 596. 
 
 Animalcules, minuteness of, 49. 
 
 Animal power, 621. 
 
 Annealing, 133. 
 
 Atoms, ultimate, 65. 
 
 Attraction, capillary, 353. 
 
 Atwood's machine, 243. 
 
 Axes, principal, 343. 
 
 Axle, wheel and, 432, ; of wheels, 554. 
 
 B. 
 
 Ball and socket. 550. 
 
 Balance, 414 ; sensibility of, 415 ; letter, 
 
 417 ; spring, 418 ; compensation, 688. 
 Balance wheel, 517. 686. 
 Barlow, table of strength of beams, 613. 
 Bevelled wheel, 450. 
 Bite of metals, 369. 
 Blood, its composition, 46; corpuscles in 
 
 a drop of, 48. 
 Bodies, their parts, iz ; falling, 237 ; 
 
 strength of, 590. 
 Boring tools, 659. 
 Brakes, 582. 
 
 Bridges, suspension, 160. 
 Brittleness, 130. 
 Buoyancy, 81. 
 
 C. 
 
 Capillary attraction, 353. 648. 
 
 Capstan, 436. 624. 
 
 Centre of gravity, 268 ; of fluids, 308 ; of 
 separate bodies, 309. 
 
 Centrifugal force, 311 ; rules for calculat- 
 ing, 314 ; examples, 318. 328; examples, 
 
 Chemical affinity, 354. 
 Chemical agency, 645. 
 Chronometers, 698. 
 Clepsydra, 668. 
 
 Clamp, 556. 
 
 Cleavage, planes of, 63. 
 
 Clocks and watches, 671 ; maintaining 
 power in, 695. 
 
 Cohesion, 262.351; manifested, 356; ex- 
 amples, 357. 
 
 Coining press, 658. 
 
 Collision, 196 ; example, 199 ; effect of, 
 203 ; of equal masses, 205 ; example, 
 206 ; apparatus to illustrate, 215. 
 
 Combustion, 68. 
 
 Compensation balance, 688. 
 
 Compression. 87 ; of wood, 90 ; of stone, 
 91 ; of metal, 92 ; of liquids, 93 ; of gases, 
 
 Composition, examples of, 170. 
 
 Congelation, 643. 
 
 Contractibility of liquids, 95. 
 
 Contraction, 109 ; effects ot, no. 
 
 Cordage, strength of, 599. 
 
 Couple, 155. 
 
 Coupling, 557. 
 
 Cradle joint, 551. 
 
 Crane, 649 ; movable, 650 ; building, 651 ; 
 
 railway, 652 ; fixed, 653. 
 Crank, 529. 
 Crown wheel, 449. 
 Crystallisation, w. 
 
 D. 
 
 Density, 76 ; example of, 80. 
 Diamond, polished surfaces of, 34. 
 Dilatation, 107; of liquids, 108 ; useful 
 
 application of, 109 ; effects of, HO. 
 Distillation, destructive, 70. 
 Divisibility, unlimited, 28 ; minute, 51 
 
 shown by taste, 54. 
 Draught, line of, 579. 
 Drops for wharves, 654* 
 Drying machine, centrifugal, 326. 
 Ductility, 135. 
 Dynamical unit, 619. 
 
 E. 
 
 Earth once fluid, 360. 
 Earth work ,628. 
 
 D D 
 
4 02 
 
 INDEX. 
 
 Elasticity of gases, 97 ; of liquids, 98 ; 
 of solids, 100 ; examples, 101 ; of tor- 
 sion, 106 ; effects of, 125 ; not propor- 
 tionate to hardness, 117 ; of metals, 129 ; 
 perfect and imperfect, 210. 
 
 Elastic force, limits of, 105. 
 
 Electro magnetic force, 644. 
 
 Endless screw, 495. 
 
 Engine, pile, 655; stamping, 656; screw 
 cutting, 662 ; lor cutting teeth of wheels, 
 663. 
 
 Equilibrium, 297 ; criterions of, 298 ; ex- 
 amples of, 301 ; not necessarily a state of 
 rest, 385. 
 
 Erard's piano forte, 428. 
 
 Escapement, connection of pendulum 
 with, 677 ; action of pendulum upon, 680 ; 
 cylindrical, 692 ; recoil, 16. ; duplex, 693 ; 
 lever, t'ft. ; detached, 694. 
 
 Evaporation, 69. 
 
 Expansibility of gases, 99. 
 
 F. 
 
 Falling body, 237. 
 
 Filtration, 83. 
 
 Fire escape, 459. 
 
 Flexibility, 130. 
 
 Flour mill, 664. 
 
 Fluids, resistance of, 585. 
 
 Fly, 518 ; position of, 540. 
 
 Fly wheel, 530. 
 
 Force, 141 ; resultant of, 145 ; in different 
 directions, 147 ; composition of, 148 ; of 
 moving mass, 185 ; examples of, 192 ; 
 centrifugal, 311 ; rules to calculate cen- 
 trifugal, 314; examples of centrifugal, 
 318. 328. 334 ; molecular, 350; resisting, 
 559 i f animals, 634 ; electro magnetic, 
 644. 
 
 Friction, 560 ; sliding and rolling, 565. 
 
 Fuel, 639 ; effects of, 640. 
 
 Fusee, 527. 685. 
 
 G. 
 
 Gaseous state, 16. 
 
 Gases, elasticity of, 97 ; expansibility of, 
 
 99 ; may be reduced to liquids or solids, 
 
 362. 
 
 Gimbal, 549. 
 Glass, its filaments, 36. 
 Glue, effect of, 370. 
 
 Gold, visible on touchstone, 35 ; leaf, 43. 
 Governor, 526. 
 Gravity, 227. 
 Guinea and. feather, 233. 
 Gun-cotton, 646. 
 Gunnery, 259. 
 
 H. 
 
 Hammer, effect of, upon a nail, 260. 534 ; 
 
 sledge, 657. 
 Hardness, 122 ; may be modified, 122 ; of 
 
 metals, 129. 
 
 Harrison's pendulum, 674. 
 He.it, effects of, 597 ; force produced bv, 
 
 642. 
 
 Hinge, 552. 
 
 Hodgskinson, researches of, 603. 
 Horse power, 629. 
 Hunter's screw, 493. 
 
 Inclined plane, 469 ; on railways, 472 -, 
 
 double, 475. 
 Inclined roads, 471. 
 Inertia, uz ; proof of, 116 ; examples of, 
 
 117. 
 
 Inking rollers, 703. 
 Insects, their wings, 42. 
 Iron, strength of, 596. 
 Isochronism, 687. 
 
 J. 
 
 Joint, knee. 289. 548 ; universal, 548 ; 
 cradle, 551 ; telescope, 555. 
 
 Knee joint, 289. 
 Knee lever, 427. 
 
 L. 
 
 Leaning towers, 282. 
 
 Level surface, 226. 
 
 Lever, 409 ; rectangular, 423 ; knee, 427. 
 
 Life preserver, 536. 
 
 Liquid state, 15. 
 
 Line of draught, 579. 
 
 Lubricants, 368 ; selection of, 572. 
 
 M. 
 
 Machine, Atwood's, 243 ; construction 
 of a, 372 ; functions of, 390 ; simple, 
 404 ; weighing, 446 ; planing, 600 ; 
 printing, 706; Times printing, 711; 
 book printing, 713 ; Marinoni's printing, 
 712. 
 
 Magic clock, 307. 
 
 Magnitude, classified, 2 ; linear, 3 ; super- 
 ficial, 4; solid, 5; unlimited, 10. 
 
 Mainspring, 684. 
 
 Maintaining'power, 695. 
 
 Malleability, 131. 
 
 Marble, pulverised, 33. 
 
 Matter inert, in. 
 
 Mechanical agents, 68. 
 
 Mercurial time measurer, 670 ; pendulum, 
 675. 
 
 Micrometer screw, 494. 
 
 Mill stones, 664. 
 
 Mirrors, 395. 
 
 Molecular forces, 350. 
 
 Molecules, 59 ; not in contact, 72. 
 
 Moments, 395. 
 
 Momentum of solids, 186 ; of liquids, 187 ; 
 of air, 188 ; arithmetical expression for, 
 194 ; reservoir of, 222. 
 
 Morm's apparatus, 244. 
 
 Motion, 161 ; direction of, in a curve, 162 ; 
 effect of force upon, 167 ; absolute and 
 relative, 181 ; examples of, 182 ; laws 
 of, Zl6; spontaneous, 220; down in- 
 clined plane, 253 ; of projectiles, 254 ; 
 retarded, 352 ; modification of, 542 ; per- 
 petual, 648. 
 
 Moving force, 190. 
 
 Musk, 52. 
 
 O. 
 
 Oblique impact, 212. 
 Observatory clock, 699. 
 Oscillation, 507. 
 
INDEX. 
 
 403 
 
 p. 
 
 Pendulum, 278 ; simple, 497 ; indicates 
 variation of gravity, 505 ; compensation, 
 510 ; a measure of force of gravity, 512 ; 
 
 variation of gravity, 505 ; compensation, 
 
 Jro ; a measure of force of gravity, 512 ; 
 ength of, 516. 672 ; compensation, 673 ; 
 
 Harrison's, 674 ; mercurial, 675 
 
 Penetrability, 25 ; examples of, 26. 
 
 Peron's table of human strength, 632. 
 
 Perpetual motion, 648. 
 
 Petrifaction, 84. 
 
 Piano forte, Erard's, 42,8. 
 
 Pile-engine, 655. 
 
 Pinion, mode of cutting, 682. 
 
 Plane, inclined, 469. 472. 475 ; self-acting, 
 476. 
 
 Planets, 360. 
 
 Planing machine, 660. 
 
 Plumb line, 224. 
 
 Pores, 73. 77, 78. 
 
 Porosity, examples of, 79, 80. 85, 86. 
 
 Power, moments of, 393 ; gained at ex- 
 pense of time, 399 > animal, 621 ; horse, 
 629 ; water, 636 ; wind. 637 ; steam, 638. 
 
 Printing, old method of, 702. 
 
 Printing machine, single, 706 ; double, 
 707 ; Applegath and Cow per 's double, 
 708; Times, 711 ; Marinoni's, 712. 
 
 Printing press, 701 ; modern, 704. 
 
 Projectiles, 254. 
 
 Pugilism, 207. 
 
 Pulley, 455 ; movable, 460 ; Smeaton's 
 and White's, 465. 
 
 R. 
 
 Reaction, 201 ; example of, zoo., 
 
 Rectangular lever, 423. 
 
 Regulators, 525. 
 
 Repose, 568. 
 
 Resistance of air, 589. 
 
 Resistance of fluids, 585. 
 
 Resisting force, 559. 
 
 Resolution of force, examples of, 170. 
 
 Retarded motion, 252. 
 
 Rollers, 575 ; inking, 703. 
 
 Rolling and punching mill, 539. 
 
 Ropes, 453. 
 
 Salt of silver dissolved, 56. 
 
 Sand-glass, 669. 
 
 Saw mill, 661. 
 
 Screw, 484 ; application of, 490 ; mode of 
 
 cutting, 491 ; Hunter's, 493 ; micro. 
 
 meter, 494 ; endless, 495 ; adjusting, 
 
 556. 
 
 Screw cutting engine, 662. 
 Screw press, 537. 
 Sheaves, 454. 
 Simple machine, 404. 
 Sledge, 574. 
 Sledge hammer, 657. 
 Soap bubble, its thinness, 41. 
 Solder, effect of, 370. 
 
 Solid state, 14. 
 
 Spade labour, 627. 
 
 Spider's web, 53. 
 
 Stability, 279 ; of vehicle, 284 ; of table, 
 
 286. 
 
 Stamping engine, 656. 
 Steam horse, 635. 
 Steam power, 038. 
 Steel -yard, 416. 
 Strength, 590 ; of timber, 598; of cordage, 
 
 599 ; of columns, 6oa ; Peron's table of 
 
 human, 632. 
 
 Strength of columns, 602. 
 Striking apparatus of clock, 700. 
 Strychnine dissolved, 55. 
 Sugar dissolved, 57. 
 Suspension, 508. 
 Sun dial, 666. 
 Swing, curious property of, 515. 
 
 T. 
 
 Teeth, 445 ; formation of, 446. 
 
 Teeth of wheels. 663. 
 
 Telescope joint, 555. 
 
 Tenacity, 137 ; of metals, 138 ; of fibrous 
 
 texture, 139. 
 
 Timber, strength of, 598. 
 Time, 260. 
 
 Times printing machine, 709. 
 Torsion, elasticity of, 106. 
 Treadmills, 437. 
 Tredgold's table of strength of beams, 
 
 6ia ; estimate of horse labour, 633. 
 Trunnions, 553. 
 
 U. 
 
 Universal joint, $46. 
 
 V. 
 
 Velocity, 163. 
 Vertical direction, 225. 
 Vis-inertias, 113. 
 
 Watches, 698. 
 
 Water power, 636. 
 
 Wedge, 477 ; theory of, 479 ; use of, 481. 
 
 Weighing machine, 426. 
 
 Weight, moments of, 393. 
 
 Welding, 134. 
 
 Wheel, 432 ; crown, 449 ; bevelled, 450 ; 
 
 balance, 517 ; fly, 530 ; position of fly, 
 
 540 ; use of, 575 ; mode of cutting, 682 , 
 
 balance, 686. 
 Whirling table, 316. 
 Windlass, 435. 
 Wind power, 637. 
 Wire in embroidery, 44. 
 Wollaston, his wire, 38 ; its minuteness, 
 
 39- 
 
 TH E END. 
 
LONDON 
 
 PJ115TEU BY SPOTTISWOODK AND CO. 
 XEW-STHEET SQUARE 
 
 
THE UNIVERSITY OF CALIFORNIA LIBRARY