ELEMENTS 
 
 OF 
 
 NATURAL PHILOSOPHY, 
 
 INCLUDING 
 
 MECHANICS AND HYDROSTATICS, 
 
 BY 
 
 JOHN, LESLIE, ESQ. 
 
 / *^<^-~' 
 
 PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF 
 
 EDINBURGH, AND CORRESPONDING MEMBER OF THE 
 ROYAL INSTITUTE OF FRANCE. 
 
 SECOND EDITION, CORRECTED AND ENLARGED . 
 
 EDINBURGH : 
 
 PUBLISHED BY 
 
 OLIVER & BOYD, TWEEDDALE COURT, AND 
 GEO. B. WHITTAKER LONDON. 
 
 1829. 
 
I 
 
 i 
 
 
 ADVERTISEMENT. 
 
 THIS volume is the first of a work designed 
 to exhibit a Comprehensive View of the Prin- 
 ciples of Natural Philosophy. Though com- 
 piled chiefly for the use of my Students, it is 
 written, I hope, with sufficient expansion, to 
 suit the taste of the public at large. I have 
 endeavoured to render it as elementary as 
 possible, without departing from the accuracy 
 of science. No previous instruction is re- 
 quired, except an acquaintance with the sim- 
 plest rudiments of Geometry and Algebra ; 
 but I would earnestly recommend the study 
 of Geometrical Analysis to all those who 
 aspire to the higher attainments. 
 
 The present volume may be considered by 
 itself as in a great measure complete ; and 
 
IV ADVERTISEMENT. 
 
 two more will conclude my plan. I intend 
 to annex copious illustrations, containing the 
 more difficult demonstrations, historical no- 
 tices and references to authors, with a cor- 
 rect set of the most useful tables. But I 
 must defer all these additions till the next 
 volume shall appear. 
 
 I have in this Edition carefully revised the 
 whole, made several important additions, and 
 given a copious table of contents. 
 
 I had designed the Second Volume of this 
 Work to appear at the same time ; but have 
 since thought it better to wait for the results 
 of a series of experiments projected on the 
 Constitution and Power of Steam. I trust, 
 however, in being enabled very soon to dis- 
 charge my promise to the public. 
 
 COATES, FlFESHIRE,} 
 
 28th October 1828. ( 
 
CONTENTS. 
 
 INTRODUCTION. 
 
 Origin of Physical Inquiries, ix. Conducted by Observation 
 vr Experiment, x. Distinction between these tico modes of In- 
 vestigation, xi. Analytical Process, xii. Accelerated by Inci- 
 dental Causes, xiii. Geometry, an important instrument in pro- 
 moting Physical Research, xiv. Advantage of a lively fancy in 
 tracing the Operations of Nature, xv. Danger of indulging Hy- 
 potheses, xvi. Science dawned in the East, but rose into splen- 
 dour among the Greeks, xvii. The general tenets of Tholes and 
 his successors, xviii. Anaxagoras, xix. Pythagoras and his 
 Doctrines, xx-xxi. Empedocles, Xenophanes and Leucippus, 
 xxii. Democritus and his Atomic Theory, xxiii. Philosophy 
 of Socrates, xxiv. Plato, the Founder of Geometrical Analysis, 
 Aristotle and his Philosophical Doctrines, xxv-xxviii. Esta- 
 blishment of the Alexandrian School, and the brilliant Galaxy 
 of Euclid, Apollonius, Pappus, and Diophantus, xxix. Voyages 
 for Geographical Discovery by Nearchus, Enthymenes, and Py- 
 theas, Archimedes and his fine discoveries in Geometry, Me- 
 chanics and Hydrostatics, xxxi-xxxii. Advances made in As- 
 tronomy by Aristarchus, Eratosthenes, and especially Hippar. 
 chus, xxxiii. Astronomical Observations and Discoveries of 
 Ptolemy, xxxiv. Mechanical Improvements of Ctesebius and 
 Hero, xxxv. Progress of the Arabians in Mathematics and As- 
 tronomy, xxxvi-xxxvii. Learning cherished by the Roman See 
 during the middle ages, xxxviii. Beneficial influence of the Cru~ 
 sades, xxxix. Invention of Paper and Spectacles, xl. The di- 
 
VI CONTENTS. 
 
 rective property of the Magnet imported from the East, Gun- 
 powder was likewise borrowed from the East, though its explosive 
 power was probably discovered by Schwartz, and was soon succeed- 
 ed by the invention of Printing, xli. Institution of Universities, 
 and the studies there pursued, xlii. The Medici Family, magni- 
 ficent patrons of learning, xliii. Revival of Science by Coperni- 
 cus, Purbach, Da Vinci, Stevinus, and Galileo, xlv. Sublime 
 discoveries of Kepler, and noble invention of Logarithms by 
 Napier, xlvi. Gilbert, the father of Experimental Science in 
 England, xlvii. Optical discoveries of Kepler, Snellius and Des 
 Cartes, xlviii. Invention of the Thermometer and Barometer in 
 Italy ; while Guricke discovers and constructs the air-pump in 
 Germany, xlix. Progress of the Higher Geometry and Mecha~ 
 nics, with the Theory of Centrifugal Force, founded by Huy- 
 gens, 1. Transcendent discoveries of Newton, li. Advances of 
 the Fluxionary Calculus, and Bradley 's nice discovery of the 
 Aberration of Light, lii. Mensuration of a degree of the Meri- 
 dian, and the investigations of Gauss, Lagrange, and Laplace, 
 liii. Discovery of Achromatic Glasses by Dolland, and of the 
 Polarity of Light by Malus, liv. Improvements in the Theory 
 of Heat, and the Laws of Electricity, Iv. Voltas formation of the 
 Galvanic Pile, and Oersted's discovery of Electro-Magnetism^ 
 Ivi. Concluding Observations on the rashness of framing Hy- 
 potheses, Quotation from the famous Pascal, Ivii.-lviii. 
 
 Natural Philosophy defined, 1. Process of investigation by 
 Observation or Experiment, 2. Division of the Course into 
 twelve heads ; enumeration of these, 4 and 5. 
 
 I. SOMATOLOGY. 
 
 Subdivision of the properties of bodies into Essential and Con- 
 tingent, 6. Character of Impenetrability, 7; elucidated by six 
 experiments, 8. Divisibility illustrated by a variety of curious 
 
CONTENTS. , Vll 
 
 facts, such as the thinness of pellicles, the tenuity of threads, the 
 extension of gold-leaf, minuteness of powders, and the evanescence 
 of animalcules, 9-16. Porosity shown by the transmission of air, 
 water, or mercury, through the substance of a body, 17. Insensi- 
 ble perspiration, 18. Bodies permeable to one kind of fluid though 
 not to another, 1 9. Measure of the diaphanous quality of air, 20. 
 Gold-leaf translucid, 21. Wood, paper, and many other substan- 
 ces sensibly diaphanous, 22. The property of contraction evinced 
 by a variety of experiments, 23-25. Condensation of wood under 
 enormous pressure, 26-27. Contraction produced by chemical 
 union, 28. Dilatation occasioned by the absorption of moisture, 
 29. Effect of wetting a cord, 30. The contingent properties of 
 Mobility and Ponderability illustrated, 31. Farther elucidation of 
 ponderability, 32. Phsenomena of Cohesion, 33. General con- 
 clusion respecting the transmutation of bodies, 34. Analysis of 
 the leading principles of the Boscovichian Theory, 35-38. 
 
 II. STATICS. 
 
 Origin of our ideas of Power and Force, 39. Experimental in- 
 vestigation of the conditions of Equilibrium, 40 and 41. Action 
 of an oblique force proportional to the cosine of its inclination, 42. 
 Hence the Parallelogram of Forces, 43 ; modifications of that 
 property, 44. Triangle of Forces, 45. Theorem of the direc- 
 tions of three balancing forces, 46 and 47. Reduction of oblique 
 forces by co-ordinates, 48. Parallel forces applied to different points 
 of a line fixed at both ends, 49. The several properties of those strains 
 or thrusts, comprising the Theory of Arches, 50 and 51. Parallel 
 forces exert the same action on a body as their compound, passing 
 through the Centre of Gravity, 52-55 ; essential property of this 
 centre, 56. Conditions of stability, 57 and 58. Methods of deter- 
 mining geometrically and practically the position of the centre of 
 gravity, 59 and 60 ; its remarkable properties, 61 and 62. Cen- 
 tre of the pressure at three points in a plane, 63. Forces applied 
 
Vlll CONTENTS. 
 
 along a flexible chain, and theory of the Catenarian Suspension, 
 64 ; this theory exemplified, 65 and 66. Approximate calcula- 
 tions, 67. Property of the Lever deduced from the composition 
 of forces, 68 ; inferred from other considerations, 69. Theory of 
 the Inclined Plane, and the original demonstration of Stevinus, 
 70 and 71. Theory of the Pulley derived from the composition of 
 Forces, 72. 
 
 III. PHORONOMICS. 
 
 Definitions, 73. Laws of motion grounded on Experimental 
 Induction, 74 and 75 ; exposition of those laws, and description 
 of Atwood's machine, 75. Nature of Inertia, 77 ; its existence 
 betrayed in various incidents of ordinary life, 78. General pro- 
 perties of uniform, accelerated, and retarded motion, 79. Theo- 
 rems which connect force, velocity, and space, 80. Properties of 
 motion uniformly accelerated or retarded, 81 and 82 ; exemplified 
 by Atwood's machine, 83 and 84. Effect of deflexion on the ce- 
 lerity of a body, 85. Descent of a body along an inclined plane, 
 and over a succession of planes, 86 and 87. Beautiful property of 
 the descent on the chord of a vertical circle, 88. Motion of a 
 body urged by a force directly as its distance from a given point, 
 89. Time of description by such a graduated force, 90 ; hence 
 the oscillations of a Pendulum in small circular arcs, 91. In- 
 fluence of the length of a pendulum, 92. Retardation from the 
 extent of the arc of description, 93. Time of descent the same 
 through every arc of a Cycloid, 94. The Cycloid likewise the 
 curve of swiftest descent, 95 and 96. Internal oscillations pervade 
 the economy of Nature, 97. Theory of compound pendulums and 
 determination of the Centre of Oscillation, 98 and 99 ; geometrical 
 properties of that centre, 100 and 101. Theory of the Metronome 
 and of Rocking Stones, 102. 
 
 Descent of a body urged by a force inversely as its distance from 
 a given centre, 103 and 104. Case of a body falling by a force di- 
 
CONTENTS. IX 
 
 rectly as its distance from the centre, 105. Those properties ex- 
 emplified in reference to the gravitation of our Earth, 106-8. 
 
 Theorems concerning Centrifugal Force, 109 and 110; expli- 
 cation of various phenomena depending on that force, 111. Coni- 
 cal Pendulum, 112; its property investigated, 113. Governor of 
 the Steam-Engine, 114. Projectiles, and Parabolic Theory, 115 
 and 1 16 ; exemplification of it, and its deviation from practice, 
 117. General theory of Central Forces, 118. Investigation of the 
 Law of Universal Gravitation, 119. Gradation of force which 
 would produce a revolution about the centre of an ellipse, 120 ; 
 transformation of the ellipse into a straight line, by the extension of 
 its eccentricity, 121. Investigation of the third law of Kepler, 
 122. Consequences of the mutations of the elliptical orbits, 123. 
 Conservative principles of the universe, with reflections on the 
 nearly circular orbits of the planets, 124 and 125. The theory of 
 projectiles, or ballistics comprehended under the general system of 
 centripetal forces, 126 and 127. Deviation of a falling body from 
 the perpendicular, 128. Descent of a cannon ball fired directly 
 upwards, 129. Motion of falling bodies considered as only the 
 extreme case of that performed in elliptical trajectories, 130 and 
 131. If any particles move in straight lines, their centre of gravity 
 will describe likewise an uniform rectilineal path, 132 and 133. 
 Principle of Virtual Velocities, 134. Limits of the position of the 
 Centre of Gravity, and property of the Conservation of Living 
 Forces, 135. Momentum of bodies, 1 36. Communication of mo- 
 tion never instantaneous, 137. The distinction of impinging bo- 
 dies into Resilient and Coalescent, more appropriate than that of 
 Elastic and Non-elastic, 138. Collision of coalescent bodies, 139. 
 Collision of resilient bodies, 140. Exemplification of the principle 
 of the Conservation of Living Forces, 141. Farther illustration of 
 collision, 142. Impulsion conveyed through several balls, 143. 
 Gradation which produces the maximum effect, 144. This pro- 
 perty exemplified, 1 45. Curious result of an inverted progression, 
 and the practical consequence deduciblc from it, 146. Oblique col- 
 lision, 147. Variable measure of resiliency, 148. Origin of the 
 
X CONTENTS. 
 
 Centre of Percussion, 149. Method of finding its position, 150. 
 Centre of Conversion, or of Spontaneous Rotation, 151. The 
 progressive motion of any system not affected by its rotation, 152 ; 
 the property exemplified, 153 ; applied to the rotation of the pla- 
 nets, 154. Centre of Gyration, 155. Momentum of Rotation, 
 157 ; influence of the position of that centre on the rolling of bodies 
 down an inclined plane, 158 ; the principle illustrated experimen- 
 tally by the rolling of loaded and hollow cylinders, 159. Investi- 
 gation of the theory of progressive motion conjoined with that of 
 revolution, 160-163. The three principal axes of rotation, 164 ; 
 properties of those axes, 165. 
 
 MECHANICS. 
 
 Origin of Mechanics, and examination of the elements of Ma- 
 chines, 166.* Concentrator of Force, 167 ; its application exem- 
 plified, 168; simple estimation of its effects, 169; rigorous for- 
 mulas for computing them, 170 ; exemplification, 171 ; diversified 
 action of the Concentrator, 172 ; application of it to practice, 173 ; 
 formula for computing its operation, abridged and exemplified, 
 174 and 175. Practical utility of the Concentrator in a variety of 
 cases, and its modification in the Fly, 176. Machine of oblique 
 action, 177 ; composed either of cords or inflexible rods, 178. 
 The five ordinary mechanical powers, 1 79. Theory of the Lever 
 as derived from the principle of virtual velocities, ] 80 ; and from 
 the property of the centre of gravity, 181 ; three kinds of levers, 
 182; their application, 183. Universal and compound lever, 184. 
 Steel Yard and Common Balance, 185. False Balance, 186. 
 Proper construction of the balance, 187. Adjustment and correc- 
 tion of the balan ce, 1 88- 1 90. Wheel and axle, 191. Wheel and 
 pinion and Double capstan, 192. Inclined plane, 193; its pro- 
 perty deduced from the consideration of virtual velocities, 194. 
 The Screw, and its modifications, 195. The Wedge, 196. The 
 Pulley, 197. The purchase deduced from virtual velocities, 198. 
 
CONTENTS. XI 
 
 Different combinations of pulleys, 199 ; form of the teeth of wheels, 
 200 ; construction of the Trundle, 201 ; modification of the forms 
 of teeth and pinions, 202 ; cogs of wheels and racks, 203 ; figu- 
 ring of a templet or pattern tooth, 204 ; accurate construction, 205 ; 
 geometrical approximation, 206 ; conditions of the number of 
 teeth in the wheel and pinion, 207. Object of Heart wheels, 208. 
 Universal joint, sun and planet wheel, and parallel motion, 209. 
 Efficacy of machines, 210. Maximum effect in the wheel and 
 axle, 211; and in the inclined plane, 212 and 213. Friction, 214. 
 Angle of repose, 215; practical observations, 216 and 217. Ge- 
 neral abstract of Coulomb's experiments on friction, 218 and 219. 
 New theory of friction, 220-223 ; its application to a variety of 
 facts, 224 ; influence of time in augmenting contact and friction, 
 225 and 226 ; explication of some remarkable effects, 227 ; friction 
 of pivots, 228 ; modified attrition of soft substances, 229. Ob- 
 struction from Rolling, 230 ; its measure determined, 23 1 ; re- 
 tardation of larger wheels, 232. Theory of friction wheels, 233 ; 
 applied to wheel-carriages, 234-236. Accurate observations, dis- 
 tinguishing attrition and adhesion, much wanted, 237 ; effects of 
 celerity in the draught of a carriage, 238. Stiffness of ropes, 239 ; 
 ascertained by the experiments of Coulomb, 240. Advantage of 
 friction in certain cases, 241. Property of the repeated coils of a 
 cord, 242. 
 
 Strength of materials, 243. Longitudinal tension, 244. In- 
 vestigation of the modulus of elasticity , 245 ; its measure in stones 
 and several kinds of timber, 246 ; tabular view of the absolute co- 
 hesion of different woods, 247 ; cohesion of hemp and metallic 
 rods, 248 and 249. Longitudinal compression, 250 ; its quantity 
 in metals, stones, and timber, 251 ; strength diminished by the 
 height, 252 ; best form of a column, 253, Transverse pressure, 
 254. Theory of Galileo, and its application to the structure of 
 herbaceous plants and animal bones, 255 and 256. Experiments 
 of Buffon on the strength of oaken beams, 257. Rennie's experi- 
 ments on bars of cast iron, 258 ; problem to cut a tree into a beam 
 of the greatest strength, 259 ; rigorous investigation of a transverse 
 
Xll CONTENTS. 
 
 strain, 260. Elastic curve, 26 1 ; estimate of the quantity of de- 
 pression, 262 and 263; variation of effect, 264; the general for- 
 mula exemplified, 265. Modulus of elasticity deduced from the 
 depression or flexure occasioned by a weight, 266 ; exemplifica- 
 tions, 267 and 268 ; quantity of depression which precedes frac- 
 ture, 269 and 270 ; strength of a hollow cylinder, 271. Nature 
 of torsion, 272 ; its quantity investigated, 273 ; exemplification, 
 and Coulomb's exquisite balance, 274. 
 
 Animal power, 275 ; diminished by prolonged exertion, 276. 
 Mode of estimating the effects of labour, and comparison of the 
 different ways of exerting human force, 277-280 ; the power of 
 animal traction reduced by acceleration, 281 ; exemplified in la- 
 bourers and horses, 282-284 ; power of the ox, the ass, the mule, 
 and the elephant, 285 ; exertion of the camel, the dromedary, and 
 the rein-deer, 286 ; action of the lama, and paco, of the dog and 
 the goat, 287 and 288. 
 
 III. HYDROSTATICS. 
 
 Nature of fluidity, 289 and 290. Exists in different degrees, 
 291. Theory of its constitution, 292; fundamental property of 
 fluidity investigated, 293-295 ; equal pressure of a fluid in every 
 direction elucidated by experiment, 296. Hydrostatical Paradox 
 of Pascal, derived also from the principle of virtual velocities, 297. 
 The weight of a fluid supported partly by the bottom and partly 
 by the inclined sides of the containing vessel, 298-301. Archi- 
 medean principle, that a body loses by immersion the weight of the 
 fluid displaced, 302 ; strict demonstration of it, 303. Hydrosta- 
 tic balance, 304. Methods of determining, in different circum- 
 stances, specific gravity, 305. Goniometer and hydrometer, 306. 
 Concise method of ascertaining the density of fluids, 307. Ar- 
 rangement of fluids having different densities, and hence the ex- 
 plication of some natural phenomena, 308 ; table of specific gra- 
 vities, 309 ; practical observations, 310. Inverted syphon and 
 
CONTENTS. Xlll 
 
 spirit level, 311. Lateral pressure of fluids, and form of embank- 
 ments, 312. Canal navigation, 313. Flood gates and their angle, 
 314. Internal strain of pipes conveying water, 315 ; formula for 
 computing it, 316; various exemplifications, 317 and 318; pres- 
 sure of water, internal or external, against a circular wall, 319. 
 Condition of a floating body ; its centre of buoyancy, line of sup- 
 port, and stability of flotation, 320 and 321 ; the floating of a 
 sphere, and of a spherical segment, 322 ; of an homogeneous sphe- 
 roid, 323 ; closer investigation of the conditions of flotation, 324. 
 Bouguer's theory of the metacentre, 325 ; application of it to an 
 homogeneous parallelepiped, 326 ; practical consequences, 327. 
 A cylinder will, according to its density, and the ratio of its diameter 
 altitude, exhibit all the three features of a floating body ; indiffe- 
 rence, instability, or permanence of equilibrium, 328-331 ; floating 
 of parabolic conoid, 332 ; application of the several formula to ex- 
 plain the oversetting of icebergs, 333 and 334) ; general investiga- 
 tion of the limits of stability, 335 ; theorem for stable flotation, 
 336. Method of computing the height of the metacentre, 337 ; 
 corrections to be made, 338 ; practical application to ships, 339 ; 
 centre of buoyancy found by gauging the immersed portion of the 
 hull, 340 ; position of the metacentre discovered by observation, 
 341 and 342. Heaving, rolling, and pitching of vessels, 343 ; in- 
 vestigation of the times of these several oscillations, 344 ; exem- 
 plification, 345 and 346. Theory of the stowage of ship's cargo, 
 347 ; oscillation by pitching calculated and applied, 348 and 349 ; 
 expedients for obtaining buoyancy, 350. Account of Bakker's 
 camel, 351 and 352. 
 
 Capillary action, 353 ; investigation of its true theory, 354 and 
 355. Ascent of water between parallel glass planes and narrow 
 tubes determined, 356 ; the suspension of the liquid depends en- 
 tirely on the smallness of the upper orifice, 357. Modifications of 
 capillary action, 358 ; the depression of mercury in narrow tubes 
 derived from the same principle, 359. Capillary action extends to 
 all substances broken with fissure, or divided by interstices, 360. 
 Error of a leading experiment of Hales, 361. Remarkable appli- 
 
XIV CONTENTS. 
 
 cation of the Aimomeler, 362. Theory of the ascent of water 
 through different soils, 363. Effect of the mutual cohesion of the 
 particles of a fluid in modifying capillary action, 364. Illustration 
 of some natural phenomena, 365 and 366. Adhesion of a disc 
 of glass to different fluids, 367. Explication of some curious facts 
 in the economy of nature, 368. Weight of a drop of water, of al- 
 cohol, and of sulphuric acid, deduced from theory, and confirmed 
 by experiment, 369 and 370. 
 
 VII. HYDRODYNAMICS. 
 
 Defined, 371. Fundamental theorem for the motion of fluids, 
 372 ; correction required in practice, 373. Popular investigation, 
 374. Some similar conclusion derived from the principle of the 
 Conservation of Living Forces, 375. Projection of a fluid from dif- 
 ferent apertures in a prismatic vessel, 376 ; consequences drawn 
 from the theory, 377. A convenient modulus proposed for mea- 
 suring the discharge of water from a conduit, 379. Theory of 
 the Clepsydra or Water- Clock, 379. Investigation of the time 
 of the efflux of water through a hole in the bottom of a cylindrical 
 vessel, 380 and 381 ; modification of the formula, 382. Recoil 
 from the effort of the projection of water, and theory of Barker's 
 mill, 383. Action of centrifugal force in raising water, 384 ; in- 
 vestigation of the effects in different circumstances, 385-387. In- 
 vestigation of the curve, in every part of which an equal ascensional 
 effort is centrifugal force, 388 ; made by appearance of the voragi- 
 nous motion of water in a glass cylinder, 389. Theory of the ori- 
 gin of whirlpools, 390. Principle of Pitot's tube, 391. Effect of 
 the contraction of a current, 392 ; exemplification, 393. Augment- 
 ed flow from an adjutage, 394. Advantage of an eductive pipe, 
 395 ; influence of the shape of the pipe, 396. Lateral action of 
 running water pipe, 397 ; its practical application, and theory of 
 the Hungarian blowing machine, 398. Investigation of the resis- 
 tance of water in passing through pipes, 399 and 400 ; correction 
 
CONTENTS. XV 
 
 from experiment, 401. Formula for computing the quantity of 
 flow through pipes, 402 ; modifications of these formula, 403. 
 Flow of water in rectangular channels, 404 ; retardation caused by 
 inflexions of the pipe or conduit, 405. Use of air-vessels in a system 
 of pipes, 406. Rule for calculating the discharge of water through 
 simple and through compound pipes, 407. Account of the former 
 and of the present supply of water for Edinburgh, 408 and 409. 
 The annual fall of rain in any city, if carefully husbanded, would 
 be sufficient for the wants of the inhabitants, 410. Aqueducts and 
 skill of the Romans in practical Hydraulics, 41 1 ; the subject co- 
 piously illustrated by quotations from the Classics, 412. Instru- 
 ments for levelling used by the Romans, 413. Water distributed 
 from the castellum or reservoir to the different quarters of the city 
 commonly by means of leaden pipes, 414 ; quantity of discharge 
 regulated by the size of the calices or spouts, 415. Pipes of un- 
 glazed earthen ware occasionally used, the pores being closed by 
 washing the inside vritbfavilla or wood-ashes, 416. Enumeration 
 of the principal aqueducts of ancient Rome, 417 and 418; their 
 aggregate discharge in a day, must have amounted to 50 millions 
 of cubic feet of water, 419. Supply of modem Rome, Paris, and 
 London, 420. Simple, but ingenious method of conducting water 
 into the splendid cisterns of Constantinople, 421. The retardation 
 of water in pipes extended to the flow of rivers along their beds, 
 422. Method of computing the celerity of a current from its de- 
 clivity and mean hydraulic depth, 423 ; exemplification, 424. Di- 
 rect impulsion of a stream against an opposing surface, 425 ; easy 
 formula for calculating that force, 426. Explication of the action 
 of a stream on stones of different dimensions, 427. Theory of the 
 washing of mineral ores, and of the gradual formation of banks, 
 428. Comparison of the effects of human labour, with the action 
 of water : The power evolved by nature in the mere suspension 
 of clouds, exceeds 200,000 times the whole accumulated toil of 
 mortals, 429. What portion of that power could be rendered 
 available over the surface of the habitable earth : The collected 
 waterfalls in this island might supply a magazine of force not infe- 
 
XVI CONTENTS. 
 
 rior to the labour of our entire male population, 430. Calculation 
 of the power that could be drawn from an extended system of tide- 
 mills, 431. Action of a stream against the float-boards of a river- 
 mill, 432. Investigation of the oblique shock of a current, 433. 
 Inclination of a rudder to the keel, when it has the greatest power 
 in turning a ship, 434. Theory of the resistance to the motion of 
 an oblique plane through water, 435 ; considerable modification 
 required in practice, 436 ; table of the actual resistance experien- 
 ced at different angles, 437. Terminal velocity r , or the limit of 
 the acceleration of a body sinking or rising in water or air, 438 ; 
 the formula applied in the case of balls, cork, drops of rain, and 
 hailstones, 439. Resistance of fluids illustrated by the gradation of 
 the descent of a series of glass-beads, 440. Retardation observed 
 in the shallow and narrow parts of a canal, 441. Measure of 
 animal force expended in the traction at different velocities along a 
 canal, 442. Advantages resulting from the slow navigation of ca- 
 nals, 443. Vibratory impressions propagated through fluids, 444. 
 Newtonian theory of undulation, 445 ; wants rectification, 446. 
 External contour of waves determined, 447. Description of the 
 origin, and of the successive rise and subsidence of waves, 448. 
 
INTRODUCTION. 
 
 OUR curiosity is first awakened by the changes in- 
 cessantly passing around us. But experience soon 
 testifies the constancy and regularity of this varied 
 spectacle. The vast movements of the Universe are 
 found all to consist in the repetition of similar events. 
 It may for ever elude the power of human ingenuity, 
 to discover the more secret springs which connect the 
 links of the extended indissoluble chain. Yet, since 
 the most complex phenomenon is always the result 
 of a very few principles, the objects of Science are at- 
 tained by distinguishing and classing those elemen- 
 tary facts. Men will seldom rest satisfied, however, 
 with such moderate advances ; and have often, in 
 their hasty and rash attempts to penetrate the arcana 
 of Nature, suffered severe disappointment. The pro- 
 VOL, i. b 
 
X INTRODUCTION. 
 
 per business of philosophical inquiry, is to study care- 
 fully the appearances that successively emerge, and 
 trace their mutual relations. 
 
 ALL our knowledge of external objects being de- 
 rived through the medium of the senses, there are 
 only two ways of investigating physical facts, by 
 Observation or Experiment. Observation is con- 
 fined to the close investigation and attentive exami- 
 nation of the phenomena which arise in the course 
 of Nature ; but Experiment consists in a sort of ar- 
 tificial selection and combination of circumstances, 
 for the purpose of searching minutely after the dif- 
 ferent results. 
 
 The range of Observation is limited by the po- 
 sition of the spectator, who can seldom expect to 
 follow Nature through her winding and intricate 
 paths. Those observations are of the most value 
 which include the relations of time and space, and 
 derive greater nicety from their comprising a multi- 
 plied recurrence of the same events. Hence Astro- 
 nomy has attained much higher degree of perfec- 
 tion than the other physical sciences. 
 
INTRODUCTION. XI 
 
 Experiment is a more efficient mean than Ob- 
 servation, for exploring the secrets of Nature. It 
 requires no constant fatigue of watching, but comes 
 in a great measure under the control of the inquirer, 
 who may often at will either hasten or delay the ex- 
 pected event. Though the peculiar boast of modern 
 times, yet the method of proceeding by Experiment 
 was not wholly unknown to the ancients, who seem 
 to have concealed their notions of it under the veil 
 of allegory. PROTEUS * signified the mutable and 
 changing forms of material objects ; and the inquisi- 
 tive philosopher was counselled by the Poets to watch 
 that slippery Daemon when slumbering on the shore, 
 to bind him and compel the reluctant captive to re- 
 veal his secrets. This gives a lively picture of the 
 cautious but intrepid advances of the skilful experi- 
 
 novit nanique omnia vates, 
 
 Quse sint, quse fuerint, qua? mox ventura trahantur. 
 
 *##**#% 
 
 Hie tibi, nate, prius vinclis capiundus, ut omnem 
 Expediat morbi causam eventusque secundet. 
 Nam sine vi non ulla dabit pfaecepta, neque ilium 
 Orando flectes : vim duram et vincula capto 
 Tende ; doli circum haec demum frangentur inanes. 
 
 GEORG. iv. 392.-402. 
 
Xll INTRODUCTION. 
 
 menter. He tries to confine the working of Nature 
 he endeavours to distinguish the several principles 
 of action he seeks to concentrate the predominant 
 agent and labours to exclude as much as possible 
 every disturbing influence. By all these united pre- 
 cautions, a conclusion is obtained nearly unmixed 
 and not confused, as in the ordinary train of cir- 
 cumstances, by a variety of intermingled effects. The 
 operation of each distinct cause is hence severally de- 
 veloped. 
 
 THE main object, therefore, in all physical inquiries, 
 is, by an Analytical Process, to separate the various 
 effects which Nature has blended together. The 
 history of Astronomy, from the earliest times, affords 
 the finest examples of such successful induction. 
 The true Philosopher labours to reduce the number 
 of principles or ultimate facts. In proportion .as his 
 views expand, he will find the relations so disclosed, 
 converging invariably to a common centre. But he 
 should avoid indulging to excess the taste for sim- 
 plification. In remounting to the source of all power, 
 it is evident that he must at last reach some impassa- 
 ble limit. Prudence will direct him when he ought 
 
INTRODUCTION. Xlll 
 
 to stop, or await the discovery of some new instru- 
 ment, to assist his ulterior inquiries. 
 
 EVERY extension of the powers of calculation, 
 every addition to our stock of philosophical appara- 
 tus, every improvement in the mode of their con- 
 struction, is a prelude to the farther advances of 
 physical science. Even slight alterations in the 
 practice of the arts have sometimes led to the most 
 important theoretical conclusions. Several disco- 
 veries in science are sometimes invidiously referred 
 to mere fortuitous incidents. But the mixture of 
 chance in this pursuit should not detract from the 
 real merit of the invention. Such occurrences 
 would pass unheeded by the bulk of men ; and it is 
 the eye of genius alone that can seize every casual 
 glimpse, and descry the chain of consequences. 
 Bodies had in all ages been viewed with indifference 
 falling to the ground ; yet the accidental sight of an 
 apple, as it dropped from the tree, was sufficient to 
 awaken, in a pensive mood, that lofty train of reflec- 
 tions, which conducted NEWTON to the system of 
 Universal Attraction. 
 
XIV INTRODUCTION. 
 
 AFTER new facts have been disclosed, either by 
 strict observation, or delicate and careful experiment, 
 the process of Synthetical Deduction should com- 
 mence. But to pursue a train of conclusions suc- 
 cessfully, requires the exercise of sound judgment, 
 and the application of vigorous and instructed intel- 
 lect, under the guidance of a sober and cautious logic. 
 The most important instrument in forwarding this 
 operation is Geometry, to which, indeed, we are in- 
 debted for whatever is most valuable in Natural Phi- 
 losophy. By this powerful aid the nobler branches 
 of science have in modern times been carried to a 
 sublime elevation. 
 
 BUT the most rigorous application of Mathemati- 
 cal reasoning will not always succeed in unveiling the 
 secrets of Nature. The philosophical inquirer must 
 often content himself at first with seeking merely to 
 approximate the truth. Analogy, in such cases, may 
 serve as a tolerable guide to assist his steps. Some 
 gifted individuals advance, indeed, with a sagacity 
 which seldom fails. They shape their path according 
 as the light successively breaks in ; and here again 
 Geometry lends its penetrating aid. 
 
INTRODUCTION. XV 
 
 NOR should Hypotheses themselves, however 
 liable to abuse, be excluded absolutely from Natural 
 Philosophy. They will often suggest new mpdes of 
 research, and in this way produce beneficial effects. 
 They may always serve as preludes of inquiry ; but 
 they are generally too fallacious and enticing, to be 
 allowed with safety to gain arty durable possession of 
 the mind. 
 
 THE gift of a lively fancy ip an important requi- 
 site to every physical observer. This faculty has 
 accordingly appeared conspicuous in all the great 
 discoverers. The imagination of the Philosopher 
 differs from that of the Poet, only because it calls 
 forth less vivid images ; but it is equally creative, 
 and equally sentient to the flitting scenes of Nature. 
 It supplies the inquirer with fresh expedients, en- 
 ables him to multiply the points of attack, and sheds 
 lustre over the scenes which he contemplates. But 
 imagination requires to be restrained, by the exer- 
 cise of a sound judgment. The Philosopher pushes 
 his researches with ardour, yet is cautious and de- 
 liberate in drawing his conclusions. His attention 
 is arrested by the appearance of anomalous facts. 
 
XVI INTRODUCTION. 
 
 While he doubts and pauses, he derives courage, 
 amidst all his perplexity, from the near prospect of 
 starting some new principle. The detection of a 
 single error is a sure step to actual discovery. 
 
 SUCH is the only successful mode of interrogating 
 Nature. But Philosophers, in all ages, have betray- 
 ed extreme impatience, in submitting to a plan of 
 investigation, apparently so timid, slow, and labori- 
 ous. It was more agreeable to human indolence 
 and presumption, at once to frame hypotheses which 
 might, in imagination at least, connect the mass of 
 preconceived opinions. But such rash attempts lull- 
 ed curiosity asleep, and fatally arrested the progress 
 of genuine science. Hence that glimmering twi- 
 light which so long overspread the world. 
 
 SCIENCE first dawned in the genial climes of the 
 East ; but its warming rays were soon absorbed in 
 the cheerless fog of despotism. The body of know- 
 ledge which had been created by the efforts of un- 
 fettered genius, became the exclusive property of the 
 order of priesthood, who rendered it an engine of 
 power, subservient to the purposes of a gloomy and 
 
INTRODUCTION. XV11 
 
 debasing superstition. The discoveries of happier 
 times were entombed in silence and darkness. 
 
 A MORE auspicious morning at length arose. 
 Greece, though but a spot on the surface of our 
 globe, began her career of glory, and gave early to- 
 kens of those eternal benefits she was destined to con- 
 fer on the human race. Her sages gleaned instruc- 
 tion by visiting foreign lands, and the seats of an- 
 cient renown. They gathered the dying embers of 
 science, and rekindled them by the breath of their 
 genius. But, quickly emerging from a state of pu- 
 pillage, they displayed the riches of a lively fancy, 
 and all the resources of a fertile invention. 
 
 THALES, the founder of the Ionic Sect, having 
 spent a large patrimony and many years of his life 
 in foreign travel, transplanted into Greece the 
 Science of the Egyptian Priests. His successors, 
 ANAXIMANDER and ANAXIMANES, taught, with a 
 few modifications, the same doctrines. Their know- 
 ledge appears to have been superficial, yet was 
 it not the less aspiring. They indulged in cosmolo- 
 gical systems, which pretend to explain the origin 
 
XV111 INTRODUCTION. 
 
 and formation of all things. Such bold speculations 
 flattered human vanity, and charmed the imagination 
 by a glittering semblance of truth. Those early 
 sages held every substance whatever to be composed 
 of four distinct elements, Earth, Water, Air and 
 Fire, merely combined in various proportions. 
 Earth and Water they viewed as naturally ponde- 
 rous and inert, while they fancied Air and Fire, en- 
 dued with elastic virtue, to possess lightness and ac- 
 tivity. While the earthy matter settled towards the 
 centre of the universe, and the aqueous fluid rolled 
 along the surface of the globe ; the Air and Fire, 
 or -<Ether, soared aloft, the former occupying the 
 whole region below the moon, and the latter stream- 
 ing through the boundless extent of space. The same 
 pure lambent fluid, being collected into globular 
 masses, formed the groupcs of stars. But portions 
 of this divine essence descended, to communicate the 
 vital sparks, and animate the subjects below. Such 
 visions, so like the picturing of fantastic dreams, 
 were yet firmly believed in former ages ; they be- 
 came afterwards incorporated with the vulgar creed ; 
 and still deeply tincture the language of Poetry. 
 
INTRODUCTION. XIX 
 
 ANAXAGORAS, who continued the Ionian line, 
 rose to higher eminence. He relinquished a splen- 
 did fortune, and dedicated the best portion of his 
 life, to the pursuit of knowledge, under the Priests 
 of Thebes and the Magi of Persia ; and, on his re- 
 turn to Greece, he was induced, from the dread of 
 foreign conquest, to transfer the School of Ionia to 
 Athens. His eloquence and learning made a power- 
 ful impression in that intellectual city. But he drew 
 upon himself the odium of the illiberal, by broach- 
 ing some opinions at variance with the received te- 
 nets, and anticipating obscurely the modern disco- 
 veries. 
 
 THE island of Samos had now the honour of giving 
 birth to PYTHAGORAS, the first sage who took the 
 modest but auspicious name of Philosopher, or 
 Lover of Wisdom. Nature bestowed on him her 
 choicest gifts, and assiduous cultivation heightened 
 all those qualities which command the minds of 
 men. PYTHAGORAS sacrificed his ample fortune, 
 and devoted the vigour of his days, to the acquisition 
 of knowledge. He was admitted into the sacred 
 college of Memphis ; he resided some time in Phce- 
 
XX INTRODUCTION. 
 
 nicia, visited Persia, and pursued his eastern journey 
 to the banks of the Indus. After an absence of near 
 thirty years, he returned home, fraught with various 
 information, and was viewed with awe and reverence 
 by his assembled countrymen at the Olympic Games. 
 He began to give instructions in his native island, 
 but soon removed to the Grecian colony planted at 
 Tarentum on the coast of Calabria. In that vo- 
 luptuous city, he collected numerous pupils, and 
 founded the powerful Italian Sect, which subsisted 
 after his death during the course of several centu- 
 ries. 
 
 NOT to shock the prejudices of his countrymen, 
 PYTHAGORAS judged it prudent to separate the doc- 
 trines he taught into two distinct sorts the Exoteric 
 and the Esoteric. The former consisted in discour- 
 ses addressed to the people in the temples and other 
 places of public resort, and calculated to reform their 
 habits and dispositions ; the latter comprised those 
 recondite principles communicated only to the very 
 few disciples who, after a very long and severe pro- 
 bation, were esteemed fit and safe depositaries of 
 such momentous truths. He strenuously urged the 
 
INTRODUCTION. XXI 
 
 study of Mathematics, especially those parts which 
 are concerned about number and proportion. His 
 imagination being full of numerical relations, he 
 founded the theory of Music, which he cultivated 
 both as an art and a science. But he transferred 
 his ideas of music to the harmony of the celestial 
 motions. Rising to the sublime conception of the 
 true system of the Universe, he seems to have veiled 
 that noble discovery in a fine allegory. Under the 
 symbol of APOLLO playing on the lyre, he taught 
 his chosen disciples, that all the planets, including 
 our Earth, are inhabited worlds, which revolve 
 about the Sun as their common centre j and he 
 farther maintained, that those bodies, while they 
 circle round the great luminary, perform a most 
 harmonious concert, though such ravishing and 
 heavenly sounds are lost to our gross ears, and 
 drowned amidst the jarring noise which prevails 
 here below. 
 
 THE most distinguished of the successors of 
 PYTHAGORAS was EMPEDOCLES, born at Agrigentum 
 in Sicily. He conceived all bodies to consist of in- 
 finitely small corpuscles in a state of perpetual mo- 
 
XX11 INTRODUCTION. 
 
 tion, held together by the inherent force of matter, 
 yet kept separate by the antagonism of its action. 
 He thus made an important step in physical science, 
 by introducing the play of two opposite principles, 
 termed in his figurative language friendship and 
 strife, very similar to the forces of attraction and 
 repulsion, which perform such important service in 
 the philosophy of modern times. 
 
 XENOPHANES transplanted most of the tenets of 
 the Pythagoreans into the small sect which he found- 
 ed at the town of Elea in Campania. He has the 
 merit of being the first to advance the important 
 principle in Geology, that the exterior crust of our 
 globe was once in a fluid state, and that the fossil 
 shells and other marine exuviae discovered in the 
 bowels of the earth, and on the tops of the highest 
 mountains, had, at some remote period, been form- 
 ed under the waters of the ocean. 
 
 LEUCIPPUS, who sprung from that school, assert- 
 ed not only the doctrine of atoms, but maintained 
 the existence of a separate plenum, and anticipated 
 the influence of centrifugal force. These doctrines 
 
INTRODUCTION. XXH1 
 
 were extended and improved by his disciple DEMO- 
 CRITUS, who flourished during the time of the Pelo- 
 ponnesian war, and may be considered as one of the 
 first geniuses that Greece, so rich in talent, ever 
 produced. To procure the higher degrees of know- 
 ledge, he visited Egypt, conversed with the Magi of 
 Babylon, and appears to have pursued his journey 
 eastwards as far as the Ganges. The expence in- 
 curred in these distant travels consumed the whole 
 of his patrimonial inheritance, and DEMOCRITUS, on 
 his return, was content to occupy a small garden 
 near Abdera, where he passed a prolonged life of 
 parsimony and seclusion, wholly intent on making 
 experiments, and investigating the operations of 
 Nature. He rectified many of the prevailing errors 
 in physics ; proved the existence of a plenum to be 
 incompatible with local motion ; rejected the notion 
 of levity \ that was attributed to the elements of Air 
 and Fire ; maintained that the weight of bodies is 
 always proportional to their mass or quantity of mat- 
 ter, and that they would hence fall in vacua with 
 equal celerities ; and he likewise entertained tolera- 
 bly correct general views concerning the properties 
 of Heat and Light. These tenets are expounded in 
 
INTRODUCTION. 
 
 the elegant poem of LUCRETIUS, who gratefully and 
 fervently styles his master in philosophy, pater et re- 
 rum inventor. 
 
 PHILOSOPHICAL sects had multiplied over Greece 
 when SOCRATES arose, to give a more useful impul- 
 sion to the human mind. Disgusted with the miser- 
 able sophisms and illusory dreams of pretenders in 
 science, he justly thought the first step towards real 
 knowledge was to discover the extent of our igno- 
 rance. His aim was not so much the elaborate 
 education of a Tew individuals, as the diffusion of 
 solid instruction among the body of Athenian youth. 
 Averse from idle speculations, he recommended that 
 kind of philosophy alone which is grounded on fact 
 and experiment. He was accustomed to pour un- 
 measured contempt on every species of hypocrisy 
 and pretension. But his intrepid conduct and high 
 celebrity created a host of secret enemies, and the 
 venerable sage at last fell a martyr to the cause of 
 truth and virtue. 
 
 PLATO, the most illustrious of all the disciples of 
 SOCRATES, after the murder of his revered master, 
 
INTRODUCTION. XXV 
 
 withdrew from the polluted city, and visited the seat 
 of the Pythagoreans at Tarentum, where he was ini- 
 tiated in the whole arcana of the Italic School. He 
 next placed himself under the tuition of THEODO- 
 RUS, an eminent mathematician of Cyrene in Lybia. 
 With a mind thus prepared, he travelled in disguise 
 over the whole of Egypt, carefully examining every 
 object, and gleaning all the information he could 
 reach. Being prevented by war from proceeding 
 farther, he returned home, and purchased the Grove 
 of Academus, in the immediate vicinity of Athens ; 
 in which sequestered spot, he delivered, under the 
 shade of spreading oriental planes, his eloquent and 
 impressive lessons, to youths of the first distinction, 
 who flocked from all parts, attracted by the fame of 
 his sublime genius. 
 
 The philosophy of PLATO was deeply imbued with 
 the mysticism of the Italic School. Yet to him we 
 are deeply indebted for the beautiful method of 
 Geometrical Analysis, which furnished so powerful 
 an instrument to direct the process of investigation. 
 This invention, in the hands of his successors in the 
 Academy, was continually extending the boundaries 
 
 VOL. i. c 
 
XXVI INTRODUCTION. 
 
 of Science. It led immediately to the discovery of 
 the Conic Sections, which, though cultivated for 
 ages merely as a fine speculation, suggested at last to 
 GALILEO the laws of motion, and aided KEPLER in 
 his search after the true orbits of the planets. Un- 
 fortunately, the soaring disposition of the Academics 
 spurned the study of material objects, and was apt 
 to exhale in airy sublimation, 
 
 ARISTOTLE, whose authority so long maintained 
 the most despotic sway in the learned world, was- 
 born at Stagira, on the confines of Thrace and Ma- 
 cedon, 385 years before the Christian sera. He 
 studied twenty years under the direction of PLATO, 
 on whose death he retired to Mysia and Mytelene, 
 till he was summoned by PHILIP of Macedon, to 
 superintend the education of the young Prince. 
 When ALEXANDER marched against Persia, his pre- 
 ceptor went to Athens, and opened the Peripatetic 
 School at a place called the Lyceum. But ARISTO- 
 TLE kept a regular correspondence with the Mace- 
 donian hero, who gave the most liberal encourage- 
 ment to his projected work on the History of Ani- 
 mals. The philosopher continued only twelve years 
 
INTRODUCTION. XXV11 
 
 more at Athens, and apprehending the tempest of a 
 persecution, prudently retired to Chalcis, where he 
 spent the last two years of his life. 
 
 THE very numerous writings of ARISTOTLE were 
 suffered to rot in a damp cellar, for an hundred 
 years after his death, and seem never to have been 
 much esteemed by the Greek and Roman authors. 
 No philosopher, either in ancient or modern times, 
 has taken such a wide range of disquisition ; and 
 yet his universal genius was marked by soundness 
 of judgment, precision of thought, and singular a- 
 cuteness, bordering occasionally in subtlety. Though 
 ambitious to maintain the character of an original 
 thinker, he endeavoured to ground all his philosophy 
 on the close observation of facts. Restraining the 
 flights of imagination, he confined his researches to 
 the circumstances of real life, and the actual consti- 
 tution of the universe. His conclusions were there- 
 fore of most value, whenever he could obtain accu- 
 rate information, on which to exercise his penetra- 
 ting sagacity. The Natural History of ARISTOTLE 
 must be deemed a wonderful production, for the 
 time in which it was composed. He was the found- 
 
XXV111 INTRODUCTION. 
 
 er of Comparative Anatomy ; and according to 
 CUVIER, a most competent judge, the divisions 
 which he adopted are still the best that could be 
 made. He was the first who distinguished nerves 
 from tendons ; he noticed the auditory and optic 
 nerves of the mole; he traced the optic and ol- 
 factory nerves in fishes ; and described with great 
 accuracy the process of the incubation of eggs, and 
 the developement of the chick. His Meteorology 
 sparkles with fine remarks and just conclusions ; 
 and even his Physics, and the book De Ccelo, con- 
 tain, amidst much idle and extravagant speculation, 
 some interesting doctrines, which might deserve to 
 be separated and abridged. His peculiar tenets like- 
 wise deserve notice, merely for their influence in the 
 History of Philosophy. We regret that ARISTOTLE 
 should have suffered his balanced mind to be drawn 
 aside, by the love of subtlety and the illusion of ge- 
 neral hypothesis, from the strict investigation of 
 facts. If the Peripatetics had cultivated Geometri- 
 cal Analysis with the same ardour as the followers 
 of PLATO, they would infallibly have made great 
 and successful progress in Natural Philosophy. 
 
INTRODUCTION. XXIX 
 
 AN event now took place, which mightily contri- 
 buted to the permanent extension of Mathematical 
 Science in all its branches. After the premature 
 death of ALEXANDER, his conquests had been shared 
 among the companions of his arms. Egypt fell by 
 lot to PTOLEMY, who, selecting for his residence the 
 city of Alexandria, occupied by a Grecian colony, 
 embellished it, and rendered it a distinguished seat 
 of learning. A magnificent edifice was erected, 
 styled the Museum, in which men of science, invi- 
 ted from all countries, were liberally entertained at 
 the public expense, and provided with books and in- 
 struments for the prosecution of their studies. It 
 was farther aggrandized by the munificence of his 
 successors, who founded a vast library, and raised a 
 spacious and well-furnished observatory. This royal 
 establishment survived all the changes of fortune 
 during nine hundred years, and conferred incalcula- 
 ble benefits on the human race. A succession of 
 the >ablest mathematicians shed lustre on the first 
 three centuries of the Alexandrian School. EUCLID 
 digested the Elements of Geometry into a System, 
 excellent at that period, though certainly not adapted 
 to the present state of the Science. APOLLONIUS 
 
XXX INTRODUCTION. 
 
 extended the Conic Sections, and improved Geome- 
 trical Analysis, in which he was likewise followed by 
 PAPPUS, and by DIOCLES and NICOMEDES, who 
 invented some of the higher curves. DIOPHANTUS 
 applied a similar investigation to arithmetical pro- 
 blems, and the few symbols he used may be consi- 
 dered as anticipations of that perfect system of cha- 
 racters, or pictured language, which modern Alge- 
 bra exhibits. 
 
 MEANWHILE our knowledge of the surface of the 
 globe was extended by the adventures of a bolder 
 navigation. The Indian Sea had been explored by 
 the small squadron of NEARCHUS, who attended the 
 expedition of ALEXANDER into the East. But the 
 republic of Marseilles, a Grecian colony settled in 
 the south of France, had the merit of fitting out the 
 first voyage of discovery. EUTHYMENES sailed to- 
 wards the equator, while PYTHIAS, an able astrono- 
 mer, shaped his course towards the north, entered 
 the Baltic, discovered Thule, and remarked the fea- 
 tures of the circumpolar climate. He noticed the 
 phenomena of tides, which are unknown along the 
 shores of the Mediterranean ; and on his return, he 
 
INTRODUCTION. 
 
 made an accurate observation of the obliquity of the 
 ecliptic, which was then 23 48', conformable to the 
 deductions of profound theory. 
 
 SICILY had the honour of giving birth to the most 
 inventive genius of all antiquity. ARCHIMEDES 
 showed from infancy a passion for science, and 
 having received what instruction his native city of 
 Syracuse could afford, he visited Alexandria, and 
 other seminaries abroad. After his return, he de- 
 voted himself entirely to the charms of abstract stu- 
 dy, and pursued his deep researches with the most 
 ardent and intense application. He gave unlimited 
 extent to the notation of numbers, and founded the 
 method of indivisibles, which led him to the finest 
 discoveries in Geometry. He assigned the quadra- 
 ture of the parabola, approximated to that of the 
 circle, and disclosed the fine relations which subsist 
 between the cylinder and its inscribed cone and 
 sphere. But ARCHIMEDES deserves to be regarded as 
 the first who really studied Natural Philosophy in the 
 right way. His advances were splendid and trium- 
 phant. He detected the fundamental principles of 
 Mechanics and Hydrostatics, and illumined those 
 
XXX11 INTRODUCTION. 
 
 branches of science by the torch of Geometry. He 
 pointed out the centre of gravity, and determined 
 its position in a variety of figures ; and he unfolded 
 the properties of floating bodies, and thus traced the 
 rudiments of naval architecture. He likewise re- 
 duced those principles into practice, and constructed 
 such powerful engines as enabled the valour of his 
 countrymen to resist for three years the whole 
 efforts of a Roman squadron and besieging army. 
 But perseverance and military discipline at last pre- 
 vailed, and one fatal night was Syracuse involved in 
 the horrors of assault. Amidst the general confu- 
 sion and carnage, an infuriated soldier entered the 
 apartment of ARCHIMEDES, and, regardless of his 
 calm occupation, massacred him on the spot, in the 
 75th year of the Philosopher's age, and 21 % before 
 CHRIST. 
 
 GEOMETRICAL science had now acquired some 
 form and consistency, and astronomy was extending 
 its domains. , ARISTARCHUS of Samos devised an 
 ingenious method of estimating the relative distances 
 of the sun and moon ; and though, with his imper- 
 fect instruments, he could obtain only vague results, 
 
INTRODUCTION. XXX111 
 
 yet were they sufficient to expand immensely our 
 conceptions of the solar system. ERASTOTHENES 
 observed, with precision, the obliquity of the eclip- 
 tic, and determined the circumference of the earth, 
 by measuring the intercepted arc of the meridian 
 between Alexandria and Syene in Upper Egypt. 
 But HIPPARCHUS was a genius of much higher or- 
 der. He found the exact length of the year, ascer- 
 tained the distance of the moon, and approximated 
 to that of the sun ; he distinguished the unequal in- 
 tervals between the equinoxes, and traced the pre- 
 cession of those points. This fine discovery suggest- 
 ed to him the scheme of ascertaining and registering 
 the positions of the principal fixed stars. But he 
 transferred the same method to terrestrial observa- 
 tions, and was the first who defined the places on 
 our globe by their latitude and longitude. Without 
 rejecting the inveterate axiom of antiquity, that an 
 uniform and circular motion was alone befitting 
 celestial bodies, he sought to explain the apparent 
 inequalities by the ingenious hypotheses of Eccen- 
 trics and Epicycles, which, being multiplied in the 
 sequel, fatally overloaded astronomical science. 
 
XXXIV INTRODUCTION. 
 
 PTOLEMY, who resided at Alexandria after Egypt 
 had become a Roman province, under the Emperors 
 ADRIAN and ANTONINUS, was one of the best and 
 most indefatigable observers that ever lived. Less 
 original than HIPPARCHUS, he laboured with equal 
 zeal to promote astronomy. He not only improved 
 every part of the science, but digested the multifa- 
 rious data into one great system. He discovered 
 the lunar evection, and celestial refraction. He like- 
 wise composed a general treatise of Geography, and 
 applied the theory of Projections, which he had in- 
 vented, to the construction of maps. The study of 
 spherical trigonometry was begun by HIPPARCHUS, 
 extended by THEODOSIUS and MENELAUS, but re- 
 duced to a practical form by PTOLEMY. 
 
 OTHER philosophers of the Alexandrian School 
 applied themselves to Mechanics. CTESEBIUS im- 
 proved the clepsydra, invented the pump, and con- 
 structed an engine for discharging arrows by the 
 force of condensed air. HERO not only formed the 
 crane, but contrived machines which acted from the 
 variable elasticity of air, as affected by the alternation 
 of heat and cold ; a principle which in after times led 
 
INTRODUCTION. XXXV 
 
 GALILEO and SANCTOHIO to the construction of our 
 Thermometers. 
 
 BUT the genius of Greece, which had been sink- 
 ing under oppression, at length evaporated in pole- 
 mical disputes. The Romans were now masters of 
 the world ; and perhaps no people deserved less the 
 favour of fortune or the gratitude of posterity. In 
 the whole range of their existence, they never made 
 a single step towards the advancement of science. 
 All the knowledge attained by them in arts or phi- 
 losophy, had been derived immediately from the 
 Greeks. Their education was entirely practical, 
 and calculated only to form orators, statesmen and 
 warriors. 
 
 ANOTHER race of men became lords of the as- 
 cendent. The Arabians, impelled by the enthu- 
 siasm of a new religion, spread the terror of their 
 arms in all directions. They subdued Egypt, Sy- 
 ria, and Persia, and compressed the limits of the 
 Eastern Empire. In the West, they occupied Spain, 
 and penetrating into the heart of France, they threat- 
 ened to extinguish the Christian name. But the 
 
XXXVI INTRODUCTION. 
 
 fervour of their zeal soon abated, and their schemes 
 of ambition were at length absorbed in the arts of 
 peace. The Arabians grew passionately attach- 
 ed to the science of the Greeks. They carefully 
 collected all the philosophical writings of that won- 
 derful people, and caused them to be translated into 
 their own language. A succession of enlightened 
 princes encouraged those efforts with unbounded 
 munificence. The Caliphs stored their palaces with 
 public libraries, and adorned them with splendid 
 observatories. 
 
 THE Arabians cultivated more especially Geome- 
 try and Astronomy ; but they likewise studied Bo- 
 tany and Chemistry. Less prone than the Greeks 
 to speculation, they directed their chief efforts to 
 practical science. They soon became most skilful 
 calculators, and accurate observers. They substitu- 
 ted the Sines instead of Chords in trigonometry, 
 and farther improved this important branch of science 
 by the introduction of Tangents, which, in allusion 
 to the art of Dialling, they termed Shadows. But 
 the greatest benefit which the Arabians rendered to 
 mankind, was the communication of the decimal no- 
 
INTRODUCTION. XXXV11 
 
 tation of numbers. With this beautiful, though very 
 simple contrivance, which they styled Indian, though 
 it be still unknown or unpractised in any part of 
 lower India, they seem not to have become acquaint- 
 ed before the end of the tenth century of our aera. 
 The use of the ten digits in arithmetic having pass- 
 ed over to the Moorish kingdom of Spain, was 
 thence transfused through the Christian nations of 
 Europe about the beginning of the fourteenth cen- 
 tury, though it was not generally adopted till near 
 two hundred years afterwards. 
 
 THOUGH the Arabians had little claim to the 
 character of original invention, they were assiduous 
 in collecting useful information from all parts. 
 They practised Brewing and Distillation, arts total- 
 ly unknown to the Greeks and Romans, but proba- 
 bly derived from the experience of the Tartarian 
 hordes. They invented other chemical processes, 
 and gave names, which are still preserved, to vessels 
 of certain forms. 
 
 MENTAL darkness, during this period, brooded 
 over the fairest regions of the Christian world. 
 
XXXV111 INTRODUCTION. 
 
 The wrecks of knowledge lay buried in the convents, 
 while the more active spirits expended all their 
 energy in violent sports or savage depredations. Yet 
 talent was not extinct in the middle ages, though it 
 unhappily ran to waste. Superstition encouraged 
 pilgrimages over Europe, and every convent opened 
 its hospitable gates indiscriminately to the weary 
 traveller. Rome was still the centre of the Chris- 
 tian commonwealth, and multitudes resorted from 
 all parts to the metropolitan city. The refinement 
 of Italy was, by this continual intercourse, partially 
 reflected to the remotest corners of Europe. 
 
 THE Crusades, undertaken against the Saracens 
 from the twelfth to the fourteenth centuries, though 
 stimulated by the wildest fanaticism and the mad- 
 dening passion for military exploits, may yet be re- 
 garded as the main cause of the renovation of the 
 human intellect. Those expensive armaments, ex- 
 hausting the fortunes of the haughty chiefs, contri- 
 buted to lighten the chains of feudal tyranny j and, 
 by giving a wider circulation to wealth, they gra- 
 dually raised into consideration that middle class of 
 men who constitute the bulwark of a free state. The 
 
JNTRODUCTION. XXXIX 
 
 Crusaders themselves, during the suspension of ho- 
 stilities, could not fail, from their intercourse with 
 the Saracens, who surpassed them so greatly in know- 
 ledge and refinement, to receive much important in- 
 formation both in arts and science. To this origin 
 many of the subsequent improvements, which advan- 
 ced the progress of society in Europe, may be dis- 
 tinctly traced, 
 
 BUT native genius was not inactive during the 
 middle ages. Some of the most valuable arts arose 
 in that benighted period. The curious process of 
 converting cotton into paper was invented about the 
 beginning of the eleventh century. Three centu- 
 ries afterwards, linen rags were manufactured into 
 a still better material, which, serving as a cheap and 
 convenient substitute for parchment or vellum, has 
 greatly promoted the practice of writing. The use 
 of letters was about this time farther assisted, by the 
 introduction of Spectacles, which SALVING DEGLI 
 ARMATI first constructed at Florence about the year 
 1285. Small spheres of crystal or glass had been 
 employed by the ancient engravers of gems, to aid 
 their sight - 9 but the transition from these globes to 
 
xl , INTRODUCTION. 
 
 mere convex lenses, though apparently trivial, led to 
 the most important consequences. 
 
 THOUGH the attractive power of the magnet was 
 known to the Greeks, they remained ignorant of its 
 more wonderful property, of pointing towards the 
 North. This directive power had perhaps been dis- 
 covered by the Chinese, and some intimation of it 
 seems to have been brought by the Crusaders from 
 the East. The magnet hence acquired the name 
 of loadstone or leading-stone, the first cpmpass in 
 Europe having been made near the close of the thir- 
 teenth century at Amalphi, near Genoa. This no- 
 ble invention give a prodigious spring to navigation 
 and commercial enterprise. 
 
 THE eastern nations had been long acquainted 
 with the deflagrating property of nitre or saltpetre ; 
 but when this wonderful substance was imported in- 
 to Europe by the Crusaders, it was confounded, from 
 its external appearance, with the natron or soda 
 brought from Egypt. More than two centuries 
 elapsed before its vast explosive force was observed. 
 This most important discovery was probably made 
 
INTRODUCTION. xli 
 
 by SCHWARTZ, a German monk, about the year 
 1382 : It has extended the empire of man over 
 Nature, by the gift of a new and tremendous power, 
 which, though commonly diverted to the work of 
 destruction, has yet rendered wars less rancorous 
 and sanguinary than before. 
 
 BUT a nobler trophy distinguished the same pe- 
 riod. The Romans had, for various purposes, used 
 metal stamps ; the Chinese employed carved blocks 
 of wood for impression ; but the modern art of print- 
 ing, by means of combined moveable types, was in- 
 vented about the middle of the fifteenth century. 
 The ingenuity and perseverance of GUTTENBERG 
 and SCHOEFFER, encouraged by the wealth of FAUST, 
 a burgess of Mentz, have conferred by far the great- 
 est benefit ever bestowed on the human race. In the 
 short space of thirty years, this invaluable art was 
 carried to its highest perfection. 
 
 SCHOEFFER likewise began the art of engraving, 
 and executed in 1491, for ARNDES, burgomaster of 
 Lubec, a series of the figures of plants and animals, 
 on wooden blocks. This emulation of genius ac- 
 
 VOL. i. d 
 
xlii INTRODUCTION. 
 
 corded with the state of society in Europe. The 
 taste for knowledge had been fast advancing. The 
 Romish clergy were anxious to promote learning, as 
 the sure means of aggrandizing their order. In all 
 the convents and cathedrals, they had opened schools ; 
 and part of the ample revenues of the Church was 
 dedicated to the gratuitous education of youth. 
 Other schools were established for communicating the 
 higher degrees of instruction, which was greatly fa- 
 cilitated, by the adoption of the Latin language as 
 the medium of intercourse over Christendom. These 
 seminaries, when they became so enlarged as to in- 
 clude all the branches of liberal knowledge, were 
 termed General Studies ; but afterwards, when 
 they had obtained the sanction of Papal bulls, and 
 the protection of legal privileges, they assumed the 
 title of Universities. An apprenticeship of seven 
 years, copied afterwards in the mechanical trades, 
 was required to complete the course of education, 
 consisting of the Trivium, followed by the Qua- 
 drivium. The tenets of ARISTOTLE were expound- 
 ed with indefatigable diligence ; but those opinions, 
 drawn from the Arabic, had been miserably corrupt- 
 ed by the severe process of a double translation. 
 
INTRODUCTION. \llh 
 
 The vigour of genius which, if better directed, 
 could have soared to sublime discovery, was consu- 
 med in idle disputations and unprofitable subtleties. 
 
 BUT a brighter prospect began to open. PETRARCH 
 resorted to the pure fountains of ancient learning, 
 and his warm enthusiasm and eloquent exhortations 
 made a very deep impression on the minds of his 
 contemporaries. The study of the Greek language 
 was gradually introduced into Italy by the frequent 
 embassies dispatched from Constantinople, to im- 
 plore the interposition of the Roman See against the 
 formidable encroachments of the Turks. The muni- 
 ficent patronage of the MEDICI Family created and 
 diffused a taste for liberal knowledge. Those princely 
 merchants spared no cost in collecting the dispersed 
 manuscripts, and invited scholars, with liberal saia:- 
 ries, from all parts of the Levant, to teach their coun- 
 trymen the refined language of ancient Greece. Tlie 
 final capture of Constantinople by the Turks, in the 
 year 1453, occasioned a general dispersion of the men 
 of letters, who transported to Italy the perishing 
 wrecks of Greek philosophy. The new art of printing 
 happily at this crisis preserved them from extinction. 
 
xliV INTRODUCTION. 
 
 EVERY thing conspired to excite a general fer- 
 mentation. The diligence of the press quickly mul- 
 tiplied the monuments of ancient literature ; but it 
 required the labour of a century, to digest and cor- 
 rect those precious remains. The veneration paid 
 to such unrivalled compositions, repressed for a time 
 the working of native curiosity. Religious contro- 
 versies too, though ultimately productive of the great- 
 est benefits to society, distracted for a long course 
 of years the proper exercise of the human faculties. 
 At length the genius of invention burst forth with 
 renovated powers. 
 
 T 
 
 THIS day-spring of reason may be dated from the 
 middle of the sixteenth century ; since which epoch, 
 the tide of discovery has flowed in a rapid and ma- 
 jestic stream. Philosophy and the arts have advan- 
 ced together, reflecting mutual lights. Little more 
 than two hundred years have yet elapsed \ but it has 
 been a period of extreme activity, investing our 
 species in a blaze of intellectual glory. 
 
 THE age of science succeeded to the age of erudi- 
 tion. The study of the ancient classics had infused 
 
INTRODUCTION. xlv 
 
 some portion of taste and vigour ; but men soon be- 
 gan to feel their own strength, and hastened to ex- 
 ert it. The bolder spirits, bursting the trammels of 
 authority, ventured to question inveterate opinions, 
 and to explore with a fearless eye the wide fields 
 of human knowledge. COPERNICUS partly restored 
 the true system of the world. PURBACH and MUL- 
 LER abbreviated the calculations of the Astronomer, 
 by their signal improvements in Trigonometry. The 
 famous painter LEONARDO DA VINCI led the way 
 to sound philosophical inquiry ; and not only urged 
 the necessity of having recourse in Physics to ex- 
 periment, but directed it successfully to the Com- 
 position of Forces. UBALDI, and more especially 
 STEVINUS, extended the principles of Mechanics and 
 Hydrostatics. The fine genius of GALILEO detect- 
 ed and applied the laws of motion ; and re-invented 
 and constructed the Telescope, which had just been 
 found out in North Holland. This truly wonder- 
 ful instrument he directed to the heavens, and thus 
 marked the varying phases of the planets, and dis- 
 covered the harmony of new worlds. 
 
 THE bold exuberant imagination of KEJPLER, 
 
INTRODUCTION. 
 
 working on the register of the accurate observations 
 of TYCHO BRAKE, and employing the most intense 
 labour in computing and combining them, at last 
 drew aside the veil, and disclosed to view those eter- 
 nal laws which govern the revolutions of the heaven- 
 ly bodies. A powerful auxiliary was yet wanting to 
 theM calculator, and our great countryman NAPIER 
 rendered himself immortal, by the sublime discovery 
 of Logarithms. 
 yj>y/ 
 
 THE Alchemists, however extravagant in their 
 pretensions, had constantly promoted experimental 
 science. BAPTISTA PORTA, not only collected all 
 the curious facts he met with in his travels, but 
 founded on his return to Naples an association of 
 individuals for the purpose of exploring Nature. 
 This obscure society was the parent of all the insti- 
 tutions and academies afterwards established in Italy, 
 for the prosecution of physical inquiries. The ex- 
 ample quickly spread over Europe. 
 
 
 
 AT length tlie light of science penetrated into 
 England ; and the seventeenth century commenced 
 with the successful labours of Dr GILBERT of Col- 
 
INTRODUCTION. xlvii 
 
 Chester, whose high merit has not yet received the 
 due meed of praise. His treatise on the Magnet 
 was a model of the application of philosophical ana- 
 lysis ; it soberly reduced the various facts to a few 
 leading* principles ; and threw occasional gleams on 
 other branches of science. GILBERT not only esta- 
 blished Terrestrial Magnetism, but laid the founda- 
 tion of Electricity *. 
 
 * It is pleasing to find the discoveries of GILBERT, mentioned 
 with applause by some of his ablest successors. The following pas- 
 sage is extracted from the English draft of the inaugural speech re- 
 cited in Latin by Mr (afterwards the famous Sir CHRISTOPHER) 
 WREN, on his installation to the Professorship of Astronomy in 
 Gresham College, in the year 1657. 
 
 " And now began the first happy appearance of liberty to phi- 
 losophy, oppressed by the tyranny of the Greek and Roman mo- 
 narchies." " Among the honourable assertors of this liberty, I 
 
 must reckon GILBERT, who having found an admirable correspon- 
 dence between his Terella, and his great magnet of the earth, 
 thought this way to determine his great question, and spent his 
 studies and estate upon this inquiry ; by which obiter he found out 
 many admirable magnetical experiments. This man would I have 
 adored, not only as the sole inventor of magnetics, a new science 
 to be added to the bulk of learning, but as the fattier of ttie new 
 philosophy, CARTESIUS being but a builder on his experiments. 
 This pei-son I should have commended to posterity in a statue, 
 that the deserved marble of HARVEY might not stand to future ages 
 without a marble companion of his own profession. He kept cor- 
 respondence with the Lyncei Academici at Rome, especially with 
 FRANCISCUS SAGREDUS, one of the interlocutors in the dialogues 
 of GALIL^US, who laboured to prove the motion of the earth ne- 
 gatively, by taking off objections, but GILBERT positively ; the 
 
INTRODUCTION. 
 
 KEPLER had reduced the ordinary principles of 
 Optics into a systematic form. SNELLIUS soon after 
 discovered the law of Incidence and Refraction, 
 which DES CARTES simplified, and employed in the 
 explication of other properties of Light, and the 
 brilliant phenomena of the Rainbow. But the same 
 penetrating genius, by applying Algebra to Geome- 
 try, effected a memorable revolution in mathemati- 
 cal investigation, attended with the most important 
 consequences. 
 
 ITALY had enriched the art of observation by the 
 Thermometer, though this instrument was not 
 brought to perfection till more than a century after- 
 wards. But the same country produced likewise the 
 Barometer, which TORRICELLI invented after the 
 death of his master GALILEO. OTTO GUERICKE in 
 
 one hath given us an exact account of the motion of gravity upon the 
 earth ; the other of the more secret and more obscure motion of at- 
 traction and magnetical direction in the earth ; the one I must re- 
 verence for giving occasion to KEPLER, (as he himself confesses,) 
 of introducing magnets into the motions of the heavens, and conse- 
 quently of building the elliptical astronomy ; the other of his per- 
 fecting the great invention of telescopes to confirm this astronomy ; 
 so that if the one be the BRUTUS of liberty restored to philosophy, 
 nTtainly the other must be the COLLATINUS." 
 
INTRODUCTION. 
 
 Germany, pursuing a different route, discovered the 
 construction of the Air-Pump, and employed it in 
 the investigation of various important phenomena. 
 Both these instruments concurred to establish the 
 existence of Atmospheric Pressure, and to refute the 
 most inveterate errors that infected the ordinary 
 creed of Philosophy. The doctrines of ARISTOTLE 
 had frequently been attacked in detail, and they no 
 longer inspired the same confidence and veneration. 
 But it was reserved for the profound and daring 
 genius of DES CARTES to demolish completely that 
 imposing fabric. Original and inventive, he soared 
 above the influence of prejudice, and ranged freely 
 through the treasures of knowledge. Unfortunately 
 he would not stoop to the slow procedure of physical 
 analysis, but suffered himself to be hurried away by 
 the ambition of erecting a grand system. The Carte- 
 sian principles, afterwards variously modified, main- 
 tained their ascendency over the greatest part of Eu- 
 rope during the space of near a century. So far they 
 no doubt obstructed the course of genuine science ; 
 but, at the same time, they certainly extinguished for 
 ever the scholastic wrangling, while they bore the 
 seeds of their own destruction. 
 
1 INTRODUCTION. 
 
 THE Higher Geometry that instrument of su- 
 blime discovery was now advancing with a rapid 
 progression. CAVALLERI had invented his Method 
 of Indivisibles ; WALLIS produced his Arithmetic of 
 Infinites ; JAMES GREGORY and MERCATOR disco- 
 vered the doctrine of Series ; and BARROW and 
 ROBERVAL, in their method of drawing tangents, 
 traced the rudiments of the Differential Calculus. 
 The theory of the collision of bodies first given by 
 DBS CARTES, had been corrected and completed by 
 HUYGENS, WALLTS and WREN. Mechanics were 
 likewise enriched by many contrivances of the inge- 
 nious Dr HOOKE. HUYGENS, who greatly surpass- 
 ed him in mathematical science, having investigated 
 the properties of oscillating bodies, applied them 
 most skilfully to regulate the movements of clocks 
 and watches, by connecting their train with a pendu- 
 lum or a spring. But this able Philosopher, pursu- 
 ing his analysis still farther, founded the prolific 
 theory of Centrifugal Force. By the help of Geo- 
 metry, he likewise discovered the law which connects 
 the density of the strata of our atmosphere with their 
 elevation. 
 
INTRODUCTION. li 
 
 SCIENCE was about to take a higher flight, when 
 NEWTON arose, and bore away the palm of triumph. 
 This immortal genius compressed the empirical laws 
 of KEPLER into the single principle of Attraction / 
 and descending again from that principle, he dedu- 
 ced, by a synthetical involution, the great phaeno- 
 mena of the universe. These conclusions were in 
 general most felicitous ; but where the powers of his 
 calculus proved insufficient, he approximated to the 
 results by some tentative process, guided by that sa- 
 gacity in which he was never excelled. The same 
 penetration that ranged through the celestial spaces 
 could define the figure of our earth, and calculate 
 the tides of the ocean. The properties of water and 
 air the motion of currents, and the propagation of 
 sound were equally brought under the dominion 
 of Geometry. But the analytical procedure, di- 
 rected to the decomposition of Light, in a series of 
 conclusive experiments, disclosed still greater won- 
 ders, and 
 
 " Untwisted all the shining robe of day." 
 
 NEWTON'S fine researches in Optics, as they be- 
 gan, so, after a loii interruption, they closed his 
 scientific labours. 
 
Hi INTRODUCTION. 
 
 NEWTON and LEIBNITZ had separately discover- 
 ed, about the same time, the method of Fluxions 
 and Fluents, or the differential and integral Calcu- 
 lus. The former stated the principles with more 
 logical strictness, but the latter adopted a far prefe- 
 rable notation. This superior algorithm has chiefly 
 contributed to the prodigious extension which the 
 Higher Analysis received on the Continent. NEW- 
 TON himself made but inconsiderable advances in 
 integration ; and the progress afterwards achieved in 
 England by TAYLOR, COTES, and MACLAURIN, how- 
 ever respectable, will scarcely bear a comparison 
 with the towering ascent of the BERNOUILLIS, of the 
 great EULER, and of D'ALEMBERT or CLAIRAULT. 
 
 THE system of mechanical philosophy was now 
 seated on a firm foundation, though many parts of 
 the structure still remained incomplete. ROMER had 
 proved that light travels with a certain prodigious 
 velocity, and BRADLEY very ingeniously applied this 
 discovery to explain the Aberration of the Fixed Stars, 
 which the delicacy of a Zenith Sector, constructed by 
 GRAHAM, had enabled him to detect. The Newto- 
 nian doctrine still experienced some opposition on 
 
INTRODUCTION. Hi! 
 
 the Continent, from the prior ascendency of the 
 Cartesian tenets. But the mensuration of a Degree 
 of the Meridian, performed within the Arctic Circle, 
 and under the Equator, between the years 1736 and 
 1742, affording results entirely conformable to the 
 Theory of Attraction, finally decided the victory. 
 The Integral Calculus, now so vastly expanded, was 
 directed to the solution of the more arduous ques- 
 tions which NEWTON had either not solved, or had 
 only sketched. The conclusions hence deduced 
 were found exactly to harmonize with observation. 
 The greatest mathematicians in Europe have since 
 exerted their skill in improving the more delicate 
 parts of the theory. GAUSS has struck out new 
 paths of investigation. The recent calculations of 
 LA&RANGE and LAPLACE have brought to light va- 
 rious important and unexpected conclusions. All 
 the anomalies in the heavens are now found to be 
 periodical. Practical Astronomy has hence acqui- 
 red remarkable precision ; and the improvements of 
 the Lunar Theory have wonderfully hastened the 
 advancement of Navigation. The various observa- 
 tions of HERSCHEL, and the late discoveries of 
 
iiv INTRODUCTION. 
 
 PIAZZI, OLBERS, and HARDING, have concurred in 
 extending our ideas of the Planetary System. 
 
 THE correction of an oversight committed by 
 NEWTON himself, in his optical experiments, ledDoL- 
 LAND, about the year 1758, to the important theory 
 of Achromatic Glasses. The manufacture of these 
 exquisite instruments has contributed essentially to 
 improve the art of observing. But Optics, that 
 beautiful and fertile subject of inquiry, has again 
 made another shoot. MALUS lately detected a very 
 singular property of the rays of light, to which he 
 gave the name of Polarity. This ingenious person 
 has been prematurely cut off in his career of disco- 
 very, leaving the subject to the zeal and perseverance 
 of other inquirers. 
 
 OF the later accessions to Natural Philosophy, 
 none appears to have made a more satisfactory 
 progress than the important Doctrine of Heat. 
 Though a subject apparently abstruse and complica- 
 ted, it has yielded to the powers of experimental 
 analysis. The Theory of Heat has already conferred 
 lasting benefits on the arts and manufactures. But 
 
INTRODUCTION, Iv 
 
 it marks likewise the character of climate, and ex- 
 plains the atmospheric phenomena under all the 
 fleeting changes displayed in Meteorology. 
 
 GILBERT had formed the earliest catalogue of 
 Electrics ; and GUERICKE, fifty years afterwards, 
 constructed the first Electrical Machine. But though 
 GRAY discovered the property of Conducting Sub- 
 stances in 1730, Electricity could not aspire to the 
 rank of a science till the year 1745, when the Ley- 
 den Jar was invented. Since that epoch, the sub- 
 ject has been cultivated with great ardour and suc- 
 cess. A multitude of brilliant facts have been elicit- 
 ed, and hence at last the explication of the appalling 
 phenomena of Thunder and Lightning. 
 
 i \ JL 
 
 AN accidental observation of GALVANI in 1791, 
 has led to the formation of a new and most interest- 
 ing branch of Electricity. But the discovery made 
 eight years afterwards, by his ingenious countryman 
 VOLTA, of that combination of alternate metallic 
 plates, called the Pile, may be deemed an epoch in 
 the history of science. Chemistry has thus been fur- 
 nished with the most powerful instrument yet known 
 for exploring the composition of bodies. In the skil- 
 
Ivi INTRODUCTION. 
 
 ful hands of DAVY, BERZELIUS and others, the Vol- 
 taic Pile has displayed the most astonishing results. 
 
 MAGNETISM has also, within these few years, been 
 advancing to maturity. The various circumstances 
 which affect the declination and depression of the 
 Needle are at length ascertained with some degree 
 of precision. Empirical laws have hence been fram- 
 ed, that seem to indicate the changes of magnetic in- 
 fluence which are going forward in different parts of 
 the surface of the earth. But the connecting prin- 
 ciples, which would harmonize the whole, remain 
 still unknown. 
 
 THE analogy between Magnetism and Electricity 
 had long been suspected ; but the very curious ex- 
 periment of OERSTED has now put the question 
 beyond all doubt. The Galvanic action, variously 
 combined with the force of Magnetism, is incessant- 
 ly bringing into view the most beautiful and surpri- 
 sing facts. The unequal distribution of heat, and 
 the rapid circulation of certain metals, are also found 
 to affect materially the needle. No conclusive ex- 
 plication has yet been offered 5 but every thing seems 
 
INTRODUCTION. Ivii 
 
 to betoken our near approach to some grand and 
 pervading discovery. 
 
 BUT amidst all this splendour, it must however be 
 confessed, that science has declined from the severe 
 majesty which distinguished the age of mechanical 
 philosophy. Patient induction, though much com- 
 mended, has very few followers at present ; and the 
 passion for hypotheses appears to have again obtained 
 ascendancy in the learned world. Vague and fan- 
 ciful images are but too often substituted for close 
 reasoning. The more popular branches of physics 
 have absolutely grown rank with metaphorical ex- 
 pression. We may therefore conclude with the judi- 
 cious admonition of the finest genius of the seven- 
 teenth century. " When the weakness of men," says 
 the acute PASCAL, " is unable to find out the true 
 causes of phenomena, they are apt to employ their 
 subtlety in substituting imaginary ones, which they 
 express by specious names that fill the ear without 
 satisfying the judgment. It is thus that the sym- 
 pathy and antipathy of natural bodies are asserted 
 
 VOL. i. e 
 
Iviii INTRODUCTION. 
 
 to be their efficient and unequivocal causes of seve- 
 ral effects, as if inanimate substances were really 
 capable of sympathy and antipathy. The same thing 
 may be said of antiperistasis, and various other 
 chimerical causes, which afford only a vain relief to 
 the avidity of men to know hidden truths, and which, 
 far from discovering them, serve only to conceal the 
 ignorance of those who invent such explications, and 
 nourish it in their followers." 
 
 
ELEMENTS 
 
 
 OF 
 
 NATURAL PHILOSOPHY 
 
 NATURAL PHILOSOPHY is the science that unfolds 
 those general principles which connect the events of 
 the material world. It assumes as a basis the con- 
 stancy and permanence of the actual state of things. 
 
 The appearances which present themselves to our 
 observation, are called PHENOMENA ; and the com- 
 mon relations which pervade these phenomena, are 
 termed LAWS. 
 
 The business of the natural philosopher is to re- 
 mount patiently from effects to causes, till he ap- 
 proach the Fountain of all power and intelligence ; 
 and from this eminence to again descend, and trace 
 the lengthened chain of consequences. This double 
 mode of procedure corresponds to the Analysis and 
 Synthesis of the Ancient Greek Geometry. 
 
 VOL. T, A 
 
2 ELEMENTS OF 
 
 The analysis or investigation of physical facts is 
 conducted, either by Observation, or by Experiment. 
 Observation is the close inspection and attentive ex- 
 amination of those phenomena which arise succes- 
 sively in the course of nature : Experiment, as the 
 term implies, consists in a sort of trial or artificial 
 selection and combination of circumstances, for the 
 purpose of searching after the remote results. The 
 main object of the philosopher is to separate always 
 the various effects which are blended together in the 
 ordinary concurrence of events. 
 
 The primary facts being once detected from close 
 observation or delicate experiment, the synthetical 
 deduction can be safely pursued, by the exercise of 
 a sober and cautious logic. But the most important 
 instrument, in forwarding this process of re-combi- 
 nation, is Geometry, to which indeed we are indebt- 
 ed for whatever is most valuable in Physical Science. 
 
 The most satisfactory mode of proceeding in the 
 exposition of the phenomena, is to consider Bodies 
 as (1.) in a state of Rest, and (2.) in a state of 
 Motion. The essential properties which belong to 
 each distinct body form a branch of science that may 
 be termed SOMATOLOGY, but which has hitherto been 
 styled incorrectly Corpuscular Philosophy. The 
 mutual action of bodies, which produces their equili- 
 brium or quiescence, constitutes STATICS. Those 
 properties, again, which bodies display while in a 
 state of motion, form the branch of science called 
 
NATURAL PHILOSOPHY.. 3 
 
 PHORONOMICS. But since all motion arises from the 
 application of force, this department has been more 
 generally termed DYNAMICS. The principles of 
 Dynamics, applied to the revolutions of the heaven- 
 ly bodies, uprear the sublime science of PHYSICAL 
 ASTRONOMY. The same principles, directed to the 
 process of art, elucidate the construction of machi- 
 nery, and constitute the Theory of MECHANICS. 
 The principles of Statics applied to liquids explain 
 the conditions of their equilibrium, and form HY- 
 DROSTATICS. The principles of Dynamics, extend- 
 ed likewise to liquids, unfold the properties of their 
 motions, and ^compose the branch of science termed 
 HYDRAULICS, or more properly HYDRODYNAMICS. 
 The same principles, extended to a far more subtle 
 fluid, constitute PHOTONOMICS, or the Theory of 
 Light, which, in reference to vision merely, has been 
 long designated by the less perfect name of OPTICS. 
 Nearly allied to Photonomics, is the science of PY- 
 RONOMICS, which treats of the properties of Heat. 
 When Pyronomics embrace likewise the affections of 
 Humidity, they comprise the science of Meteorology, 
 and explain the character and condition of Climate. 
 The principles of Dynamics are applied, besides, 
 though as yet in a very limited extent, to MAGNET- 
 ISM and ELECTRICITY ; and as these interesting 
 branches of science shall advance to perfection, they 
 will assuredly come more within the range of calcu- 
 lation. 
 
4 ELEMENTS OF 
 
 A Course of Natural Philosophy may hence be 
 rightly distributed under twelve distinct heads. 
 
 1. SOMATOLOGY, which includes the exposition of 
 the "general Properties of Bodies that are essential 
 to their separate existence. 
 
 2. STATICS, which explains the Equilibrium of 
 bodies, as resulting from their mutual action, or from 
 combined pressure and divellence. 
 
 3. PHORONOMICS, or DYNAMICS, which explores 
 the Laws of Motion, and traces the flux of changes 
 produced by the application of Farce. 
 
 4. PHYSICAL ASTRONOMY, which is the extension 
 of Dynamics to develope the great Phenomena of 
 the Heavens. It explains the motions and figures 
 of the Planets, and deduces the various consequent 
 effects. 
 
 5. MECHANICS, in which the principles of Dyna- 
 mics descend to improve the vulgar arts, and to ex- 
 plain the composition and arrangement of the various 
 Machines contrived to assist the labour of man. 
 
 6. HYDROSTATICS, which consists in the applica- 
 tion of the principles of Statics to explain the Equi- 
 librium of Liquids or of Fluids in general : It treats 
 likewise of the construction of Works depending on 
 such properties. 
 
 7- HYDRODYNAMICS, or HYDRAULICS, which con- 
 sists in applying Dynamics to the Motion of Li- 
 quids : It consequently investigates the construe- 
 
NATURAL PHILOSOPHY. 5 
 
 tion and performance of the various Engines em- 
 ployed to raise Water, or which are driven by the 
 impulsion of that Fluid. 
 
 8. PNEUMATICS, which includes the application of 
 Statics and Dynamics to Air and other Gaseous 
 Fluids : It explains the constitution, the operations, 
 and general phenomena of our Atmosphere. 
 
 9. PHOTONOMICS, which treats of the properties 
 and operations of Light. 
 
 10. PYRONOMICS, which explores the properties 
 and operations of Heat. 
 
 11. MAGNETISM, which investigates the properties 
 of the Loadstone, and their application to the sus- 
 pended Needle. 
 
 12. ELECTRICITY, which explains all the brilliant 
 phenomena derived from those first produced by 
 the rubbing of Amber. 
 
 Such appears to be the systematic arrangement of 
 those subdivisions ; but it will admit of being con- 
 veniently modified. I shall therefore dispose them 
 more nearly in the order of difficulty. 
 
ELEMENTS OF 
 
 1. SOMATOLOGY * 
 
 Comprehends our knowledge of Bodies, or External 
 Substances. 
 
 The properties of body are detected by the senses 
 either from immediate observation or through 
 the application of experiment and the aid of instru- 
 ments. The more obvious properties are revealed 
 to us merely by sight or touch ; but the penetration 
 of the telescope has enabled us to survey vast systems 
 of worlds dispersed through the remotest heavens, 
 while the opposite power of the microscope has 
 brought within our view, from the very verge of ex- 
 istence, a miniature creation of organised beings. 
 Again, the most careless observer can hardly fail to 
 perceive that Air is a compressible fluid, while it re- 
 quires a very delicate experiment to discover the 
 same property in Water. 
 
 The properties of body are either essential and 
 permanent, or they are contingent and susceptible 
 of change or variation. 
 Body is essentially, 
 
 1. Extended. 4. Divisible. 
 
 2. Figured. 5. Porous. 
 
 3. Impenetrable. 6. Contractile or disten- 
 
 sible. 
 
 From zZpat, a Body. 
 
NATURAL PHILOSOPHY. 7 
 
 It is contingently, 
 1. Moveable. 2. Ponderable. 
 
 Of these, the two first properties essential to bo- 
 dies, EXTENSION and FIGURE, belong equally to 
 Space, and hence constitute the foundation of Geo- 
 metry. 
 
 HI. IMPENETRABILITY forms the third discrimi- 
 nating feature of body. This principle is common- 
 ly regarded as an axiom, but it is a truth only de- 
 rived from early and invariable experience. It rests 
 on this incontrovertible fact, that no two bodies can 
 occupy the same space in the same precise instant of 
 time. Had the case indeed been otherwise, each 
 body might be successively absorbed into the sub- 
 stance of another, till the whole frame of the uni- 
 verse, collapsing into a point, were lost in the vortex 
 of annihilation. 
 
 But although the most palpable observation attests 
 sufficiently the impossibility of the mutual compene- 
 tration of bodies, yet this property may be farther 
 illustrated and confirmed by a few simple experi- 
 ments. 
 
 1. A vessel being filled to the brim with water, 
 if any solid, incapable of solution in that liquid, be 
 plunged in it, a portion of the water will overflow, 
 which is exactly equal to the bulk of the wood or 
 metal immersed. 
 
8 ELEMENTS OF 
 
 2. If a cylinder be gradually pressed downwards 
 into a glass cylinder partly filled with water, the 
 liquid will rise proportionally, till the space mounted 
 over by its surface shall be equal to the portion of 
 the cylinder introduced. 
 
 . These two experiments show that water opposes 
 the entrance of a solid substance, and retires on all 
 sides to give room for its advance. Simple as the 
 fact now appears, it was first distinctly noticed by 
 Archimedes, who made it the ground of his Hydro- 
 statical Theory. 
 
 The same truth is evinced by other experiments. 
 
 3. If a cork be rammed hard into the neck of a 
 v phial full of water, the phial will burst, while its neck 
 
 remains entire. 
 
 4. Bladders filled with water and disposed upon 
 a table will support very large weights placed on a 
 board that has been laid over them. 
 
 5. The same experiment will succeed equally with 
 bladders blown full of air. 
 
 6. But the disposition of the air to resist all pene- 
 tration is made conspicuous in another way. Let a 
 large and very tall glass vessel be nearly filled with 
 water, on the surface of which a lighted taper is set 
 to float ; if over this glass, a smaller cylindrical ves- 
 sel, likewise of glass, be inverted and pressed down- 
 wards, the contained air maintaining its place, the 
 internal body of the water will descend, while the 
 rest will rise up at the sides, and the taper will appear 
 
NATURAL PHILOSOPHY. 9 
 
 for some seconds to burn, encompassed by the whole 
 mass of liquid. 
 
 iv. DIVISIBILITY is another essential property of 
 bodies. Every substance with which we are ac- 
 quainted is capable of being separated into parts, and 
 each of these again may be repeatedly subdivided. 
 Nor has any limit ever been assigned to this process 
 of continual subdivision, though it seems probable 
 that, at some term, however distant, the resulting 
 particles may lapse into simple atoms, incapable of 
 any farther resolution. 
 
 The actual subdivision of bodies has, in many cases, 
 been carried to a prodigious extent. A slip of ivory, 
 of an inch in length, is frequently divided into an 
 hundred equal parts, which are distinctly visible. 
 But, by the application of a very fine screw, five 
 thousand equidistant lines, in the space of a quarter 
 of an inch, have been traced on a surface of steel or 
 glass with the fine point of a diamond, producing 
 delicate iridescent colours. Common writing paper 
 has a thickness of about the 500th part of an inch ; 
 but the pellicle separated from ox-gut, and then 
 doubled to form gold-beaters' skin, is six times thin- 
 ner. A single averdupois pound of cotton has been 
 spun into a thread 76 miles in length ; and the same 
 quantity of wool has been extended into a thread of 
 95 miles ; the diameters of those threads being hence 
 only the 350th and 400th parts of an inch. 
 
10 ELEMENTS OF 
 
 The finest flaxen thread drawn by machinery at 
 Leeds has only the 15,5th part of an inch in thick- 
 ness, so that a single pound of it would reach 17| 
 miles. At Dunfermline, yarn, of the 200th part 
 of an inch in diameter, is now manufactured from 
 the best foreign flax. But lately, an old woman, 
 near Belfast, span Irish flax into yarn, a pound 
 of which would extend to 30 miles, its diameter be- 
 ing only the 250th part of an inch. Yet this effort 
 of skill and patience comes far short of the fineness 
 of the thread sometimes used in Flanders, for the 
 fabric of lace ; a specimen of which, from St Quen- 
 tin, but spun at Catillon sur Sambre, had only the 
 700th part of an inch in diameter, a pound of it be- 
 ing sufficient to extend to 342 miles. 
 
 But the ductility of some metals far exceeds that 
 of any other substance. The gold-beaters begin 
 their operations with a ribband an inch broad and 
 150 inches long, which had been reduced, by passing 
 it through rollers, to about the 800th part of an 
 inch in thickness. This ribband is cut into squares, 
 which are disposed between leaves of vellum, and 
 beat by a heavy hammer, till they acquire a breadth 
 of more than three inches, and are thus extended to 
 ten times their former surface. These are again 
 quartered, and placed between the folds of gold- 
 beaters' skin, and stretched out, by the operation of 
 a lighter hammer, to the breadth of five inches. The 
 same process is repeated, sometimes more than once, 
 by a succession of lighter hammers j so that 376 
 
NATURAL PHILOSOPHY. 11 
 
 grains of gold are thus finally extended into 2000 
 leaves, of 3.3 inches square, making in all 80 books, 
 containing each of them 25 leaves. The metal is con- 
 sequently reduced to the thinness of the 282,000th 
 part of an inch, and every leaf weighs rather less 
 than the fifth part of a grain. A particle of gold, 
 not exceeding the 500,000th part of a grain, is hence 
 distinctly visible to the naked eye. 
 
 Silver is likewise capable of being laminated, but 
 will scarcely bear half the extension of gold, or the 
 150,000th part of an inch thick. Copper and 
 tin have still inferior degrees of ductility, and can- 
 not be beat thinner perhaps than the 20,000th part 
 of an inch. These form what are called Dutch 
 Leaf. 
 
 In the gilding of buttons, five grains of gold, 
 which is applied as an amalgam with mercury, is al- 
 lowed to each gross ; so that the coating left must 
 amount to the 110,000th part of an inch in thick- 
 ness. If a piece of ivory or white satin be immersed 
 iij a nitro-muriate solution of gold, and then exposed 
 to a current of hydrogen gas, it will become covered 
 with a surface of gold hardly exceeding in thick- 
 ness the ten millionth part of an inch. 
 
 The gilt wire used in embroidery, is formed by 
 extending gold over a surface of silver. A silver 
 rod, about two feet long and an inch and half in dia- 
 meter, and therefore weighing nearly twenty pounds, 
 is richly coated with about 800 grains of pure gold. 
 In this country the lowest proportion allowed is 100 
 
12 ELEMENTS OF 
 
 grains of gold to a pound of silver. This gilt rod is 
 then drawn through a series of diminishing holes, 
 till it has stretched to the vast length of 240 miles, 
 when the gold has consequently become attenuated 
 800 times, each grain being capable of covering a 
 surface of 9600 square inches. This wire is now 
 flatted, the golden film suffering a farther extension, 
 and having its thickness reduced to the four or five 
 millionth part of an inch. 
 
 It has been asserted, that wires of pure gold can 
 be drawn, of only the 4000th part of an inch in dia- 
 meter. But Dr W. H. Wollaston, by an ingenious 
 procedure, has lately advanced much farther. Tak- 
 ing a short cylinder of silver, about the third part of 
 an inch in diameter, he drilled a fine hole through 
 its axis, and inserted a wire of platinum only the 
 100th part of an inch thick. This silver mould was 
 now drawn through the successive holes of a steel 
 plate, till its diameter was brought to near the 1500th 
 part of an inch, and consequently the internal wire, 
 being diminished in the same proportion, was reduced 
 to between the 4 and 5000th part of an inch. The 
 compound wire was then dipped in warm nitric acid, 
 which dissolved the silver, and left untouched its 
 core or the wire of platinum. By passing the in- 
 crusted platinum through a greater number of holes, 
 wires still finer were obtained, some of them only 
 the 30,000th part of an inch in diameter. The te- 
 nacity of the metal, before reaching that limit, was 
 even considerable ; a platinum wire of the 18,000th 
 
NATURAL PHILOSOPHY. IS 
 
 part of an inch in diameter supporting the weight of 
 one grain and a third. 
 
 Such excessive fineness is hardly surpassed by the 
 filamentous productions of nature. Human hair 
 varies in thickness, from the 250th to the 600th part 
 of an inch. The fibre of the coarsest wool is about 
 the 500th part of an inch in diameter, and that of 
 the finest only the 1500th part. The silk line, as 
 spun by the worm, is about the 5000th part of an 
 inch thick ; but a spider's line is perhaps six times 
 finer, or only the 30,000th part of an inch in dia- 
 meter, insomuch, that a single pound of this atte- 
 nuated substance might be sufficient to encompass 
 our globe. 
 
 The red globules of the human blood have an ir- 
 regular roundish shape, from the 2500th to the 
 3300th of an inch in diameter, with a dark central 
 spot. 
 
 The trituration and levigation of powders, and the 
 perennial abrasion and waste of the surface of solid 
 bodies, occasion a disintegration of particles, almost 
 exceeding the powers of computation. Emery, after 
 it has been ground, is thrown into a vat filled with 
 water, and the fineness of the powder is distinguish- 
 ed by the time of its subsidence. In very dry si- 
 tuations, the dust lodged near the corners and cre- 
 vices of ancient buildings is, by the continual agita- 
 tion of the air, made to give a glossy polish to the 
 interior side of the pillars and the less prominent 
 parts of those venerable remains. 
 
14 ELEMENTS OF 
 
 So fine is the sand on the arid plains of Arabia, that 
 it is carried sometimes 300 miles over the Mediter- 
 ranean sea, by the sweeping and violent Sirocco. 
 Very lately, the deck of a ship was covered with im- 
 palpable dust, while navigating the Atlantic, at the 
 distance of 600 miles from the western coast of Afri- 
 ca. The rocks are peopled, along the shores of the 
 Mediterranean, by the pholas, a testaceous and edible 
 worm, which, though very soft, yet, by unwearied 
 perseverance, works a cylindrical hole into the heart 
 of the hardest stone. The marble steps of some of 
 the great churches in Italy are worn by the inces- 
 sant crawling of abject devotees ; nay, the hands and 
 feet of bronze statues are, in tKe lapse of ages, wasted 
 away by the ardent kisses of innumerable pilgrims 
 that resort to those shrines. What an evanescent 
 pellicle of the metal must be abraded at each succes- 
 sive contact ! 
 
 The solutions of certain saline bodies, and of other 
 coloured substances, exhibit a prodigious subdivision 
 and dissemination of matter. A single grain of the 
 sulphate of copper, or blue vitriol, will communicate 
 a fine azure tint to five gallons of water. In this 
 case, the copper must be attenuated at least ten mil- 
 lion times ; yet each drop of the liquid may contain 
 so many coloured particles, distinguishable by our 
 unassisted vision. A still minuter portion of cochi- 
 neal, dissolved in deliqueate potash, will strike a 
 bright purple colour through an equal mass of water. 
 
 A small portion of iodine, heated within a sealed 
 

 NATURAL PHILOSOPHY. 15 
 
 flask, will fill the space with a beautiful purple steam, 
 which in cooling condenses into minute acicular 
 crystals ; and the fumes of ammonia, however ex- 
 panded, display their efficacy by the coloration of a 
 bit of test paper, the moment it is introduced below 
 the neck of the phial. But odours are capable of a 
 much wider diffusion. A single grain of musk has 
 been known to perfume a large room, for the space of 
 twenty years. Consider how often, during that time, 
 the air of the apartment must have been renewed, 
 and have thus become charged with fresh odour ! At 
 the lowest computation, the musk had been subdi- 
 vided into 320 quadrillions of particles, each of them 
 capable of affecting the olfactory organs. The vast 
 diffusion of odorous effluvia may be conceived from 
 the fact, that a lump of assafatida, exposed to the 
 open air, was found to have lost only a grain in the 
 space of seven weeks. Yet, since dogs hunt by the 
 scent alone, the effluvia emitted from the several 
 species of animals, and even from different indivi- 
 duals of the same race, must be essentially distinct. 
 
 The vapour of pestilence conveys its poison in a 
 still more subtle and attenuated form. The seeds 
 of contagion are known to lurk for years in various 
 absorbent substances, which, on exposure to the air, 
 scatter death and consternation. 
 
 But the diffusion of the particles of light defies all 
 powers of calculation. A small taper will, in a 
 twinkling, illuminate the atmosphere to the distance 
 of four miles ; yet the luminous particles which fill 
 
1'6 ELEMENTS OF 
 
 that wide concavity cannot amount to the 5000th 
 part of a grain, which may be the whole consump- 
 tion of the wax in light, in smoke, and ashes. 
 
 Animated matter likewise exhibits, in many in- 
 stances, a wonderful subdivision. Between the Tro- 
 pics, the small marine polypi, by the immensity of 
 their combined numbers, speedily raise up clusters 
 of coral reefs, so dangerous at present to the naviga- 
 tion of those seas, but which are destined at no very 
 remote period, to form groupes of inhabited and cul- 
 tivated islands. The milt of a codfish, when it be- 
 gins to putrefy, has been computed to contain a bil- 
 lion of perfect insects ; so that thousands of these 
 living creatures could be lifted on the point of a 
 needle. But the infusory animalcules display, in 
 their structure and functions, the most transcendent 
 attenuation of matter. The vibrio undula, found 
 in duck-weed, is computed to be ten thousand mil- 
 lion times smaller than a hemp seed. The vibrio 
 lineola occurs in vegetable infusions, every drop con- 
 taining myriads of those oblong points. The monas 
 gelatinosa, discovered in ditch-water, appears in the 
 field of a microscope a mere atom endued with life, 
 millions of them playing like the sun-beams in a 
 single drop of liquid. 
 
 v. POROSITY is also an essential property of Bo- 
 dies. It is not confined to the animal and vegetable 
 compounds, which have an evident vascular structure, 
 
NATURAL PHILOSOPHY. 17 
 
 but is found in every substance that we are able to 
 explore. This may be shown from inspection with 
 the microscope, by the passing of different fluids 
 through solid bodies, <>r by the transmission of light 
 itself in all directions through their internal sub- 
 stance. 
 
 1. The porosity of wood is so remarkable, that 
 air may be blown, by the action of the mouth and 
 lungs, in a profuse stream, through a cylinder two 
 feet long, of dry oak, beech, elm or birch. If a 
 piece of wood, or a lump of marble, granite, or 
 other compact stone, be plunged under water, and 
 placed within the receiver of a pneumatic machine ; 
 on withdrawing the external pressure, the air which 
 had been dispersed through their interior cavities, 
 will issue from every point of their surface, and rise 
 in a torrent of bubbles. A similar appearance is 
 exhibited when such bodies are sunk under mercury, 
 which in some cases becomes finely injected through 
 all the ramifying pores, on restoring the pressure of 
 the atmosphere. In like manner, mercury is forced 
 through a piece of dry wood, and made to fall in 
 the form of a fine divided shower. 
 
 If a few ounces of mercury be tied in a bag of 
 sheep skin, it may be squeezed through the leather 
 by the pressure of the hand, in numerous minute 
 streamlets. This experiment illustrates the porosity 
 of the human cuticle. From microscopic observa- 
 tions, it has been computed that the skin is perfo- 
 
 VOL. I. B 
 
18 ELEMENTS OF 
 
 rated by a thousand holes in the length of an inch. 
 If we estimate the whole surface of the body of a 
 middle-sized man to be sixteen square feet, it must 
 contain no fewer than 2304 million pores : and 
 these pores are the mouths of so many excretory ves- 
 sels, which perform that important function in the 
 animal economy Insensible Perspiration. The 
 lungs discharge every minute 6 grains, and the sur- 
 face of the skin from 3 to 20 grains ; the average 
 over the whole body being 15 grains of lymph, con- 
 sisting of water, with a very minute admixture of 
 salt, acetic acid, and a trace of iron. At this rate 
 of transpiration, the loss would amount to 3f 
 pounds troy, in the space of twenty-four hours. If 
 we suppose this perspirable matter to consist of glo- 
 bules only about 30 times smaller than the red par- 
 ticles of blood, or about the 10,000th part of an inch 
 in diameter, it would require a succession of three of 
 them to issue from each orifice every minute. 
 
 The permeability of a solid body to any fluid, de- 
 pends however on its peculiar structure and its re- 
 lation to the fluid. A compact substance will some- 
 times oppose the entrance of thin fluid, while it gives 
 free passage to a gross one. Thus, a cask, which could 
 hold water, will permit oil to ooze through it ; and 
 a fresh humid bladder, which is air-tight, will yet, 
 when pressed under water, imbibe a notable portion 
 of that liquid. If a cylindrical piece of oak, ash, 
 elm, or other hard wood, cut in the direction of its 
 
NATURAL PHILOSOPHY. 19 
 
 fibres, he cemented to the end of a long glass tube, 
 water will How freely through it, in divided stream- 
 lets ; but a soft cork inserted into a similar tube will 
 effectually prevent all escape of the liquid. Mercury 
 may be carried in a small cambric bag, which would 
 not retain water for a moment. If a circular bottom 
 of close-grained wood, divided by a fine slit from the 
 30th to the 100th part of an inch wide, be cemented 
 to the end of a glass tube, it will support a column 
 of mercury, from one to three or more inches high, 
 the elevation being always proportional to the nar- 
 rowness of the slit. Hence a cistern of box-wood is 
 frequently used for portable barometers, the fine 
 joints admitting the access and pressure of the air, 
 but preventing the escape of the mercury. Yet a 
 sufficient force might overcome this obstruction ; 
 and, in the same manner, the air which is confined 
 in the common bellows under a moderate pressure, 
 will, by a more violent action, be made to tran- 
 spire copiously through the boards and the leather. 
 The transmission of a fluid through a solid substance 
 shows the existence of pores ; but the resistance, in 
 ordinary cases, to such a passage, is insufficient, there- 
 fore, to prove the contrary. 
 
 The permeability of translucent substances to the 
 rays of light, in all directions, evinces the most ex- 
 treme porosity. But this inference is not confined 
 merely to the bodies usually termed diaphanous j f p. 
 the gradation towards opacity advances by insensible 
 
20 ELEMENTS OF 
 
 shades. The thin air itself is not perfectly translu- 
 cid, nor will the densest metal absolutely bar all pas- 
 sage of light. The whole mass of our atmosphere, 
 equal to the weight of a column of 34< feet of water, 
 transmits, according to its comparative clearness, 
 only from four-fifths to three-fourths of the perpen- 
 dicular light, and consequently retains or absorbs 
 from a fifth to a fourth part of the whole. But this 
 absorption is greatly increased by the accumulation of 
 the interposed medium. When the sun has approach- 
 ed within a degree of the horizon, and his rays now 
 traverse a tract of air equal in weight to a column of 
 905 feet of water, only the 212th part of them can 
 reach the surface of the earth. 
 
 By a peculiar application of my Photometer, I 
 have found that half of the incident light, which 
 might pass through a field of air of the ordinary den- 
 sity and 1 5 % miles extent, would penetrate only to 
 the depth of 15 feet in the clearest sea- water, which 
 is therefore about 5400 times less diaphanous than 
 the ordinary atmospheric medium. But water of 
 shallow lakes, though not apparently turbid, betrays 
 a greater opacity, insomuch, that the perpendicular 
 light is reduced one half, in descending only through 
 the space of six, or even two feet. The same mea- 
 sure of absorption would take place in the passage 
 of light through the thickness of two or three inches 
 of the finest glass, which is consequently 500,000 
 times more opaque than an equal bulk of air, or 
 
NATURAL PHILOSOPHY. 21 
 
 300 times more opaque than an equal weight or mass 
 of this fluid. 
 
 But even gold is diaphanous. If a leaf of that 
 metal, either pure or with only an 80th part of alloy, 
 and therefore of a fine yellow lustre, but scarcely ex- 
 ceeding the 300,000th part of an inch in thickness, 
 and inclosed between two thin plates of mica, be held 
 immediately before the eye, and opposite to a win- 
 dow, it will transmit a soft green light, like the colour 
 of the water of the sea, p.r of a clear lake of moderate 
 depth. This glaucous tint is easily distinguished 
 from the mere white light which passes through any 
 visible holes or torn parts of the leaf. It is indeed 
 the very colour which gold itself assumes, when pour- 
 ed liquid from the melting pot. A leaf of pale gold, 
 or gold alloyed with about the 80th part of silver, 
 transmits an azure colour ; from which we may, with 
 great probability, infer, that if silver could be redu- 
 ced to a sufficient degree of thinness, it would dis- 
 charge a purple light. These noble metals, there- 
 fore, act upon white light exactly like air or water, 
 absorbing the red and orange rays which enter into 
 its composition, but suffering the conjoined green 
 and blue rays to effect their passage. If the yellow 
 leaf were to transmit only the tenth part of the whole 
 incident light, we should only conclude, that pure 
 gold is 250,000 times less diaphanous than pellucid 
 glass. 
 
#2 ELEMENTS OF 
 
 The inferior ductility of the other metals will not 
 allow that extreme lamination, which would be re- 
 quisite, in ordinary cases, to show the transmission 
 of light. But their diaphanous quality may be in- 
 ferred, from the peculiar tints with which they af- 
 fect the transmitted rays, when they form the alloy 
 of gold. 
 
 Other substances, which are commonly reckoned 
 opaque, yet permit in various proportions the passage 
 of light. The window of a small apartment being 
 closed by a deal board, if a person within shut his 
 eyes a few minutes to render them more sensible, he 
 will, on opening them again, easily discern a faint 
 glimmer issuing through the window. If this 
 board be plained thinner, more light will successive- 
 ly penetrate, till the furniture of the room becomes 
 visible, and perhaps a large print may be distinctly 
 read. 
 
 Writing paper transmits about the third part of 
 the whole incident light, and when oiled it often 
 supplies the place of glass in the common work-shops. 
 The addition of the oil does riot, however, materially 
 augment the diaphanous quality of the paper, but 
 renders its internal structure more regular, and more 
 assimilated to that of a liquid. The rays of light 
 travel, without much obstruction, across several folds 
 of paper, and even escape copiously through paste- 
 board. 
 
NATURAL PHILOSOPHY. 2$ 
 
 Combining these various facts, it follows, that all 
 bodies are permeable, though in extremely different 
 degrees, to the afflux of light. They must therefore 
 be widely perforated, and in every possible direction. 
 The porosity of bodies is consequently so diffuse, 
 that the bulk of their internal kernel, or of the ul- 
 timate obstacles which they present, may bear no 
 sensible proportion to the space which they occupy. 
 
 VI. The last essential property which belongs to 
 all bodies, is that of CONTRACTION or DILATATION. 
 
 Though absolute penetration is impossible, yet 
 every substance, however dense or compact, can 
 yet have its volume enlarged or diminished. This 
 change of bulk is in some instances quite apparent, 
 while, in other cases, to render it visible, requires 
 either the agency of a vast force, or the application 
 of some very delicate measure. The effect is pro- 
 duced on fluids, either by increasing or diminishing 
 their ordinary external compression ; but solid sub- 
 stances will have their volume contracted or distend- 
 ed by squeezing or pulling. A few experiments will 
 confirm and illustrate the general propositipn. 
 
 1. If a flaccid bladder be placed within the recei- 
 ver of an air pump, it will gradually swell as the ex- 
 haustion advances, but, on restoring the external 
 pressure, it will again shrink into its former dimen- 
 sions. 
 
ELEMENTS OF 
 
 2. If a tall flask, nearly filled with water, be in- 
 verted in a jar of water, and placed likewise under 
 a pneumatic receiver, the air collected near the top 
 of the flask will visibly expand as the operation of 
 pumping proceeds, till it presses down the water, 
 and begins to make its escape from below in the form 
 of rarefied bubbles. 
 
 3. But air is easily compressed, by the opposite 
 action of a syringe. In the vault or chamber of the 
 air-gun, it is often condensed fifty or even eighty 
 times. Nearly the half of that charge may be thrown 
 into the pneumatic blow-pipe, from which, on par- 
 tially opening the valve, it will again continue to 
 flow for the space of a quarter of an hour. 
 
 Other gases are likewise notably contracted or di- 
 lated, by the increase or diminution of external pres- 
 sure. But liquid substances themselves manifest a 
 similar property, though in a much lower degree. 
 If a large glass-ball, terminating in a long narrow 
 and open stem divided into minute spaces correspond- 
 ing to the millionth parts of the whole capacity, be 
 filled with distilled water carefully purged of air, and 
 introduced under the receiver of a pneumatic ma- 
 chine ; as the exhaustion advances, the water will 
 proportionally expand and rise near fifty divisions ; 
 but, on admitting the atmosphere again to compress 
 the water, it will sink to its former place in the stem. 
 The contraction which the water suffers, at every in- 
 crease of pressure, exceeds not indeed the 20,000th 
 
NATURAL PHILOSOPHY. 25 
 
 part of what air would undergo in like circumstan- 
 ces ; but it is equally real, and evinces an inherent 
 property of that liquid. Mercury treated in the 
 same way shows a contraction three times less than 
 water. Alcohol, ether, oils, and the various acids 
 and saline solutions, are all condensed or expanded, 
 though in different degrees, by the change of at- 
 mospheric pressure. 
 
 If the stem of the instrument now mentioned were 
 made to screw to the ball at a wide aperture, frag- 
 ments of solid bodies could be easily introduced, and 
 the vacant space filled up with water ; the contrac- 
 tion of this portion of water being deducted from 
 the contraction of the mixture, would give the dis- 
 tinct condensation of the hard materials. In this 
 way, the compressibility of the various stones and 
 metals could be accurately examined. A series of 
 experiments of this nature would essentially contri- 
 bute to the improvement of the mechanical arts. 
 
 The contraction and distension produced by ex- 
 ternal or internal pressure on glass is quite visible, 
 in a thermometer with a large bulb and very long 
 tube. When the mercury stands near the top of 
 the scale, it will immediately rise on reclining the 
 tube, and will continue to flow till the thermometer 
 has been reversed j but the mercury will again re- 
 treat, as the instrument is brought back to its ver- 
 tical position. This experiment proves that the bulb 
 has its capacity sensibly enlarged by the thrust of 
 the mercurial column. 
 
26 ELEMENTS OF 
 
 When long bars of wood, iron, or other metals 
 are laid horizontally on supports, they bend down- 
 wards by their own weight, and this depression is in- 
 creased by augmenting the incumbent pressure. See 
 fig. 1. The upper fibres are therefore drawn into a 
 narrower curve than those in the middle of the bar, 
 while the under fibres are extended into a wider con- 
 vexity : the particles of the former are thus contract- 
 ed, and those of the latter distended. It is likewise 
 obvious, that in this incurvation, the contraction or 
 dilatation occasioned will be proportional to the 
 thickness of the bar. 
 
 The various kinds of wood are far more compress- 
 ible than water, and suffer, hence, a very consider- 
 able degree of condensation, on being let down to 
 great depths in the ocean. Pieces of oak, ash or 
 elm, plunged two or three hours in a calm sea, at 
 the enormous depth of a thousand fathoms, and then 
 drawn up, have been found to contain four-fifths of 
 their weight of water, and to acquire such increase 
 of density, as indicates a contraction of the wood in- 
 to about half its previous volume. The specimens 
 which have undergone this singular change, if thrown 
 into a pail of water, will sink like a stone. Hence 
 probably the reason why barks lost near the shore are 
 afterwards discovered by their timbers breaking up 
 and floating to the surface ; while the ships which 
 founder in the wide ocean, acquiring permanent den- 
 sity from the vast compression they sustain, remain 
 
NATURAL PHILOSOPHY. 
 
 motionless at the bottom, and never rise again to dis- 
 close their fate. 
 
 But pieces of wood, even of the softer kinds, whe- 
 ther round or square, may be easily squeezed in the 
 direction perpendicular to their fibres, by the action 
 of a common vice. If allowed to stand only a few 
 minutes under that compression, and then quickly 
 thrown into a jar of water, they will sink and remain 
 at the bottom. If the wood be kept much longer 
 under the vice, it will take a set, and become con- 
 stitutionally denser. Even cork may, by compres- 
 sion, be made to sink in water ; but as its texture 
 is nearly uniform, the force must be exerted on all 
 sides. Into a thick and very strong glass cylinder, 
 having a syringe adapted to it, introduce a large cork 
 ball, and inject the air by smart and powerful strokes ; 
 the ball will gradually shrivel, till it has contracted 
 even to less than one-third of its usual bulk ; but, 
 on allowing the charge of air to escape, the cork 
 will speedily resume its former shape and dimen- 
 sions. If the condenser be partly filled with wa- 
 ter, on which the ball of cork is set to float, it will, 
 under a like compression, though it has a minute 
 portion of the liquid driven into its substance, shrink 
 to nearly the same size as before, and soon fall 
 to the bottom. Hence the success of the com- 
 mon experiment at sea, of letting down, in calm wea- 
 ther, to the depth of twenty or thirty fathoms, an 
 empty corked bottle, and then drawing it up full of 
 water, though the cork still remains in the neck. 
 
ELEMENTS OF 
 
 The water is not in this case forced through the 
 pores of the cork as generally supposed, but the cork 
 itself, being condensed by the lateral pressure of the 
 incumbent mass of liquid, allows it to enter by the 
 sides and dislodge the air. 
 
 Bodies are also contracted or dilated, from the 
 operation of some internal cause. Thus, dry air is 
 visibly expanded by its union with moisture. If, in 
 a warm room, the inside of a tall flask be wetted by 
 a few drops of water, and the mouth inverted in a 
 bason of water, the contained air, in proportion as it 
 becomes humified, will discharge a copious stream of 
 bubbles. On the other hand, a notable contraction 
 of their joint volume is produced in the absorption 
 of water by saw-dust, linen, or bibulous paper. The 
 combination of equal measures of water and alcohol 
 is accompanied by a contraction amounting to the 
 fiftieth part of the whole bulk. A similar effect 
 results from the solution of the sulphate of soda, 
 and other highly soluble salts. 
 
 The alliage of different metals often betrays a large 
 contraction. The power of tin to condense copper 
 in the composition of bronzes was even remarked by 
 the ancients, who combined these metals in various 
 proportions, to form their knives, chisels or hatchets. 
 Equal bulks of tin and copper are found to suffer a 
 contraction amounting to not less than the fifteenth 
 part of their whole volume. 
 
 A liquid, in joining any solid substance, commonly 
 
NATURAL PHILOSOPHY. 29 
 
 occasions a general contraction ; but the solid itself 
 may yet by this accession expand with prodigious 
 force. Thus, dry peas being rammed into a gun barrel, 
 and their interstices filled with water, will in a short 
 time burst the barrel. In like manner, if wedges of 
 soft dry wood be driven into slits made with a saw 
 in blocks of freestone or marble, and then have water 
 poured upon them, these wedges will quickly swell 
 and rend the rock. Such was the ancient mode of 
 quarrying, before the explosive power of gunpowder 
 came to be introduced. It is still practised in the 
 art of cutting mill -stones in France, holes being bo- 
 red at intervals in a line drawn across the block, and 
 wooden plugs driven into them, and then wetted. 
 
 Hence, if the side of a thin piece of wood be 
 moistened, it will bend backwards, the humidity in- 
 sinuating itself into the soft parenchymatous matter 
 between the fibres, and therefore enlarging the circle 
 of flexure. The thinner the wood is sliced, the 
 greater will evidently be the incurvation produced by 
 the wetting of its convex surface. The fibres of hair 
 and wool, by the unequal rubbing and moistening of 
 their sides, are made to curl up, and to condense 
 like a clue. On this property seems to be really 
 founded the very important process of milling, full- 
 ing or felting, by which a raw web of woollen cloth 
 is thickened, and its texture rendered firm and com- 
 pact. 
 
 A leathern thong is extended by wetting, and so 
 
 
30 ELEMENTS OF 
 
 are the filaments of hemp or flax when laid parallel. 
 But the moisture spreads chiefly through the longi- 
 tudinal interstices of those filaments, and conse- 
 quently enlarges their thickness. As a hempen cord 
 is shortened by twisting, so is it likewise contracted 
 by wetting, the diameter of the coil being thereby 
 increased, and the extension of the overlapping fibres 
 hence proportionally curtailed. If a spunge dipp- 
 ed in hot water be drawn more than once along a 
 well-spun rope, it will in dry weather occasion, du- 
 ring the space perhaps of an hour, a contraction 
 amounting to the fifteenth or twentieth part of the 
 whole length. This remarkable property has in 
 some instances been employed as a very efficient 
 mechanical agent. 
 
 The most powerful principle of internal expan- 
 sion, is the introduction of Heat. This energy 
 varies exceedingly, however, in different substances, 
 but it can always be reduced to calculation, by com- 
 paring its effects with the opposite influence of ex- 
 ternal compression. Thus, the same absolute por- 
 tions of heat communicated to cylinders of one inch 
 diameter and height, of air, alcohol, water, mercury 
 and copper, would enable those columns respectively 
 to sustain the weights of 10, 12, 3 and 2 pounds. 
 
 The properties which may be regarded as only 
 Contingent, and not Essential to the Constitution 
 of Bodies, are MOBILITY and PONDEROSITY. 
 
NATURAL PHILOSOPHY. 31 
 
 I. Every body at rest can be put in motion, and 
 if no impediment intervenes, this change may be 
 effected by the slightest external impression. Thus, 
 the largest cannon-ball, suspended freely by a rod 
 or chain from a lofty cieling, is visibly agitated by 
 the horizontal stroke of a swan-shot, which has gain- 
 ed some velocity in its descent through the arc of a 
 pendulum. In like manner, a ship of any burthen 
 is, in calm weather and smooth water, gradually pull- 
 ed along even by the exertions of a boy. A cer- 
 tain measure of force, indeed, is often required to 
 commence or to maintain the motion ; but this con- 
 sideration is wholly extrinsic, and depends on the 
 obstacles at first to be overcome, and on the resis- 
 tance which is afterwards encountered. If the ad- 
 hesion and intervention of other bodies were abso- 
 lutely precluded, motion would be generated by the 
 smallest pressure, and would continue with undimi- 
 nished energy. 
 
 II. The other Contingent Property of Bodies is 
 PONDERABILITY. Every substance within our sphere 
 of observation is found to possess weight, or a dispo- 
 sition to gravitate towards the centre of tJie earth. 
 But to constitute gravity, it is not required that a 
 body should invariably fall to the ground. Smoke 
 ascends in the atmosphere, and a lump of lead rises 
 in a tube of mercury, from the same cause that a pine 
 tree, plunged into a lake, mounts again to the sur- 
 face. Withdraw the air, the mercury, and the wa- 
 
ELEMENTS OF 
 
 ter, which supported those comparatively lighter sub- 
 stances, and the smoke, the lead, and the timber, 
 will immediately descend. Pour mercury over a 
 smooth piece of cork applied to the bottom of a glass, 
 and it will remain in the same situation, while an 
 iron-ball can be set to float on the liquid metal. The 
 order of Nature might here seem to be reversed. But 
 since mercury does not insinuate itself through a very 
 narrow interstice, it merely rests on the upper side, 
 without pressing against the under side, of the cork. 
 If Levity, however, as the Schoolmen asserted, had 
 been a real property belonging to certain bodies, the 
 smoke and the cork would, in every instance, have 
 occupied the lower stations. 
 
 But the weight of a body is not the same in all 
 places and situations. A lump of lead, which weighs 
 a thousand pounds at the surface of our globe, would 
 lose two pounds as indicated by a spiral spring, if 
 carried to the top of a mountain four miles high ; 
 and, if it could be conveyed as deep into the bowels 
 of the earth, it would lose one pound. The same 
 mass transported from Edinburgh to the Pole would 
 gain the addition of three pounds ; but if taken to 
 the Equator, it would suffer a loss of four pounds 
 and a quarter. 
 
 The variable, and therefore contingent, weight of 
 bodies, is only the gradation of that mutual and uni- 
 versal tendency, diminishing as the square of the dis- 
 tance, which retains the Moon in her orbit, and up- 
 
 1 
 
NATURAL PHILOSOPHY. 33 
 
 holds the circulation of the whole system of Planets 
 around the Sun. The gravitation even of small 
 masses towards each other, such as balls of lead se- 
 parated only by the interval of a few inches, has 
 been detected by delicate experiment, and reduced 
 to rigorous calculation. But when the approxima- 
 tion is closer, this force acquires a modified charac- 
 ter, and passes into Cohesion. Thus, if two leaden 
 bullets have a little portion of the surface of each 
 pared clean, and be then pressed together with a 
 slight twist, they will cohere firmly into one mass. 
 In the same manner, gold or silver foliage and other 
 ornaments, struck with a heavy hammer into the sur- 
 face of polished iron or steel, become permanently 
 united. 
 
 Within other limits, the tendency to mutual ap- 
 proach is changed into an opposite quality. Thus, 
 drops of rain or dew run along the smooth and glos- 
 sy surface of a cabbage leaf without spreading. If 
 the dust of the Lycopodium, or Club-Fern, or even 
 fine pounded rosin, be strewed on water contained in 
 a glass, any smooth rounded piece of soft wood will 
 float upon it, or may be immersed in the liquid, with- 
 out being wetted, the powder preventing, by its re- 
 pulsion, all contact of the water. A fine needle 
 laid on the surface of water makes a dimple in which 
 it swims. On the same principle, the slender limbs 
 of insects, and the minute down which covers their 
 wings, protects them from the penetration of humi- 
 
 VOL. i. c 
 
ELEMENTS OF 
 
 dity. If the hand be rubbed with linseed oil, it may 
 be plunged with impunity for a few seconds in boil- 
 ing water, the oil repelling the water, and conse- 
 quently checking the communication of heat. The 
 application of palm-soap to the skin is still more ef- 
 fectual. 
 
 It thus appears that bodies are indefinitely porous, 
 compressible without limits, and capable of assuming 
 all varieties of form. How different is the constitu- 
 tion of ice, of water, and of steam ? Examine the 
 mutable aspects which mercury exhibits. Beginning 
 at a low degree of cold, and ascending through the 
 gradations of heat, we find it a friable solid, next a 
 shining liquid, then a penetrating vapour, and lastly 
 reduced to a fine red powder. A bright piece of 
 any of the ductile metals passes successively into an 
 earthy oxide and a pellucid glass. Charcoal is ab- 
 solutely of the same nature as the diamond ; yet what 
 a contrast between the dingy appearance of the one 
 and the dazzling lustre of the other ? How various- 
 ly are substances transformed by the operations of 
 art ? The skins of animals become changed into 
 parchment and different kinds of leather, and its 
 shreds into glue. The vegetable fibres are converted 
 into matting, cordage, and linen cloth ; and the rags 
 of this again, reduced to a pulp, and manufactured 
 into paper. 
 
 How diversified appear the compounds of the fa- 
 rinaceous substances ! By a distinct operation, the 
 
NATURAL PHILOSOPHY. 3,5 
 
 same grain produces gruel, bread, biscuit, starch, and 
 u hard pellucid concretion resembling mother of 
 pearl. But the plastic powers displayed in the pro- 
 cess of vegetation and animal life infinitely surpass 
 the resources of art. Many plants are fed by water 
 and air alone, and consequently these fluids are ca- 
 pable of being transmuted into all the various pro- 
 ducts. In short, Nature exhibits only a chain of 
 endless metamorphoses : The substance or material 
 remains unchanged, but its form undergoes conti- 
 nual mutations. 
 
 The properties of bodies result from those of their 
 component particles. At certain mutual distances 
 they remain quiescent ; but, at other distances, they 
 show a disposition either to approach or to recede. 
 Such opposite tendencies are commonly referred to 
 the principles of attraction and repulsion. But all 
 those diversified effects may be comprehended under 
 a general law, which connects the mutual action of 
 particles with their relative distance. In the lan- 
 guage of modern analysis, the corpuscular energy is 
 always some function of the distance ; and it may be 
 represented by an extended curve, of which the 
 abscissae mark the distances, and their ordinates 'ex- 
 press the corresponding forces. Fig. 2. exhibits this 
 curve of primordial action ; in which A denotes an 
 action or ultimate particle, and B, C, D, E, &c., the 
 successive positions of another particle, the perpen- 
 diculars CM, EN, GO, IP, LQ, &c., representing 
 
 c 2 
 
36 ELEMENTS OF 
 
 their mutual action attractive between B and D, F 
 and H, K and X, when above the axis AX, and re- 
 pulsive between D and F, H and K, below it. The 
 final branch of the curve must gradually assimilate 
 itself to the law of universal gravitation. But the 
 primary branch of the curve must, in like manner, 
 continually approach AY, the perpendicular to the 
 axis ; and since no pressure or impulsion can ever ac- 
 complish the penetration of matter, it follows, from 
 the principles of Dynamics, that the space included be- 
 tween the curve and that asymptote must be infinite. 
 Where the curve repeatedly crosses the axis, are so 
 many quiescent positions, B, D, F, H, K, &c. in 
 any one of which a particle would continue in equi- 
 libria. But this equilibrium is of two kinds, the 
 stable or the instable ; the former easily recovering 
 itself from any slight displacement, and the latter 
 when once disturbed being irremediably dissolved. 
 If the curve in its progress cross the axis from the 
 side of repulsion to that of attraction, its intersec- 
 tion will evidently be a point of stability ; for if a 
 particle be pushed inwards, it will then be repelled 
 back again ; and if it be pulled outwards, it will ex- 
 perience an attractive force, which will recall it to 
 its first position. But if the curve pass from attrac- 
 tion to repulsion, its intersection with the axis is a 
 point of instable equilibrium ; for, in proportion as 
 a particle is pressed inwards, it will be pulled forci- 
 bly from its position ; and if it be drawn outwards, 
 
NATURAL PHILOSOPHY. 3^ 
 
 the repulsion, now conspiring, will bear it along with 
 accumulating power. Thus, B, F and K, the tran- 
 sitions from repulsion to attraction, are points of sta- 
 bility ; but D and H, the opposite transitions, are 
 points of instability. 
 
 According as the ordinates, near the points of 
 transition, increase less or more rapidly, the tendency 
 of the particles to coalesce, or to separate, will be 
 proportionally feeble or intense. If the curve cut 
 the axis very obliquely, therefore, it will mark a limit 
 of languid cohesion, as at the point F ; but if it shoot 
 nearly at right angles across the axis, as at B or K, 
 it will indicate a limit of powerful cohesion. 
 
 Those atoms or ultimate particles have no sensible 
 magnitude. But though the range of our concep- 
 tions may be unbounded, every thing in the material 
 world appears to be distinct and determinate, Ex- 
 perience indeed informs us to what astonishing de- 
 gree matter can be attenuated ; but philosophy de- 
 scries the existence of certain fixed or impassable li- 
 mits at which the capability of farther subdivision 
 utterly ceases. The primordial line of action is hence 
 a physical, and not a mathematical, curve ; or it is 
 not strictly incurvated at every point, but proceeds 
 by successive minute deflections, corresponding to 
 the breadth of the elementary particles. Such a mo- 
 dification of the curve is represented by fig. 3. ; be- 
 ing a serrated line, whose gradations answer to the 
 successive stages of corpuscular action. Continuous 
 
38 ELEMENTS OF 
 
 shades, indeed, exist only in our modes of concep- 
 tion ; and Nature exhibits always individual objects, 
 and advances by finite steps. 
 
 The material world is thus reducible to atoms, 
 actuated by forces depending merely on their mutual 
 distances. From such simple elements the diffe- 
 rent arrangements of the particles and their mul- 
 tiplied interior combinations, this sublime scene of 
 the universe derives all its magnificence and splen- 
 dour. 
 
NATURAL PHILOSOPHY. 39 
 
 II. STATICS 
 
 Imports the stability resulting from the balance of 
 connected bodies. It therefore explains the con- 
 ditions that must determine any material system, 
 among the several parts of which a mutual action is 
 exerted, to maintain the state of quiescence. 
 
 Our ideas of Power, Farce, Action, Energy, and 
 other similar objects of contemplation, seem all to be 
 derived from the muscular effort which we find, in 
 our operations, required to precede or produce every 
 external change. This feeling we spontaneously 
 transfer to inanimate bodies themselves ; and while 
 the effects are the same, we associate likewise their 
 origin with the same terms. Force, in its simplest 
 form, may hence be represented by weight, whether 
 this be made to act by pressure or by traction, or 
 whether it pushes or draws the point of attachment. 
 Power and energy are more complex conceptions, 
 and may be regarded as the results or modifications of 
 the application of force. 
 
 The fundamental principles of Statics, however 
 few and simple, are to be discovered only by experi- 
 ment, or an appeal to the actual constitution of Na- 
 ture. But the primary relations being once detect- 
 
40 ELEMENTS OF 
 
 ed, their various combinations, and the whole train 
 of derivative properties, are easily traced by the help 
 of Geometry. 
 
 To exclude all unnecessary complication, I shall 
 consider, in this inquiry, the forces as directed to 
 single particles or physical points. These forces may 
 be conveniently measured by help of a spiral steel 
 spring lodged in a thin cylindrical case, the traction 
 or weight applied to it being indicated by the pro- 
 trusion of a divided scale fixed to the remote end of 
 the spiral. It will be sufficient to examine, 1, The 
 equilibrium of two forces ; and, 2. The equilibrium 
 of three forces, in the limited case where two of these 
 are equal. 
 
 1. Let (fig. 4.) the spring A, holding a little ball 
 P, be suspended from a hook, and to the lower side 
 of this ball append another similar spring JB. If now 
 a weight of 10 pounds be attached to the end of B, 
 both the scales will descend till the ball P comes to 
 rest, when they will stand opposite in the same ver- 
 tical line, and each mark 10 divisions. If 10 pounds 
 more be added, the scales will lengthen still farther 
 in the same directions, and indicate the strain of 20 
 pounds. If, instead of hooking the ball P immedi- 
 ately to springs, a piece of small cord be fastened to 
 each side of it, and then attached to those springs, 
 the result will be precisely the same ; the tension of 
 any weight applied being still directed in the per- 
 pendicular, and indicated alike by both scales. The 
 
NATURAL PHILOSOPHY. 41 
 
 same effects will take place, if, instead of weights, an 
 exertion of animal power had been applied. 
 
 It hence follows, that a body will remain at rest if 
 it be urged by equal and opposite forces. This truth 
 is commonly assumed as intuitive ; but if the pro- 
 position appears now so simple and natural as readily 
 to command our assent, it is only because it accords 
 with the information of our earliest experience. 
 The untutored mind will hastily admit things that 
 are most erroneous. 
 
 2. Let two similar springs be attached at the 
 points A and B, (fig. 5.), situate in a horizontal line 
 at the interval of twenty-four inches, and connect 
 their ends by a silk thread capable of being length- 
 ened at pleasure, and from the middle of which, at 
 the point C, successive weights can be appended. 
 It will soon be perceived that the direction CW of 
 the weight will bisect the angle ACB, and occupy 
 the same vertical plane. When the weight is in- 
 creased from 2 to 4 pounds, the corresponding de- 
 pression DC will increase likewise from 1 to 2 inches, 
 while the strain at A and at B will still measure 12 
 pounds. But on farther augmenting the weight, 
 though the quantity of depression will continue to 
 follow the same ratio, yet the force of oblique trac- 
 tion will now begin to increase. If the weight be 
 10 pounds, the depression being 5 inches, the points 
 A and B will be drawn in the directions AC and 
 BC, each with a force equal to 13 pounds. Again, 
 
42 ELEMENTS OF 
 
 if the weight applied at C amounted to 18 pounds, 
 the depression DC would be 9 inches ; but the strain 
 at A and at B wonld reach 15 pounds. And, fi- 
 nally, on augmenting the weight to 32 pounds, and 
 lengthening the thread, the depression will increase 
 to 16 inches, while the strain of each spring will 
 amount to 20 pounds. 
 
 It is easy to see, that since the angle ACB is al- 
 ways bisected, one-half of the weight at C is sup- 
 ported by the oblique traction of the spring A, while 
 the other half is supported by the equal and similar 
 traction of the spring B. But when half the weight 
 was 5 pounds, and the depression DC 5 inches, the 
 hypotenuse AC must have been 13 inches, while the 
 strain in that direction measured 13 pounds J In 
 like manner, when DC was 9 and 16 inches, the 
 hypotenuse has been 15 and 20 inches, and the strain 
 at A and at B indicated so many pounds. The force 
 by which the point C is pulled in the direction C A, 
 is to the weight by which it is drawn in the direction 
 CW or DC, as AC to CD, or as radius to the cosine 
 of the angle ACD. In general, therefore, it fol- 
 lows from this simple experiment, that the oblique 
 action of any force is proportional to the cosine of 
 its inclination. 
 
 Not to embarrass the investigation, I have thrown 
 out of sight the weight of the spiral springs, which 
 could be rendered very inconsiderable. But the 
 experiment is greatly simplified, by substituting for 
 
NATURAL PHILOSOPHY. K3 
 
 those springs, fine pulleys, over which a long thread 
 is passed, holding an equal weight at each end ; for 
 it is readily perceived that the only effect of the pul- 
 ley is to change the direction, without impairing its 
 intensity, of any force applied at the circumference. 
 The principle now derived from observation will 
 enable us to assign the conditions under which any 
 two forces are balanced by a third force. Let a 
 physical point A (fig. 6.) remain at rest, while it is 
 drawn by three forces in the several directions AB, 
 AC and AD, expounded or represented by the 
 lengths^ of those lines. Produce DA till AE be 
 equal to it ; and the force AE, being thus equal and 
 opposite to AD, must exert the same effect as the 
 joint action of AB and AC. Through A, draw 
 GAH at right angles to DE, meeting the parallels 
 BG and CH. Wherefore the oblique action of the 
 force AB on the point A is expressed by AG ; and, 
 for the same reason, the oblique action of AC is de- 
 noted by AH. But since the point A continues at 
 rest, the equal forces AG and AH, by which it is 
 urged, must likewise be exactly opposite. Conse- 
 quently the lines AG and AH lie in the same plane, 
 and therefore the force AD, or its extension AE, 
 must always act in the plane of its balancing forces 
 AB and AC. But the force AB draws the point 
 A in the direction AE, with a force represented by 
 AF ; and the force AG draws it in the same direc- 
 tion, with a force expressed by A I. Wherefore 
 
44 ELEMENTS OF 
 
 these two forces AF and A I are equivalent to the 
 resulting force AE, and hence the segment FE 
 must be equal to AL Join EB and EC ; and the 
 triangles EBF and ACI, having the side FE equal 
 to IA, the side FB or AG or AH equal to 1C, and 
 the contained angle EFB equal to the right angle 
 A 1C, are equal. Whence the angle FEB is equal 
 to IAC, and these angles being alternate, the side 
 BE is parallel to AC ; but they are equal, and 
 therefore AB and CE are likewise parallel. Where- 
 fore, if the lines expressing any two forces be form- 
 ed into a parallelogram, its diagonal will expound 
 the resulting force, or that which is equal and op- 
 posite, to the third or balancing force. 
 
 This remarkable property is called the Parallelo- 
 gram of Forces ; and all the problems in Statics are 
 reducible to the composition and resolution of forces, 
 or to the finding of the sides of a parallelogram from 
 its diagonal, and of the diagonal from the sides of 
 the figure. But the solution is often simplified, by 
 means of other derivative properties. 
 
 In the triangle ABE, it is evident from Plane 
 Trigonometry, that AB : BE or AC : : sin AEB 
 or CAE : sin BAE. When two forces, therefore, 
 balance any third force, they are inversely as the 
 sines of the angles which they make with it. 
 
 If AE (fig. 7.) be made radius, the perpendi- 
 culars EK and EL will evidently be the sines of 
 the angles BAE and CAE. Wherefore, When two 
 
NATURAL PHILOSOPHY. k) 
 
 forces balance a third force, tliey are inversely pro- 
 portional to the perpendiculars let fall from any 
 point in its extension upon the lines representing 
 them. 
 
 From the point A, (fig. 7.) draw AM and AN 
 perpendicular to AB and AC, and terminated by 
 MON, a perpendicular to AE. The exterior angle 
 MAE is equal to both the interior angles AMO 
 and MOA ; and taking away the right angles M AB 
 and MOA, and there remains the angle BAE equal 
 to AMO or AMN. In like manner, the angle 
 CAE or AEB is proved to be equal to ANM. 
 Wherefore the triangle MNA is similar to ABE, 
 and AB : BE or AC : : AM : AN. Whence three 
 forces in equilibrio are proportional respectively to 
 the several sides of a triangle drawn perpendicular 
 to their directions. A triangle so constructed is 
 sometimes called the Triangle of Forces. 
 
 To find the force resulting from the combination 
 of several forces acting upon a point in the same 
 plane. Let the point A (fig. 8.) be drawn by the 
 forces AB, AC, AD, and AE, extended all in one 
 plane. Complete the parallelogram C ABF, and the 
 diagonal AF will exhibit the result of the forces AB 
 and AC. Complete the parallelogram DAFG, and 
 its diagonal AG will express the result of the three 
 forces AB, AC, and AD. In like manner complete 
 the parallelogram EAGH ; and the diagonal AH 
 will represent the force compounded of all the four 
 
46 ELEMENTS OF 
 
 forces AB, AC, AD, and AE. But the construc- 
 tion may be simplified by drawing the lines BF, FG 
 and GH equal and parallel to AC, AD and AE, 
 and finally joining AH, which will express the re- 
 sulting force. 
 
 If the directions of the forces AB, AC, and AD, 
 (fig. 9.) lie in different planes, complete the paralle- 
 logram BACE, and the diagonal AE will express 
 the result of AB and AC. Next, in the plane of 
 AD and AE complete the parallelogram DAEF, 
 and the diagonal AF will represent the compound 
 force, which thus forms the diagonal of the paralle- 
 lepiped. 
 
 If three connected points be urged by three ba- 
 lancing forces in the same plane, their directions 
 will always tend to the same centre. Suppose the 
 physical points A, B and C (fig. 10. and 11.) to be 
 disposed into a triangle, either by threads or inflex- 
 ible wires, and let them remain at rest, while they 
 are pulled by forces acting in the directions AG, 
 BH and CI, these lines when produced will meet in 
 the same point O, either within or without the trian- 
 gle. For from O, the concurrence of GA and HB, 
 let fall the perpendiculars OD, OE and OF upon 
 the sides of the triangle, and join OC. The force 
 AG is held in equilibria by the retraction of the 
 threads in the directions AB and AC ; whence the 
 tension AB is to the tension AC, as OE to OD. 
 For the same reason, the action of the force BH is 
 
NATURAL PHILOSOPHY. 47 
 
 kept in check by the threads pulling the point B in 
 the directions BA and BC ; consequently the ten- 
 sion BA is to the tension BC as OF to OD. But 
 since a general equilibrium obtains, the point B 
 must be drawn with a force equal and opposite to 
 that with which the point A is drawn in the direc- 
 tion AB. Wherefore, by equality of ratios, the ten- 
 sion AC is to the tension BC, as OF to OE. But 
 these tensions must be equal to the opposite tensions 
 CA and CB, by which the point C is pulled by its 
 attached threads. The balancing force CI is, there- 
 fore, an extension of OC, and passes through the 
 same centre O. 
 
 If the centre of concurrence O lies without the 
 triangle, (fig. 11.), the points A and B must be 
 connected by an inflexible rod ; for the force AG 
 is checked by a force proportional to OD, drawing 
 the point A in the direction AC, and by another 
 force proportional to OE, pushing it in the direction 
 BA. If the point O coincides with the vertex C 
 of the triangle, the forces AG and BH will be di- 
 rected along the sides AC and BC, and no power 
 will be exerted at the base, either to distend or con- 
 tract the mutual distance of the points A and B. 
 
 It is obvious that the three balancing forces AG, 
 BH, and CI will have the same intensity as if they 
 originated in O, the centre of concurrence. If, 
 therefore, KOL (fig. 12.) be drawn perpendicular to 
 CO, and KM and LM perpendicular to BO and 
 
48 ELEMENTS OF 
 
 AO, the sides KL, KM, and LM of the triangle 
 thus formed will represent the forces CI, BH and 
 AG. 
 
 Any oblique force AB (fig. IS.) may be decom- 
 posed into a force BC perpendicular to a given plane, 
 and another force AC coincident with the plane. But 
 such a force may likewise be resolved into three given 
 forces at right angles to co-ordinate planes. Thus, 
 let three planes mutually perpendicular pass through 
 the point A (fig. 14.) ; the oblique force AB may 
 be reduced to CB perpendicular to the plane AEC, 
 and CA lying in that plane. Again, the force CA 
 may be reduced into CE perpendicular to the plane 
 AEF, and CD perpendicular to ADG. Wherefore 
 the oblique force AB is decomposed into the per- 
 pendicular forces CB, CD, and CE. 
 
 Suppose it were required to find the figure which 
 would be assumed by several connected inflexible 
 lines attached by the ends to two fixed points, and 
 having parallel forces applied at their several junc- 
 tions. Let the lines AB, BC, CD, DE and EF, 
 consisting either of threads, as in fig. 15., or of 
 slender wires, as in fig. 16., tied or jointed at B, C, 
 D, and E, and fastened to the points A and F, have 
 the weights or parallel forces BG, CH, DI and EK 
 applied to them, and pulling downwards or upwards. 
 To maintain the equilibrium of these forces, it 
 is evident that the compound or polygonal line 
 
NATURAL PHILOSOPHY. Mj 
 
 - 
 
 A BCDEF must take some determinate form. Thus, 
 the point B (fig. 1,5.) is balanced by three forces, or 
 by the weight BG, the strain BA, and the strain 
 BC ; and the point C again is kept at rest by the 
 strain CB, the strain CD, and the weight CH. In 
 like manner, the points D and E are each maintain- 
 ed in equipoise by the action of triple forces, com- 
 posed of vertical weights and oblique strains. The 
 same forces are exerted in fig. 16., only in opposite 
 directions, the strains being there converted into 
 thrusts. 
 
 But the strains BA and BC, (fig. 15.), balancing 
 on both sides of the force BG, are as the sine of the 
 angle CBG to the sine of the angle ABG ; and for 
 the same reason, the strains CB and CD, exerted at 
 C, are as the sine of the angle DCH to the sine of 
 the angle BCH. Now, the points B and C being 
 both at rest, the strain BC must evidently be equal 
 to the opposite strain CB, while the sine of CBG is 
 equal to the sine of its supplemental angle BCH, 
 Wherefore, by compounding the analogies, the strain 
 BA is to the strain CD, as the sine of the angle 
 DCH is to the sine of the angle ABG. In fig. 1 6. 
 the same forces act as thrusts. Wherefore, gene- 
 rally, at any point It, C 9 D and E, the strain 
 or thrust is inversely as the sine of the angle which 
 this makes with a vertical line, or directly as the se- 
 cant of the angle which it forms with the horizon. 
 
 It is evident that, in both figures, an oblique strain 
 
 VOL. i. D 
 
50 ELEMENTS OF 
 
 or thrust AB, if reduced to the horizontal direction 
 AL, will be diminished in the ratio of AB to AL, 
 or of the radius to the cosine of the inclination LAB. 
 Wherefore, since the strain or thrust at A is propor- 
 tional to the secant of that angle, its action in the 
 line of the horizon AF is always as the radius mere- 
 ly. The horizontal strain or thrust, as might be ex- 
 pected, is hence the same through the whole system 
 of points. 
 
 Supposing the junctions to be multiplied indefi- 
 nitely, the series of weights may be conceived as dif- 
 fused over the chain, disposing it into a regular curve. 
 If the base AL (fig. 1? '. and 18.) be therefore di- 
 vided into equal parts, and the verticals MB, NC, 
 OD, &c. drawn, the intercepted portions AB, BC, 
 CD, &c. of the curve, being evidently proportional 
 to the secants of the angles of deflection from the ho- 
 rizon, must express the strains or thrusts exerted at 
 the several points A, B, C, frc. 
 
 By a similar decomposition of forces, it is easy to 
 discover the relation of the weights necessary to form 
 a given polygon. In fig. 15. and 16., the strain or 
 thrust BA is to the weight acting at B, as the sine 
 of the angle CBG is to the sine of the angle ABC ; 
 but if the polygon be supposed to be contained in a 
 given arc of a circle while its sides are given, this 
 angle ABC must have a given magnitude ; conse- 
 quently the strain BA is to the weight BG, as a con- 
 stant quantity to the secant of the deflection at B. 
 
NATURAL PHILOSOPHY. M 
 
 But the strain at B was already shown to be propor- 
 tional to that secant, and therefore the weights at B 
 must be as the square of the secant of deflection. 
 This elegant property can easily be exhibited expe- 
 rimentally. 
 
 Suppose the junctions to be indefinitely multipli- 
 ed, the several weights being disposed in the arc 
 AFL of a circle, (fig. 19- and 20.), which has O for 
 its centre. Let this arc be distinguished into minute 
 equal portions AB, BC, CD, c. and having ap- 
 plied the tangent FQ, draw the secants OM, ON, 
 OP, &c. It follows, from what has been demon- 
 strated, that, in order to maintain an uniform incur- 
 vation, the weights of the portions FE, ED, DC, 
 &c. of the circumference must be respectively pro- 
 portional to OF>, OM% ON*, OP*, &c. From C 
 and P, draw CR and PS perpendicular to OB ; 
 from similar triangles, OC : OP : : CR : PS, and 
 OF or OC : OP : : PS : PQ ; whence, by composi- 
 tion, OC* or OF- : OP* : : CR or FM : PQ. If 
 FM, therefore, denote the weight of an element of the 
 circumference at F, PQ must express the weight of 
 an equal portion at C. The weights of the successive 
 portions FE, ED, DC, &c. are hence proportional 
 to the segments FM, MN, NP, &c., or to the dif- 
 ferences of the tangents of the corresponding angles 
 of deflection. This proposition includes the whole 
 Theory of Arches, whether suspended or incumbent. 
 
 
52 ELEMENTS OF 
 
 Suppose two connected physical points A and B, 
 (fig. 21.), which maintain invariably the same re- 
 lative position, to be drawn by the forces AP and 
 BP to the centre P : complete the parallelogram 
 PAQB, and the diagonal PQ will evidently bisect 
 AB in O ; wherefore a force applied at O in the di- 
 rection PO, and equal to 2PO, will balance the two 
 forces AP and BP. But AB may be made the dia- 
 gonal of any other parallelogram P'AQ'B ; and con- 
 sequently the middle point O is a centre at which 
 the forces from A and B are held in equilibrium by 
 another force tending to the same point P or P'. 
 If these points be removed to an indefinite distance, 
 the forces AP and BP will become equal and pa- 
 rallel, and the opposing force PQ will be double of 
 either of them. 
 
 Conceive now three connected physical points A, 
 B, and C, (fig. 22.), which constantly maintain their 
 relative position, to be drawn towards a centre P in 
 the same plane by the forces AP, BP, and CP. 
 Construct the parallelograms PAQB and PQRC, 
 and draw the diagonals AB, QC, PQ and PR. It 
 is evident that PR will express the resulting force, 
 and the problem is to assign in this line a point O, 
 which may be independent of the variable position of 
 P. The points K and L bisect the diagonals PQ 
 and PR, which are the sides of the triangle PQC, 
 and consequently the base PC is double of KL ; but 
 the converging lines PL and CK must cut each 
 
NATURAL PHILOSOPHY. 53 
 
 other in the some ratio, and therefore PO 2OL 
 and CO zr 2OK. Hence PL = 3OL, and PR = 
 6OL = 3PO ; and CK = 3OK, or OK = * CK. 
 The force PR, which is triple of PO, balances, there- 
 fore, the three forces AP, BP and CP ; but that 
 force may be conceived to be attached at the centre 
 O, which is determined directly from the points A, 
 B and C since K bisects AB, and KO is the 
 third part of KC this centre is consequently in- 
 dependent of the position of P. If P be removed 
 to an indefinite distance, the several forces acting at 
 A, B and C will become equal and parallel, and the 
 counterbalancing force at O will be equal to their 
 sum, and tend in the same direction. 
 
 In like manner, it may be shown, that if any num- 
 ber of atoms be attracted in the same plane by equal 
 and parallel forces, a certain point may be found at 
 which they will be balanced by an opposite aggregate 
 force. Suppose four points A, B, C and D (fig. 23.) 
 to be drawn towards a centre P by the forces AP, 
 BP, CP and DP ; it is evident, from the theory of 
 their composition, that those tendencies will be coun- 
 teracted by a single force in the intermediate direc- 
 tion PO. But, having let fall upon this the perpen- 
 diculars Aa, B&, Cc and Dd ; the force AP may be 
 decomposed into aP and A, the force BP into bP 
 and B&, the force CP into cP and Cc, and the force 
 DP into dP and T>d. Of these, the forces aP, bP, 
 cP and dP, all in the same direction, must be coun- 
 
ELEMENTS OF 
 
 terpoised by the force extending to P through O. 
 The remaining forces a A, &B, cC and dT), being 
 opposed to each other, must produce a mutual ba- 
 lance. Wherefore the perpendiculars Aa and B& 
 on the one side of PO must be equal to Cc and Dd, 
 the perpendiculars on the other side. Bisect AB in 
 H, and CD in I, and let fall the perpendiculars Hh 
 and Li. It is obvious, that Aa -f- B& %Hh, and 
 Cc + Dd 2K ; consequently %Rh = 2li, and 
 HA Li. Wherefore the oblique line HI must be 
 bisected in the centre O. This point is hence deri- 
 ved merely from A, B, C and D, independent alto- 
 gether of the position of P. Whatever be the place 
 of P, the resulting force will pass through O ; and 
 if it be thrown to an indefinite distance, the several 
 atoms will be urged in parallel directions, their 
 forces AP + BP + CP + DP being then equal to 
 the aggregate force # P -(- bP + cP -|- dP exerted 
 at O. 
 
 If the particles have a permanent arrangement, 
 though not in the same plane, the resulting force 
 will in every position pass through the same indivi- 
 dual centre. For suppose that, beyond those parti- 
 cles, a plane were drawn parallel to another touching 
 the resulting force, it is evident that the sum of all 
 the perpendiculars let fall upon it from each of the 
 particles, divided by their number, will be equal to 
 the mutual distance of the two planes. Hence the 
 position of a plane passing through P in the direction 
 
NATURAL PHILOSOPHY. 55 
 
 of the resulting force is given. But if P' be assumed 
 in another place, the position of a second plane pass- 
 ing through P', and touching the direction of the 
 resulting force, is likewise given. The intersection 
 of these two planes is consequently a straight line 
 given in position. Lastly, if the centre of the at- 
 tractive forces be changed to P", a third plane may 
 be assigned, which shall indicate the direction of the 
 resulting force. But this plane must cut the former 
 line of intersection in a given point, which is the 
 centre at which all the forces would be balanced. 
 This centre therefore depends on the mutual arrange- 
 ment of the particles, and has no reference whatever 
 to the position of the variable concourse of those 
 forces. 
 
 But it may be likewise proved, that, about the cen- 
 tre thus found, the system of particles will in every 
 position maintain their equilibrium. For if any 
 fourth plane be made to pass through the common 
 intersection of the three given planes, the sum of 
 the perpendiculars let fall upon it from the several 
 particles on the one side must be equal to the sum 
 of the perpendiculars let fall on the other side. To 
 prove this, it will be sufficient to show, that of lines 
 drawn at given angles to the three planes, the seg- 
 ments intercepted by any fourth plane are mutually 
 balanced. Let (fig. 24.) AK, AL and AM be the 
 three planes, meeting in the lines AB, AC and AD, 
 and let them be cut obliquely by a fourth plane 
 AMK. Conceive another plane AN passing through 
 
56 ELEMENTS OF 
 
 AD, and meeting the plane AMK in K ; if from 
 any point P of the body or system of particles, the 
 lines PF, PG and PH be drawn to the three fixed 
 planes parallel to AD, AB and AC, their sums will 
 be severally balanced. Produce GP to meet the 
 plane AN in E, and a plane passing through GE 
 will cut the planes AN, AL and AK in the lines 
 EQ and GI, parallel to AD, and QI parallel to 
 AB. Whence, employing the symbol f to denote 
 summation, f, PH ~f, GS zz ; but the triangle 
 GSR is evidently given in species, and consequent- 
 ly PG has a given ratio to GS ; wherefore f, PG 
 zz 0, and since, from the general condition,^ EG 
 zz 0, it follows that /, PE zz 0. The plane AN 
 has thus the same property as AK, AL, and AM. 
 Again, the oblique plane AO standing in the same 
 relation to the planes AN, AK and AM, must like- 
 wise produce a balance in the summation of the lines 
 drawn parallel to AB, AC and AD, the axes of the 
 three given planes. 
 
 Every system of particles has therefore a certain, 
 constant and individual point of equilibrium, which 
 is called the Centre of Gravity. If that point be 
 supported, the whole system must hence continue in 
 a state of rest. To preserve this condition, it is ne- 
 cessary that an impediment should be opposed in the 
 line of gravitation, or in the vertical drawn through 
 the centre of gravity. The obstacle may be placed 
 either above or below that centre, or the falling of 
 
NATURAL PHILOSOPHY. 57 
 
 the body may be prevented either by suspension or 
 by support. By the former, a permanent stability 
 is procured ; for if the system or body be drawn 
 aside, the oblique action of gravity will pull it again 
 into the vertical position. The lower the centre of 
 gravity is placed, and the wider is the base, the firm- 
 er will the body stand. But if the support be not 
 of sufficient breadth, the equilibrium will be preca- 
 rious and unstable ; since the moment the vertical 
 line projects beyond the base, the body must totter 
 and irretrievably tumble down. 
 
 In certain cases, however, a body resting upon a 
 single point, may yet have a disposition to recover 
 from any partial derangement, and to resume its 
 vertical position. Thus, if the base be a plane, and 
 the bottom of the body rounded, but such that the 
 centre of gravity lies below the centre of curvature, 
 tfre mass may rock backwards and forwards, but will 
 soon regain its erect site. Let O (fig. 25.) be the 
 centre of the incurvation at the end of the body, and 
 G or g its centre of gravity lying in the axis AO. 
 Conceive the body to be rolled on its horizontal plane 
 from A to A', the point which touched A will merge 
 into a, and the axis will come into the position aO'. 
 Now, if the centre of gravity G stood above O, it 
 would evidently in the position G' lean beyond the 
 vertical A'O', and the body would fall over ; but if 
 the centre of gravity were at g below O, it would 
 still in changing to </, lie within the vertical A'O', 
 
58 ELEMENTS OF 
 
 and consequently the body would roll back to its 
 first position. 
 
 Again, suppose the round end of the body to be 
 implanted upon a circular base, (fig. 26.), while the 
 contact shifts from the point A to A', the radii AO 
 and A'O will obviously converge to P, the centre of 
 the arc A A'. Having drawn A'M parallel to AP 
 to meet aO, if the centre of gravity should lie any 
 where below M, its vertical will fall within the base 
 AA', and the body will, therefore, have a constant 
 tendency to redress itself and recover its first posi- 
 tion. But from the property of diverging and pa- 
 rallel lines, OP : AP : : aO : aM, or AP AO 
 : AP : : AO : a M, the height of what may be called 
 the metacentre, or extreme vertical limit of the centre 
 of gravity. If the base be convex, the first term of 
 the analogy will be the sum, instead of the diffe- 
 rence, of the radii AO and AP. On this princi- 
 ple seems to depend the curious phenomenon of 
 Rocking or Laggan Stones, the natural joints of the 
 columns being, in the course of ages, worn by the 
 agitation of the wind into regular curved surfaces. 
 
 The centre of gravity of any body is determined, 
 by dividing the sum of the distances of its consti- 
 tuent particles from any plane by the number of those < 
 particles. The surface of a triangle, its three sides, 
 and its angular points, have all the same centre of 
 gravity, which is situate at two-thirds of the length 
 
NATURAL PHILOSOPHY. 59 
 
 of a straight line drawn from the vertex to bisect 
 the base. The centre of gravity of a parabola lies 
 at the distance of three-fifths of the axis from the 
 vertex ; that of a cone is placed at three-fourths of 
 the length of the axis, and that of an hemisphere at 
 five-eighths of the radius. A pyramid and its four 
 terminating points have the same centre of gravity. 
 When the body consists of two parts, their com- 
 mon centre of gravity will be found, by dividing the 
 distance between the centre of each reciprocally as 
 their corresponding weights. The principle may be 
 extended to bodies which are more complex. Thus, 
 the centre of gravity of two portions being deter- 
 mined, they may be conceived to be united in that 
 point and connected with a third portion, and the 
 centre of the three portions found. All these again 
 may be supposed collected in this last point and made 
 to balance against a fourth portion, and their com- 
 mon centre of gravity computed. In this way, the 
 process may, by successive steps, be carried to any 
 extent, and whatever order is followed, the result 
 will be always the same. Thus, in fig. 27., the cen- 
 tre of gravity of four points A, B, C and D in the 
 same plane, is found by a progressive procedure. 
 Join AB, and bisect it in K ; conceive the points 
 A and B to be collected at K, and trisect CK in L, 
 the centre of these, and the third point C ; lastly, 
 suppose A, B and C to be condensed at L, cut off 
 
60 ELEMENTS OF 
 
 OL, a fourth part of DL, and O will be the com- 
 mon centre of gravity of all the four points. 
 
 The same method is applicable either to surfaces 
 or solids, the former being subdivided by triangles, 
 and the latter decomposed into wedges. The cen- 
 tre of gravity of the larger portion of any figure 
 maybe conveniently found, by estimating from the de- 
 fect. This centre is in a trapezium, deduced from the 
 two triangles formed by producing its oblique sides, 
 and in the frustum of a cone, from the cone itself 
 and its upper portion. For the excess of the mass 
 A above B is to B, as the mutual distance BA to 
 the interval in this direction of the fragment beyond 
 A. 
 
 Of irregular bodies, the centre of gravity may be 
 discovered practically in various ways. If the body 
 be poised, for instance, in two different positions on 
 a sharp edge, the vertical drawn from the point of 
 intersection will pass through the centre of gravity. 
 Or, if a loose thread or string have its ends fastened 
 to two distinct points of the body, and suspended in 
 two 'different positions from a fixed pivot, the verti- 
 cals let fall from this will cross in the centre of gra- 
 vity. 
 
 Many singular and paradoxical appearances are 
 dependent on the properties of the centre of gravity. 
 Hence a ball may be suspended beyond the border 
 of a table ; hence an eccentric loaded cylinder will 
 
NATURAL PHILOSOPHY. ()1 
 
 roll partly up an inclined plane ; and hence also a 
 double cylinder will seem to advance along two ri- 
 sing and spread edges. Various toys are constructed 
 on the same principles, which likewise direct the skill 
 of the balancer. 
 
 It is a remarkable property of the centre of gravity 
 of any number of points, that forces directed from 
 them to this centre, and indicated by the several lines, 
 will maintain a mutual equilibrium. Thus, if O 
 (fig. 28.) be the centre of gravity of the points A, 
 B, C, D, E, &c., the forces AO, BO, CO, DO and 
 EO will all balance at O. For if a plane were sup- 
 posed to pass through O, the several oblique forces 
 O A, OB, OC, OD, OE, &c. might be reduced into 
 forces perpendicular to that plane and other forces 
 directed along it. But since an equilibrium obtains, 
 the perpendiculars on the one side of the plane must 
 be together equal to those on the other, and conse- 
 quently O is the centre of gravity of all the points 
 A, B, C, D, E, &c. 
 
 A still more distinguished property of the centre 
 of gravity of any line or plane is, that the circum- 
 ference traced by this about its boundary as an axis, 
 drawn into the line or plane of revolution, is equiva- 
 lent to the surface or solid thus described. In short, 
 the space involved by circumvolution is the same, as 
 if the line or plane had advanced directly forwards 
 over the same extent of description. Hence the sur- 
 
62 ELEMENTS OF 
 
 face and contents of solids of revolution are easily 
 discovered. 
 
 But the most elegant property of the centre of 
 gravity is, that, in any given plane, the sum of the 
 squares of the distances of physical points from their 
 centre of gravity is less than the aggregate of the 
 squares of their distances from any point in the cir- 
 cumference of a circle described about that centre, 
 by the square of the radius multiplied by the num- 
 ber of those points. It hence follows that the sum 
 of the squares of the distances of points from their 
 centre of gravity is always a minimum. 
 
 If three fixed points A, B and C (fig. 29.) in a 
 horizontal plane be drawn by the forces AO, BO 
 and CO to a remote centre O, the tension may be 
 balanced, by triple the vertical force PO acting at 
 an intermediate point P of the same plane. For this 
 force PO being perpendicular to the plane, the forces 
 expressed by the bases AP, BP and CP of the se- 
 veral triangles AOP, BOP and COP must be in 
 equilibrium, and consequently P is the centre of gra- 
 vity of the three points A, B and C. If we suppose 
 m particles to be collected at A, and n particles at 
 B, p particles at C, the resulting vertical must still 
 pass through P, the common centre of gravity of the 
 clusters or weights m, n and p at the points A, B 
 and C. Wherefore a force m x AP must balance 
 n X BP and jt? X CP, and it will thence follow that 
 m is to n, as the area of the triangle BPC is to that 
 
NATURAL PHILOSOPHY. ) 
 
 of APC, and that m is top as the triangle APB is 
 to BPC. 
 
 Conceive the centre O to be now removed to a 
 vast distance, and the parallel forces directed to it 
 will express the pressures which a horizontal plane 
 resting upon the angles at A, B, C, would exert 
 against those supports when a weight is laid at the 
 point P. These pressures would be equally shared 
 if the weight were incumbent on the centre of gra- 
 vity of the triangle. In other positions the distribu- 
 tion of pressure would be unequal, depending on the 
 relative proportions of the interior triangles APB, 
 BPC and CPA. 
 
 Let ACB (fig. 30.) be a perfectly flexible chain, 
 whose ends are attached at the points A and B. If 
 we consider any portion CF of this chain, intercepted 
 from the vertex or lowest point C, its weight must 
 be supported by the oblique tensions exerted at C 
 and F in the directions of CL and'FL, the tangents 
 at those points. The strains at C and F may there- 
 fore be conceived as acting at the common point L, 
 to which likewise the action of the weight GL must 
 be directed ; or the vertical LG will pass through 
 G, the centre of gravity of the arc CF, where all 
 its efforts are united. Wherefore the strain at the 
 lowest point C is to the weight of CF, as the sine of 
 the angle GLF, or the cosine of FLM, is to the sine 
 of FLM, that is, as radius to the tangent of the an- 
 
64 ELEMENTS OF 
 
 gle FLM, which a line touching the curve at F 
 makes with the horizon. Hence the tension at C 
 being constant, the weight, and therefore the length, 
 of any portion CF of the equable chain must be pro- 
 portional to the tangent of the deflection of the curve 
 at F ; which is the geometrical definition of the CA- 
 TENARY. (See Geometry of Curve Lines, p. 383.) 
 
 Again, from the same equilibrium of forces, about 
 the point L, the strain LC is to the strain LF, as 
 the sine of the angle FLG, or the cosine of FLM 
 is to the sine of CLG or the radius, that is, as the 
 radius to the secant of the angle FLM, which the 
 curve makes with the horizon. The strain exerted 
 at any point F is thus proportional to the secant of 
 deflection, as was formerly shown. 
 
 If therefore the vertical line CO, or parameter of 
 the catenary, represent the strain at the lowest point 
 C, and the generating circle be described about the 
 centre O ; a straight line EH, drawn to touch this cir- 
 cle, will be parallel to the tangent FT applied to the 
 curve, and equal to the length of the intercepted seg- 
 ment CF. Hence the tension exerted at any point 
 F will be expressed by OE, which is the secant of 
 deflection to a radius OC or the parameter : It thus 
 exceeds the tension at the lowest point C by the ab- 
 sciss CE. A tangent DI to the circle will hence 
 be equal to the length of half the curve CB, and will 
 indicate the flexure of its extremity B, while OD 
 will denote the tension at that point. 
 
NATURAL PHILOSOPHY. O.J 
 
 1. Suppose the parameter of the Catenary, or the 
 measure of tension at its lowest point C, to be equal 
 to the depression DC. Since OC, being denoted by 1, 
 it is evident that the tangent DI, which is equal to 
 BC, half the curve, will be expressed by ^/ 3 = 
 1.7320508. Wherefore, by Prop. IV. of the Cate- 
 nary, the hyperbolic logarithm of the ratio of OC 
 to OC + CD + CB or of 1 to 3.7320508 will re- 
 present the ordinate DB, or half the width AB ; con- 
 sequently DB= 1.3 169578, or AB = 2.6339156, 
 the entire length of the chain ACB being 3.4641016. 
 In this case, the strain at the lowest point is express- 
 ed by 1, and at the ends A and B by 2. 
 
 2. Next, suppose that the strain at the lowest 
 point C is equal to the whole weight of the chain. 
 
 It is obvious that OC being still denoted by 1, Dlis -, 
 
 and OD = / = 1.1180340, or CD = .1180340 ; 
 
 ^r At 
 
 whence DB = hyperbolic logarithm of 1.6180340 
 or .4812117, the whole span AB being .9624234, 
 and the length of the chain itself 1. Here the 
 width is 8.1538 times, and the extent of the curve 
 8.4724 times, the depression : The tension at C is 
 1, and at A or B it is 1.118034. 
 
 3. Again, suppose the tension at the lowest point 
 to be double the weight of the chain. The para- 
 meter being expressed by unit as before, DI will 
 
 VOL. i. E 
 
66 ELEMENTS OF 
 
 bei and consequently OD = J ^| = 1.0307764, 
 4 ID 
 
 or the depression CD = .0307764. The ordinate 
 CB is therefore equal to the hyperbolic logarithm 
 of 1.2807764 or .2474664, and the whole span 
 AB = .4949328. Here then the span is 16.0816 
 times, and the extent of the chain 16.2462 times, 
 the depression CD. The strain at the lowest point 
 is 1, and that at both ends 1.0307764. 
 
 When the quantity of depression of the catenary 
 is comparatively small, as happens in all practical 
 cases, an easy approximation may be found. Thus, 
 if the span AB be denoted by b, the depression DC 
 by d, and the whole length ACB of the chain by I ; 
 it may be shown, from the general nature of curve 
 lines, that the semiarc CB is very nearly equal to 
 
 2DC* 8d* 
 
 DB+HWfy- ; and consequently l=b+- . But the 
 
 semiarc is equal to the tangent DI, and its square is 
 consequently equal to OD*-OC* = DC(OD+OC) = 
 CD.DNj orp denoting the parameter OC, it fol- 
 
 b* 4td* 
 lows approximately, that - \- -y- = d (%p + d) 
 
 JP 3c? 
 
 = 2pd + d* 9 and therefore/? =+77-, or the strain 
 
 o(Z> O 
 
 at the lowest point. The strain at the extremities 
 A and B will hence be expounded by o^+zp 
 
 These expressions may be converted into o- 
 thers involving the length of the chain. Thus, 
 
NATURAL PHILOSOPHY. (>7 
 
 I* d /* d 
 
 which last expressions hence represent the ten- 
 sions at the lowest point C, and at both extremi- 
 ties A and B. As examples of approximation, some 
 of the former instances may be resumed. Thus, let 
 the strain at the lowest point be equal to the whole 
 weight of the chain ; it was then found that I = 1 , 
 
 8,7* 
 
 and d= .1 18034. Wherefore b = I 57-= .9628, 
 
 ol 
 
 true to every place of figures ; and the strains at C, 
 and at either extremity A or B, are respectively 
 
 =r-j and /ry+-^> or 1 and '1.118034, likewise 
 
 <j(l- /w OCl/ Xi 
 
 the same as before. Again, when the strain at the 
 lowest point is double of the weight of the chain ; 
 in which case, I zz. 5, and d zz .0307764. Here 
 b = .4949483, which approximates extremely near 
 to the correct measure .4949328. The strains 
 come out exactly as in the rigorous calculation. In 
 practice, where the depression seldom exceeds the 
 fourteenth part of the entire length of the chain, this 
 very simple approximation will, therefore, be suffi- 
 ciently accurate for every purpose. The engineer 
 has only to recollect, that when the depression 
 amounts to an eighth part of the length of the chain, 
 the strain at each end is just equal to the whole of 
 the weight sustained ; and, in other cases, the quan- 
 tity of depression will be inversely as the load. 
 
t)8 ELEMENTS OF 
 
 Since three balancing forces AG, BH and CI 
 (fig. 31.) tend always to the same centre, however 
 oblique is the triangle ACB, this property must 
 likewise hold in the extreme case, when the points 
 A, B, and C may be considered as ranging in a 
 straight line. Let ACB, therefore, be a rod or 
 inflexible line, and suppose the extremities A and 
 B to be drawn by forces tending to O, while an in- 
 termediate point C is sustained by a force in the di- 
 rection CI. But this direction must proceed from 
 O, the concurrent centre of the three forces. Draw 
 CE and CF perpendicular to AO and BO, the di- 
 rections of the other J;wo forces ; the force acting 
 at A must be to the force AB, as CF to CE. If 
 the point O be removed to an indefinite distance, 
 the lines EA and BF will become parallel, and con- 
 sequently CF is to CE, as CB is to CA. Where- 
 fore, in the case of parallel forces in equilibria, the 
 force at A is to the force at B, as CB is to CA. 
 Instead of supporting the point C by a force CI, it 
 may be made to rest on a firm obstacle or fulcrum, 
 which would exert an equal force upwards ; the 
 rod ACB would then become a lever, having the 
 arms CA and CB, at the extremities of which 
 the power and weight are applied. Hence the 
 general principle of the lever, that the power and 
 weight are inversely as their distances from the 
 fulcrum or point of suspension. 
 
 Because the forces acting in the directions AO 
 
NATURAL PHILOSOPHY. ()<) 
 
 and BO are inversely as the perpendiculars let fall 
 from C upon them, the equilibrium will be main- 
 tained at whatever points in the lines AO and BO 
 those forces be applied. Wherefore the same forces 
 attached at the extremities N and P of the bent le- 
 ver NCP, would likewise produce a mutual balance. 
 
 But the property of the straight lever, that the 
 power and weight are inversely as their distances 
 from the point of suspension, may be deduced from 
 considerations of the most familiar kind. Let AB 
 (fig. 32.) represent a cylinder of homogeneous mat- 
 ter, such as wood or metal : It could evidently be 
 supported at the middle point O, where the stress 
 will be the same if the whole weight had been con- 
 centrated in that point. But the cylinder might be 
 conceived to consist of two distinct portions, AC 
 and BC, which would be separately sustained at 
 their middle points D and E. Wherefore, the 
 weight of AC attached at D, and the weight of BC 
 attached at E, would balance the inflexible line DE, 
 if upheld at the centre O by the whole weight of 
 AB. But OD = AO AD = ^AB JAC=BC, 
 and OE = OB EB = AB CB = AC ; 
 consequently OD is to OE, as BC to AC, or as the 
 weight concentrated at E is to the weight concen- 
 trated at D. This in substance is the elegant de- 
 monstration of the property of the lever, given by 
 Stevinus and Galileo. 
 
70 ELEMENTS OF 
 
 The next object is, from the decomposition of 
 forces, to assign the portion of any Weight which is 
 supported by the action of an inclined plane. Every 
 plane, it should be remarked, is in some degree flex- 
 ible. Under the point where pressure is applied, the 
 fibres or connected particles bend into a concave sur- 
 face, till their mutual approximation creates a re- 
 pulsive energy sufficient to withstand the force di- 
 rected to the centre of concavity, and therefore per- 
 pendicular to the plane. Let the horizontal and 
 vertical lines AB and BC (fig. 33.) determine the 
 oblique position of the plane AC, on which a weight 
 is supposed to be placed at D. Let FD denote the 
 measure or vertical pressure of that weight, and de- 
 compose this into HD perpendicular to the plane, 
 and FH or ED parallel to it. But HD is the only 
 part supported by the plane, and the other part DE 
 would require to be sustained by a force acting along 
 AC. The triangle DEF is obviously similar to ABC, 
 and therefore DF is to DE, as AC to CB ; that is, 
 the whole weight is to the force of descent on the 
 oblique plane, as the length of the plane to its height, 
 or as radius to the sine of its elevation. 
 
 The true proportion of any weight to its tendency 
 of descent along an inclined plane was first establish- 
 ed by Stevinus, who derived it from a very simple 
 but extraneous consideration. Suppose a chain or a 
 ehaplet composed of equal attached balls were thrown 
 loosely over the triangle ABC, (fig. 34.) and the 
 
NATURAL PHILOSOPHY. 71 
 
 ends united a little below the base AB. It is evi- 
 dent this chaplet would remain at rest ; for were it 
 drawn partly round, each ball would always be suc- 
 ceeded by a similar one in the same situation ; if it 
 began of itself to move, therefore, it would move for 
 ever, which is a proposition inadmissible. But the 
 branch of the chaplet below the base, being suspend- 
 ed equally at both extremities, must evidently be in 
 a state of equilibrium. Wherefore the remainder of 
 the chaplet would by itself maintain a balance, or 
 the part of it which hangs along the perpendicular 
 must form an equipoise to the part that leans against 
 the inclined plane ; and consequently a weight pla- 
 ced on the plane, would be to the force which in 
 the same direction sustains it, as the length, to the 
 height, of the plane. 
 
 From the resolution of forces, may likewise be dedu- 
 ced the Theory of the Pulley. This elementary 
 machine consists of a circle moveable about an axis, 
 and having a very flexible cord applied in the groove 
 of its circumference. Let a pulley, from whose cen- 
 tre D (fig. 35.) hangs a weight P, be supported by 
 the bend of a cord which has its ends attached at A 
 and B. It is evident that the tension of this cord at 
 A may be considered as acting at any point in the 
 direction of the tangent AC, while the tension at 
 B is exerted in the direction of the tangent BC. 
 These oblique forces may, therefore, be conceived to 
 
72 ELEMENTS OF 
 
 unite their efforts at the point of concourse C. But 
 the weight D, acting in a vertical line from the cen- 
 tre or axis of the pulley, must pass through the same 
 point C. Wherefore, this weight is to the strain of 
 the cord at A or B, as the sine of the angle ACB, 
 is to the sine of its half ACE or BCE ; or, having 
 let fall the perpendicular AF upon BC, the weight 
 P will be to the strain at A, as AF to AE. Hence, 
 if AF were to represent the whole weight applied, 
 AE and BE would express the portions supported 
 by the joint action of the cord at A and B. The 
 loss of force occasioned by this obliquity is, therefore, 
 in the ratio of AB to AF, or as radius to the cosine 
 of the angle ACE. But if the points A and B be 
 brought nearer, and consequently the angle ACB 
 diminished, AF will obviously approximate to AB ; 
 and when the opposite folds of the cord are parallel, 
 they will each sustain exactly one half of the attach- 
 ed weight. 
 
 Hence, if the end of the cord be fastened at B, 
 the J vrce at the other end pulling in the direction 
 CB, and requisite to sustain the given weight P, is 
 to that weight^ as the sine of the angle A CE to the 
 sine of its double A CB 9 which approximates as these 
 angles diminish, to the ratio of 1 to 2, the limit of 
 parallel action. 
 
NATURAL PHILOSOPHY. 
 
 III. PHORONOMICS 
 
 Explores the properties of Moving Bodies. It is 
 often comprehended under Dynamics, which traces 
 the series of changes arising from the application of 
 Force. This most important branch of physical 
 science is entirely of modern origin ; and though it 
 rests on the simplest principles, it has yet, by the aid 
 of the higher analysis, been pushed to an amazing 
 extent. 
 
 It would be superfluous to attempt any definition 
 of motion, and quite futile to answer the sophisms 
 that have been urged against its very existence. The 
 Aristotelian distinction of motion into Natural and 
 Constrained was rightly conceived j but, from want 
 of close observation, it was inaccurately applied. 
 
 All motion is performed either in a straight line 
 or in a curve ; and it flows uniformly, or with an 
 accelerating or retarding pace. The velocity or 
 celerity of a moving body is the minute space which 
 it describes in a given instant of time. 
 
 If any body be held in equipoise by the action of 
 combined forces, it. will continue at rest. Thus, a 
 book laid upon the table will press down the surface, 
 till the approximation of the succumbent particles 
 
74 ELEMENTS OF 
 
 exerts a repulsive effort just equal to the weight of 
 the book, and sufficient therefore to sustain it in the 
 same position. All stability is hence produced, by 
 the mere balance of opposing forces. 
 
 But if this equilibrium be deranged, the body will 
 advance in the direction of the preponderating force. 
 Supposing that, after having thus acquired motion, 
 it be liberated from external influence, and abandon- 
 ed to its mere impulsion, the object is now to ascer- 
 tain experimentally, as the only basis of Phorono- 
 mics, what course and rate the body will pursue. 
 Unfortunately these conditions are strictly unattain- 
 able in our place of the universe. The action of gra- 
 vity is never one moment suspended, and no motion 
 upon earth is exempt from the operation of some re- 
 tarding causes. We can only approximate to the 
 undisturbed effect, by neutralizing the constant in- 
 fluence of gravitation, and diminishing the obstruc- 
 tion created by friction and atmospherical resistance. 
 If a body move along a perfectly horizontal plane, 
 its weight being exerted vertically, will neither in- 
 cite nor impede its progress. Let a ball be propel- 
 led along a spacious level floor, and its motion will, 
 in every case, appear rectilineal. An ivory ball, roll- 
 ed with a certain velocity on a Turkey carpet, will 
 suffer a visible relaxation in its course ; but, with the 
 same impulsion, it will advance farther on a surface 
 of baize, still farther on smooth oaken planks, and, 
 on a sheet of pure ice, it will scarcely seem at all to 
 
NATURAL PHILOSOPHY. 7 
 
 abate its celerity. A leaden ball, in similar circum- 
 stances, will maintain its motion through a much 
 greater extent. The gradual diminution of velocity 
 being thus proportioned to external impediments, 
 must be attributed solely to their influence. We are 
 entitled, therefore, to conclude, that any body, after 
 having received an impulsion, would, if left wholly 
 to itself, continue to move in a straight line, and with 
 an uniform celerity. Every deviation from the rec- 
 tilineal course betrays the influence of some extrin- 
 sic deflecting force ; while the relaxation of motion, 
 in any case, intimates the accumulated effect of in- 
 cessant impediments. 
 
 If a sling be whirled and its pellet discharged, it 
 will no longer describe a circle, but will shoot off* in 
 a tangent. The rectilineal is hence that motion 
 which the missile naturally takes, its previous circu- 
 lation having been merely constrained by the con- 
 tinued retraction exerted by the muscles of the arm. 
 In all the circular movements of machinery, the de- 
 flection from a rectilineal course is maintained by an 
 attractive or centripetal force, occasioned by the dis- 
 tension of the particles from the axis to the circum- 
 ference. 
 
 When friction is greatly diminished, a piece of fine 
 mechanism, set once in motion, will continue to cir- 
 culate for a very long time. This may be effected, 
 either by adapting friction-wheels, or by making the 
 pivots turn in fine holes of agate or diamond. 
 Under the exhausted receiver of an air-pump, the 
 
ELEMENTS OF 
 
 motion is found to last much longer. In short, the 
 more the friction is diminished and the resistance 
 of the air abstracted, the nearer will the machine 
 approach to perfection, and to the constant main- 
 tenance of its celerity. It is the impossibility of 
 absolutely excluding those impediments on the sur- 
 face of our globe, that must prevent any efforts of 
 ingenuity from ever achieving a perpetual motion. 
 But, in the celestial spaces, no such obstructions oc- 
 cur, and the revolution of the planetary system, with 
 all its balanced cyclical variations, bears the stamp 
 of ETERNITY. 
 
 If the boxes of Atwood's machine, (which turns 
 smoothly upon friction wheels, (see fig. 35.) in which 
 those wheels are, for the sake of clearness, however, 
 omitted, and likewise the attached pendulum,) be 
 equally loaded, and a small bar laid across one of 
 them to generate an impulsion ; after this bar has 
 been intercepted in the passage through a stage placed 
 a few inches lower, the progress of both boxes up- 
 wards and downwards will be found scarcely to dif- 
 fer from a perfectly uniform motion, especially if the 
 velocity incited be not too feeble. This motion is 
 only vertical indeed, but the same experiment could 
 easily be performed in any given direction, with equal 
 rollers running along an oblique plane. 
 
 We are, therefore, entitled to infer, that all bodies 
 are quite passive or indifferent to a state of rest or 
 of motion, and would continue for ever at rest, or 
 
NATURAL PHILOSOPHY. 77 
 
 persevere in the same rectilineal and uniform motion, 
 unless disturbed by the operation of some extrinsic 
 force. This absolute inertia of matter is the great 
 law on which rests the whole Doctrine of Phorono- 
 mics. Notwithstanding its extreme simplicity, it lay 
 hid for ages, and was first discovered by the sagacity 
 of Galileo. The ancients attributed to matter a cer- 
 tain inaptitude, reluctance, or renitency to motion. 
 Nor was such a conclusion at all singular, at a period 
 when only random observations were made, and the 
 mechanical arts had not yet attained any precision. 
 Every motion experienced upon earth insensibly re- 
 laxes and becomes extinguished ; and it required 
 mature reflection to discriminate this effect as a mere 
 casual result, and not an inherent property of mat- 
 ter. 
 
 On the birth of accurate science, the term inert- 
 ness or inertia, which had been borrowed from the 
 indolence or sluggishness of animal effort, totally 
 changed its meaning. Kepler, conceiving the dis- 
 position of a body to maintain its motion as indicat- 
 ing an exertion of power, therefore prefixed the word 
 vis ; and this compound expression, vis inertia, has 
 been long retained. But the single term inertia, 
 signifying merely passive indifference, should now be 
 preferred, as more correct and appropriate. 
 
 The inertia of matter is, in every incident of life, 
 unceasingly recalled to our observation. Thus, a per- 
 son sitting erect in a chaise is thrown backwards, the 
 
78 ELEMENTS OF 
 
 moment the horses start, or rather he retains his rest* 
 ing place for an instant, while the vehicle begins to 
 advance ; but having once acquired the progressive 
 motion, he is pitched forward if the horses suddenly 
 stop, that is, he holds on his course for an instant 
 after the chaise has come to rest. If two wine-glas- 
 ses, placed on a table at the interval of about half a- 
 foot, have a thin narrow ruler laid across them, with 
 a piece of money above the mouth of each, on strik- 
 ing a smart blow against the end of the ruler, it will 
 shoot forwards, while the coins, not having time to 
 acquire this impulsion, will merely drop into the 
 glasses. Let two or three different kinds of smooth 
 worn coins, or of the same kind only distinguished 
 by marks, be piled on the bottom of an inverted egg- 
 cup, which is then pushed suddenly along the table, 
 it will be found that the coins will be thrown for- 
 ward always exactly in the order in which they lay, 
 the lowermost, receiving the greatest share of the im- 
 pulsion, being projected farthest, and the uppermost, 
 being the least affected, scarcely advancing at all. 
 In like manner, while a ship glides swiftly through 
 smooth water, a ball dropped from the topmast 
 will strike the same spot of the deck, as if it had 
 fallen when the vessel was at anchor. On this 
 principle, the balancer easily tosses up oranges and 
 catches them again, as he ambles in the ring ; 
 for these fruits acquire the motion of his ^palfrey. 
 Hence the rotation of our globe about its axis does 
 
NATURAL PHILOSOPHY. 79 
 
 not prevent a stone, falling from the top of a high 
 tower, from alighting at the bottom. The minute 
 deviation lately detected in this perpendicular de- 
 scent, arising from the excess of the circular descrip- 
 tion of the top above that of the bottom, as we shall 
 soon find, affords the most direct proof of the revo- 
 lution of the Earth. 
 
 Every motion, then, is Rectilineal or Curvilineal. 
 Rectilineal motion, again, is either uniform, or sub- 
 ject to acceleration or retardation. 
 
 In whatever way motion be generated, it will, if 
 left free and undisturbed, persevere uniformly in the 
 same straight line. In this simplest kind of motion, 
 three things are to be considered : The velocity 
 with which the body moves, the portion of time 
 elapsed, and the corresponding space described. 
 These may be denoted by the letters v, t, and s. 
 It is evident that s = vt, and, consequently, that 
 
 rt . . 
 
 v = - and t = -. Hence the space of description 
 
 is compounded of the time and velocity, and this 
 velocity is directly as the space, and inversely as the 
 time ; while the time itself is directly as the space, 
 and inversely as the velocity. 
 
 In the case of accelerated or retarded motions, 
 the inciting or obstructing force is especially to 
 be distinguished : It may be generally denoted 
 by the letter /. Force and Velocity, Time and 
 
80 ELEMENTS OF 
 
 Space, are quantities in a state of continual flux, 
 and passing through a series of proximate stages. 
 These contiguous steps may be marked by re- 
 peated accents, and their successive intervals will 
 form the minute differences or differentials of the 
 varying quantity which are expressed by the pre- 
 fix ?). According to this notation, D f ' = f f f, 
 7) v = v f v, D t =s p t, and 7) s = s' s. Now, 
 the velocity of a body being the result of the ac- 
 cumulated impression of the accelerating force, 
 every minute accession which it receives must 
 evidently be proportional to that intensity and to 
 the duration of the instant in which it is exerted. 
 
 Hence, Dv = / fit, and/ = . Again, though the 
 
 velocity be continually varying, it may be view- 
 ed as constant at least during an instant of time ; 
 the element of the space described will therefore 
 be proportional to the velocity, and to the dura- 
 tion of its momentary flight. Whence !3s = vdt, 
 
 Ds 
 
 and v = . Combine this expression with the for- 
 mer, and iTds = v~dt. = vdv. Thus, the acce- 
 
 ot 
 
 lerating force multiplied into the variation of the 
 space, is equal to the velocity multiplied into its cor- 
 responding variation. But this expression admits 
 of a simpler form. Since (v r + t?) (v r v) = v'* v z , 
 
 d 
 
NATURAL PHILOSOPHY. 81 
 
 and v' may be considered as approximating indefi- 
 nitely to v, it follows that 2 v^d v = ^d.v* and v~d v 
 = D.u*, or the product of the velocity into its va- 
 riation is equal to half the variation of the square of 
 the velocity. Wherefore,/^* = D.v% or the ge- 
 nerating force multiplied by each successive element 
 of the space described, will give an amount equal to 
 half the variation of the square of the corresponding 
 velocity. 
 
 Suppose this generating force to be constant, and 
 the sum of all the fds must evidently be the pro- 
 duct off into the aggregate elements of s, whence in 
 the case of a constant acceleration, fs = v*. But 
 it was shown generally, th&tfDt = ^dv, and conse- 
 quently f t SB v ; by substitution, therefore, f s = 
 ^f*t* 9 OTS=^ft z . Now, abstracting from the resis- 
 tance of the air, bodies near the surface of the earth 
 are observed to fall about sixteen feet during a se- 
 cond. In round numbers, f, denoting the power of 
 gravitation, is hence expressed by 32. Substitute 
 this number, and the several formulas will become 
 definite. Wherefore, 1, 32 t = v, or the velocity 
 acquired in falling, is equal to thirty-two feet multi- 
 plied by the number of seconds elapsed ; 2, s = I6t* 
 = (4)% or the space described, is equal to sixteen 
 multiplied into the square of the time, or to the 
 
 square of quadruple the time ; 3, s = ** ' 
 
 64 
 
 or the space of descent, is equal to the square of the 
 VOL. i. F 
 
08 ELEMENTS OF 
 
 eighth part of the acquired velocity ; 4, t = 
 or the number of seconds in falling, is the fourth 
 part of the square root of the height j 5, v Sy/^, 
 or the velocity acquired, is equal to eight times the 
 square root of the height ; and since v = S ^/s, and 
 t = \ ^/s, therefore 6, vt=2 s, or the body with its 
 acquired velocity, would, in the time of its descent, 
 shoot through double the space. 
 
 The same formulae apply in the case of bodies 
 thrown directly upwards, gravitation acting then as 
 a retarding force. But similar results are deduced, 
 from the inspection of a simple geometrical diagram. 
 Since quantities may be expressed by lines or surfaces, 
 let (fig. 37.) the equal portions AB, BC, CD, DE, 
 &c. denote the successive instants of time elapsed, 
 and the perpendiculars BL, CM, DN, EO, &c. 
 equal to the whole segments AB, AC, AD, AE, &c., 
 will express the velocities acquired, from the accu- 
 mulated action of the same inciting force. The 
 spaces described, in those several instants will hence 
 be represented by the elementary rectangles CL, 
 DM, EN, FO, c. Wherefore, the whole space of 
 descent will be expressed by the compound or ser- 
 rated triangle KAT. But the indented boundary 
 will obviously pass into a straight line, if the inter- 
 vals be indefinitely multiplied, and the impulses gra- 
 duate into a continued force. 
 
 Since the circumscribing square AKTV is double 
 
NATURAL PHILOSOPHY. 83 
 
 the area of the triangle, it follows, as already stated, 
 that a falling body, if no longer accelerated, would, 
 with its acquired velocity, describe double the space 
 in the same time. Hence, also, in constantly acce- 
 lerated motions, the space described is always pro- 
 portional to the square of the time, or to the square of 
 the velocity. The inciting force is here conceived 
 to be denoted by unit; but if it undergo a change, the 
 annexed triangle will be made to correspond, by in- 
 creasing or diminishing its base KT into KT', in the 
 same ratio. Hence, during the same interval, the 
 acquired velocity, and the space of description, will 
 be as the inciting force. 
 
 These theorems respecting uniform motion and 
 motion constantly accelerated or retarded, are il- 
 lustrated and confirmed by Atwood's machine. 
 Thus, each box being loaded with 15 J ounces, if a 
 bar equal to one ounce be laid over the box A (fig* 
 36k), this excess of weight is the 32d part of the 
 compound mass, which will therefore acquire that 
 share of the entire celerity produced by the accele- 
 ration of gravity. The box A will hence, in a se- 
 cond, drop through the space of 6 inches, and then, 
 leaving its surplus load at the stage, it will conti- 
 nue to descend uniformly, at the rate of a foot each 
 second. Suppose the boxes A and B to be loaded 
 respectively with 16 and 15 ounces, the motion 
 will be constantly accelerated, and the inciting force 
 
84 ELEMENTS OF 
 
 being the same as before, the descents will form a 
 series of squares, 32 times less than those of gravity. 
 The box A will fall through 6 inches in one second, 
 through 24 in two seconds, and through 54 in three 
 seconds. Again, let each box hold 15^- ounces, while 
 A has a bar laid across it ; after A has, in three 
 seconds, reached 54 inches, and dropt the bar, let 
 B take up another bar, and it will perform an op- 
 posite retarded motion, rising again to the point 
 from which it fell. This alternate ascent and de- 
 scent may continue for some time. 
 
 Such examples are confined to uniformly acce- 
 lerated or retarded motions. But the principles 
 may be extended geometrically to all cases where the 
 inciting force is some function of the distance. Let 
 AB (fig. 38.) express the space described, and the or- 
 dinate BC of the curve denote the corresponding force 
 which urges the body. The element BCcb =fljs 9 
 we have already seen, is proportional to the acces- 
 sions of the square of the velocity ; and consequent- 
 ly the square of the velocity acquired in passing 
 through AB, is represented by the area ABC, while 
 the square of the final velocity attained is measured 
 by the whole curvilineal space ACEDA. 
 
 If a body be suddenly turned aside from its course, 
 it will suffer some loss of impulsion. Thus, if a ball, 
 rolling on a horizontal plane, be diverted from the 
 line AB, (fig. 39.) into the direction BC, the im- 
 pulsion, which was the result of a force expressed by 
 
NATURAL I'lIILOSOrHY. S3 
 
 AB or BC, is resolved into a force BD in the ex- 
 tension of AB, and another force DC perpendicular 
 to this. In turning from B into the line BC, the 
 body has, therefore, its velocity reduced, in the ratio 
 of BC to BD, or of radius to the cosine of the angle 
 of deflection CBD. The loss of velocity sustained at 
 B, or DE, is hence, if the radius be reckoned unit, 
 as the versed sine of CBD, or in small angles, as 
 half the square of the sine CD. It may be found 
 very nearly, by dividing the square of the number of 
 degrees contained in the angle by 6566. 
 
 Conceive the change of direction from AB (fig. 
 40.) to DE to be divided by a series of intermediate 
 deflections ; the loss of impulsion will then be as the 
 aggregate of half the squares of the sines of those de- 
 flections. Let their number be doubled, and the 
 square of each sine will be reduced to one-fourth, 
 and their sum to one-half. Multiply the bendings 
 indefinitely, and the loss of velocity will thus become 
 insensible. The gradually deflected motion in a 
 curve is hence absolutely unimpaired. 
 
 Suppose a body, by the action of gravity, to slide 
 down an oblique plane AC (fig. 41.) : Draw the 
 horizontal line CB and the vertical AB. It has been 
 already proved, that the weight of the body is to its 
 effort to descend in the direction of the plane, as the 
 length AC is to the altitude AB. Wherefore, from 
 what has been shown, the square of the time of fall- 
 
86 ELEMENTS OF 
 
 ing through AB is to the square of the time of slid- 
 ing along AC, in the compound ratio of AB to AC, 
 and again of AB to AC, or as the square of AB 
 to the square of AC. Consequently the time of fall- 
 ing through the perpendicular AB is to the time 
 of the oblique descent through AC, as AB to AC. 
 Again, the velocity acquired in the direct fall is to 
 the velocity gained during the oblique descent in the 
 ratio compounded of AB to AC, the times of descrip- 
 tion, and of AC to AB, the accelerating forces, 
 
 that is, in a ratio of equality. 
 
 * 
 
 Let a body descend by its weight through a suc- 
 cession of planes AB, BC, CD, &c. ffig. 42.) Draw 
 the vertical AG, and the horizontal lines AI, BE, CF 
 and DG, and produce CB and DC to meet AI in H 
 and I. When the body arrives at B, it is turned into 
 the direction BC ; and when it has reached C, it is 
 deflected along CD. The loss of force occasioned 
 by these deflections may be overlooked when the an- 
 gles ABC, CBD, &c. are very obtuse, since they are 
 only as the versed sines of the supplements. The 
 velocity which the body gains in sliding down AB is 
 the same as it would have acquired in descending 
 over HB, or by a perpendicular fall from A to E. 
 In descending through ABC, therefore, it will ac- 
 quire the same velocity as if it had falleh from A to 
 F. The body hence enters on the plane CD with 
 the velocity which it would have gained in sliding 
 from I to C, and the velocity which it accumulates. 
 
NATURAL PHILOSOPHY. 87 
 
 at D will be the same as in a perpendicular fall from 
 A to G. In a series of many planes, therefore, the 
 velocity acquired is always the same as in an equal 
 vertical descent. If we substitute curves, the con- 
 clusion must evidently still hold. Hence, balls roll- 
 ing along different curves, from the horizontal line 
 ACE (tig. 4)3.) to a lower level BDF, will acquire 
 equal velocities at B, D and F. Hence also, if a ball 
 suspended by a fine thread, after descending through 
 the arc of a circle, meet, in its subsequent ascension, 
 with some obstacle which gradually shortens the 
 thread, it will still attain the level from which it fell. 
 
 The same conclusion is derived likewise from an- 
 other principle, and rendered even more general. 
 The vertical inciting force at the horizontal line 
 GHIK, (fig. 43.) is to the accelerating force in the 
 oblique direction H^, as H/a to Qy. Wherefore, 
 ghik being an adjacent parallel, the increase of the 
 square of the velocity in falling from G to y will be 
 to the corresponding increase of the square of the 
 velocity during the oblique descent through Hh in 
 the compound ratio of HA to Gy, and of Qg to Hh, 
 that is, in, a ratiq of equality. A body descending 
 along the curve, AHB, must thus receive the aug- 
 mentation of velocity at each successive parallel, that 
 another body dropped from A will acquire in the 
 course of its fall. To produce this effect, it is not 
 even requisite that the inciting force should remain 
 constant, the essential condition being merely that it 
 
88 ELEMENTS OF 
 
 should have the same intensity at every point of any 
 parallel. 
 
 Let AB (fig. 44.) be an oblique plane, and BC its 
 altitude ; draw AD perpendicular to AB, meeting the 
 vertical BD in D. The time of descending through 
 AB is to the time of falling from C to B as AB to 
 BC, and therefore the square of the time of descent 
 through AB is to the square of the time of the ver- 
 tical fall CB, as the square of AB to the square of 
 BC, that is (Geom. vi. 15,) as BD to BC. But this 
 ratio is the same with that of the squares of the times 
 of falling through DB and CB ; and hence, the fall 
 through DB is performed in the time of the oblique 
 descent along AB. Now, BAD, being a right 
 angle, is contained in a semicircle, and therefore ge- 
 nerally the times of descent through different chords 
 AJB and EB are equal. 
 
 Let a body be urged by a force whose intensity is 
 directly as its distance from a given point. Thus, if 
 the motion be performed from B to A (fig. 45.) 
 under the exertion of a force which varies with the 
 distance from the point A ; having erected the per- 
 pendicular BD equal to AB and joined AD, it is 
 evident that any other perpendicular CE must ex- 
 press the intensity of the inciting force at C. Where- 
 fore, the elementary rectangle CEec, or the product 
 of that force into the element of the space, must re- 
 present the increment of the square of the velocity ; 
 
NATURAL PHILOSOPHY. 8Q 
 
 and, consequently, the trapezoid BDEC will repre- 
 sent the square of the velocity acquired in passing 
 from B to C. But this trapezoid is the excess of the 
 triangle BAD above CAE, or half the excess of the 
 square of AB above the square of AC. With the 
 radius AB describe the quadrant BG, produce EC 
 and ec to F and/, and join AF. The square CF is 
 obviously equal to the excess of the square of AF or 
 AB above the square of AC, and consequently in 
 this species of accelerated motion, the perpendicular 
 CF bounded by the circumference will denote the 
 velocity acquired at C, the final velocity being ex- 
 pressed by AG or AB. 
 
 The same conclusion is derived in another way. 
 Draw F<p parallel to AB and join AF ; the ele- 
 mentary triangle ~F(pf being evidently similar to 
 ACF, it follows that AC : CF : : <pf : F<p or Cc, and 
 AC.Cc=CF,<g/*. But, apply ing the formula, f$s=vdv, 
 the accelerating force f being here denoted by the 
 space AC or #, and Cc by ds, the ordinate CF, and 
 its increment <g/* are evidently expressed by the ve- 
 locity v y and its increment 8v. 
 
 Other leading properties are easily deduced from 
 the same figure. For, CF : AF : : F<p or Cc : F/; 
 
 Cc F/ F/ 
 
 and hence = f But - evidently denotes 
 
 pp 
 
 the minute portion of time in which the elementary 
 arc Yf is described by the final velocity AF or AB, 
 
90 ELEMENTS OF 
 
 while ^ denotes the time in which the elementary 
 
 space Cc would be described by the velocity acquir- 
 ed at C. Wherefore, collecting those elements, the 
 accelerating body passes over the space BC in the 
 same time that another body, moving uniformly with 
 the final velocity, would describe the arc BF. 
 
 This final velocity is the same as what would be 
 acquired by a constant acceleration from the half of 
 only the first intensity ; for bisect BD in H, and 
 describe the rectangle HBAI, which will be equal 
 to the triangle ABD ; a body urged by the inva- 
 riable force BH will hence acquire at A- the same 
 square of velocity as another which is incited by a 
 force varying from the initial intensity BD in the 
 ratio of the distance from A. But the velocity ac- 
 quired by an uniform acceleration would carry the 
 body through double the space AB during the de- 
 scent from B to A. Wherefore the time elapsed 
 during the uniform acceleration through AB, is to 
 the time required for its description by a force of a 
 regularly diminishing intensity, as the diameter of a 
 circle to the length of the quadrant, or as a square 
 to the inscribed circle, that is, as 1 to .785398, or 
 nearly in the ratio of 14 to 11. 
 
 It thus appears, that the celerities generated by an 
 uniform acceleration from B to C and to A, are 
 measured by the chords BM and BG in the circle, 
 or the ordinates CM and AL of a parabola, which 
 
NATURAL PHILOSOPHY. "91 
 
 has B A for half its parameter. The celerities, again, 
 derived from the action of a force, decreasing as its 
 distance from A, are denoted, at the same points, by 
 CF and AG. In like manner, while the chords 
 BM and BG express the times of falling to C and 
 A by an uniform acceleration, the corresponding 
 arcs AF and AFG of the circle will express the 
 times of descent, when the body is urged by a force 
 proportional to the distance from the centre A. 
 
 Let a body slide down a small circular arc FB 
 (fig. 46.) The accelerating force at any point G, 
 will be as the sine of the corresponding obliquity, or 
 as the sine of the arc BG, which differs not sensibly 
 from the length of this arc itself. Wherefore, the 
 body will be urged in its descent by a force very 
 nearly proportional to its distance from the lowest 
 point B. The final velocity attained at B in the cir- 
 cular sweep, will evidently be the same as what would 
 be acquired in sliding through the chord EBFB. 
 The inciting force at F in the arc is likewise double 
 the constant force exerted in the chord ; for the 
 oblique angle FHI is double of declivity FBH. 
 Wherefore, the time of descent in the small arc is to 
 the time of descent in its chord, as a square to its in- 
 scribed circle. It is obvious that a ball, suspended by 
 a thread OB from the centre O, and therefore called 
 a Pendulum, would oscillate in arcs of the circle. ~ 
 
 Since the time of oscillation has a constant ratio 
 
!)2 ELEMENTS OF 
 
 to that of descent through the chords or the diame- 
 ter, its square must be proportional to the length of 
 the pendulum. On the supposition that the diame- 
 ter BF of the circle is 16.0953 feet, the descent 
 through any chord will be performed, on the paral- 
 lel of London and at the level of the sea, in a second, 
 and consequently each half vibration would occupy 
 .785398", or a whole vibration 1.570796", the ex- 
 pression for the semicircumference of a circle of 
 which the diameter is unit. 
 
 Such is the time in which a pendulum of 8.04765 
 feet length must perform its vibrations. Hence the 
 measure of a pendulum oscillating single seconds 
 may be easily found. For the square of the circum- 
 ference of a circle is to quadruple its circumscribing 
 square as 8.04765 to 3.2616 feet, or 39.1393 inches, 
 the length of a seconds' pendulum for the latitude 
 of London. In round numbers, the analogy 
 10 : 4 : : 8 : 3.2 affords a very tolerable approxima- 
 tion ; in short, the fifth part of the fall in any place 
 on the earth's surface during a second, is nearly the 
 corresponding length of a pendulum. 
 
 Hence, a clock will be made to go faster or slow- 
 er, by raising or depressing the bob of the pendu- 
 lum. Let I and I be two approximate lengths of its 
 rod, and 31 + I' : 31' + I: : 86400" : to the number 
 very nearly of beats in a day. The shortening of 
 the rod of a seconds' pendulum, only by the tenth 
 part of an inch, would therefore occasion an accele- 
 
NATURAL PHILOSOPHY. 9J 
 
 ration of 11 4", in the space of 24 hours. A single 
 turn of the nut-screw, of which the thread is one- 
 thirtieth part of an inch, will correspond to 34". 
 
 In very small arcs, the vibrations of the pendulum 
 may be viewed as isochronous. But the time of de- 
 scribing a large arc, such as BE, (fig. 46.) is consi- 
 derably longer ; for the inciting force at any point 
 E being proportional only to the sine EK, and not 
 to the arc itself, is now much inferior to what would 
 be necessary to produce a descent in the same time. 
 A seconds' pendulum would require 1.18" to describe 
 the full semicircle, and 1.0736" to sweep over an 
 arc of 120. In small arcs, the daily retardation in 
 seconds will be expressed nearly by five-thirds of the 
 square of the number of degrees through which the 
 pendulum traverses on either side of the vertical. 
 Thus, if the vibration should alternate the arc of 3, 
 a clock will lose in the space of 24 hours a quarter 
 of a minute, or a whole minute if this arc were dou- 
 bled. The correction for the going of a clock, how- 
 ever, may be simplified by drawing, across the case, 
 an horizontal line divided into inches and tenths, 
 36f inches below the point of suspension. The 
 square of .the number of inches through which the 
 rod of the pendulum oscillates, will express in seconds 
 the alteration of the rate in the course of a day. 
 
 To the Cycloid belongs the remarkable property, 
 that t/ie time of descent through every arc is abso- 
 
94 ELEMENTS Of 
 
 lutely the same. For let ADB (fig. 57.) be a cycloid, 
 of which CGD is the generating circle. From the 
 property of the curve, (see Geometry of Curve 
 Lines), the tangent at any point E is parallel to the 
 chord GD of the generating circle ; and conse- 
 quently the inclination of descent to the horizon is 
 equal to the angle DGH or DCG. But the sine 
 of this angle to a radius CD is DG, of which the 
 cycloidal arc DE is double. The accelerating force 
 at each point of the cycloid is therefore proportional 
 to the arc intercepted from D, and consequently the 
 time of descent must, in all cases, be the same. 
 But a pendulum KPE, suspended from K, and vi- 
 brating between the inverted semicycloids AK and 
 BK, would describe the cycloid ADB. The oscil- 
 lations of such a constrained pendulum would hence 
 be perfectly isochronous. This cycloidal pendulum 
 will, near the lowest point D, evidently coincide in 
 its motion with that of a circular pendulum having 
 the same length CD, which is the radius of curva- 
 ture. But, in the large aberrations from the verti- 
 cal, while the warping pendulum maintains its iso- 
 chronous oscillation, the inflexible pendulum will 
 suffer a visible retardation. 
 
 If the cycloid ADB were extended into a straight 
 line equal to twice DN, and, with this radius, a ce- 
 micircle described upon it, a perpendicular erected 
 from any point, as now shown in reference to fig. 
 45., would denote the acquired velocity, and would 
 
NATURAL PHILOSOPHY. 95 
 
 cut off an arc proportional to the time elapsed. But 
 the section of a semicircle, constituted on the expan- 
 sion of a portion of the cycloid, situate equally on 
 both sides of D, will intimate the same proportional 
 quantities. 
 
 The Cycloid is likewise distinguished as the curve 
 of swiftest descent. Thus, a body will slide from 
 B to E, (fig. 57.) along the cycloidal arc BDE in 
 a shorter time than by any other track, whether 
 straight or curvilineal. To demonstrate this beauti- 
 ful property, it will be sufficient to show that a body 
 descending in the curve, would sweep over each 
 portion of it in the shortest interval. Suppose, in 
 general, that a body falling from A to B, (fig. Q5.) 
 makes its quickest transit in a vertical plane from A 
 to B, passing by some point C in a given inter- 
 mediate horizontal line CD. Let the velocity of 
 description between A and C be denoted by E, 
 and that between C and B by F. If an adjacent 
 point c be assumed, it is obvious, that in the position 
 of a minimum transit, the increment of the time of 
 passage from A to c must be just equal to the de- 
 crement of passing from c to B. Wherefore, de- 
 scribing the arcs, or letting fall the perpendiculars, 
 Ca and cb 9 the elemental space ca must be described 
 with a velocity E, in the same instant that cb is 
 traced with the velocity F. Hence ca is to C&, or 
 the cosine of the angle ACD to the cosine BCD, 
 as E to F. The condition of the curve of swiftest 
 
9() ELEMENTS OF 
 
 descent, is therefore such, that the sine of the angle 
 which each portion of it makes with a vertical is pro- 
 portional to the celerity of descent. But this is a dis- 
 tinguishing property of the cycloid ; for the celerity 
 acquired by a body in descending from the extre- 
 mity A to E (fig. 57.) is measured by the chord CG, 
 which likewise expresses the sine of the angle CDG, 
 made by the curve at E with a vertical line. 
 
 It hence follows, that the diameter CD, denoting 
 the time of falling through this height, the semicir- 
 cumference CND will express the time of descending 
 through the semicycloidal arc from B to D, while 
 the chord DB itself will measure the descent over 
 this inclined plane. The time of arriving in the 
 cycloid from B to I will likewise be expressed 
 by CNDK or D, the semicircumference with the 
 arc K intercepted by the parallel IKN. The time 
 of sliding down the inclined plane BI is evidently 
 measured by L, the fourth term of this analogy, 
 CK : CD : : BI : L. 
 
 To find the track of swiftest descent from B to 
 any point M, it is only required to produce BM till 
 it meet a given cycloid in I, and make BI to BM as 
 CD to the radius of a generating circle, which, roll- 
 ing along BA, would trace another cycloidal arc 
 passing this M. 
 
 On this property of the cycloid are founded some 
 applications, very useful in practical mechanics. 
 Thus, in Switzerland and several parts of Germany, 
 
NATVliAL PHILOSOPHY. 97 
 
 slides have been constructed along the sides of moun- 
 tains, by which the timber felled near their summits 
 is conducted with extreme rapidity to the distant 
 valleys. In such cases the angle of descent is taken 
 greatest at first, and afterwards gradually diminished. 
 A similar incurvation should be traced in laying the 
 slips for launching vessels from the stocks. 
 
 Oscillations in different arcs of the cycloid are al- 
 ways performed in the same time, because the inci- 
 ting forces, as we have proved, are exactly propor- 
 tional to the spaces described. But the same pro- 
 perty must obtain, whether the tracks of motion be 
 curved or rectilineal. The only condition required, is, 
 that the body should be urged by a force directly as 
 its actual distance from the middle point of the path 
 traced. This principle has a most extensive applica- 
 tion in the economy of Nature. It includes not only 
 the isochronous vibrations of pendulums and springs* 
 but comprehends all those internal tremors excited in 
 any system of atoms upheld by the antagonism of at- 
 traction and repulsion. Hence the regular propaga- 
 tion of pulses through air and through water ; and 
 hence likewise the equable transmission of the tre- 
 mulous impressions along beams of wood and bars 
 of iron. 
 
 Weights attached at different distances from the 
 point of suspension, constitute a Compound Pen- 
 dulum. But since short pendulums vibrate faster 
 
 VOL. i. G 
 
98 ELEMENTS OF 
 
 than long ones, the nearer weights must conspire to 
 accelerate the motion of such as are remote ; while 
 these again retard the former. There is hence a 
 certain neutral point entirely exempt from the op- 
 posite impressions. This point is called the Centre of 
 Oscillation, and its distance from that of suspension 
 is equal to the length of a simple pendulum, which 
 vibrates in unison with the compound one. Suppose 
 (fig. 47.) the balls B, C, and D were fixed to a very 
 slender inflexible rod, their centre of oscillation 
 being O. It is obvious, that in every position of 
 the pendulum, the tangents to the arcs of descrip- 
 tion will, at the several points B, C, O and D, make 
 equal angles with the horizon, and consequently those 
 points are all urged by the same measure of force. 
 To render the descents in the arcs BE, CF, ON and 
 DH isochronous, it would have required, however, 
 the inciting energies to be proportional to them, or 
 to the respective distances AB, AC, AO and AD. 
 If AO, therefore, express the accelerating power at 
 O, BO and CO will denote the excess of accelera- 
 tion at B and C, and DO the retarding influence at 
 D. But, from the principle of the lever, the effect 
 of these disturbing forces in moving the pendulum 
 must be as their distances from the point of suspen- 
 sion. Wherefore the accelerating action at the cen- 
 tre O must be equal to that of retardation ; or 
 B.BO. AB + C.CO. AC = D.DO. AD. But BO = 
 AO AB,CO = AO AC, and DO = AD AO ; 
 
NATURAL PHILOSOPHY. 99 
 
 by substitution consequently B (AO AB) AB + 
 C (AO AC) AC = D ( AD AO) AD, or 
 B.AO.AB + B.AB* + C.AO.AC C.AC* = 
 D.AD* D.AO.AD ; whence AO (B.AB+ C.AC 
 + D.AD) = B.AB* + C.AC* + D.AD and 
 
 rn- 
 B.AB+C.AC+D.AD 
 
 distance of the Centre of Oscillation from the point 
 of suspension, is found by multiplying each weight 
 into tJie square of its distance from that point ', and 
 dividing by the sum of the products of those weights 
 into their distances simply. 
 
 The same procedure will apply, should any of the 
 weights occupy a position above the point of suspen- 
 sion ; only, in this case, their products into the se- 
 veral distances must be taken as subtractive. The 
 general formula will admit of being modified also, in 
 relation to G the Centre of Gravity. For, from the 
 property of the lever, 
 B.AB+C.AC+D.AD = (B + C + D) AG ; and 
 
 A B. AB* + C.AC * + D. AG* 
 consequently AC (B + C + D) AG 
 
 Again, since AB = AG BG, AC = AG CG, 
 and AD = AG + DG, it follows, by involution 
 and substitution, that AO = 
 
 (B + C + D) AG 
 
 , ~ r B.BG' + C.CG* + D.DG* 
 and therefore OG = (B + C + D) AG 
 
100 ELEMENTS OF , 
 
 Whence the interval of the centre of oscillation be- 
 low the point of suspension will be found, by divi- 
 ding the sum of the products of the weights into the 
 squares of their several distances from the centre of 
 gravity, by the product of the sum of all the weights 
 into the distance of the centre of gravity from the 
 point of suspension. 
 
 By either of these theorems, the position of the 
 centre of oscillation is computed. Hence, an even 
 slender rod, viewed as a mathematical line, and sus- 
 pended from the end, has its centre of oscillation 
 situate at two-thirds of its length \ but, in an isosce- 
 les triangle vibrating flatwise, it is three-fourths be- 
 low the vertex ; and, in a parabola, the position is 
 intermediate, being at the five-seventh part of the 
 axis. Again, the distance of the centre of oscillation 
 
 . . 
 
 of a cone, suspended from its apex, is - - , where 
 
 o a 
 
 a denotes the altitude of the cone, and r the radius 
 of its base ; and consequently this distance becomes 
 equal to four-fifths of the altitude, when the cone is 
 viewed as indefinitely slender. Of a hemisphere vi- 
 brating from its pole, the centre of oscillation divides 
 the radius, in the ratio of 26 to 9 ; but, of a whole 
 sphere, it occupies seven-tenths of the diameter. 
 
 The centre of oscillation being found for one point 
 of suspension may be readily discovered for any other, 
 since the product of the distances of both those points 
 from the centre of gravity remains always the same. 
 
NATURAL PHILOSOPHY. 101 
 
 Thus, if a slender cylinder be supported by a line, 
 the depression of its centre of oscillation below the 
 middle point will be equal to the square of the length 
 of the cylinder divided by twelve times the distance 
 of that middle from the point of suspension. 
 
 Again, let a sphere be suspended by a fine thread. 
 Then the length of this thread, together with the 
 radius, is to the radius, as two-fifths of this radius, to 
 the depression of the centre of oscillation below the 
 centre of the sphere. Suppose a ball, or a lenticular 
 bob flattened in the same proportions and 6 inches 
 in diameter, to hang by a fine wire or very slender 
 rod of 36 inches in length. Then 36 + 3 : 3 : : f- : &. 
 So that in this case the centre of oscillation would 
 be 39/j inches below the point of suspension. 
 
 In general, the distance of the centre of oscillation 
 below that of gravity, is inversely as the distance of 
 the point of suspension from the same centre j for 
 the system of atoms being unaltered, the product of 
 the distances of the centre of gravity from those op- 
 posite points is constant. Hence the centre of os- 
 cillation and the point of suspension are interchange- 
 able. Thus, if the compound pendulum ABCD 
 (fig. 470 were suspended from O, the centre of oscil- 
 lation would occur in the point A. This pendulum 
 would therefore perform its vibrations in the . same 
 time, either about A or O. 
 
 Let two equal balls B and C (fig. 48.) be at- 
 tached to the inflexible line AC, which swings 
 
102 ELEMENTS OF 
 
 from the end A ; since BC is now bisected by the 
 
 f u nn BG'+GC* BG* 
 centre of gravity, GO - = - , and 
 
 GO : BG : : BG : AG. Whence, by mixing, 
 BG GO : BG + GO : : AG BG : AG + BG, 
 
 or OC : OB : : AB : AC j the interval BC be- 
 tween the balls is, therefore, divided internally and 
 externally in the same ratio, by the centre of oscil- 
 lation and the point of suspension. In (fig. 49.) the 
 ball B is placed as much above the point of suspen- 
 sion as in fig. 48. it lies below it ; the centre of os- 
 cillation O is now thrown to a considerable distance, 
 dividing BC externally in the ratio of AB to AC. 
 Hence, by proportioning the distances of the balls of 
 a short pendulum on both sides of the point of sus- 
 pension, it may be made isochronous with one of any 
 given length. Slow vibrations are thus obtained 
 within narrow bounds. On this principle, is con- 
 structed a small instrument called a Metronome, for 
 marking conveniently the beats in music. 
 
 The position of the centre of oscillation will de- 
 termine the character of Hocking Stones, for we may 
 consider them as only rolling about their metacentre. 
 Suppose a wooden model, as in fig. 26., consisting 
 of a cylinder 9 inches high and 3 inches in diame- 
 ter ; if a thin convex segment with a radius AO of 
 4 inches were attached to the base, it would not 
 stand upon a plane, but, placed in a cavity less than 
 three feet in diameter, it would rock. Thus, if the 
 
NATURAL PHILOSOPHY. 103 
 
 radius AP of the concavity were 6 inches, AM would 
 
 f\ zL 
 
 be 77- 1 - =12 inches ; or if AP were 24 inches, AM 
 0-4 
 
 24 4 
 
 would be ' A = 4.8. Now the centre of oscillation 
 24-4 
 
 of the cylinder being at two-thirds of its height, is 6 
 inches from the top, and its distance from the point 
 
 of suspension when AM = 12, is 10 *A =:: 12 ; but 
 
 
 
 when AM = 4.8, it is = 30. Wherefore, 
 
 since 30 = 1.2 X 25 and ^/ 25 = 5, the latter ar- 
 rangement will make the vibrations five times slower 
 than the former. 
 
 Suppose a body, urged by a force inversely as the 
 square of its distance, to fall towards a given centre, 
 its celerity and time of descent may be easily deri- 
 ved from the general principles of phoronomics. Let 
 the path traced reach from A to O (fig. 43 *.), bi- 
 sect AO in I, and describe a semicircle, draw the 
 tangent AL, and taking an ordinate BC, extend the 
 chord OC to L, and assuming an adjacent ordinate 
 be, prolong the chord Oc to meet the tangent in /. 
 Make OB : OA : : OI : OM : : OM : ON ; whence 
 OB a : OA 3 : : OI : ON, and OI being constant, 
 ON must represent the force of acceleration at B. 
 Now, from similar triangles, C or B& : Cc : : BC : 1C, 
 or OI. Again, the elementary triangles COcand LO/ 
 
ELEMENTS OF 
 
 being similar, Cc : U : : BC : AL : : OB : OA : : 
 OI : OM, and by composition of ratios, BC : OM : : 
 Kb : U. But BC : AL : : OB : O A : : OM : ON, 
 and alternately, BC : OM : : AL : ON. Whence 
 AL : ON : : B6 : U and AL.L/ = ON.B6. In 
 the formula, /() s = v~d v, /and 7) s being denoted 
 by ON and B, it follows that AL will represent the 
 velocity acquired at B, the increment being L/. 
 
 To find the time of fall due to this law of 
 acceleration. Since AL : BC : : OA : OB, it fol- 
 
 O A OR 
 
 lows, that -jy . B zz -op.B. But OA being con- 
 
 OA 
 
 stant, the expression, -^ T .'Bb denotes the element 
 
 of the time, which, by decomposing the fraction 
 
 OB BI , CI 
 
 -, is therefore equal to-.C^ + "nTs* V*- Now, 
 
 from the similar triangles CBI and C#c, BC : Bl : : 
 
 T>T 
 
 Cz : zc, and BC : CI : : CK : Cc ; whence rr^.Cx, zz c, 
 
 )C 
 
 CI 
 
 -n^.C%> zz Cc, and the element of the time is com- 
 
 pounded of #c and Cc, or of the increment of the 
 sine BC and of its arc AC. The time of falling 
 from A to B is, therefore, denoted, by the sum of 
 BC and AC ; and that of the whole descent to the 
 centre O, by the semicircumference. It is hence 
 evident, that the ordinate BCD of a cycloid which 
 has AO for its altitude, must likewise express the 
 
NATURAL PHILOSOPHY. 105 
 
 time of passing from A to B. The properties of 
 these three modes of acceleration are all exhibited 
 collectively in fig. 43 *. The velocity which a bo- 
 dy uniformly accelerated acquires in falling from 
 A to B, is represented by the chord AC, while the 
 velocity accumulated by an acceleration inversely as 
 the square of OB is denoted by the tangent AL. 
 But the former is to the latter as OC to OA, or in 
 the subduplicate ratio of OB to OA, for by similar 
 triangles OC : OA : : AC : AL. The ultimate ce- 
 lerity of the one being as OA, that of the other be- 
 comes infinite. 
 
 The time of initial descent through a minute space 
 being obviously the same in all the modes of accele- 
 ration, while the arc, together with its sine, is at first 
 double of the chord, it follows that the time of falling 
 to B, under an uniform action, must be represented 
 by twice AC ; the ordinate of the cycloid denoting 
 the corresponding time elapsed, when the body is 
 urged by a force inversely as the square of OB. The 
 whole times of descent are hence respectively as 2O A 
 to OF or AGO ; that is, as a square is to its inscri- 
 bed circle. 
 
 In the case of a body urged by a force directly 
 proportional to the distance from a given centre, 
 the velocity acquired at B will be represented on 
 the same scale, by diminishing the ordinate BH of 
 the quadrant in the ratio of AG to AO. The time 
 elapsed during the fall to B will be expressed by 
 the arc AH. The final velocity at O is therefore 
 
106 ELEMENTS OF 
 
 denoted by OP perpendicular to AG, and the whole 
 time of descent by the quadrantal arc AHG, or the 
 semicircumference AGO. 
 
 These properties furnish the solution of some cu- 
 rious problems. Thus, suppose the axis of the earth 
 were perforated from pole to pole : a body falling 
 through the perpendicular hole, being attracted on 
 all sides, would be urged downwards only by a predo- 
 minating force proportional to its distance from the 
 centre. The velocity acquired at this centre, reckoning 
 the length of the axis 7900 miles, would hence be 
 
 = 25834 feet each second. The 
 
 . 1.5708 
 time of descent is 
 
 //3950.5280\ 
 7 f -- - - J z= 1 268"^ = 
 
 21' 8", and the whole time of passage to the op- 
 posite pole 42', 16"J. 
 
 Conceive a body under the mere influence of ter- 
 restrial attraction, to fall from the orbit of the moon 
 to the earth's surface. At the mean distance of 60 se- 
 midiameters, the initial force would be diminished 
 3600 times : with the same continued acceleration, 
 therefore, it would consume the period represented by 
 
 7(59.3956.5^80) = 526578", or 6 days, 2 hours, 
 
 T ^r 
 
 16 minutes, and 18 seconds, in performing the whole 
 descent. The final velocity, on this supposition, 
 
 would be ^^(59.3956.5280) = 4680,69 feet each 
 second. 
 
NATURAL PHILOSOPHY. 10? 
 
 It is easily computed, that a tangent at the earth's 
 surface would intercept, from a semicircle described 
 on the radius of the lunar orbit, the arc of 165 9' 54". 
 But as twice the chord of this arc, or 3.9854020, is 
 to the sum of the arc and its sine, or 3.1386812, so 
 is the time of descent under an uniform acceleration, 
 to the time required with an acceleration inversely 
 as the square of the distance from the centre ; which 
 is hence 414645", or 4 days, 9 hours, 10 minutes, 
 and 45 seconds. Again, the final velocity, being aug- 
 mented in the subduplicate ratio of 1 to 60, is 36256,45 
 feet, or about seven miles each second. 
 
 In general, r denoting in feet the radius of the 
 Earth, the velocity which a body would acquire in 
 falling with an uniform acceleration from the height 
 
 o 
 
 nr to the surface will be expressed by -^/(n !)/; 
 
 and consequently the velocity acquired under an ac- 
 celeration inversely as the square of the distance from 
 
 the centre will be denoted by 8^//\/ - . When 
 
 n is indefinitely great, the final celerity is hence 8^/r, 
 or 36562.43 feet, the mean radius being 3956 miles. 
 Abstracting, then, from the resistance of the atmos- 
 phere, a body shot directly upwards with a velocity 
 of 36256.45 feet each second, would mount to the 
 orbit of the moon ; but, with the additioa of one 
 120th part more, or 305 feet each second, it would 
 reach the sun, and the farther acceleration of less than 
 one foot would have enabled it to continue its flight 
 into the regions of boundless space. 
 
108 ELEMENTS OF 
 
 The same results are obtained more easily. Thus, 
 r denoting in feet the radius of the earth, the ex- 
 
 pression for a body falling from the height x is 
 _ ~ft x 
 
 64<r z . =%vdv=^d.v* 9 which may be written thus, 
 x 
 
 and 
 
 > (- \ = 64r> (-, -} = ^ . v ; 
 
 ^XX XX J \X X ) 
 
 hence, by the summation of the differences of the suc- 
 cessive fractions, supposing a to be the initial height, 
 
 64r 2 f --- ) = vv. When a? becomes equal to r, 
 and the body reaches the surface, Sr v ( --- J = v 9 
 
 or 8V (- - ). If a be assumed indefinitely great, 
 the final velocity is simply 
 
 If a body describe a curvilineal path, it must be 
 continually deflected from its impulsive motion, by 
 the influence of some regular force. When this 
 force is directed to a fixed point, it is called Centri- 
 petal, and the antagonist effort of the body to fly off 
 in a tangent, is termed Centrifugal. To begin with 
 the simplest case, let the body A (fig. 50.) revolve 
 in the circumference of a circle, whose centre is C. 
 If it were abandoned to its mere impulsion, it would 
 shoot forwards in the tangential line AB j but, in 
 the time of describing the small portion of the 
 curve Ab, it is constrained to sink towards the cen- 
 tre from B to b. Extend BcfcE across the circle, 
 
NATURAL PHILOSOPHY. 10 ( J 
 
 and (Geometry, III. 26.) AB'z: EB.Bd ; where- 
 fore, since EB may be considered as equal to the 
 
 AB* 
 
 diameter, the measure of deflection B& zz 
 
 Let the radius AC in feet be denoted by r, the time 
 of revolution in seconds by t, the velocity by v, the 
 power of gravity by g, the centrifugal force by^ 
 and the ratio of the circumference to the diameter 
 
 of a circle by K. Then AB zz -, and the mea- 
 
 L 
 
 sure of deflection Bb zz r zz . But this is 
 
 2r t* 
 
 the momentary descent which would be caused by 
 the force which confines the motion to a circle ; 
 
 f ~ j ,. 
 
 wherefore, -^-zz %f, or -^-zz^/, and/zz - . 
 
 Hence the centrifugal force is directly as the radius 
 of the circle, and inversely as the square oftlie time 
 of revolution. Assuming, for the sake of round num- 
 
 , 5r . 
 
 bers, ** = 10 ; and / = *= a Ver 7 simple ex- 
 
 pression. Suppose the centrifugal force were equal 
 to the action of gravity, and 5r = 4#* = (2^)*, or 
 t = ^5r. 
 
 These formulae may be changed into terms of 
 the velocity, for vt = 2^-r, and consequently v*t z = 
 
 4-/r z r z , and = - =^. But this quantity, which 
 t * 
 
 marks the deflection, is equal to I6f> and therefore 
 
110 ELEMENTS OF 
 
 V* 
 
 f=. -. When the centrifugal force is equal to 
 
 the power of gravity, v* = 32r, or v = Sy/^r ; that 
 is, the velocity is the same as what the body would 
 acquire in falling through half the radius. 
 
 Hence the centrifugal effort on the tension of 
 a pendulum at the lowest point, after descending 
 through a quadrant, would be just double its weight ; 
 for, in this case, v = Sy/r, and consequently f = 
 
 64/ 1 
 
 = 2. Let d denote generally the vertical de- 
 
 / -, _c T 
 
 scent, and v = 8^/d ; whence / =- = . In 
 
 or r 
 
 small arcs, the vertical tension may be found, by di- 
 viding the square of the number of degrees described 
 in each oscillation, by 13153. 
 
 Hence, also, a sling two feet long, circling verti- 
 cally, with a celerity of 8 feet each second, would 
 just sustain its load ; if it were accelerated so as to 
 perform a revolution in one second, the tension of 
 the string would exceed that weight 2J times. A 
 tumbler full of water is therefore easily whirled 
 about the head, without spilling a single drop. 
 
 From the same formula, it is likewise computed, 
 that, at the equator, the diminution of gravity oc- 
 casioned by the centrifugal force arising from the 
 rotation of the earth, amounts to about the 289th 
 part. But since this number is the square of 17 ; it 
 follows, that, if our globe turned more than seven- 
 
NATURAL PHILOSOPHY. Ill 
 
 teen times faster about her axis, or performed the 
 diurnal revolution within the space of 84 minutes, 
 the centrifugal force would have predominated over 
 the power of gravitation, and all the fluid and loose 
 matters would, near the equinoxial boundary, have 
 been projected from the surface. On such a suppo- 
 sition, the waters of the ocean must have been drain- 
 ed off, and an impassable zone of sterility interposed 
 between the opposite hemispheres. By a similar cal- 
 culation, combined with the decreasing force of gravi- 
 ty at great distances from the centre, it may be in- 
 ferred, that the altitude of our atmosphere could 
 never exceed 26,000 miles. Beyond this limit, the 
 equatorial portions of air would have been shot into 
 boundless space. 
 
 The action of centrifugal force is familiar to us in 
 the trundling of a wet mop. It likewise supports 
 the vertiginous motion, and prevents the fall, of a 
 spinning-top. In the practice of the arts, centrifugal 
 force is often advantageously employed. By a rapid 
 whirl, the globules of air, which are apt to form in 
 filling a spirit-of-wine thermometer, are easily detach- 
 ed ; and, in like manner, different liquids intermixed, 
 but not chemically combined, are separated into dis- 
 tinct masses. Hence the flour, as fast as it is ground, 
 is thrown out from the rim of the revolving mill- 
 stone. Several manipulations of the potter and the 
 glass-blower depend on the same principle. Hence 
 the clay, under a gentle pressure, swells out regular- 
 ly during the rotation of the wheel. Hence also the 
 
ELEMENTS OF 
 
 lengthening of a rod of glass by whirling, and the 
 spreading of a sheet by the process called flashing. 
 But this extension is produced uniformly, j For let 
 OH (fig. 51.) represent a rod, or some radiated por- 
 tion of the soft material, which is whirled about O 
 as a centre, and conceive it to be distinguished into 
 equal spaces OA, AB, BC, CD, &c. The centri- 
 fugal forces exerted at A, B, C, D. &c. are as the 
 radii OA, OB, OC, OD, &c., the distances of those 
 points from O. Wherefore the force at B exceeds 
 that at A by a force AB ; the force at C exceeds 
 that at B by a force BC ; and so, through the whole 
 of the chain, the successive differences being all equal. 
 The part A is therefore drawn from O, the part B 
 from A, the part C from B, and so forth, all by the 
 same excess of force. Consequently the whole co- 
 lumn is stretched uniformly, and extended to the 
 same thickness. 
 
 If a ball suspended obliquely from a point by a 
 thread or thin rod, describe a circle in the horizon- 
 tal plane j it will form what is called a Conical Pen- 
 dulum. Let AC (fig. 52.) circulate about the axis 
 AB, it is evident that a strain CH in the direction 
 of the pendulum must be compounded of CG the 
 vertical power of gravity, and of CF the horizontal 
 centrifugal force. Wherefore, the action of gravity 
 will be to this centrifugal force, as CG to CF, that 
 is, from similar triangles, as AB to BC. But when 
 
 the time of revolution remains constant, the centri- 
 
 i 
 
NATURAL PHILOSOPHY. \13 
 
 fugai force is proportional to the radius BC, and 
 hence the perpendicular AB, representing the power 
 of gravity, must continue likewise unvaried ; conse- 
 quently, (fig. 53.) if a number of balls B, C, D, at- 
 tached by wires AB, AC and AD, of unequal 
 lengths, be carried round in the same time, they will 
 always range themselves in a horizontal plane. 
 
 Let AC (fig. 52.) be expressed by /, AB by h, 
 BC by r, and GH by/J the power of gravity CG 
 
 ft* r 
 being represented by unit. Theny= ; but, 
 
 since AB : BC : : CG : GH, or h : r : : 1 :/, 
 
 T TT 2 V T 
 
 therefore, f = -~> and by substitution = - , or 
 
 /> O t ft 
 
 <~ ft* 
 
 Now, -- is constant, and hence the 
 
 axis of the conical pendulum is proportional to the 
 square of the time of revolution. In round num- 
 
 5 19 
 
 bers, -.A = t* 9 and t = ^jjh. But if the correct 
 
 result be preferred, 1 ,2337.^=*% or* = 1,1 1072 </, 
 and h = .81057.**. It may be sufficiently near, 
 
 however, to assume t = ^/A, and/i = ~ . t 2 . 
 
 y Ou 
 
 Hence the altitude of a pendulum, which circulates 
 in a second, is 9-72684 inches ; the altitude, corre- 
 sponding to half seconds, is 2.4S171 inches, and that 
 answering to a circumvolution in the third of a se- 
 cond, 1 .08076 inches. 
 
 H 
 
114 ELEMENTS OF 
 
 The conical pendulum is applied to regulate the 
 movements of a delicate piece of mechanism, which 
 exhibits the continuous flow of time, and measures 
 intervals less than the tenth part of a second. It is 
 likewise adapted, with complete effect, as a Governor 
 in the steam-engine and other rotative machines. 
 For when the circulation exceeds a certain rapidity, 
 the balls B and C (fig. 54.) flying out, shorten their 
 perpendicular distance below the centre A, and con- 
 sequently pull down the opposite point F of the 
 parallelogram DAEF, and thus open the principal 
 valve, or otherwise regulate the general movement. 
 
 If a body be thrown directly upwards into the air, 
 it will, abstracting the resistance of that medium, 
 ascend with an uniformly retarding velocity, till its 
 motion is overcome by the action of gravity, when 
 it will again descend, and acquire in the fall a cele- 
 rity equal and opposite to its previous impulsion. 
 The height to which it will rise must therefore be 
 equal. to the square of the eighth part of the velocity 
 of projection ; and the whole time spent during this 
 ascent, and the subsequent descent, is the sixteenth 
 part of that velocity. But if the body be thrown 
 obliquely, it will be incessantly bent from its recti- 
 lineal flight, and brought sooner to the ground. 
 The deflected space being, from the action of gravi- 
 ty, as the square of the time elapsed, must conse- 
 quently be proportional to the square of the tan- 
 
NATURAL PHILOSOPHY. 115 
 
 gents of description. Wherefore (Geometry of 
 Curve Lines, p. 235.) the trajectory, or the curve 
 traced by the projectile, is a PARABOLA. 
 
 Let v denote the velocity with which the projec* 
 tile is discharged, and t the entire duration of its 
 flight. Since it would be carried by its oblique im- 
 pulsion, through the straight line AB, (fig. 58.) 
 in the time which it would require to drop from 
 B to C, therefore AB = vt, and BC = 16**. But, 
 from the property of the Parabola, the square of 
 the tangent AB is equal to the rectangle under 
 BC, and the parameter of the diameter at A< 
 Wherefore, v* t 1 16 t* X parameter, and the pa- 
 
 v 2 
 rameter itself = -^. Hence the fourth part of this, 
 
 or the focal distance AF, is equal to AG the verti- 
 cal descent ; and, consequently, the projectiles dis- 
 charged from A, at whatever elevation, but with the 
 same velocity, must trace parabolic paths, which have 
 all the same directrix GHI, and whose foci F or F' 
 occupy the circumference of a circle described with 
 the radius AG. 
 
 Now, from the property of the parabola* the tan- 
 gent AD bisects the angle FAG, and consequently 
 AFE, or its alternate angle GAF, is double of 
 GAD ; wherefore the supplemental angle AFD is 
 double of the complemental angle E AD, or the ele- 
 vation. Hence, in the right-angled triangle AEF 
 radius is to the sine of AFE, which is the same as 
 
 H 
 
116 ELEMENTS OF 
 
 the sine of AFD, or of twice EAD, as AG or AF 
 is to AE, the half of AC the horizontal range. Re- 
 suming the former notation, and e expressing the 
 elevation CAB, the radius being unit ; then 
 
 sin 2e (~) = AE, and sin 2e. = AC. Again, 
 
 sin CAB.AB = BC, or sine.vt = 16*% and there- 
 
 sin e.v 
 fore t = __g_. 
 
 Hence the greatest horizontal range is attained 
 by an elevation of half a right angle, for the sine of 
 double of this elevation is equal to the radius itself. 
 In this case, the focus of the parabola occurs at F. 
 It likewise follows, that two elevations, such as CAB 
 and CAB', equally distant on either side of this 
 middle angle, will have the same horizontal range, 
 because their doubles would evidently be supplemen- 
 tal angles having equal sines. Here the point E 
 bisects the base AC of both those parabolas. 
 
 Suppose a bomb were thrown with a velocity of 
 only 24>0 feet each second, the height to which it 
 would rise, if thrown perpendicularly upwards, would 
 be (SO)* or 900 feet ; if projected at an angle of 45, 
 the horizontal range would be the double of 900 or 
 1800 feet, and the time of flight 10.6 seconds. But 
 if the shell had been thrown at an elevation either of 
 75 or 15, it would have reached only the horizon- 
 tal distance of 900 feet, the time of describing the 
 
NATURAL PHILOSOPHY. 117 
 
 higher path being 14.5 seconds, while the lower path 
 is described in 3.9 seconds. 
 
 The elevation and horizontal range of a projectile 
 being known, let it be required to find where the 
 ball will strike a given oblique plane. Suppose AB 
 (fig. 59.) to represent the line of impulsion, AC the 
 horizontal range, and AD the oblique plane. Erect 
 the perpendicular CD meeting AD in D, draw DE 
 parallel to AB, and EG parallel to CD ; and the 
 parabola will cut AD in the point G. By the ap- 
 plication of such successive parallels, the curve might 
 also be traced. (See Geometry of Curve Lines, pp. 
 297 and 447). 
 
 In the practice of Gunnery, however, unless in 
 the case of very small velocities, the Parabolic Theo- 
 ry is of little avail, the resistance of the air being 
 so prodigious as completely to derange all the effects. 
 Thus, a 24 pound shot, discharged at the elevation 
 of 45, and with a velocity of 2000 feet each second, 
 would, in vacuo, reach the horizontal distance of 
 125,000 feet, but has its range through the atmo- 
 sphere confined to 7300 feet. Accurate experiments 
 on the resistance of the air will introduce the pro- 
 per corrections, and lead to the determination of the 
 Ballistic Curve. 
 
 Suppose a projectile to be now deflected from its 
 tangential course, not in the same constant direction, 
 but in lines tending to a given point or centre of ac- 
 tion. Let the body A, (fig. 57 *.) which, abandoned 
 to itself, would travel uniformly along the path ABC, 
 
118 ELEMENTS OF 
 
 after the lapse of the first instant it has arrived at B, 
 instead of advancing through an equal space to C, is 
 drawn aside to D, by a force directed towards O. 
 But the segments AB and BC, assumed as extremely 
 small, in comparison of the distance of the centre, 
 the line CD may be considered as parallel to BO, 
 and hence the elemental triangle OBD is equal to 
 OBC, which again is equal to AOB. The series of 
 triangles formed about the centre O, during succes- 
 sive instants, are hence all equal, and consequently 
 the sector comprising them is proportional to the 
 whole time elapsed. But those instants being taken 
 indefinitely small, the polygon ABD, &c. will merge 
 into a continuous curve. In all motions, therefore, 
 controuled by the action of a central force, the space 
 traced over by a radiant is proportional to the time 
 of description. Such is the primary law of our Solar 
 System, first detected by Kepler, and demonstrated 
 by Newton. 
 
 Conceive a body to revolve in the periphery of an 
 ellipse, urged by a force directed to the centre of the 
 curve. In moving through the elemental arc Ee 
 (fig. 69 *)> the area of the triangle OE<s, measuring 
 a constant portion of the time, the deflection et will 
 mark the corresponding centripetal force. This arc 
 Ee, or the velocity at E, is therefore inversely as 
 the perpendicular OI upon the tangent ; or, from 
 the property of the ellipse, it is directly as the semi- 
 conjugate diameter OL. But the square of Ee in 
 the circle is equal to the rectangle under eg and 
 
NATURAL PHILOSOPHY. 119 
 
 chord of the equicurve circle or twice EK. 
 Wherefore the square of OL, or the rectangle un- 
 der OE and EK, is proportional to the rectangle un- 
 der et and the same EK, and consequently eg, the 
 measure of the centripetal force at E 9 is directly as 
 the distance OE. It likewise follows, that, in the 
 revolution about the centre of an ellipse, the velocity 
 at any point E is always proportional to OL, the se- 
 miconjugate diameter. 
 
 Next, suppose a body to describe an ellipse, under 
 the action of a force emanating from one of its foci. 
 This force being measured by ez in the direction of 
 EF, is therefore inversely as EH half the chord of 
 the equicurve circle, and directly as the square of 
 the velocity Ee. But Ee being inversely as the al- 
 titude FG of the elemental triangle EFe, it follows 
 that the constraining force must be inversely as EH 
 and the square of FG. Now, from the property of 
 the ellipse, the square of OL is equal to the rectangle 
 under EH and the semitransverse axis OA, and 
 therefore Eg is inversely as the combined squares of 
 OL and FG ; but OD : OL : : FG : FE, and 
 OL.FG = OD.FE ; whence, in this case, the cen- 
 tripetal force is inversely as the squares of OD and 
 FE, or OD being constant, it is as the square of FE 
 simply. Such is the law of Universal Gravitation, 
 that a body is attracted to another by a force reci- 
 procally as the sOjUare of their distance a fun- 
 damental principle, which Newton deduced from the 
 
120 ELEMENTS OF 
 
 second great discovery of Kepler, respecting the 
 figure of the planetary orbits. 
 
 When the revolution is performed about the centre 
 of the ellipse, (fig. 69 *.) the element EOe of the 
 time being equal to half the rectangle under Ee and 
 the perpendicular OI, the celerity Ee is consequent- 
 ly inversely as OI. But, from the property of the 
 curve, the rectangle under OI and the semiconju- 
 gate diameter OL is equal to the constant rectangle 
 under the semiaxes OA and OE, and therefore OL 
 is the reciprocal of OI ; whence the celerity of E 
 must be directly as OL ; and thus the celerity at A 
 or B is as OC, and C or D as OA. 
 
 From the relation of the ellipse to the circle, the 
 area IBX (fig. 63 *.) is to IBS as YX to YS or as 
 IP to IQ, or as the area of the ellipse to that of the 
 circumscribing circle. But in the same ratio are 
 the initial velocities at the vertex B, and consequent- 
 ly the time which a body, drawn by a force direct- 
 ly as its distance from the centre I, takes to de- 
 scend in the ellipse, or the circle to an ordinate 
 YXS is measured by the arc AS, or generally by 
 the angle BIS. Hence the time of revolution is 
 the same in all the ellipses constructed on the dia- 
 meter BW. Nay, this period will yet continue 
 the same while the length of the diameter is aug- 
 mented, since the initial velocity in a circle must 
 be proportionally increased. 
 
 These properties will hold, even though the curves 
 
NATURAL PHILOSOPHY. 121 
 
 should collapse into straight lines. Hence the de- 
 scents from A to O are isochronous, as in the case 
 of cycloidal pendulums j and hence, likewise, the 
 time of performing a complete oscillation, and of re- 
 turning to the same point, is measured by the cir- 
 cumference of a circle described with the radius O A. 
 Again, from the property of the ellipse, OA 2 +OC* 
 (fig. 69 *:) = OE* + OL 2 ; and supposing the curve 
 to be compressed into its diameter AB, then would 
 OA 2 = OM* + OL 2 , and therefore OL*, which 
 represents the square of the velocity at M, becomes 
 YS 2 (in fig. 53 *.) the time of falling to M being 
 likewise indicated, as before, by the arc BS. 
 
 Let the attractive force be now directed to the 
 focus F (fig. 69 *) of the ellipse ; the element EFe 
 of the time is equal to half the rectangle under Ee 
 and the perpendicular FG drawn to the tangent, and 
 consequently Ee is in the inverse ratio of FG. But, 
 having let fall the perpendicular fE, from the other 
 focus, the rectangle FG, fR is equal to the square of 
 OC, and is therefore constant ; whence the celerity at 
 E is directly as this perpendicular fR. The celeri- 
 ties at the vertex A, at C or D, and at the opposite 
 extremity B, are thus respectively proportional to 
 A/, OC, and/B or AF. 
 
 Since the elliptical sector IBX (fig, 53 *.) is to the 
 circular one IBS, as YX to YS, or as IP to IQ, and 
 the triangle OIX is to OIS in the same ratio, it fol- 
 lows, that the areas OBX and OBS, which, in the 
 ellipse or the circle, measured the time elapsed, are 
 
ELEMENTS OF 
 
 likewise proportional to IP and IQ, or the velocities 
 at the vertex, and consequently those spaces must 
 indicate equal intervals. The body descends in the 
 same time, therefore, to any perpendicular YS, whe- 
 ther by an elliptical or a circular arc. Hence, the 
 periods of revolution are equal in all those ellipses, in- 
 cluding the circle itself, which have the same trans- 
 verse axis or diameter. 
 
 In general, let r denote the radius of any circle, 
 v the velocity, and t the time of circumvolution. 
 
 The centripetal force being now expressed by ~, it 
 
 is evident, that v .r ~ -, and v = V-, or the 
 r* r v r 
 
 velocity is in the inverse subduplicate ratio of the 
 diameter of the orbit. Hence t =. ^ and* = r 3 , 
 
 V/r 
 
 or tJie square of the period of revolution is as the 
 cube of the principal axis the third great law of 
 the planetary motions, brought to light by the inge- 
 nuity and indefatigable research of Kepler. 
 
 Whether the controuiing force be directly pro- 
 portional to the distance of the centre, or inversely 
 as the square of that distance, the body might have 
 acquired its velocity of projection by falling with an 
 uniform acceleration through a determinate space. 
 In the case of circular revolutions, / denoting the 
 vertical lapse, corresponding to the initial sweep of 
 the trajectory, 4/.B2 (fig. 53 *.) will express the 
 
NATURAL PHILOSOPHY. 
 
 square of the velocity gained in descending to Z un- 
 der a constant force ; which, from the property of 
 the circle, is likewise equal to /.BW, and, conse- 
 quently, the measure BZ is always half the radius. 
 This point Z coincides, therefore, with /the farther 
 focus. If the curve described pass into an ellipse, the 
 index of the primary velocity must still be half the 
 radius of osculation at its vertex. To generate a cir- 
 cular description, it is hence requisite that the body 
 should have the precise celerity due to a fall through 
 half that radius. But, projected with inferior celerity, 
 it would, about the same axis, trace elliptical orbits 
 more or less compressed ; the foci mutually retiring 
 towards the extremities, while the conjugate diame- 
 ter contracts. When the primary impulsion becomes 
 extinct, the elongated ellipse merges in a straight 
 line. On the other hand, if the projectile motion 
 should exceed the limit of circular velocity, a new 
 series of curves will arise. But when the accelera- 
 tion is directly as the distance, on passing the limit 
 of celerity, the foci will first unite in the centre, and 
 the axis suddenly turning at right angles, the foci will 
 now gradually dispart in this transverse position. 
 As the velocity of impulsion is successively augment- 
 ed, the ellipse will assume all the degrees of oblate- 
 ness, till it finally vanishes in two parallel lines. 
 
 If the centripetal force be inversely as the square 
 of the distance, the moment the primary impulsion 
 transcends the limit due to a circular trajectory, the 
 
ELEMENTS OF 
 
 orbit would suddenly change into an equilateral hy- 
 perbola ; the farther focus, which had come to coalesce 
 with the attractive one, now flying to the opposite 
 side beyond the diameter. When the celerity of 
 projection receives a farther increase, the incurvation 
 at the vertex will be proportionally diminished. 
 With an extreme impulsion, the body would shoot oft 1 
 in a straight line perpendicular to the axis. 
 
 It thus appears, that a circular revolution, which 
 the ancients so fondly contemplated as the perfec- 
 tion of the celestial movements, is incompatible with 
 the stability of the universe. The most absolute 
 precision of impulse would have been necessary, and 
 the very slightest subsequent addition of celerity, 
 from the incidental influence of those disturbing 
 forces which are incessantly in operation, would at 
 once have transformed the circle into an hyperbola, 
 and have carried the planet away for ever into the 
 boundless expanse of heaven. In viewing the grand 
 phenomena of nature, our admiration is drawn to 
 those conservatory principles, which, in shorter or 
 longer periods, correct every occasional deviation 
 from the general balance of the system. 
 
 It is curious, however, to remark how very nearly 
 the planetary orbits approximate to circles. In that 
 of our earth, the two axes differ only by the 7086th 
 part. In the trajectory of Mars, this difference 
 amounts to the 231st part. It was accordingly the 
 greater eccentricity of that orbit which led Kepler to 
 
NATURAL PHILOSOPHY. 1 2 J 
 
 detect its elliptical form. The group of small kin- 
 dred planets lately discovered revolve in curves still 
 more elongated, the diameters of those of Juno and 
 Pallas being nearly in the ratio of 30 to 29. These 
 singular bodies might seem to rank between the or- 
 dinary planets and the comets, which wander in 
 ellipses of extreme elongation, scarcely distinguish- 
 able from parabolas, during a great part of their 
 visible track. In another circumstance, too, the 
 analogy obtains ; for while the larger planets de- 
 viate not more than 2 or 3 degrees from the plane 
 of the ecliptic, Pallas crosses it at an angle of 35, 
 and the paths of the comets have every possible in- 
 clination. It remains to be discovered, whether 
 diversified bodies, travelling in the celestial spaces, 
 may not fill up more completely that prolonged gra- 
 dation of existence, which appears so conspicuous in 
 other parts of nature. Supposing the projection of 
 the planets to be the result of some more general law, 
 those which had an excess of impulsion would totally 
 escape our range of observation, being swept away in 
 hyperbolas into boundless space, there perhaps to 
 form other stellar systems. 
 
 Abating the resistance of the air, a projec- 
 tile discharged horizontally, with the velocity of 
 25853 feet in a second, would circulate around our 
 earth. For half the radius being 1978 miles, 
 8^(1978.5280) = 25853.6, and hence thetimeofre- 
 
126 ELEMENTS OF 
 
 volution, on this hypothesis, would have been 5109*8 
 seconds, or 85 minutes. The deflective force be* 
 ing inversely as the square of the distance from the 
 centre of gravitation, the smallest increase of celerity 
 beyond the limit assigned would have led the body 
 away in a boundless hyperbolic orbit* But, with a 
 smaller velocity than 25853, the trajectory would 
 contract into an ellipse, of which the projectile, whe- 
 ther discharged in a horizontal or an oblique direction, 
 must describe a portion, till it strikes the earth's sur- 
 face. The theory of Ballistics is thus comprised in the 
 general system of centripetal forces. When the cele- 
 rity of impulsion is moderate, as in the case of all ar- 
 tificial projectiles, the portion of their elliptical orbit, 
 traced within our atmosphere, will not sensibly de- 
 viate from a parabola* Thus, reckoning the utmost 
 rapidity of a ball, shot from the mouth of a cannon, 
 to be 2000 feet each second, the focus would be on- 
 
 (2000^ * 
 } = 62500 feet, or about 11 J miles below 
 
 the vertex of the curve. The semiconjugate diame- 
 ter of the ellipse would have extended no farther 
 than 305J miles, had the intervention of the body 
 of the earth not opposed the complete description of 
 the curve. 
 
 Assuming, therefore, the extreme portion of the 
 trajectory as really parabolic, the horizontal velocity 
 at the vertex V (fig. 58.) must be to the velocity of 
 projection, as FV to the perpendicular FL ; but, 
 
\ATUIIAL PHILOSOPHY. 
 
 from similar triangles, FV : FL : : FL : AG, and 
 consequently FV : AG or AF : : FV> : FL*. 
 Now, this is the ratio of the spaces through which a 
 body must fall, to acquire the velocities at V and A, 
 and hence the projectile would have been carried 
 directly upwards to G, the position of the directrix. 
 Again, the time of descent, under an uniform acce- 
 leration, is the same as that of the passage from A 
 to V, and therefore a body would have fallen through 
 BC, during the whole sweep from A to V, and 
 thence to C. The parabolic theory of ballistics is 
 thus derived, by a slight modification, from the pro- 
 perties of an elliptical trajectory. 
 
 It will be more accurate, in some cases, however, 
 to take into account the convergency of the radiant 
 lines of attraction. Every body, for instance, is 
 at the equator, carried round by the rotation of the 
 earth, with a celerity of 15&5J feet each second. 
 A ball therefore dropped from a high tower, instead 
 of falling through a mere vertical line, must really 
 describe a portion of an ellipse, of which the focus 
 lies 6.8888, or veiy nearly 7 miles below its vertex. 
 Let BG (fig. 53 *.) represent the tower, and O, be- 
 ing the centre of the earth, the ball dropped from B, 
 having the celerity of rotation due to the radius OB, 
 will describe the small portion BC of the ellipse, and 
 strike the surface at C. Divide the arc BK into 
 equal parts, and draw from O the elementary tri- 
 angles. It is evident that the sector BOC will de- 
 
128 ELEMENTS OF 
 
 note the time of descent from B to C, and OBK the 
 time which the tower takes to gain the position KC. 
 But the curved space GBC, considered as parabolic, 
 being two-thirds of GBKC, is equal to a segment 
 GgkC, bounded by a concentric arc at two-thirds of 
 the altitude CK. At the instant when the ball a- 
 lights at C, the top of the tower has only travelled 
 through an arc equal to gk, and, consequently, the 
 bottom G is retarded by an interval equal to two- 
 thirds of the excess of BK above GC. Thus, sup- 
 pose at the equator the tower to be 576 feet high, 
 the ball would fall to the ground in 6" ; but the sum- 
 mit would, during this lapse, describe an arc exceed- 
 ing that passed over the base by - zr 3.024 
 
 14'OuU 
 
 inches, and two-thirds of this, or 2.016 inches, indi- 
 cates the deviation of the falling body to the east of 
 the tower. Experiments of this kind, which have 
 been successfully made at Pisa, and at Hamburg, 
 prove indisputably the rotation of the earth about 
 her axis. 
 
 Again, suppose a ball were fired at A from a can- 
 non directly upwards ; though it reached the same 
 absolute height, yet spending a longer time in the at- 
 mosphere, it would fall considerably behind the point 
 of discharge. The vertical impulsion, combined with 
 the transverse rotation of the earth, would really 
 give the projectile an oblique direction along the tan- 
 gent AD. While it therefore describes the portion 
 
 i 
 
NATURAL PHILOSOPHY. 
 
 of an ellipse ABC, the point A travels through Gand 
 beyond C to x ; the circular sector AOCG marking 
 the time of rotation from A to C, and the elliptical 
 sector AOCB the time of the flight in the curve till 
 its descent at C. Whence |OG is to f GB, or 3OG 
 to 4GB, as AC to C, the space through which the 
 point CA is carried beyond C during the descrip- 
 tion of the trajectory. Thus, suppose as before, that 
 the ball were shot vertically at the equator, with a ve- 
 locity of 2000 feet in a second; the altitude to which 
 it would mount is 62,500 feet ; and the whole time 
 of ascent, and of the subsequent descent, if the earth 
 were at rest, would have been 125". Wherefore 
 3 X 3962 X 5280 : 4 x 62500 : : AC : C*, that is, 
 125" to .4977", the interval during which the pro- 
 jectile is detained in the air, after the point A has 
 come into the position C. It amounts to the 25 1 , 1 6th 
 part of the whole time of the flight ; and being multi- 
 plied by 1525f , the velocity at the equator, gives 
 759,36 feet for the distance C, where the ball must 
 drop to the westward of the mouth of the cannon, 
 which in the meanwhile has travelled over 36^ miles. 
 It would be easy to show that this retardation of the 
 ball is proportional to the cube of its velocity. 
 With an impulsion, therefore, of 1000 feet each se- 
 cond, the western deviation in falling would be only 
 95 feet. 
 
 Though rising apparently in a vertical line, these 
 projectiles really take an oblique course ; for, in the 
 
 VOL. I. I 
 
130 ELEMENTS OF 
 
 first supposition, AC : CD : : 15&5f : 2000, and 
 the resulting velocity in the direction of the tangent 
 AD is 2515^- feet, the angle of elevation DAG be- 
 ing 52 39' 40". To make the ball strike exactly 
 at the same spot, it would require the cannon to be 
 pointed eastwards at the very small declination of 
 3' 27". 
 
 The ascent and descent of bodies subject to the 
 law of gravitation, may be viewed as only an extreme 
 case of the motions performed in an elliptical trajec- 
 tory, the curve being compressed into its axis, of 
 which the foci come to occupy the opposite ends. 
 The time of falling from A to B, or of describing 
 the circular arc AC, will now be measured by the 
 area OAC, composed of the sector AIC, and of the 
 triangle ICO, both having the altitude of the radius 
 1C, but the arc AC and its sine BC for their bases. 
 The time of descent through AB is hence propor- 
 tional to the sum of AC and BC, or to the ordinate 
 BD in the cycloid, as was already shown. Again, the 
 velocity acquired at B must be as a perpendicular 
 from A, the farther focus, to the ultimate position of 
 the tangent CT. Now, from the property of the tan- 
 gent to the circle or ellipse, IB : IA::IA : IT, and by 
 division, IB : IA : : IA-IB, or AB : IT-IA or AT, 
 and thence OB : IA : : BT : AT : : BC : AR ; 
 again OB : OA : : BC : AL, and, therefore, by 
 equality, IA : OA : : BC : AL, and IA being the 
 half of OA, the velocity at B indicated by AR is 
 
NATURAL PHILOSOPHY. 
 
 likewise the half of AL, or is always proportional to 
 this line, as was formerly investigated. 
 
 It was found that, the attractive force being in- 
 versely as the square of the distance from the centre 
 of motion, the velocity in a circle follows the inverse 
 subduplicate ratio of the same radiant. The initial 
 
 velocity at A being denoted by ^7yT ^ e velocity 
 acquired at S will be proportional to / QO*./ prs 
 
 01 / rK*\y O~A' ^ ^^ ^ e assumed indefinitely 
 
 great, and the point S taken near the centre O, the 
 ratio of SA to OA will be that of equality, and 
 consequently the velocity at S is expressed simply by 
 
 v/OS* 
 
 It may be calculated, that near the surface of the 
 sun, a body would fall through 455 feet in a se- 
 cond. Hence, likewise, the celerity acquired in this 
 short interval of time, by descending from the remo- 
 test regions of space, would be 390J miles. The same 
 rapidity would have enabled a projectile to escape 
 for ever frem the attractive force of the sun. Hence 
 had Light darted with less than the 500th part of its 
 actual velocity, it would have been recalled in its 
 journey, by the predominant power of that great 
 luminary. 
 
ELEMENTS OF 
 
 An atom or physical point, moving uniformly in 
 a straight line, will equally advance to any plane, or 
 recede from it. Let AB (fig. 55.) be the direction 
 of the motion, and conceive a perpendicular AP to 
 be let fall upon a plane passing through B ; in a cer- 
 tain portion of time, the point A arrives at the po- 
 sition C, when its distance from the plane will be re- 
 duced from AP to CQ. In the plane of the trian- 
 gle ABP, draw CD parallel to BP, and AD will 
 mark the corresponding advance of the point. But 
 the right-angled triangle APB is evidently given in 
 species ; and its hypotenuse AC being given, the 
 base AD must likewise be given. At each interval 
 of time, therefore, the point A will make a constant 
 approach to the plane BP. Should this plane lie on 
 the opposite side, the point A will, for the same rea- 
 son, uniformly recede from it ; or if occupying a po- 
 sition parallel to AB, the point will neither approach 
 nor retire from it. The recession, indeed, may be 
 viewed as merely a modified case of advance. 
 
 But the converse of this proposition likewise holds 
 or a point which advances equally to any plane 
 has an uniform and rectilineal motion. To simplify 
 the demonstration, we may suppose the point to glide 
 along one plane, and to make regular advances to- 
 wards two other planes which are right angles to 
 this. Let AP (fig. 56.) be perpendicular to the 
 plane BP, and BP another plane at right angles to 
 
NATURAL PHILOSOPHY. 133 
 
 the plane of the triangle ABP, in which the perpen- 
 dicular AP is let fall. The path of the point A ly- 
 ing in the plane of ABP, or ABF, the advances in 
 the position C to the planes BP and BP' will be in- 
 dicated by AD and AD', as limited by the parallels 
 CD and CD'. Join DD' and PF ; since the seg- 
 ments AD and AD' correspond to the same interval 
 of time, they are obviously proportional to AP and 
 AP', and consequently DD' is parallel to PP'. 
 Wherefore the triangle DCD' is equiangular to 
 PBF, and DD' : DC : : PF : BP, or alternately 
 DD' : PF : : DC : BP ; but, from the property of 
 diverging and parallel lines, AD : AP : : DD 7 : PF, 
 and hence AD : AP : : DC : BP, or alternately 
 AD : DC : : AP : BP. The right-angled triangle 
 ADC is therefore similar to APB, and the angle 
 CAD equal to the given angle BAP ; whence the 
 locus of C is a given straight line, or the point A 
 must describe a rectilineal course ; but its motion is 
 likewise uniform, since AC has a given ratio to AD. 
 Now, suppose any system of atoms or physical 
 points to move each of them in a straight line, and 
 with an uniform celerity, their several advances to 
 any given plane will be equable, and consequently 
 the sum of those divided by the number of particles, 
 or the measure of the approach of the common 
 centre of gravity must likewise be proportional to the 
 time. But the same property belongs to every plane, 
 
134 ELEMENTS OF 
 
 and therefore the centre of the system travels uni- 
 formly in a rectilineal path. 
 
 From this proposition, it follows, that if any group 
 of atoms impressed with uniform rectilineal motions, 
 advance as much collectively on one part to a given 
 plane, as they recede from it on the other, their 
 centre of gravity must remain at rest, and the system 
 will maintain its equilibrium. Suppose these atoms, 
 when thrown into motion, to preserve their mutual 
 arrangement, they must each of them, in the mo- 
 mentary effort, describe minute portions of arcs of 
 circles, about the common centre of gravity. But 
 such elementary arcs may be viewed as coincident 
 with their tangents, which evidently measure the ve- 
 locities impressed. In the case of an equilibrium, 
 therefore, the sum of these velocities, which are call- 
 ed virtual, estimated in one direction, must be equal 
 to their aggregate in the opposite direction ; or if 
 the weight of each distinct group, or the number of 
 atoms which it contains, were multiplied into its ve- 
 locity of aberration reduced to a given direction, the 
 different products collected together would extin- 
 guish each other. Such is the general principle in 
 Dynamics relative to virtual velocities, which deter- 
 mine the condition of the equilibrium of a system 
 from the minute alterations which would follow the 
 disturbing of it. 
 
 The quiescence of the centre of gravity is essential 
 
NATURAL PHILOSOPHY. 
 
 to the equilibrium of any system of atoms. But this 
 quiescence may be viewed as either absolute rest, or 
 as only the stationary limits of extreme evagation, or 
 when the centre of gravity occupies the higJiest or 
 lowest position. This maximum likewise forms an- 
 other condition of the balance of a body, which is 
 tottering, however, if the centre of gravity be above 
 the point of support, and stable only when below 
 it. 
 
 Suppose the motion of a system of atoms to be 
 gradually stopped by the influence of a certain ob- 
 structing force, the same force which restrained the 
 farther advance of the system, repeating its operation 
 again through the same space, must generate an 
 equal square of velocity in the opposite direction ; 
 for the area of the curve ACDE, (fig. 370 which 
 represents the square of the velocity extinguished in 
 the approach from A to D, will also express the 
 square of the velocity accumulated during the sub- 
 sequent retreat from DA. The like effect must 
 evidently take place, however short the space may 
 be in which this change of impulsion is produced. 
 In all the graduated mutations of any system of 
 atoms, therefore, the sum of the squares of their ve- 
 locities, estimated in any direction, continues still 
 the same. This principle, which has a very exten- 
 sive application, is usually termed the Conservation 
 of Living Forces. This figurative expression had 
 
136 ELEMENTS OF 
 
 better been avoided ; but it implies merely the per- 
 manence of the amount of the products of the squares 
 of the velocity of each cluster of physical points in- 
 to their number or weight. 
 
 Impulsion is never instantaneous. It is created, 
 but in a shorter or a longer interval of time, only 
 from the accumulation of some inciting energy. 
 This accumulation, which carries the body for- 
 ward, has been denominated Moving Force. It will 
 hence, in each accession, be the combined result of 
 the duration and intensity of action. The moving 
 force of a body is therefore measured, by multiplying 
 its mass or number of atoms into its acquired velo- 
 city. This product is concisely termed the momen- 
 tum. In any system of matter, the momentum is 
 composed of the advances of all the different portions 
 of atoms during a certain portion of time to a given 
 plane, and is consequently the same as the momen- 
 tum of the mass, if the whole had been condensed 
 in the centre of gravity. Through all the changes, 
 therefore, which ensue in the motion of bodies, from 
 congress or mutual action, their momenta, estimated 
 in the same direction, remain still unaltered. 
 
 The communication of motion, or the transfer of 
 impulse, likewise requires a finite portion of time, 
 so small, however, as often to elude the most atten- 
 tive observation. Whether I strike or push the end 
 
NATURAL PHILOSOPHY. 
 
 of a rod in the direction of its length, the remote 
 extremity will not advance simultaneously. View 
 it as a series of connected atoms : The first is im- 
 pelled towards the second, till the shock is extin- 
 guished by the accumulating powers of repulsion ; 
 but, in this constrained position of proximity to the 
 second atom, the first repels and causes it to make 
 a similar approach towards the third. By successive 
 partial oscillations, the original impulse will thus be 
 transferred along the whole chain of atoms. This 
 internal process may be rendered more familiar to 
 the imagination, by examining the mode in which 
 any stroke is propagated through a spiral or helical 
 spring. If I give a twitch, near the end of a very 
 long cord stretched tight, the jerk, forming a slight 
 sinuosity, will visibly dart along the whole line. In 
 the ordinary cases of impact, motion becomes gene- 
 rated or transferred in a portion of time that is scarce^ 
 ly at all discernible. The very same velocity is pro- 
 duced, if the pressure be augmented just in propor- 
 tion as the duration of action is diminished. But, 
 even in the most extreme case, the influence of time 
 must never be overlooked. 
 
 Bodies, with regard to their collision, are com- 
 monly divided into elastic and non*elastic. This 
 distinction, however, is not well founded ; for though 
 some of them, approaching by their softness to the 
 nature of fluids, are nearly indifferent to the change 
 
138 ELEMENTS OF 
 
 of figure, yet they all recover from any compression 
 of volume with unabated vigour. But if such bodies 
 be struck with immense rapidity, they will even re- 
 sist a change of figure. Thus, a stone thrown ob- 
 liquely and with great velocity, will be made to re- 
 bound from the surface of water, just as if it had im- 
 pinged against a sheet of ice ; because the shock is 
 then confined to a narrow spot, where the repulsive 
 force, occasioned by the compression of the proxi- 
 mate particles of the fluid, heaves back the stone, 
 before they had time to retire and compel the parti- 
 cles below them, by a diffusive motion, to give room 
 for the entrance of that missile. Under such circum- 
 stances, therefore, every substance may display the 
 properties of perfect elasticity. 
 
 It is hence more philosophical to distinguish im- 
 pinging bodies into Coalescent and Resilient. These 
 we shall treat separately. 
 
 I. COLLISION OF COALESCENT BODIES. Let the 
 ball A (fig. 60.) advance with the velocity AO in 
 the same direction as B, which moves with only the 
 velocity BO ; it will evidently overtake the ball B 
 at O, and, having coalesced or united with this, they 
 will both travel forward in the same rectilineal path. 
 Divide AB in G, so that AG be to BG as the weight 
 of B is to the weight of A, and the point G will be 
 the position of the centre of gravity, when they began 
 to move. This centre must consequently have ad- 
 vanced from G to O at the moment of their collision ^ 
 
NATURAL PHILOSOPHY. 
 
 but its uniform motion is not affected by the mutual 
 action of the balls. Wherefore their combined mass 
 will, after impact, proceed with the common velocity 
 GO. 
 
 Since, by construction, A : B : : BG : AG, it is 
 obvious that A. AO : B.BO : : BG. AO : AG.BO ; 
 whence the momenta of the balls A and B will be 
 expressed by BG. AO, and AG.BO. But BG.AO = 
 BG (GO+AG), and AG.BO= AG (GO BG) ; 
 these two momenta are therefore equal to (BG+ AG) 
 GO, the momentum of the compound after the col- 
 lision has taken place ; and thus an essential princi- 
 ple is maintained. Next, suppose the balls are im- 
 pelled in opposite directions. Make (fig. 61.) AO 
 to BO as the velocity of A is to the velocity of B, 
 and they will evidently meet in the point O. Let 
 G be their centre of gravity, which must lie either 
 between A and O, as in fig. 61., or between B and 
 O, as in fig. 62. After collision, therefore, the two 
 balls, coalescing in a single mass, will proceed in the 
 direction and with the velocity GO. 
 
 In this case, likewise, the collective momenta of 
 the balls will continue the same after the shock. 
 For since A : B : : BG : AG, the products A.AO 
 and B.BO may be expressed by BG.AO and 
 AG.BO ; wherefore the momenta, estimated in 
 the same direction, will be BG.AO AG.BO = 
 BG (AGGO) AG (BG==GO) = (AG+BG) 
 GO n: AB.GO, or the momentum of the joint mass 
 
140 ELEMENTS OF 
 
 after collision. Its motion also will be progressive 
 or retrograde, according as the centre of gravity lies 
 between A and O or B and O, that is, according as 
 the momentum of the ball A or that of the ball B 
 predominates. 
 
 II. RESILIENT BODIES. Let (fig. 63.) a glass or 
 ivory ball A, advancing with the velocity AO, strike 
 a similar ball B, which moves in the same direction, 
 but with the velocity BO. The collision will evi- 
 dently take place at the point O. The first effect 
 of this shock is to produce a momentary union of 
 the two balls, the impinging surfaces in both being 
 partially flattened. The next act is to recover their 
 globular figure by an elastic or resilient effort, which, 
 if exerted, in an equal instant of time, would exact- 
 ly redouble the change impressed ; for, as in fig. 
 37., whatever velocity is gained or lost by either 
 ball during their congress, must, by the reaction of 
 the same repulsive forces, be repeated again. If the 
 energy of recoil were exactly equal to the power of 
 compression, the elasticity would be perfect. Let 
 it be assumed as such in the collision now under re- 
 view. 
 
 At the moment, therefore, when the ivory balls 
 come into the closest union, they take the common 
 velocity GO, (fig. 63.) and consequently A loses the 
 velocity AG, while B gains the velocity GB. But, 
 in the subsequent act of recovering their figure, by 
 a mutual exertion of elastic force, the loss of A's 
 
NATURAL PHILOSOPHY. 141 
 
 velocity and the gain of B's velocity, are each doubled. 
 Hence, after their separation has been effected, the 
 velocity of A is AO 2AG, and the velocity of B 
 is BO+2GB. Make GP = GO ; and, it will fol- 
 low, that PA = PG AG = GO A G = AO 2 AG, 
 and PB = PG+GB = GO+ GB = BO + 2GB ; 
 whence PA and PB will express the resulting velo- 
 cities of A and B. It also appears that PG expres- 
 ses the velocity of the centre of gravity, being equal 
 to GO, the velocity which it had before collision. 
 
 Next, let the balls A and B, whose centre of gra- 
 vity is G, meet in opposite directions at O (fig. 64.) ; 
 GP being taken equal to GO ; the velocity of A will, 
 in the act of approach, have its velocity diminished 
 from AO to AG, and by the subsequent recoil redu- 
 ced to AP, while the velocity of B is augmented to 
 BC and next to BP. In fig. 65., a similar, though 
 modified, result takes place. 
 
 The case of perfectly resilient bodies admits the 
 application of the principle of the Conservation of 
 Living Forces. Thus, (in fig. 63, 64, and 65.) the 
 weights of the balls A and B, multiplied into the 
 squares of their velocities before and after collision, 
 will form the same amount. In other words, 
 AO'.BG + BO*.AG = PA'.BG + PB'.AG. To 
 prove this, it will be sufficient to show that AO*.BG 
 exceeds PA*.BG, as much as BO*.AG is less than 
 PB*. AG. Now (Geom. II. 17.) AO 2 PA* = 
 f AO + PA) (AO PA) = PO.2AG ; and, for the 
 
H'2 ELEMENTS OF 
 
 same reason, it follows that PB* BO* = (PB+BO) 
 (PB BO) = PO.2BG. Wherefore (AO PA*) 
 BGzzPO.2AG.BG=(PB a BO) AG. 
 
 If the balls A and B be of equal weights, they 
 will interchange their velocities ; for G the centre 
 of gravity occupying now the middle point, it is evi- 
 dent that PA, the velocity of A after collision, is 
 equal to BO, the velocity which B had previously ; 
 and likewise that PB, the velocity acquired by B, 
 is equal to AO, the velocity which A brought into 
 action. This exchange of mutual condition takes 
 place equally, whether the balls move in the same or 
 in opposite directions. 
 
 Let the ivory ball A (fig. 66.) strike another B 
 at rest. The point O must evidently coincide with 
 B, and G being the centre of gravity, make GP= BG. 
 The velocity of A after the shock will be denoted 
 by PA, and that of B by PB or 2CB. If the balls 
 have equal weights, (fig. 67.) the point P will obvious- 
 ly fall on A, and consequently the ball A will stop 
 and transfer all its motion to B. Hence, having 
 placed any number of ivory balls in mutual contact 
 along a straight line or horizontal groove, if the first 
 be struck in the same direction, the last one only 
 will fly off, leaving all the rest apparently unmoved. 
 The impulse here is conveyed along the whole chain, 
 neutralizing, in succession, each impinging ball, till 
 it seizes and transports the extreme one. 
 
 Suppose B to be a firm obstacle (fig. 68.), or a 
 
NATURAL PHILOSOPHY. 143 
 
 mass of indefinite extent at rest, all the points O, B, 
 G, and P will then coincide, and therefore the ball 
 A will be made to rebound in the opposite direction 
 with a velocity BA equal to that with which it im- 
 pinged. This property has a very general influence 
 in the operations of nature. 
 
 The interposition of a third ball C between two 
 unequal balls A and B may augment the velocity 
 communicated. Thus, let the ball A of nine ounces 
 weight, and velocity of one foot a second, strike di- 
 rectly a ball B of an ounce. The velocity which B 
 
 will receive is fTT^iT^lj* ^ ut su PP ose a 
 
 of four ounces were interposed ; the velocity which 
 B would then acquire by a double transfer is 
 
 . = ' = 2 ^' Let the intervenin g ballbe 
 
 now only two ounces, and B will, through this me- 
 
 2.9 2.2 
 
 dium, obtain a velocity equal to < ~ . ..-7-5 = 
 
 ^T~y VT** 
 
 - -=* 
 
 n's 
 
 The intervention of the third ball thus augments 
 very considerably the impulse delivered to B. The 
 effects, too, are evidently not quite the same in both. 
 Assume, therefore, an intermediate ball of three 
 ounces. On this supposition, the velocity acquired 
 
 , , 2.9 2.3 18 6 
 by B would be ~ . l =4 i wmch 1S evi " 
 
144 ELEMENTS OF 
 
 dently somewhat greater than either of the former re- 
 sults. The intervening ball C, being now three ounces, 
 is evidently a mean proportional between nine and 
 one, the weights of the balls A and B. It will be 
 found in general, that a maximum velocity is com- 
 municated, by interposing a ball which is a mean 
 proportional to both extremes. 
 
 To investigate this curious property, let (fig. 
 69.) AB and BC represent the weights of the 
 perfectly elastic balls A and B, and the perpen- 
 dicular BD, the weight of the inserted ball C. 
 The velocity communicated to B will be expressed 
 
 2AB 2BD 
 by AB+ BD'ED+BC* and lts recipr cal 1S c n - 
 
 (AB+BD)(BD+BC) 
 sequently proportional to - A R RD - > 
 
 but this expression may be expanded into 
 
 AB.BC+AC.BD+BD* 
 
 AT> PT ^ - . About the triangle ADC 
 AIJ..DJL/ 
 
 describe a circle, and produce DB to meet the 
 circumference in E, and the velocity of the ball B 
 BE.BD+AC.BD+BD* AC+DE 
 AB.BD ~AB~~' 
 
 which must therefore be a minimum. But AC and 
 AB are both of them constant quantities, and hence 
 the chord DE must be a minimum. Bisect AC in 
 O, and the distance BO from the centre of the circle 
 is consequently given. The circle itself must there- 
 fore be the least possible : Now, of all the circles 
 
NATURAL PHILOSOPHY. 145 
 
 which can pass through the points A and C, the 
 smallest is obviously that which has AC for its dia- 
 meter. Hence ADC, being contained in a semi- 
 circle, is a right angle, and the perpendicular BD, 
 or the weight of the intermediate ball, is (Geom. VI. 
 15.) a mean proportional between AB and BC, the 
 weights of the balls A and B. 
 
 The velocity communicated to the extreme ball 
 is likewise increased, by multiplying the interposed 
 balls. Thus, a ball of 64 ounces with a velocity of 
 one foot each second, striking another ball of only 
 1 ounce, will impress the velocity Iff. If a ball of 
 8 ounces be interposed, the velocity will become 
 3|f ; if two balls of 16 and 4 ounces be inserted, 
 the velocity will amount to 4<i l /j -, but, if four in- 
 termediate balls be placed in the continued propor- 
 tion of 32, 16, 8, 4, 2 ounces, this velocity will rise 
 
 to 5M-9- 
 
 Suppose a ball of 32 ounces, and with a velocity 
 of one foot each second, to strike against a series of 
 balls of 16, 8, 4, 2, and 1 ounces ; the velocity com- 
 municated to the last one will be 4 2 \%. But if the 
 order of these balls were reversed, and the smallest 
 impinged with the same momentum, or a velocity 
 of 32 feet , the velocity transferred to the largest 
 ball will be 2\ 2 7 .32 == 4)^ 9 the same as before. 
 With the same original impulsion, therefore, a ball 
 is, by this arrangement, made to move as fast as an- 
 other, only the thirty-second part of its weight. 
 
 VOL. i. K 
 
146 ELEMENTS OF 
 
 This curious result might appear at first to con- 
 tradict the principle, that, in every communication 
 of impulse, the same momentum is maintained. But 
 when the collision commences with the smallest ball, 
 the rest of the range, except the last one, are thrown 
 partially backwards, and consequently their several 
 momenta are to be deducted from the final momen- 
 tum of the projected ball. On the other hand, 
 again, when the collision is propagated through a 
 descending series of balls, they all advance with cer- 
 tain velocities after the shock has been transmitted 
 through them. The momenta thus acquired and 
 retained should consequently be joined to the mo- 
 mentum of the final ball, in estimating the full ag- 
 gregate effect. Computed in this way, the results 
 will be found exactly to correspond. 
 
 In regard, however, to the impulsion communicated 
 through the ascending series to the extreme ball, an 
 obvious and important advantage is gained. The 
 final momentum, in the example which we have 
 chosen, is, without any sensible expense of time, 
 more than quadrupled. Were this idea pursued, it 
 might, in various instances, suggest the means of 
 saving labour, and improving the performance of 
 machinery. 
 
 Suppose a ball of perfect elasticity to impinge ob- 
 liquely against another at rest. If the ball A, (fig. 
 70.) moving in the direction CA, strike B at the 
 
NATURAL PHILOSOPHY. 14,7 
 
 point D, the plane of collision will evidently be per- 
 pendicular to the line BDA passing through both 
 centres. Parallel to that plane, draw CE meeting 
 the extension of BA in E. The oblique velocity 
 CA may be decomposed into CE, parallel to the 
 tangent at D, and EA perpendicular to it. But 
 CE has no concern in the collision, while EA is di- 
 rectly exerted against the ball B. Divide EA in 
 G, so that EG shall be to AG, as the weight of the 
 ball A is to the weight of the ball B, and make GF 
 equal to EG. Since AG is the velocity lost by the 
 ball A in the first act of approach, 2AG AE AF 
 must be the velocity of its recoil. But the velocity 
 CE remaining unaltered, draw the perpendicular 
 FH equal to it, and the oblique line AH will ex- 
 hibit the velocity and direction of the resilient ball 
 A. The larger ball B which is struck will move 
 with the force and direction EF. 
 
 When the two balls are equal, the point G will 
 bisect E A, and the point F coincide with A. The 
 velocity EA will therefore be transferred from the 
 ball A to the ball B, which will move in the perpen- 
 dicular direction DB, while A, retaining the veloci- 
 ty CE, will proceed in the parallel direction AI. 
 
 If the ball B be considered as indefinitely large, 
 the points G and F will coincide with E ; this ball 
 will continue at rest, while the ball A will be reflect- 
 ed in the line AK, making an angle EAK equal to 
 
 K2 
 
148 ELEMENTS OF 
 
 the incident angle EAC. .Such is the effect of any 
 firm obstacle extended in the plane of collision. 
 
 The propositions respecting the impact of Coales- 
 cent and Resilient balls are easily verified by experi- 
 ment. If they be suspended at the same height by 
 parallel threads, they will acquire velocities very 
 nearly proportional to the arcs which they describe, 
 and, being dropped at the same instant, they must 
 always impinge at the lowest point. But were they 
 made to oscillate in the arcs of a cycloid, their mo- 
 tions would be absolutely isochronous, and their ac- 
 quired velocities exactly as the spaces passed over. 
 
 In treating of the impact of elastic bodies, I have 
 confined myself to the case of perfect resiliency, 
 where the partial change of figure occasioned by the 
 shock is recovered with the same energy and rapidi- 
 ty. But such accurate and complete development 
 of force never occurs, except in the reflection of light 
 and the expansion of air. In other instances, a de- 
 ficiency of reaction lessens the velocity of recoil. 
 Thus, a marble ball dropped from the height of 16 
 inches upon a marble floor, will perhaps rebound 
 only 9 inches. On this supposition, the resilient 
 energy being in the subduplicate ratio of the space, 
 is only three-fourths of the force of impact. Were 
 the proportion constant in the same bodies, those 
 'nodified effects could easily be computed. But the 
 reaction is rendered more complete, by the great ra- 
 
NATURAL PHILOSOPHY. 149 
 
 pidity of the shock. Had the same ball fallen from 
 the height of 16 feet, it might perhaps have rebound- 
 ed 12 feet, instead of 9- Such perfection of resilien- 
 cy is hence the limit to which all substances tend, 
 as their motions approach to extreme celerity. 
 
 If a system of atoms, turning about a fixed axis, 
 impinge at a given point against a firm obstacle, the 
 shock will either be completely extinguished in that 
 point, or partly spent upon the axis' of motion. 
 When this impulsion is only partially checked by the 
 obstacle, a jarring stroke will be communicated to 
 the axis ; but when the whole of it is absorbed in 
 reaction, the axis will suffer no tremor. The point 
 of impact in which this balance of moving forces oc- 
 curs, is therefore called the Centra of Percussion. 
 
 To simplify the investigation, let it be restricted 
 to the case of a compound pendulum or inflexible 
 rod, with the balls B, C and D attached to it, and 
 swinging from the point A (fig. 71-)- If P* the 
 centre of percussion, be stopped, those balls must ex- 
 pend their momenta against the lever AD, to which 
 P may be considered as a fulcrum. The velocities 
 of impulsion at B, C and D are evidently as the se- 
 veral distances BA, CH and DA from the point of 
 suspension ; but these velocities must all be extin- 
 guished by the obstacle in the same moment of time ; 
 and consequently during the short lapse preceding 
 the extinction, they will act in the same proportion, 
 
150 . ELEMENTS OF 
 
 as inciting forces or pressures. The ball B will 
 hence press against the lever for a moment with a 
 force B. AB, the ball C will press with a force C.AC, 
 and the ball D, during the same instant, will press 
 with the force D.AD. Now, from the property of 
 the lever, the forces at B and C will cause the strains 
 
 BP CP 
 
 .-, and C.AC.- at the point of suspen- 
 
 sion A, while the force at P, on the other side of 
 
 DP 
 
 the fulcrum, will exert an opposite strain D.AD. -rp. 
 
 Wherefore, since an equilibrium of impression must 
 
 obtain, B.AB.^ + C.AC.^? = D.DA.^? and 
 
 J\.r 
 
 thence B.AB.BP + C.AC.CP = D.AD.DP. But 
 
 such was likewise the condition that determined 
 the Centre of Oscillation, which in this case, there- 
 fore, must coincide with the Centre of Percussion. 
 Both these centres are indeed the same, unless the 
 dimensions of the compound body should extend 
 considerably in width in proportion to its height. In 
 all the ordinary cases, they may be assumed as iden- 
 tical. 
 
 If the compound pendulum ABCD, (fig. 710 in 
 the state of rest, were struck perpendicularly to AB 
 at the centre of percussion P, it is evident that 
 the whole would be made to turn freely about A, the 
 point of suspension. If BCD were conceived to be 
 
NATURAL PHILOSOPHY. 1,51 
 
 merely a rigid system of clustered atoms, the direct 
 impact at P would cause it to revolve about A, with- 
 out jarring against that point. The first effect of 
 the shock, therefore, would be the same, whether the 
 system were fixed at A or not. The several balls 
 B, C and D would begin by describing the minute 
 arcs B&, Cc, T)d, and then shoot forwards with ve- 
 locities proportional to those spaces. The point A, 
 about which the system thus begins its motion, is 
 called the Centre of Conversion or Spontaneous 
 Rotation. Let Gg be the initial arc described by 
 the centre of gravity, and draw Afb'c'gd' parallel to 
 AD ; the velocity Bb is composed of the direct ve- 
 locity B&' or Gg and a retrograde velocity b'b ; the 
 velocity Cc likewise consists of the direct velocity 
 Cc' or Gg and the retrograde velocity c'c ; but the 
 velocity Dd is compounded of the direct velocity T)d' 
 or Gg and of the direct velocity d'd. While the 
 centre of gravity or the whole system is carried for- 
 wards with the common velocity Gg, the balls B 
 and C retreat with the velocities b'b and c'c, and the 
 ball D advances with the velocity d'd. These se- 
 veral velocities are thus proportional to the distances 
 from the point G or^, and consequently proportion- 
 al to the centrifugal forces occasioned by the balls 
 continuing after impact to revolve in the same time 
 about that centre. But, since B.BG + C.CG =: 
 D.DG, these divellent forces are exerted equally on 
 both sides, and can have no influence whatever in 
 
152 ELEMENTS OF 
 
 distraining the rectilineal advance of the centre of 
 gravity, which must therefore pursue an undeviating 
 and rectilineal course. Every other point of the 
 system will describe a species of cycloid, either cur- 
 tate or prolate, (See Geometry of Curve Lines, 
 p. 3570 
 
 From the equilibrium of the opposite momenta 
 of the balls, or B.BG+C.CG=DO.DG, it likewise 
 follows, that the progressive motion of the whole 
 system, or of the centre of gravity, is neither aug- 
 mented nor diminished at all by their rotation. 
 The rectilineal and uniform advance of this centre, 
 and the constant circulation maintained around it, 
 are therefore two distinct and independent move- 
 ments. 
 
 Hence, while a rigid system of atoms performs 
 a single revolution, its centre of gravity will travel 
 over a space equal to the circumference of a circle 
 described about the centre of spontaneous rotation. 
 Thus, if the narrow cylinder AB (fig. 72.) were 
 struck perpendicularly at the point P, situate at 
 two-thirds of the whole length from the end A, the 
 cylinder would begin to move about that extremity, 
 and consequently the centre of gravity, or the mid- 
 dle point G, must advance in the direction of the 
 stroke over a line that is equal to 3.1416 times the 
 length AB, during the time in which the cylinder 
 performs a complete revolution. But if the point of 
 impact be taken nearer G, the corresponding centre of 
 
NATURAL PHILOSOPHY. 163 
 
 rotation will be thrown beyond A. Suppose the cy- 
 linder to be struck at/?, the distance Ap, being three- 
 
 is AI f AT* nv AB * AB 5AB 
 
 fifths of AB ; then GV == _=-_, or 
 
 V, the centre of rotation, lies at one-third of the 
 length beyond A, and consequently G will advance 
 fX 6.2832 or 5.236 times AB, while the cylinder 
 performs a revolution. But if the stroke were aim- 
 ed exactly at the centre of gravity, the point of con. 
 version V would be thrown to an indefinite distance, 
 and the cylinder would hence be carried directly for- 
 ward, without any sort of rotation, merely in a pa- 
 rallel position. 
 
 Again, suppose two equal balls A and B (fig. 73.) 
 to be connected by an inflexible straight line. If 
 this pendulum were suspended from A, the centre 
 of oscillation would evidently occur in B, and there- 
 fore the rectangle under the distances of the point 
 of the suspension and of the centre of oscillation 
 from the middle point G must, in every case, be 
 equal to the square of GB. Hence, if the line AB 
 were struck at P, which bisects GB, the centre of 
 rotation would be transferred to V, at a distance 
 AV equal to AG. The centre of gravity G would, 
 therefore, at each revolution, describe a rectilineal 
 path, equal to the circumference of a circle whose 
 diameter < AB. Let AP be only two-thirds of 
 AB, and the point of conversion V will be thrown 
 to a distance Gv, equal to the triple of AG ; the cen- 
 
154 ELEMENTS OF 
 
 tre of gravity would hence travel over 4.7124 times 
 AB, during each circumvolution of the connected 
 balls. 
 
 If, conformably to that simplicity which pervades 
 all the works of Nature, the planets derived their 
 motions both of rotation and revolution at once from 
 the impression of the same original force, our earth 
 must have received the initial impact at the distance 
 of the 1 65, 6th part of the equatorial radius from the 
 axis, or very nearly 24 miles. The centre of con- 
 version would be distant 66 radii, a little beyond 
 the orbit of the moon. In like manner, it may be 
 computed, that Jupiter had received the stroke which 
 impressed his very disproportionate diurnal and an- 
 nual revolutions, at the .351th part of his radius, or 
 14040 miles from the centre. 
 
 If a rigid system of atoms, impelled by a progres- 
 sive and revolving motion, be stopped at the centre 
 of gravity, its rotation about that point will evident- 
 ly continue the same as before. The shock being 
 extinguished by the resistance of the obstacle, the 
 circulation of the system is maintained, without any 
 effort exerted against the axis, which may be hence- 
 forth either fixed or loose. But if the centre of ro- 
 tation remain fixed, the circumvolution of the system 
 can also be arrested at any point in the radius of de- 
 scription. This absolute cessation of motion must 
 consequently be transfused through the whole mass* 
 
NATURAL PHILOSOPHY. 
 
 during the same very small portion of time. Let the 
 radius ABCD (^fig. 7^.) advance into the proximate 
 position Abed, while the revolving impulsion is ex- 
 pended. The atoms at B, C, and D therefore pass 
 over the minute spaces B, Cc, and Dd, with velo- 
 cities proportional to these, or to the distances AB, 
 AC, and AD, from the centre of rotation. Conse- 
 quently, the retarding forces that extinguish the mo- 
 menta, and which, multiplied into the spaces, are as 
 the squares of the velocities, must be proportional to 
 AB, AC, and AD. The pressures exerted during 
 impact by the groups or balls at B, C, and D, are 
 hence as B.AB, C.AC, and D.AD ; but, from the 
 property of the lever, the powers exerted by those 
 pressures, being augmented in the ratio of the dis- 
 tances AB, AC, and AD, are proportional to B.AB*, 
 C.AC*, and D.AD*. There is consequently a cer- 
 tain point R in the radius, at which, if all the balls 
 B, C, and D were conceived to be collected, the 
 shock of circumvolution or angular motion would 
 be the same. This point has been therefore call- 
 ed the Centre of Gyration. From its condition, 
 
 and 
 
 ^ 
 ) 
 
 therefore AR= . . B+C+D 
 
 The centre of gyration, however, is not strictly 
 confined to a mere point, but includes the whole cir- 
 cumference of a circle described at the assigned dis- 
 tance about the centre of rotation. It has been al- 
 
156 ELEMENTS OF 
 
 ready shown, that G being the centre of gravity, the 
 distance of the centre of oscillation or percussion is 
 
 B. AB* + C. AC* + D. AD* 
 
 ^ -, and thus AR* = AG.AO, 
 
 or AR = V /(AG.AO.) Consequently, the distance 
 of the centre of gyration from the point of suspen- 
 sion, is a mean proportional between the distances 
 of the centre of gravity and of oscillation. 
 
 Hence, the distance of the centre of gyration of a 
 straight line revolving about one extremity, is a mean 
 proportional between the whole length and its third 
 part ; and hence also the distance of the centre of 
 gyration of a circle, or of a circular sector, is a mean 
 proportional between the radius and its half. If 
 two equal balls B and C (fig. 48. and 49.) be at- 
 tached to an inflexible line, either on the same or 
 opposite sides of A the centre of rotation, the dis- 
 tance of the centre of gyration will in either case be 
 
 / A T* 4- A P* 
 /( HF -)=V(AG.AO). In a sphere, the 
 
 distance of the centre of gyration from that of mo- 
 tion is a mean proportional between the radius and 
 two-fifths of it ; but, in a hollow sphere, this dis- 
 tance is a mean proportional between the radius and 
 two-thirds of the radius. 
 
 The power of turning any rigid system of atoms 
 is thus proportional to the square of its distances 
 from the axis of motion. This power may therefore 
 be termed the Momentum of Rotation, and its total 
 
NATURAL PHILOSOPHY. , 157 
 
 amount in any revolving body is expressed, by the 
 mass multiplied into the square of the distance of 
 the centre of gyration. 
 
 The descent through a declivity is performed 
 either by sliding or rolling. In the former case, the 
 body maintains the same position, and is immediate- 
 ly urged forwards by that part of its weight which 
 comes into action. But when the body rolls down 
 any path, the accelerating force has not only to in- 
 cite the progressive advance of the centre of gravity, 
 but to create a corresponding rotation in the whole 
 mass. By this distribution of force, the action ex- 
 erted on the centre of gravity becomes diminished, 
 and its velocity, generated in a given space, is hence 
 reduced in a subduplicate ratio. Thus, if a thin 
 hollow cylinder were set to roll along an inclined 
 plane, the matter being all collected to the circum- 
 ference, the rotatory impulsion must be equal to the 
 progressive momentum of the centre ; wherefore, it 
 will descend in a given time only through half the 
 space over which it would slide lengthwise. On the 
 other hand, if the matter of the cylinder had been 
 concentrated in the axis which is fixed to two very 
 thin circular ends, no sensible portion of the inciting 
 force would be lost in producing rotation, and a body 
 of such a spindle shape would in equal times travel 
 twice as far as the former drum. 
 
 These different effects are rendered very conspi- 
 
158 ELEMENTS OF 
 
 cuous, by having two cylinders of light wood, but 
 loaded equally with lead, the one near the centre, 
 and the other about the circumference, each of them 
 being terminated by narrow protuberant rings or 
 beads. The cylinder that has its mass approximated 
 to the axis of motion will be found to roll down any 
 inclined plane almost as fast as if it had merely slid, 
 and to acquire, in the same time, nearly double the 
 velocity which is attained by the cylinder that had 
 its weight thrown to the circumference. 
 
 In a homogeneous cylinder, the distance of the 
 centre of gyration being a mean proportional be- 
 tween the radius and its half, the inciting force will 
 be divided into three shares, two of these employed 
 in generating the progressive motion, and the other 
 consumed in producing that of revolution. It there- 
 fore descends with only two-thirds of the power of 
 gravity. A hollow cylinder has its centre of gyra- 
 tion situate in the circumference, and consequently 
 one half of the inciting force urges the progressive 
 motion, while the other half merely supports the 
 rotation. In a solid sphere, the force of accelera- 
 tion being distinguished into seven shares, five of 
 these are exerted in propelling the centre, while the 
 remaining two are spent in maintaining the conjoin- 
 ed rotation ; its descent is consequently urged by 
 five-sevenths of the power of gravity. But in a hol- 
 low sphere or very thin spherical shell, the inciting 
 force is allotted in the ratio of three to five, so that 
 
NATURAL PHILOSOPHY. 1,59 
 
 the propelling energy is only three-fifths of gravita- 
 tion. 
 
 Since the square of the velocity acquired, in de- 
 scending through the same path, is proportional to the 
 accelerating force, the time of descent must evident- 
 ly follow its inverse subduplicate ratio. Hence the 
 time of a semi- vibration of a pendulum in a cycloidal 
 and circular arc is to the time in which a solid cy- 
 linder would descend through the same arc by roll- 
 ing, as ^/% to y/3, or very nearly, as 9 to 11 ; 
 but this same vibration would be to the time spent 
 in rolling of a hollow cylinder, as 1 to y/2, or near- 
 ly as 5 to 7 Again, the semi-vibration of a pen- 
 dulum is to the time of rolling over the same arc by 
 a solid sphere as ^/5 to Y/7, or nearly, as 11 to 13 ; 
 but to the time in which a hollow sphere would roll 
 through it, as ^/3 to ^/5, or as 7 to 9- 
 
 These proportions are reversed when the rolling 
 bodies are by their acquired celerities again carried 
 up an inclined plane. The hollow ball or cylinder, 
 which in their descent had been retarded by the 
 large share of their inciting force consumed in ge- 
 nerating the rotation, are now, by this accumulated 
 momentum, borne with greater rapidity in rising 
 over a similar plane. If they roll in a cycloid or in a 
 small arc of a circle, their motions will be recipro- 
 cated in the same periods as those of sliding bodies, 
 but very unequally divided. The hollow spheres 
 and cylinders will in their descent always consume 
 
160 ELEMENTS OF 
 
 a longer interval of time than is required for their 
 subsequent ascent. The contrary would take place 
 if the matter were collected near the centre of these 
 bodies. 
 
 If a body be struck in a direction which passes 
 through its centre of gravity, it will advance in that 
 line with an uniform celerity, maintaining inva- 
 riably the same parallel position. But if the scope 
 of impulsion should incline on either side of the 
 centre of gravity, the body will, besides its progres- 
 sive and rectilineal motion, acquire about that point 
 a coexistent and independent rotation. To illus- 
 trate farther this important property, let the body 
 ACBD, (fig. 75.) having the perpendicular planes 
 AB and CD crossing in the centre of gravity 
 G, be struck at L with a force expressed by LO, 
 the double of LN, and at right angles to AGB. 
 Join LG, and produce till LGziGP, and let fall the 
 perpendiculars LM, PR and PQ. The force LN, 
 which is half of the original impulse, may be decom- 
 posed into the forces LG and GN or DL. But the 
 force LG, viewed as acting upon any point P in that 
 direction, is equivalent to GP, which again may be 
 decomposed into the forces QP and GQ or RP. 
 But QP is evidently equal to LN or NO, while GQ 
 is equal to DL or GN. The equal forces QP and 
 NO, thus exerted at equal distances on both sides of 
 the centre of gravity, must produce an equilibrium, 
 and urge forward that point by their joint action. 
 Wherefore the centre of gravity is borne along in 
 
NATURAL PHILOSOPHY. 161 
 
 the same or parallel direction DC, with the entire 
 force of impulsion LO. But there still remain the 
 forces DL and RP, which may be conceived to act 
 at the equidistant points M and R at right angles 
 to the diameter MR, and hence generate a revolu- 
 tion about the centre G, with a force represented 
 by 2ML or QN. This rotatory motion, since the ef- 
 forts on opposite sides are equal, cannot impair, or 
 in any way disturb the progressive momentum. 
 
 It is evident that, every part of the plane perform- 
 ing its revolution in the same time, the centrifugal 
 force exerted by the point P will have its intensity and 
 direction denoted by GP. But this oblique force may 
 be resolved into the forces PQ and PR, at right an- 
 gles to the cross diameters. Now, from the property of 
 the centre of gravity, the perpendicular distances PQ, 
 of every point on both sides of the line AGB, ba- 
 lance or extinguish each other ; and, for the same 
 reason, the perpendicular distances PR on both sides 
 of CGD are exactly balanced. Consequently the 
 centrifugal forces GP of all the points about the 
 centre of gravity, completely extinguish each other 
 at the axis, and leave it free and undisturbed. 
 
 This axis would evidently pass through the cen- 
 tres of gravity of every similar parallel plane which 
 has the same position. In each of these, an equili- 
 brium of centrifugal action must likewise obtain. 
 Wherefore, the axis of a symmetrical body, which 
 
 VOL. I. L 
 
ELEMENTS OF 
 
 may be conceived to be composed of such planes, 
 will continue permanent. 
 
 An irregular body may commence a rotation about 
 any axis which passes through its centre of gravity, 
 because the aggregate of all the centrifugal forces, 
 estimated in the direction of perpendiculars to a 
 plane touching the axis, must produce a mutual ex- 
 tinction. But though this collective action would, 
 if exerted at a single point, maintain a perfect ba- 
 lance ; yet being unequally distributed over the axis, 
 it may, from the principle of the lever, predominate 
 upon one side of the primary more than upon an- 
 other, and therefore continue to bend the axis with 
 greater or less effect in some particular direction. 
 Thus, let the body be cut by a plane through any 
 point H (fig. 76.) above or below the common cen- 
 tre of gravity. Suppose g to be the centre of the 
 particular section, and through H, where the axis 
 pierces it, draw the perpendicular diameters aELgb 
 and cHd. The centrifugal force of any point P ex- 
 erted against the axis at H, being expressed by Hp, 
 is decomposed into the forces pq and pr, at right 
 angles to H and He ; but from the property of the 
 centre of gravity, the sum of all the pq's on both 
 sides of a1A.gb is nothing, and so is likewise the sum 
 of all the pi's perpendicular to the cross diameter gL 
 Wherefore, there is an excess ir of the lateral force 
 pr, which for the whole plane will amount to the num- 
 
NATURAL PHILOSOPHY. 168 
 
 her of points multiplied into *> or ^H. This pre- 
 ponderance will consequently be exerted at H, in 
 bending the axis in a direction parallel to a 
 and with an efficacy proportional to the distance 
 from the centre of gravity. Let the parallel sec- 
 tions, therefore, be multiplied, corresponding to dif- 
 ferent points along the axis ; the several excesses of 
 centrifugal action being thus variously combined, 
 and their effects estimated by the application of the 
 principle of the lever, will give a certain resulting 
 impression, which tends to push the centre or axis 
 about the fulcrum, (fig. 770 with the force and in 
 the direction OZ. 
 
 Conceive such parallel sections at equal intervals, 
 multiplied indefinitely in various directions, and at 
 the distances 1, 2, 8, 4, 5, &c. from the centre O ; 
 and let their corresponding efforts to derange the 
 axis, when all reduced to the same vertical plane, be 
 denoted by , b, c, d, e, fyc. It is evident, that 
 the aggregate power exerted to bend the axis in this 
 plane, will be expressed by a + 2b + 3c+4<d+5e + 
 c. From the property of the centre of gravity, all 
 the different efforts about the axis must counterba- 
 lance each other, or a+b+c+d+e, &c. =0. But, 
 that the axis should remain undisturbed, the action 
 of those powers must produce a mutual balance, or 
 a+ 2b = 3c +4>d+5e+ &c. =o. . These are hence the 
 only two equations for determining the innumerable 
 quantities a, b, c y d, e, &c., which will therefore ad- 
 
 L 2 
 
ELEMENTS OF 
 
 mit of values indefinitely varied, and subject only to 
 such very general conditions. Suppose those values 
 to be represented by a countless multitude of curves 
 applied to the axis ; their mutual intersections will 
 form an infinite variety of dispersed points. Among 
 such immense diversity, therefore, it would be pos- 
 sible to trace a succession of them in a straight line. 
 This marks the direction of a permanent axis, which 
 unites the conditions implied in the two fundamen- 
 tal equations. 
 
 It may therefore be inferred, that every rigid body, 
 however irregular in its form, has at least one axis 
 about which the cumulative centrifugal action would 
 be exactly balanced. This axis is called the Prin- 
 cipal Axis ; but, depending on it, there are always 
 two other axes which have the same denomination. 
 For let OB and OC (fig. 78.) be drawn through the 
 centre of gravity perpendicular to the principal axis 
 OA and to each other. Let the plane AOB cut 
 the section, which passes through H in HL ; from 
 any point P in that section draw PL at right angles 
 to HL, and LK parallel to AO, and join PK. The 
 centrifugal force of the point P about the axis OB 
 being proportional to its distance PK, this force is 
 reduced to KL acting in the direction of OA, and 
 therefore bending OB with an energy KLxOK. 
 The combined efforts to change the position of 
 this new axis will consequently be expressed by 
 /, OKxKL, or/, HLxOH; but the latter expres- 
 
NATURAL PHILOSOPHY. lf)5 
 
 sion was shown to produce accumulatively a mutual 
 balance about the axis OA, and hence the disturbing 
 impressions exerted on OB will extinguish each 
 other, and leave it perfectly free in its rotation. The 
 same reasoning will apply to the third axis OC, 
 which is at right angles both to OA and OB. Every 
 body, however irregular, has consequently those 
 three Principal Axes. In the case of a symmetrical 
 body generated by the revolution of a plane, the 
 primary axis evidently is an individual line, but the 
 other two principal axes are indeterminate, being 
 any perpendicular diameters of the circle of the equa- 
 tor. 
 
 The momentum of rotation about the primary axis 
 is always a minimum. For this axis, in a symme- 
 trical body, must pass through the centre of gravity 
 of each vertical plane, and consequently, by Prop. 21. 
 Book III. of Geometrical Analysis, the sum of 
 the squares of all the lines drawn from that centre 
 to every particle of each plane, which constitutes the 
 momentum of rotation, is less than the sum of the 
 squares of their distances from any other point. In 
 the case of irregular bodies, the primary axis will 
 evidently cut the successive planes as near as possi- 
 ble to their centres of gravity, and therefore the ag- 
 gregate squares of the distances of all the physical 
 points will yet be a minimum. It can likewise be 
 proved, that about one of the remaining principal 
 axes the momentum of rotation is a maximum. 
 
1()6 ELEMENTS OF 
 
 V. MECHANICS, 
 
 which consists in the application of the Principles 
 of Dynamics to the construction and composition of 
 Machinery. 
 
 The various wants of society are supplied by the 
 operations of human industry on the surface of the 
 globe. It was necessary to divide and re-unite bo- 
 dies, and to transport portions of matter from one 
 place to another. But those labours are abridged, 
 and rendered greatly more productive, by the proper 
 exercise of skill. The incessant efforts to augment 
 our powers by the aid of tools, called forth the ear- 
 liest germs of ingenuity and invention. Without 
 the simpler implements of art, mankind could never 
 have emerged from the savage state ; and to the 
 prodigious improvement and extension of machinery 
 in modern times, we are indebted for all the com- 
 forts, enjoyments, and delicacies of highly civilized 
 life. 
 
 The elements of Machines may be ranked under 
 two distinct classes those of a general, and those 
 of a particular nature. To the former belong what 
 I should call the Concentrator of Force, and the 
 Engine of Oblique Action, which, when composed 
 of connected cords, has been named the Funicular 
 
NATURAL PHILOSOPHY. 
 
 Machine ; the latter include the five ordinary me- 
 chanical powers the Lever, the Wheel and Axle, 
 the Inclined Plane, the Screw, the Wedge, and 
 the Pulley. We shall consider these several in- 
 struments in their order. 
 
 1. The Concentrator of Force. This engine ex- 
 hibits, in a very striking manner, the accumulation 
 and transfer of impulsion among bodies, and may 
 therefore be regarded as, next to Atwood's inge- 
 nious machine, a most important addition to our 
 stock of illustrative philosophical apparatus. It not 
 only sheds a clear light on some abstruse parts of 
 mechanical theory ; but may with advantage be di- 
 rected, in a variety of important cases, to the prac- 
 tice of the arts. 
 
 This Concentrator consists of a ponderous wheel, 
 composed of a thin circle of iron, loaded at the cir- 
 cumference with a broad swelling ring of lead, and 
 fixed to a strong steel axle, to which is likewise at- 
 tached three or more barrels or short cylinders, of 
 different diameters, the smallest formed of brass, and 
 divided in two parts that are capable of locking to- 
 gether at pleasure. The axle, placed in a horizon- 
 tal position, runs upon gudgeons on the top of a 
 high and solid frame ; and the machine may be set 
 in motion, either by turning a winch, or more com- 
 monly by the descent of a small weight fastened to 
 a silk line, which passes over a pulley, and is lapped 
 
168 ELEMENTS OF 
 
 round one of the barrels. Fig. 79. represents the 
 only model which I have yet used, the frame being 
 about 5 feet high, the wheel has 18 pounds weight, 
 and is 17 inches in diameter, while the diameters of 
 the successive barrels are only 6, 4, and 2 inches. 
 The principal application of this engine is to raise 
 from its platform any great weight. If 1 pound, 
 for instance, in descending through 30 feet, gra- 
 dually communicate its impression to the wheel, and 
 the instant it reaches the ground, the detached part 
 of the brass barrel should lock and catch hold of the 
 loop of a cord holding a half-hundred weight or 56 
 pounds, this mass will be almost immediately lifted 
 up near 6 inches, and there suspended. But what 
 is remarkable, and appears at first sight paradoxical, 
 the effect is precisely the same, about whatever bar- 
 rel the line be wound. The result, however, is quite 
 altered, when different descending weights are used, 
 the elevation produced being always proportional to 
 theni. Thus, the descent of 2 and of 4 pounds 
 through 30 feet, will respectively raise 56 pounds 
 to nearly 1 and C 2 feet. 
 
 It is not difficult to explain generally these effects. 
 Let the descending power be very small compared 
 with the weight of the ponderous wheel, and sup- 
 pose its action at first to be exerted at the rim. This 
 rim, in which is condensed the entire mass, will now 
 describe a space equal to the measure of descent, and 
 the whole power may be considered as inciting its 
 
NATURAL PHILOSOPHY. If)!) 
 
 revolution. Consequently the square of the velocity 
 acquired by the wheel must be proportional to the 
 descending power multiplied into the space through 
 which it falls. When the accelerating force acts on 
 a barrel smaller than the wheel, the energy exerted 
 at the rim is proportionally diminished, but the space 
 described by it is augmented in the same ratio, and 
 hence the square of the velocity resulting from those 
 combined causes will continue unaltered. This 
 square of the velocity, or measure of impulsion, is 
 extinguished by the efforts expended in raising the 
 weight. Wherefore the power multiplied into the 
 quantity of descent is equal to the weight multiplied 
 into its corresponding ascent. The power and the 
 weight are thus inversely proportional to the spaces 
 which they severally describe. 
 
 This general proposition must be viewed, how- 
 ever, as only a very near approximation to the 
 truth, since the small quantities which would af- 
 fect the result are, for the sake of simplicity, re- 
 jected. Thus, a minute portion of the inciting 
 force is wasted in generating the slow descent of 
 the falling body, while the final impulsion of the 
 wheel incurs a slight loss merely in sustaining the 
 momentary gravitation of the weight to be raised. 
 To investigate the rigorous formula, let p denote 
 the falling body, and s the space which it de- 
 scribes ; m the mass of the wheel, g the distance 
 of its centre of gyration, . arid , the radius of the 
 
170 ELEMENTS OF 
 
 barrel round which the thread is lapped. Since 
 the circumference of this barrel must move just as 
 fast as the descending weight, the effect would be 
 the same if, instead of the wheel, a mass express- 
 
 ed by ^- had been collected at the distance a 
 a 
 
 from the axis. The whole inciting force p is 
 hence shared proportionally between p and ~, 
 
 and therefore the part of it exerted on the barrel 
 
 , . , . . ,. ., , . 
 whlch > bem dlvlded b 
 
 |- - - for the accelerating energy. The velocity 
 
 communicated to the circumference of the barrel, 
 by the descent of the weight p through the space s, 
 
 ^ \ 
 
 *j 
 
 is hence = 
 
 \jg 
 
 The velocity communicated to the centre of gy- 
 
 ration is therefore Sg^/t ~ - X But this fi- 
 
 \rn*+a 2 ' 
 
 nal velocity may be again extinguished, by the 
 operation of an opposite retarding force. Let w 
 denote the height to be raised, h the height of its 
 elevation, and b the radius of the barrel which lifts 
 
 it up; 
 
 ps wh 
 
 and consequently f- t = - , , , ilence, 
 * mg*-\-a?p mg*+b*tv 
 
NATURAL PHILOSOPHY. 
 
 the other quantities remaining, the value of h is 
 easily detern lined from that of w. 
 
 As an exemplification of this formula, let the 
 
 measures of the model be substituted. Here 
 
 p = 1 lb., ,v = 30 feet, m=lS lb., w = 56 lb., g = f , 
 
 and a and b the smallest barrels = T V j whence 
 
 30 56.h 
 
 and ' b reductlon > we have 
 
 4320 8064.A 
 
 = 1Qntt -> and An: .56 13 feet or 6.735 inches. 
 llOo 1/cOo 
 
 If the barrels of 2 and 3 inches radii were employ- 
 ed, h would be respectively 6.717 and 6.688 inches ; 
 but, if the accelerating weight had been 4 pounds, 
 the ascents corresponding to the smallest, the middle, 
 and the largest barrel, would be 26,868, 26,589, 
 and 26,124 inches. 
 
 In general, the quantities a*p and b*w are very 
 inconsiderable in comparison of mg* 9 and may be 
 therefore safely omitted. Whence ps = wh, and 
 
 t) s 
 
 h = .s,orM?=.-r. There is no limit, then, but 
 w n 
 
 the strength of the axle, to the weight which may 
 be raised, by the action of the Concentrator of Force ; 
 the only indispensable condition being, that the 
 height is always inversely as the load. The fall of 
 2 lb. through 30 feet would lift five hundred weight 
 or 560 lb., over one inch and a third part. Such a 
 small primary force might hence, when accumulated 
 
ELEMENTS OF 
 
 in this way, prove sufficient to start the greatest 
 load, or overcome the most powerful obstacle. 
 
 Instead of raising great weights, the Concentra- 
 tor might be adapted to tear asunder thick wires or 
 metallic rods. The power exerted will then be in- 
 versely as the spaces through which those rods 
 stretch, before they suffer fracture. The effects will 
 consequently be augmented, by shortening the lengths 
 of the rods. To the bottom of the engine, screw a 
 strong bar above four feet in height, and to this 
 fasten rods from six inches to a foot in length. If 
 the limit of extension, which precedes the final dis- 
 ruption, were only half an inch, the power exerted, 
 though produced by the descent of a single pound, 
 would amount to about 1500 Ib. A rigid and un- 
 yielding body is hence the most easily torn or bro- 
 ken. 
 
 But the impetus accumulated by the Concentra- 
 tor may be wholly consumed, in merely stretching 
 a very elastic substance which has a sufficient length. 
 If a light contorted spring, for instance, be opposed 
 to the rotation of the mass, it will, by its large 
 though languid extension, gradually destroy the 
 motive energy. A thick woollen cord, loosely plait- 
 ed, and tied to a ring at the bottom of the machine, 
 will produce a similar effect. A slender hempen 
 string, though possessing little of a stretching qua- 
 lity, may still serve the same purpose, if it be taken 
 of an adequate length. It is only required that 
 
NATURAL PHILOSOPHY. 
 
 half the final strain multiplied into the correspond- 
 ing extension should be equal to the product of the 
 falling weight by its quantity of descent. A ten- 
 sion of 60 lb., acting through a height of one foot, 
 would be sufficient to muffle and extinguish the 
 momentum of rotation generated by the descent of 
 one pound through 30 feet. To produce this effect, 
 therefore, it is only wanted to select such a length 
 of cord as will extend one foot, by the application 
 of a strain of 120 pounds. A more slender sub- 
 stance, if proportionally more stretching, would have 
 the same effect. In either case, a weight exceed- 
 ing the absolute tension, and attached to the end of 
 the string, would not, during the moderated con- 
 sumption of the shock, be stirred in the slightest 
 degree from its place. 
 
 The strength of a cord depends on its thickness, 
 but the power to resist impulsion is determined by its 
 elasticity and its length. This principle, which has 
 been much overlooked, enters largely into the consi- 
 deration of practical mechanics. Hence the practice 
 of stemming a ship's way into a harbour by the fric- 
 tion of a long rope, the momentum being thus gra- 
 dually spent. A short rope, firmly fastened to the 
 pier head, so far from staying the vessel, would in- 
 stantly snap. For the same reason, a ship riding at 
 anchor is obliged to lengthen her cables. When 
 these are composed of chains, the tension resulting 
 from a diminution of curvature is precisely the same 
 
174 ELEMENTS OF 
 
 as if a contractile force had been exerted. It is per* 
 haps a general error in civil architecture to aim at 
 mere solidity. Lightness combined with elasticity 
 will often resist the shocks of ages, while stiff and 
 ponderous materials are crumbled into ruins. 
 
 It may be curious to mark the time of accumula- 
 tion of momentum in the Concentrator, and that 
 of its subsequent expenditure. The force accele- 
 
 CL 2l/ p 
 
 rating the descent of p is - 2 , and conse- 
 
 mg* + a?p 
 
 quently the time of this descent in seconds is 
 
 time required to lift the weight will hence be always 
 
 But these expressions may be abbreviated, by 
 omitting the quantities a z p and b*w. Wherefore, 
 
 the time of descent is nearly ~, / , and the 
 
 J 4#V p 9 
 
 time of the subsequent ascension of the weight 
 
 . g imh a fmps . 
 
 18 TZV or ~- / ^r, because ps=wh. Hence, 
 46 v w *b v w* 
 
 g and m remaining the same, the time of genera- 
 ting the impulsion is compounded of the inverse 
 ratio of the radius of the barrel, and the inverse 
 subduplicate ratio of the falling body, with the di- 
 rect subduplicate ratio of the space of descent ; 
 while the time of expending this impulsion is in- 
 
\ 
 
 NATURAL PHILOSOPHY. 175 
 
 versely as the weight to be raised, and directly in 
 the subduplicate ratio of the falling body and of its 
 descent. Thus, one pound, having a line coiled 
 
 about the smallest barrel, will descend through 30 
 
 g 
 feet in -7 V( 18.30) seconds or 46" ; connected 
 
 with the middle barrel, it would descend in C 2S\" ; but 
 applied ta the largest barrel, it would require only 
 15|-". The succeeding act of lifting an half-hundred 
 
 o 
 
 weight would be performed in 77; y/( 1 8. 30) se- 
 conds, or five-sixths of a second. If the descending 
 power had been four pounds, the times spent by the 
 successive barrels in generating the momentum of 
 rotation would be llf", 5|f", and 2" ; but the 
 time of hoisting the weight of 56 pounds would be 
 
 If". 
 
 Let a=b, and the time of descent will be to 
 
 that of the subsequent ascension as / to / ~-, 
 
 or as ,/ ^r t /-^j-, that is, inversely as/>to w. 
 
 Hence the greater is the weight to be raised, the 
 more rapid is the act of its ascension ; and it is the 
 same thing, whether an obstacle be overcome, or a 
 disruption effected. The impulsion which required 
 46^- seconds to accumulate, now exerting a strain 
 equal to 1500 pounds, would tear a rod of metal 
 asunder in less than the thirtieth part of a second. 
 
170 ELEMENTS OF 
 
 The slowness with which this impulsive energy col- 
 lects, is remarkably contrasted with the rapidity of 
 its subsequent discharge. The greatest effects may 
 be concentrated within such a portion of time as 
 eludes the observation of the senses, and appears 
 really instantaneous. 
 
 The Concentrator of Force thus finely elucidates 
 the acquisition and the transfer of impulsive energy. 
 The same accumulated momentum produces diver- 
 sified effects, according to the way in which it is dis- 
 posed. It will raise a ponderous mass, tear asunder 
 a solid body, or will expend all its action in merely 
 stretching a substance of a very distensible quality. 
 These different purposes are attained in the opera- 
 tions of the mechanical arts ; but sound theory is 
 yet required to guide and improve the practice. 
 
 This engine, constructed on a large scale, might 
 hence, with obvious advantage, be adopted as a most 
 powerful auxiliary in various operations of art. 
 Many situations occur which require an immense 
 effort to be made on a sudden, and within a very 
 limited space. This can be accomplished only, by 
 storing up, as it were, a magazine of force, which 
 may be opened and discharged at some precise mo- 
 ment. Even moderate animal exertion, if applied 
 during any considerable time, will communicate to 
 the Concentrator an impulsion sufficient to burst the 
 firmest obstacles, and to lift, through a short space, 
 the most enormous loads. The only thing wanted 
 
NATURAL PHILOSOPHY. 177 
 
 is, by the application of geometrical principles, to 
 graduate at pleasure the transfer of impulsion. 
 
 This engine involves likewise the theory of the 
 Fly, which is annexed to various machines, not to 
 augment their power, but merely to equalise their 
 motion. The variable inciting forces are thus, by 
 the intervention of a heavy wheel, blended together 
 in creating one great momentum, which afterwards 
 maintains a nearly uniform action. The use of the 
 Fly in mechanics hence resembles that of a reservoir, 
 which collects the intermitting currents, and sends 
 forth a regular stream. 
 
 2. Machine of Oblique Action. This depends 
 on the theory of the composition of forces. A force 
 exerted in the proper direction will balance any two 
 forces ; but if one of these be sustained by some fix- 
 ed point, the first force may be considered as acting 
 only against the other. Let the force OB (fig. 80.) 
 be supported at B, produce BO till OD be equal to 
 it, and complete the parallelogram AOCD ; the 
 force OA will counteract a force OC, or produce an 
 equal force in the opposite direction. If the angle 
 BOC be very oblique, OC will be much greater than 
 OA, for O A is to OC as the sine of the angle COD 
 to the sine of AOD. Suppose AO, BO, and CO 
 to be cords tied at the point O ; the force OA would 
 occasion at C a strain OC. If, therefore, while the 
 cord OB is fastened at B, an extension of the cord 
 
 VOL. I. M 
 
178 ELEMENTS OF 
 
 OC were passed over a pulley at C, the force OA 
 would cause the partial elevation of a weight repre- 
 sented by OC. This arrangement has been called 
 The Funicular System. It is, to a certain extent, 
 familiar to the seaman, who has often recourse to 
 that mode in bracing the sails. He pulls at P la- 
 terally the rope AP'C (fig. 81.) into the oblique po- 
 sition AOC, and his power being at first indefinitely 
 augmented, enables him to start the yard, and move 
 it through a minute space \ and by fastening the 
 slackened rope again at B, he can renew the pro- 
 cess. 
 
 If, instead of cords, there be substituted steel bars, 
 all jointed at O, while OB turns about the fixed 
 point B, a most powerful thrust will be produced in 
 the direction OC. This thrust would even be in- 
 creased in a slight degree, if exerted in the direction 
 BC, by making the end of the bar OC to slide along 
 the line BC ; for OA will be to CP, as the pres- 
 sure at O to the thrust at C. But the angle BOC 
 being very obtuse, the ratio of the pressure to the 
 thrust may be considered as the same as that of the 
 sine of its supplement COD to the radius. If this 
 angle COD were five degrees, the thrust would 
 hence be 11 \ times the pressure ; but if the angle 
 were only the tenth part of a degree, the thrust 
 would exceed the pressure 573 times. At the mo- 
 ment, therefore, of the collapse of the bars BO and 
 
NATURAL PHILOSOPHY. 179 
 
 CO into a rectilineal position, the power is infinitely 
 augmented. 
 
 The end C of the bar OC may be easily made to 
 move in the direction BC, by combining (fig. 82.) 
 two opposite bars BN and NC similar to BO and 
 OC. The thrust will, in the case of great obliquity, 
 be inversely as the angle NCO. Suoh a combina- 
 tion of bars, variously modified, has lately been adopt- 
 ed, with great effect, in several operations of art 
 for opening the steam-valve of Watt's engine for 
 the construction of the printing-press and for ex- 
 tracting the steel core from the hollow brass cylinder 
 now used as a roller in the printing of cotton. In 
 all these instances, a vast momentary effort only is 
 wanted. 
 
 The common mechanical powers are the chief ele- 
 ments of all machinery. They amount to five, but 
 may be ranged in three divisions : 1. The Lever y 
 and the Wheel and Axle, which is only an exten- 
 sion of it ; 2. The Inclined Plane, and its modifi- 
 cations, The Screw and The Wedge ; and, 3. The 
 Pulley, which admits of multiplied combinations. 
 Of these instruments, the theory has been already 
 given, in as far as regards equilibrium ; and to pro- 
 duce motion, it is only required to apply additional 
 force. But it will be satisfactory to arrive at the 
 same conclusions, by employing the principle of vir- 
 
 M 2 
 
180 ELEMENTS OF 
 
 tual velocities. We shall consider these elementary 
 engines in their order of succession. 
 
 1. The Lever ) which consists of an inflexible bar 
 ACB, (fig. 83.) either straight or bent, resting on 
 a point C called the fulcrum, the power being ap- 
 plied at the end A of the arm AC, to raise a weight 
 at the end B of the other arm CB. Throwing out 
 of view the weight of the lever itself, and supposing 
 it to be at first horizontal ; let it shift into the proxi- 
 mate position A'CB'. The minute arcs AA' and 
 BB' thus described may be regarded as tangents, 
 and, consequently, while the power P descends ver- 
 tically through a space AA', the weight W rises 
 through a space BB'; wherefore the opposite momenta 
 being equal, P X A A' = Wx BB', and P : W : : 
 BB' : AA' : : BC : AC, or the power and weight 
 are inversely as their distances from the fulcrum. 
 
 Let ACB (fig. 84.) represent a bent lever, and 
 from its extremities A and B, let fall the perpendi- 
 culars AH and BI, upon the vertical CH passing 
 through the fulcrum C. Conceive the lever to move 
 into the proximate position A'CB', and parallel to 
 CH draw A'L meeting AH in L and BM meeting 
 B'M a parallel to AH in M. From similar trian- 
 gles, A'L : AA' : : AH : AC, AA' : BB' : : AC : 
 BC, and BB' : BM : : BC : BI ; whence, by com- 
 position of ratios, A'L : BM : : AH : BI. But 
 AL is the minute descent of the power, and BM the 
 
NATURAL PHILOSOPHY. 181 
 
 corresponding ascent of the weight ; consequently 
 P x A'L = W x BM, and P : W : : BM : A'L, or, 
 by equality of ratios, P : W : : BI : AH. The 
 power and weight are, therefore, inversely as the 
 perpendiculars let fall upon the vertical CH. 
 
 The same result is deduced from the property 
 which belongs to the centre of gravity, of occupying 
 the lowest place possible, whenever an equilibrium is 
 attained. Join AB, and AH : BI : : AG : BG ; 
 whence Px BG= Wx BG, and the power and weight 
 being supposed to be attached at A and B, the point 
 G would be the centre of gravity of the loaded lever. 
 This centre would, therefore, while the lever turns 
 about its fulcrum, describe an arc of a circle, and 
 hence falls into the lowest position in crossing the 
 vertical CH. 
 
 Levers are usually distinguished into three kinds, 
 according to the relative position of the power, the 
 weight, and the fulcrum. 1. When the fulcrum (fig. 
 So.) lies between the power and the weight. This 
 kind includes the crow and handspike, pincers, and 
 scissars. The toothed hammer is only a bent lever of 
 this kind. Its invention was in mythology ascribed 
 to Neptune, his trident being only a three-pronged 
 crow. The arm PC is commonly longer that WC, 
 and consequently the weight exceeds the power. 
 The number of times which the weight contains the 
 power, is always called the mechanical advantage or 
 purchase. 2. When the weight lies between the 
 
ELEMENTS OF 
 
 fulcrum and the power (fig. 86.). This kind in- 
 cludes the crow in its more general application, the 
 baker's and druggist's knife, the common door, the 
 wheel-barrow, nut-crackers, and oars. 3. When the 
 power is applied (fig. 87.) between the fulcrum and 
 the weight. To this kind belong the sheep-shears. 
 It has a mechanical disadvantage, but admits of a 
 proportionally wider motion. The bones of animals 
 are therefore levers generally of this sort, pulled by 
 the moderate contraction of muscles inserted near 
 their joints. 
 
 Conceive the forces represented by the power and 
 the weight to be exerted upwards, (fig. 88.) in a 
 lever of the first kind, their joint efforts will be sus- 
 tained by an opposite fulcrum at C. Instead of the 
 fulcrum, substitute a load suspended from the same 
 point, and this will be shared at the ends P and W 
 inversely as the distances PC and WC. Such is the 
 distribution of pressure in the case of a pole bearing 
 an intermediate weight. If it hang from the middle, 
 the carriers will always share the burden alike ; but 
 if the load be only placed upon the pole, as in fig, 
 89. a vertical from the centre of gravity will, in going 
 up hill, divide the space unequally at C, and the 
 lower person must therefore suffer a greater strain. 
 
 Two horses of unequal strength may yet be yoked 
 to draw equally, by a proportionate division of the 
 bar. This is partly effected in the ordinary way, by 
 attaching the perch to a short projection from the 
 
NATURAL PHILOSOPHY. 183 
 
 middle of the bar. To range a number of men 
 along the amis of a pole, for the purpose of trans- 
 porting heavy loads, is very unskilful, though fre- 
 quently done in this country. Those who are near- 
 est must evidently take the greatest share of the 
 burthen, while the remote bearers have not the 
 means of exerting their strength. The method prac- 
 tised in the East is much preferable, the strain being 
 successively subdivided by a system of levers crossing- 
 each other. In China the same object is obtained 
 still more simply, by placing the load on the middle 
 of long bamboos, which cross at different angles, each 
 end of them being borne by a labourer. 
 
 When a load is laid upon a plane whose weight 
 may be neglected, the pressures sustained by three 
 points have been already assigned. But they may 
 be derived more easily perhaps from the property of 
 the lever. For suppose the triangle ABC, ( fig. 90.) 
 bearing the weight P, to rest upon the angles A, B 
 and C. Through P draw BPD, CPE, and APR 
 Conceive the plane to be lifted by the point B about 
 AC as a fulcrum ; BPD being then a lever, the 
 load P would be to the force exerted, as BD to PD, 
 or as the triangle ABC to APC. In the same man- 
 ner, were the triangle lifted by the point A, the 
 load P would be to that strain, as the triangle ABC 
 to BPC. Wherefore, collectively the forces requir- 
 ed to raise the triangle from the several points A, B 
 and C, or the pressures which these exert are pro- 
 
184 ELEMENTS OF 
 
 portional to the opposite triangles BPC, APC, and 
 APB. When B coincides with the centre of gravi- 
 ty, the several internal triangles are equal, and con- 
 sequently the strain is equally distributed. 
 
 The action of the simple lever is evidently con- 
 fined within a very narrow space. But, by means 
 of a small addition, it can be rendered capable of 
 repeating its operation to an unlimited extent. This 
 is effected by annexing to the end of the short arm 
 two claws, which work alternately in the teeth of a 
 ratchet wheel. Such a machine is called the Uni- 
 versal Lever. It is used occasionally for raising 
 weights, but more commonly for dragging logs of 
 timber to the saw-mill. 
 
 The mechanical advantage may be repeatedly mul- 
 tiplied, by a combination of levers of the first kind. 
 Thus, the power being applied at (fig. 91.) A, let 
 AB act upon the end C of another lever, and this 
 upon E ; the termination F will lift a very great 
 weight. Suppose the purchase of the lever AB to 
 be 4, that of CD, 3, and that of EF, 2 ; then one 
 pound applied at A will support 4x3x2, or 
 pounds at F, and a slight additional force will set 
 it in motion. The space of ascent, however, must 
 evidently be very confined. But the wheel and 
 pinion being only an extension of the lever, the same 
 system of combination may have its action continued 
 or incessantly renewed in a train of toothed wheels. 
 
NATURAL PHILOSOPHY. 185 
 
 One of the most ordinary but useful applications 
 of the lever is to weigh substances, or rather to com- 
 pare their weights by some standard. This can be 
 accomplished, by employing either a single weight 
 or a set of subdivided weights. When the former 
 mode is adopted, the long arm of the lever is divid- 
 ed into portions equal to the short one, which has a 
 counterpoise appended to it. The standard consist- 
 ed of a weight resembling in shape the pomegra- 
 nate, and hence called in the East, Romman ; 
 which was fixed to a ring that could slide along 
 the successive divisions, and mark the relative weight 
 of the substance examined. In allusion to this cir- 
 cumstance, the Steel Yard, as it is now termed, was 
 formerly, through misconception, named the Roman 
 Statera. If the fulcrum or point of suspension be 
 taken still nearer the end, the scale of weights will 
 be proportionally augmented. The small balance of 
 the Chinese consists of a tapering rod of ivory, like 
 a quill, perforated in four points, to be suspended 
 by different threads, the subdivisions being extended 
 along each corresponding side. 
 
 The common balance, though less expeditious, is 
 capable of greater accuracy than the steel yard. As 
 it has equal arms, it requires a series of intermediate 
 weights. For philosophical purposes, the easiest 
 way is to reckon always by grains. The geometri- 
 cal progression 1, 2, 4, 8, 16, 32, 64, &c. forms the 
 simplest arrangement ; but it will be found more 
 
18() ELEMENTS OF 
 
 convenient to follow the decimal division, and the 
 successive sets of 1, 2, 3, and 4 ; 10, 20, 30, and 
 40 ; 100, 200, 300, and 400 ; 1000, 2000, 3000, 
 and 4000, &c. would save much trouble in adding 
 up the weights. 
 
 A False Balance has one arm somewhat longer 
 than the other. But the fraud is easily detected, by 
 interchanging the places of the weight and the sub- 
 stance to be weighed ; for the weight assigned will 
 then be diminished, in the same proportion as it 
 was before augmented. It perhaps deserves remark, 
 that the true weight is rather less than half the sum 
 of those opposite indications. Suppose the perpen- 
 dicular BD (fig. 92.) to represent the true weight, 
 and let BD be to BA in the ratio of the arms of the 
 fraudulent balance ; having completed the semicircle, 
 it is evident that AB will indicate the weight of the 
 substance when placed in the scale of the longer arm, 
 and BC its weight when suspended from the shorter 
 arm. But BD is less than the radius, or the arith- 
 metical mean between AB and BC. 
 
 Since, from the property of the semicircle, BD is 
 a mean proportional or a geometrical mean between 
 AB and BC, the true weight of any body might be 
 discovered by means even of a false balance. For 
 let it be weighed first in one scale, and then in the 
 other ; and, the results being multiplied together, 
 the square root of their product will give the accu- 
 rate value. 
 
NATURAL PHILOSOPHY. 187 
 
 Every correct balance must thus have amis of pre- 
 cisely equal lengths, or its fulcrum placed equally 
 distant from the extreme points at which the scales 
 are suspended. But the delicacy of the instrument is 
 derived from the proximity of its fulcrum to the 
 straight line joining those two points. To preserve 
 a stable equilibrium, however, the fulcrum must oc- 
 cupy a position somewhat above that line. The 
 beam should be strong and light, the preferable form 
 consisting of two hollow cones : it should turn with 
 a fine knife-edge upon a plate of agate, polished 
 crystal, or hard steel ; and the scales should likewise 
 be hooked from sharp edges. The sensibility is far- 
 ther augmented, and the risk of injury obviated, by 
 various other contrivances. To such perfection have 
 the arts been carried in this country, that Ramsden's 
 famous balance would turn with only the seven-mil- 
 lionth part of its load. 
 
 In treating of the properties of the common ba- 
 lance, it will be requisite to consider the beam as a 
 bent lever. We shall assume it as devoid of weight, 
 and then apply the small correction that such an 
 omission renders necessary. Let AC and CB (fig. 
 93.) be the equal arms, C the fulcrum, and A and 
 B the points of suspension. Suppose the mass in 
 the scale attached at B to exceed somewhat the 
 weight applied at A ; the beam will move into 
 an oblique position, so that the vertical line CE 
 will divide AB into segments AE and BE, which 
 
188 ELEMENTS OF 
 
 are inversely proportional to the forces acting at A 
 and B. Let W denote the weight with its scale, 
 W' the opposite scale and mass to be weighed, while 
 a expresses the length of the arm CA or CB, d the 
 perpendicular CD, and I the angle of declination 
 DCE. Since W : W : : BE : AE, by composition 
 : W' W : : AB : 2DE : : AD : DE, and 
 
 consequently Tjr^irwTTy ButDEziCD. tan I, 
 
 and W' + W may be viewed as merely equal to 
 2W ; wherefore, by substitution, we have W W= 
 
 ..,sd.tanl\ rpn j.f . , 
 
 2W I 1. Ine difference of weight is thus, 
 
 in the same beam, proportional to the tangent of de- 
 clination. The value of d in relation to a may be 
 easily determined by experiment ; and, for the con- 
 venience of calculation, a line of tangents, divided 
 centesimally, might be annexed to the index of the 
 balance, instead of a graduated arc. 
 
 Let the weight of the beam itself be now taken 
 into account. If G (fig. 94.) be the place of the 
 centre of gravity, from which a vertical cuts AB in 
 F ; the effort of the beam to recover its horizontal 
 position would, from the property of the lever, be 
 the same as if a part of its weight, diminished in the 
 ratio of BE to EF, were applied at B. Let m de- 
 note the weight of the beam, and g the depression 
 CG of its centre of gravity ; then EF=g.tan I, and 
 consequently the additional pressure at B, indicated 
 
NATURAL PHILOSOPHY. 189 
 
 by such declination, must be ~.tanl. Thevalues 
 
 a 
 
 of a and m being once known, that of g can easily 
 be discovered experimentally, from observing the 
 angle of deviation occasioned by the appending of a 
 single grain at B. But the position of the centre 
 of gravity, and consequently the value of g y is altered 
 at pleasure, by merely screwing a nut or bob higher 
 or lower on the index. If this bob were so adjusted, 
 that the whole weight of the beam is to one grain, 
 as the length of an arm to the depression of the 
 centre of gravity below the fulcrum, the tangential 
 line of the index would mark the quantity of correc- 
 tion in centesimal parts of a grain. 
 
 The value of d and g may be readily and accurate- 
 ly found, by observing the times of the oscillations 
 of the loaded beams. Let fig. 93. represent the 
 oblique lever, in its deviations on either side of its 
 position of equilibrium. The tendency to redress 
 itself is measured by DE, which is proportional to 
 the tangent of the angle DCE, or to the small arc 
 AA'. The oscillations of the beam are hence, un- 
 der the same circumstances, all isochronous. 
 
 But the time of those oscillations may be deduced 
 from theory. Since the accelerating force is to the 
 weight appended at A, as DE to CA ; it is hence to 
 the action of gravity, in the oscillation of a pendu- 
 lum CA from A to A 7 , as DE to AA' or CD, or as 
 CD to CA. Wherefore, CD is to CA, as CA to 
 
190 ELEMENTS OF 
 
 the length of a pendulum which would vibrate in con- 
 cert with the loaded beam. The length of this iso- 
 chronous pendulum is consequently the diameter of 
 a circle (fig. 95.) described through the fulcrum C 
 and the points of suspension A and B. Conversely, 
 the quantity of depression CD may be found, by di- 
 viding the square of C A by CF, the length of a pen- 
 dulum corresponding to the observed time of vibra- 
 tion. Thus, if the arm C A were 8 inches long, and 
 
 64 
 oscillated in 12 seconds ; CD would be - , 
 
 or the 80th part of an inch. 
 
 In this investigation, I have thrown out of 
 view the weight of the beam, which can have lit- 
 tle influence when charged with its full load in the 
 scales. But this slight discrepancy might easily 
 be corrected, for let the vibrations of the naked 
 and loaded beam correspond to those of the pen- 
 
 .11 w i TT A 
 dulums CK and CL and 
 
 , ni? 
 or very nearly ( F zz 
 
 The delicacy of a balance may hence be inferred, 
 from the slowness of its oscillations. When the 
 arms extend almost in a straight line, and the cen- 
 tre of gravity is brought near to the fulcrum, the 
 beam will turn with the smallest additional weight. 
 This extreme sensibility, however, proves inconve- 
 
NATURAL PHILOSOPHY. 191 
 
 nient in practice, by requiring the protection of a glass 
 case, and rendering the process of accurate weighing 
 very tedious. It were perhaps better generally to 
 give the beam a greater flexure that it may vibrate 
 more quickly, and to apply a certain correction, 
 which is easily computed from the tangent of decli- 
 nation. 
 
 II. TJie Wheel and Axle is reckoned the next 
 mechanical power. It consists of a wheel fixed to 
 a smaller cylinder moving about the same centre ; 
 the power is applied to the circumference of the 
 wheel, and the weight to be raised is attached by a 
 cord lapped about the cylinder. This instrument 
 may be regarded as a continued lever. The power 
 is, therefore, to the weight, as the radius of the axle 
 to that of the wheel ; or, if the principle of virtual 
 velocities be preferred, the power and weight are 
 inversely as the circumferences of the wheel and of 
 its axle. If spokes or arms be applied to the wheel, 
 the circumference described by these must be con- 
 sidered as the path of action. 
 
 As varieties of the same instrument, we may 
 name the Capstan or Windlass, and the Common 
 Gin. In the latter, the rope which draws up the 
 weight is wound about a drum, with a long project- 
 ing arm, which a horse puts in motion by treading 
 round a circular path. Of a similar nature is the 
 
192 ELEMENTS OF 
 
 crane, driven by men or cattle, walking within its 
 circumference ; but here the purchase is only in the 
 ratio of the distance of the point of impulsion from a 
 vertical through the centre of the wheel, to the ra- 
 dius of the axle. 
 
 The wheel and axle may turn also on different 
 centres, and have their circumferences connected in 
 mutual action, either by means of a belt or strap, 
 or by the indentation of a system of cogs or teeth. 
 This latter arrangement is usually called Wheel and 
 Pinion. 
 
 The wheel has sometimes its axle of a tapered or 
 conical shape, which gives it a varying purchase. 
 This construction is adopted in the fusee of a watch, 
 and is employed likewise advantageously in raising 
 minerals, by an uniform pull from very deep pits, the 
 rope at its greatest length being coiled about the 
 narrow end of the axle, and advancing towards the 
 wide end as it gradually shortens. 
 
 The Double Capstan is a very ingenious contri- 
 vance, originally brought from China, for augment- 
 ing in any degree the efficacy of the Wheel and 
 Axle, without reducing its strength. It consists of 
 two conjoined cylinders of nearly equal diameters, 
 turning about the same axis, the weight being sup- 
 ported by the loop of a very long cord, of which one 
 end uncoils from the smaller cylinder, while the 
 other end laps constantly about the larger cylinder. 
 (See fig. 96.) The elevation of the weight at each re- 
 
NATURAL PHILOSOPHY. 
 
 volution is therefore equal to half the difference be- 
 tween the two circumferences, and the effect of the 
 arrangement is the same as if the cord sustaining the 
 weight had been wound about a cylinder, which has 
 a circumference merely equal to that quantity. The 
 mechanical advantage of the instrument, combined 
 with its pulley, is hence in the ratio of the diame- 
 ter of the larger cylinder to half its excess above 
 that of the smaller one. Nearly the same effect is 
 procured, by employing a single conical axle ; one 
 part of the train of cord unrolling itself from the 
 smaller end, while the other part is coiled up towards 
 the larger end. 
 
 III. THE INCLINED PLANE is accounted the third 
 Mechanical Power. It has been already shown that 
 an equilibrium would be produced, if the force ex- 
 erted were to the weight to be raised, as the height 
 is to the length of the plane. But the same con- 
 clusion is derived, from the consideration of Virtual 
 Velocities. For, suppose a weight were drawn up 
 the plane AC (fig. 33.) by a cord passing over a 
 pulley at C. While this weight, in moving from A 
 to C, acquires a vertical elevation BC, the power ex- 
 erted will descend in the opposite direction through 
 a space equal to AC. The power and weight must 
 hence be inversely, as AC to BC. 
 
 The Inclined Plane is often combined with a cir- 
 cular motion. Suppose ABE (fig. 970 to De a 
 
 VOL. I. N 
 
194 ELEMENTS OF 
 
 spiral fixed to a perpendicular axis at P. As it 
 turns round, it will push in the direction PB ; and 
 the proximate radius PD being drawn, and the small 
 arc BC described, the thrust exerted at D must evi- 
 dently be to the rotatory power applied at B, as CB 
 to CD. If the curve be the Equable Spiral, it will 
 maintain an uniform pressure, and the mechanical 
 advantage will then be as the circumference of de- 
 scription to the quantity of protrusion PC, during a 
 single revolution. But a power may be thus evol- 
 ved, with an intensity varying as circumstances shall 
 require. If a circle, for instance, be made to turn 
 about an eccentric point P (fig. 98.), it will produce 
 an effort that increases continually, from the inter- 
 mediate position C, till it becomes infinite at the re- 
 mote extremity B of the diameter. The heart-shape 
 (fig. 99) * s employed in the composition of many of 
 the most useful machines. A similar arrangement 
 will conveniently change a revolving into an inces- 
 sant reciprocating motion. 
 
 The Inclined Plane is employed chiefly in facilita- 
 ting excavations, and raising the materials for the con- 
 struction of edifices. On the same principle, the 
 draught on roads over a hilly country is diminished, 
 by conducting them with a circuitous but gentle as- 
 cent. 
 
 IV. The SCREW is a most efficient Mechanical 
 Power. It consists of a ridge or groove winding 
 about a cylinder, and cutting at the same angle every 
 
NATURAL PHILOSOPHY. 
 
 line on the surface drawn parallel to the axis. The 
 Screw is called exterior or interior, according as it is 
 formed on the outside or inside of the cylinder. The 
 acting cylinder has always a handle annexed to it, 
 and the screw is therefore really a compound of the 
 Lever and Inclined Plane. As a machine, it is 
 commonly employed for producing compression, but 
 it may be likewise used in lifting heavy weights. The 
 power applied to the screw is to the pressure exerted 
 or the load sustained, as the interval between the 
 adjacent threads, to the circumference described by 
 the point of impulsion. 
 
 The action of the screw may be rendered far more 
 intense, by applying to it the principle of the Double 
 Capstan. Suppose a hollow screw adapted within 
 a firm nut, should work upon an exterior screw with 
 a finer thread, the remote end of this would evident- 
 ly be pushed forward at each revolution, and by 
 the difference only between the intervals of the 
 threads. Such an instrument was proposed by the 
 late Mr Hunter to be applied as a Jack, for the 
 moderate elevation of great weights, and also to serve 
 as a delicate micrometer. 
 
 An endless screw, working in the teeth of a 
 wheel, has its power multiplied by their number. 
 But it may act at once on two rachet wheels divided 
 almost alike, the difference in breadth between the 
 parallel teeth being the space of exertion. By such 
 a contrivance, which also reckons the revolutions, 
 
196 ELEMENTS OF 
 
 the mechanical purchase might be enlarged to any 
 extent. 
 
 The coining engine consists of a screw carrying 
 ponderous arms. The impulsion accumulated by 
 the swing, produces a stroke similar to the Concen- 
 trator of Force ; but the violence of the blow is soft- 
 ened, and the shock partly consumed, by the pro- 
 longed friction of the slanting grooves of the screw, 
 by which the stamper advances to the die. 
 
 V. The WEDGE, is sometimes employed in raising 
 bodies, but more commonly in dividing and cleaving 
 them. As an elevator, it resembles exactly the in- 
 clined plane, for the action is obviously the very 
 same, whether the wedge be pushed under the load, 
 or the load be drawn over the wedge. But when 
 the wedge is driven forward, the percussive tremor 
 excited in the block destroys for an instant the ad- 
 hesion or friction at its sides, and augments prodi- 
 giously the penetrating effect. From this principle 
 chiefly is derived the power of the wedge in rending 
 wood and other substances. It then acts besides as 
 a lever, insinuating itself into the cleft as fast as the 
 parts are opened by the vibrating concussion. To 
 bring the action of the wedge, therefore, under a 
 strict calculation, would be extremely difficult, if 
 not impossible. Its peculiar operations must be dis- 
 covered by experience. All the various kinds of 
 cutting tools, such as axes, adzes, knives, chisels, 
 
NATURAL PHILOSOPHY. 197 
 
 saws, planes, and files, are only different modifica- 
 tions of the wedge. 
 
 VI. The PULLEY is a very useful auxiliary in 
 the composition of the generality of machines. In 
 its simplest form it has already been considered un- 
 der the head of Statics. But whatever addition is 
 made to the power of equilibrium must generate 
 motion. 
 
 The Purchase, however, procured by the various 
 combination of pulleys, may be computed from other 
 principles. We may either estimate the subdivision 
 of primary strain, or compare the celerity of the 
 weight to that of the power. Pulleys occur some- 
 times singly, but oftener conjoined in blocks, and 
 then they turn on the same or different centres, and 
 have their diameters either equal or unequal. 
 
 The usual combination is represented in fig. 100. 
 Suppose n to be the number of pulleys in each block, 
 the weight will be supported by %n cords, each bear- 
 ing an equal share. The last moveable cord must 
 therefore sustain'the 2/i th part of the weight. Such 
 is the power required to maintain the equilibrium, 
 and consequently the purchase obtained by the 
 blocks is 2 n, or the number of extended folds of 
 cord. 
 
 The principle of Virtual Velocities gives the same 
 result. For, suppose the weight to rise one inch j 
 each of the cords would slacken an inch j and the 
 
198 ELEMENTS OF 
 
 last one to which the power is applied, would be 
 lengthened 2 n inches. The power is consequently 
 to the weight, as 1 to 2 n. 
 
 If a weight be attached to a pulley which is sup- 
 ported by a cord having one end fastened to a fixed 
 obstacle, the loose end will evidently sustain only 
 half of the weight, or, for every inch which the 
 weight rises, the power which draws it upwards will 
 move over two. But instead of this power, we may 
 apply another pulley mounted in the same way. 
 (See fig. 101.) The sustaining power will now be 
 only a quarter of the weight. The similar application 
 of a third pulley, would reduce the load borne up, to 
 one-eighth part. The action is thus doubled, by each 
 additional pulley. The purchase finally obtained in 
 this system of pulleys is therefore denoted by 2 n . 
 The line of traction is here directed upwards ; but 
 it may be reversed and adapted to perpendicular pres- 
 sure, by passing the loose cord over a fixed pulley. 
 
 Another combination of pullies may be noticed, 
 which, though less convenient, has still greater effi- 
 cacy. In this system, each cord is fastened to the 
 weight, and passing over a pulley, has its end tied 
 to another pulley. (See fig. 102.) The first pulley, 
 fixed to a firm obstacle, merely changes the direc- 
 tion of the force ; but the other pullies in succes- 
 sion greatly augment its action. Thus, while the 
 weight rises one inch, the moveable pulley will sink 
 an inch, and each of the folds of the cord bent over 
 
NATURAL PHILOSOPHY. 
 
 it, will shorten as much, permitting, therefore, the 
 second moveable pulley to sink three inches. This 
 pulley, now, by its descent, lets the folded cord 
 on each to slacken six inches, which, joined to the 
 inch which the weight mounts, gives 7 inches for 
 the shifting of the third moveable pulley. This, 
 again, giving 14 inches of extension to its adapted 
 cord, and the inch derived from the ascent of the 
 weight, leaves the next pulley to drop 15 inches. 
 Hence, one moveable pulley will, in this way, enable 
 one pound to support 3 pounds ; two moveable pullies 
 will enable it to support 7 pounds ; three such pul- 
 lies will augment its action to 15 pounds ; and a 
 fourth pulley would make it uphold 31 pounds. In 
 general, if n be the number of moveable pullies com- 
 bined in this manner, the purchase obtained by the 
 
 system will be denoted by 2* 1 1. 
 
 In the composition of machines, it is often requi- 
 red to augment or diminish, in a given ratio, the 
 celerity first impressed. This object is effected by 
 means of a train of wheels, put in motion by the ap- 
 plication of a strap, or by the action of engrained 
 teeth. It is essential, however, that the force should 
 be exerted perpendicular to the impelling surfaces. 
 Let A (fig. 103.) be the centre of the pinion or 
 smaller wheel, which drives another wheel about the 
 centre B. Suppose, first, that this pinion acts upon 
 
200 ELEMENTS OF 
 
 the teeth of the larger wheel, merely by implanted 
 pins or cylinders of insensible diameters, either per- 
 pendicular to its plane or radiating from its centre. 
 The point E, which represents one of those pins, 
 must therefore meet the curved surface of the tooth 
 DE at right angles, and consequently press against 
 it in the direction of the normal CE. But the right 
 angle CEA is contained in a semicircle, and hence 
 the curve DE is only a portion of the exterior epi- 
 cycloid, described by the revolution of the circle AC, 
 which has half the diameter of the pinion, about the 
 circumference of the primary circle. Produce EC 
 to meet BF drawn parallel to AE. The force with 
 which the pinion presses against the tooth at E has 
 the same effect, in turning the wheel about the centre 
 B, as if it acted upon the point F ; and, being pro- 
 portional to the radius AC, it urges the circulation 
 of that wheel, by an energy proportional to the dis- 
 tance BF. But AE is to BF, as the radius AC to 
 the radius BC ; and consequently the equable pres- 
 sure exerted by the pinions will maintain the uniform 
 circumvolution of the wheel. 
 
 The pinion will communicate its entire impulsion 
 to the wheel, evidently, whether it acts on the con- 
 cave or the convex surface of each tooth. In the 
 former case, it must turn the wheel to the left ; but 
 in the latter, to the right. Its action is in either 
 way exerted uniformly. 
 
NATURAL PHILOSOPHY. 201 
 
 Suppose now the pinion, instead of carrying lineal- 
 pins, should be armed with thick cylinders or spindles, 
 a form of construction which gives it the name of 
 LantJwm, Trundle, or Wallower. Divide the dis- 
 tance AB (fig. 104.) between the centres, into seg- 
 ments proportional to the number of the spindles 
 and of the teeth, and AC and BC will be the pri- 
 mitive radii or pitch-lines, whose circles mark by 
 their contact the path of action. Since the spindle 
 must press perpendicularly against the surface of the 
 tooth, it is evident that the line OC drawn through 
 the centre O, and the point E of impulsion, must be 
 a normal to the curve. But the centre O of the 
 spindle describes the arc GO of an exterior epicy- 
 cloid, and consequently the tooth is only a parallel 
 curve, shifted backwards by an interval equal to the 
 radius OE or GD. 
 
 Let the pinion be furnished with teeth which act 
 upon those of the wheel. The line of pressure EC 
 (fig. 105.) must be necessarily perpendicular to both 
 curves at their point of contact E. Wherefore, 
 while the tooth DE of the wheel is traced by the 
 rolling of a circle on the outside of the circumference 
 of the wheel, the tooth GE of the pinion is described 
 by the revolution of the same circle within the cir- 
 cumference of the pinion. The corresponding teeth 
 are thus only portions of an exterior and an interior 
 epicycloid generated by the same primitives. The 
 diameter, however, of the revolving circle is left in- 
 determinate. If it be taken equal to half the pinion, 
 
ELEMENTS OF 
 
 it will, instead of a curve, describe internally the 
 mere diameter. Consequently, on this assumption, 
 the teeth of the pinion will have straight edges, and 
 appear only as projecting portions of the radii. 
 
 In this investigation, we have supposed each tooth 
 of the pinion to come first in contact with the cor- 
 responding tooth of the wheel at the point C in the 
 line of the common centres. But if they locked be- 
 fore, and only parted from each other at this point, 
 the formation of their curvature would be reversed. 
 The teeth of the wheel would be portions of the in- 
 terior epicycloid, marked by the rolling of a circle 
 within its circumference, while the teeth of the pi- 
 nion would be portions of an exterior epicycloid de- 
 scribed by the revolution of the same circle on the 
 outside of the circumference of the pinion. Conse- 
 quently, if this tracing circle had just half the di- 
 mensions of the wheel, the teeth of the wheel would 
 become straight edged, being merely projecting radii 
 impelled by the exterior epicycloidal teeth of the 
 pinion. 
 
 But, by enlarging the surface of contact, the 
 teeth of the pinion and of the wheel may at once 
 have a compound form, the lower part being straight 
 and the upper part curved. In this arrangement, 
 the curved end of a tooth of the pinion seizes the 
 straight root of a tooth of the wheel, and pushes 
 this forwards, till in its turn, it brings its straight 
 edge to press against the rounded termination of the 
 tooth. The curves are, in both cases, exterior epicy- 
 
NATURAL PHILOSOPHY. 
 
 cloids, but differently described. A circle of half 
 the diameter of the pinion is made to revolve about 
 the wheel ; and a circle of half the diameter of the 
 wheel is constrained to revolve about the circumfer- 
 ence of the pinion. 
 
 Conceive the pinion to be a circle of indefinite 
 extent, bearing straight teeth. It will then merge 
 into a rack with perpendicular studs. The corre- 
 sponding teeth of the wheel will be portions of epicy- 
 cloids described about its circumference, by a circle 
 of half the diameter of the pinion, and therefore still 
 of unlimited magnitude. The curves which it de- 
 scribes are hence the evolutes of the wheel, and might 
 be traced by the unfolding of a thread or a flexible 
 spring. Such is the form of the cams or lifting 
 cogs of forges, or those of the stampers for bruising 
 stones and beating flax or hemp. 
 
 Suppose both the opposite indentations to be cur- 
 ved, but let the wheel, by its unbounded extent, be- 
 come a toothed rack. In this case, the generating 
 circle will trace interior epicycloids, for the teeth of 
 the pinion ; but rolling along the straight line which 
 now represents the wheel, it will describe mere cy- 
 cloids, for the form of the parallel teeth. 
 
 By changing the size and the application of the 
 generating circle, a considerable variety may thus be 
 produced in the mutual configuration of the teeth 
 of pinions, wheels, or racks. Nor is the rolling of 
 a circle an essential requisite, since any other curve, 
 
204 ELEMENTS OF 
 
 treated in the same way, will describe the forms 
 which are capable of accurate contact, and fitted 
 therefore to communicate the full energy of impul- 
 sion. 
 
 To form a Templet or Pattern Tooth for wheels, 
 the easiest mode of proceeding is by means of a me- 
 chanical construction. Let a small segment FCG 
 (fig. 106.) of a circle denoted by the pitch lines, 
 including the space of at least two teeth, be descri- 
 bed on a smooth board, which is then accurately 
 rounded ; and to this apply a corresponding segment 
 DCE of the generating circle, bearing a pencil or 
 tracer at D, both circumferences being rubbed with 
 rosin or chalk to prevent sliding. The point D 
 being made to apply at F, let the other points in 
 DCE be successively brought in contact with FCG ; 
 the tracer, as it recedes from D, will mark out 
 the required exterior epicycloid FD. The interior 
 epicycloid of the pinion is described in a similar 
 manner, only the generating circle must evidently 
 be less than the circle which contains it. For the 
 sake of accuracy, the circular segments may consist 
 of steel, and have the surfaces divided by fine flu- 
 tings. 
 
 If the rolling arc DCE belong to a large circle, 
 the portion FD of its epicycloid will not differ sen- 
 sibly from the evolute of the circle of which FCG is 
 a segment. This discrepancy is less perceptible 
 when the teeth are small. In ordinary cases, there- 
 
NATURAL PHILOSOPHY. 
 
 fore, it will be sufficiently accurate, to assume the 
 initial part of ane volute of the primary circle FCG 
 for the form of the teeth. The curve is traced by 
 the end of a thread, as it unfolds from the circum- 
 ference between F and G, or more correctly by a 
 fine straight saw, applied successively along the arc 
 FCG. 
 
 But a more accurate mode of construction may 
 be derived from the property, that an epicycloid 
 is the evolute of a similar curve. (See Geometry 
 of Curve Lines, p. 372.) For let R and r be the 
 radii of the fundamental and generating circles ; 
 
 and it follows, from Prop. V, that ^ ~JL J^.Ar 
 
 (li-r^/V 
 
 will denote the radius of curvature at the beginning 
 of the epicycloid. Instead of R, and r, the number 
 of teeth of the wheel and half of those of the pinion 
 may be adopted, as having the same ratio. Find, 
 by calculation, therefore, CE ffig. 10?.) the value 
 of this expression, and from E describe an arc CF, 
 which may be viewed as coincident with a portion of 
 the involute of the exterior epicycloid. Unfold a 
 thread or fine steel spring from CF, and its extremi- 
 ty will describe the tooth CG. 
 
 When an internal epicycloid is wanted, the ra- 
 dius of curvature of its opening involute will be- 
 come /P ITy 4r. The point E will hence lie 
 without the fundamental circle, and the evolution of 
 
206 ELEMENTS OF 
 
 the arc CF (fig. 108.) will now produce an internal 
 tooth CG. 
 
 If the generating, had half the diameter of the 
 fundamental, circle, the radius of the involute for 
 
 3 
 the exterior epicycloid would be -r, but infinite 
 
 for the interior. In the latter case, the first part 
 of the involute becomes a straight line, the evo- 
 lution of which traces a perpendicular, the teeth 
 here being only the extremities of the several 
 diameters. Let R iz 3 r, and the radii of the in- 
 ner and outer involutes will be r and 24 r ; and 
 
 ?o 
 
 20 
 
 they will become ^- r and 1 2 r, when Rzz4r. 
 y 
 
 It would be easy to give an approximative geome- 
 trical construction. Bisect CB (fig. 109.) in H, and 
 make AH : CB : : AB : CI, and CB : AC : : Cl 
 : CE, from E describe the small arc CF, and in it 
 assume any points K, L, &c. either equidistant or 
 brought closer together as they approach to F. From 
 these points, describe the successive arcs CM, MN, 
 NO, &c., which will nearly coalesce into the form of 
 a tooth, and, if necessary, their junctions may be 
 somewhat rounded. 
 
 If the fundamental circle be indefinitely large, its 
 circumference will become a straight line, and the 
 epicycloid will pass into the common cycloid. Hence 
 the mode of tracing the initial portion of this curve, 
 
NATURAL PHILOSOPHY. 207 
 
 which suits the form of cams. Let CP (fig. 110.) 
 be the path of rotation, and A the centre of the ge- 
 nerating circle ; make CE equal to twice its diame- 
 ter, from E describe an arc CF, in which assume any 
 points, and from these describe the minute arcs CL, 
 LM, MN, &c. terminated by the successive tangents 
 IL, KM, FN, &c. The compound arc CP will, for 
 a considerable space, scarcely differ from a cycloid. 
 
 In crown wheels and bevelled geer, the faces of 
 the teeth must be all tapered, as if they proceeded 
 from a common centre. The pinion may be consi- 
 dered as a portion of a cone, which, rolling along 
 the horizontal rim of the wheel, traces the contour 
 of the teeth, by describing a modified epicycloid. 
 The same general principles will indicate all the va- 
 riations. 
 
 It is of practical consequence that the teeth of 
 wheels should be all equally worn. In their mutual 
 congress, therefore, they ought to produce a series 
 of the most diversified contacts. To effect this ob- 
 ject, the numbers of teeth in the pinion and the 
 wheel must be as discordant as possible. If the one 
 were any aliquot part of the other, the same inces- 
 sant coincidence would soon recur, occasioning a par- 
 tial and disproportionate attrition. Prime numbers 
 should be preferred ; and the larger they are, the 
 smoother will the wheels work. The odd tooth in 
 
208 ELEMENTS OF 
 
 the pinion is hence by our mill-wrights very appo- 
 sitely called the hunting cog. 
 
 The effects produced by machines are extremely 
 diversified. To extend action to a distance, or to 
 divert its direction ; to reduce or multiply the celerity 
 impressed ; to modify an uniform progression into an 
 accelerating or retarding one ; to maintain a paral- 
 lel motion ; to change a rectilineal into a circular mo- 
 tion, and the reverse ; to convert a reciprocating play 
 into a constant circulation or equable rectilineal flow ; 
 these are a few of the objects generally aimed at. 
 Pressure is conveyed to any distance, by a system of 
 parallel levers, supported at certain intervals, the ex- 
 tremities of their arms being connected by a train of 
 beams. The direction of any power is easily chan- 
 ged, by means of a crank or bent lever. The same 
 contrivance, aided by the action of a fly, converts a 
 reciprocating into a circular motion. This effect is 
 likewise produced by a rack, working alternately on 
 the opposite sides of a toothed wheel. The celerity 
 is modified at pleasure, by affixing to the axle solid 
 blocks, sometimes called heart wheels, (see fig. 99.) 
 and fashioned like spiral or eccentric curves. These 
 lines may be traced with such accuracy as to evolve 
 the precise succession of impulse required. But all 
 the transitions should, if possible, be gradual, any 
 sudden change occasioning a concussion which wastes 
 the force, and tends to disjoint and shatter the parts 
 of the machine. 
 
NATURAL PHILOSOPHY. 
 
 f 
 
 The most elegant mode of transmitting force in any 
 direction, is by a contrivance termed the Universal 
 Joint. (See fig. 111.) Instead of the cross, a ring 
 or a ball is often preferred. But the angular motion 
 communicated in this way is not quite equable, and 
 becomes even irregular* in the case of great obliquity. 
 By doubling, however, the combination of axes, as 
 in fig. 112, the impression may be turned aside al- 
 most into an opposite direction, and may be render- 
 ed uniform, by a compensation of irregularities. 
 
 An alternate motion ivS beautifully converted into 
 a revolving one, by a vertical rod which carries affix- 
 ed to it a wheel indented to another wheel of the 
 same size. The central wheel, describing at each 
 reciprocation a double circumference, must turn twice 
 round. This mechanism is, from a vague analogy, 
 called the Sun and Planet Wheel. 
 
 A reciprocating beam can be made to raise and 
 depress a rod very nearly in a vertical line, by means 
 of a regulated parallelogram. Thus, C (fig. 113.) 
 being the centre of the beam, let an arm BE turn 
 about the firm joint B, and guide the end E of the 
 parallelogram ADEF ; if CD be taken a mean pro- 
 portional to AD and BE, the other end F, carrying 
 the rod of a piston, will travel in a path which is 
 very nearly rectilineal and perpendicular. These 
 two fine contrivances were happily combined in 
 Watt's Steam Engine. 
 
 VOL. i. o 
 
 . 
 
ELEMENTS OF 
 
 The efficacy of any complex machine is generally 
 computed from the statical relation of the power to 
 weight to be raised. This, however, is merely the 
 proportion which would maintain a state of quies- 
 cence ; and to produce actual motion, it requires the 
 application of more force. Nor does the velocity 
 thus created correspond to the simple ratio of the ad- 
 ditional power, but follows a modified and much slower 
 law of increase. The performance relatively becomes 
 greatest, when the force exerted has attained a cer- 
 tain limit of intensity. It is of the utmost conse- 
 quence, therefore, in the economy of machines, to 
 approximate at least to this measure of advantageous 
 exertion. But the problem involves so many and 
 such intricate considerations, that theory can seldom 
 furnish a direct solution, and requires all the aid of 
 experience. I shall select only two instances illus- 
 trative of the general principle ; the first relating to 
 the Wheel and Axle, and the next to the Inclined 
 Plane. 
 
 I. Let it be required (fig. 114.) to find the 
 radius OB of a circle, from whose circumference 
 the descent of an appended weight shall raise, 
 with the greatest possible celerity, another equal 
 weight attached to a given circle AC fixed on the 
 same axis. These weights being each of them 
 regarded as unit, the force at B which would 
 
 oc 
 
 maintain the equilibrium, is denoted by 7, and 
 
NATURAL PHILOSOPHY. 
 
 consequently the inciting force by ^75. But, both 
 
 circles being supposed devoid of weight and ha- 
 ving the same angular motion, this motive power 
 will be shared as the velocity of OC* to OB*, and 
 therefore the acceleration of the weight from B is 
 CB OB* >, CB.OB 
 OB 
 
 velocity generated at that point, and consequently 
 
 CB.OC _. 
 the velocity at A must be QT* i r)p * * nis ex- 
 
 pression is hence a maximum, or its reciprocal 
 
 OB*4-OC* 
 
 - ,^nr\^ a minimum. But for OB*, substitut- 
 
 CB 2OC 
 
 ing (CB + OC)*, it becomes + 2 + --. Omit- 
 
 ting, therefore, the constant number 2, and mul- 
 tiplying the rest by OC, which is likewise a con- 
 
 2OC* 
 
 stant quantity, and the result CB+ T must be 
 
 a minimum. Let OC (fig. 115.) and 2OC to- 
 gether, form the diameter of a circle, and draw 
 the chord BCF ; it is evident that the rectangles 
 
 BC.CF = OC.2OC = 20C*, and CF = 
 
 Wherefore, BCF is the shortest chord passing 
 through C ; it consequently approaches nearest to 
 the centre G, and must be perpendicular to GC 
 
ELEMENTS OF 
 
 or to the diameter OE. Hence JBC = CF, and 
 BC> = 2OC>, or BC = O(V % 
 
 In approximative numbers, therefore, OC be- 
 ing reckoned 5, CB will be 7, and OB 12. The 
 
 r ,. T> 7 !44 84 
 
 accelerating force exerted at 13 is . -77^- = -77-, or 
 
 very nearly half the power of gravity ; but this ac- 
 tion would cause the weight from A to mount only 
 
 .8 feet, or 40 inches in a second. 
 
 2. Suppose a weight, acting over a pulley at A 
 (fig. 1 16.), to be just sufficient to support a load on 
 the inclined plane AC ; it is required to substitute 
 another more sloping plane AD, along which this 
 load would be drawn or made to rise from the level 
 DB to the elevation A, by the same power and in 
 the shortest time possible. The urging weight 
 being reckoned unit, the load which it sustains 
 
 AC 
 
 on the plane AC will be denoted by -^-^; but 
 
 this load, transferred to the plane AD, would be 
 held at rest by a weight or vertical force expressed 
 i AJt> AC/ AO rp,, r . , 
 
 y "AT) " "A"R' Or "AD* excess of weight en- 
 
 gaged in producing the compound motion, is there- 
 
 fore AP ~ AC , or , AE being cut off equal to 
 AJD 
 
 AC. This force, communicating equal velocities to 
 the load and the primary weight, must be shared be- 
 
NATURAL PHILOSOPHY. 
 
 tween them according to their respective masses. 
 But these remaining constant, the accelerating power 
 
 DE 
 
 in both is proportional to -^-^. Now, since the 
 
 space described is as this force multiplied into the 
 square of the time, the load will be carried from D 
 to A in the shortest time, when AD divided by 
 
 DE AD* . 
 
 or --, is 3, minimum. Suppose Ar were 
 
 made equal to EA, and DG found a third propor- 
 tional to DE and DA ; while EA is given, the point 
 D, and hence G must in its extension be assigned, 
 so that DG shall have the least possible extent. By 
 division, DE : DA : : EA : AG, and DE : EA : : 
 EA : FG ; consequently, the rectangle DE, FG, be- 
 ing equivalent to the square of EA, is constant. But 
 EF, or the double of E A, is likewise constant ; and 
 therefore, the sum of those extreme segments DE 
 and FG must be the least possible. As they con- 
 tain a given space within the shortest perimeter, they 
 must consequently form a square, and thus DE, 
 EA, AF, and FG are all equal. Wherefore AD, 
 the length of the plane required, is double of AE 
 or AC. 
 
 Suppose BA were 3 feet and AC 5 feet, one 
 pound suspended vertically would support 1 pounds 
 leaning upon AC ; but this load, when transferred 
 to AD, which is 10 feet, would be upheld by half a 
 pound, and, consequently, the remaining half pound 
 must supply the accelerating force. The energy ex- 
 
ELEMENTS OF 
 
 13 3 
 erted is therefore -. - = -^, and the load would be 
 
 drawn from D to A, or lifted vertically in / -5 
 
 seconds, or 1,825". The direct action of gravity 
 would have produced the same effect in 
 
 But such investigations are unfortunately render- 
 
 ed in a great measure superfluous, by those unavoid- 
 
 able imperfections which are incident to all machines. 
 
 A very large portion of the force applied is in most 
 
 cases consumed, by the various obstructions which it 
 
 has to encounter during the course of its transmission. 
 
 The resistance of the air causes a certain retardation ; 
 
 but the chief impediment arises from the rubbing or 
 
 attrition of the surfaces which come incessantly into 
 
 contact. Most solid bodies, when brought close to- 
 
 gether, are disposed to cohere mutually, and with 
 
 various degrees of tenacity. This peculiar force, be- 
 
 ing exerted perpendicular to the surface of contact, 
 
 can evidently have no influence whatever in impe- 
 
 ding a lateral traction. But all substances appear to 
 
 possess likewise a certain adhesive property, which 
 
 opposes any change of mutual contact, and retards 
 
 even the horizontal passage of one plane along an- 
 
 other. This latent obstructing power constitutes 
 
 Friction, which has such extensive influence in di- 
 
 minishing the performance of all machinery. The 
 
 effect, however, is much diversified, according to cir- 
 
NATURAL PHILOSOPHY. 
 
 cumstauces, and can bo discovered in any particular 
 case, only by experiments conducted in a scale of 
 sufficient magnitude. 
 
 There are two general modes of exploring the na- 
 ture and operation of Friction. The first ascertains 
 the weight required to draw a body under the pres- 
 sure of a given load along the horizontal surface of 
 another. The second method is still simpler, and 
 consists merely in raising the end of the upper plane 
 till it acquires the declination at which the succum- 
 bent load begins to slide. This extreme declination is 
 hence called the Angle of Equilibrium, Quiescence, 
 or Repose. Let CA (fig. 33.) represent the position 
 of the plane at the moment the load slips from its 
 place. If the vertical FD represent the weight, FE 
 will express the perpendicular pressure sustained by 
 the plane, and OED must denote the corresponding 
 friction, which is just balanced here by the tenden- 
 cy to descend. Wherefore, the Pressure is to the 
 Friction, as FE to ED or as AB to BC, that is, as 
 radius to the tangent of the angle of repose. 
 
 These two modes of experimenting, however, will 
 seldom give precisely the same results. Most bodies 
 require a greater force to pull them from their con- 
 tact, than what is afterwards sufficient to maintain 
 their adhesive progression. But the obliquity of the 
 plane of descent evidently marks only the initial ob- 
 struction, and not the subsequent unfeebled, yet un- 
 ceasing, action of Friction. 
 
 
C 21() ELEMENTS OF 
 
 Thus, if the plane CA were a pine board, upon 
 which is laid a block of oak with a smooth surface, 
 it may be elevated to an acclivity of 30 degrees be- 
 fore it disengages its load. But should the angle 
 of declination exceed only 10 degrees, on striking 
 the side of the plane with a smart blow 7 , the oaken 
 block will start from its seat, and then glide down- 
 wards. While the plane retains this lower inclina- 
 tion, the load will not rest, on being again replaced, 
 unless it be a few seconds held in contact. 
 
 It was inferred from the earlier observations, that 
 Friction is a constant retarding force, proportioned 
 nearly to the pressure, but commonly varying from 
 the third to the sixth part of this quantity. In po- 
 lished freestone, it exceeds the half of the pressure, 
 and the rough surface of bricks augments it to three- 
 fourths. This great obstruction contributes most 
 essentially to the stability of arches and vaults. 
 
 A body subjected to no obstruction will descend, 
 it was demonstrated, in the same time, through 
 any chord to the lowest point of a vertical circle. 
 But even when retarded by the influence of friction, 
 it will yet slide down equally through the chords of 
 a certain determinate arc. For the whole weight be- 
 ing represented by CF (fig. 9<5.), the pressure on 
 the oblique plane AF is denoted by AC, and con- 
 sequently the measure of the friction by AM, a 
 definite portion AC. The remaining force of ac- 
 celeration is hence expressed by MF, which will be 
 
NATURAL PHILOSOPHY. 217 
 
 described under the attrition, in the same time that 
 AF is traced by a free descent. But the sides AC 
 and AM of the right angled triangle CAM having a 
 given ratio, that triangle is given in species ; conse- 
 quently the angle AMC is given, and likewise the 
 exterior angle CMF, which must therefore be con- 
 tained in a given segment of a circle. The body 
 would hence slide down the planes MF or PF, in the 
 time that it would have fallen directly from C to F. 
 It likewise follows that the tangent NF to the arc 
 AMF will mark the position of the plane of repose. 
 
 The angle of repose often determines the con- 
 tour of natural objects. Thus, tine sand slides more 
 easily than ordinary mould, and hence sand-hills 
 have generally a softer ascent than the grassy flanks 
 of mountains. The latter, without being broken 
 into precipices, may rise at an angle of 40 degrees ; 
 but the former will seldom support an acclivity above 
 25 degrees. Again, the angle of repose of iron press- 
 ing upon iron being 16 degrees, if the threads or spi- 
 rals of a vice wind closer than this inclination, the 
 screw must hold at any place to which it is carried. 
 
 By removing the visible asperities from the sur- 
 faces of bodies, their mutual attrition is diminished. 
 But any higher polish than what merely prevents 
 the grinding and abrasion of the protuberant particles, 
 has no material effect in reducing the measure of 
 Friction. Other peculiar circumstances appear to 
 have more extensive influence. 
 
218 ELEiMENTS OF 
 
 The experiments on Friction which Coulomb 
 made between the years 1779 and 1781, being de- 
 vised with ingenuity, performed with great care, and 
 executed on a large scale, are the most original, and 
 by far the most important that have been made. 
 His general method was to draw a sort of sledge, 
 mounted on different sliders and variously loaded, 
 along a large horizontal bench, fitted likewise with cor- 
 responding slips of wood or metal. This sledge was ei- 
 ther allowed to rest a certain time, or it was instantly 
 put into motion. Some idea of the diversified results 
 may be conceived, by referring them to a common stan- 
 dard of comparison. Assuming the pressure as equal 
 to one hundred parts ; the friction of oak against fir 
 was 66 in the direction of the fibres, but amounted 
 only to 16 when moved with the velocity of a foot 
 each second ; the friction of oak against oak, in the 
 direction of the fibres, was 45, and across them on- 
 ly 27 > the effect being still reduced by motion to 10 ; 
 the friction of fir against fir, in the direction of the 
 fibres, was 56, which sunk to 17 during motion ; 
 the friction of elm against elm, in the direction of 
 the fibres, was 46, which motion reduced to 10 ; the 
 friction of copper upon oak was lengthwise 18, and 
 only 8 when kept in motion ; the friction of iron 
 upon oak was 28, and diminished during motion to 
 11. But the mutual friction of metals appeared in 
 general to be less affected by motion. Thus, iron 
 against iron had its friction, in passing from rest to 
 
NATURAL PHILOSOPHY. 219 
 
 motion, diminished only from 8 to 25 ; again, the 
 friction of brass upon iron, which reached 25, was 
 contracted by motion to 17 ' 
 
 The application of grease, soap, or other soft lini- 
 ments, to the rubbing surfaces, generally diminishes 
 the friction, though in very different degrees. Oak, 
 fresh greased, has its friction upon oak, only depress- 
 ed from 4-5 to 43 ; but this was reduced to 26, af- 
 ter the tallow had been thoroughly imbibed into the 
 wood and left a glossy smoothness. The greasing 
 of the surfaces of copper and iron reduces their fric- 
 tion to 10. 
 
 When a coat of tallow is applied to metals, the 
 friction attains almost instantaneously its limit. But 
 the different sorts of wood, treated in the same way, 
 suffer a slow change in the condition of their sur- 
 faces, which continues for a considerable time to aug- 
 ment the intensity of friction. A piece of greased 
 oak, the moment it is laid upon another, is drawn, 
 with a force of 4, but would have this friction aug- 
 mented in three seconds to 10, in one minute to 16, 
 in two hours to 28, and in the space of five or six 
 days to 44. The friction of wood against metal in- 
 creases likewise during a sensible time. A closer 
 approximation of surfaces must evidently be produced 
 S by some gradual process, which is enfeebled by the 
 softness of the substance impressed. Hence the fric- 
 tion of wood is greatly diminished by rapid motion, 
 while that of metals continues with very little alter- 
 
ELEMENTS OF 
 
 ation, whether they be drawn along slowly or ex- 
 tremely fast. 
 
 This general view of the phenomena of Friction 
 affords a glimpse of its origin. Matter, however 
 passive to external impressions, is not strictly inert. 
 Its attractive and repulsive energies, modified only 
 by the mutual distance of the particles, are inces- 
 santly in action, and produce varied effects and in 
 different times. Nor are such operations confined 
 to the substances commonly termed chemical ; they 
 occur to a certain extent, though more concealed 
 from observation, in other bodies which manifest no 
 peculiar signs of activity. " Nature speedily extin- 
 guishes every motion upon earth, and seems to diffuse 
 a principle of silence and repose ; which made the 
 ancients ascribe to matter a sluggish inactivity, or 
 rather an innate reluctance and inaptitude to any 
 change of place. We shall perhaps find, that this 
 prejudice, like many others, has some semblance of 
 truth ; and that even dead or inorganic substances 
 must, in their recondite arrangements, exert such 
 varying energies, so like sensation itself, as could 
 not fail, if completely unveiled in our sight, to strike 
 us with wonder and surprise. 
 
 " When one solid is drawn along another, if the 
 opposing surfaces be rough and uneven, there is 
 a necessary waste of force, occasioned by the grind- 
 ing and abrasion of their prominences. But 
 Friction subsists after the contiguous surfaces are 
 
NATURAL PHILOSOPHY. 
 
 worked down as regular and smooth as possible, 
 In fact, the most elaborate polish can operate no 
 other change than to diminish the size of the na- 
 tural asperities. The surface of a body, being 
 moulded by its internal structure, must evidently be 
 furrowed, or toothed, or serrated ; and such is the 
 information derived from close inspection with the 
 Microscope. Friction is, therefore, commonly ex- 
 plained on the principle of the inclined plane, from 
 the effort required to make the incumbent weight 
 mount over a succession of eminences. But this ex- 
 plication, however currently repeated, is quite in- 
 sufficient. The mass which is drawn along is not 
 continually ascending ; it must alternately rise and 
 fall, for each superficial prominence has a corre- 
 sponding cavity ; and since the horizontal boundary 
 of contact is supposed to be horizontal, the total ele- 
 vation will be equal to their collateral depressions. 
 Consequently, though the actuating force might suf- 
 fer a perpetual diminution in lifting up the weight, 
 it would, the next moment, receive an equal increase 
 by letting it down again ; and those opposite effects, 
 destroying each other, would have no influence what- 
 ever on the general motion. 
 
 " Adhesion appears still less capable directly of 
 explaining the source of Friction. A perpendicu- 
 lar force acting on a solid, can evidently have no 
 effect to impede its advance ; and though this late- 
 ral force, owing to the unavoidable inequalities of 
 
222 ELEMENTS OF 
 
 contact, must be subject to a certain irregular obli* 
 quity, the balance of chances must on the whole have 
 the same tendency to accelerate as to retard the mo- 
 tion. If the conterminous surfaces were hence to 
 remain absolutely passive, no Friction could ever 
 arise. Its existence betrays an unceasing mutual 
 change of figure, the opposite planes, during the 
 passage, continually seeking to accommodate them- 
 selves to all the minute and accidental varieties of 
 contact. The one surface, being pressed against the 
 other, becomes, as it were, compactly indented, by 
 protruding some points and retracting others. This 
 adaptation is not accomplished instantaneously, but 
 requires very different periods to attain its maximum, 
 according to the nature and relation of the substan- 
 ces concerned. In some cases, a few seconds are 
 sufficient ; in others, the full effect is not produced 
 till after the lapse of several days. While the in- 
 cumbent mass is drawn along, at every stage of its 
 progress, it changes its external configuration, and 
 approaches more or less to a strict contiguity with 
 the under surface. Hence the effort required to 
 put it first in motion, and hence too the decreased 
 measure of Friction, which, if not deranged by ad- 
 ventitious causes, attends generally an augmented 
 rapidity. 
 
 " Friction, then, consists in the force expended 
 in raising continually the surface of pressure by an 
 oblique action. The upper surface travels over a 
 
NATURAL PHILOSOPHY. 
 
 perpetual system of inclined planes ; but the system 
 is ever changing, with alternate inversion. In this 
 act, the incumbent weight makes incessant, yet un- 
 availing efforts to ascend ; for the moment it has 
 gained the summits of the superficial prominences, 
 these sink down beneath it, and the adjoining cavities 
 start up into elevations presenting a new series of ob- 
 stacles which are again to be surmounted ; and thus 
 the labours of Sisyphus are realized in the phenome- 
 na of Friction. 
 
 " The measure of friction must depend merely 
 upon the angles of the natural protuberance, which 
 are determined by the elementary structure or the 
 mutual relation of the two approximate substances. 
 The effect of polishing is only to diminish those as- 
 perities and augment their number, without altering 
 in any respect their curvature or inflexions. The 
 constant or successive acclivity produced by the ever- 
 varying adaptation of the contiguous surfaces, re- 
 mains therefore the same, and consequently the ex- 
 pense of force will still amount to the same share of 
 the pressure. The intervention of a coat of oil, 
 soap, or tallow, by readily accommodating itself to 
 the variations of contact, must tend to equalize it, 
 and therefore must lessen the angles, or soften the 
 contour of the successively emerging prominences, 
 and thus diminish likewise the friction which hence 
 results *." 
 
 * Experimental Inquiry into the Nature and Propagation of 
 Heat, pp. 298-303. 
 
ELEMENTS OF 
 
 The Theory now quoted is the result of strict in- 
 duction, and seems quite accordant with all the ob- 
 servations hitherto made on the subject of Friction. 
 This obstructing power must, in the attrition of 
 the same surfaces, be proportional to the pressure, 
 since a certain share of the incumbent weight is re- 
 quired to surmount the prominences or natural ac- 
 clivities which rise in perpetual succession along the 
 range of contact. But the friction becomes sensibly 
 diminished, if the surface in action be reduced to the 
 smallest dimensions. Thus, while the friction of a 
 ruler of brass against a similar one of iron, is ex- 
 pressed by %6 ; it was found to be only 17, after the 
 sledge had been mounted upon four round-headed 
 brass nails. The reason of this diminution appears 
 to be, that the contact is not so close, when the nails 
 are drawn along the iron surface, the great convex- 
 ity towards the edges being less affected by the pro- 
 jecting eminences. The same consideration will ex- 
 plain the diminished friction of an axle against its 
 box. Thus, an iron-axle turning in a brass socket 
 has its friction reduced from 26 to 16, or lessened 
 by more than one-third part. The curvature of the 
 axle produces nearly the same effect as the softening 
 of the natural acclivities of its box. Hence this pe- 
 culiarity of loose contact is lost, if a coat of tallow be 
 interposed. When fresh greased, the friction be- 
 tween the axle and socket is 9, the same as in the 
 case of plane surfaces. So likewise axles of lignum 
 
NATURAL PHILOSOPHY. 
 
 vitae running in boxes of oak or elm, have only a 
 friction from 3 to 4, but increased to 6 after tbe 
 unguent is rubbed off. 
 
 In the mutual contact of metals, the friction 
 attains, almost instantaneously, its maximum. But 
 when metal rubs against wood, or one piece of wood 
 against another, the friction always augments by 
 resting, and reaches not its full measure till after the 
 lapse of some portion of time. The duration of this 
 progressive increase appears to depend on certain pe- 
 culiar circumstances, but chiefly on the pliancy of the 
 conterminous surfaces. Two pieces of wood acquire 
 the utmost friction in the lapse of an hour or two ; 
 while iron laid upon oak will have its friction still 
 augmenting for the space of five or six days. The 
 application of a coat of tallow seems to protract the 
 limit of friction. This limit is attained by the grea- 
 sed surfaces of iron and copper in four minutes ; 
 while pieces of wood, treated in the same way, will 
 have their friction gradually augmented during nine 
 or ten days. 
 
 If two planed boards of oak be coated with tallow, 
 and rubbed against each other till they become smooth 
 and shining, the friction will at first increase most 
 rapidly, and still continue progressive for a very long 
 time. This friction, which at the instant of contact, 
 is only the twelfth part of the pressure, becomes 
 one-half more in the space of a single minute, is 
 doubled in eleven minutes, and tripled in twenty-four 
 
 VOL. i. p 
 
ELEMENTS OF 
 
 hours. Amounting now to the fourth part of the in- 
 cumbent weight, it acquires not sensibly any farther 
 augmentation. If t represent in minutes the whole 
 time elapsed, the friction of such finely greased oak 
 
 will be expressed nearly by 77,+^ log. (1 + 10 ). 
 
 A & &\J 
 
 This gradual augmentation is still farther pro- 
 longed by applying a soft layer of tallow. If this 
 were spread to the thickness of the twentieth part of 
 an inch over polished boards of oak, it would pene- 
 trate into the wood about the eighth part of an inch, 
 and the adaptation of the impregnated surfaces would 
 be effected with extreme slowness. The friction would 
 be doubled in very little more than a second, tripled 
 in 6^-", quadrupled in 36-", quintupled in 3'.9"> and 
 would continue its progress nearly nine days, beyond 
 which no farther increase would be perceived, the fric- 
 tion being then augmented tenfold, and equal to two- 
 fifths of the whole pressure. If t denote the time in 
 minutes, the friction will be expressed with tolerable 
 
 accuracy by this formula : ^+^Q % (1+240 0- 
 
 Atd 1O 
 
 Where friction is augmented by the duration 
 of contact, it must evidently be greater in slow 
 than in rapid motions. Hence, it is not affected at 
 all by the celerity of the attrition of metals. In all 
 other cases, however, the influence of friction de- 
 creases in some proportion to the quickness of trac- 
 tion. This diminished obstruction, resulting from 
 
NATURAL PHILOSOPHY. 
 
 a swift congress, is visible in the mutual attrition of 
 pieces of wood : it is still more prominent in the he- 
 terogeneous contact of metal with wood, but is most 
 remarkable when a coat of tallow has been interposed. 
 A ship is launched along sliders, which commonly 
 slope only from 4 to 5^-. The lowest friction is here 
 exerted, all previous adhesion being destroyed by 
 blows of the mallet, and shocks given in the act 
 of withdrawing the wedges. The momentary fric- 
 tion being 4, leaves an accelerating force of 3, that 
 hurries the vessel forwards, notwithstanding its im- 
 mense pressure of perhaps thirty-five tons on every 
 square foot of the slide. If any imperfection in the 
 track should arrest the progress of the ship, it will 
 soon gain such adhesive power as to render its re- 
 moval extremely arduous. A tremulous agitation is 
 the only expedient to urge forward the ponderous 
 mass. Hence the reason of the sudden falling down 
 
 o 
 
 of weak or decayed structures. They are upheld 
 long beyond the term of equilibrium, by the rooted 
 adhesion of their parts ; but any ^accidental shock 
 dissolves this union, and the whole pile is precipita- 
 ted to the ground. 
 
 The adhesion arising from prolonged contact, ex- 
 plains likewise some striking facts in the practice of 
 the mechanical arts. Hence an axe, struck into a 
 heavy block of wood, will, notwithstanding its wedge 
 form, take such a firm hold as to lift it up. For 
 
228 ELEMENTS OF 
 
 the same reason, a round iron bolt, with a slight 
 degree of taper, being driven into a hole bored in a 
 stone, will safely raise the ponderous mass, and it 
 is only disengaged by the smart blow of a hammer. 
 The binding action of nails depends on the same 
 principle. Their adhesion to wood is not exactly 
 as the quantity of surface compressed, but seems to 
 be nearly proportional to the square root of the cube 
 of the depth of insertion. Thus, to extract an 
 iron nail from a fir plank, requires a force denoted 
 by QQO^d, in pounds averdupois ; d marking in 
 inches the depth of penetration. In dry elm, the 
 tenacity of cohesion is about a fourth part greater. 
 
 The friction of pivots is peculiar. Being so near 
 the axis, it is never affected in any degree by the 
 swiftness of rotation. This friction is not exactly 
 proportional to the pressure, but nearly a mean be- 
 tween the pressure and its square root. Thus, if 
 one pivot weighs 10 and another 810 grains, the fric- 
 tion of the latter will be 28, and not 81 times great- 
 er. The probable reason of this discrepancy is, that 
 the loaded pivot occasions a depression at the spot 
 on which it turns, which incurvation of the plate di- 
 minishes the friction. The tapering of the pivot 
 lessens this still more. It should not exceed an 
 angle from 30 to 45 degrees, and the cone may be re- 
 duced even to 10 or 12 degrees, if the weight should 
 not exceed an hundred grains. The retardation 
 
NATURAL PHILOSOPHY. 229 
 
 which a steel pivot suffers, varies according to the 
 nature of the substance in which it turns. On gar- 
 net it loses only 10 parts of its motion, on agate 12, 
 on rock-crystal 13, on glass 18, and on steel 23. 
 
 If a hard body rub against a very soft substance, 
 the friction will no longer continue uniform, but in- 
 crease remarkably with the celerity. Thus, when 
 the sliders of the bench are covered with lists of 
 cloth, the loaded sledge being mounted as before 
 upon rules of wood or metal, the motion incited by 
 a predominant weight, so far from accelerating, is 
 quickly retarded, and would soon become entirely 
 extinguished. But the impediment now created 
 partakes more of the nature of the resistance of fluids ; 
 it consists in the consumption of force occasioned by 
 the continual depressing of the spongy and elastic 
 substance of the cloth with the celerity of the pas- 
 sage of the sledge. The effect of such obstruction, 
 being compounded of the quantity of matter dis- 
 placed and of the velocity of its removal, must there- 
 fore be proportional to the square of that velocity. 
 If a denote the incipient friction, and v the velocity, 
 the corresponding friction will be expressed by a + mv* y 
 where m is a coefficient to be determined by experi- 
 ment. 
 
 When a cylinder is made to roll upon a plane 
 surface, it encounters a new sort of obstruction, 
 
230 ELEMENTS OF 
 
 quite distinct in its character, and generally much 
 inferior to that of Friction. These retarding forces 
 are strikingly contrasted in the rolling and sliding 
 of different cylinders of wood down inclined planes 
 The cylinders will not begin to slide lengthwise, 
 though disengaged by percussion, unless the slope of 
 the plane exceeds 10 degrees ; but they will roll 
 easily at much smaller angles. The inclination of 
 4 degrees will be sufficient to enable a cylinder of 
 elm, one inch in diameter, to roll down an oaken 
 board, and an angle of 3 degrees will decide the roll- 
 ing of a cylinder of lignum vitae of the same dimen- 
 sions. But a single degree of slope will make an 
 elm cylinder of four inches diameter begin to roll, 
 and three-fourths of that angle will occasion the roll- 
 ing of a like cylinder of lignum vita?. The loss of 
 force in the act of rolling is hence inversely propor- 
 tional to the diameter of the cylinder. 
 
 If adhesion were confined to the mere line of con- 
 tact, it could have no effect whatever in hindering, 
 on a horizontal plane, the revolving motion of a cy- 
 linder. But this power, subsisting an instant after 
 contact has taken place, may be conceived as con- 
 stantly drawing it down at a certain distance behind. 
 Thus, if A (fig. 1 1 70 be the contact of the cylinder 
 and B the point where the adhesion is mainly exert- 
 ed, its efficacy to restrain the rolling of the cylinder 
 will evidently be as AB to AC. On the supposi- 
 tion that AB is constant, this retarding force is in- 
 
NATURAL 'PHILOSOPHY. 
 
 versely as the radius AC. In the case of elm, the 
 distance AB of vigorous cohesion, must be the 28th 
 part of an inch ; but in that of lignum vitas it is on- 
 ly the 40th part. 
 
 Rollers, by removing friction, are extremely use- 
 ful in practice of Mechanics. The only obstacle is 
 the necessity of frequently replacing them under the 
 load ; but this inconvenience may, in a great mea- 
 sure, be obviated by substituting cylindrical axles 
 fixed to large wheels. 
 
 Balls appear to roll still more easily than cylinders 
 of the same diameter, though they have never been 
 subjected to any nice experiments. On a large scale, 
 however, they are often employed most advantage- 
 ously in aiding the removal of enormous masses. 
 
 If the diameter of a cylinder in inches be de- 
 noted by d, that portion of the incumbent weight 
 which is required to maintain the rolling of a cylin- 
 der of elm upon a plane of oak, will be expressed by 
 
 3, but the retardation of a cylinder of lignum 
 vita3 is only 3. Balls of the same dimensions 
 
 and materials would, in rolling upon such a plane, 
 probably consume only two-thirds of those measures 
 of force. 
 
 From some trials made by Tredgold with small 
 models, it would follow, that the obstruction which 
 cast-iron wheels encounter in rolling along an iron 
 
232 ELEMENTS OF 
 
 railway, employing the same notation, amounts only 
 to the g- part of the load. By the application of 
 
 conditional equations, we are enabled to infer, from 
 three measures incidentally furnished by Wood, of 
 the retardation along a cast-iron railway, encoun- 
 tered by an ordinary waggon carriage, with pon- 
 derous wheels of 35 inches diameter, that the ob- 
 struction from the rolling alone amounts to the 
 235th part of the whole incumbent pressure, which 
 is more than three times greater than what might 
 be expected from the action of small models. But 
 when the rims of the wheels are case-hardened, 
 this obstruction is reduced to about the 650th part 
 of the load, and thus very nearly corresponds with 
 the experiments in miniature. The difference of ce- 
 lerity probably modifies the effect of rolling. 
 
 In the composition of machines, we should avoid 
 attrition as much as possible, and prefer rolling 
 movements, wherever circumstances will admit. But 
 friction itself may be diminished indefinitely, by 
 bringing its action nearer to the centre of revolution, 
 or by transferring its influence from the circumfe- 
 rence, to the axle, of a wheel. Thus, the axle of 
 the wheel AB (fig. 118.), sustaining a pressure, 
 being made to roll on the summit of the circumfe- 
 rence of CB, its friction against the gudgeon at A 
 
NATURAL PHILOSOPHY. 233 
 
 will be transferred to the axle C. Supposing the 
 axles to have the same diameters, the influence of 
 this friction in retarding the machinery must, from 
 the principle of the lever, be now diminished in the 
 ratio of CB to CD. For the sake of convenience 
 the axle of the incumbent wheel A is generally 
 planted at the intersection of two equal wheels C 
 and E. The weight, considered as pressing verti- 
 cally, becomes hence shared between them, and the 
 effect of friction is reduced, in the ratio of their 
 diameter, to that of their axles. These axles may 
 yet be made to roll each of them on the circumfe- 
 rence of an equal pair of wheels, as in fig. 119, and 
 the friction will be again diminished in the same ratio. 
 The wheels thus introduced, for the purpose of ap- 
 proximating the operation of friction to the centre 
 of motion, are called Friction- Wheels. If their 
 diameters, or rather the vertical chords, were ten 
 times greater than the diameters of their axles, two 
 wheels would reduce the friction to one-tenth, four 
 additional wheels to one-hundredth, and eight wheels 
 more would contract the friction to the thousandth 
 part. Scarcely any obstruction would then be left 
 but that of rolling, which is comparatively inconside- 
 rable. 
 
 Since the primary axle does not rest precisely on 
 the summit of the friction-wheel, but leans against a 
 point beyond this, and therefore divides its oblique 
 pressure between two equal wheels, the friction is 
 
ELEMENTS OF 
 
 really diminished in a ratio somewhat less than that 
 of the diameters of the wheel and axle. Let A and 
 B be the centres of those wheels, and C the centre 
 of the incumbent axle : join ADC, and let fall the 
 perpendicular DE upon AB. If AD express half 
 of the load borne by the axle, DE will denote the 
 pressure exerted at D, or by the secondary axle at 
 F. Wherefore the friction will be diminished in 
 the ratio of the perpendicular DE to CD. This 
 perpendicular, in most cases, however, differs little 
 from the radius AD itself. 
 
 The theory of friction wheels explains likewise the 
 advantages derived from the construction of wheeled 
 carriages. In these machines, the draught is facili- 
 tated by three distinct circumstances : 1. The ex- 
 cessive obstruction, which the rim of the wheel 
 would encounter if dragged along the road, is chan- 
 ged to the very inferior friction of the axle against 
 the bush of the nave. 2. This reduced friction has 
 its influence in impeding the progress of the carriage, 
 still farther diminished, in the ratio of the diameter of 
 the wheel to that of the axle. And, 3. The dimen- 
 sions of the wheel enable it to surmount easily any 
 obstacle which may occur ; the effect being the same 
 as if it were drawn over an inclined plane, from its 
 point of contact to the top of the prominence. The 
 friction of the rim of a locked wheel, even on the 
 smoothest road, might perhaps exceed the half of 
 the whole load j but the friction of the axle in its 
 
NATURAL PHILOSOPHY. 235 
 
 box, which is substituted for it, would amount only 
 to the eighth part when iron is inserted in a copper 
 bush, and scarcely the seventh part when oak turns 
 in lignum vitse, the grease in both cases being worn 
 smooth. The power of this friction within the nave, 
 in retarding the motion of the carriage, must be di- 
 rectly as the diameter of the axle, and inversely as 
 the height of the wheel. A large wheel and a small 
 axle are hence the most advantageous. For this 
 reason, an iron-axle, though it has twice as much 
 friction as an oaken one of the same dimensions, may 
 be preferred for its smallness. In carriages rightly 
 fitted and carefully greased, the whole friction sel- 
 dom exceeds the thirtieth, but need not amount to 
 the hundredth part of the load. 
 
 To assign the force expended in overcoming an 
 obstacle, let the wheel (fig. 120.) touch A the hori- 
 zontal line of traction ; if it meet a protuberance BD, 
 it must be lifted over this with the progressive motion 
 AB : the draught is therefore to the load, as AB to 
 BD, or, from the property of the circle, as BE or 
 AF to AB. But AF* : AB* : : AF : BD, and 
 consequently AF is to AB in the subduplicate ratio 
 of AF, the diameter of the wheel, to BD the height 
 of the obstacle. Large wheels are hence best adapt- 
 ed, not only for diminishing the effect of friction, 
 but for surmounting the inequalities of the road. 
 
 Cylindrical wheels will answer best on level roads ; 
 for their breadth, though it has no effect on the 
 
236 ELEMENTS OF 
 
 quantity of friction, may yet lessen their sinking in- 
 to the ground. In hilly and uneven roads, a slight 
 declination of the spokes, called dishing, will, like 
 an arched vault, give strength to the wheel, and 
 prevent its accidental twisting. 
 
 The obstruction which a loaded carriage has to 
 overcome, when drawn along a smooth level road, is 
 always composed of two very distinct portions ; first, 
 the attrition of the axle against the box of the nave, 
 and, secondly, the adhesion of the rim of the wheel 
 as it rolls over the yielding surface of the ground. 
 These elements of retardation, though quite different 
 in their nature, have been often confounded under 
 the general term friction. But it would evidently be 
 rash to infer the properties of adhesion from experi- 
 ments made on ordinary friction ; nor is it ascertain- 
 ed whether this adhesion be constant or variable, and 
 whether, by great celerity, it lessens, like attrition, 
 or augments, as in the resistance of fluids. 
 
 The attrition of the nave is proportional to the 
 weight of the carriage and its load, but the ad- 
 hesion to the ground is proportional to the whole ag- 
 gregate mass, including the wheels and axle. Both 
 of these sources of retardation are diminished with 
 the height of the wheel, but the attrition is like- 
 wise diminished by the small ness of the axle. In 
 ordinary cases, we may reckon the effect of adhesion 
 at least five times greater than that of attrition. 
 Thus, a horse travelling at the rate of three miles 
 
NATURAL PHILOSOPHY. 237 
 
 an hour, exerts a pull of 81 Ibs. in drawing a load- 
 ed cart : of this force only 12 Ibs. are spent on the 
 attrition of the axle, but the remaining 69 Ibs. are 
 consumed on the adhesion of the rim of the wheel 
 to the level road. Supposing the absolute friction 
 of the nave to be one-eighteenth of the incumbent 
 weight, and the diameter of the axle the fifteenth part 
 of that of the wheel ; then 18 X 15 X 12 = 3240 Ibs. 
 or 28| cwts, leaving about 25 cwts. for the load. The 
 attrition being thus equal to the 270th part of the 
 pressure, the adhesion along the road may be taken 
 at the 50th part of the entire charge ; which gives 
 50 X 69 = 3450 Ibs., being an excess of 210 Ibs. 
 for the additional weight of the wheels and axle. 
 
 It is matter of just regret, that the relation be- 
 tween the influence of attrition and of adhesion, in 
 retarding the motion of wheel-carriages, has never 
 been ascertained with any degree of precision ; yet 
 nothing could be more easily determined than the 
 comparative expenditure of power, occasioned by the 
 operation of those distinct causes : Let the retarda- 
 tion of a loaded carriage, with ordinary wheels, be 
 found ; and next that of a light carriage, mounted 
 on very heavy wheels, perhaps of solid cast iron. 
 This obstruction might be examined on different 
 roads and railways, the rims of the wheels being shod 
 with a thin ring of wood, or of iron, cast or hammer- 
 ed, or case-hardened. Since the attrition is pro- 
 portional only to the weight of the carriage, and of 
 
ELEMENTS OF 
 
 its load, while the adhesion is produced by the whole 
 collective pressure of carriage and wheels, the several 
 results would afford data for distinguishing the 
 blended shares. These experiments, performed on 
 a scale of sufficient magnitude, would throw a clear 
 and steady light on the principles of the formation of 
 roads, and the construction of carriages ; and might 
 therefore be deemed of national importance. Simi- 
 lar inquiries ought certainly to have been instituted 
 by the proprietors of railways, before they ventured, 
 with such limited and imperfect information, to em- 
 bark in their expensive schemes. 
 
 Though ordinary friction is generally diminished 
 by rapid motion, the attrition of the nave cannot be 
 sensibly affected by the swiftness of rotation, since 
 the rubbing surface being so near the centre, its ce- 
 lerity is proportionally reduced. It is different al- 
 together from the adhesion of the rim of the wheel 
 in rolling along the road, which bears an affinity to 
 the resistance of soft and yielding substances, and 
 therefore increases probably after some ratio of the 
 velocity. 
 
 In the drawing of carriages, it is of the utmost 
 consequence to blunt or avoid the shocks occasioned 
 by the inequalities of the road. Such concussions 
 waste the power of draught, in proportion to the 
 celerity of their deranging influence. The applica- 
 tion of springs to a carriage, by restraining and en- 
 feebling those irregular movements, must therefore 
 
NATURAL PHILOSOPHY. 239 
 
 be an unprofitable expenditure of power. The ad- 
 vantages thus procured are greater in rapid travelling. 
 It has been found that a carriage mounted on springs 
 may be drawn along a rough road at the rate of two 
 miles an hour, with three-fourths of the force neces- 
 sary to effect it without springs. At the rate of 
 three miles and a half an hour, the corresponding 
 force is only two-thirds ; but, at the velocity of five 
 miles, it becomes reduced to one half. 
 
 But the motion of machines, in which pulleys are 
 concerned, suffers another obstruction, arising from 
 the stiffness of the ropes, or the force consumed in 
 continually bending them into a new direction. 
 This impediment, in the case of a single strand, 
 must depend on the rate of inflexion, and will there- 
 fore follow the inverse proportion of the diameter of 
 the pulley or revolving cylinder. If those strands 
 were laid parallel, their combined stiffness would be 
 as their number, or as the square of the diameter of 
 the rope which they form. In practice, however, 
 this obstructing force appears adapted to a slower 
 ratio. A certain portion of the stiffness of a rope 
 is owing to its peculiar tenacity, but the greatest 
 part of it proceeds from the tension occasioned by 
 the appended weight. But this is liable to much 
 diversity, which can be ascertained only by experi- 
 ment. 
 
 Two methods have been adopted for exploring 
 
#40 ELEMENTS OF 
 
 the stiffness of ropes. The first was proposed by 
 Amontons, which consists in a strong beam, having 
 two pulleys fixed to it, and over them a long rope is 
 passed ; and both ends being lapped about a cylin- 
 der, are then fastened to a frame loaded with heavy 
 weights. See fig. 121. A slender silken line is 
 next wound about the cylinder, having an appended 
 scale to hold small weights. These being added, 
 till the cylinder begins to descend, will evidently 
 measure half the stiffness of the rope, since they are 
 exerted in turning it not about the centre, and at 
 the distance of the radius, but at the whole length of 
 the diameter. 
 
 This mode of experimenting was pursued on a 
 large scale by Coulomb, and the results were con- 
 firmed by a simpler method of investigation which 
 he afterwards employed. The cylinder was now 
 placed upon two parallel and horizontal edges* 
 having the rope coiled about it, and equally loaded 
 at both ends ; a weight was now added on the one 
 side, just sufficient to put the cylinder in motion ; 
 and the small obstruction in rolling obtained by a 
 previous experiment, being deducted from this, 
 leaves an accurate measure of the stiffness of the 
 rope. See fig. 122. An approximation, reduced to 
 the English standard, may easily be deduced from 
 the different facts ascertained by Coulomb. Let 
 D express in inches the diameter of the pulley, d 
 that of the rope, and P in pounds averdupois the 
 
 
NATURAL PHILOSOPHY. 
 
 5 ( 1404-3? 
 weight applied to it \ then will a 7 ^ To75 
 
 denote very nearly the stiffness of a new hempen 
 rope. In the case of old ropes, the fraction J is 
 more correctly the index of d. Small cords have 
 their stiffness diminished by wetting, but large cords 
 are thus rendered somewhat stiffen The applica- 
 tion of tar adds about one-sixth part more to the 
 stiffness of ropes. 
 
 But we sometimes need to create friction, as well 
 as to lessen or extinguish it. All machinery is gra- 
 dually set in motion, and cannot be stopped suddenly 
 without incurring imminent danger. As the cele- 
 rity was progressively acquired, so to insure safety it 
 must be slowly retarded. Friction is the simplest 
 and most effectual means of arresting all motion. 
 For this purpose, its action is augmented, by trans- 
 ferring it in complex machines from the centre to 
 the circumference of revolution. But simpler modes 
 of exerting the retarding force of friction are fre- 
 quently adopted in practice. The attrition of a rope 
 against a round post is the common way of stemming 
 the motion of a ship. 
 
 This peculiar species of friction deserves a dis- 
 tinct investigation. Let a flexible cord ABC (fig. 
 1 23.), supporting a weight P, and drawn by a force 
 Q, wind about the circumference of a fixed pulley or 
 cylinder, whose centre is O. This cord must press 
 against the cylinder at any point, in proportion to 
 
ELEMENTS OF 
 
 its degree of inflexion. Conceive the arc ABC to 
 be divided into elementary portions, and the tension 
 of the cord in the direction of a tangent at B will be 
 to the pressure of the element bB, as the radius OB to 
 B. If 1 : m denote the ratio of the pressure of 5B to 
 
 . . m.bB . 
 its friction, -o~ W1 ^ express the proportional 
 
 increase of tension from b to B. Suppose the 
 successive tensions to be represented by P, P', P", 
 
 i T r> ., j , , v 
 &c. thenF P = P. -- and P" F = 
 
 &c. Wherefore, the hyperbolic logarithm of 
 
 Q ?w. AC , A ^^ v T 
 
 p -7y^ = w . angle AOL. Hence, assuming 
 
 equal angles about the centre O, the corresponding 
 tensions will form a geometrical progression. Thus, 
 assuming the arc AC to be a quadrant, if a weight 
 P of one pound balanced a traction of two pounds 
 at C, it would uphold four pounds at the end of the 
 semicircumference, eight pounds at three quarters 
 of a circuit, and sixteen pounds at a complete cir- 
 cumvolution ; but the same power would, at the end 
 of two, three, and four turns, &c. sustain no less than 
 lb., 4096 lb., 65,536 Ib. The progression thus 
 
 augments with extreme rapidity j and after a few 
 turns of the cord, a very small weight will be suf- 
 ficient to support the most enormous load. This 
 efficacy is nowise modified by the size of the cylin- 
 der, about which the line is coiled, but depends en- 
 
 
 
NATURAL PHILOSOPHY. 
 
 tirely on the number of its circumvolutions. Hence 
 the firmness procured by the lapping of cordage ; 
 and hence, likewise, the principle of various fire- 
 escapes, by which the celerity of descent from a great 
 height is retarded, and the shock against the ground 
 rendered soft and easy. 
 
 In all mechanical structures, each member should 
 not only be able to resist the strain under which it is 
 constantly subjected, but should also be capable of 
 withstanding those occasional shocks to which it is 
 ever exposed. The stability of the fabric depends 
 no doubt mainly on the strength of its several parts ; 
 but the same degree of strength will exert very dif- 
 ferent forces, according to its direction, and the elas- 
 ticity that accompanies it. In combining the mate- 
 rials, therefore, the power brought into action should 
 always be proportioned to the respective strain. 
 The skilful disposition and arrangement of the com- 
 ponent members, hence contribute often more essen- 
 tially than their size or absolute strength to the se- 
 curity and duration of any structure. 
 
 In estimating the strength of materials, they are 
 generally considered as having a prismatic or colum- 
 nar form. But they may likewise be exposed eithei 
 to a longitudinal or a transverse strain. Their 
 strength is therefore exerted in four different ways : 
 1. In sustaining a longitudinal tension ; 2. In with- 
 standing a longitudinal compression ; 3. In resisting 
 
ELEMENTS OF 
 
 a transverse pressure ; and, 4. In opposing the act 
 of twisting or wrenching. 
 
 1. Longitudinal Tension. The tension which 
 a stone pillar, a bar of metal, a beam of wood, or 
 even an hempen rope, can bear when pulled length- 
 wise, must evidently depend on the cohesion of any 
 cross section. As the material stretches out, the 
 longitudinal attraction of the particles becomes aug- 
 mented. This increase, at first, is proportional to 
 the dilatation, but it afterwards advances very slow- 
 ly, and a small additional strain is then sufficient to 
 produce that limit of extension which occasions total 
 fracture or disruption of the column. Its length 
 will nowise affect the utmost strain which it can bear, 
 this being determined merely by the smallest cross 
 section where the dislocation of the particles will 
 take place. 
 
 Let ZA (fig. 124.) be a prism stretched in the 
 direction of its length. The particles of the sec- 
 tion JB will be pulled from those of the section C, 
 till an, attraction be generated equal to the whole 
 tension. But the particles of the section C must 
 settle in equilibrium, and are consequently drawn 
 back by an equal force. In this acquired position, 
 therefore, they must attract the particles of the sec- 
 tion D with the same force. The original tension 
 is thus transferred successively to the extremity Z, 
 where it is finally exerted, the effects of its action be- 
 ing neutralized in all the intermediate sections. 
 
NATURAL PHILOSOPHY. 245 
 
 The cohesive power thus evolved, is hence the 
 accumulative attraction of all the particles in any 
 section. The corresponding longitudinal distension 
 is at first proportional to it, but afterwards increases 
 in a more rapid progression. Thus, a bar of soft 
 iron will stretch uniformly, by continuing to append 
 to it equal weights, till it be loaded with half as much 
 as it can bear ; beyond that limit, however, its exten- 
 sion will become doubled by each addition of the 
 eighth part of the disruptive force. Supposing the 
 bar to be an inch square, and 1000 inches in length, 
 36,000 Ib. averdupois would draw it out 1 inch \ but 
 45,000 Ib. will stretch it 2 inches, 54,000 Ib. 4 in- 
 ches, 63,000 lb. 8 inches, and 72,000 Ib. 1 6 inches, 
 when it would finally break, the extension being now 
 increased eight times beyond its ordinary rate. 
 
 Let AB (fig. 125.) be a prism or bar of any ma- 
 terial, and suppose its prolongation BC to express 
 the whole longitudinal force exerted, in causing the 
 small extension Aa. While the length of this bar 
 BC continues the same, it is evident that aA must 
 be proportional to B, the distraining weight BC. 
 Make, therefore, a A : BC : : AB : CD ; or, alter- 
 nately, Aa : AB : : BC : CD, and CD must be 
 constant. Since BC now bears the same relation to 
 CD as aA to AB, any portion of CD will, by its 
 weight, produce a corresponding distension of AB. 
 Thus, a column of the thousandth part of the length 
 of CD would extend AB one thousand part, and 
 
ELEMENTS OF 
 
 the same weight acting by compression, would oc- 
 casion an equal contraction. The column CD thus 
 found, is called the Modulus of Elasticity ; it de- 
 pends entirely on the nature of the cohesive substance, 
 and may be determined by a single experiment. 
 
 The Modulus of Elasticity, though of great im- 
 portance, has been ascertained but in a few instances. 
 That of white marble is 2.150,000 feet, or equal to 
 the weight of 2,520,000 pounds averdupois on the 
 square inch ; while that of Portland stone is only 
 1,570,000 feet, corresponding, on the square inch, 
 to the weight of 1,530,000 Ib. 
 
 White marble and Portland stone are found to 
 exert, on every square inch of section, a cohesive 
 power of 1811 Ib. and 857 Ib. ; wherefore, suspend- 
 ed columns of these stones, with the altitudes of 
 1542 and 945 feet, or only the 1394th and 1789th 
 parts of their respective measure of elasticity, would 
 be torn asunder by their own weight. 
 
 Of the principal kinds of timber employed in 
 building and carpentry, the annexed table will ex- 
 hibit the respective Modulus of Elasticity, and the 
 portion of it which limits their extreme longitudinal 
 cohesion. 
 
 Teak, 
 
 6,040,000 feet. 
 
 _ 168th. 
 _ 144th. 
 _ 108th. 
 ... 107th, 
 
 Oak, 
 
 4,150,000 feet 
 
 Sycamore, 
 
 3,860,000 feet _ 
 
 Beech 
 
 4,180.000 feet. 
 
NATURAL PHILOSOPHY. 
 
 A.sh, srrwm , 
 
 ~ 4,617,000 feet. 
 
 _ 109th. 
 _ 146th, 
 ~~ 205th. 
 ~~ 146th. 
 - 121th. 
 
 Elm, 
 
 5,680,000 feet. 
 
 Memel Fir, Jj ^^ 
 
 ~ 8,292,000 feet. 
 
 Christiana Deal, 
 Larch , 
 
 8, 11 8,000 feet. _ 
 , 5.096.000 feet 
 
 A Tabular View may be likewise given of their 
 absolute cohesion, or the load which would rend a 
 prism of an inch square, and the altitude of the 
 prism which would be torn asunder by the action of 
 its own weight. 
 
 Teak, 
 
 _, 12,915 Ib. . 
 
 36,049 feet. 
 32,900 feet. 
 35,800 feet. 
 38,940 feet. 
 42,080 feet. 
 
 
 11,880 Ib. 
 
 Sycamore, 
 B*ech, ___^- 
 
 9,630 Ib. L 
 
 12,225 Ib. ., 
 
 Ash, _ J^_*~ 
 
 _ 14,130 Ib. , 
 
 Elm, M M^MMrfMMMta 
 
 ._ 9,720 Ih. i. 
 
 39,050 feet. 
 ~ 40,500 feet. 
 
 Memel Fir, 
 
 9,540 Ib. : 
 
 Christiana Deal, 12,346 Ib 55,500 feet. 
 
 Larch,^ 12,240 Ib 42,160 feet. 
 
 It is singular, that woods of such diversified 
 structure should yet differ so little on the whole in 
 the measures of their elasticity and cohesion. Spe- 
 cimens of the same sort will occur, which are some- 
 times as much varied as the several kinds them- 
 selves. 
 
 The modulus of the elasticity of hempen fibres 
 has not been determined, but may probably be rec- 
 
ELEMENTS OF 
 
 koned about ,5,000,000 feet. Their cohesion is, for 
 every square inch of transverse section, 6,400 Ibs. 
 The usual mode of estimating the strength of a cable 
 or rope of hemp, is to divide by five the square of its 
 number of inches in girth, the quotient expressing 
 in tons the utmost strain it could bear. But a sim- 
 pler computation is to double the square of the dia- 
 meter of the rope. This estimate, however, applies 
 only to new ropes formed of the best materials, not 
 much twisted, and having their strands laid even. 
 Yarns of 180 yards long are usually worked up into 
 a rope of only 120 yards, and lose one-fourth part 
 of its strength, the exterior fibres alone resisting the 
 strain. But the register cordage of the late Captain 
 Huddard exerts nearly the whole force of its strands, 
 since they suffer a contraction of only an eighth part 
 in the process of intertwisting. 
 
 The metals differ more widely from each other in 
 their elastic force and cohesive strength, than the 
 several species of wood or vegetable fibres. Thus, 
 the cohesion of fine steel is about 135,000 Ib. for the 
 square inch, while that of cast lead amounts only to 
 about the hundred and thirtieth part, or 1800 Ib. 
 
 According to the very accurate experiments of 
 Mr George Rennie in 1817, the cohesive power of 
 a rod an inch square of different metals, in pounds 
 averdupois, with the corresponding length in feet, 
 is as follows : 
 
NATURAL PHILOSOPHY. ' 24-9 
 
 Cast steel, 134,256 Ib 39,455 feet. 
 
 Swedish Malleable Iron, 72,064 Ib 19,740 feet. 
 
 English ditto 55,872 Ib 16,938 feet. 
 
 Cast Iron, 19,096 Ib 6,110 feet. 
 
 Cast Copper, 19,072 Ib 5,003 feet. 
 
 Yellow Brass, 17,958 Ib 5,180 feet. 
 
 Cast Tin, 4,736 Ib. ~ ' 1,496 feet. 
 
 Cast Lead, 1,824 Ib 348 feet. 
 
 It thus appears, that a vertical rod of lead 348 
 feet long, would be rent asunder by its own weight. 
 The best steel has nearly twice the strength of Eng- 
 lish soft iron, and this again is about three times 
 stronger than cast iron. Copper and brass have al- 
 most the same cohesion as cast iron. This tenacity 
 is sometimes considerably augmented by hammering 
 or wire-drawing, that of copper being thus nearly 
 doubled, and that of lead, according to Eytelwein, 
 more than quadrupled. The consolidation is pro- 
 duced chiefly at the surface, and hence a slight notch 
 with a file will materially weaken a hard metallic 
 rod. Hence the advantage of case-hardening. Eng- 
 lish malleable iron has 7,550,000 feet for its mo^ 
 dulus of elasticity, or the weight of 24,920,000 Ib. 
 on the square inch, while cast iron has 5,895,000 
 feet or 18,42 1 ,000 Ib. Of other metals, the modulus 
 of elasticity is probably smaller, but has not yet been 
 well ascertained. 
 
 2. Longitudinal Compression. The compres- 
 sion which any column suffers, is at first equal to the 
 
250 ELEMENTS OF 
 
 dilatation occasioned by an equal and opposite strain, 
 being in both cases proportional to the modulus of 
 elasticity. But while the incumbent weight is in- 
 creased, the power of resistance likewise augments, 
 as long as the column withstands inflexure. After 
 it begins to bend, a lateral disruption soon takes place. 
 A slender vertical prism is hence capable of supporting 
 less pressure than the tension which it can bear. 
 Thus, a cubic inch of English oak was crushed by 
 the load of only 3860 lb., but a bar, of an inch square 
 and 5 inches high, gave way under the weight of 
 2572 lb. It would evidently have been still feebler 
 if it had been longer. On the other hand, if the 
 breadth of a column be considerable in proportion to 
 its height, it will sustain a pressure greater than its 
 cohesive power. Thus, though the cohesion of a rod 
 of cast iron, of the quarter of an inch square, is only 
 300 lb., a cube of that dimension will require' 1440 
 lb. to crush it. 
 
 In general, while the resisting mass preserves its 
 erect form, the several sections are compressed and 
 extended by additional weight, and their repellent 
 particles are not only brought nearer, but multiplied. 
 This repulsion is likewise increased, by the lateral 
 action arising from the confined ring of detrusion. 
 The primary resistance becomes hence greatly aug- 
 mented, in the progress of loading the pillar. 
 
 The most precise experiments on this subject are 
 those of Mr Rennie. The weights required to crush 
 
NATURAL PHILOSOPHY. 
 
 cubes, having the dimension of the quarter of an 
 inch of certain metals, are as follow : 
 
 Iron cast vertically, 11,140 Ib. 
 
 Iron cast horizontally, 10,110 Ib. 
 
 Cast Copper, 7,318 Ib. 
 
 Cast Tin, 966 Ib. 
 
 Cast Lead, 483 Ib. 
 
 Cubes of an inch are crushed by the weights below : 
 
 Elm, 1,284 Ib. 
 
 White Deal, 1,928 Ib. 
 
 English Oak, 3,860 Ib. 
 
 Craigleith freestone, 8,688 Ib. 
 
 Cubes of an inch and a half, and consequently 
 presenting a section of 2^ times greater than the 
 former, might be expected to resist compression in 
 that ratio. They are crushed, however, with loads 
 considerably less. 
 
 Red brick, **. ~~~~~ **** ^ 
 
 1,817 Ib. 
 
 Yellow baked brick, -~~~*~L~*i*+ fss s 
 
 ~*~~ 3^54 Ib. 
 
 Fire brick, M, * J ^- J -^^^^^^^ 1V ^^ J - J - 
 
 3,864 Ib. 
 
 Craigleith stone, with the strata, 
 Ditto, across the strata, 
 White Statuary Marble, 
 
 15,560 Ib. 
 12,346 Ib. 
 _*_ 13,633 Ib. 
 
 \Vlii fr.e-veinfrl Italian TVTa.rblp, ^^^^^^ 
 
 21,783 Ib. 
 
 Pnrhfck T/imf,ston^, MMM4*MMM*MMMM 
 
 20,610 Ib. 
 
 Cornish Gran it* 1 , ~~~-~^~ ^ ***^**, 
 
 14,302 Ib. 
 
 Pp.tprhparl frrnriitp, . mj ., .... v 
 
 18,636 Ib. 
 
 Aberdean R]iift Oranifp, -^r^r,^^^ 
 
 24,536 Ib. 
 
ELEMENTS OF 
 
 These facts show the comparative firmness of dif- 
 ferent materials ; but it is to be regretted, that such 
 results are not of much practical use, since they are 
 confined to a very narrow scale, and applicable only 
 to cubical blocks. While the breadth remains the 
 same, the resistance appears to depend on some unr 
 ascertained ratio of the altitude of the column. Nay, 
 the absolute height itself has probably a material in- 
 fluence on the effect. Thus, from some experiments 
 made in France, it appears, that prisms of seasoned 
 oak, two inches square, and two, four or six feet 
 high, would be crushed by the vertical pressures of 
 17,500 lb., 10,500 lb., and 7,000 Ib. ; but, if four 
 inches square, and of the same altitudes, they would 
 give way under loads of 80,000 lb., 70,000 lb., 
 and 50,000 lb. In the first set of trials, the mean 
 cohesive power amounts to 130,000 lb., and in the 
 second to 520,000 lb. The vertical support is there-, 
 fore greatly inferior to these limits. When the 
 length of the pillar exceeds 36 times its breadth, the 
 resistance to longitudinal compression appears to be 
 diminished 18 times, Mr Rennie estimates, that 
 the granite which composes the great arch of the 
 new London Bridge, would be capable of supporting 
 four tons upon every square inch of its upper sur- 
 face, 
 
 TO ascertain the form of a vertical column, the 
 best fitted to support a load, is a problem in arch^ 
 
NATURAL PHILOSOPHY. 
 
 lecture of very considerable difficulty, which seems 
 to involve the peculiar structure and internal com- 
 position of the material. If the pressure were ap- 
 plied to the extremity, and in the direction of the 
 axis, the particles under it would become condensed 
 perpendicularly, and likewise accumulated towards 
 the sides. The repulsive energy opposed to the in- 
 cumbent weight would come to be augmented by a 
 certain oblique or lateral action. The column is 
 therefore made to swell out below, the curve of an 
 ellipse forming the preferable outline, though the 
 enlargement of the diameter at the bottom seldom 
 amounts to a fourth part. Practice has varied, how- 
 ever, according to the different perceptions of the 
 beauty of form. 
 
 If the centre of pressure should act beyond the 
 axis, it must evidently have a tendency to bend the 
 column. But the flexure having once begun, the 
 deranging influence will, from the property of the 
 lever, be exerted with increasing energy. The alti- 
 tude of a column thus only contributes to its weak- 
 ness. For the same reason, the walls of a house 
 should taper towards the top, forming a sort of in- 
 verted wedge, as the chief strain is accumulated near 
 the bottom. 
 
 3. Transverse or Lateral Pressure. In estima- 
 ting the strain occasioned by a load resting upon a 
 horizontal beam, this may be viewed as composed 
 either of longitudinal fibres, which oppose a greater 
 
54 ELEMENTS OF 
 
 resistance, if set on its edge, than when laid on its 
 face. Suppose the beam AB (fig. 1 26.) to have one 
 end firmly implanted in a wall GH, while a vertical 
 pressure is applied at the other end. This beam, 
 sinking under the load at B, may be conceived to 
 turn on the lowest point at A as a fulcrum ; conse- 
 quently the particles of the vertical section AC will 
 be forced into the oblique position AD, each of them 
 being turned aside through a space proportional to 
 its distance from A. The strain exerted at the end 
 AC will therefore be the result of the aggregate dis- 
 locations of all the particles of the section. When 
 the breadth and length of the beam remain the same, 
 this accumulate strain must evidently be proportional 
 to the area of the triangle CAD, and consequently 
 to the square of the depth AC. But when the 
 breadth of the beam is taken likewise into account, 
 the triangle of tension becomes converted into a 
 wedge, and the strain hence follows the direct ratio 
 of that breadth. Omitting the weight of the beam, 
 and assuming its depth and breadth as constant, the 
 tension of any particle at C may be considered as 
 acting against the short arm AC of the rectangular 
 beam CAB, and withstanding the load suspended 
 at B. The weight thus resisted by the cohesion of 
 the beam, is therefore inversely as the length AB. 
 Combining all the circumstances now together, we 
 may conclude, that the strength of a beam firmly 
 inserted in a wall, or its power to resist a pressure 
 
NATURAL PHILOSOPHY. 
 
 at its remote extremity, is compounded of the direct 
 ratios of its breadth and of the square qf its depth, 
 and the inverse ratio of its length. Thus, a beam 
 having the same length and breadth as another, but 
 twice the depth, is four times stronger ; and a beam 
 of the same depth and breadth, and double the 
 length, is only half as strong. Hence also, a beam, 
 whose depth is triple its width, will sustain a load 
 three times greater. For the same reason, a square 
 prism will have its strength inversely as its length 
 and the cube of its thickness. 
 
 In general, the resistance of a beam of any form, 
 but of a given length, to a cross strain, will be the 
 same as if the whole power exerted were collected in 
 the centre of gravity of each section. Thus, the 
 strain of a triangular prism may be conceived as con- 
 centrated in a point at one-third of the distance of 
 the perpendicular from the vertex to the base. Such 
 a prism is therefore twice as strong set on its edge as 
 when laid on its side.. 
 
 This simple investigation, which we owe to the il- 
 lustrious Galileo, though partly hypothetical, may be 
 regarded as a near approximation to the truth. It 
 is not only of essential service in improving the prac- 
 tice of carpentry, but sheds a clear light over the 
 economy of nature in the structure of animals and 
 vegetables. Reeds and other herbaceous plants de- 
 rive their power of resisting the force of the wind 
 from the subdivision of their length into moderate 
 intervals by hard knots. But they acquire still great- 
 
256 ELEMENTS OF 
 
 er strength from their hollow or tubular form ; for 
 the matter which they contain, being thus removed 
 to a greater distance from the fulcrum, exerts its co- 
 hesion with proportionally more effect, in withstand- 
 ing any lateral impulsion. The bones of animals 
 are likewise rendered much stronger by their fistular 
 structure, and their partition into short members 
 connected by large compact joints. Hence, in the 
 construction of fine mechanical and astronomical in- 
 struments, hollow brass cylinders are now preferred, 
 on account of their stiffness and lightness, to solid 
 pillars. 
 
 If a beam be supported horizontally at both ends, 
 and loaded in the middle, the pressure will be equal- 
 ly shared between the props, and the effect must evi- 
 dently be the same as if it had been fixed at the mid- 
 dle, and each end pulled upwards by half its load. 
 The breaking weight is consequently double of what 
 would be required to tear a beam, of half the length, 
 implanted in a wall. According to the principle of 
 Galileo, therefore, this limit is inversely as the length 
 of the beam, and directly as the breadth and the 
 square of the depth. 
 
 This result is, on the whole, confirmed by the nu- 
 merous experiments which the celebrated Buffon per- 
 formed between the years 1738 and 1746. Thus, 
 reducing all the quantities to English measures, an 
 oaken beam, 4 inches square and 10 feet long, broke 
 
NATURAL PHILOSOPHY. 
 
 under the weight of 4015 Ib. ; and another beam of 
 the same wood 8 inches square and 20 feet long, was 
 broken by a load of 16,700 Ib. The latter, being 
 twice as thick, should have been eight times stronger 
 with the same length ; but the length being doubled, 
 again reduced the excess to four times. Now, 
 
 4 X 4015 = 16,060, or very nearly 16,700. Beams of 
 
 5 inches square, and of 7, 14, and 28 feet long, were 
 broken with loads of 16,060 Ib., 7460 Ib., and 2472 Ib., 
 being nearly inversely as the respective lengths. In 
 general, if a denote the depth, b the breadth, and 
 / the length of a beam of oak in feet, those experi- 
 ments will be represented with tolerable accuracy by 
 
 
 the formula \ 1,200,000 Ib. If, instead of a 
 
 load exerted, the weight of the beam itself be sup- 
 posed to act at half the distance, then, .52 Ib. being 
 
 a*b 
 
 the weight of a cubic foot of oak, -^ zz 1,200,000 
 
 i* 
 
 zz 52abl, or V zz 46923.a and / zz 217 y/ a. A 
 horizontal plank of oak, 3 inches deep, and 108 feet 
 long, would hence sink under its own weight. 
 
 The same agreement may be remarked in 
 Mr Rennie's experiments on the transverse strain of 
 cast-iron bars 32 inches long, and 9^- Ib. weight. 
 Thus, a bar of an inch square, resting on horizontal 
 props at its end, bore a load of 1086 Ib. at the mid- 
 dle, yet the half of it supported 2320, or a little more 
 than double. Another bar of the same weight, but 
 
 R 
 
258 ELEMENTS OF 
 
 2 inches deep, and half an inch thick, sustained, at 
 the same interval between the props, likewise the 
 double, or 2186 Ib. Now, this bar had its strength 
 quadrupled by doubling the depth, but reduced again 
 to the double by the bisection of its breadth. A bar 
 of still the same section, but having a depth of 3 in- 
 ches combined with a breadth of one-third of an inch, 
 supported, as might be expected, three times the load 
 borne by a square inch bar, or 3588 Ib. A bar 
 which had an equilateral triangle of an inch for its 
 section sustained the weight of 840 Ib. when plant- 
 ed on its angle ; but when rested on its base, it bore, 
 as theory would indicate, 143? Ib., or very nearly the 
 dxmble. 
 
 If the dimensions of a rectangular bar of cast iron 
 be expressed in feet, a, b, and /, denoting as before 
 the depth, breadth and length ; those experimental 
 deductions will be represented with sufficient preci- 
 
 sion by the formula, (^- 5,000,000 Ib. 
 
 From these principles, we derive the solution of a 
 useful problem in carpentry, to cut the strongest 
 rectangular beam out of a given cylindrical tree. 
 This, it may be shown, will be effected, by making 
 the square of the depth of the beam double the square 
 of its breadth. An easy construction is hence ob- 
 tained. Let AB (fig. 127.) be an oblique diameter 
 of a circular section ; trisect it in the points C and 
 
NATURAL PHILOSOPHY. 
 
 D, draw the opposite perpendiculars Ml) and FC, 
 and complete the rectangle AEBF, which will re- 
 present the end of the beam. For it may be proved, 
 that, of all the rectangles which could be inscribed 
 in the given circle, the one now determined gives the 
 solid EG*.GH, expressing the strength of any sec- 
 tion, a maximum. 
 
 Suppose a beam AB (fig. 128.) laid in a hori- 
 zontal position upon two props, had a load applied 
 at an intermediate point C. From what was de- 
 monstrated in Statics, the pressure must be shared 
 between those props in the inverse ratio of its dis- 
 tance from them. The pressure exerted at B may 
 therefore be represented by AC, and the effect is 
 just the same as if the extremity B were pushed up- 
 wards by a force AC. Hence the strain at C, com- 
 municated by the lever BC, is as the rectangle under 
 AC and BC, the two segments. If a semicircle 
 were described upon AB, the square of the perpen- 
 dicular CD would thus be proportional to the strain 
 at C ; or if A B be bisected in O, the excess of the 
 square of OA above OC would be proportional to 
 that strain. 
 
 When a uniform horizontal beam AB (fig. 128.) 
 bends merely under its own weight, the strain ex- 
 erted at any point C is the same as what woidd be 
 produced, by half the weight of one segment BC 
 acting at the end of a lever equal to the other AC. 
 It is obvious, that the beam must press equally at 
 
260 ELEMENTS OF 
 
 both ends, and that the loads of the segments AC 
 and BC may be viewed as concentrated at their mid- 
 dle points I and K. Suppose the beam were sus- 
 tained at C by an opposite pressure from below ; it 
 would now be held in equipoise against this fulcrum 
 by vertical forces exerted at I and K, which are there- 
 fore inversely as CI and CK, or directly as BC to 
 AC. The weight of the segment BC, acting at A I 
 half the length of AC, or half the weight of BC 
 acting at the whole length AC, will hence occasion 
 half the strain at C, which results from the aggre- 
 gate pressure of the beam itself, the other half being 
 furnished by the equal effort of half the weight of 
 the segment AC acting at the distance BC. 
 
 In the hypothesis of Galileo, the prism or beam 
 is conceived to be absolutely inflexible, and to give 
 way only at the section of fracture. But all mate- 
 rials are, at least within certain limits, really elastic, 
 and bend progressively under an increasing load, till 
 they finally break. To determine with accuracy, 
 therefore, the effect of lateral or transverse pressure 
 against a bar, it becomes necessary to take the in- 
 fluence of incurvation into the estimate. 
 
 The principles of Dynamics show, that any weight 
 or vertical pressure, acting upon a small portion of 
 an elastic plate, is to the longitudinal strain which it 
 occasions, as the length of that portion to the radius 
 of curvature. Thus, if T)d (fig. 129.) the element 
 
NATURAL PHILOSOPHY. '2() 1 
 
 of the curve, represent the share of the weight which 
 this minute part sustains, the normals DO and dO 
 being drawn, the common distance DO of the centre 
 of osculation will express the direct tension exerted 
 at C. Any inflected beam would consequently form 
 itself into an arc of a circle, if the action of the pres- 
 sure were conceived to be equally distributed over 
 it. In that case, the length of the arc would be to 
 the radius of its circle, as the whole weight borne is 
 to the uniform longitudinal tension. 
 
 Suppose a thin bar (as in fig. 129.) to have one 
 end firmly fixed in a wall, and some weight attached to 
 the other. The action of this power in bending the 
 plate, will evidently not be diffused equally over its 
 length. The inflecting energy must, from the pro- 
 perty of the lever, be here proportional to the hori- 
 zontal distance of any point. Consequently, the 
 curvature or degree of incurvation at D will vary 
 with the ratio of DE. The plate, therefore, bends 
 at first quickly, then more slowly, and approaches to 
 a straight line at its extremity A. Such is the na- 
 ture of the Elastic Curve , which has its radius of 
 curvature DO always inversely as the corresponding 
 ordinate DE. 
 
 The Elastic Curve at its origin coincides nearly 
 with an Harmonic Curve ; but if its inflexure be 
 small, it may be viewed as a Cubic Parabola, the dis- 
 tance DE or FB being always proportional to the 
 cube of DF or BE. 
 
262 ELEMENTS OF 
 
 If the projecting plate, however, be only bent by 
 its own weight, the effect of this pressure, at any 
 point, will evidently be the same, as if half the weight 
 of the remote portion were collected at the end. 
 Wherefore, the tension exerted at D, which marks 
 the incurvation, being produced by a weight DB or 
 DE acting on the lever DE, may be considered as 
 proportional to the square of DE. Jn the curve 
 thus formed, the ordinate BE will hence be as the 
 fourth power of the absciss BE. It might also be 
 shown, that the extreme depression caused by this 
 diminishing incurvation is, in the case of an append- 
 ed load, the third part ; but in the case of its own 
 weight, only the sixth part of what an extension of 
 the same curvature from the fixed end would have 
 occasioned. 
 
 When a horizontal beam ABC (fig. 128.) is free- 
 ly supported at both ends, each portion of it, though 
 pressing equally downwards, must yet produce a 
 vertical stress, proportional to the rectangle under 
 the corresponding segments, and consequently the ra- 
 dius of curvature at C will be inversely as ACxCB, 
 or the square of CD. Find CE a third propor- 
 tional to some multiple of the diameter and CD, 
 and connect the several points E by a curve. This 
 curve will represent very nearly the form which the 
 beam would spontaneously assume. The depression 
 CE, being hence proportional to the square of CD, 
 will express the effort of an uniform tension to with- 
 
NATUUAL PHILOSOPHY. 263 
 
 stand the lateral stress. This same line CE will 
 nearly mark the curvature at E, which thus conti- 
 nually diminishes or flattens from the middle point 
 F to the extremities A and B. The curve hence 
 traced approximates to the Harmonic Curve, which 
 likewise has an equable tension, its graduating cur- 
 vature being proportional to the ordinates. 
 
 If we view this curve as only slightly bent from its 
 horizontal base, and estimate the amount of its suc- 
 cessive inflexions, it will follow, that the depression 
 OF of the middle of the bar is equal to the five-sixth 
 parts of what would have obtained, had the same cur- 
 vature extended through the whole length. From 
 the actual depression, therefore, the radius of oscu- 
 lation is easily computed. Thus, in fig. 1 30, if the 
 incurvation of the beam were uniform, the radius of 
 its circle would be equal to the square of HB, di- 
 vided by double the corresponding depression. But 
 the real depression HK being only five-sixths of that 
 quantity ; wherefore the radius of osculation at H 
 
 5 HB' 5HB* SAB* 
 = G X iHK = 12HK I: 48HK' r 
 times the square of the length of the beam, divided 
 by forty-eight times its middle depression. 
 
 In the case of an horizontal beam supported at 
 both ends, but depressed by its own weight, the up- 
 per surface becomes concave, and the under surface 
 convex. The particles of the upper surface EF (fig. 
 
2()4< ELEMENTS OF 
 
 130.) are therefore mutually condensed, while those 
 of the under surface CD are distended ; in a certain 
 intermediate curve AB, the particles are not affected 
 longitudinally, though bent from their rectilineal 
 position. This curve of neutral action may be as- 
 sumed in the middle of the beam. The attractive 
 effort of the particles stretched over the under surface 
 CD to approximate and regain their usual position, 
 produces a vertical thrust measured by the radius 
 GO. On the other hand, the repulsion exerted by 
 the particles compressed along the upper surface EF, 
 occasions a perpendicular detrusion marked by the 
 radius IO. The excess GI of the former force above 
 the latter, therefore, indicates the weight which 
 would be sustained at I, by the predominating action 
 of the lower surface. 
 
 If M denote the modulus of elasticity, it is evi. 
 dent, that the longitudinal tension of the fibre GD 
 
 GL GH - 48GH.HK 
 
 the vertical thrust produced by the strain is 
 M. 5!?. Of this, the variable part GH% 
 
 or the square of the distance from the neutral position 
 H, represents the element of the momentum of the 
 triangle GHL, which may be considered as accumu- 
 lated in the centre of gravity. The whole amount 
 then of the expression, taking the action of the 
 fibres equally distant on either side from H, is 
 
NATURAL PHILOSOPHY. $65 
 
 IT IT 
 
 effort is eual to the 
 
 . T 
 
 5A13 
 
 pressure which would be produced by the weight of 
 HD, or the fourth part of the beam, acting at a dis- 
 
 T. r 32CjrH 3 . HlV S^T-T TTTX 
 
 tance HB ; whence M. = GH.HDs or 
 
 oop T-T 
 
 M - -ATI* =HD, and doubling GIJ and HU, 
 
 , , 32GPHK 5AB 4 
 
 M 
 
 The modulus of elasticity may thus be found, by di- 
 viding five times the fourth power of the length of 
 a beam, by thirty -two times the product of its spon- 
 taneous depression into the square of the depth. 
 
 In the Experimental Inquiry into the Nature of 
 Heat, it is incidentally observed, that a white deal 
 138 inches long, and A5 of an inch deep, suffered 
 a depression of Z\ inches, by its own weight. Here 
 
 M. = .A V which in round numbers is 
 
 111,936,000 inches, or 9,328,000 feet. Hence the 
 spontaneous depression of any horizontal beam is di- 
 rectly as the fourth power of its length, and inverse- 
 ly as the square of its depth. Thus, a fir plank of 
 10 feet long, and 1 inch deep, will bend T 3 oth of an 
 inch ; another of the same depth, but 20 feet long, 
 would bend 4.8 inches ; while a third beam, of still 
 the same depth, but 30 feet, would sink no less than 
 24.3 inches. If this beam had the depth of 3 inches, 
 
266 ELEMENTS OF 
 
 the depression would be diminished nine times, and 
 would still have been three times less than the pro- 
 portion of the general dimensions. The depression 
 of a beam hence increases faster than its length. 
 
 It likewise follows, that the radius of spontaneous 
 curvature is directly as the square of the depth, and 
 inversely as the square of the length. Similar beams, 
 therefore, assume always the same curvature. 
 
 The spontaneous depression being in most cases 
 very small, the Modulus of Elasticity may be com- 
 puted more easily and correctly, from the aug- 
 mented depression occasioned by suspending a load 
 at the middle of the beam. Let X denote the length 
 of a bar of an inch square that weighs one pound, / 
 expressing the length of the beam, and a its depth 
 in inches. The weight of the beam corresponding 
 
 to the breadth of an inch, is hence = , and the 
 
 A 
 
 fourth part of this, or , applied at the middle, 
 
 TPA. 
 
 would produce the same strain and incurvation as the 
 mere pressure of the beam. Putting, therefore, 
 
 al 4to . ., A/r 51*- 
 
 = r l = ~ m the equatlon M : 
 
 where d indicates the depression, and it becomes 
 
 5/3.4X* 5l*fa> T M 
 
 M = _ ,, = - 3. Let P = express the 
 3 Sa d \ 
 
NATURAL PHILOSOPHY. 
 
 value of the modulus of elasticity in pounds, dr the 
 weight of a bar of an inch square, and of the altitude 
 
 of that modulus ; then P = . 
 
 When the length and depth of the beam remain 
 the same, the depression being inversely as the ra- 
 dius of curvature, will evidently be directly propor- 
 tional to the load supported by the strain. The for- 
 mula now given, in strictness, though applicable only 
 to the case where the weight was equal to that of 
 the fourth part of the beam, is hence quite general. 
 As an exemplification, an experiment of Mr EbbePs 
 may be taken. A bar of cast iron exactly an inch 
 square, and supported at the interval of 3 feet, suf- 
 fered a depression of T %ths of an inch, from a 
 load of 308 lb., suspended at the middle. Here 
 
 5(36-) (308) 
 P = - -^-j ' equal in round numbers to 
 
 17,900,000 lb. Another bar of the same dimen- 
 sions, but supported at the interval of 31 inches, 
 seems to have been carefully observed by Mi* Tred- 
 gold. With a load of 80 lb., it bent ^th of an 
 inch, with 180 lb. T V, and with 380 lb. ; the de- 
 pressions being thus very nearly proportional to 
 weights, till the elasticity began to give way. Here 
 
 P = 
 
 Since P is constant in bars of the same 1 materials, 
 it follows., that the depression is directly as the cube 
 
268 ELEMENTS OF 
 
 of the length, and inversely as the cube of the depth. 
 When the breadth is increased, a proportionally 
 greater weight is required to produce the same de- 
 gree of depression, which is therefore inversely as the 
 breadth of the bar. These results are sufficiently 
 confirmed by the experiments of Buffon, and the re- 
 cent observations of Dupin and Barlow. Thus, of 
 oaken beams of 5.35 inches square, one of 7 feet 
 bent 2.67 inches, under a weight of 12,400 Ib. ; 
 another of 15 feet long, bent 8.7 inches, under 
 5.700 Ib. ; and a third of 30 feet long, bent 21.7 
 inches, under 1910 Ib. Now, if the second beam had 
 been loaded as much as the first, the depression would 
 have augmented eight times, and amounted to 17.36 ; 
 but diminished again in the ratio of 12,400 to 5,700, 
 gives 8 inches, differing very little from 8.7. Again, 
 the third beam, if loaded as much as the second, 
 would have sunk 8x8.7 or 69.6 inches; but this 
 quantity is reduced to one-third or 23-2 inches, 
 from the inferior weight sustained. The discrepancy 
 is inconsiderable, and owing to that irregularity 
 which precedes the rupture of cohesion. 
 
 Another example may suffice. A beam of 4.28 
 inches square, and 12,86 feet long, was bent 7.5 
 inches under a load of 3,228 Ib., while another of 
 the same length and thickness, but double the 
 breadth, was bent 3.16 inches under 12,068 Ib. 
 The depression under the same weight should have 
 been only .9375 inches, or eight times less, and this> 
 
NATURAL PHILOSOPHY. 269 
 
 augmented in the ratio of 3,228 to 12,068, gives 3.8, 
 not materially different from 3.16. 
 
 When the load depressing a horizontal beam is in- 
 creased till the under side becomes overstrained, or 
 its particles are distended beyond a certain limit, 
 their elasticity is then destroyed, and a fracture en- 
 sues. This disunion commences at both the sur- 
 faces, being produced equally by the tearing of the 
 particles or fibres of the under layer, and, by the 
 crushing of those of the upper one. Let the limit 
 of cohesion be the w th part of the length of any line 
 or chain of particles ; the radius, of curvature OH, 
 at the moment of disruption, must hence be equal 
 
 to w.GH, or -.GI. But OH = - OTJT> , and con- 
 
 sequently =w.GI, or 
 
 Hence, likewise, n = ' T . Thus, in the 
 
 example already quoted, where a bar of cast iron of 
 an inch square, bent one-fifth of an inch, and then 
 
 broke, n = - = = 1204 ; or the cohe- 
 
 
 sion of the upper and under surfaces was destroyed, 
 when their contraction or distension came to exceed 
 this portion of the whole length. Now, though n 
 may vary in different instances, even from 150 to 
 1500, it is always constant in reference to the same 
 material. Hence the square of the length of a beam 
 
270 ELEMENTS OF 
 
 is in the compound ratio of the depth, and of the 
 quantity of depression which precedes fracture. 
 Hence, likewise, when the length remains the same, 
 this final depression is inversely as the depth of the 
 beam. Thus, Mr Barlow found a fir batten, 2 feet 
 long and an inch deep, to break with a depression of 
 1.25 inches, while another of the same depth, but 3 
 feet long, sunk 2.6 inches before it broke. Now, the 
 square of 2 is to that of 3, or 4 to 9, as 1,25 to 2.8, 
 differing very little from 2.6. Again, the depression 
 preceding the fracture of a fir batten, 3 feet long 
 and 2 inches deep, was only 1.12, being nearly the 
 half of 2.6. 
 
 But the breaking load, or the transverse pres- 
 sure which would overcome the strength of a 
 beam, may be derived from the same principles. 
 
 Resuming the former notation 3rf = -^> anc ^ su ^~ 
 
 stituting %4<n.ad for /*, the expression becomes 
 3lw , I7 , c 4 a* ,. 
 
 =- Wherefore ' =- p - 
 
 and since P is constant, the strength of the beam, 
 or the weight which it is capable of sustaining, is 
 directly as the square of its depth, and inversely as 
 its length. This power of resistance must also be 
 directly proportional to its breadth. The conclu- 
 sions are hence the same as those of Galileo. 
 
 Suppose the beam were a square prism, having a 
 for its side and / for its length ; the strength would 
 
NATURAL PHILOSOPHY. 271 
 
 It 1 5 
 
 then be denoted by- . P, being thus as the cube 
 
 of its thickness. From the same principles, it might 
 be shown, that the transverse strength of a cylinder is 
 only two-thirds of that of a square prism of the same 
 
 thickness, and is consequently denoted by =-. P. In 
 
 the case of a hollow cylinder, let a' express the dia- 
 meter of the internal cavity, and the resistance of the 
 
 8P 
 
 fistular column will be represented by -^- (a 3 a' 3 ). 
 
 If the matter were condensed into a solid cy- 
 linder, the strength would be reduced to 
 
 fij-( a* a'*}*' Suppose the hollow part to have 
 
 nine-tenths of the diameter of the cylinder, the 
 strength would be diminished in the ratio of 1 to 
 I (.9) 3 or .271 ; but had this tube been formed 
 into a solid rod, the strength would have amounted 
 only to .08. A cylinder having half its core hol- 
 lowed out, should be rendered an eighth part weak- 
 er, which agrees with an experiment of Barlow. 
 
 The lateral strength of any beam thus depends 
 mainly on the distance and cohesion of the upper 
 and under surfaces. Whatever stiffens the exterior 
 layers, contributes greatly to strengthen the whole. 
 A small incision drawn across the under side weak- 
 ens a bar essentially ; while a notch cut near the 
 middle of the upper side will not impair the strength, 
 
ELEMENTS OF 
 
 but, if filled up with a harder material, will even 
 sensibly augment it. Thus, Duhamel found, that a 
 bar of willow, cut through one-third its depth, the cut 
 being filled up with a thin slip of hard wood, was 
 thereby rendered about one-sixth part stronger than 
 before. It was even remarked, that the incision 
 could be carried much farther, without injuring the 
 strength of the bar. 
 
 The last application of the strength of any mate- 
 rial, consists, 
 
 4. In its resistance to the effort of twisting or 
 torsion. A cylindrical body suspended vertically, 
 but having its upper end fixed, may be wrenched 
 or turned round through any angle, by the exertion 
 of some lateral force ; and if its elasticity be not thus 
 impaired, it will, after the deranging influence has 
 ceased, return to its former position, and perform 
 this retrocession in a certain time. Many substances 
 may be considered as only bundles of parallel fibres, 
 which by twisting exert an augmented longitudinal 
 force. It will be more accurate, however, to view 
 v materials in general as composed of particles equally 
 distributed through the mass. We may hence con- 
 ceive any cylinder to consist of a series of thin discs, 
 as represented in fig. 131. When twisted, each 
 successive disc will make a small angular advance, 
 till this accumulation, at a certain distance, amounts 
 
NATURAL PHILOSOPHY. 
 
 to a complete circuit, and such revolutions will be 
 repeated at every like interval. 
 
 Let A denote the total angle of torsion, and h 
 the height of the cylinder ; the angular derange- 
 ment of any disc, as in fig. 132, will be expressed by 
 
 A 
 
 j- . Ifd represent the diameter of this disc, d* will 
 /it 
 
 mark the number of particles, and consequently d 4 
 must indicate their force of gyration. The torsion 
 corresponding to the angle A and the height h, is 
 
 hence denoted by -r : It is thus directly propor- 
 tional to the angle of deviation and the fourth 
 power of the diameter of the cylinder, and inversely 
 as its height. 
 
 This result is confirmed by nice experiments on 
 the torsion of slender metallic cylinders or wires. It 
 hence appears, that a wire of half the thickness is 
 twisted to the same angle with the 16th part of the 
 force, and a wire of the third of the thickness is 
 twisted to the same extent with only the 81st part 
 of the force ; and if the length were tripled, 243d 
 part only of such force would be required to produce 
 the effect. From a comparison of several experi- 
 ments, it appears, that the weight acting at the dis- 
 tance of a foot, required to wrench asunder a cy- 
 linder of an inch diameter, and 3 inches high, is 
 200 lb., and, consequently, the power of any other 
 
 VOL. i. s 
 
ELEMENTS OF 
 
 d* 
 
 cylinder to resist torsion is denoted by 7-. 600. To 
 
 twist an iron wire of the tenth of an inch diameter, 
 and 10 inches long, complete through a revolution, 
 would require the action of a weight of 80 grains at 
 the distance of an inch. 
 
 A fine wire will bear repeated circumvolutions, 
 and yet return to its original position. But if the 
 exterior particles be distrained beyond certain limits, 
 then cohesion is impaired and finally dissolved. 
 Within those limits, however, the elastic force evol- 
 ved is quite regular and directly proportional to the 
 angle described. Hence, under the same circum- 
 stances, all the oscillations of a contorted wire are 
 perfectly isochronous. 
 
 The power of torsion being denoted by j- , and 
 
 the quantity of matter in each disc by d* 9 the acce- 
 
 d z 
 
 lerating force is j- ; whence the time of vibration, 
 
 when the wire carries merely its own weight, will be 
 -J But if the wire be loaded, by a cylinder 
 
 whose diameter is constant, the weight W. of the 
 mass to be now moved will greatly diminish the ac- 
 celerating energy. Omitting, therefore, the weight 
 of the wire itself, as inconsiderable, the force of ac- 
 celeration will be denoted by rrr r , and hence the 
 
NATURAL PHILOSOPHY. 
 
 time required for performing an oscillation is v - 
 
 From the observed oscillations again, the power of 
 torsion is, by an inverted process, readily computed. 
 The principle of torsion was happily applied by 
 the ingenious Coulomb to the construction of an ex- 
 quisite balance, for detecting and measuring with ac- 
 curacy the smallest forces. By help of this very de- 
 licate instrument, he investigated the laws of mag- 
 netical attraction and repulsion ; and a similar com- 
 bination enabled our illustrious countryman Caven- 
 dish to determine, in the seclusion of his apartments, 
 the density of our globe. 
 
 All machines are impelled, either by the exertion 
 of Animal Force, or by the application of the Powers 
 of Nature. The latter comprise the potent elements 
 of Water, Air, and Fire. The former is more com- 
 mon, yet so variable as hardly to admit of calculation; 
 it depends not only on the vigour of the individual, 
 but on the different strength of the particular 
 muscles employed. Every animal exertion is attend- 
 ed by fatigue ; it soon relaxes, and would speedily 
 produce exhaustion. The most profitable mode of 
 applying the labour of animals, is to vary their mus- 
 cular action, and revive its tone by short and frequent 
 intervals of repose. 
 
 The decrease of animal power from continued ex- 
 ertion is exemplified in the race-horse. At his 
 
 s 2 
 
ELEMENTS OF 
 
 greatest speed, he has been known to go over three 
 miles and a half in six minutes, at the rate, there- 
 fore, of 35 miles an hour, or 50 feet each second. 
 But he could not travel more than 20 miles in the 
 course of an hour, or perhaps 100 miles during a 
 whole day. This gradual tendency to exhaustion is 
 more precisely ascertained in the case of expert 
 couriers. A swift runner may spring over the space 
 of 150 yards, at the rate of 20 miles an hour, or 28 
 feet each second ; but his speed is reduced to 21 
 feet in passing over a quarter of a mile, and to 18 
 feet in completing the whole mile. If he runs over 
 six miles, his average rate of going will only be 11 
 feet in a second ; and to travel 600 miles, he will 
 take ten days, which, reckoning half the time to be 
 spent in rest, gives an average pace of 7i feet, or 
 about the fourth part of the swiftness with which he 
 could start. 
 
 The ordinary method of computing the effects of 
 labour is, from the weight which it is capable of ele- 
 vating to a certain height, in a given time, the pro- 
 duct of these three measures expressing the absolute 
 quantity of performance. But these measures have 
 evidently a mutual dependency, which in all cases 
 brings the final result nearer to equality. Thus, 
 great speed wilj occasion a waste offeree, and abridge 
 the period of its continuance. It was reckoned by 
 Daniel Bernoulli and Desaguliers, that a man could 
 raise two millions of pounds averdupois one foot 
 
NATURAL PHILOSOPHY. 
 
 in a day. But our civil engineers have gone much 
 farther, and are accustomed, in their calculations, to 
 assume, that a labourer will lift ten pounds to the 
 height of ten feet every second, and is able to con- 
 tinue such exertion for ten hours each day, thus ac- 
 cumulating the performance of 3,600,000. But 
 this estimate seems to be drawn from the produce of 
 momentary exertions, under the most favourable 
 circumstances ; and it therefore greatly exceeds the 
 actual results, as commonly depressed by fatigue, 
 and curtailed by the unavoidable waste of force. 
 
 Coulomb has furnished some of the most accurate 
 and varied observations on the measure of human 
 labour. A man will climb a stair, from 70 to 100 
 feet high, at the rate of 45 feet in a minute. Rec- 
 koning his weight at 155 lb., the animal exertion 
 for one minute is 6,97^, an d would amount to 
 4,185,000 if continued for ten hours. But such 
 exercise is too violent to be often repeated in the 
 course of a day. A person may clamber up a rock 
 500 feet high by a ladder-stair in twenty-minutes, 
 and consequently at the rate of 25 feet each minute ; 
 his efforts are thus already impaired, and the per- 
 formance reaches only 3,875 in a minute. 
 
 But, under the incumbrance of a load, the quan- 
 tity of action is still more remarkably diminished. 
 A porter weighing 140 lb. was found willing to 
 climb a stair 40 feet high 266 times in a-day ; but 
 he could carry up only 66 loads of fire-wood, each 
 
ELEMENTS OF 
 
 of them 163 Ib. weight. In the former case, his 
 daily performance was very nearly 1,500,000 ; 
 while, in the latter, it amounted only to 808,000. 
 The quantity of permanent effect was hence only 
 about 700,000, or scarcely half the labour exerted 
 in mere climbing. In the driving of piles, a load 
 of 42 Ib., called the ram, is drawn up 3J feet high 
 20 times in a minute ; but the work has been con- 
 sidered so fatiguing as to endure only three hours a- 
 day. This gives about 530,000, for the daily per* 
 formance. Nearly the same result is obtained, by 
 computing the quantity of water, which by means of 
 a double bucket, a man drew up from a well. He 
 lifted 36 Ib. 120 times in a-day from a depth of 120 
 feet, the total effect being 518,400. A skilful la- 
 bourer working in the field with a large hoe, creates 
 an effect equal to 728,000. When the agency of a 
 winch is employed in turning a machine, the per- 
 formance is still greater, amounting to 845,000. 
 
 In all these instances, a certain weight is hea- 
 ved up, but a much smaller effort is sufficient to tran- 
 sport a load horizontally. A man could, in the 
 space of a day, scarcely reach an altitude of two 
 miles by climbing a stair ; though he will easily 
 walk over thirty miles on a smooth and level road. 
 But he would in the same time carry only 130 Ib. 
 to the fourth part of that distance, or 7i miles. 
 Assuming his own weight to be 140 Ib., the quan- 
 tity of horizontal action would amount to 42,768,000, 
 
NATURAL PHILOSOPHY. 
 
 or twenty-eight times the vertical performance ; 
 but the share of it in conveying the load is 
 ^0,9^1,780, or about thirty times what was spent in 
 its elevation. The greatest advantage is obtained 
 by reducing the burthen to 120 lb., the length of 
 journey being augmented in a higher ratio. 
 
 These results are apparently below the average 
 labour performed in England, which is not only most 
 vigorous, but, in many cases, quite overstrained. Mo- 
 derate exertion of strength, joined to regularity and 
 perseverance, would be more conducive to robust 
 health, and the comfortable duration of human life. 
 It is painful to remark, that English labourers are 
 sooner worn out than those of most other nations. 
 
 A porter in London is accustomed to carry a 
 burthen of 200 lb. at the rate of three miles an hour. 
 In the same metropolis^ a couple of Irish chairmen 
 continue at the pace of four miles an hour, under a 
 load of 300 lb. These exertions are greatly infe- 
 rior, however, to the labour performed by the sinewy 
 porters in Turkey, the Levant, and generally on the 
 shores of the Mediterranean. At Constantinople, 
 an Albanian will carry 800 or 900 lb. on his back, 
 stooping forward and assisting his steps by a short 
 staff. At Marseilles, four porters commonly carry 
 the immense load of nearly two tons, by means of 
 soft hods passing over their heads, and resting on 
 their shoulders, with the ends of poles, from which 
 the goods are suspended. 
 
280 ELEMENTS OF 
 
 According to some experiments of the late 
 Mr Buchanan, the exertions of a man in working a 
 pump, in turning a winch, in ringing a bell, and in 
 rowing a boat, are as the numbers 100, 167, 2#7> 
 and 248. But those efforts appear to have been con- 
 tinued for no great length of time. The Greek sea- 
 men in the Dardanelles are esteemed more skilful 
 and vigorous in the act of rowing, than those of any 
 other nation. Even the Chinese, by applying both 
 their hands and their feet, are said to surpass all 
 people in giving impulsion to boats by sculling. 
 
 The several races of men differ materially in 
 strength, but still greater diversity results from the 
 constitution and habits of the individual. The Eu- 
 ropean is, on the whole, decidedly more powerful 
 than the inhabitants of the other quarters of the 
 globe ; and man, reared in civilized society, is a 
 finer, robuster, and more vigorous animal than the 
 savage. In the temperate climates, likewise, men 
 are capable of much harder labour, than under the 
 influence of a burning sun. Coulomb remarks, that 
 the French soldiers employed on the fortifications 
 of the Isle of Martinique became soon exhausted, 
 and were unable to perform half the work executed 
 by them at home. 
 
 The most violent and toilsome exertion of human 
 labour is performed in Peru, by the carriers or car- 
 gueros, who traverse the loftiest mountains, and 
 clamber along the sides of the most tremendous 
 
NATURAL PHILOSOPHY. 281 
 
 precipices, with travellers seated on chairs strapped 
 to their backs. In this manner, they convey loads 
 of twelve, fourteen, or even eighteen stones ; and 
 possess such strength and action, as to be able to 
 pursue their painful task eight or nine hours, for 
 several successive days. These men are a vagabond 
 race, consisting mostly of mulattoes, with a mixture 
 of whites, who prefer a life of hardship and vicissi- 
 tude to that of constant, though moderate labour. 
 
 When a man stands, he pulls with the greatest ef- 
 fect ; but his power of traction is much enfeebled 
 by the labour of travelling. A valuable set of ex- 
 periments on this subject, made by Schultze of Ber- 
 lin, seem to confirm the second formula proposed 
 by Euler. It may hence be stated generally, that 
 if v denote the number of miles which a person walks 
 during an hour, the force which he exerts in dragging 
 a load, by means of a rope passed over his shoulders, 
 will be expressed in pounds averdupois by 2(6 1>). 
 Thus, when standing still, he pulls with a force of 
 about 72 Ib. ; but if he walks at the rate of two 
 miles an hour, his power of traction is reduced to 
 32 Ib. ; and if he quicken his pace to four miles an 
 hour, he can draw only 8 Ib. There is consequent- 
 ly a certain velocity which procures the greatest ef- 
 fect, or when the product of the traction by the ve- 
 locity becomes a maximum. This takes place when 
 ,he proceeds at the rate of two miles an hour. The 
 utmost exertion which a man walking might conti- 
 
 o 
 
 , ; : ; 
 
282 ELEMENTS OF 
 
 nue to make in drawing up a weight by means of a 
 pulley, would amount, therefore, in a minute, only 
 to 5,632 ; but if he applied his entire strength, 
 without moving from the spot, he could produce an 
 effect of 12,672. 
 
 The labour of a horse in a day is commonly rec- 
 koned equal to that of five men ; but he works 
 only eight hours, while a man easily continues his 
 exertions for ten hours. Horses likewise display 
 much greater force in carrying than in pulling ; and 
 yet an active walker will beat them on a long jour- 
 ney. Their power of traction seldom exceeds 144* 
 pounds, but they are capable of carrying more than 
 six times as much weight. The pack-horses in the 
 West Riding of Yorkshire are accustomed to tran- 
 sport loads of 420 Ib. over a hilly country. But, in 
 many parts of England, the mill-horses will carry 
 the enormous burthen of 910 Ib. to a short distance. 
 The action of a horse is greatly reduced by the du- 
 ration of his task. Though not encumbered at all 
 with any load or draught, he would be completely 
 exhausted perhaps by a continued motion for 20 
 hours in a day. The rate in miles each hour, which a 
 good horse could thus travel, will be represented near- 
 ly by the formula (20 1\* 9 where t denotes the 
 
 number of hours during which he trots or walks. 
 Thus, though the horse might start with a celerity 
 of 16 miles, this would be reduced in 4 hours to 10, 
 and in 8 hours to 5f . Hence the great advantages 
 
NATURAL PHILOSOPHY. 
 
 resulting from short stages, lately adopted for the 
 rapid conveyance of the mail. 
 
 The force which a horse exerts in pulling is dimi- 
 nished likewise by the quickness of his pace. A 
 strong horse may have his power of traction express- 
 ed, in pounds averdupois, by the formula (15 v*)*, 
 where v denotes the velocity in miles during an hour. 
 But (12 vy will represent more nearly the ordi- 
 nary draught. Thus, a horse beginning his pull 
 with the force of 144 lb., would draw 100 Ib. at a 
 walk of two miles an hour, but only 64 lb. when 
 advancing at double that rate, and not more than 
 36 lb. if he quickened his pace to six miles an hour. 
 His greatest performance would hence be made with 
 the velocity of four miles an hour. The accumula- 
 ted effort in a minute will then amount to 22,528. 
 
 This formula gives results somewhat greater than 
 the performance of the ordinary horses, employed in 
 tracking along the English canals. They generally 
 go at the rate of two miles and a half in an hour, with 
 a pull of only 81 lb. But in several places, and parti- 
 cularly between Manchester and Liverpool, they move 
 at only half that pace, and therefore exert a force of 
 104 lb. The measure generally adopted in compu- 
 ting the power of steam-engines is much higher, the 
 labour of a horse being reckoned sufficient to raise, 
 every minute, to the elevation of one foot, the weight 
 of 33,000 lb. But this estimate is not only greatly 
 
284 ELEMENTS OF 
 
 exaggerated, but should be viewed as merely an ar- 
 bitrary and conventional standard. 
 
 Wheel carriages enable horses, on level roads, to 
 draw, at an average, loads about fifteen times greater 
 than the power exerted. The carriers between 
 Glasgow and Edinburgh transport, in a single-horse 
 cart, weighing about 7 cwt., the load of a ton, and 
 travel at the rate of 22 miles a-day. At Paris, one 
 horse, in a small cart, conveys along the streets, half 
 a cord of wood, weighing two tons ; but three horses 
 yoked in a line are able to drag 105 cwt., or that of 
 a heavy cart loaded with building stones. The Nor- 
 mandy carriers travel, from 14 to 22 miles a-day, 
 with two-wheeled carts, weighing each 11 cwt., and 
 loaded with 79 cwt. of goods, drawn by a team of 
 four horses. 
 
 The French draught horses, thus harnessed to 
 light carriages, are more efficient perhaps than the 
 finer breeds of this country. They perform very 
 nearly as much work as those in the single horse 
 carts used at Glasgow, and far greater than those 
 heavy animals which drag the lumpish and towering 
 English waggons. The London dray-horses, in the 
 mere act of ascending from the wharfs, display a 
 powerful effort, but they afterwards make little ex- 
 ertion, their force being mostly expended in tran- 
 sporting their own ponderous mass. 
 
 Oxen, on account of their steady draught, are 
 in many countries preferred for the yoke. They 
 
NATURAL PHILOSOPHY. 
 
 Were formerly employed universally in the various 
 labours of husbandry. The tenderness of their 
 hoofs, however, makes them unfit for pulling on 
 paved roads, and they can work only with advan- 
 tage on soft grounds. But they want all the plian- 
 cy and animation, which are the favourite qualities 
 of the horse. 
 
 The patient drudgery of the ass, renders him a 
 serviceable companion of the poor. Though much 
 inferior in strength to the horse, he is maintained at 
 far less cost. In this country, an ass will carry about 
 two hundred weight of coals or limestone, twenty 
 miles a-day. But, in the warmer climates, he be- 
 comes a larger and finer animal, and trots or ambles 
 briskly under a load of 150 pounds. 
 
 The mule partakes of the distinct qualities of the 
 ass and of the mare. This cross-breed is much esteem- 
 ed in Italy and Spain, being equally fit for draught 
 and burthen. It is a stronger and a hardier animal 
 than even the horse, performs longer tasks, is less sub- 
 ject to disease, and lives to double the age. 
 
 In the hotter parts of Asia and Africa, the pon- 
 derous strength of the elephant has been long turn- 
 ed to the purposes of war. He is reckoned more 
 powerful than six horses, but his consumption of 
 food is proportionally greater. The elephant carries 
 a load of three or four thousand pounds, his ordi- 
 nary pace is equal to that of a slow trot, he travels 
 easily over forty or fifty miles in a day, and has been 
 
286 ELEMENTS OF 
 
 known to perform, in that time, a journey of an 
 hundred and ten miles. His sagacity and intelli- 
 gence direct him to apply his strength according to 
 the exigency of the occasion. 
 
 The camel is a most useful beast of burthen in 
 the arid plains of Arabia. The stronger ones carry 
 a load of ten or twelve hundred weight, and the 
 weaker ones transport six or seven hundred ; they 
 walk at the rate of two miles and a half in an hour, 
 and march regularly about thirty miles every day. 
 The camel travels often eight or nine days, without 
 any fresh supply of water ; when a caravan encamps 
 in the evening, he is perhaps turned loose, for the 
 space of an hour, to browze on the coarsest herbage, 
 which serves him to ruminate during the rest of the 
 night. In this manner, without making any other 
 halt, he will perform a dreary and monotonous jour- 
 ney of two thousand miles. 
 
 The dromedary, a small species of camel, though 
 less fitted for bearing loads, travels with great speed, 
 at a very hard trot, which is extremely fatiguing to 
 the rider. Yet a Bedouin Arab, mounted upon one 
 of those animals, lately conveyed an express from 
 Cairo to Mecca, a distance exceeding 7^0 miles, in 
 the space of five days. 
 
 Within the Arctic Circle again, the rein deer 
 is a domesticated animal, not less valuable. He serves 
 to feed and clothe the poor Laplander, and trans- 
 ports his master with great swiftness, in a cover- 
 
NATURAL PHILOSOPHY. 
 
 ed sledge, over the snowy and frozen tracts. The 
 rein deer subsists on the scanty vegetation of moss 
 or lichens, and, though very docile, he is not power- 
 ful. Two of them are required to draw a light 
 sledge : but so harnessed, they will run fifty or sixty 
 miles on a stretch, and perform sometimes a journey 
 of a hundred and twelve miles in the course of a 
 day. Such exertions, however, soon wear them out. 
 
 A sort of dwarf camel was the only animal of bur- 
 then possessed by the ancient Peruvians. The Lama 
 is indeed peculiarly fitted for the lofty regions of 
 the Andes. The strongest of them carry only from 
 150 to 200 pounds, but perform about fifteen miles 
 a-day over the roughest mountains. They generally 
 continue this labour during five days, And are then 
 allowed to halt two or three days, Before they renew 
 their task. The Paco is another similar animal, 
 employed likewise in transporting goods in that 
 singular country ; it is very stubborn, however, and 
 carries only from fifty to seventy pounds. 
 
 The stag-hound is capable of running three or 
 four hours on a stretch, and at the rate of 15 miles an 
 hour. Mastiffs are sometimes, in our large towns, 
 made to assist in drawing light carts. The rough 
 shaggy dog of the Esquimaux is a most hardy 
 and powerful animal. He is early trained by those 
 forlorn tribes to carry burthens, and generally to 
 drag their loaded sledges over the hard but uneven 
 surface of the snow. The dogs are harnessed in a line, 
 
288 ELEMENTS OF 
 
 sometimes to the number of eight or ten, and they 
 perform their task with great rapidity, steadiness, and 
 perseverance. They can draw a heavy sledge to 
 a considerable distance, with the swiftness of 13 or 
 14 miles an hour ; but they will travel long journeys 
 at half that rate, each of them pulling the weight of 
 130 pounds. 
 
 Even the exertions of goats have, in some parts of 
 Europe, been turned to useful labour. They are made 
 to tread in a wheel which draws up water, or raises 
 ore from the mine. In Holland, the young goat, 
 gaily caparisoned, is yoked to the ornamental minia- 
 ture chariots of the children of the wealthy burghers. 
 Though a very light animal, the goat is nimble, 
 and climbs at a high angle. Supposing this soar- 
 ing creature, though only the fourth part of the 
 weight of a man, to march as fast along an ascent of 
 40 degrees, as he does over one of 18 degrees, 
 the sine of the former being double that of the latter, 
 it must yet perform half as much work. 
 
NATUHAL PHILOSOPHY. \>S<) 
 
 VI. HYDROSTATICS 
 
 explains the Equilibrium and Pressure of Fluids. 
 It is, indeed, only the application of the principles 
 of Statics to the peculiar constitution of Fluids. 
 Every substance appears to be composed of an as- 
 semblage of atoms, connected together by a system 
 of mutual attraction and repulsion. In solid bo- 
 dies, these integrant molecules affect a certain ar- 
 rangement, and resist every change of figure. But 
 fluids are distinguished, by the loose aggregation of 
 their particles, which yield to the smallest external 
 impression. 
 
 This indifference of the particles to arrangement 
 or configuration, is, however, insufficient alone to 
 constitute Fluidity. The disintegration of a Solid 
 approximates it to such a property, but without con- 
 ferring at all the distinctive character of a Fluid. 
 When a solid body is reduced to a very thin plane, 
 it will easily fold up in one direction ; if it be di- 
 vided into slender filaments, it will bend every way 
 freely, and still betray no disposition to fracture. 
 Crumbled to dust or powder, the aggregate parti- 
 cles now give way to any external force, yet with- 
 out diffusing this impression through the mass. 
 
 VOL. i. T 
 
t?9O ELEMENTS OF 
 
 Among such detached portions of matter, there ex- 
 ists no sympathy or mutual concatenation. 
 
 A solid body, subjected to a compressing force, 
 undergoes a proportional contraction, but recovers 
 its volume after this contraction is withdrawn. But 
 if the compression be urged beyond a certain limit, 
 the substance will be crushed under the load, and 
 will suffer a complete dissolution. A fluid, how- 
 ever, inclosed in the chamber of a very strong me- 
 tallic vessel, is found to be capable of withstanding 
 the greatest pressure we can command, while its 
 contractions are comparatively more extensive ; and 
 it always returns with unimpaired vigour to its for- 
 mer condition, the moment that such external force 
 has ceased to act. The constitution of a fluid re- 
 mains unaltered under the most enormous loads, and 
 its several portions at all times separate and reunite 
 with extreme facility. Whether the fluid has a li- 
 quid, or assumes a gaseous, form, it can equally bear 
 any compression, the corresponding contraction in 
 the latter being only much greater than in the for- 
 mer. 
 
 If the minutest subdivision of a Solid really con- 
 tributes nothing to Fluidity, neither will any suppos- 
 ed smoothness or globular shape of the particles con- 
 fer that character. Exact spherules might procure 
 lubricity on a plane, but, among one another, they 
 will become implanted, and affect certain configura- 
 tions, as manifested in the common piling of round 
 
NATURAL PHILOSOPHY. 5291 
 
 shot. The absolute contact of the particles is be- 
 sides inadmissible, since all bodies whatever seem to 
 be capable of indefinite condensation. 
 
 But Fluidity occurs in very different degrees. 
 Any disturbing impression is more quickly obeyed 
 by one liquid than by another ; by water, for in- 
 stance, than by oil or treacle. All the possible 
 shades of softness, in fact, might often be traced, 
 from a solid to a fluid substance. The application 
 of heat generally promotes fluidity. Thus, honey 
 in winter is a candied or friable solid ; in the spring 
 it melts down ; but, as the summer advances, it gra- 
 dually loses its viscidity, and flows more freely. Oil 
 seems affected in the same way, and hence the prac- 
 tice in Italy of depositing it in casks for exportation 
 during the winter season, when it is thickest and 
 least penetrating. But water itself, and other li- 
 quids, not excepting mercury, have their fluidity 
 likewise augmented by heat, as evinced in their 
 quickened flow through capillary tubes. 
 
 If an angular bit of glass be held in the flame of 
 a blowpipe, it will become gradually rounded, as 
 the heat penetrates and softens the mass. In like 
 manner, a fragment of sealing-wax loses its rough 
 exterior, and assumes a regular curved surface, on 
 approaching it to the fire. But the centre of cur- 
 vature is the point to which the combined attractions 
 of the outer range of particles must be directed. 
 Since this point retires, therefore, from the surface 
 
 T 2 
 
ELEMENTS OF 
 
 in the progress of Softness to Fluidity, the mutual 
 connection of the integrant molecules must be ex- 
 erted over a much greater extent in Fluids than in 
 Solids. This inference appears to afford likewise an 
 explanation of the facility of their internal motion, 
 so conspicuous in Liquids. When the sphere of 
 activity is confined, as in solid substances, to a very 
 narrow spot, a few particles only are connected, by 
 their sympathetic tendencies. Any internal dislo- 
 cation will in this case require the approximation of 
 certain particles and the recession of others, which 
 must hence occasion the exercise of corresponding 
 repulsive and attractive forces. But these forces, 
 being directed to a small number of centres, will 
 constitute a very unequal group, incapable of gain- 
 ing a smooth and graduated equilibrium. Every 
 change of inclination would produce a violent effort 
 to attain a new position of repose. On the other 
 hand, when the sphere of activity embraces a multi- 
 tude of particles, as in the composition of Fluids, the 
 slightest mutual derangement will be sufficient to ac- 
 commodate every variation of external impression. 
 The most minute deviation of each particle, would, 
 by such prodigious repetition, amount to any re- 
 quired change of direction. In their dislocations, 
 the particles are almost unconstrained, and need 
 scarcely approach or recede ; the repulsive and at- 
 tractive forces evolved become, therefore, extremely 
 small, and by their multitude produce in every po- 
 
NATURAL PHILOSOPHY. 
 
 sition nearly a perfect counterbalance. A Fluid is 
 hence fitted to obey any impression with the utmost 
 facility. 
 
 But whatever may be the system of forces that 
 connects the fluid atoms, the properties of the com- 
 pound are deducible, from the absolute facility with 
 which those ultimate particles receive and transmit 
 external impressions. Suppose a Fluid, either gase- 
 ous or liquid, to be inclosed in a very broad and ex- 
 tremely shallow cylinder of glass or metal AB (fig. 
 135), at the top of which is inserted a tall narrow 
 tube CD of the same material, having a plug or 
 piston E nicely fitted into it. On pushing down 
 this piston, the fluid particles immediately under it 
 will at first give way, and in receding, must approach 
 closer to each other. They will, in consequence, 
 display a repulsive force proportional to their mu- 
 tual approximation. Now, it may be proved that 
 points or atoms can never be ranged through space 
 in perfect straight lines ; wherefore, the pressure 
 communicated, will not be confined to the vertical 
 column, but will insensibly diverge in every direc- 
 tion. The particles at F will repel obliquely those 
 at G or at H, and these again will distribute the 
 oblique impression to other adjacent particles. The 
 whole stratum of fluid, from the orifice C to the 
 boundaries A and B, must hence continue to recede 
 from the piston, and contract its volume, till the in- 
 tegrant particles have all of them attained the same 
 
ELEMENTS OF 
 
 mutual distance, and exert the corresponding repul- 
 sive energy. An uniform condensation must thus 
 be diffused through the compressed mass, before an 
 equilibrium can take place in it. This condition ob- 
 tains equally in liquids and in gaseous fluids, only the 
 contraction produced by a compressing force, which 
 is so visible in the former, can seldom be distinguish- 
 ed by ordinary perception in the latter. But a very 
 minute alteration of the volume of a liquid evolves as 
 much force, as an extensive change in the mass of a 
 gaseous fluidl 
 
 The pressure exerted by the piston E at the 
 orifice C is hence diffused equally through the 
 whole of the fluid, every particle of which acquires 
 the same intensity of repulsion. Each point on the 
 surface of the vessel must therefore sustain an equal 
 effort. If a wide cylinder IK, fitted likewise with a 
 piston, were inserted at I, the compression opposed 
 would be proportional to the space of action or the 
 circle of the orifice. On the supposition that the 
 surface of the piston I were ten times greater than 
 that of E, it would likewise support a load ten times 
 greater than the pressure applied at E or C. 
 
 Let the piston I be now removed, while the pis- 
 ton E is conceived to act as before. The liquid 
 must evidently rise in the cylinder IK (fig. 136), 
 till its weight becomes equal to the pressure exerted 
 at the orifice I. In like manner, if another cylin- 
 der LM were inserted, the liquid would rise till its 
 weight was equal to the pressure at L. But the 
 
NATURAL PHILOSOPHY. 
 
 pressures at I and L being proportional to those 
 orifices, or the circular sections of the cylinders IK 
 and LM, the altitudes IK and LM of the columns 
 themselves must be equal. If a cylinder NO were 
 inserted obliquely, the liquid would still rise to the 
 same level ; for in this case, the column being partly 
 sustained by the under side, its weight would, from 
 the property of the inclined plane, be to its pres- 
 sure at N, as the length ON to the perpendicular 
 OP. 
 
 Let the piston E itself be withdrawn, and the li- 
 quid will mount in the cylinder CD, till its weight 
 becomes equal to the pressure applied at C, and con- 
 sequently will attain the same altitude as in the 
 other communicating cylinders. Suppose the cylin- 
 ders to be enlarged and brought nearer to each 
 other, and still the same level will be maintained 
 among them. Conceive they were even united, so 
 as to compose a single cylindrical vessel, and the 
 contained liquid would always assume a level sur- 
 face. Such is the distinguishing character of fluids. 
 
 It hence follows, that the condensation accumu- 
 lated at any point of an open liquid mass, and there- 
 fore the actual pressure exerted there, is proportional 
 to the altitude of the incumbent column of a sup- 
 posed vertical tube, or to the depth of the point be- 
 low the surface of the fluid. The pressure of water 
 against the perpendicular sides of any cistern must 
 thus increase regularly, from the top to the bottom. 
 
ELEMENTS OF 
 
 111 a cubical vessel, the pressure borne by each side 
 would be just half the weight supported at the bot- 
 tom, and consequently the pressure sustained by all 
 the four sides would be double of this weight. 
 
 That the pressure of a fluid is exerted equally 
 
 every way in proportion to its depth, may be con- 
 
 firmed by various experiments. Thus, having fas- 
 
 tened a long narrow glass tube to the neck of a thin 
 
 bladder, fill this with water till it stand perhaps an 
 
 inch higher, and plunge the whole in a tall jar of wa- 
 
 ter ; the liquid will be seen to rise in the tube, and 
 
 maintain the same altitude, exactly in proportion as 
 
 the bladder descends. Again, if a tall glass tube* 
 
 spreading below into a wide funnel-mouth, to which 
 
 a loaded plate of brass has been ground and closely 
 
 fitted, were let down and held in a body of water, 
 
 at the depth where a cylindrical column of fluid, in- 
 
 cumbent upon its broad base, has a weight equal to 
 
 that of the plate, this would remain supported. 
 
 But if a hole were pierced in the side of the tube 
 
 admitting a small portion of the water to fill up the 
 
 funnel, its load would quickly be precipitated to the 
 
 bottom. On the other hand, if the tube had its 
 
 funnel-mouth turned upwards, and fitted with a thin 
 
 brass plate surmounted by a very thick cylinder of 
 
 cork ; the buoyancy of this cover would be over- 
 
 come at a certain depth below the surface of the wa- 
 
 ter. But, on letting water into the funnel, the pres- 
 
NATURAL PHILOSOPHY. 
 
 sure now exerted under the plate would iin mediate- 
 ly float it up. 
 
 The fundamental principle, that a fluid compress- 
 ed in a close shallow vessel exerts the same effort 
 upon every equal portion of the confining surface, 
 was first distinctly stated by the famous Pascal, who 
 even proposed it as a new mechanical power of great 
 efficacy and ready application. If the piston I 
 (fig. 135.) were, for instance, an hundred times 
 larger than the piston E, the force of one man push- 
 ing down the former would be sufficient to withstand 
 the action of an hundred men exerted upon the latter. 
 Nor did another feature of resemblance escape this 
 acute philosopher, that, as in all other mechanical com- 
 binations, what is here gained in power is lost in cele- 
 rity, or, in other words, that the height to which a 
 load is raised is still inversely as the purchase. Thus* 
 while the piston E descends through one inch, the 
 piston I ascends only the hundredth part of an inch. 
 This consideration is essentially the same as the 
 principle of virtual velocities, which affords another 
 demonstration of the equality of pressure diffused 
 through a fluid. For, if one pound depress the pis- 
 ton E one inch, the piston I will lift an hundred 
 pounds over the hundredth part of an inch, the mo- 
 mentum of the weight at E being still equal to the 
 opposite momentum of the load incumbent at I. 
 The pressure exerted by the fluid upon every point 
 
ELEMENTS OF 
 
 of the surface of the shallow vessel is hence the 
 same. 
 
 The project originally started by Pascal, of apply- 
 ing this capital property of fluids to the construction 
 of a powerful mechanical engine, has been reduced 
 to convenient practice in our own time. The pro- 
 gress of the arts has now begun to realize all the de- 
 licacy of theory. Watt employed, to a certain ex- 
 tent, the soft compression of air, as the chief agent 
 in his Coming Engine ; and Bramah has most suc- 
 cessfully availed himself of the constrained energy of 
 water, in the composition of his Hydraulic Press. 
 
 It will be easy, from the principles already stated, 
 to determine what share of the weight of a liquid 
 contained in a vessel of any form, is supported by the 
 sides, and what part rests upon the bottom. Sup- 
 pose a wide-spreading vessel, of which CABD 
 (fig. 137.) represents a vertical section, to be filled 
 with some liquid. The pressure, which a very nar- 
 row column of fluid FEe/* exerts against the side of 
 the vessel at E may be decomposed into a force act- 
 ing vertically, and another acting horizontally. But 
 those three forces, being proportional to perpendicu- 
 lars to their several directions, are as the lines E, 
 e<p, and E<p, and consequently, are as the elementary 
 rectangles EexEF, epxEF, E<pxEF. Now, the 
 whole pressure exerted upon the portion Ee of the 
 side of the vessel, is evidently, as the depth EF of 
 
NATURAL PHILOSOPHY. 
 
 the liquid film multiplied into Ee the breadth of its 
 base ; and therefore this base will sustain a vertical 
 pressure denoted by epxEF, or the weight itself of 
 the elementary column EF/e, and will also resist a 
 horizontal thrust expressed by E<pxEF. But, for 
 the same reason, the other side of the vessel at G 
 supports the load of the elementary column GHhg, 
 and pushes horizontally with the force GHxGy, 
 which is equal to the opposite thrust EF X E<p. 
 These horizontal forces are hence mutually balanced 
 or extinguished ; while the sides of the vessel sustain 
 the whole weight of the series of vertical columns, 
 from C to AI, and from D to BK. The same pro- 
 perty belongs to every parallel section, and, conse- 
 quently, the bottom of the vessel supports only its 
 incumbent prism or cylinder IABK, while the rest 
 of the surrounding liquid is upheld wholly by the re- 
 action of the sides. 
 
 The pressure exerted by a fluid against the bottom 
 of a vessel is thus nowise augmented by its spreading 
 shape ; nor shall we find it at all diminished, on the 
 other hand, by any degree of contraction. Let CABD 
 (fig. 138.) represent such a tapered form of vessel. 
 Assuming a vertical section as before, the perpendi- 
 cular thrust against the element Ee of the side must 
 be compounded of FE, the depth, under the surface 
 of the fluid, and the base Ee. But this force may be 
 decomposed into a vertical and a horizontal pressure, 
 in the ratio of Ee to e(p and E<p, or as FE X Ee to 
 
ELEMENTS OF 
 
 FE X e<p and FE X E<p. The space Ee is therefore 
 pushed outwards horizontally by a force denoted by 
 FE X E<p, and directly upwards by the force FE x e<p, 
 or the weight of an exterior column FE<?/*. In the 
 same manner, it is shown that the pressure exerted 
 against the element Gg of the opposite side of the 
 vessel may be resolved into a horizontal thrust 
 GH X Gy, and a thrust GH X gy from below, in 
 the direction GH, which is equal therefore to the 
 weight of the exterior column GH%. But those ho- 
 rizontal thrusts EF X E<p and GH X Gy, being evi- 
 dently equal and opposite, are mutually extinguished. 
 The vertical thrusts EF X e<p and GH X gy from 
 below are resisted by the elasticity of the elementary 
 spaces Ee and Gy of the vessel ; or, what is the same 
 thing, those spaces press downwards the columns LEe/ 
 and MG^m, by forces equal to the weights of the 
 exterior columns EF/e and GH//^. The conjoined 
 pressures exerted at U and Mm are hence repre- 
 sented by the columns LF/7 and MHkm. The pres- 
 sure of the whole section upon AB is, therefore, the 
 same as that of the rectangle IABK. Collecting 
 then all the sections, their total pressure upon the 
 bottom will be equal to the weight of an uniform 
 prismatic or cylindrical column IABK. 
 
 This very singular and important conclusion, that 
 the pressure of a liquid upon the bottom of any ves- 
 sel depends merely on its altitude and the surface of 
 its base, might be derived from simpler but indirect 
 
NATURAL PHILOSOPHY. 
 
 considerations. Suppose the portion of the liquid, 
 which in fig. 13y. encircles the vertical cylinder 
 IABK, to be frozen or converted into a solid sub- 
 stance, but without changing its density ; the pres- 
 sure of this congealed mass would evidently continue 
 the same as before, and therefore the bottom has only 
 to sustain the weight of the cylindrical column which 
 rests immediately upon it. If the lateral ice were 
 again melted, it would only resume its horizontal 
 thrust, which could in no degree alter the vertical 
 pressure of the liquid. 
 
 In like manner, if IABK in fig. 138. were a cy- 
 linder of liquid resting upon a circular base ; con- 
 ceive the portions I AC and KBD about the sides 
 to be rendered solid or congealed without alteration 
 of density, and the remaining portion ACDB will 
 evidently exert the same pressure as before. The 
 incurved sides AC and BD of this tapering vessel 
 hence produce an effect analogous to the load of the 
 supposed frozen masses I AC and KBD. 
 
 The property now demonstrated is commonly 
 termed the Hydrostatic Paradox. The smallest 
 portion of a liquid is thus capable of producing a 
 pressure equal to that of the largest mass. The 
 fact is confirmed and elucidated by a variety of strik- 
 ing experiments. In all water-works, it forms an 
 essential consideration. But the principle extends 
 its influence likewise to remoter objects. No ma- 
 sonry is safe which neglects it. If the smallest 
 
302 ELEMENTS OF 
 
 quantity of water should lodge to a considerable 
 height in the gravel, sand, or loose earth close be- 
 hind a wall or embankment, it would exert a lateral 
 pressure sufficient to push the solid materials from 
 their base. Hence, a sudden shower often occasions 
 great devastation. The thinnest vein of water, col- 
 lected in a perpendicular crevice, will split the hard- 
 est rock, and hurl its fragments down the precipice. 
 This hydrostatic pressure is the chief agent which 
 nature employs, in the silent and gradual demolition 
 of mountains. 
 
 Suppose any interior portion of a liquid to become 
 solid ; it would evidently remain in the same state 
 of indifference or equilibrium as before. It must 
 therefore be borne up by the vertical pressure of the 
 fluid with a force just equal to its weight, or the 
 weight of the liquid whose place it occupies. Con- 
 ceive this congealed mass to have its gravity aug- 
 mented or diminished ; it will be pulled downwards 
 or upwards by the difference between this force and 
 the weight of an equal bulk of the liquid. Substi- 
 tute any solid body instead of this block of ice, and 
 ike loss of weight by the immersion it sustains will 
 be equal to that of the volume of fluid which it dis- 
 places. 
 
 But this fundamental property, first detected by 
 the genius of Archimedes, may be demonstrated by 
 a stricter process of reasoning. Let a cylindrical 
 body FEGH (fig. 139.) be plunged vertically in a 
 
NATURAL PHILOSOPHY. 303 
 
 vessel C ABD, filled with water or any other liquid ; 
 the lateral pressure, acting equally around the axis, 
 will evidently produce a complete balance of efforts. 
 But the cylinder will be pushed upwards by a co- 
 lumn of fluid having IE for its altitude, and the cir- 
 cle EG for its base, and pressed downwards again by 
 the circular column IFHK. The solid is, therefore, 
 buoyed up by the excess of the former force above 
 the latter, or by the weight of an equal cylinder 
 FEGH of the liquid. 
 
 Suppose a solid of any shape were immersed in a 
 liquid, and let fig. 140. represent a vertical section. 
 The pressure exerted perpendicular to the element 
 of the surface E<? is resolved into a horizontal and a 
 vertical force, and these three forces are as the lines Ee, 
 e<p and E<p, or as the rectangles GE x E, GE x e$ 9 
 and GE x E<p. Wherefore, the horizontal impres- 
 sion at Ee is represented by GE x e<p. Draw the 
 horizontal- lines EH and eh, and the corresponding 
 verticals HI and hi. It is evident that the portion 
 of the perpendicular pressure against H7* acting in 
 the horizontal direction HE will be denoted by 
 IH x hi, which is equal to the opposite exertion 
 GE x etp. In every horizontal space EHAe, there- 
 fore, the lateral efforts are balanced, and hence the 
 whole section has no tendency towards either side. 
 This must be the case likewise in all the collective 
 sections which compose the submerged solid, which 
 has no disposition to move to the one side of the 
 vessel or to the other. But the section is pushed 
 
304 ELEMENTS OF 
 
 directly upwards at Ee by a force represented by 
 GE X Ee, or the elementary film of liquid 
 while it is pressed downwards only by the film 
 It is consequently buoyed up by their difference, or 
 by the film FEef, and the buoyant effort of the ag- 
 gregate parcels, therefore, is equal to the whole sec- 
 tion. The collective sections, again, form the solid 
 body, which hence loses, by immersion, just as much 
 weight as that of an equal volume of the surrounding 
 fluid. 
 
 On this principle is founded the method of ascer- 
 taining the density of a body, or the relation of its 
 weight to its bulk, which, in reference to some com- 
 mon standard, is termed its specific gravity. Wa- 
 ter, at its state of greatest contraction, is preferred 
 as the most convenient unit of comparison, the den- 
 sity of other bodies being reckoned in decimal parts. 
 The Hydrostatic Balance is generally used for this 
 purpose. The substance to be examined may be 
 either liquid or solid. The specific gravity of li- 
 quids is easily determined, by means of a ball or pear- 
 shaped lump of glass or crystal, either partly hollow 
 or loaded. This poise, suspended by a hair or fine 
 thread, is weighed in vacuo or air, then in pure wa- 
 ter, and next in the particular liquid ; the loss of 
 weight which it suffers in water is to its loss in the 
 fluid under trial, as unit to the specific gravity of this 
 fluid. The calculation is sometimes carried to five 
 places of decimals ; though it is seldom safe or expe- 
 dient to push them beyond three figures. If the 
 
NATURAL PHILOSOPHY. 3()5 
 
 lump of glass were ground to such a size as to lose 
 exactly a thousand or ten thousand grains in distilled 
 water, no computation would be required, its loss of 
 weight in the liquid indicating at once the specific 
 gravity. 
 
 In the case of solid substances, they may be either 
 denser than water or rarer, and they may be inso- 
 luble in it or capable of solution. The mode of 
 determining their specific gravity will accordingly be 
 different. 
 
 1. Insoluble solid bodies, denser than water, are 
 weighed in vacua, and then in distilled water ; and 
 the loss of weight, which they suffer by immersion, 
 is to their whole weight, as unit is to their specific 
 gravity* 
 
 2. Insoluble solid bodies, lighter than water, re- 
 quire to be joined to some heavier substance, in or- 
 der to make them sink in that liquid. The weight 
 of the ballast, when immersed alone, must be pre- 
 viously ascertained. This weight, diminished by the 
 weight of the compound after immersion, will, there- 
 fore, give the buoyant power, or the weight of a mass 
 of water equal to the bulk of the rare substance. The 
 weight thus corrected is hence to the weight of the 
 substance in vacua, as unit is to its specific gravity. 
 
 3. When the solid substances to be examined are 
 saline, or liable to solution in water, they may be gent- 
 ly heated and covered with a thin coat of melted bees- 
 wax. Thus defended, they may now be plunged 
 without any risk in distilled water. A slight allow- 
 ance should be made for the buoyant influence of 
 
 VOL. i. u 
 
306 ELEMENTS OF 
 
 the film of wax itself, which, however, must be very 
 minute, since wax has very nearly the density of 
 water itself. This mode of finding specific gravity 
 is evidently very imperfect, and quite inapplicable to 
 substances that have a powdery form. The instru- 
 ment which I term a Goniometer, founded upon the 
 dilatation of air, may supply the defect. 
 
 A solid substance, rarer than the fluid medium, 
 must evidently sink, till it displace an equal weight 
 of the fluid. The submerged part of the solid 
 hence always marks the volume of this equiponde- 
 rant mass. If the floating body have a globular 
 shape, terminated by a long slender stem, its de- 
 pression in any liquid will measure the smallest dif- 
 ferences of specific gravity. The stem may be made 
 exactly cylindrical, for instance, and divided into 
 portions which correspond to the thousandth parts 
 of the bulk of the ball. Such is the general con- 
 struction of the Hydrometer, a very convenient in- 
 strument for examining readily the densities of dif- 
 ferent liquids. The stem will scarcely bear more 
 than an hundred distinct subdivisions ; but the range 
 can be easily enlarged, by attaching, as circumstances 
 may require, loads answering to 100, 200, 300, &c. 
 
 One of the easiest and simplest methods of deter- 
 mining the densities of different liquids, is by a set 
 of small glass beads, previously adjusted, and nume- 
 rically marked. Thrown into any liquid, the heavier 
 balls sink, till they approach the required density 
 and become gradually buoyant, and the one which 
 first rises to the surface indicates, in thousandth 
 
NATURAL PHILOSOPHY. J07 
 
 parts, the specific gravity of the fluid. These balls 
 are adapted for examining liquids, whether lighter 
 or heavier than water. 
 
 But the most accurate and concise mode of ascer- 
 taining the density of liquids, is to employ a small 
 glass measure with a very short narrow neck, and 
 adjusted to hold exactly a thousand grains of 
 distilled water. The vessel being filled with any 
 other liquid, the weight of it is observed, and 
 thence its relative density to water may be found, 
 by merely striking off three decimal places. At 
 each operation, the glass must be carefully rinsed 
 with pure water, and again dried, by heating it, and 
 then sucking out the humified air, for a few minutes, 
 by help of a slender inserted tube. 
 
 If fluids of various densities, and not disposed to 
 unite in any chemical affinity, be poured into a ves- 
 sel, they will arrange themselves in horizontal strata, 
 according to their respective densities, the heavier 
 always occupying a lower place. This stratified ar- 
 rangement of the several fluids will succeed, even 
 though a mutual attraction should subsist, provided 
 only that its operation be feeble and slow. Thus, a 
 body of quicksilver may occupy the bottom of a glass 
 vessel, above it a layer of concentrated sulphuric 
 acid, next this a layer of pure water, and then an- 
 other layer of alcohol. The sulphuric acid would 
 scarcely act at all upon the mercury, and a consider- 
 able time would elapse before the water sensibly pe- 
 netrated the acid, or the alcohol the water. Bodies 
 of different densities might remahrsuspended in those 
 
308 ELEMENTS OF 
 
 strata. Thus, while a ball of platinum would lie at 
 the bottom of the quicksilver, an iron ball would 
 float on its surface ; but a ball of brick would be 
 lifted up to the acid, and a ball of beech would swim 
 in the water, and another of cork might rest on the 
 top of the alcohol. 
 
 Hence the reason of various natural phenomena. 
 Thus, the fresh water discharged by a river into the 
 sea continues to float upon the surface of the denser 
 mass of salt water, till the several strata become in- 
 termixed by the action of winds, or the commotion 
 of tides and currents. For the same reason, the 
 brackish water collected near the mouth of a great 
 river must rise proportionally above the general le- 
 vel. Hence, likewise, certain wells and springs near 
 the coast are observed to swell and sink regularly 
 with the flow and ebb of the tide, from the mere varia- 
 tion of hydrostatic pressure of the adjacent sea acting 
 on the veins of fresh water. Fountains occur some- 
 times on the very summits of detached rocks and islets. 
 The same principle appears to furnish the simplest 
 and most probable explication of the gradual subsi- 
 dence of the Baltic Sea, which has not at present the 
 fifth part of the saltness of the Ocean. It must there- 
 fore stand at a higher level in proportion to its depth ; 
 but if we suppose its waters to have worked into 
 certain submarine beds of salt, they must contract 
 and sink down, as they dissolve it and become den- 
 ser. To verify this hypothesis, it is only requisite 
 to ascertain whether the Baltic be actually growing 
 salter. 
 
NATURAL PHILOSOPHY. 
 
 309 
 
 It may be convenient here to state, merely in 
 round numbers, the specific gravities of the more 
 remarkable substances. 
 
 
 Platinum, purified, 19.50 
 
 . hammered, 20. 34 
 
 laminated, 22.07 
 
 drawn into wire, 21.04 
 
 Gold, pure and cast, 19.26 
 
 hammered, 19.36 
 
 Mercury, 13-57 
 
 Lead, cast, 11.35 
 
 Silver, pure and cast, 10.47 
 
 hammered, 10.51 
 
 Bismuth, cast, 9-82 
 
 Copper, cast 8.79 
 
 wire, 889 
 
 Brass, cast, 8.40 
 
 wire 8.54 
 
 Cobalt and Nickel, cast, 7.81 
 
 Iron, cast, 7.21 
 
 malleable, 7.79 
 
 Steel, Soft, 7.83 
 
 hammered, 7.84 
 
 Tin, cast, 7.30 
 
 Zinc, cast, 7.20 
 
 Antimony, cast, 4.95 
 
 Molybdaenum, 4.74 
 
 Sulphate of barytes, 4,43 
 
 Zircon of Ceylon, 4.41 
 
 Oriental ruby, 4.28 
 
 Brazilian ruby, 3.53 
 
 Bohemian garnet, 4.19 
 
 Oriental topaz, 4.01 
 
 Diamond, 3.50 
 
 Crude manganese, 3.53 
 
 Flint glass, 2.89 
 
 Glass of St Gobin, 2.49 
 
 Fluorspar, 3.18 
 
 Parian Marble, 2.34 
 
 Peruvian emerald, 2.78 
 
 Jasper, 2.70 
 
 Carbonate of lime, 2.71 
 
 Rock crystal, 2.65 
 
 Flint, 2.59 
 
 Sulphate of lime, 2.32 
 
 Sulphate of soda, 2.20 
 
 Common salt, 2.13 
 
 Native sulphur, 2.03 
 
 Nitre, 2.00 
 
 Alabaster, 1.87 
 
 Phosphorus, 1.77 
 
 Plumbago, 86 
 
 Alum, 72 
 
 Asphaltum, 40 
 
 Jet, 24 
 
 Coal, from 1.24 to .30 
 
 Sulphuric acid, 84 
 
 Nitric acid, 22 
 
 Muriatic acid, 19 
 
 Equal parts by weight of water and 
 
 alcohol, 93 
 
 Ice, 92 
 
 Strong alcohol, 82 
 
 Sulphuric aether, 74 
 
 Naphtha, 71 
 
 Sea water, 1.03 
 
 Oil of Sassafras, 1.09 
 
 Linseed oil, 94 
 
 Olive oil, 91 
 
 White sugar, .. : 1.61 
 
 Gum Arabic and honey, ....... 1.45 
 
 Pitch 11.5 
 
 Isinglass, 1.11 
 
 Yellow amber, 1.08 
 
 Hen's egg, fresh laid, 1.09 
 
 Human blood 1.05 
 
 Camphor, 99 
 
 White wax, 97 
 
 Tallow, 94 
 
 Pearl, 2.75 
 
 Sheep's bone, 2.22 
 
 Ivory, 92 
 
 Ox's horn, 84 
 
 Lignum vitae, 33 
 
 Ebony, 18 
 
 Mahogany, 06 
 
 Dryoak, 93 
 
 Beech, 85 
 
 Ash, 84 
 
 Elm, from 80 to .60 
 
 Fir, from 57 to .50 
 
 Poplar, 38 
 
 Cork, 24 
 
 Chlorine, 00302 
 
 Carbonic acid gas, 00164 
 
 Oxygen gas, 09134 
 
 Atmospheric air, 00121 
 
 Azotic gas, 00098 
 
 Hydrogen gas, 00008 
 
310 ELEMENTS OF 
 
 One hundred cubic inches of chlorine, carbonic- 
 acid gas, oxygen gas, atmospheric air, azotic gas, and 
 hydrogen gas, weigh respectively 76.2, 46.6, 33.9, 
 30.5, 29.6, and 2.1 grains Troy. One cubic inch 
 of distilled water, at the ordinary temperature of 62 
 degrees Fahrenheit, weighs in air 252.5 grains. 
 Hence the weight of a cylindrical inch of water 
 is 198.3 grains ; but the same measure of quicksil- 
 ver would weigh 2691 grains, which affords a ready 
 method for ascertaining the diameters of very narrow 
 or capillary glass tubes. 
 
 The averdupois pound is now enacted to contain 
 7000 grains Troy, and must hence be equal to the 
 weight of 27,724- cubic inches of water. Wherefore, 
 a cubic foot of water weighs 62.331 pounds, or al- 
 most exactly one thousand ounces averdupois. The 
 weight of an Imperial gallon is fixed at ten such 
 pounds. A cubic foot of water seems to have an- 
 ciently corresponded to the weight of a bushel or fir- 
 lot (fourthlet) of wheat, four of which make a boll, 
 eight times this a ton, and the double of this again 
 a chalder. The ton is therefore 2000 lb., or more 
 generally reckoned twenty hundred weight, each 
 hundred consisting of 112 lb. 
 
 The weight of a cable of 720 feet, or 120 fathoms 
 length, worked up from strands 180 fathoms long, 
 is found, by multiplying the square of the girth or 
 circumference in inches by 19^- lb. The square of 
 half the diameter will hence nearly express in pounds 
 the weight of a foot of cord. A chaldron of coal at 
 Newcastle is reckoned equal to 53 cwts., or 5936 lb. ; 
 
NATURAL PHILOSOPHY. 311 
 
 but in London the chaldron contains 36 bushels, 
 each weighing 84 Ib. 
 
 A hollow glass cylinder, open at top, and fitted 
 below with a loaded plate of brass, being immersed 
 in a vessel of water, till the quantity of liquid thus 
 displaced shall have a weight equal to that of the 
 compound, the plate will not only float, but sustain 
 the incumbent glass. If a solid cylinder were push- 
 ed vertically into the water, it would press down the 
 vessel by a load equal to the weight of an equal bulk 
 of the fluid, which is readily indicated by suspend- 
 ing the vessel from one of the ends of a balance. 
 
 A glass tube of considerable length being laid 
 horizontally, with both ends bent upwards, and now 
 filled with any liquid, such as water or quicksilver, 
 the opposite terminating surfaces must always keep 
 the same level. If small sights with cross wires were 
 hence set to float in the vertical tubes, this combina- 
 tion might serve as a levelling instrument. Such a con- 
 struction has accordingly been sometimes employed, 
 though it seems not susceptible of much accuracy. 
 
 The Spirit Level is far more delicate. This va- 
 luable instrument consists of a short glass tube having 
 a very slight curvature, and almost filled with alco- 
 hol, leaving but a small bubble of air. Alcohol is 
 preferred to every other liquid, because it moves rea- 
 dily, and is .not liable to freeze. The horizontal 
 line is indicated by a tangent to the upper and con- 
 vex side of the tube, at the middle of the floating 
 bubble. 
 
ELEMENTS OF 
 
 If the two vertical branches of a tall recurved tube 
 be filled with liquids of different densities, the oppo- 
 site surfaces will no longer maintain the same eleva- 
 tion. The lighter fluid will rise proportionally high- 
 er, so that their altitudes will be inversely as their 
 respective densities. Such a form of tube might 
 therefore serve as an imperfect Hydrometer. 
 
 Since the pressure of any fluid is proportional 
 merely to the depth below the surface, the strain 
 borne by a sluice or the sides of a canal must increase 
 uniformly from the top to the bottom. The centre 
 of pressure is hence not in the middle, but at one- 
 third of the entire altitude. To this point, there- 
 fore, if more strength be wanted, the additional prop 
 should be applied. 
 
 If water be confined in a canal or basin by a wall 
 or embankment, the thickness of the dike must in- 
 crease regularly in proportion to its depth. The ad- 
 hesion of any materials or their resistance to a hori- 
 zontal thrust, may be estimated as a certain propor- 
 tion of their weight, commonly the half or the third 
 part. Let AB (fig. 141.) be the height of the wall, 
 and make its density to three times that of water, as 
 AB to BC, and join AC, which will represent the 
 proper slope. If the dike be composed of stones or 
 bricks, the base BC must be at least equal to the al- 
 titude AB ; but if it consist of earth, BC should be 
 one half more. When the embankment is formed 
 of earth, its side must not be perpendicular ; it should 
 form an inclined plane, not exceeding 35 degrees, 
 
NATURAL PHILOSOPHY. 315 
 
 or the angle of repose, lest the softened parts should 
 slide down ; the outside, being more solid, may be 
 steeper. 
 
 In canal navigation, a boat is raised from a lower 
 to a higher level by a series of locks, a portion of 
 the water somewhat exceeding the length of the ves- 
 sel being inclosed at the sides by walls, and at both 
 ends by masonry, and opposite flood-gates. As soon 
 as it passes these gates, they are shut behind it ; 
 and a small lateral or superior sluice being opened, 
 the water rushes into the inclosure, and quickly 
 mounts to the higher level, thus enabling the vessel 
 again to proceed. A similar operation is perform- 
 ed at each successive lock. In descending the ca- 
 nal, the procedure is exactly reversed, the water 
 contained in the series of inclosures being allowed 
 to flow out, and thus lower by degrees the level of 
 the boat. 
 
 The flood-gates are contrived to shut at a cer- 
 tain angle. If this angle be very acute, they sus- 
 tain too great a pressure, and yet close feebly. Let 
 AC and BC (fig. 142.) represent the flood-gates of 
 a canal, which are opened by help of the extended 
 arms AE and BF. When shut, the gate AC is 
 pressed at right angles by the water, with a force as 
 AC itself, which, from the principle of the lever, 
 must exert a perpendicular effort at the end C, as 
 the square of AC. The thrust thence produced in 
 the direction AB will be as AC x CD, and will en- 
 counter an equal and opposite thrust from the gate 
 BC. These two forces constitute the power which 
 
314 ELEMENTS OF 
 
 closes the gates. The force with which they are 
 made to cohere, thus increasing with AC and CD, 
 must augment rapidly when the angles BAC and 
 ABC are enlarged, or their mutual inclination ACB 
 becomes diminished. 
 
 If the angle ACB were very obtuse, those con- 
 joined gates would, like a low roof, occasion a great 
 thrust against the walls of the canal, or the centres 
 of the gates at A and B. The thurst in the direc- 
 
 tion CA might be shown to be rr . . It will 
 
 hence be easy to determine the angle ACB, with 
 the centres of the flood-gates which suffer the small- 
 
 AC 2 
 
 est strain ; for AD being constant, TTFT must be a 
 
 minimum. But this quantity is evidently the dia- 
 meter of a circle circumscribing the triangle ACB ; 
 and since the least circle is that described about the 
 point C, the angle ACB, at which the gates lock, 
 should be a right angle. 
 
 Water may be contained in circular basins or cy- 
 lindrical vessels, and conveyed in pipes, which there- 
 fore bear a lateral pressure proportioned to the alti- 
 tude of the column. But this pressure will, in con- 
 sequence of the curvature, occasion likewise a longi- 
 tudinal strain or distension. Let fig. 143. repre- 
 sent a section of the tube ; and assume, in the in- 
 terior circumference, the proximate points B and C, 
 equally distant on either side from A. But three 
 forces may be conceived to act at these points ; the 
 perpendicular thrust of the water against A, and 
 
NATURAL PHILOSOPHY. 315 
 
 longitudinal tensions at B and at C. These forces 
 being proportional to the sides of a triangle perpen- 
 dicular to their several directions, are consequently 
 as the lines BC, BO and CO. But the pressure 
 exerted by the fluid at A, is as the little space BC ; 
 wherefore the tension of the ring at the points A, 
 B, C must be expressed by the radius BO. The 
 same distending energy is hence exerted around the 
 whole internal circumference. The lateral pressure 
 of the water against each ring of the cylinder, must 
 thus produce the same effect, as a longitudinal force 
 applied to it, equal to the weight of a prism of the 
 fluid, resting on a base which has the breadth of 
 that ring, with the radius for its thickness. The 
 strength of the cylinder must consequently be in 
 the compound ratio of the altitude of the water, and 
 of its own radius or diameter. Similar pipes will 
 therefore bear equal pressures, their thickness being 
 proportioned to the diameter. Thus, a pipe of only 
 a foot in diameter, and half an inch thick, will with- 
 stand the thrust of the same altitude of water, as a 
 pipe of the same materials two feet wide and a whole 
 inch in thickness. But since lead has only the tenth 
 part of the tenacity of cast-iron, a pipe of that soft 
 metal will require, in similar circumstances, to have 
 ten times greater thickness in proportion to its dia- 
 meter, than one of cast-iron. Such is the case with 
 a pipe of elm, while one of free-stone would still 
 need to have its thickness doubled. 
 
 Let h denote the height of the water in inches, 
 d the diameter of any pipe, t its thickness, and 
 
316 ELEMENTS OF 
 
 c the longitudinal cohesion of a square inch bar of 
 its material expressed in pounds averdupois. The 
 longitudinal strain occasioned by the internal pres- 
 sure against each ring of an inch breadth, is in cubic 
 
 hd , . , hd 1 2hd 
 
 inches-, which corresponds to . = - 
 
 in pounds averdupois. Wherefore, jry^ cf y 
 
 h = - . -7 ; or, if H express the altitude in feet, 
 cl 
 
 37 c t 
 
 H = . T-. Substituting the observed cohesion 
 o d 
 
 f . ' A IT 37.19096 t t 
 
 of cast-iron, and H = -7-= 88319 .7-. 
 
 o a a 
 
 But this result must exceed the actual strength 
 of such pipes. In the first place, they will evident- 
 ly give way, as soon as the elasticity becomes im- 
 paired, and the smallest fissures begin to open. This 
 happens in malleable iron, when only half its whole 
 cohesion is exerted, and probably occurs sooner in 
 cast-iron. But, in the next place, the longitudinal 
 tension occasioned by the perpendicular pressure of 
 the water being proportional to the radius of cur- 
 vature, must increase, with the thickness of the pipe, 
 from the inner to the outer surface. The particles 
 are hence not pulled by equal and parallel forces, as 
 in the case of a vertical bar of iron ; but must be 
 distrained by an oblique action, which will accele- 
 rate their separation. We may therefore estimate 
 the actual cohesion as reduced nearly one-half^ which 
 
NATURAX PHILOSOPHY. 317 
 
 gives in round numbers H = 50,000 .^-, or it will be 
 
 safer in practice to reckon the co-efficient 30,000, or 
 even 25,000 feet. Thus, for a cast-iron pipe of the very 
 best quality a foot in diameter, and 3-4ths of an inch 
 
 thick ; H = 30000 . ^ = 1500 feet. Again, let 
 a pipe of the same diameter have a thickness of an 
 inch and quarter ; and H = 30,000 . ~ =2500 feet. 
 
 Such are the dimensions of the cast-iron pipes now 
 laid, for conveying water from the Pentland Hills 
 to Edinburgh ; and such was the strength of some of 
 them which had been formed carefully of the best ma- 
 terials, ascertained by means of a large forcing pump, 
 it being sufficient, however, that the pipes should 
 withstand the pressure of a column of water respec- 
 tively of the altitudes of 400 and 800 feet. 
 
 Following the same analogy, the former leaden 
 pipes laid at Comiston, which were but four inches 
 and a half wide and three-fifths of an inch thick, 
 could have sustained only the thrust of 400 feet in 
 
 height of water : For 3000 . -= 3000. = 400. 
 
 4 4 o 
 
 Drawn leaden pipes are, on the principle of case- 
 hardening, considerably stronger. Thus, Mr Jardine 
 has found that such a pipe 2 inches wide, and 2-7ths 
 of an inch thick, bears a pressure equal to that of a 
 vertical column of water of 600 feet ; but stretched 
 about the three hundredth part of an inch under that 
 of 650, and finally burst after the columnar pressure 
 
318 ELEMENTS OF 
 
 had been gradually raised to 1200 feet. But the for- 
 mula for cast lead would here give only 428 feet. 
 
 According to Tredgold, the lateral cohesion of 
 oak is 2316 Ib. for each square inch. Hence a pipe 
 formed of this tough wood, 15 inches in diameter 
 and 2 inches thick, would sustain a column of water 
 
 of the height of only - . 1500 . = ~ . 100, or 
 
 <j L t.) 4* 
 
 875 feet. It would have required twice that thick- 
 ness to enable a pipe of larch to bear the same strain. 
 Elm, which is more commonly used for this purpose, 
 approaches to the strength of oak. 
 
 The cohesion of Portland stone being only 857 lb., 
 a pipe formed of that material, and one foot in dia- 
 meter and an inch and half thick, would be sufficient 
 to bear a pressure of water, of no greater altitude 
 
 than ?? . 500 . *| =|? . 500, or 288 feet. Those 
 
 elegant pipes lately cut out of a block of free stone, 
 in a series of cores, by the application of a circular 
 saw, have hence failed, from their weakness and their 
 disposition to chip and crack. 
 
 The same principles regulate the strength of a cir- 
 cular basin confining water. The perpendicular 
 pressure against the wall depends indeed merely on 
 the altitude of the fluid, without being affected by 
 its volume ; but the longitudinal effort of the thrust, 
 or its tendency to open the joints of the masonry, is 
 measured by the radius of the circle. To resist that 
 action, in very wide basins, the range or course of 
 
NATURAL PHILOSOPHY. SIQ 
 
 s, along the inside of the wall, must be propor- 
 tionally thicker. On the other hand, if any oppo- 
 sing surface present some convexity to the pressure 
 of water, the resulting longitudinal strain will now 
 be exerted in closing the joints and consolidating the 
 building. Such reversed incurvation is hence gene- 
 rally adopted in the construction of dams, the bend 
 inwards of the arc being about the eightieth part of 
 the length of its chord, while the exterior boundary 
 is made rectilineal. 
 
 If a solid body, not subject to solution, but 
 placed in a liquid denser than itself, it will sink 
 till the pressure which it encounters from below 
 becomes sufficient to support its entire load, which 
 is then just equal to the weight of the portion of 
 fluid displaced. While the body floats, therefore, 
 it is at the same time drawn downwards, and push- 
 ed directly upwards, by the action of two equal and 
 opposite forces. The incumbent weight may be con- 
 sidered as collected in its centre of Gravity ', and the 
 sustaining efforts as united in the centre of Buoy- 
 ancy, which is evidently the same as the centre of 
 gravity of the water displaced, or of the immersed 
 portion of an uniform solid. To these two points, 
 therefore, the antagonist forces are directed ; and 
 the line which joins them, called the Line of Sup- 
 port, will have constantly a vertical position, in the 
 case of equilibrium. 
 
 The centre of gravity of the whole mass, about 
 which it turns in the water, must evidently continue 
 
320 ELEMENTS OF 
 
 invariable ; but the centre of buoyancy will change 
 its relative place, according to the situation of the 
 immersed portion of the solid. If those two centres 
 should coincide, the body will float indifferently in 
 any position. It will likewise float, as often as a verti- 
 cal line, drawn from the centre of buoyancy, shall 
 pass through the centre of gravity. But this will 
 obtain whenever the line of support becomes perpen- 
 dicular to the horizon. The equilibrium, however, 
 may be either permanent or instable. It is per- 
 manent, if, on pulling the body a little aside, it has 
 a tendency to redress itself, or to recover its original 
 position ; it is instable^ when the body, on being 
 slightly inclined, tumbles over in the liquid, and as- 
 sumes a new situation. These 'opposite conditions 
 will occur in a body of irregular form, when the 
 centre of gravity occupies the highest or the lowest 
 possible position ; for, though the volume of im- 
 mersion remains the same, the solid will evidently 
 be less or more depressed in the fluid medium, ac- 
 cording to the width of its section or water-line. 
 
 If the centre of buoyancy stand higher than the 
 centre of gravity, the floating body will, in every 
 declination, maintain its stability, and regain its per- 
 pendicular position j for, though made to lean to- 
 wards either side, the vertical pressure exerted 
 against that variable point, will soon bring it back 
 again into the line of support. But the elevation of 
 the centre of buoyancy above that of gravity, is by 
 no means an essential requisite to the stability of 
 flotation ; on the contrary, it falls in most cases con- 
 
NATURAL PHILOSOPHY. 
 
 siderably below the centre of gravity about which 
 the body rolls. The buoyant efforts may be con* 
 sidered as acting upon any point in the vertical line, 
 and, consequently, as united in the point where this 
 line crosses the axis of the floating body. If the 
 point of concourse thus assigned should stand above 
 the centre of gravity, the body will float firmly, and 
 will right itself after any small detrusion ; if it coin- 
 cide with the centre of gravity of a homogeneous 
 body, this will continue indifferent with regard to po- 
 sition ; but if the vertical should meet the axis below 
 the centre of gravity, the body will be pushed for- 
 wards, its declination always increasing till it finally 
 oversets. 
 
 To begin with simple and regular bodies : Let a 
 wooden sphere, of uniform consistence, be set to 
 float in water. It will sink, till the weight of the fluid 
 displaced by the immersed portion HFI (fig. 144.) 
 shall be equal to its own load. The centre of gra- 
 vity of this body is obviously C, the centre itself of 
 the sphere. But the centre of buoyancy B must be 
 the centre of gravity of the volume of immersion 
 HFI, and will therefore lie below C, in the axis KCF 
 perpendicular to the water-line or plane of flotation 
 HI. The ball is hence pressed down by its own 
 weight collected at C, and pushed up, in the opposite 
 direction, by an equal force combined at B, both of 
 the forces, however, concurring in the same point C. 
 Wherefore, being always held in equilibrium by those 
 
 VOL. r. x 
 
ELEMENTS OF 
 
 antagonist forces, it will remain still in any position 
 which it happens to occupy. But this indifference 
 to floating will obtain only, when the sphere is per- 
 fectly homogeneous, and its centre of gravity coin- 
 cides with the centre of magnitude ; for otherwise 
 the former, descending as low as possible, would al- 
 ways assume a determinate position. 
 
 Suppose next a mere segment of the sphere to 
 float in water. The centre of gravity G (fig. 145.) 
 of this homogeneous body, will now lie below the cen- 
 tre of the sphere, and in the axis AF at right angles 
 to its base DE ; but B, the centre of buoyancy, or 
 the centre of gravity of the immersed segment HFI, 
 must, in every situation of the floating mass, occur 
 in a perpendicular bisecting the water-line HI, and 
 consequently passing through the centre of the 
 sphere. In the case of equilibrium, this perpendi- 
 cular must have a vertical position, or the inverted 
 base of the segment must form a horizontal plane. 
 Suppose the body to be drawn aside into the posi- 
 tion D'F'E' ; it will then be pressed down by its 
 own weight collected at G, and pushed upwards by 
 an equal buoyant power exerted at B, in the oppo- 
 site direction BC. But this force may be conceived 
 to act upon any point in the line BC, and therefore 
 at the centre C, the concourse of the two axes CF 
 and CF'. The buoyancy transmitted to C hence 
 pushes the axis CF' obliquely, the greater part of it 
 heaving the point C in the direction FC, and an- 
 
NATURAL PHILOSOPHY. 
 
 other small part pressing C perpendicular to CF' 
 or CG', and making the body turn about its centre 
 of gravity G, from E' towards E. Every derange- 
 ment is thus corrected by a restoring energy, which 
 maintains a permanent equilibrium. 
 
 Suppose an oblate homogeneous spheroid (fig. 
 146.) were now substituted for the entire sphere. 
 Since its transverse sections are proportional to the 
 corresponding sections of a sphere described about 
 the shorter axis, it will sink to the same depth AF 
 as before, and carry the centre of buoyancy B into 
 a like position. The declination of the axis from 
 CF to CF', occasions that point to shift its relative 
 place from B to B'. The change, however, is pro- 
 duced, by the wedge EAE' being joined to the one 
 side of the immersed segment, and the wedge DAD' 
 taken away from the other side. Wherefore the 
 bulk of the half segment AFE is to that of the ac- 
 crescent wedge EAE', as AO, the distance of the 
 centre of gravity of the former, to BB', the lateral 
 evagation of the centre of buoyancy. That propor- 
 tion between the segment and its wedge is the same, 
 both in the sphere and the spheroid ; but the distance 
 AO, being increased by the oblateness of the figure, 
 the variation BB' is augmented in the like ratio. 
 Now the triangles BcB' and EAE' tfeing similar, it 
 follows that Be must be greater than BC, and that 
 the effort of buoyancy is exerted at a point above 
 the centre of gravity of the spheroid. This verti- 
 
324 ELEMENTS OF 
 
 cal thrust tends consequently to redress the floating 
 body, and to secure its stable equilibrium. 
 
 On the other hand, let a prolate spheroid (fig. 147.) 
 be immersed in water or any other liquid. Its plane 
 of flotation, and its centre of buoyancy will each 
 have the same height, as in a sphere described on 
 the longer axis. But the shifting of that point from 
 B to B' will be diminished in proportion to the nar- 
 rowness of the spheroid. Wherefore the vertical B'c 
 will meet the principal axis below the centre of gra- 
 vity of the solid, and will push it still more aside, till 
 the spheroid falls, and extends its longer diameter in 
 a horizontal position. It may then roll indifferent- 
 ly upon that line, as the sphere itself turns about its 
 centre. 
 
 But to investigate the subject more generally, let 
 HAIF (fig, 148.), a solid of any form, not ab- 
 ruptly irregular, be set to float in water. The 
 principal axis AF will divide it into correspondent 
 equal portions, and will cross the plane of flotation 
 HNI at right angles. Conceive the body to be now 
 slightly inclined, the water-line being moved into 
 the position H'NP, and the corresponding vertical 
 into AF' : The change AB' of the centre of buoy- 
 ancy or of the centre of gravity of the submerged 
 part HFI, will be to the distance NO of the lateral 
 wedge INr, as the contents of this wedge is to the 
 capacity of the segment NIF. Wherefore BB' is 
 proportional to the sine of the angle INF or of 
 
NATURAL PHILOSOPHY. 
 
 BNB', and consequently JBM is proportional to the 
 sine of the angle BB'M or to the radius, and must 
 thus be constant. While the floating solid thus 
 varies its inclination, and the centre of buoyancy 
 shifts its place, the point M in the axis where the 
 effort to redress the body is exerted remains unalter- 
 ed, like the centre of gravity itself. Such, at least, 
 is its constancy in all moderate oscillations j and, 
 from the relative position it holds, all the properties 
 of floating bodies are easily derived. That charac- 
 teristic point, standing always above the centre of 
 gravity of the mass, and limiting its greatest eleva- 
 tion in the case of permanent stability, was hence 
 called Jby Bouguer, to whom we are indebted for all 
 this fine theory, the Metacentre. 
 
 Let the floating body be a homogeneous parallele- 
 piped (fig. 149). If placed vertically in the liquid, 
 it will evidently sink, till the immersed part NF 
 shall be to its whole height AF, as its density is to 
 that of the fluid. The centres of gravity and of 
 buoyancy will evidently be the points C and B, in 
 the middle of the axis AF and of its depressed por- 
 tion NF. Conceive the solid to be now inclined to 
 one side, shifting its water-line from the position 
 HNI into H'NT. The centre of buoyancy, in 
 making a corresponding change from B to B', will 
 describe a small arc of a circle ; for B' will be raised 
 in relation to the altitude OP of the centre of gra- 
 
ELEMENTS OF 
 
 vity of the triangle INI', as the area of the rectangle 
 NIF to that of the triangle, while B' is carried la- 
 terally in the same ratio. But the rectangle NIF is 
 to the triangle INI', as NF to 11', and therefore 
 NB=5NI ; whence NF : 11' : : |NI : BB', and 
 
 NLIF 
 BB' = ' . But, from similar triangles, 
 
 BB' : BM : : II' : NI, and consequently the height 
 
 NP HP 
 BM of the metacentre is ^ or .-. Let AF, 
 
 the altitude of the parallelepiped, be denoted by a, 
 its breadth or thickness HI by b, and its density by 
 x. When the metacentre coincides with the centre 
 of gravity, and the solid floats passively and indif- 
 ferent to its position, BM is equal to BC or to 
 
 AF BF b* a ax w 
 
 - - - , and therefore = 77. Whence, 
 
 by reduction, we obtain %b* = 12#*# 1 &**#*, and the 
 
 b z 
 quadratic equation x z x = , ^ -. Of which, the 
 
 two roots are ^ \/ v~ 9 * /" ^ ^e paral- 
 
 lelepiped become a cube, then a=b, and the two den- 
 sities of indifferent flotation are irty/yV , or express- 
 
 ed in approximate numbers ^- and ^-. Between 
 
 j \) y^-' 
 
 these limits, there can be no stability ; but above, or 
 below them, the floating again acquires permanence. 
 Hence a cube of beech will float erect in water, 
 
NATURAL PHILOSOPHY. 327 
 
 while one of fir or cork will overset ; yet all these 
 three cubes will stand firmly when set upon the sur- 
 face of mercury. 
 
 Let the radical part of this expression vanish, and 
 3a z = 2&% or, in approximate numbers* lla = 9b- 
 Hence a parallelepiped, of half the density of water, 
 and having 9 inches for its altitude and 11 inches 
 for the side of its square base, would float indifferent- 
 ly. But if its density were either increased or di- 
 minished, it would gain stability. Thus, if the den- 
 sity were two-thirds of that water, the metacentre 
 would stand jf parts of an inch above the centre of 
 gravity, and ^f above it, were the density reduced 
 to one-third. A parallelepiped, with such propor- 
 tions, might therefore in every case continue erect. 
 
 It might likewise be proved, that if the parallele- 
 piped were set upon water with one of its solid an- 
 gles upwards, the stability would be limited within 
 the densities of & and If. In short, if the specific 
 gravity were greater than A, or less than f f , the so- 
 lid would permanently float in that position ; but if 
 this were either less than the former, or greater than 
 the latter, it would overset. 
 
 A cylinder will, according to its density and the 
 proportion of its diameter and altitude, exhibit the 
 three features of a floating body, indifference, in- 
 stability, or permanence of equilibrium. When this 
 uniform solid is placed with its axis horizontal, it 
 
ELEMENTS OF 
 
 will sink always to the same depth in the water, and 
 the centre of buoyancy will occur likewise at a given 
 depth in a perpendicular from the middle of that 
 line to the place of flotation, and therefore in the 
 vertical passing through its centre. The cylinder 
 thus levelled may hence repose in a horizontal posi- 
 tion, the opposite forces being constantly balanced. 
 Let the cylinder be now set perpendicularly upon 
 its end in the water. It will evidently sink, till its 
 depth NF (fig. 150.) of immersion shall be to its 
 whole height AF, as its density is to that of the 
 fluid. The point C in the middle of the axis will 
 be the centre of gravity of the cylinder, and B, the 
 middle of the immersed portion, will mark the cen- 
 tre of buoyancy. Suppose the floating body to be 
 pressed to one side, so that the water-line shall 
 change its position from HNI to H'NJ' ; the centre 
 of buoyancy, from the accession of the circular 
 wedge INP and the defect of H'NH', must likewise 
 proportionally shift its place, from B to B'. But it 
 may be shown that, T denoting the circumference of 
 
 a circle of which the diameter is unit, . NI 3 . II' 
 
 o 
 
 will represent the product of the contents of the 
 circular wedge INT into the distance of its centre 
 of gravity from N. Consequently the quotient of 
 
 . , T.NP.NF ^ ,., 
 
 this expression by , the solidity of the 
 
NATURAL PHILOSOPHY. 
 
 half cylinder NFI, will give the evagation BB' of the 
 centre of buoyancy, which is therefore = ' ^ 
 
 But the triangles INI' and BMB' being evidently 
 similar, II' : NI : : BB' : BM. Whence, by substi- 
 
 NI* 
 tution, the altitude BM of the metacentre is 
 
 HP 
 or TiMrF' -Let the height of the cylinder be de- 
 
 noted by h, its diameter by d y and its specific gra- 
 
 fi m^m //'/* 
 
 vity by x ; then NF = hx, BC = ^ , and 
 BM = rr- Wherefore an equilibrium of indiffe- 
 
 a $o i hhx . 1 . d* 
 
 rence will obtain, when - is equal to >-> or 
 
 8h*x 8h z x*=d z , and hence the quadratic equa- 
 tion x* x = T Of which, the two roots are 
 
 If 2h*=d\ the part af- 
 
 fected by the radical sign vanishes, and these two 
 roots become equal. When the density of the cy- 
 linder, therefore, is .5, and the ratio of its diameter 
 to its altitude is that of 1 to ^/ f Z y the solid will float 
 indifferently in its vertical position. Suppose, for 
 the sake of round numbers, the height of the cylin- 
 der to be 12 inches, and the diameter of its base 17 
 inches : It will sink 6 inches in the water, and con- 
 
330 ELEMENTS OF 
 
 sequently the centre of buoyancy will lie 3 inches 
 below the centre of the mass. But the metacentre 
 
 (17V 289 
 will meet in the same point, for -j^g- = -gg- = 3. 
 
 If the density were increased to .J5, the depression 
 would be 9 inches, and the distance of the centre 
 of buoyancy from the general centre of gravity only 
 
 I- inches. Now- = = 2 so that the 
 
 metacentre would stand half an inch above the cen- 
 tre of the cylinder, and would therefore secure its 
 equilibrium. Were the density of the solid reduced, 
 however, suppose to one-third, the depression would 
 be only four inches, leaving the centre of buoyancy 
 likewise 4 inches below the centre of gravity. But 
 
 (17V 289 
 
 Jv. , =-7rr = 4i = the distance of the metacen- 
 
 16x4 64 
 
 tre. The cylinder will hence, with this diminished 
 density, yet maintain a stable equilibrium. 
 
 Let the diameter and altitude of the cylinder be 
 equal ; then x ^-dby/^, and the two values of the 
 density, expressed in approximate fractions, are 
 
 T7 ^ 
 
 and . Any intermediate density will be at- 
 
 tended with instability. Thus, suppose the density 
 of the cylinder were three-fifths, its height being 
 20 inches, it would sink 12 inches, and bring the 
 centre of buoyancy 6 inches below the general centre 
 
NATURAL PHILOSOPHY. $31 
 
 of gravity. But = = 2 T V, the altitude 
 
 of the metacentre, which therefore lies below the 
 centre of gravity, and must, from its position, over- 
 set the cylinder. On the other hand, if the density 
 transgress those limits, whether in excess or defect, 
 a permanent equilibrium will result. Supposing the 
 density to be .9, the depression is 18 inches, and 
 (20V 400 
 
 7 
 
 ix 18 288 y * ' S lat> te metacentre 
 stands T 7 ^ of an inch above the centre of gravity. 
 Again, let the density be .1, the corresponding de- 
 
 (20V 400 
 
 pression is 2, and ^ J = -^ = 12? ; the metacen- 
 J.O X ^ o/4 
 
 tre being now 3 inches above the centre of gravity. 
 Hence a wooden cylinder of equal height and dia- 
 meter, which cannot stand erect in water, will yet 
 maintain a vertical position when set upon quick- 
 silver. 
 
 Suppose, lastly, the floating body to be a parabo- 
 lic conoid, (fig. 151.) with its base upwards and its 
 axis vertical. The part immersed having only half 
 the contents of the corresponding portion of the 
 
 NI* 
 
 cylinder, the height BM of its metacentre is ^ 
 
 or half the parameter of the generating parabola. 
 But the centres of gravity of the whole and of the 
 submerged part occur at two-thirds of the respective 
 altitudes, while the bulks or weights are as the 
 
332 ELEMENTS OF 
 
 squares of those altitudes. Wherefore, assuming 
 the former notation, it follows that the equation 
 
 ^(h Ti^/x) zz T- will correspond to the case of 
 indifferent equilibrium. Now, by reduction, 
 
 <*f7z 
 
 = 3d\ and Jx = 1 ^7- or 
 
 When 3d 21 = l6A a , the va- 
 
 lue of x vanishes ; but so long as the ratio of y/ 3 to 
 4, or that of 13 to 30 nearly, subsists between the 
 altitude and the diameter of the solid, its equilibrium 
 will remain firm at every given density. If the al- 
 titude and diameter become equal, the specific gra- 
 vity which would occasion indifference of flotation 
 
 169 
 is TT or .66. Any increase of density beyond this 
 
 limit would procure stability. Let/? denote the para- 
 meter of the generating parabola, and 4<ph = d*, or 
 
 ph = 3d* -, whence 1 ~r-=.^/x. Thus, while 
 
 the parameter continues invariable, the altitude of 
 the parabola may be made to procure indifference of 
 floating, for every degree of density. The corre- 
 sponding depth of immersion is xh, or the portion 
 of the axis above the water-line is expressed by 
 
 These remarks will explain the cause of the over- 
 
NATURAL PHILOSOPHY. 333 
 
 setting of the huge masses of ice which float within 
 the Arctic Circle. Such enormous blocks are gene- 
 rally of a columnar shape, approaching to the form 
 of a parallelepiped or of a cylinder, though perhaps 
 occasionally contracted below. The upper surface 
 thaws slowly, from the action of the atmosphere, as 
 the summer advances ; the under side likewise melts 
 at first, but becomes soon protected by a stratum of 
 fresh water of the same temperature, consisting of 
 the dissolved portion of the ice which is upheld by 
 the superior density of the surrounding medium. 
 The principal waste of the icy mass taking place 
 along its submerged sides, the current of melted wa- 
 ter continually rises upwards, and leaves a new sur- 
 face to the attack of a warmer current. After the 
 width of the vast column has become thus diminish- 
 ed beyond a certain ratio to its height, it tumbles 
 over upon its side. 
 
 To determine this limit, suppose the floating 
 block to be first a parallelepiped. Let the height a 
 be denoted by h, and the side b of the base by hy ; 
 since the density of ice compared with that of sea- 
 water is .89, the depth of submersion will be .89 h. 
 Wherefore the metacentre will coincide with the 
 
 h z y* 
 
 centre of gravity, when y = hx.055. Con- 
 sequently, by reduction, y 2 " = .5874, and y = .766. 
 Hence the icy parallelepiped will overset, the mo- 
 ment its breadth approaches to three-fourths of its 
 
334? ELEMENTS OF 
 
 altitude. Thus, if the whole height of the mass 
 were 1000 feet, 890 feet would sink, leaving 110 
 feet to tower above the surface. The elevation of 
 the centre of gravity beyond that of buoyancy would 
 hence be 55 feet, which is the limit of the metacen- 
 tre after the base of the column has been reduced to 
 a breadth of 766 feet. 
 
 Again, suppose the ice to have a cylindrical form. 
 Assuming the same data as before, we shall have 
 
 = . 7832. Wherefore y = .885. A cylindrical 
 mass of ice 1000 feet altitude would sink 889 feet 
 in sea-water ; but when the diameter of its base 
 was reduced to nearly the same quantity, or 885, 
 it would overset. The instability of the cylinder 
 thus begins sooner than that of the parallelepiped, 
 taking place when the width below becomes eight- 
 ninths instead of three-fourths of the whole height. 
 But the greater extension of the summit, owing to 
 its slow waste, may in every case hasten the period 
 of overwhelming the icy column. 
 
 Lastly, conceive the block of ice to be wasted and 
 rounded below into the shape of a parabolic conoid. 
 At the moment when its equilibrium fails, we have 
 
 3d 2 
 seen that ^/ x = 1 fprr. But tne density of ice 
 
 in relation to sea-water being .89, it follows that the 
 
NATURAL PHILOSOPHY. 
 
 3d* 
 
 square root of this, or .9435 = 1 T^TI all( l hence 
 
 = .0565 ; wherefore J = .30133, or -, = 
 
 n h 
 
 ^/ .30133 = .549- Whenever the mass has its 
 base reduced in the ratio to its depth of about 11 
 to 20, it will suffer a total inversion, its lowest point 
 being whelmed uppermost. This form of a body of 
 ice would, therefore, suffer a greater previous waste ; 
 but its balance is at last more completely destroyed. 
 Stability in every case becomes precarious, after the 
 breadth of a block is inferior to its depth. 
 
 To investigate generally the conditions of a floating 
 body, let HAIF (fig. 148.) represent the vertical 
 section, the point C being the centre of gravity of 
 the whole, and situate in the principal axis ANF, 
 while B indicates the centre of gravity of the immer- 
 sed portion HIF, or the centre of buoyancy. When 
 the solid, declining from its vertical position, changes 
 the water-line from HNI to H'NF, the centre of 
 buoyancy shifts from B to B'. It is evident that the 
 area of the segment HIF will be to the areas of both 
 the accrescent triangle Nil' and the deficient trian- 
 gle HNH', as NO, the distance of the centre of gra- 
 vity of the former, is to BB'. But the areas of those 
 triangles being equal to NI x IF, and the distance 
 NO equal to two- thirds of NI ; it follows that the 
 
33d ELEMENTS OF 
 
 area of HIF is to NI x II, as NI to BB'. Again, 
 from the similar triangles IAI' and BNB', II' is to 
 NI as BB' to BM ; wherefore, by compounding 
 those analogies, the area of HIF is to the square of 
 NI, as NI is to BM, the height of the metacentre 
 above the centre of buoyancy. Hence, the area of 
 the section HIF being denoted by A, and the width 
 
 of the water-line HNI by b ; A : (\ V: : | : BM, 
 
 V^ J o 
 
 or A : ^ : :| : BM = --- The Flotation will 
 
 hence be stable, only when the quotient of the cube 
 of the breadth HI by twelve times the area of the 
 segment HIF exceeds the interval BC between the 
 centre of gravity and that of buoyancy. The stabi- 
 lity of a ship is hence greatly augmented by increa- 
 sing the dimension of the water-line. 
 
 The area of any section, and consequently the po- 
 sition of its centre of gravity, may be determined, to 
 any degree of accuracy, by the method proposed by 
 Newton, and improved by Stirling and Simpson, of 
 drawing a line of the parabolic kind through any gi- 
 ven system of points. If the parallel sections were 
 all equal, the calculation of the altitude of the meta- 
 centre would be concise and direct. When those 
 sections are unequal, the area of each is to be multi- 
 plied into the corresponding height of the metacentre, 
 and the sum of the products divided by the aggre- 
 gate contents of the immersed part of the solid. In 
 
NATURAL PHILOSOPHY. 337 
 
 the case of a merchant ship, it may furnish a tole- 
 rable approximation to take the section near to the 
 prow, where the girth is commonly largest. 
 . It will not differ much from the truth, to assume 
 the cross section of the hull of a ship as of the form 
 of a parabola. Wherefore, the height of the meta- 
 centre above the centre of buoyancy, or BM (fig* 
 152.) is equal to the cube of HI, the breadth of 
 the water-line, divided by twelve times the area of 
 HFI, or of two-thirds of the rectangle under HI, 
 and the depth of immersion NR Consequently, 
 m* HP HP NP p,. i , 
 
 = fflENF = SOT = *NF 
 
 equal to half the parameter of the parabola, or to the 
 double of its focal distance. But FB, the altitude of 
 the centre of buoyancy, is equal to f NF, and there- 
 fore FB, the whole height of the metacentre above 
 
 .1 1 1 5MTT . NL 6NP 
 
 the keel, is |NF + = 
 
 Such is the height of the metacentre above the 
 keel, on the supposition that the vertical sections of 
 the hull are all equal, as may be nearly the case in 
 long track-boats. But the figure of the keel in most 
 vessels fitted for sailing approaches to a semi-ellipse, 
 which is likewise the general form of an horizontal 
 section. Owing to these conjoined modifications, the 
 metacentre is on the whole depressed by one-fourth 
 part, and consequently its altitude above the centre 
 
 VOL. i. y 
 
3vS8 ELEMENTS OF 
 
 11 1 J l - HI * 3NI 
 
 of buoyancy will be expressed by f . == 
 
 Suppose, for the sake of illustration, a ship whose 
 water-line is 40 feet wide, and its depth of immer- 
 
 3(20 V 1 200 
 sion 1.5 feet. Here ^ ^ = =10 feet ; but 
 
 the centre of buoyancy, or the centre of gravity of 
 the immersed portion of the hull, falls 6 feet, or 
 f X 15 below the water-line, and hence the metacen- 
 tre stands four feet above it. The ship will there- 
 fore float securely, so long as the general centre of 
 gravity is kept under that limit, or less than four 
 feet above the plane of flotation. In loaded vessels, 
 the centre of gravity has been found to be common- 
 ly higher than the centre of buoyancy, by about the 
 eighth part of the extreme breadth. Wherefore, 
 in the present instance, the centre of the whole mass 
 would lie still one foot below the surface of the wa- 
 ter, or five feet lower than the metacentre, leaving 
 ample space for maintaining the stability of the 
 ship. 
 
 Such is the position of that metacentre situate in 
 the vertical plane at right angles to the longitudinal 
 axis, and which regulates the rolling of a vessel from 
 side to side. But there is another similar point in 
 the plane of the masts and keel, which determines 
 the pitching, or the movement of alternate rising 
 and sinking of the prow. The altitude of this 
 metacentre is derived from the same formula,, by 
 
NATURAL PHILOSOPHY. 
 
 \nerely substituting the length for the breadth of the 
 vessel. Thus, let the keel measure 180 feet, and 
 
 3(90)* _ 202 ^ feet> W j th such power f iu stability, 
 
 o.lo 
 
 therefore, in the direction of its course, a ship can 
 scarcely ever founder in consequence of pitching at 
 sea. 
 
 The formula now given for computing the height 
 of the metacentre above the centre of buoyancy, 
 may, with some modification, be deemed sufficiently 
 accurate in practice. It is best adapted, however, 
 for cutters or frigates, and will require to be some- 
 what diminished in the case of merchant vessels. 
 The late Mr Atwood performed a laborious calcula- 
 tion on the hull of the Cuffnells, a ship built for the 
 service of the East India Company, having divided 
 it into 34 transverse sections, of five feet interval. 
 The result was, that the metacentre stood only 4 
 feet 3 inches above the centre of buoyancy. But 
 that ship, being designed chiefly for burthen, ap- 
 pears from the drawings to have been constructed 
 after a very heavy model, its vertical sections ap- 
 proaching much nearer to rectangles than parabolas. 
 To suit it, the formula above given would have re- 
 
 NP 
 
 quired to be reduced two-thirds, or rj^p Now, the 
 
 breadth of the principal section was 43 feet and 
 2 inches, and its breadth 22 feet 9 inches. Whence 
 
340 ELEMENTS OF 
 
 (21.6V 
 
 -^p- 5.1 feet, differing little from the conclu- 
 sion of a stricter but very tedious procedure. 
 
 Since the height of the metacentre is inversely as 
 the draught of a vessel, and directly as the square 
 of its breadth, its stability depends mainly on its 
 spreading shape. This property is an essential con- 
 dition in the construction of life-boats. But the 
 lowering even of the centre of gravity has been 
 found to be sometimes insufficient to procure stabi- 
 lity to new ships, which, after various ineffectual at- 
 tempts, were rendered serviceable, by applying a 
 sheathing of light wood along the outside, and thus 
 widening the plane of flotation. 
 
 It is not very difficult to determine the centre of 
 buoyancy, by guaging the immersed part of the hull. 
 A cubic foot of sea-water weighs 64 Ib. averdupois, 
 and thirty-five feet, therefore, make a ton. The 
 load of the vessel corresponding to every draught of 
 water may be hence computed. But to find the 
 place of the centre of gravity is more difficult ; since 
 this depends less upon the figure of the hull than 
 upon the disposition of its internal burthen. Con- 
 ceive a horizontal plane to touch the bottom of the 
 vessel, and let the weight of each part of the timbers 
 and of the cargo be multiplied into their height 
 above it ; the sum of all these products, divided by 
 the accumulated load, will give the altitude of the 
 
NATURAL PHILOSOPHY. 341 
 
 centre of gravity. Again, suppose another plane 
 at right angles to the longitudinal axis of the ship, 
 to touch its bow ; the perpendiculars drawn to this, 
 from every part of the hull and cargo, are to be 
 multiplied into the several weights, and the quotient 
 of the total amount by the general load will give 
 the distance of the common centre of gravity. Such 
 a calculation, however, would involve a tedious mul- 
 tiplicity of details. 
 
 The height of the metacentre above the centre of 
 gravity in a loaded vessel, may be discovered by a 
 simple observation. Let a long, stiff, and light 
 beam be projected transversely from the middle of 
 the deck, and a heavy weight suspended from its 
 remote end, inclining the ship to a certain angle, 
 which is easily measured. Thus, if NL (fig. 151.) 
 represent this lever, P the weight attached, M the 
 metacentre, and CMQ the decimation produced, C 
 being the centre of gravity, and CR a perpendicu- 
 lar drawn from it to the vertical LP. The power 
 of the weight P to redress the vessel will be express- 
 ed by P x CR ; but, W denoting the entire load, 
 the effort exerted at the metacentre to keep the 
 mast erect, will be represented by W x CQ, or 
 W x CM x sin CMQ. Wherefore P x CR = 
 W x CM x sin CMQ, and consequently the ele- 
 vation CM above the centre of gravity is expressed 
 
 P , 
 
 by WlmCMQ' N W CR may ' vvithout any sen " 
 sible error, be assumed as equal to the length LM 
 
ELEMENTS OF 
 
 of the beam from the middle of the deck. Suppo- 
 sing the height of the metacentre to be 3 feet 10^ 
 inches above the centre of gravity, a weight equal to 
 the two hundredth part of the burthen or tonnage 
 of the ship, and acting on a lever of 50 feet in length, 
 would occasion a declination of five degrees. If 
 the experiment were performed in a wet-dock, or on 
 a smooth calm sea, such small angle could be mea- 
 sured with sufficient accuracy. In calculating the 
 effect of this disturbing influence, it is easy to per- 
 ceive, that half the weight of the beam should be an- 
 nexed to P. The distance MB of the metacentre 
 from the centre of buoyancy, having been ascertain- 
 ed by a previous computation, the height BC of that 
 unvarying point above the centre of gravity is thence 
 deduced. A trifling correction may be likewise 
 made, for assuming CR as equal to NL ; by dimi- 
 nishing NL first, by its product into the versed sine 
 of the inclination, and next augmenting it, by the 
 product of CN into the sine of that angle. The 
 same result might be obtained, in ships of war, by 
 a more complex calculation, indeed, from the ordi- 
 nary process of heeling, when the guns are all run 
 out upon one side. 
 
 A similar observation might discover the altitude 
 of the longitudinal metacentre of the ship, above 
 the common centre of gravity. But, acting in this 
 direction, a greater load will be required to produce 
 a sensible depression. Let such a load be carried to 
 the prow of the vessel, and again transferred to the 
 
NATURAL PHILOSOPHY. Ho 
 
 xtern. The intermediate place of the centre of 
 gravity is hence determined, for its distances from 
 those opposite points of pressure must evidently be 
 inversely as the corresponding angles of inclination. 
 The small aberration of the centre of gravity occa- 
 sioned by the interchange of these loads, may like- 
 wise be computed. Finally, therefore, the product 
 of either load into its distance from .the centre of 
 gravity, being divided by the product of the whole 
 burthen of the ship into the sine of the inclination, 
 will give the height of the metacentre of the longitudi- 
 nal section on which depends the motion of pitching. 
 When a floating body has its equilibrium disturb- 
 ed, it brings into action a redressing force, which is 
 always proportioned to the quantity of derangement. 
 It is hence made to oscillate like a pendulum about 
 its general centre of gravity ; and, for the same 
 reason, its alternate movements, whether of less or 
 greater extent, are still performed in equal times. 
 But any ship is liable to three distinct kinds of oscil- 
 lations the vertical, the transverse, and the longi- 
 tudinal, or what is called heaving, rolling, and 
 pitching. Thus, if a vessel were lifted a little above 
 its seat of flotation, and then allowed to sink into 
 the water, it would continue for some time after- 
 wards alternately to rise and fall. If the vessel were 
 suddenly pushed to one side it would roll upon its 
 longitudinal axis ; but, if it were depressed at the 
 head or the stern, it would begin to pitch or spring 
 about the transverse axis. 
 
344 ELEMENTS OF 
 
 1. The alternate heaving and subsiding of a ship 
 in the vertical direction, is occasioned by the fluc- 
 tuating excess or defect of the buoyant effort of the 
 water. The time of oscillation is therefore the same 
 as that of a pendulum, whose length is equal to the 
 mean depth of immersion, or to the quotient of the 
 capacity of the submerged portion of the hull, by 
 the horizontal surface of flotation. This medium 
 draught of water may be reckoned at about two- 
 third parts of the extreme depth of the keel. Thus, 
 a ship which draws 20 feet may have 13 feet allow- 
 ed for its mean depth, which is four times the length 
 of a second's pendulum. Each oscillation upwards 
 and downwards of such a vessel will hence take two 
 seconds. 
 
 2. The time required for the rolling of any ship, 
 though performed about its general centre of gravity, 
 must be the same as that of its vibration, if it had 
 been suspended like a pendulum from its metacentre. 
 From the principle investigated in p. 99, the dis. 
 tance of this centre of gravity from the centre of os- 
 cillation, is equal to the quotient of the momentum 
 of rotation, or of the sum of the products of all the 
 different bodies into the squares of their several dis- 
 tances from the longitudinal axis, divided by the 
 product of the whole mass into the interval between 
 their common centre of gravity and the metacentre. 
 It hence follows, that while the distribution of the 
 cargo remains the same, the time of rolling, or of 
 
NATURAL PHILOSOPHY. 34t5 
 
 lateral oscillation, will be inversely as the square of 
 the elevation of the metacentre above the centre of 
 gravity. Again, those alternating motions may be 
 rendered proportionally slower, by removing the 
 various articles of the cargo to a greater distance 
 from the longitudinal axis, above and below the 
 centre of gravity, to both the sides of the vessel. 
 If the lateral distances were doubled, the oscillation 
 would be rendered twice as slow ; and if tripled, 
 they would become three times slower. 
 
 Suppose the floating body were an homogeneous 
 parallelepiped, of which the height is a, and the 
 breadth b : It may be shown that the momentum of 
 rotation about the centre of gravity, is expressed 
 
 by -- , and consequently the distance of the 
 centre of oscillation below the centre of gravity is 
 equal to r divided by the relative elevation of 
 
 the metacentre. Again, if the transverse section of 
 the floating body were considered as a parabola, of 
 which the height and breadth are denoted as before 
 by a and b : It may be computed that the momen- 
 
 , .. . - 1440 '+1052**' 
 
 turn of rotation is then equal to - ^rr- 
 
 3150 
 
 whence, dividing this by - r-, or the area of the pa- 
 
 rabola, the quotient - -- will express the pro- 
 
346 ELEMENTS OF 
 
 duct of the distances of the common centre of gra- 
 vity from the metacentre above, and from that point 
 below it, which represents the centre of oscillation. 
 
 Suppose, for example, a parallelepiped 80 feet 
 broad and 48 feet deep, of an uniform density, but 
 three times less than that of water : It would evi- 
 dently sink 16 feet, and then float. The centre 
 of buoyancy lies hence 8 feet below the water- 
 line, while the centre of gravity stands 8 feet 
 above it. But the height of the metacentre above 
 
 f 80V 
 the centre of buoyancy being ^ = 33? feet, it 
 
 must exceed by 17^ feet the altitude of the cen- 
 tre of gravity. Now the momentum of rotation, 
 divided by the weight of the floating body or 
 
 80 ';t 48 * = ^ = 725^ feet ; and the quotient of 
 
 1^ 1/w 
 
 this, by 17 i feet, gives 41|^ feet, for the length of 
 an isochronous pendulum. The floating parallele- 
 piped would consequently roll or oscillate, in 3.58", 
 or in about 3^- seconds. 
 
 Conceive a body of the same density and dimen- 
 sions, but of a parabolic form, to float in water. Its 
 depth of immersion will be 23.08 feet, and there- 
 fore the centre of buoyancy will fall 9.23 feet be- 
 low the plane of flotation, while the common centre 
 of gravity stands 5.720 feet above it. The breadth 
 of the water-line being now 55.47, tne altitude 
 of the metacentre above the centre of buoyancy is 
 
NATURAL PHILOSOPHY. 347 
 
 hence ~ Q = 16.86 feet ; so that the meta- 
 
 o 
 
 centre is only 1.91 feet higher than the centre of 
 
 gravity. Now *&&&& = ** feet, the 
 
 / 00 
 
 quotient of the momentum of rotation by the weight 
 of the parabolic section ; this number, again, being 
 divided by 1.91, gives &50.26 feet, the length of a 
 pendulum which would oscillate in concord with the 
 rolling of the body. Those vibrations are hence 
 performed very slowly, each taking 8.76'', or about 
 8f seconds. 
 
 In the cases now computed, the transverse sec- 
 tions are all equal. But, to determine the rolling 
 of a ship, it would be requisite to combine the mea- 
 sures of different sections of the hull, and thence de- 
 duce a medium result. To abridge this calculation, 
 however, it may perhaps be sufficiently correct, if 
 we consider the distance of the general centre of os- 
 cillation as equal to two-thirds of the distance which 
 corresponds to the greatest cross section of the 
 vessel. 
 
 On the stowage of a ship's cargo, depends chiefly 
 the character of the rolling. If the goods be placed 
 nearer the longitudinal axis, the momentum of ro- 
 tation being diminished, the rolling will in conse- 
 quence be quickened. But if the various articles of 
 loading be removed, as far laterally as possible from 
 the position of the centre of gravity, the oscillations 
 
348 ELEMENTS OF 
 
 will be rendered proportionally slower. When the 
 cargo consists of light goods of the same kind, the 
 hold is quite filled up, and no room left for skilful 
 stowage. But if it include many ponderous articles, 
 the rolling may be damped, and the motion of the 
 vessel rendered easier, by bringing those loads near 
 to its sides. In ships of war, such a change of trim- 
 ming, by the removal of the guns and shot, is occa- 
 sionally practised, with the greatest advantage. 
 
 3. The time of oscillation about the transverse 
 axis, or that of pitching, may be found by a similar 
 calculation. The various loads in each vertical sec- 
 tion parallel to the keel, are to be multiplied into the 
 squares of their several distances from the centre of 
 gravity ; and the quotient of the collective aggre- 
 gate by the whole burthen of the vessel, will express 
 the product of the distances of the general centre of 
 gravity from the longitudinal metacentre and the 
 centre of oscillation. It may be sufficiently near the 
 truth, to take two-thirds of the result of a computa- 
 tion grounded on the chief section in the plane of 
 the masts and of the keel. The momentum of ro- 
 tation of the longitudinal section is to that of the 
 cross section, nearly as the square of the length of the 
 ship is to the square of the breadth ; but the alti- 
 tudes of the corresponding metacentres being like- 
 wise approximately in that ratio, the times of rolling 
 and pitching will in all cases of uniform stowage ap- 
 proach|to an equality. The disposition of a ship to 
 
NATURAL PHILOSOPHY. 349 
 
 violent plunging may often be corrected, by convey- 
 ing the heaviest part of the cargo towards the head 
 and the stern. The remoteness of the lading from 
 the common centre of gravity, serves materially to 
 retard and soften the oscillatory movements, both 
 about the longitudinal and the transverse axis. 
 
 Let the parallelepiped of 80 feet wide and 48 
 feet deep be resumed as an example, its length 
 being now considered as 480 feet. Here the al- 
 titude of the longitudinal metacentre above the 
 
 (480)* 
 
 centre of buoyancy is -j ^ zz 400 feet, and con- 
 sequently its height above the common centre of 
 gravity must be 400 16 = 884 feet. But the 
 quotient of the momentum of rotation of each 
 longitudinal section divided by its weight, is 
 
 (48 )3 .t (48) * = 19392 feet, which being divided by 
 l/o 
 
 984, gives 50.5 feet for the length of an isochro- 
 nous pendulum. The time of oscillation is hence 
 3.935", or very nearly 4 seconds. It is obvious 
 that such problems could easily be reversed, and 
 that the height of the metacentre above the centre 
 of gravity, could be computed from the different 
 oscillations in smooth water. 
 
 While the shape and the lading of a vessel con- 
 tinue the same, the height of its centre of buoyancy 
 and the derivative position of the metacentre must 
 likewise remain unaltered. By lowering the places 
 of the more ponderous articles, the common centre 
 
350 ELEMENTS OF 
 
 of gravity may be depressed or brought nearer to 
 that of buoyancy ; and by dispersing them towards 
 the sides and extremities of the hull, the oscillations 
 of rolling and pitching may be retarded and render- 
 ed smoother. Such are the only internal changes 
 which can be made ; but it is of great importance in 
 many cases to lessen the draught of water, or to lower 
 the elevation of the centre of gravity. This can only 
 be effected, however, by supplying externally the re- 
 quisite additional buoyant force. 
 
 Since a body submerged in water is pushed up- 
 wards by the effort of the volume of fluid which it 
 displaces, its whole power of buoyancy must be equal 
 to the excess of that pressure above its own weight. 
 So long, therefore, as the bulk remains unaltered, 
 this power will increase with every diminution of the 
 weight. Hence the buoyancy procured to a floating 
 body by the application of inflated bladders or bottles 
 of skin, empty casks or hollow chests. Bladders are 
 sometimes employed to assist swimmers, and they 
 support the fishermen's nets. Bags of goats skins, 
 covered with boards, have from the earliest times 
 been used as floating bridges among rude nations for 
 crossing rivers. The Egyptians have been accus- 
 tomed in all ages to descend the Nile on slender 
 rafts, carrying their produce, though supported only 
 by empty earthen jars. A hollow wooden box, of a 
 conical shape, and attached by an iron chain, serves 
 at present as a buoy, for marking the position of 
 sunken rocks or sand banks. Vessels which have 
 
NATURAL PHILOSOPHY. 351 
 
 been stranded are often raised up again, by fastening 
 at ebb tide, a row of empty casks along each side. 
 Long boxes or chests, lightened after their fixing, 
 by having the water pumped out, have likewise been 
 employed to lessen the draught of ships of burthen. 
 In this way, the Hollanders, in the year 1672, when 
 their commercial prosperity was at its height, dis- 
 patched to various climes, numerous heavy laden 
 vessels, thus conducted from the harbour into deep 
 water. But the raising of the hull in this method 
 w r as found to occasion an overstraining of the tim- 
 bers, the external pressure of the water being sud- 
 denly withdrawn from the bilging sides. But in the 
 year 1688, Bakker, an ingenious burgomaster of 
 Amsterdam, obviated that objection, by a very use- 
 ful contrivance, which, from its property of trans- 
 porting immense bodies, was termed a Camel. It 
 consisted of two huge chests, so formed as to em- 
 brace closely under water the hull of the largest 
 ships : Its length was 127 feet, its breadth at the 
 one end 22, and at the other 13 feet. These chests 
 were securely fastened by ropes passed under the 
 keel, and stretched by horizontal windlasses. Ob- 
 lique props or stays were then applied, and wedged 
 firmly, to support the ship's sides. The water was 
 now vigorously pumped out of the hold of these 
 chests, which were divided into several compart- 
 ments, for the greater convenience of adjusting, du- 
 ring that operation, the balance of the ship. Thus 
 lightened, an Indiaman which drew 15 feet water, 
 
352 ELEMENTS OF 
 
 had its draught reduced to 11 feet. The largest 
 vessels were hence enabled to effect the passage of 
 the Pampus, between two sand banks opposite the 
 mouth of the river Y, twenty-five miles below the 
 city of Amsterdam. This simple but valuable in- 
 vention, though scarcely known in England, has long 
 been adopted at Venice and other parts of the Con- 
 tinent. It has likewise been introduced into the 
 rivers and ports of Russia, where some of those 
 Camels are constructed of the enormous dimensions 
 of 217 feet long, and 36 feet broad. 
 
 From their aptitude to receive and propagate every 
 impression, fluids derive the capital property of main- 
 taining the same level in any system of connected 
 vessels. But this general principle is subject to cer- 
 tain modifications. Thus, water is observed always 
 to stand somewhat higher in narrow glass tubes than 
 in those of greater width. Alcohol manifests a si- 
 milar disposition, inferior only in degree. Neither 
 of these liquids, however, appears to rise above the 
 level in the finest metallic pipes, but rather betrays 
 an opposite tendency. Quicksilver, on the other 
 hand, suffers some depression in glass tubes of nar- 
 row bores ; yet recovers its elevation, or even mounts 
 higher, when the inside is lined with the thinnest 
 film of bees-wax or tallow. 
 
 This occasional rise of water, and depression of 
 mercury, in very narrow tubes, must evidently pro- 
 ceed from the operation of some peculiar attractive 
 
 1 
 
NATURAL PHILOSOPHY. 353 
 
 t>r repulsive force existing among the particles of the 
 fluid, or between them and the surface of the glass. 
 As the effect appears the most conspicuous, when 
 the width of the bore is so small as to resemble that 
 of a hair, the cause of the phenomenon has been 
 termed Capillary Action. The popular mode of 
 explaining the fact, is to refer the suspension of the 
 slender column of water to the attraction of the in- 
 terior ring of glass immediately above it. But why 
 should not the ring just below the summit of the 
 column attract it equally downwards ? And such op- 
 posite forces producing a perfect equilibrium, the wa- 
 ter would merely preserve its level, and show no dis- 
 position to rise in the tube. 
 
 The chief obstacle in explaining the mode of ca- 
 pillary action, comes from the prejudice, that a ver- 
 tical attraction is necessary to account for the ele- 
 vation of the liquid. Yet such undoubtedly is not 
 the primary direction of the force evolved j for the 
 action of the glass being evidently confined within 
 very narrow limits, this virtue must be diffused over 
 the internal surface of the tube, and must hence ex- 
 ert itself laterally, or at right angles to the sides. 
 Nor is it difficult to conceive how a lateral action 
 may yet cause the perpendicular ascent. It is a 
 fundamental property of fluids, that any force im- 
 pressed in one direction, may be propagated equally 
 in every direction. The tendency of the fluid, then, 
 to approach the glass, will occasion it to spread over 
 
 VOL. i. z 
 
354 ELEMENTS OF 
 
 the internal cavity of the tube, and consequently to* 
 mount upwards. 
 
 But to view the matter a little more closely : 
 Suppose a drop of water were laid upon a clean ho- 
 rizontal plate of glass ; it will change its globular 
 form, adhere to the glass, and spread out, till it has 
 covered the whole surface with a thin aqueous film. 
 What is the cause of this result? Surely not the 
 mere incumbent weight of the water, for it would 
 have been insufficient to surmount even the mutual 
 adhesion of the particles of the fluid, or their natural 
 tendency to conglomerate. But the same effect will 
 take place, if the drop be applied to the under side 
 of the plate. The water, therefore, diffuses itself 
 over the glass, just in the same manner as if it were 
 urged by the pressure of a column of its own sub- 
 stance acting against that surface. Its attraction to 
 the glass produces the lateral motion of the fluid, 
 since the remoter particles cannot approach without 
 spreading themselves and extending the film. The 
 cohesive power will consequently augment with the 
 gradual approximation, till this has attained the term 
 of closest union. 
 
 Let the plate be dipped perpendicularly in a ba- 
 son of water, and the film will suffer a very consider- 
 able modification. A new portion of liquid will 
 greedily attach itself to the film, and draw it down- 
 wards by this additional load. A fluid margin is 
 formed at the line of junction, with a depressed in- 
 curvation, extending to the distance of about the 
 
NATURAL PHILOSOPHY. 355 
 
 sixth part of an inch. Suppose next that two glass 
 plates, held parallel and adjacent, were set perpendi- 
 cular to a body of water, the liquid would mount in 
 the included space to a certain altitude. The aque- 
 ous protuberance being now confined, the ascent of 
 the column must be proportionally greater ; and this 
 effect is also doubled, by the joint action of the two 
 opposite surfaces. Each surface acts only upon its 
 proximate thin film ; but the force being spent in 
 supporting the ulterior particles which adhere to this, 
 the weight of the aqueous column suspended on both 
 vsides must remain constant, and hence its altitude 
 will be inversely as its thickness, or the distance be- 
 tween the two plates. The power of efficient sus- 
 pension corresponding to each inch square of surface 
 may be estimated at 2 grains, or the hundredth part 
 of the weight of a cubic inch of water. Suppose 
 the interval between those plates to be the 300th part 
 of an inch ; then each column may be considered 
 as acting against a column of the thickness of the 
 600th part of an inch, and hence the corresponding 
 ascent must be six inches. In general, if d denote 
 
 the interval of the plates, - will express the height 
 
 of the column suspended. This result corresponds 
 perfectly with observation. If two rectangular plates 
 of glass, having their edges joined in a vertical line, 
 be opened at a very acute angle, and set upright in a 
 shallow basin of water, the liquid will rise between 
 the converging surface, and trace out the boundar^ 
 
356 ELEMENTS OF 
 
 of a rectangular hyperbola, referred to its asymp- 
 totes. For perpendiculars being inversely to the 
 surface of the water as the thickness of the inserted 
 column, or the separation of the plates, must like- 
 wise bear the same inverse ratio to their distances 
 from the centre, which is a distinguishing property 
 of that curve. 
 
 In capillary tubes, the attraction of the internal 
 surface is exerted on a thin circular lining ; but 
 this force is dilated and consumed by the pressure 
 of the cylinder which adheres to the film, and occu- 
 pies the interior cavity of the tube. Now, the area 
 of a circle is equal to the rectangle under its cir- 
 cumference and half the radius ; and therefore the 
 attractive power of the glass will produce the same 
 effect as if it acted merely against a column whose 
 thickness is one-fourth part of the bore of the tube. 
 
 The measure of that force being represented by 
 
 - , if d denote the diameter of the cavity, the alti- 
 
 tude at which the water will be suspended in the 
 tube is expressed by ^J-^= J-^. A tube with 
 
 the 150th part of an inch bore would hence be ca- 
 pable of elevating water six inches. The altitude 
 of suspension in capillary tubes is thus the double of 
 what obtains with parallel glass plates, which have 
 their mutual distance equal to the diameter of the 
 bore. This altitude is likewise in general inversely 
 as the width of the tube. 
 
NATURAL PHILOSOPHY. 357 
 
 The suspension of a fluid in capillary tubes must 
 depend entirely on the smallness of the upper ori- 
 fice. If the bore should swell out below, the water 
 will not rise indeed spontaneously within the cavity ; 
 but, by plunging the tube into the bason till the li- 
 quid reach its capillary part, the whole included mass 
 may be lifted up to the former elevation. The 
 central column of the water, which has the same 
 width as the bore, being sustained by its adhesion 
 to the film at the top, the pressure of the parallel 
 columns of fluid surrounding it below can have no 
 influence in disturbing the equilibrium. The same 
 effort of pressure extends through each horizontal 
 stratum. The centre column of the fluid, being 
 wholly supported by attraction, does not press a- 
 gainst the bottom ; but the particles in the higher 
 parts of it are actually pushed by the excess of that 
 force above their weight, and thus bear the load of 
 the lateral mass. The lower and wide part of the 
 vessel may consist of metal or any other substance 
 different from glass. It is sufficient that the cavity 
 terminate above in a fine capillary tube. By this 
 arrangement a very large body of water may be kept 
 suspended. 
 
 If the capillary tube be dipped in another narrow 
 tube holding water, this will evidently stand at a 
 lower altitude than before, since the opposite action 
 of the outer cavity, though much inferior, is exerted 
 in pulling back the liquid. Let d and D denote the 
 diameters of the interior and exterior bores, and 
 
358 ELEMENTS OF 
 
 ^rz( -, rrr ) w *^ ^ x P r ess in inches the ascent above 
 
 the under surface. 
 
 If a capillary tube be lifted from the bason of wa- 
 ter in which it was dipped, a drop will adhere to the 
 lower orifice, and the column will remain at the same 
 height in the bore. But if this drop be absorbed by 
 the contact of bibulous paper, the column will sub- 
 side ; for the tendency of the water to agglobulate 
 then counteracts, in some degree, its capillary diffu- 
 sion within the tube. Having covered the level 
 surface of a piece of glass with a fine film of water, 
 bring the tube with its charge to touch it, and the 
 fluid will immediately desert the cylindrical cavity, 
 and spread over the film. The attraction which the 
 cylindrical column of water, joined to its weight, 
 bears to the expansive horizontal film, overcomes 
 that of the narrow film which lines the inside of the 
 tube. This may be viewed as an extreme case ; but 
 the mutual attraction of the particles of water or 
 other fluids must always diminish their ascent in ca- 
 pillary tubes ; for such a force, tending to concen- 
 trate and agglobulate the mass, will evidently op- 
 pose any filamentous ramifications of the fluid. Thus, 
 into a wide cylinder glass, terminating in a fine ca- 
 pillary orifice, mercury may be poured to the height 
 of several inches, without betraying any disposition 
 to run out. But, if the projecting point be likewise 
 immersed in mercury, the moment a junction or 
 contact takes place, the whole of the fluid mass will 
 
NATURAL PHILOSOPHY. 
 
 hasten to subside. Mercury is, in the same manner, 
 upheld to a considerable altitude in wide glass tubes, 
 fitted with bottoms of iron or box-wood that have 
 delicate slits. 
 
 The depression of mercury in capillary tubes may 
 
 be estimated at ?r:r- 7 , if d denote the diameter in 
 Ooa 
 
 inches. Thus, when the width of the bore is one- 
 seventeenth part of an inch, the mercurial column 
 stands a quarter of an inch below its level ; it would 
 sink even five inches, if the bore were contracted to 
 the 340th part of an inch. The convexity of the 
 surface of the mercury in the tube indicates here the 
 excess of the mutual attraction of its particles above 
 their adhesion to the sides of the glass. If that sur- 
 face should ever appear flattened or hollow, it only 
 betrays the impurity of the mercury, or the soiling 
 of the inside of the tube. The notion which has 
 sometimes prevailed, that such convexity marks a 
 tendency of the mercurial column in the barometer 
 to fall, is merely a vulgar error. Great cautiqn be- 
 comes requisite, however, to boil mercury in those 
 tubes, lest a slight and scarcely visible oxydation 
 should line their inner surface. 
 
 It hence follows that, if the product of the ele- 
 vation into the corresponding width of the bore ex- 
 pressed in inches, should, at any part of the tube, be 
 
 less than or .04, the water will ascend still higher. 
 
 <d 
 
 Suppose the bore were tapered, therefore, like the 
 spindle formed by the revolution of a rectangular 
 
360 ELEMENTS OF 
 
 hyperbola about its asymptote, when the diameter 
 exceeds not four-tenths of an inch, the fluid would 
 remain suspended indefinitely at any altitude. If 
 the upper orifice were sufficiently small, it is ob- 
 vious that capillary action might even surpass the 
 power of atmospherical pressure. But every prac- 
 tical effect is produced within this limit. Assu- 
 ming it as equal to a column of 34 feet of water, or 
 nearly 400 inches, the width of bore answering such 
 an ascent would be only the ten thousandth part of 
 an inch. 
 
 Capillary action is not confined to glass tubes ; 
 but is exerted among all substances which are per- 
 forated by pores, or subdivided by fissures or inter- 
 stices. On this power, depend chiefly the functions 
 of the excretory vascular system in plants and ani- 
 mals j and hence likewise the ascent of humidity 
 through the shivered fragments of rocks, broken 
 potsherds, gravel, earth, and sand. Thus, if the 
 pores of the human skin were no finer than the three 
 thousandth part of an inch in diameter, they would 
 yet be sufficient to support lymph to an altitude of 
 120 inches, or 10 feet, or much higher indeed than 
 is required for any individual. The rejection of the 
 perspirable matter from those external mouths, must 
 occasion a continued flow of the liquid from the 
 lower and wider trunks of the capillary vessels, aided 
 no doubt by a connected chain of alternate contrac- 
 tions and dilatations, extending through their mus- 
 
NATURAL PHILOSOPHY. 
 
 cular structure. The pores in the leaves of trees 
 and tall plants must be still finer, seldom perhaps 
 exceeding the ten thousandth part of an inch. As 
 fast as the humidity is exhaled into the atmosphere, 
 it is constantly supplied by the ascent of sap from 
 the roots. 
 
 The ingenious Dr Stephen Hales, whose experi- 
 ments throw so much light on the vegetable econo- 
 my, attempted to measure the force with which the 
 sap mounts in the ramifying vessels of plants. He 
 cemented a living branch into a glass tube filled with 
 water and planted in a small cistern of mercury, 
 and he exposed the whole to the action of the sun 
 and air. As the process of vegetation advanced, the 
 water continued to rise in the tube, and was followed 
 by the quicksilver. In the space of two or three 
 days, a column of several inches was often drawn up, 
 though the effect varied considerably in different cir- 
 cumstances. But this experiment, however striking, 
 was entirely fallacious. It did not indicate, as 
 Dr Hales imagined, the power of the vegetating prin- 
 ciple ; for a dead branch would have given the very 
 same results, had the evaporation from its extreme 
 surface been sufficiently copious. 
 
 The vast energy which capillary action derives 
 from the combination of numerous and very minute 
 orifices, is illustrated and confirmed in a striking 
 manner, by an application of the Atmometer, an in- 
 strument designed to measure with accuracy the 
 
362 ELEMENTS Of 
 
 quantity of evaporation in a given time. A thin 
 ball of porous earthen-ware, about three inches in 
 diameter, is cemented by its narrow neck to a glass 
 tube, a quarter of an inch wide, and three feet long. 
 The whole cavity of the ball and of the tube being 
 now filled with recently distilled water, is inverted, 
 and set upright in a small cistern of quicksilver. 
 The surface of the ball soon loses its glistening hu- 
 mid appearance ; but the evaporation continues just 
 as at first, and the water rises gradually in the tube, 
 to supply the incessant consumption. The quicksil- 
 ver follows the water, and mounts, with greater or 
 less rapidity, according to the slenderness of the 
 tube, and the dryness of the encircling air of the 
 room, from 2 to 5 or even 8 inches every day. This 
 ascension is nearly uniform for some time, till the 
 quicksilver has gained an elevation of perhaps 20 
 inches. The water having then its atmospheric 
 pressure lessened, yields a portion of its contained 
 air, which, collecting within the ball into diffuse va- 
 pour, checks by its elasticity in some degree the 
 rise of the water and quicksilver in the tube. How- 
 ever, this ponderous fluid still continues to mount, 
 though more slowly than before ; and, in the space 
 perhaps of ten days, it stands scarcely an inch below 
 the level of the barometric column. In this situa- 
 tion it remains, till the whole of the water within 
 the ball has evaporated, and the pores are again open- 
 ed to the admission of the external air. When this 
 
NATURAL PHILOSOPHY. 3f)3 
 
 event takes place, the mercury in the tube suddenly 
 falls down into its cistern. 
 
 The attraction of the very fine pores of the At- 
 mometer is thus more than sufficient to support a 
 load of mercury equal to that of 400 inches of wa- 
 ter. Those pores are hence smaller than the ten 
 thousandth part of an inch. Hollow balls made of 
 coarser clay would no doubt have less effect, and a 
 gradation of power might probably be traced among 
 them. 
 
 We may hence conceive the rise of water 
 through successive strata of gravel, coarse sand, 
 fine sand, loam, and even clay. If the gravel were 
 divided into spaces of the 100th part of an inch, the 
 water would ascend more than four inches ; but sup- 
 posing the particles of the coarse sand to be the 
 500th part of an inch, it would mount through a bed 
 of sixteen inches of this material. Assuming the 
 fine sand to have interstices of the 2,500th part of 
 an inch, the humidity would be drawn up through a 
 new stratum of seven feet thick ; and if the pores of 
 the loam were only the 10,000th part of an inch, it 
 would gain the farther height of twenty-five feet and 
 a half through that soft mass. The clay would re- 
 tain the moisture at a greater altitude, and in this 
 way each stratum of finer pulverization might suc- 
 cessively raise the moisture still higher. But though 
 the extreme subdivision of the particles may carry 
 the elevation almost to unlimited extent, yet will it 
 
364 ELEMENTS OF 
 
 also retard the insinuation of the water. Hence 
 the use of clay in puddling or choking up the gross- 
 er pores, which might favour the efflux from a dam. 
 
 Hitherto we have considered the action of the in- 
 terior surface of the tube upon the proximate film of 
 liquid, which occasions this to spread laterally, and 
 therefore to mount upwards, as counterbalanced by 
 the mere weight of the adherent and included co- 
 lumn of water. But the mutual cohesion of the par- 
 ticles of the fluid brings another force into play, ex- 
 erted in producing precisely the opposite effect. The 
 column contained within the tube, being thus entire- 
 ly detached from the general fluid mass, wants the 
 lateral attraction which held it before in equipoise. 
 This deficient power must hence occasion the same 
 result as if an equal and opposite pressure from the 
 sides of the tube had come into action, causing the 
 column to retreat and subside towards the body of 
 the fluid. 
 
 It is hence not the whole attraction of the glass 
 to water, but merely its excess above half the cohe- 
 sion subsisting among the particles of the liquid it- 
 self exerted internally, or on the one side, that is 
 employed in producing the capillary ascension. If 
 this attraction were equal to its antagonist cohesion, 
 the fluid would remain balanced at the common level. 
 On the other hand, if the attraction of the sides of 
 the tube be less than the corpuscular cohesion of the 
 
NATURAL PHILOSOPHY. 365 
 
 fluid, it will sink proportionally below the general 
 surface. The depression of mercury in capillary 
 tubes is not occasioned therefore by any supposed re- 
 pulsion of the glass, but is the necessary result of the 
 preponderating cohesion of its own particles. 
 
 Hence dew or rain collects into globules, or 
 spreads into rounded heaps, on the glossy surface 
 of a cabbage leaf. Water likewise runs into such 
 drops, like the globules of quicksilver, when thrown 
 upon a red-hot surface of glass, or brick, or metal, 
 the vapour forming around it a sort of atmosphere, 
 which prevents its contact and adhesion. In this 
 way the surface of copper is congealed into a thin 
 plate, which being removed is immediately succeed- 
 ed by another concretion derived from the same af- 
 fusion of water. In like manner, quicksilver affects 
 a globular shape, when poured upon a table of wood 
 or marble. A ball of crude platinum having a spe- 
 cific gravity of 14.3, or little more than that of mer- 
 cury, will even float upon it, being indirectly sus- 
 tained by the cohesion and retreat of the liquid me- 
 dium. When this floating mass, however, is once 
 pressed down, the mercury closes over it, and pre- 
 vents it from rising again. 
 
 To the powerful cohesion of their particles, is 
 owing the property possessed by all liquids, of re- 
 maining heaped up to a sensible height above the 
 brims of the vessels which contain them, whether 
 formed of glass or of metal. Small bits of cork and 
 
366 ELEMENTS OF 
 
 other light substances set to float in water are ob- 
 served gradually to approach ; a fact which has often 
 been vaguely ascribed to the distant attraction of 
 those bodies. This very minute force, however, is 
 scarcely appreciable, and the tendency of the corks 
 to unite is really caused by the portion of water ad- 
 hering to their sides, which presents a continuous 
 sinuosity, pressing the prominent liquid to a mutual 
 junction. 
 
 The contour of the top of the included column 
 likewise indicates the relations of those opposite 
 forces. When they are equal, it has no curvature 
 at all ; but if the attraction of the glass be superior 
 to half the cohesion of the particles of the fluid, the 
 upper surface becomes concave ; and if this cohesive 
 force prevail, it will assume a convex outline. The 
 curvature of the boundary has, at its middle point, a 
 radius proportional to the width of the tube. This 
 external curve may be considered, when the bore is 
 small, as merely an arc of a circle ; and consequent- 
 ly its margin will, in every change of dimensions, 
 meet the inner surface at the same angle. 
 
 The attraction which fluids have to solids, differs 
 little in general from half the cohesion of their mo- 
 bile particles. Hence, of those forces, a very small 
 portion is ever exerted in elevating or depressing the 
 fluid columns within capillary tubes. The attrac- 
 tion to a solid is ascertained, by observing the weight 
 required to detach a given circular disc from the ho- 
 
NATURAL PHILOSOPHY. 367 
 
 rizontal surface of the fluid. Thus, I find, that, at 
 the ordinary temperature, a piece of plate-glass ad- 
 heres to water, with a force equal to 40 grains for 
 every square inch. But the mutual cohesion of the 
 aqueous particjes in the same space must amount to 
 75 grains. Water is hence pressed upwards in a ca- 
 pillary tube, by the excess of 40 above the half of 
 75, or 2^- grains for each inch of internal surface, as 
 already stated. 1 have likewise found, that a force 
 equal to 144 grains for each square inch is required 
 to detach a plate of glass from a surface of mercury. 
 The mutual attraction of the particles of this fluid, 
 must therefore be measured by 31 3f grains. The 
 mercury is hence depressed in a capillary tube, by 
 the defect of attraction, or the force of 12f grains, 
 which corresponds to the 267th part of a cubic inch. 
 
 The attraction between solids and fluids is great- 
 ly diminished by heat. Thus, at the temperature 
 of 48 by Fahrenheit's scale, water adheres to glass 
 with the force of 42 grains for each square inch, but 
 at 97 with only 32.9 grains. The attraction of 
 mercury to glass at 51 amounts to 148.4 grains, 
 and at 81 only to 130 grains. A similar alteration, 
 differing only in proportion, is manifested by other 
 fluids. Thus, the attraction of sulphuric acid to 
 glass at the temperature of 48 is 48.2 grains, but 
 reduced to 40.1 grains at 140. 
 
 It may be hence inferred, that the mutual cohe- 
 sion of the particles of fluids is likewise affected by 
 
S6S ELEMENTS OF 
 
 change of temperature. But what appears very re- 
 markable, the capillary action is not subject to any 
 sensible influence from that cause. The alterations 
 which heat induces, on those powers of attraction 
 and repulsion, must therefore be such^as to leave al- 
 ways the same differences, whether of excess or de- 
 fect. 
 
 On these principles we may explain why a fine 
 needle will swim on the surface of water, which al- 
 ways forms a small pit for it. The attraction among 
 the particles of the fluid occasions the same effect as 
 an actual repulsion, pressing upwards with a force 
 inversely proportional to the breadth of the needle 
 or the separation of the water. Hence the various 
 slender insects which commonly frequent pools easily 
 walk on the surface without being wetted, the water 
 retreating from the contact of their glossy and fila- 
 mentous limbs. The same property defends the 
 feathered tribe against the injury of the rain ; and 
 the dust of the butterflies wings affords those slender 
 membranes a similar protection. We may remark 
 that the needle can hardly be made to swim on al- 
 cohol or strong ardent spirit. The only way of 
 succeeding is, to set the first needle afloat on a small 
 body of water in a glass, and afterwards introduce 
 the alcohol gently by means of a tube at the bottom. 
 
 It is difficult to measure directly the cohesion of 
 fluids ; but an approximation may be derived from 
 the magnitude of drops and the thickness of liquid 
 
NATURAL PHILOSOPHY. 3()<J 
 
 sheets, heaped upon an horizontal surface. In this 
 view, let us trace the formation of a drop of water, 
 as it slowly collects at the end of a capillary syphon. 
 The mutual attraction of the particles always rounds 
 the under part of the pendant fluid, which continues 
 to lengthen till its accumulating weight begins to 
 overcome the cohesion of the particles. But this 
 force being 75 grains for each horizontal square inch, 
 while a cubical inch of water weighs 252.5 grains, 
 must correspond to the pull of a cylinder of .18 inch 
 high, which will influence also the breadth of the 
 pendant liquid. Beyond this limit a separation will 
 ensue, when the cylinder merges into a sphere a 
 little wider, or about two-tenths of an inch diame- 
 ter. 
 
 The cohesion among the particles of alcohol and 
 of sulphuric acid, being respectively the fifth part of 
 215 and of 460 grains, the weight of a cubic inch 
 of each of those fluids, a drop of them should mea- 
 sure .17 of an inch in diameter. 
 
 Again, since the cohesive power of quicksilver, 
 at the ordinary temperature, amounts to 312 grains 
 on each horizontal inch, while a cubic inch of that 
 ponderous liquid weighs 3424 grains, the drop must 
 separate when its mean depth approaches to the nine- 
 hundredth part of an inch. 
 
 These results agree sufficiently with observation. 
 They give 91, 23, 40, and 76 grains, for the se- 
 veral weights of an hundred drops of water, of al- 
 VOL. i. 2 A 
 
370 ELEMENTS OF 
 
 cohol, of sulphuric acid, and of mercury. In the 
 higher temperatures, the drops of all liquids must, 
 from their decreasing cohesion, be considerably di- 
 minished. A drop, or, in the language of the apo- 
 thecaries, a gutty is hence incorrectly assumed as, in 
 all cases, equivalent to the weight of a grain. In- 
 dependent of the influence of heat, we have seen that 
 they differ very considerably, and that a drop of al- 
 cohol, the most ordinary solvent in tinctures, is not 
 only much lighter, but even smaller, than one of 
 water. 
 
NATURAL PHILOSOPHY. 371 
 
 VII. HYDRODYNAMICS 
 
 Consists in the application of the principles of Dy- 
 namics to the impulsion and flow of water and other 
 liquids : It therefore explains all the various mo- 
 tions of such liquids, and of fluids in general. This 
 important branch of science has, in reference to the 
 construction and performance of water-works, been 
 commonly termed Hydraulics ; but it should then 
 comprise, in strictness, the joint operation both of air 
 and of water. 
 
 The theory of Hydrodynamics is entirely of mo- 
 dern origin, and still wants that perfection which, in 
 unfolding the simple pressure of fluids, Hydrosta- 
 tics has attained. The forces evolved in the in- 
 ternal motion of such a system of particles are so 
 excessively complicated, that it becomes necessary 
 to adopt some auxiliary hypothesis or abbreviation, 
 in order to obtain even approximate results. Theory 
 must therefore, as in the practice of the mechanical 
 arts, be rectified, by a constant appeal to experience. 
 In this way, the most important accessions have been 
 lately made to the science of Hydrodynamics. 
 
 The fundamental problem in the motion of fluids 
 is to determine the efflux from a small aperture in 
 the bottom or the sides of the containing vessel. 
 Suppose a cylinder ABCD (fig. 152.) of the thin- 
 
ELEMENTS OF 
 
 nest materials, holding water, to have a very minute 
 hole opened at F, giving passage to the liquid. The 
 whole body of the water may be conceived to remain 
 at rest, while each particle in succession, as it reaches 
 this opening, suffers the momentary pressure which 
 causes its expulsion. The action of the accelerating 
 force which incites the motion, is hence confined to 
 the mere interval between the several particles. The 
 force itself is evidently proportional to the number of 
 particles arranged in the vertical line EF : Each par- 
 ticle is at first urged equally on every side ; but when 
 it comes within the limits of the aperture, the pressure 
 in front ceases, and it is pushed forwards, by the whole 
 weight of the incumbent row. But, from the prin- 
 ciples of dynamics, the square of the velocity generat- 
 ed, is proportional to the product of the accelerating- 
 force into the space. If h denote the height of the 
 cylinder of water, and n the number of particles in 
 
 the perpendicular range j then - will be the mutual 
 
 interval, and Xn, or h, will express the square of 
 
 the velocity of ejection. But 1 X h, or A, repre- 
 sents likewise the square of the velocity, which a par- 
 ticle would acquire by its mere gravity, in falling 
 through the whole altitude of the fluid. The velocity 
 of expulsion in feet each second is therefore express- 
 ed by S^/h. If the aperture be enlarged beyond 
 a mere point, each particle in its parallel access will 
 
NATURAL PHILOSOPHY. 
 
 evidently escape with the same celerity, and hence 
 the quantity of discharge will be proportional to 
 the surface of the section. The effect must be 
 likewise the same, whether the aperture be made at 
 the bottom, or in the side of the vessel, or be placed 
 with any obliquity ; since the impelling force is ulti- 
 mately directed at right angles to it. If the jet be 
 thrown exactly vertical, it ought evidently to mount 
 to the level of the cistern. 
 
 The accuracy of this general conclusion, however, 
 is affected by several assumptions. It supposes the 
 aperture to be suddenly opened, and indicates only 
 the first ejaculation of the water. But the whole 
 mass of fluid is, in consequence of this act, put in mo- 
 tion ; the particles advance slowly to the orifice, but 
 are not urged now in their exit by the same vigor- 
 ous pressure. The mutual attraction of the liquid 
 particles, besides, greatly impedes their separation 
 and escape, especially in the case of a very small aper- 
 ture. But, what more than any other cause dimi- 
 nishes the quantity of efflux, is the tendency of the 
 various streamlets towards a common centre, which 
 occasions them to shoot beyond the edge of the ori- 
 fice, and therefore to contract the channel of the cur- 
 rent. The effect of these different distributing 
 causes can only be deduced, from a comparison with 
 actual experiment. 
 
 A simple consideration leads to the same result : 
 Suppose a narrow vertical tube were inserted in the 
 
ELEMENTS OF 
 
 orifice at the side of the cylinder. The water would 
 mount to the elevation of the opposite surface ; but 
 the addition of the tube could have no effect what- 
 ever in promoting the ascent of the fluid, since the 
 resistance of its perpendicular sides could furnish 
 only an horizontal pressure. If the tube were anni- 
 hilated, the column of water would still remain the 
 same as before. That pressure which sustained the 
 fluid at rest, might be expected to raise it into the 
 like position. 
 
 This reasoning seems plausible, but is perhaps not 
 very conclusive. But the same inference can be 
 drawn from a stricter view of the question. Con- 
 ceive the jet to be directed perpendicularly : The 
 motion of the water is then exactly reversed, and its 
 descent is followed by a corresponding ascent. Hence 
 the common centre of gravity must rise just as much 
 in the second instant, as it fell during the first. 
 Thus, while the surface of the water in the cylinder 
 sinks from AB (fig. 153.) into the position GH, 
 an equal portion IKLM is projected upwards. The 
 centre of gravity of the mass first sinks from O to P, 
 and then springs up again to O. Hence OP is half 
 the depth of subsidence AG, and the product of 
 whole ABDC into OP is equal to the product of 
 the portion IKLM into its elevation above O ; 
 wherefore the product of AC into OP is equal to 
 the product of AG into the altitude of the project- 
 ed portion of fluid above the common centre of gra- 
 
NATURAL PHILOSOPHY. 375 
 
 Yity. This altitude is consequently the half of AC. 
 The jet should therefore attain the height of the 
 head AB. 
 
 A similar conclusion is deduced from the principle 
 of the Conservation of Living Forces, which is pe- 
 culiarly applicable to such an elastic body as water. 
 In subsiding from AB to GH, the fluid mass de- 
 scribes a space AG, with an accelerating velocity, the 
 square of which must be proportional to AG. But 
 the portion of water projected upwards from the ori- 
 fice must have risen with a retarding celerity, the 
 square of which, at its commencement, being pro- 
 portional to the altitude obtained. Now AC X AG = 
 IKx AC, and therefore the jet should spring to the 
 height of the surface, AB. 
 
 It is obvious that pressure might be substituted 
 for the weight of the incumbent cylinder of water. 
 The jet thus produced will rise to the altitude of 
 an equiponderant column of fluid. If the density 
 of the compressing mass be different from that of 
 the projected liquid, the velocity of discharge must 
 likewise vary. Thus, a column of mercury acting 
 upon water, will expel it with a celerity nearly four 
 times greater, or what is due to an altitude repeated 
 13^ times. On the contrary, the inferior action of 
 a cylinder of water upon mercury, could only raise 
 this to the 13 part of its own height. 
 
 Suppose water to spout at right angles from a 
 small aperture E (fig. 154.) in the side of an up- 
 
ELEMENTS OF 
 
 right cylindrical or prismatic vessel A BCD. It 
 will be projected with a velocity which would carry 
 it horizontally through double the space BE, in 
 the time that a body takes to fall from B to E. 
 But it now descends with an accelerated motion 
 to the level of DC, while it continues its uni- 
 form horizontal flight, thus describing a parabo- 
 la which meets the ground in G. Wherefore 
 v/ BE : y/ EC : : 2BE : CG, and BE : EC : : 
 4BE* : CG*; consequently CG z =4BEx EC, and 
 having, upon the diameter BC, described a semi- 
 circle, CG* = 4EF% and CG = 2EF. Thus, the 
 horizontal range of the jet is double of the ordinate 
 EF. This range is hence the greatest when the 
 aperture is in E, the middle of the altitude BC be- 
 ing then equal to the altitude itself. In other 
 cases there are two apertures, Land H, equidis- 
 tant from the middle, which give the same range 
 CK=2LM=2HL 
 
 Let the aperture be placed at the side close to the 
 bottom of the vessel, but directed at different eleva- 
 tions. It is evident, from the theory of projectiles, 
 that the extension of the level AB (fig. 155.) will de- 
 fine the directrix of all the parabolas described by the 
 jets. If a circle be described from the centre C with 
 the radius CB, it will trace the foci. Let the tan- 
 gent CE indicate the elevation, and draw CF ma- 
 king the angle ECF equal to BCE ; let fall the per- 
 pendicular FH, and take HG equal to it ; the dis- 
 
NATURAL PHILOSOPHY. 
 
 tance CG will mark the range of the jet. But 
 CG=2CH=2CFx cosFCH = <2CB x sin 2ECH. 
 Hence, when the elevation is 45, the focus of the 
 parabola falls on the horizontal line at F, and the 
 range CK is then the greatest possible, being double 
 of the altitude CB. 
 
 Since the velocity of emission from any orifice is 
 8^/k, the square of it is 647*. If a parabola, there- 
 fore, be described upon the altitude BC (fig. 156.) 
 as an axis, and with a parameter of 64- feet, the ordi- 
 nate GH will express, in feet each second, the flow 
 from the aperture GC. Suppose the side of the ves- 
 sel were slit down its whole height, the efflux of wa- 
 ter would then be denoted by the area of the parabola. 
 But if an opening equal to that slit were made at 
 the bottom, the discharge of fluid would be re- 
 presented by the circumscribing rectangle BIHC. 
 Wherefore the discharge by a perpendicular slit, is 
 only two-thirds of what is effected by an horizontal 
 aperture of the same dimensions. 
 
 The efflux from an orifice at the bottom of a ves- 
 sel being always as the square of the altitude, it 
 may, by the accumulate elevation of the fluid, be 
 brought to correspond with any given measure. 
 When a stream of water flows into the cistern just 
 as fast as it is again discharged, the fluid must evi- 
 dently maintain a constant level, from which the 
 celerity is easily computed. Suppose this altitude 
 to be three inches, the velocity of discharge would 
 
378 ELEMENTS OF 
 
 be Sy/^, or four feet each second, that is 240 feet 
 every minute. The delivery would, therefore, be at 
 the rate of a cubic foot a minute, if the aperture 
 were only the 240th part of a square foot, or three - 
 fifths of a square inch. This opening corresponds 
 to a circle of which the diameter is .874 parts of an 
 inch. 
 
 Hence, a very convenient modulus or measure, 
 for ascertaining readily the quantity of water deli- 
 vered in a minute by any pipe or conduit. Let a 
 cylinder of tin, above four feet high and one foot 
 wide, be fitted with a float, bearing a slender divi- 
 ded rod, which passes through a hole in the middle 
 of a bar stretched across the open top. This rod is 
 marked 1, 2, 3 5 4 at the distances of 3, 12, 27 and 
 48 inches, being the fourth parts of the squares of 
 the first numbers in feet. The divisions may like- 
 wise be quartered by beginning with the sixteenth 
 part of 8, 12, 27 and 48 inches ; and continuing 
 the process by multiplying the squares of 5, 6, &c., 
 as far as 16 by 3, and dividing the products again 
 by 16. The cylinder, having a circular hole with a 
 fine edge cut in the bottom, being now set upright 
 at some distance from the ground to receive the cur- 
 rent, the water mounts gradually in the vessel, till 
 it has gained an altitude sufficient to cause a dis- 
 charge equal to the whole influx. The rod then 
 indicates the number of cubic feet received every 
 minute. 
 
NATURAL PHILOSOPHY. 379 
 
 But the application of this modulus may be ex- 
 tended, by adapting to it a series of apertures 
 which screw into the bottom of the cylinder. Let 
 them have successively these diameters in inches 
 .437 _ .874 1.748 and 3.496. The rod 
 would then be square, and marked on all its four 
 sides ; the divisions on the first side corresponding 
 to a discharge of , , f , and one cubic feet, those 
 on the next to 1, 2, 3, 4, those on the third side to 
 4, 8, 12, and 16, and those on the fourth side inti- 
 mating a flow of 16, 32, 48 and 64 cubic feet every 
 minute. From these data, it is easy to construct 
 other moduli suited to particular limits. The cor- 
 rection required in the size of the apertures, to ac- 
 commodate theory with practice, will be afterwards 
 explained. 
 
 The native troops in India are accustomed to re- 
 lieve guard, on the sinking of a perforated metallic 
 cup in a vase of water. As a converse of this, the 
 ancients, instead of a sand-glass, employed a cistern, 
 from which the water trickled through a small hole 
 at the bottom, under the name of a Clepsydra or 
 water -clock, to measure time. In a cylinder, the 
 flow would evidently diminish, as the level of the sur- 
 face is incessantly lowered. To procure an uniform 
 descent of the water, it would be necessary to adopt 
 the figure of a conoid of the parabolic kind, each 
 circular section of which is proportional to the 
 square root of the corresponding altitude : Suppose 
 this were 24 feet, and the diameter at the top 13.28 
 
'380 ELEMENTS OF 
 
 feet, the diameters of the successive sections being 
 always six times the fourth root of the altitude. 
 The velocity of efflux would then be 8^23 = 39.192 
 feet each second. If the water sank at the rate of 
 a foot every hour, the width of the orifice would 
 be to the extreme diameter or 13.28, as 1 to 
 60^/89.192. This gives an aperture of .424 parts 
 of an inch. A conoid of such dimensions would, as 
 represented in fig. 156, therefore, answer correctly 
 as a Clepsydra, the equable subsidence of a float 
 marking the series of twenty-four hours in a natu- 
 ral day. This float being fastened to a thread 
 wound about a cylindrical barrel, of a foot in cir- 
 cumference, would carry the index of a dial regular- 
 ly round. 
 
 Let it be required to find the time which water 
 takes to flow out of a cylindrical vessel, through a 
 hole in the bottom. Since the velocity of projec- 
 tion, after the level has subsided to G, (fig. 157.) is 
 in the subduplicate ratio of the altitude GC, it may 
 be represented by the ordinate GH of a parabola 
 applied to the axis CB. Wherefore, the time of 
 discharging the thin stratum Gg is expressed by 
 
 ; but, having drawn the normal HN to the 
 
 curve, the elementary triangle Hy^ is obviously si- 
 milar to NGH, and Hy or Gg : yh, : : GH : GN ; 
 
 whence the element of the time or TY = f 
 
 Now, the subnormal GN is constant, being equal to 
 
NATURAL PHILOSOPHY. 381 
 
 half the parameter of the parabola ; consequently 
 the portions of time ~ 9 taken collectively during 
 
 the descent of the surface of the water from B to G, 
 
 ICT? 
 will be denoted by n, the whole time of the flow 
 
 BE 
 being -~. Hence, if the altitude BC were divided 
 
 into equal parts by a number of ordinates, the dif- 
 ferences between these would mark the intervals of 
 the times of descent. Suppose an extreme tangent 
 were applied at E, meeting the extension of DC in 
 L, from which is drawn LM parallel to the axis CB. 
 The time of the whole discharge, if the water had 
 continued to flow with its initial velocity, will be de- 
 noted by EM. But the tangent would cut BC pro- 
 duced at an equal distance beyond C, and conse- 
 quently LM must bisect EB. The discharge of the 
 fluid requires, therefore, double the time which 
 would have sufficed, if the efflux had maintained its 
 first intensity. 
 
 In general, let H and h denote in feet the alti- 
 tudes of the column of water in two different situa- 
 tions, and n the ratio of the diameter of the cylinder 
 to that of the aperture. The initial velocity is 8^/H, 
 and hence the whole time of discharge from the ves- 
 sel, had the flow continued uniform, would have been 
 
 o But the time of actual discharge 
 
 o 
 
382 ELEMENTS OF 
 
 H* 
 
 is the double of this quantity, or ^/H. Now the 
 
 4* 
 
 intermediate time during which the surface of the 
 water descends from B to G, being proportional to 
 EK, the excess of the ordinate BE above GH, 
 which are themselves in the subduplicate ratio of the 
 corresponding heights AC and CGH ; it follows 
 that </H ^/h may be substituted instead of 
 
 n 
 
 The expression then becomes (^/H ^/h,) which 
 
 denotes, in seconds, the time of the subsidence of 
 the level of the water in the cistern from B to G. 
 
 In these investigations, we have considered, for 
 the sake of simplification, the aperture of projec- 
 tion as extremely small. But when it bears a sensi- 
 ble proportion to the width of the vessel, the celeri- 
 ty of discharge will evidently be diminished. The 
 accelerating pressure may be conceived as only the 
 excess of the weight of the contained water above 
 that of the column incumbent over the orifice. 
 The velocity of emission is hence expressed by 
 
 -l). Where- 
 fore the time of subsidence from the level H to h 
 
 win be 
 
 When an aperture is made in the side of a vessel 
 holding any fluid, the equilibrium of hydrostatic 
 pressure must evidently be disturbed. This force, 
 
NATURAL PHILOSOPHY. 383 
 
 which was exerted equally all around the inner sur- 
 face, now ceases to act at the opening ; or rather it 
 expends its action there, in projecting the uncovered 
 portion of the fluid. The reaction, or incessant re- 
 coil sustained by the general mass, will hence be the 
 same, as the external pressure of a column of the 
 fluid of equal altitude directed against the section of 
 the orifice. If the vessel have therefore a cylindri- 
 cal form, and turn freely about a vertical axis, while 
 it carries under it two horizontal branches, extend- 
 ing both ways, each of them perforated near the end 
 at right angles to their plane with the axis, but in 
 opposite sides ; the machine will be driven back- 
 wards by the spouting fluid, and forced to revolve as 
 long as the action continues. The force thus exerted 
 is equal to the weight of a column of the fluid ha- 
 ving the general altitude, with the two apertures for 
 its base. But the effect becomes augmented in this 
 case, by the length of the arms or levers to which 
 the power is applied. 
 
 This principle has been long successfully adopted 
 in the construction of a very simple but efficient wa- 
 ter-mill, commonly designated Barker's mill. It is 
 capable, however, of various other useful and im- 
 portant applications, which remain still to be carried 
 into execution. 
 
 The action of centrifugal force may be advantage- 
 ously employed, either in raising water to a higher 
 level, or in urging its flow from an orifice. Let a 
 cylindrical vessel, (fig. 158.) carrying an horizontal 
 
ELEMENTS OF 
 
 arm or tube DE, turned up at the end into a per- 
 pendicular branch EG, be made to revolve quickly 
 about its axis AB. If while at rest, it contained 
 water to the height C, this would obviously stand at 
 the same level in the opposite tube at F. But after 
 the circulation of the system commenced, a differ- 
 ent arrangement would take place : The several 
 particles in the horizontal column will be driven for- 
 ward by the centrifugal effort, which, as they re- 
 volve all in the same time, is therefore in the com- 
 pound ratio of their gravity and of their distance 
 from the axis of motion. This force will conse- 
 quently increase uniformly from B, along the whole 
 range, to the extremity E. Wherefore the action ex- 
 erted at this limit will be the same as the weight or 
 pressure of a vertical column, whose altitude P is to 
 the length of the arm BE, as the centrifugal force 
 at the middle point O is to the power of gravitation. 
 This incumbent load hence raises and supports an 
 equal column above the level F, the small vertical 
 portion FE being sustained merely by CB, and not 
 affected by any centrifugal tendency. Let BE be 
 denoted by r, and the time of revolution in seconds 
 
 5.1-r 
 
 by t ; the centrifugal force at O is, therefore, -~- 
 
 4 1 * 
 
 5r 
 
 , and the altitude of the equivalent column, or 
 
 , is . r = y . If the vertical tube EG 
 
NATURAL PHILOSOPHY. 385 
 
 were shut above F, and an aperture made at that 
 point, the rush of the fluid along the pipe DE, which 
 corresponded to the pressure of GF, would be ex- 
 erted in projecting the fluid. The velocity of ex- 
 pulsion would hence be . But if the orifice were 
 
 * 
 
 opened at E, the celerity of emission would be far- 
 ther augmented by the pressure of FE. In Barker's 
 mill, the action is hence apparently augmented by 
 the centrifugal force brought into play by the revo- 
 lution of the machine. But this force being gained 
 only at the expense of the pressure of the water in 
 the cistern AB, cannot affect the absolute quantity 
 of performance. 
 
 Suppose perpendicular tubes (fig. 159.) HL, IM, 
 KN, &c. were erected at equal distances from the 
 axis AB along the horizontal area BK. Since the 
 time of revolution is constant, the ascents PL, AM, 
 and RN, &c. of the fluid must be proportional to r*, 
 or to the squares of the several distances BH, BI, 
 and BK, &c. PL, QM, and RN, &c., are hence as 
 the numbers 1, 4, 9, &c. A parabola described from 
 the axis AB through the vertex C, would, therefore, 
 pass through the several points L, M, N, &c. Con- 
 ceive the vertical pipes to be united into one mass, 
 and their summits will trace the curve of a parabola. 
 Obliterate the sides of those tubes altogether, and 
 the surface of the fluid, extending (as in fig. 160.) 
 on every side of the axis, will, by its circumvolution, 
 
 VOL. i. 2s 
 
386 ELEMENTS OF 
 
 form the cavity of a parabolic conoid. The curve 
 will not be affected by the depth of the water in the 
 cistern, since of each perpendicular column all the 
 particles situated below the vertex must receive the 
 same horizontal thrust, derived from centrifugal ac- 
 tion, to enable them to support the additional verti- 
 cal pressure. The parameter of the parabola de- 
 pends merely on the time of circulation, being equal 
 in feet to eight-fifths of the square of the number of 
 
 Qtf.r 
 seconds. Employing the usual notation, t~ - and 
 
 
 , 
 , or very nearly ; whence the para- 
 
 meter is denoted by or ( - | . Wherefore, 
 J v* \v J 
 
 the central depression of the water, being from the 
 property of the curve equal to r z , divided by the 
 
 parameter, is -rrr or ( - J . The depth of the cavi- 
 
 ty is thus independent of its width, being propor- 
 tional to the square of the exterior velocity. 
 
 Next, let a reflected pipe rise obliquely, as in 
 fig. 162. The horizontal pressure at F is denoted 
 
 by , and that exerted at G by - . But as 
 
 ot of 
 
 long as the continuity of the water included between 
 F and G is maintained by its cohesion, the centrifu- 
 gal force directed to the middle point O may be 
 considered as what sustains the oblique column FG, 
 
NATURAL PHILOSOPHY. 3SJ 
 
 or an equiponderant vertical column GH. Whence 
 ^r.PO* = GH, or A.. PO = FG . sin GFH ; 
 but PO = CF + FG . cos GFH, and therefore 
 -^ (CF + FG . cos GFH)* = FG . sin GFH. 
 
 Let FG be supposed to rise from the bottom of the 
 axis AB, near which level the water of the cistern 
 has likewise subsided ; the expression will be- 
 
 come ~ (FG . cos GFH)- zz FG, . sin GFH, 
 or _A_ . FG*. cos* GFHzzFG.sin GFH. Where- 
 
 fore FG . cos* GFH = - . sin GFH, or FG= 
 
 5 
 
 sin GFH 32t> tan GFH 
 
 A he centntu- 
 
 . - rTT^TT z -- TTrvrr. 
 5 cos* GFH 5 cos GFH 
 
 gal action is thus unequally exerted along a straight 
 tube obliquely placed. 
 
 Let it now be required to find a curve, in every 
 part of which that force has the same efficacy. We 
 may here exclude the influence of hydrostatic pres- 
 sure, and consider smooth balls as occupying the 
 sides of the cavity. The centrifugal force acting 
 against the curve at C (fig. 163.) is denoted by DC, 
 and may be decomposed into EC perpendicular to 
 the curve, and DE parallel to it. This last is the 
 only force exerted in sustaining a ball at C. But 
 part of the weight of the ball being upheld by the 
 
f 
 
 388 ELEMENTS OF 
 
 curve, the efficacy of the direct centrifugal action 
 DE is proportional to DN, which must hence be 
 constant. And such is the property of the subnor- 
 mal DN in the parabola, which therefore answers 
 the conditions of the problem. Let P denote the 
 parameter of the curve, of which ADN is the axis ; 
 then DN = P, and consequently the centrifugal 
 force due to a radius DN becomes equal to the 
 power of gravity, while a ball at C is just supported, 
 
 or ^i- = 1, and 5P = St*. Wherefore if t, the 
 
 TrC 
 
 time of revolution, were equal to ^/ . P, or very 
 
 o 
 
 4 
 
 nearly- ^/ P, a ball would be upheld in any part 
 5 
 
 of the cavity of a parabolic conoid, and the slight- 
 est acceleration would be sufficient to roll it up to 
 an indefinite height. But the circumvolution of a 
 vessel of this form would have the same effect in 
 raising a sheet of water along its inner surface. If 
 the parabolic conoid were perforated at the ver- 
 tex, and a little immersed in a standing pool, it 
 would at first produce no sensible effect ; but after 
 it had quickened the time of its revolution beyond 
 the precise limit, it would draw up the water at 
 once to the brim, and disperse that liquid in a co- 
 pious horizontal shower. A machine of this kind, 
 driven by a small wind-mill, might be applied ad- 
 vantageously to the draining of marshes. 
 
NATURAL -PHILOSOPHY. 38$ 
 
 If a cylindrical glass vessel, partly filled with wa- 
 ter, be set on the middle of a whirling table, the de- 
 pression of the liquid will increase rapidly as the cir- 
 cumvolution is accelerated. With double the velo- 
 city, the depth of the cavity will be quadrupled ; 
 and with triple the velocity, it will be rendered nine 
 times greater. In urging this voraginous motion, 
 the water will continue to descend in the centre, 
 while it rises up the sides of the vessel : It will then 
 divide at the bottom, and spread more upwards, 
 forming always a parabolic conoid, but with a di- 
 minishing parameter. See fig. 161, which exhibits 
 three successive states of circumvolution. If a thin 
 ring were applied round the top of the cylinder, the 
 water in its extreme celerity would cover the inside 
 of the cylinder with a stratum of almost equal thick- 
 ness. If a hollow glass globe be partly filled with 
 water, and made to whirl very rapidly, the fluid will 
 separate, leaving a vacant cylindrical space around the 
 axis ; and if a portion of quicksilver be introduced, 
 it will rise up along the inside, mounting to the wi- 
 dest part of the sphere, and there forming a beauti- 
 ful zone. Suppose a cylinder 6 inches wide and 16 
 inches high, filled with water to the altitude of 4 in- 
 ches : When the circumvolution is performed in .68", 
 the water is depressed an inch, the central part stand- 
 ing at 3f and the sides at 4 ; but when the cylinder 
 circulates in .24", the cavity sinks 8 inches in the bot- 
 tom, and rises at the sides to the height of 8 inches. 
 
390 ELEMENTS OF 
 
 If the revolution were achieved in .152", the water 
 would mount to the lips, leaving a dry circle at the 
 bottom of the vessel of 2 J inches in diameter. 
 
 Hence the origin of dimples on the surface of 
 small streams, and of whirlpools in mighty rivers 
 and narrow seas. The effect is produced by the 
 lateral attrition of adverse currents, which, extending 
 their influence from the neutral point or centre of 
 equal and opposite action, gradually convert their 
 parallel motions into a combined system of circum- 
 volution. Suppose each current had a velocity of 9 
 miles an hour, or 13.2 feet each second, and the ra- 
 dius of the whirlpool to be 100 feet : The time of 
 circulation would then be 47.6", and the depression 
 in the centre of the gulf only 2.72 feet, the parame- 
 ter of the parabola being 3672 feet. 
 
 Since the pressure of a column of water occasions 
 a corresponding flow, every current may be viewed 
 as originating from the action of such a force, and 
 can therefore be determined by the altitude of the 
 incumbent fluid. Hence the construction of Pitot's 
 tube, a very convenient small instrument for mea- 
 suring the velocity of any stream. It consists of a 
 recurved tube of glass, of which the one branch, 
 however, is much taller than the other ; the short 
 branch is bent at the top into a spreading funnel- 
 shaped mouth, which receives the direct shock of the 
 
NAT.URAL PHILOSOPHY. 39 1 
 
 water, and communicating this impression, causes a 
 proportional elevation above the common level. See 
 fig. 164. To prevent the irregular oscillation of the 
 liquid in the syphon, it may be proper to have the 
 bore much contracted in the whole of the under part 
 of either branch. This object is farther promoted 
 by covering the funnel orifice by a thin circular piece 
 of brass with a very small hole in the centre. The 
 pressure will be still propagated as before, but with 
 more steady effect. The divisions on the scale are 
 reckoned upwards from the surface of the stream. 
 To ascertain them, let the expression 8Vh~v be re- 
 
 W* S"1' ~\. * 
 
 sumed, and 647* v*, or h ^- ( ^ J , where h 
 
 denotes in feet the height of the column, and v 
 the velocity each "second. Hence the rise corre- 
 sponding to the rate of a mile an hour would be 
 
 ^ -7r; = T7^i> or 4-10ths of an inch : The 
 15 J 64 1200 
 
 scale would hence be marked 1, 2, 3, 4, 5, 6, 7 8, 9, 
 &c. miles, at the respective heights of .4-1.6-3.6 
 -6.4 -10.0 -14.5 -19-7 -25.8 -32.7, &c. inches. 
 Few rivers, therefore, would require the glass tube 
 to rise six feet above the surface of the water. The 
 instrument may be composed of a long narrow tube 
 of brass or tin, cemented to a wide cylinder of glass, 
 carrying the divisions. 
 
 A construction somewhat different might answer 
 for measuring a ship's way at sea. A long tin pipe 
 
 ( 
 
392 ELEMENTS OF 
 
 presenting its trumpet orifice under water at the 
 prow could be conducted along the side of the keel, 
 and brought up into the cabin, there to be cement- 
 ed into a wider perpendicular tube of glass. The 
 scale could be made to slide along, having been adjust- 
 ed to the level of the external water during a calm ; 
 or this level might be found, by placing another pa- 
 rallel glass tube, which has cemented to it a narrow 
 tin pipe running out below towards the bottom of 
 the vessel. 
 
 If the channel of any stream be suddenly con- 
 tracted, the water will be forced to rise above its 
 ordinary level. Suppose, for instance, that the flow 
 was at the rate of a mile an hour ; this might be con- 
 sidered as the effect of an incumbent column of four- 
 tenths of an inch. But where the section of the 
 current is reduced to one half, the resulting celerity 
 of two miles an hour would correspond to the pres- 
 sure of an altitude of 1 .6 inches. An accumulation 
 of the water to the height of the difference, or 1.2 
 inches, would hence be occasioned by this obstruc- 
 tion. In general, if v denote the velocity of a river 
 in miles every hour, a the measure of its vertical 
 section, before it suffers any impediment, and b that 
 of its contracted course ; the rise of the level will be 
 
 v* /a z b z \ 
 expressed in feet by ^ g*~~/- Jt ma y* in P rac " 
 
 tice, however, be more accurate to admit a slight 
 
NATURAL PHILOSOPHY. 3<|3 
 
 t; /Sa-_^*\ 
 
 modification, and assume-- ( j^ / A his for- 
 
 ou * o 
 
 mula was remarkably exemplified in the ruinous pile 
 of the old London Bridge, the piers of which were 
 so massive as to reduce the breadth of the water-course 
 from 14 parts to 3. The velocity in spring-tides is % 
 
 miles an hour ; and hence -- (~~Q / = ' ^ ee ^* 
 
 Such then was the fall, and this with a rapidity of 
 9y miles an hour. No wonder that the shooting of 
 the Bridge was sometimes attended with so much 
 hazard and danger. 
 
 All the preceding investigations are grounded on 
 legitimate theory, but some reduction is necessary in 
 bringing the results to conform with actual practice. 
 If a wide orifice at the side, but close to the bottom, 
 of a tall cylindrical vessel, containing water, be co- 
 vered by a thin plate of metal perforated by a fine 
 round hole ; on removing the finger from this aper- 
 ture, the first emission of the fluid will exactly agree 
 with calculation. But after the whole mass has ac- 
 quired motion, the various streamlets, all directed 
 towards the centre of the hole, bend at a little dis- 
 tance from its sharp edges, and escape in a contract- 
 ed current. It has been ascertained by experiment, 
 that, through such a simple aperture, the flow is on- 
 ly five-eighth parts of what theory would indicate. 
 If, therefore, we substitute five instead of eight in a 
 
394 ELEMENTS OF 
 
 preceding formula, an expression will be obtained 
 for the quantity of real water discharged from a given 
 orifice in any vessel. Let A denote the area of this 
 orifice, and h the height of the fluid ; then 5^/h.A 
 will express the efflux, in cubic feet every second, or 
 300 A^/h the quantity delivered in a minute. If, 
 while h marks the altitude of the water in feet, d 
 should measure the diameter of the circular hole in 
 inches, the discharge every minute will be represent- 
 ed by %.65%6d z ^/h, which corresponds very nearly 
 
 to 159d*v/^ m an nour - 
 
 If a cylindrical tube, whose length is rather more 
 than twice its width, be adapted to this aperture, 
 the discharge will become augmented from five parts 
 to six and a half, and will consequently amount to 
 thirteen-sixteenths of the quantity assigned by theo- 
 ry. The formula in this case will therefore be 
 390 A^/h, for every minute ; or the diameter of this 
 annexed tube or adjutage being denoted in inches by 
 d, the discharge will be 3.41 d z ^/h. 
 
 Prony has lately proposed, as a modulus for Hy- 
 draulic Operations, the quantity of water discharged 
 in the space of twenty -four hours from a wide shal- 
 low cistern, kept constantly full by an ingenious me- 
 chanical contrivance. At the depth of 5 centime- 
 tres, or 1.9685 English inches below the surface, is 
 the centre of a circular hole of 2 centimetres or 
 .7874 inches diameter, perforated through the thin 
 side of the vessel j and to this aperture a cylindrical 
 
NATURAL PHILOSOPHY. :>\)C) 
 
 tube of only .6693 inches in length must be fitted. 
 The measure which is thus delivered in a natural 
 day amounts to two moduli or 20 cubic metres, cor- 
 responding to 7H24 cubic feet, being very nearly 
 three-fourths, instead of five-eighths, of the quantity 
 assigned by rigorous theory. 
 
 The addition of the projecting pipe has in all 
 cases really the effect of extending the action of the 
 hydraulic pressure from the inner to the outer aper- 
 ture, while the cavity is kept always full by the in- 
 cumbent weight of the external atmosphere, aided 
 likewise by the adhesion of the water to the sides of 
 the tube. Accordingly, if small perforations be 
 made in this eductive pipe, the air will enter, and 
 flow along the contracted surface of the current, 
 hence occasioning a proportional diminution of the 
 quantity of discharge. On a small scale, the con- 
 traction and separation of a jet from the sides of its 
 tubulated orifice may be remarked under the ex- 
 hausted receiver of an air-pump. The efficacy of 
 the pipe thus proceeds entirely from its continuing 
 always full, while the celerity of the stream is still 
 maintained or even augmented. The quantity of 
 discharge is likewise increased with the length of 
 the pipe, at least to certain limits. Thus, from an 
 orifice of an inch diameter, the quantity of water 
 delivered, in a given time, through a spout of an 
 inch long, is to the expenditure by one of a whole 
 inch in length, as 6 to 7 
 
396 ELEMENTS OF 
 
 If the anterior part of the cylinder, which is an- 
 nexed to the aperture, be contracted in the middle, 
 so as to consist of two reversed frusta of a cone, 
 the discharge of fluid will be still exactly the same. 
 Suppose (fig. 165.) AB and EF to be each two 
 inches wide, the contraction CD 1.6 inches, the 
 length IL 2.3 inches, and LM zz 4.1 inches : 
 The effect is found to be the same as if a regular 
 cylinder of six inches length had been applied at 
 AB. But if the simple tapered projection ACDB 
 (fig. 166.) were affixed, the measure of discharge 
 would be diminished in the ratio of 4 to 3. Now, 
 the section CD being to EF as (4)* to (5) s , or as 
 16 to 25, the velocity of the stream, even at the 
 wider aperture of its exit, is consequently augment- 
 ed in the ratio of 21 \ to 25, or of 64 to 75. This 
 compound pipe, ACE, FDB, (fig. 165.) therefore, 
 not only accelerates the flow through its narrowed 
 throat CD, but even quickens the emission from the 
 wide orifice EF. 
 
 When water issues through a simple circular a- 
 perture in a thin plate, it forms a contracted vein, of 
 a conical or rather tapering funnel shape, its section 
 being reduced to very near five-eighth parts at a dis- 
 tance little more than half its diameter. If a tube 
 of this figure be annexed, the quantity of discharge 
 will come within the thirtieth part of the result of 
 theory, as computed for the exterior aperture. (See 
 fig. 167). But the adjutage being directed upwards, 
 
NATURAL PHILOSOPHY. 397 
 
 the jet will rise to fourteen-fifteenth parts of the 
 whole height of the incumbent column. 
 
 If to this tapered spout another conical tube five 
 times longer, and opening to the original width, be 
 joined, the discharge of water will be augmented 
 from 21 to 38 parts. (See fig. 168.) The effect of 
 the adjutage is now the greatest possible, the abso- 
 lute flow from the anterior aperture being only 40 
 parts according to theory. This pipe widens at an 
 angle of about three degrees ; but when the annex- 
 ed piece diverges at a greater angle, its influence be- 
 comes diminished, and appears to cease altogether at 
 an angle of 16 degrees, as in fig. 169- The stream 
 has now ceased to fill up the whole of the cavity, 
 and is consequently no longer augmented by adhe- 
 sion to the sides of the spout. 
 
 The lateral action of water flowing through a pipe 
 is evinced in a more striking manner. Let a cylin- 
 der one inch wide and three inches long be adapted 
 to an orifice at the bottom of a cistern ; and on the 
 upper side, at the distance of half an inch from its 
 origin, let a narrow arched glass tube be inserted 
 and carried down to a bason of water three feet 
 lower. When the stream is projected with a velo- 
 city of nine feet in a second, it will draw up water 
 to the height of two feet ; but if the tube be short- 
 ened within that limit, the slender column will mix 
 with the body of the current, and soon drain the 
 contents of the bason. When a conical tube, open- 
 
398 ELEMENTS OF 
 
 ing with a considerable angle of divergence, is sub- 
 stituted, a series of slender glass tubes inserted at 
 different distances from the interior aperture will be 
 found to raise the water to several heights, which 
 diminish as the stream begins to separate from the 
 sides of the spout. 
 
 This property of running water may, in various 
 situations, be turned to useful account. By con- 
 necting the edge of the stream, for instance, by 
 means of a small slanting pipe with a collection of 
 water at a lower level, this will be gradually drawn 
 up and carried away in the general current. If a 
 swift descending rivulet be made to shoot across any 
 small pool, it will sweep the water over its opposite 
 bank. Venturi, to whom we are chiefly indebted 
 for these remarks, availed himself of the rapidity 
 and lateral draught of a mill-race, to drain a marsh 
 situate considerably below it, near the city of Mo- 
 dena. 
 
 The same principle is likewise the principal cause 
 of the action of the Hungarian Blowing Machine, 
 which consists of a very tall perpendicular pipe, ter- 
 minating below in a close wide box. A stream of 
 water rushes down this shaft, drawing along with it 
 the air which enters the small holes pierced along 
 the sides, and becomes accumulated and condensed 
 in the chamber, whence it again issues in a powerful 
 blast on opening a cock. The blowing and disper- 
 sion of the spray on all sides from water-falls have a 
 
NATURAL PHILOSOPHY. 399 
 
 like origin. A body of air is involved in the broken 
 descending current : collected in the recess or bro- 
 ken cavity, it exerts its elastic efforts in every direc- 
 tion. The adhesion of running water to the sides 
 of its channel is also the main cause of those eddies, 
 which impede the general motion. 
 
 If a fluid suffers impediment while escaping at a 
 small aperture, it encounters much greater obstruc- 
 tion in affecting its passage through a train of pipes, 
 or flowing over an extended channel. This retar- 
 dation, however, is quite distinct in its nature from 
 ordinary friction. When a solid is drawn along the 
 surface of another solid body, it is virtually made to 
 ascend over a series of inclined planes, and conse- 
 quently the impediment it meets with, may be view- 
 ed as* merely equal to a certain portion of its weight, 
 independent of the rapidity or slowness of the mo- 
 tion. But a fluid, in its passage over a resisting sur- 
 face, needs not have its whole mass either elevated 
 or depressed. Those particles only which come in 
 succession to touch the solid boundary, are impeded 
 and detained. The pressure of the incumbent fluid 
 cannot affect the other particles, which are thence 
 urged equally in every direction. The loss of im- 
 pulse which a current sustains from the attrition of 
 the sides of a pipe or of a canal, is occasioned by the 
 incessant stoppage of the extreme particles. This 
 consumption of force must hence be compounded of 
 
400 ELEMENTS OF 
 
 the number of particles arrested in a given time and 
 of their velocity, and is therefore proportional to the 
 square of the velocity. But the retardation of the 
 current must likewise depend on the extent of im- 
 peding influence, that is, on the length of the pipe, 
 and on the relation which the interior surface bears 
 to its whole capacity. But the square of the velo- 
 city, with which the stream first issues from the cis- 
 tern, being proportional to the altitude of the in- 
 cumbent column, a certain part only of this constant 
 inciting force is employed in generating the initial 
 velocity, while the rest is expended in renewing the 
 velocity as fast as it expires, along the sides of the 
 channel, or in maintaining through the mass a gene- 
 ral uniform flow. 
 
 It results from some accurate experiments of Bos- 
 sut, that water has its celerity diminished eight times, 
 by passing through a tube of an inch in diameter 
 and 204 feet long. Of sixty-four parts of compres- 
 sion, one part only must therefore have created the 
 motion of the fluid, while sixty-three parts were re- 
 quired to support it. This motion is hence renewed 
 eight times during the passage of the water, or at 
 the interval of every &5 feet through the whole ex- 
 tent. In each of these successive transits, all the 
 central particles must be thrown towards the sides 
 of the pipe, whence they are again drawn into the 
 body of the stream, and there acquire new celerity. 
 The celerity is lost and regained close to the inte- 
 
NATURAL PHILOSOPHY. '101 
 
 rior surface within a very small but limited space. 
 Wherefore, from the fundamental principle of dy- 
 namics, the square of the velocity acquired or'extin- 
 guished must be as the product of the inciting force 
 into the limit of its action. The square of the ve- 
 locity hence suffers a diminution proportioned to the 
 extent of surface compared with the capacity of the 
 pipe, or it is directly as the length, and inversely as 
 the diameter. But the pressure of the water has no 
 effect whatever in causing this reduction. Thus, 
 resuming the former example, let the pipe of an inch 
 wide and 204 feet long be fed by a cistern of ten 
 inches altitude, and the quantity of discharge noted. 
 Raise this cistern now to twenty inches, and turn up, 
 by a soft bend, the farther extremity to the height of 
 ten inches, and the corresponding flow will be still 
 the same. Increase the altitude of the cistern to one 
 hundred inches, while the remote end is bent with 
 an elbow to the height of ninety inches, leaving still 
 the same excess or exciting force, and the quantity 
 of discharge will be found not to vary. 
 
 Let V denote the velocity with which water 
 would issue from a simple orifice near the bottom of 
 a cistern, and v the reduced velocity in consequence 
 of passing through an extended horizontal pipe, d 
 being the diameter of the pipe and L its length. 
 From what has been shown, it follows that v* = V* 
 
 ' and w=v v 7 The den - 
 
 VOL. T. 2 C 
 
402 ELEMENTS OF 
 
 initiator <50d?+L of the radical fraction expresses the 
 consumption of the pressure or inciting force, the 
 part 5Qd indicating the length of track which cor- 
 responds to the initial velocity. The absolute waste 
 of such force follows the measure of internal surface, 
 while the quantity of impulse must depend on the 
 charge of the pipe ; and hence the relative influence 
 
 will be computed as nd (50f/+L) to - .d, that is, as 
 
 4< 
 
 (5(k/-|-L) to \d) and is therefore expressed generally 
 
 by 1 **- 
 y 50^+L* 
 
 Since H denoting the height of the cistern, the 
 primary velocity of projection is equal to Sy/H, or 
 
 V- = 64H ; whence v* = 64H 
 
 and " = ' 
 
 this pipe were curtailed to a mere adjutage, then v* = 
 /\ii 
 
 64. pp = 64H. This coefficient 64 might, there- 
 ou 
 
 fore, be adapted to the formula ; or, if some allow- 
 ance were made for the unavoidable expense of force 
 in the transition of the water from the cistern to 
 the pipe, the round number 50 may be preferred. 
 The modified expression for the velocity of the final 
 
 ft FT 
 
 discharge will then become 50^ ^ , which a- 
 
 grees well with observation. In very long pipes, 
 
NATURAL PHILOSOPHY. 
 
 the lirst part of the denominator may be omitted, 
 and the expression for the final velocity will be 
 
 rl FT 
 
 simply 50 ^/-j . It may hence be sufficiently ac- 
 curate in most cases to assume that V* is to v z 9 as 
 64H to 2500.^, or as 64L to 2500d / and sup- 
 
 posing the coefficient 64 to be reduced to 50 in 
 the act of entering the train of pipes, then will 
 
 v* = V* . -y , and v= V^/ -^ . Wherefore, of the 
 
 V U 
 
 altitude H of the incumbent column, the part 
 
 50d 
 H . - only is exerted in creating the flow of the 
 
 water, the remainder of the inciting force being 
 wholly consumed in overcoming the obstruction the 
 current meets with along the internal surface. Every 
 length of tube amounting to fifty times its diameter, 
 thus occasions a waste of power equal to that which 
 produced the general impulsion. 
 
 Let AB (fig. 166.) represent the altitude of the 
 cistern, of which the small part AC causes the pri- 
 mary motion of the water, and the remainder CB 
 maintains its lateral attrition through the extended 
 horizontal pipe BD. It is evident, that if an incli- 
 ned pipe CD were substituted for BD, the length 
 would not be sensibly increased, while the inciting 
 force CB, now diffused, will exert still the same ac- 
 tion as before. The velocity which the fluid must 
 
404 ELEMENTS OF 
 
 acquire in its gradual descent along CD is hence 
 equal to what it receives from the constant pressure 
 during a horizontal progress. If the train of pipes be 
 sufficiently prolonged, the points C and A may be 
 viewed as coincident, and the angle ADB consider- 
 ed as the slope of uniform flow. Suppose the decli- 
 vity from A to D to be the n ih part of the distance ; 
 
 then L = n H and t> = 5(\/ = 50^7 -. The 
 
 n 
 
 celerity of the flow thus depends as much on the 
 width of the pipe as on the rate of its descent. Let 
 
 wzilOOO, and v will be ^/ d. With such a 
 
 gentle slope, therefore, it would require a pipe of 
 1.582 of a foot in diameter, to give the flow of a 
 single foot each second ; and a pipe of one foot wide 
 would discharge only 1.243 cubic feet of water in the 
 same time. 
 
 Conceive a rectangular channel to have a section 
 equal to the circle of the pipe, while its bottom and 
 sides are equal to the circumference ; the retarda- 
 tion which this would occasion to the current must 
 evidently be the same. Hence, the fourth part of 
 the diameter will be equal to the quotient of the sec- 
 tion by the compound measure of its bottom and 
 sides, which is called the mean hydraulic depth. 
 This depth being denoted by a y therefore 4a=d, 
 
 and by substitution v = 50 y ^ = 100V -y- 
 
NATURAL PHILOSOPHY. 405 
 
 
 
 Hence much less obstruction is encountered along 
 an open canal than within a close pipe. Let, as be- 
 fore, the slope be the n ih part of the distance ; then 
 
 ^ = 100^-. If n = 1000, the expression will be- 
 n 
 
 come v = lOO-- = /Wa = 3.1623 a. 
 
 In these estimates of the velocity and discharge of 
 water through pipes and conduits, the inside is pre- 
 sumed to be smooth, the width uniform, and every 
 sudden bending avoided. The want of evenness of 
 surface impedes the motion of the fluid, which is far- 
 ther obstructed by any violent change of celerity or 
 direction. Whether the channel be contracted or 
 enlarged, the change is unavoidably attended by a 
 proportional loss of impulsion. Any sharp flexure 
 of the pipe or conduit will occasion a still greater 
 waste of the inciting force. The diminution of the 
 square of the velocity is expressed by the product of 
 that square into the square of the sine of the angle 
 of deflexion, divided by the constant number 270. 
 With a deflexion of 30 degrees, the velocity would 
 therefore lose only the 2160th part ; but if the tube 
 branched off at right angles, the retardation would 
 amount to the 540th part. Every contraction or 
 enlargement of the pipes, requiring a corresponding 
 change in the celerity of the water, must likewise 
 create an expense of force, though this effect could 
 scarcely be reduced to calculation. 
 
406 ELEMENTS OF 
 
 
 
 Water is subject in its motion through pipes to 
 another impediment, owing to the air which con- 
 stantly separates from it and collects in all the upper 
 sinuosities of the train. This accumulation is most 
 copious, whenever the supply of water happens to be 
 insufficient to fill the whole extent of cavity. To 
 remedy the defect, boxes of cast-iron are fixed above 
 the principal incurvations of the pipe, to receive the 
 compressed air, and by the operation of a valve or of 
 a cock, gradually to discharge it, without allowing 
 any of the water to escape. Of such air-vessels, 
 with a cylindrical form, four feet high, and eighteen 
 inches wide, fourteen have been made, for the pipes 
 which are to supply Edinburgh. (See fig. 167.) 
 These being screwed at the summit of each declivity, 
 will be opened, every two or three days, by the sur- 
 veyor of the works. 
 
 It must be observed, that the emission of fluids 
 from a very minute orifice, or through a capillary 
 tube, is not reducible to the foregoing principles. 
 The obstruction is then proportionally far greater ; 
 but the peculiar quality and condition of the fluid 
 materially derange the whole result. The internal 
 motions of the particles of any liquid, which in this 
 case mainly determine .its flow, are extremely re- 
 tarded by any tendency to a viscuous state. Heat, 
 therefore, as it brings them nearer to the condition 
 of perfect fluidity, promotes greatly the celerity of 
 their course. Thus, pure water near the boiling 
 
NATURAL PHILOSOPHY. 407 
 
 point is found to run about five times faster than at 
 the verge of freezing. Alcohol, again, has its con- 
 stitution so much altered by the rise of only 124 
 of temperature, as to flow six -times quicker than 
 before. Quicksilver is indeed less affected in this 
 way ; but it endures heat through a far wider range. 
 
 To supply the inhabitants of great cities, water is 
 often conveyed from distant but high -seated springs, 
 by a long train of pipes. The velocity of the flow 
 
 may be determined from the formula, v=50 / -j > 
 whence the quantity of water delivered every mi- 
 nute is 2356^*7 -y- cubic feet, or 2356 <$ / -. 
 
 Let m denote the altitude of the source above the 
 reservoir in hundredth parts of the whole distance, 
 and the discharge of water in a minute will be ex- 
 
 pressed by 235.6 d* ^/ m. 
 
 Suppose a pipe were composed of two portions ha- 
 ving different widths. Let the length and diameter 
 of the first part, and the height and velocity of the 
 contained water, be denoted by /, d y k and v ; while 
 these measures in the second part are represented by 
 capitals. The whole accelerating force is therefore 
 and the retardation = 
 
 D* 
 
 whence, by substituting V*.- for t? 1 , we have 
 
408 ELEMENTS OF 
 
 .~.V> + LD.V* = M* + HD>, and 
 
 hd*+HD*d //M*+HD* d\ 
 
 ' or V = W ("70+15- ' D /' 
 
 quantity of water delivered in a minute is therefore 
 
 The former supply of Edinburgh was brought by 
 two trains of cast-iron pipes ; one from Green Craig 
 to the Castle Hill, 26,980 feet long, and 7 inches 
 in diameter, under a head pressure of 404 feet ; the 
 other from Comiston to Heriot's Reservoir, 13,520 
 feet long, and 5 inches wide, under a load of 88 
 feet. The first, when fully charged, is found to de- 
 liver 46, and the second only 10 cubic feet, every 
 minute, making together 56 feet, which furnishes a 
 supply of 80,640 cubic feet in the space of twenty- 
 four hours. This amounts to scarcely three-fifth 
 parts of the quantity assigned by the formula. The 
 deficiency must be attributed wholly to the imper- 
 fect execution of those pipes, their uneven interior 
 surface, and their frequent abrupt and sudden bend- 
 ings. 
 
 The water-works lately designed for the complete 
 supply of Edinburgh, were conducted in a much 
 finer style, but at vast expense. The several pieces 
 of pipe were nicely fitted together by spiggot and 
 
NATURAL PHILOSOPHY. 409 
 
 faucet, all the accidental prominences along the in- 
 side being carefully removed by chiseling. The 
 pipes showed no visible incurvation, and were gene- 
 rally laid with a gentle regular slope, the ground 
 on which they rest being lowered in some places 
 and raised in others. From the Crawley Spring to 
 Straiton March Fence, the distance is 18,300 feet, 
 with a fall of 65 feet ; and, in this line, the pipes 
 vary from 20 to 18 inches in diameter ; their great 
 width being intended to meet the exigency of having 
 lateral branches extending to other remote springs 
 on the north side of the Pentlands, as the increase of 
 population may afterwards require. The next train 
 has a diameter of only 1 5 inches diameter, but runs, 
 with a fall of 286 feet, through an extent of 27,900 
 feet, being conducted by a tunnel of 360 fathoms 
 length through Heriot's Ridge, and by another of 
 290 fathoms through the Castle Hill, till it reaches 
 the level of Prince's Street. It may be computed, 
 that the first train should deliver 442.93 cubic feet, 
 and the second 416.71, or the compound system 
 425.31 every minute. Their discharge, however, 
 has not yet exceeded 300 cubic feet in a minute, 
 though, no doubt, they could easily convey one- 
 third more. 
 
 It has been computed that, the quantity of rain 
 which falls annually over any city, if carefully col- 
 lected and deposited to purify in cisterns, would be 
 
410 ELEMENTS OF 
 
 sufficient for the supply of the inhabitants, at least 
 in all the essential domestic and culinary purposes. 
 Venice has abundance of fine soft water procured in 
 this way ; and the store seldom fails, except in dry 
 seasons, when it is recruited from the river Brenta. 
 The roof of a lofty house at Paris, containing at an 
 average 25 lodgers, might deliver annually 1800 cu- 
 bic feet of rain-water, which would furnish each in- 
 dividual daily the fifth part of a cubic foot, or about 
 thirteen pounds averdupois, rather a scanty provi- 
 sion, to be sure, according to our modern ideas of 
 comfort ; yet Prony reckons ten litres, or the thou- 
 sandth part of his modulus, as a sufficient supply, 
 amounting only to about twenty-two pounds. 
 
 Since, from a pipe of the same diameter, the dis- 
 charge in every case depends on the relation of the 
 altitude of the source to the length of track, a lower 
 elevation may frequently be preferred in conjunction 
 with a shorter train. The diminished obstruction, 
 in such instances, compensates for the inferior pres- 
 sure. From any point of an inclined plane, the pipe 
 would convey exactly an equal body of water. In 
 the same train, the quantity of discharge, being as 
 d? 9 must increase in a faster ratio than the mere sec- 
 tion of the pipe. Hence the manifest advantage of 
 employing large pipes. For the same reason, aque- 
 ducts or open conduits are in many situations to be 
 preferred. When these convey large streams of wa- 
 
NATURAL PHILOSOPHY. 411 
 
 ter, the attrition of the sides and bottom is compa- 
 ratively small, and they require very little descent. 
 Such durable structures are common in the South of 
 Europe, and often display much architectural sym- 
 metry in their extended and imposing ranges of ar- 
 cades. 
 
 It is a very prevailing opinion, that the Romans, 
 amidst all their magnificence, were ignorant of the 
 simplest elements of Hydrostatics, and therefore en- 
 tirely unacquainted with the method of conducting 
 and raising water by a train of pipes, Nothing, 
 however, can be worse founded than this notion. 
 The ancient writers, who either treat of the subject, 
 or incidentally mention it, are clear and explicit in 
 their remarks, while many vestiges of art still attest 
 the accuracy of those statements. Pliny, the natu- 
 ral historian, lays down the main principle, that 
 " water will invariably rise to the height of its 
 " source :" Subit altitudinem exortus sui. He 
 subjoins, that leaden pipes must be employed, to 
 carry water up to an eminence *. Palladius, in his 
 treatise De Re Rustica, teaches how to find springs, 
 by observing immediately before sunrise in the month 
 of August, the vapours which hover above particular 
 spots ; and having there dug a well, he directs the 
 water to be conducted to the farm or villa, either by 
 
 * Laminae esse debebuut, per quas surgere in sublime opus 
 fuerit, e plumbo. PLIN. xxxvi. 7. 
 
ELEMENTS OF 
 
 a channel constructed of masonry, or by means of 
 pipes of lead, of wood, or even of earthenware *. 
 He allows one foot in sixty, or in a hundred, for an 
 uniform descent. But if the ground should after- 
 wards rise, he says the conduit must be supported 
 on piles or arches, or the water must be inclosed in 
 leaden pipes, when it will mount just to the level of 
 its head t. But Palladius testifies his aversion to 
 the use of lead, as apt to become covered with ce- 
 russ, and thereby rendered unwholesome, or even 
 poisonous. This consideration had no doubt served 
 to restrain the general adoption of leaden pipes a- 
 mong the Romans. Still, however, we may infer, 
 from the allusions of the Poets, that such pipes had 
 come into very common use. They were not cast 
 tubular as at present, but consisted of thin plates 
 bent up into the form of a cylinder, and soldered 
 along the edge. They must frequently have given 
 way, therefore, at this seam. Horace asks, if the 
 water which threatens in the streets to burst its 
 lead, be purer than the rivulet that trembles and 
 murmurs as it flows t ? Ovid compares the gush of 
 
 * Cum vero ducenda est aqua, ducitur aut forma structili, aut 
 canalibus ligneis, aut fictilibus tubis. PALL AD. ix. 11. 
 
 f Sed si se vallis interserat, erectas pilas vel arcus usque ad 
 aquae justa vestigia construemus, aut plumbeis fistulis clausum de- 
 jici patiemur, et explicata valle consurgere. IBIDEM. 
 f Purior in vicis aqua tendit rumpere plumbum, 
 
 Quam quae per pronum trepidat cum murmure rivum ? 
 
 EPIST. I. x. 20. 
 
NATURAL PHILOSOPHY. 413 
 
 blood from the mortal wound which Pyramus, in 
 the agony of despair, had inflicted upon himself, to 
 the accidental rupture of a leaden-pipe *. Statius 
 speaks^ no doubt with poetical exaggeration, of whole 
 rivers being discharged by such conduits t. Vitru- 
 vius describes the three principal modes of conveying 
 water ; but directs, as the previous operation, to 
 trace a level (libramentum) on the ground. This 
 libration was performed, by the dioptron, the water- 
 level, or the chorobates. The dioptron seems to 
 have been a sort of quadrant fitted with sights ; the 
 water-level consisted of a tube, probably of copper, 
 five feet long and an inch wide, turned up an inch 
 and half at both ends, and was adjusted till water 
 rose equally in them ; the chorobates, or perambula- 
 tor, which he considered as the most accurate instru- 
 
 * Non aliter, quam cum vitiato fistula plumbo 
 Scinditur, et tenues stridente foramine longe 
 Ejaculatur aquas, atque ictibus aera rumpit. 
 
 METAM. iv. 120* 
 Thus translated : 
 
 As out again the blade he, dying, drew, 
 Out spun the blood, and streaming upwards flew ; 
 So if a conduit- pipe e'er burst you saw, 
 Swift spring the gushing waters through the flaw ; 
 Then spouting in a bow they rise on high, 
 And a new fountain plays amid the sky. 
 
 \ Terque per obliquum penitus quae laberis amnem 
 Martia, et audaci transcurris flumina plumbo. 
 
 STATIUS, i, SYLV. 
 
414 ELEMENTS OF 
 
 ment, was composed of a rod twenty feet long, ha- 
 ving a square and plummet attached at each extre- 
 mity. Vitruvius allows only half a foot in the hun- 
 dred, for the slope of an aqueduct. After the water 
 had reached the walls of a city, it was admitted into 
 a reservoir or castellum, divided into three distinct 
 and equal compartments, one to feed the pools and 
 fountains, another to supply the public baths, and a 
 third for the accommodation of palaces and private 
 houses. The distribution of the water was effected 
 commonly by means of leaden pipes. The smallest 
 of these was called a denaria, being ten feet in 
 length, the sixteenth part of this in breadth or girth, 
 and weighing 120 Roman pounds. This gives, for 
 the thickness of the lead, exactly the quarter of an 
 English inch. In lower situations, where the stress 
 against the sides was greater, the pipes appear to 
 have been made proportionally stronger *. 
 
 * In the Physical Cabinet of the University of Edinburgh, is 
 now deposited a specimen of ancient leaden pipe, lately brought from 
 Rome, where it had been dug up among the ruins of the Palace of 
 the Csesars. It bears an inscription in raised letters, intimating 
 the name of the plumber, and the year of the reign of the Empe- 
 ror Domitian. Though only 16 inches long and 9^ in girth, it 
 weighs 22J Ibs. averdupois ; so that the lead must be very nearly 
 half an inch thick. The pipe is slightly curved and rudely form- 
 ed into merely a flattened oval, 2 \ inches broad, and 1| wide; the 
 joining at the edge being filled by a quantity of melted solder run 
 along both inside and outside. The section corresponds to a cir- 
 cular orifice of If inches in diameter. 
 
NATURAL PHILOSOPHY. 415 
 
 The quantity of water delivered from the cisterns 
 was regulated by the dimensions of the spouts, term- 
 ed calices. These formed a series of twenty-five dif- 
 ferent kinds, which served as moduli. Their dia- 
 meters were sometimes reckoned by ounces, or the 
 twelfth parts of a Roman foot, but more commonly 
 by quarter digits, or the sixty-fourth part of a foot. 
 The quinaria seems to have been considered as the 
 standard, and its width must have hence correspond- 
 ed to the .906 part of an English inch. The adju- 
 tage or length of all those spouts was the same, be- 
 ing twelve digits, or three-fourths of a Roman foot, 
 and therefore equal to 8.7 English inches. Prony 
 conjectures, from very probable grounds, that such 
 was also the altitude of column of pressure above the 
 middle of each orifice. This estimate gives 1979 
 cubic feet, for the quantity of discharge of a denaria, 
 in the space of twenty-four hours. 
 
 Leaden pipes were likewise employed, to carry 
 water across vales and other eminences: But it be- 
 hoved to erect, at the several incurvations, columna- 
 ria, or chimneys, to give vent to the air which might 
 collect and gorge up the passage of the water. Such 
 funnels required to be raised to near the height of 
 the fountain-head. 
 
 Vitruvius, however, joins with Palladius and 
 Columella, in recommending pipes of earthenware, as 
 not only cheaper, but more wholesome than those of 
 lead. They could be formed thicker if necessary, 
 
416 ELEMENTS OF 
 
 and might be farther strengthened and secured, they 
 said, by an outer coating of lime worked up with 
 oil. But such pipes, not being glazed, it became ne- 
 cessary, before using them, to fill up the pores by a 
 sort of puddling, that is, to wash their inside with 
 favilla, or fine wood-ashes. 
 
 No wonder, therefore, that leaden pipes were held 
 in little estimation among the ancient Romans. They 
 seem to have been seldom used indeed beyond the 
 limits of the imperial city, except as auxiliaries in 
 the smaller aqueducts. When such conduits hap- 
 pened to be interrupted by a deep narrow vale, in- 
 stead of joining them by an arch thrown over the 
 gap, the connexion was sometimes formed by an in- 
 verted syphon of lead, carried on the one side down 
 to the bottom, and brought up on the other. 
 
 Rome was supplied by nine great aqueducts, ac- 
 cording to Frontinus, who had been appointed cu- 
 rator of those magnificent works by the Emperor 
 Nerva. He added five more ; and the number was 
 afterwards augmented, by successive emperors, to 
 twenty. Of these, the most remarkable were, 1. The 
 Aqua Appia, so named from its having been con- 
 structed by the Censor Appius Claudius in the 442d 
 year of Rome, began between the 6th and 8th mile- 
 stone, made a circuit of 880 paces, and then proceed- 
 ed by a deep subterranean drain of more than 11 
 miles, delivering the main body of its water in the 
 Campus Martius. 2. The Old and New Anio, con- 
 
NATURAL PHILOSOPHY. 447 
 
 duits so called from their bringing into Rome the 
 waters of that river. The former began above the 
 Tiber at the 30th mile-stone, and consisted mostly 
 of a winding drain carried through an extent of 
 about 43 miles. The latter, constructed under Nero, 
 took a higher level, running 7>543 paces above 
 ground, and then pursuing a subterranean passage 
 of 54,267 paces in length. 3. The Aqua Martia, 
 which owed its erection to Quintus Martius, rose 
 from a spring, distant 33 miles from Rome, made a 
 circuit of 3 miles, and afterwards, forming a vault of 
 16 feet diameter, it ran 38 miles along a series of 
 arcades at the elevation of 70 feet. It had vents 
 perforated at certain intervals, for disgorging the 
 collected air j and the conduit was occasionally in- 
 terrupted by deep cisterns, in which the water set- 
 tled and deposited its sediment. It was hence re- 
 markable for its clear green colour *. The Aqua 
 Julia and the Aqua Tepula were brought by the 
 same aqueduct, only in two lower conduits. 4. The 
 Aqua Virginia, conducted by Agrippa, the patrio- 
 tic lieutenant of Augustus, who laboured to improve 
 and beautify Rome, and who, according to Pliny, con- 
 structed in one year 70 pools, 105 fountains, and 130 
 reservoirs. It commenced at a very copious spring, 
 
 * Pliny celebrates particularly the coolness and salubrity of this 
 water. " Clarissima aquarum omnium in toto orbe frigoris salu- 
 " britatisque palma prseconio urbis, Martia est reliqua deum mu> 
 " nere urbi tributa," &c. Lib. xxx. 3. 
 VOL, I. 2 D 
 
4<1H ELEMENTS OF 
 
 in the midst of a marsh, at the distance of 8 miles 
 from the city, and ran about 12 miles, passing through 
 a tunnel of 800 paces in length. 5. The Aqua 
 Claudia, begun by Nero and completed by Claudius, 
 took its rise 38 miles from Rome ; it formed a sub- 
 terranean stream 36 \ miles in length, run 10 J miles 
 along the surface of the ground, was vaulted for the 
 space of 3 miles, and supported on arcades through 
 the extent of 7 miles, being carried along so high a 
 level as to supply all the hills of Rome. It was 
 built of hewn stone, and still continues to furnish 
 the modern city with water of the best quality, which 
 has hence procured it the name of Acqua Felice. 
 
 The practice of tunnelling was begun under 
 Augustus, who greatly extended the aqueducts. 
 Other emperors likewise directed their attention to 
 that important object. Trajan showed particular soli- 
 citude in improving the aqueducts. Those works were 
 executed in the boldest manner ; nothing could re- 
 sist the skill and enterprise of the Romans ; they 
 drained whole lakes, drove mines through mountains, 
 and raised up the level of valleys by rows of accumula- 
 ted arcades. The water was kept cool by covering it 
 with vaults, which were often so spacious, that, accord- 
 ing to Procopius, who wrote in the time of Belisarius, 
 a man on horseback might ride through them. So 
 abundant indeed was the supply, as to induce Strabo 
 to say, that whole rivers flowed through the streets 
 of Rome. Contemplating the utility, the extent. 
 
NATURAL PHILOSOPHY. 419 
 
 and grandeur of those aqueducts, Pliny justly regard- 
 ed them as the wonder of the world *. The same 
 admiration is expressed by the poet Rutilius t. 
 
 According to the enumeration of Frontinus, the 
 nine earlier aqueducts delivered every day 14,018 
 quinaria. This corresponds to 27,743,100 cubic 
 feet. We may therefore extend the supply, when 
 all the aqueducts were in action, to the enormous 
 quantity of 50,000,000 cubic feet of water. Reckon- 
 ing the population of ancient Rome at a million, 
 which it probably never exceeded, this would furnish 
 no less than fifty cubic feet, for the daily consump- 
 tion of each inhabitant. 
 
 In modern Rome, three aqueducts, the Acqua 
 Felice, Juliana and Paulina, with some additional 
 sources, deliver in twenty-four hours, according to 
 the calculation of Prony, 5,305,000 cubic feet. This, 
 shared among a population of 130,000, gives about 
 
 * Si quis diligentius aestimaverit aquarum abundantiam in Pub- 
 lico, Balneis, Piscinis, Domibus, Euripis, Hortis suburbanis, Villis, 
 spatioque adrenientes, exstructos arcus, monies perfossos, con- 
 valles aequatas ; fatebitur, nihil magis mirandum fuisse in toto orbe 
 terrarum. PLIN. xxxvi. 15. 
 
 f Quid loquar aerio pendente fornice rivos, 
 
 Qua vix imbriferas tollere Iris aquas, 
 Hos potius dicas crevisse in sidera monies : 
 Tale Giganteum Graecia laudat opus. 
 
 RUTILIUS IN ITIN. 
 
420 ELEMENTS OP 
 
 forty cubic feet for each individual, being nearly the 
 same comparative supply as in the period of Roman 
 splendour. 
 
 Such profusion of water altogether transcends our 
 conceptions. The supply of London in the year 
 1790 was only 2,626,560 cubic feet daily ; but late- 
 ly, when the rivalship of the several water-companies 
 almost deluged the streets, it amounted to 3,888,000 
 cubic feet. It has again, by the mutual understand- 
 ing of those associations, been reduced to about three 
 millions of cubic feet ; and this quantity may be 
 sufficient for all the wants of a luxurious mass of 
 inhabitants, equal certainly to the population of an- 
 cient Rome, where the consumption, however, was 
 still sixteen times greater. How paltry then appears 
 the actual supply of Paris, amounting only to 293,600 
 cubic feet of water in a day ! It affords scarcely half 
 a cubic foot, or thirty pounds averdupois, to each in- 
 habitant, in a population of upwards of 600,000. 
 
 The Greeks of the Lower Empire had simplified 
 the general mode of conducting water. This ap- 
 pears evident from the practice which now prevails 
 in supplying the city of Constantinople. The ground 
 is levelled by means of the Terazi, a sort of invert- 
 ed mason's plummet, which hangs from the middle 
 of a cord stretched between two rods divided into 
 inches and parts, set upright and removed succes- 
 sively from one station to another. But the chief 
 improvement consists in substituting, for the colum- 
 
NATURAL PHILOSOPHY. 
 
 naria of the Romans, the Souterazi or water- 
 balance, a sort of hydraulic obelisk or pyramid. By 
 this ingenious contrivance, the expense of aqueducts 
 is reduced to a fifth part. The water runs down 
 with a gentle slope in covered drains, till it reaches 
 an obelisk constructed of masonry ; and rising up 
 the one side, by a narrow channel, discharges itself 
 into a basin at the top, from which again, at a level 
 only 8 inches lower, it descends by a similar chan- 
 nel on the other side. The form of this hydraulic 
 pyramid is shown in fig, 172, and the upper part of 
 it is enlarged in fig. 173. Such auxiliary machines, 
 which facilitate the escape of the air and allow the 
 water to settle, are commonly erected at distances of 
 about two hundred yards. The system is in fact 
 only a repetition of conduits. From each separate 
 basin, the water is distributed, by orifices of different 
 diameters, but having their centres all in the same 
 horizontal line, three inches beneath the brim. 
 
 The charge of the water-works at Constantinople 
 is entrusted to a body of 300 Turks and 100 Alba- 
 nese Greeks, who form almost an hereditary profes- 
 sion. According to the interesting work of General 
 Andreossy on the Bosphorus, the whole supply 
 of a population of 600,000 is only two-thirds of a 
 cubic foot, or about forty pounds of water every clay. 
 There still remain at Constantinople two ancient 
 cisterns : 1. The Subterranean Palace, built of hard 
 brick, vaulted and resting on marble columns : and 
 2. The cistern of one hundred and one columns, 
 
422 ELEMENTS OF 
 
 called anciently Philoxene ; it consists of three tiers 
 of columns, one above the other, and is capable of 
 holding five days' supply, for the whole inhabitants 
 of that spacious city. 
 
 The same principles which regulate the motion of 
 water in pipes and along canals, are likewise appli- 
 cable to the flow of rivers in their beds. Since the 
 propelling power is proportional to the elevation of 
 the main source, the celerity acquired by those de- 
 scending streams would become enormous, if their 
 force were not gradually absorbed by the operation 
 of some constant impediments. Suppose such a river 
 as the Rhone to receive its principal waters at the 
 altitude of 900 feet above the level of the sea, and 
 that no system of obstruction had intervened in its 
 course, it would have shot into the Bay of Marseilles, 
 with the tremendous velocity of 240 feet in a second, 
 or at the rate of 164 miles every hour. Even an in- 
 ferior stream, such as the Thames, fed at the height 
 of only an hundred feet, would still, if not retarded 
 by the attrition against its bottom and sides, have rush- 
 ed into the sea with a velocity of 54^ miles in an hour. 
 
 The resistance of fluids, like the friction of solids, 
 thus enters largely into the economy of nature. As 
 the latter is the great principle of stability and con- 
 solidation, so the former serves most essentially to 
 restrain the accumulation of celerity, and to mode- 
 rate all violent motions. A current presses forwards 
 with increasing rapidity, till the obstruction which 
 
NATURAL PHILOSOPHY. 423 
 
 it encounters becomes at last equal to the inciting 
 force ; and having attained this limit, the water then 
 continues to flow in a uniform stream. The main- 
 taining power is proportional to the quantity of de- 
 scent in a given space ; but the impeding influence 
 depends on the surface of the bed of the river com- 
 pared with its volume of water. This obstruction 
 must at first augment very fast, being as the square 
 of the celerity. 
 
 Let n y as before, denote the measure of declivity, 
 and a the mean hydraulic depth or the depth which 
 a river would take if it stood upon an even base 
 equal to the bottom and sides of its channel j then 
 
 100 I -- will express the resulting velocity in feet 
 
 each second. Hence, if the rate of descent were 
 only one part in ten thousand, the stream would 
 acquire a velocity represented simply by the square 
 root of the mean hydraulic depth. Let/* denote the 
 fall in feet each mile, and the formula will change 
 
 into v = /5 ^ 8Q \/ a f=-$\/ a f- Hence the velo- 
 city, reckoning in miles every hour, is expressed by 
 . <Jafr^/af. This result is quite conform- 
 able to actual observation. The square root of the 
 product of the hydraulic depth into the fall each mile 
 in feet being diminished by one-sixteenth part, will 
 hence represent the mean velocity of a river in 
 
ELEMENTS OF 
 
 inches each hour. This gives the central velocity of 
 any section of the stream ; but the flow is the most 
 rapid in the middle of the upper surface, and dimi- 
 nishes regularly as it approaches the bottom and sides 
 of the channel, where the retardation originates. 
 The extreme difference between the velocity at the 
 top and near the bed of a river appears by obser- 
 vation to amount to half the square root of the 
 computed mean velocity. Thus, the Ganges, in 
 its ample tide, has only a fall of 4 inches in 
 a mile, with an hydraulic depth of SO feet ; but 
 Y/ (30 X i) zz 3.16, which diminished by one six- 
 teenth part, is g.96, or very nearly three miles an 
 hour. Again, y/2.96 = 1.70, the half of which is 
 .85 ; whence the celerity at the surface of the stream 
 is 3.81, and at the bottom only 2.15. 
 
 These calculations proceed oa the supposition that 
 the river holds nearly a straight course. If it should 
 wind considerably, the multiplied deflections which 
 it suffers must still farther impede its motion. In 
 every bend which it makes, part of its impulse will 
 be spent against the concave side of the channel ; 
 the centrifugal effort will likewise raise the surface 
 of the water in those sinuosities, and therefore aug- 
 ment the abrasion of the banks. Hence no stream 
 can be long confined to a rectilineal channel." If an 
 accidental swell should once effect a breach, the sweep 
 of the current must necessarily tend to enlarge the 
 concavity by an accelerating progression j the oppo- 
 
NATURAL PHILOSOPHY". 425 
 
 
 
 site shore, from the accumulation of gravel and other 
 deposits, gradually advancing into the channel. Ri- 
 vers thus naturally form sinuosities ; they seek to 
 meander over the plains ; and they would incessant- 
 ly change their beds, if not restrained by sedulous 
 attention and skilful hydraulic operations. In such 
 a country as Italy, whose rich plains are swept by tor- 
 rents from the Alps and Appenines, the superinten- 
 dance of water-courses constitutes an important de- 
 partment of government. 
 
 "' tv * 
 
 If a flat surface be directly opposed to the action 
 of a stream as it shoots from the side of a vessel, it 
 must evidently sustain a pressure just equal to that 
 which actually projected the fluid, or the load of the 
 incumbent column. In every case, therefore, the im- 
 pulsion of any current against a perpendicular plane, 
 may be estimated by the weight of a body of the fluid 
 standing upon that surface, and having the altitude 
 due to the velocity. Resuming the former notation, 
 
 since v=8^/h, it follows that h -r- =(? ) This 
 
 expression corresponds, in the case of a stream of 
 water, very nearly to one pound averdupois for every 
 square foot of the obstacle, multiplied into the square 
 of the velocity in feet each second. Let A denote 
 
 62 1 
 
 the area of the opposing surface, and -f.A.v z , or 
 
 OT 
 
 250 
 
 jrp. A.v , will express more accurately the mea- 
 
4,26 ELEMENTS OF 
 
 sure of the shock. If the stream were to consist of 
 sea-water, the fractional coefficient might be omitted, 
 and A.v* simply will represent the impelling force. 
 Let V denote the velocity of the current in miles 
 each hour, and the former expression must be mul- 
 
 22 
 tiplied by the square of the fraction . This gives 
 
 , ,- , 250 484 . Tr , 605 . ^ Tz 
 for the shock . . A.V>.= . A.V% which 
 
 21 
 is very nearly , A.V*. In practice, it may be 
 
 sufficiently accurate, to reckon, for every square foot 
 of opposing surface, the product of two pounds 
 averdupois into the square of the celerity of a stream 
 of water expressed in miles an hour. The pressure 
 of a river against the piers of a bridge may be hence 
 computed. The shock becomes augmented in a 
 high ratio during floods ; for not only is a greater 
 extent of surface then opposed to the current, but 
 the effort on every given space follows also the 
 square of the increased velocity. 
 
 The mighty rush of a torrent, carrying along with 
 it fragments of rocks, stones, or gravel, depends on 
 the same principle. Let w=6%^ lb., and a denote 
 in feet the side of any cubical block, of which the 
 specific gravity is g ; its weight under water will 
 evidently be expressed by a 1 (1 -g) w 3 . The 
 impulse of the stream would therefore be sufficient 
 
NATURAL PHILOSOPHY. 
 
 to support the load, when 2a*.V* zna 3 (1 #)M> 3 , 
 or V*=-(l #)w. In the case of stones, the value 
 
 of g lies between 2 and 3, and consequently 
 2V*zil.6x62 Xz=100, and V =^/50.a. The 
 force required to overcome the friction of a stone at 
 the bottom of a current, or to urge it forward, will 
 generally be less than this quantity, seldom perhaps 
 exceeding the half of it. Hence we may assume 
 V~4<^/a. The effect will be nearly the same, if 
 the block should approach to a round shape. A 
 torrent with the celerity of eight miles an hour 
 would therefore be capable of rolling a stone of four 
 feet in diameter. But a stream gliding at the rate 
 of two miles an hour would only be sufficient to car- 
 ry along with it a pebble of three inches in diameter. 
 With lower velocities, the current will scarcely move 
 gravel. If the particles of sand were supposed to 
 have a diameter equal to the twenty-fourth part of 
 an inch, it would require a flow of a quarter of a 
 mile in an hour to bear them along. A velocity of 
 the tenth part of a mile in an hour would be suffi- 
 cient to carry sandy particles of only the hundred and 
 twenty-third part of an inch in diameter. 
 
 Hence, the theory of the washing of metallic ores, 
 and the deposition of gold dust in the beds of rivers. 
 The ores being broken into very small fragments by 
 means of a stamper, these are laid upon an inclined 
 plane, and exposed to the action of a descending 
 stream of water, which sweeps away all the lighter 
 
4/28 ELEMENTS OF 
 
 earthy particles. In like manner, the pellicles of 
 gold, adhering commonly to minute portions of 
 quartz, being at length detached by incessant roll- 
 ing, are left in the little pools, while the sandy par- 
 ticles are carried still farther. Hence also the rea- 
 son why the bottoms of rapid rivers are covered 
 with large round stones, or at least with rolled peb- 
 bles. Where the celerity of the current becomes 
 moderated, gravel and coarse sand begin to appear. 
 But when the flow is sluggish, the bed of the river 
 is always covered with fine sand or mud. Such de- 
 posits occur chiefly in the pools and near the influx 
 into the sea. Hence. likewise the gradual formation 
 of banks, a process which is constantly going on over 
 all the stagnant parts of water, and along the limits 
 of opposite currents. 
 
 Water is the readiest and most powerful agent 
 that can be directed by human skilL A mill-race, for 
 example, three feet broad, and two feet deep, and 
 running at the rate of four miles an hour, would 
 communicate an impulsion equal to the fall through 
 .538 parts of a foot j whence the action thus crea- 
 ted, during the space of a minute, is expressed by 
 the product 3 X 2 X 352 X 6%- X .538 = 70,966, 
 which being incessant, amounts to the ordinary la- 
 bour of an hundred men. If this current had then 
 fallen 26} feet, its quantity of operation would 
 have been augmented fifty times more. But such 
 .streams are easily collected and formed in mime- 
 
NATURAL PHILOSOPHY. <l<2() 
 
 rous situations over the undulating face of the 
 country. 
 
 It will expand our conceptions, if we survey the 
 great laboratory of Nature, and calculate the enor- 
 mous extent of power, displayed in elevating the 
 watery stores into the lofty regions of the atmo- 
 sphere. Between the tropics, the annual fall of rain, 
 and consequently the measure of evaporation which 
 supplies it, amounts to about 10 feet ; and estimating 
 this in other countries, as nearly proportional to the 
 cosine of the latitude, the quantity of moisture ex- 
 haled in the course of a year, over the whole surface 
 of the globe, would form a shell of five feet deep. 
 The number of cubic feet of water turned into vapour, 
 and dispersed through the mass of atmosphere every 
 minute, would hence be 5 X 10,424,000,000, or 
 52,120 millions. But this quantity is to be multiplied 
 by 18,000, the mean height of the atmosphere in feet, 
 and again by 62^-, the weight in pounds averdupois 
 of a cubic foot of water ; the final measure of effect 
 is therefore expressed by 58,635,000,000 millions, 
 and equal to the labour of about 80,000,000 millions 
 of men. Now, the whole population of the globe be- 
 ing reckoned 800 millions, of which only the half is 
 capable of labour ; it follows, that the power exerted 
 by Nature, in the mere formation of clouds, exceeds, 
 by two hundred thousand times the whole accumu- 
 lated toil of mortals. 
 
 A considerable portion of the power thus expend- 
 ed, might be directed to useful purposes, by inter- 
 
430 ELEMENTS OF 
 
 cepting the water again in its descent towards the 
 ocean. Suppose one-sixth of all the exhalations to 
 return to this great gulf, and that half of the falls in 
 the rivers and streams over the habitable earth, com- 
 prising the fifth part of the whole surface of the 
 globe, are detained from an elevation of 600 feet ; 
 there would be drawn from those mighty stores a force 
 eleven times greater than the aggregate of human 
 labour. 
 
 It may be satisfactory, however, to take a more de- 
 finite illustration. The surface of this island is com- 
 puted at 67,243 square miles, or 1,874,627,000,000 
 square feet. But reckoning the annual measure of 
 rain 36 inches, of which the sixth part may flow to- 
 wards the sea, and supposing one half of this surplus, 
 or three inches, to be intercepted at an elevation of 
 100 feet, it would require to multiply the former 
 number by 100 X J X 62, or 1562*-, and divide 
 the product by 525949, the number of minutes in 
 a year, to obtain the quantity of performance. The 
 result is 4,423,760,000, equivalent to the action of 
 6,703 steam-engines, of what is called twenty-horse 
 power. It may hence be fairly estimated as not in- 
 ferior to the ordinary labour of the whole of our male 
 population. Such is the vast magazine of force, 
 which a rigid economy might command. 
 
 The power exerted by the moon and sun to heave 
 the tides of the ocean, is only about the eigh- 
 tieth part of the action of the atmosphere in produ- 
 cing the train of meteorological phenomena. Rec- 
 
NATURAL PHILOSOPHY. 431 
 
 koning the swell at the equator to be 4 feet, this 
 gives a mean elevation of 2 feet over the whole fur- 
 face. Wherefore, a body of water, 2 feet deep, is 
 raised to the intermediate height of one foot twice in 
 the lunar day, or 706 times in the course of the year. 
 The relation of the force thus employed to that of the 
 general exhalation over sea and land, is hence as 
 2 x 1 X f X 706, or 1129.6 to 5x18000, or 90000, 
 that is, as 1 to 80 ; and therefore still two thousand 
 five hundred times greater than the aggregate labour 
 of the human race. 
 
 But the rise and fall of the tide, along our ex- 
 tended shores, would be sufficient to drive numerous 
 mills. Suppose a basin were inclosed only a chain 
 or 66 feet in width and 10 chains in length, and 
 containing therefore aji acre of salt water, this 
 would give an impulse equal to the flow of 43,560 
 cubic feet in 12 hours and 25 minutes, or about 
 5S feet every minute, with a fall of 5 feet. The 
 expression for the force evolved, is hence the conti- 
 nued product of 5 X 5S X 64 zz 18660. 
 
 The performance of this tide-mill might hence be 
 equal to that of 25 common labourers. But eight 
 such basins could be included in each mile of coast ; 
 and therefore, estimating the circuit of the island 
 at 1750 miles, there might be formed no fewer 
 than 14,000 mills, by drawing a sea-wall 66 feet 
 from the shore. In this way, a saving of power 
 might be effected, equal to the labour of 350,000 
 men. The expence of erecting such a dam, would 
 
432 ELEMENTS OF 
 
 probably defeat the object as a general scheme of 
 improvement ; but there occur very many creeks and 
 bights along our indented coast, which could be 
 most profitably inclosed as reservoirs for large tide- 
 mills. 
 
 The float -board of a river-mill is impelled, not by 
 the whole velocity of the stream, but only by the 
 excess of this above the velocity of the board itself. 
 Let A denote in square feet the surface immersed 
 in the water, v the velocity of the current, and v' 
 that of the middle of the float ; then 2A (v v}* 
 will express in pounds averdupois the pressure which 
 turns the wheel. The momentum thus acquired 
 must hence be proportional to v' (y v')*, an expres- 
 sion, which being assimilated to the property of a para- 
 bola, must become a maximum when v' v. The 
 greatest impulsion communicated merely to a wheel 
 
 14 4f 
 is therefore only - . - or of the whole power of 
 
 the stream. But it would be more advantageous to 
 make the float-boards turn slower, and to multiply 
 their velocity afterwards by means of a train of in- 
 ternal machinery. The current might then strike 
 with nearly its full celerity. In the case of under- 
 shot wheels, a great loss of power is occasioned by 
 the accumulation of the dead-water^ or of the water 
 which, having impinged against a float-board, re- 
 mains nearly stagnant, and therefore impedes the 
 advance of the next float-board. 
 
 3 
 
NATURAL PHILOSOPHY. 433 
 
 The shock will evidently be the same whether a 
 current strikes against a fixed plane, or the plane it- 
 self moves with an equal and opposite velocity 
 through a fluid at rest. The formula of impulsion 
 already given must hence include likewise the case 
 of resistance, which is therefore proportional to the 
 square of the celerity. But this conclusion might 
 be derived from direct considerations. As the plane 
 advances, it receives the stroke of all the particles 
 in its progress. The quantity of momentum thus 
 consumed is evidently compounded of the number 
 of particles encountered and the impetus of each ; 
 but the number and individual force of those parti- 
 cles being as the velocity, the combined effect must 
 be proportional to the square of the velocity. 
 
 If a fluid strike a plane obliquely, its action is to 
 be detennined from the decomposition of forces. 
 Let the oblique plane AB (fig. 174.) receive the 
 shock of a current perpendicular to AC. If DB, 
 assumed equal to AB, denote the direction and in- 
 tensity of the stroke, this force may be resolved into 
 BE, parallel to AB, and DE at right angles to it. 
 It is the latter only that can have any effect upon 
 the plane AB. But the force DE, again, is decom- 
 posed into DF, perpendicular to AC, and FE pa- 
 rallel to it. FE therefore urges the plane in the di- 
 rection CA ; but the breadth of the stream beini; 
 likewise proportional to CA, the combined action 
 must be represented by the rectangle FE, CA, or by 
 VOL. i. 2 E 
 
434 ELEMENTS OF 
 
 
 
 the solid AB.FE.CA, since AB is constant. Now 
 the right angled triangle ABC, being obviously 
 equal to DBE, the sides BC and CA are equal to 
 BE and DE. But AB : AC : : BE or BC : EF, 
 whence AB.EFzz AC.BC, and the solid AB. AC.EF 
 zr AC*.BC. The force which impels the plane 
 AB in the direction CA is therefore exerted to the 
 utmost when AC*.BC is a maximum. But AC 
 and BC are chords in a given semicircle, described 
 on AB ; and from what was shown in p. 258, the 
 greatest solid requires that the square of AC should 
 be double the square of BC. Whence the tangent 
 of the angle BAC zz ^/ % = tan. 35 16'. Such 
 is the angle that the rudder of a ship makes with a 
 perpendicular to the keel, when it exerts the greatest 
 power in turning the vessel. 
 
 Suppose an oblique plane AB (fig. 175.) or a 
 wedge CAB to move in the direction CA against a 
 quiescent fluid. Let BD be taken equal to AB and 
 parallel to AC, draw DE and EF perpendicular to 
 AB and BD. After repeated decomposition, the 
 only portion of the impinging force DB opposed to 
 the advance of the plane AB, is DF. The whole 
 resistance is therefore expressed by BC.DF, or by 
 BC.BD. But BD : DE : : DE : DF, or BG, if 
 CG were let fall perpendicular to AB. Again, 
 BD* : DE* : : BD : DF; whence the resistance of the 
 wedge CAB, while its base CB, or the transverse sec- 
 tion of the fluid remains constant, is proportional to 
 
NATURAL PHILOSOPHY. [;>J 
 
 the square of the sine of the angle BAC. But if 
 AB, the side of the wedge, or the length of the ob- 
 lique plane, should be the constant quantity, the re- 
 sistance will follow the cube of the inclination. 
 
 These investigations, however, assume, for the 
 sake of simplicity, that every particle, the instant it 
 has impinged, retires into the general mass of the 
 fluid, without consuming any force. But such a sup- 
 position deviates widely from the truth. The fluid con- 
 tinues, after collision, to press against the sides of the 
 wedge, and therefore impede its advance. To over- 
 come the inertia of this accumulation, will necessari- 
 ly require the expenditure of a large share of the im- 
 pulsion. The fluid particles which have met the 
 shock of the oblique plane, must be pushed off la- 
 terally, by some force depending on the cosine of the 
 angle of incidence. 
 
 The resistance which wedges experience in mo- 
 ving through water is accordingly much greater than 
 what mere theory indicates. It diminishes indeed 
 with their angle, but not in so fast a ratio as the 
 square. The acuteness of that angle occasions a 
 greater drag of the fluid, and a corresponding re- 
 tardation. When the floating body is very long, 
 the nature of the resistance becomes totally changed. 
 Thus, a ship's mast, though it has a conical shape, 
 is more easily drawn through the water with its broad, 
 than with its narrow, end foremost. In this case, 
 the primary obstruction is no doubt greater ; but 
 
436 ELEMENTS Ot 
 
 the water heaped on the front, being made to stream 
 off with a slight divergency, does not hang on the 
 sides of the mast. For the same reason, whatever 
 tends to weaken the adhesion, serves likewise to di- 
 minish resistance. A wedge which has its sides 
 rubbed with grease, is found to move more freely 
 through the water. Hence the great benefit derived 
 from sheathing the bottom of a ship with copper. 
 
 To lessen the resistance, it is of more consequence 
 that the prow of a vessel should be gently incurved 
 than have a sharp outline. The roundness of the 
 prow assists in gradually turning aside the current. 
 Even the breadth of the bow of the vessel is some- 
 times advantageous ; but the tapering of the hull is 
 always an important requisite. The bilged form of 
 the stern, which the Dutch have long adopted, 
 appears at last to be judged preferable, not only 
 strengthening the ship, but materially lessening its 
 resistance in the water. Whenever an eddy occa- 
 sioned by the abruptness of outline whirls under the 
 stern, or a portion of fluid follows the vessel, its 
 progress must evidently be retarded. In general the 
 shape of the hinder part of any floating body has a 
 very considerable influence on the quantity of resis- 
 tance which it must encounter. To determine all 
 these circumstances from strict theory, would require 
 an investigation of the most repulsive and hopeless 
 intricacy. It would not only be necessary to calcu- 
 late the intensity and direction of the shock of each 
 
NATURAL PHILOSOPHY. 
 
 43? 
 
 particle of Huid, but to trace the diverging streamlets 
 with which they again recede, and to reduce into a 
 single amount the effect of all the diversified elements 
 of resistance. The higher calculus might compress 
 those conditions under the powers of its notation j 
 but the resulting equations would still admit of no 
 complete integration. It is a much easier and a 
 safer procedure, to resume the limited and imperfect 
 deduction already given, and to correct it by a series 
 of accurate experiments. Many of these have been 
 made, but not on a scale, it must be confessed, at all 
 suited to the importance of the subject. The reports 
 of different observers are, besides, not very consistent. 
 We may therefore assume, for an oblique plane 
 whose inclination is the angle a, the expression 
 sin a 31 -(- ^ cos a, as a tolerably near approximation to 
 the resistance, that of its base being reckoned unit. 
 The same result may be given geometrically for the 
 wedge CAB (fig. 175.) by making BH = AC, 
 when GH will indicate the resistance. 
 
 It may be worth while, however, to exhibit the 
 gradation, by a tablet of the resistance for each five 
 degrees, that of the base being unit. 
 
 Angle. 
 
 Resist. 
 
 Angle. 
 
 Resist. 
 
 Angle. 
 
 Resist. 
 
 5 
 10 
 15 
 20 
 25 
 
 .255 
 
 .276 
 .308 
 .362 
 .385 
 
 30 
 35 
 40 
 
 45 
 50 
 
 .466 
 .534 
 .605 
 .677 
 .748 
 
 55 
 
 60 
 65 
 70 
 
 75 
 
 .814 
 
 .875 
 .927 
 .968 
 .998 
 
438 ELEMENTS OF 
 
 If a stone be dropped into a deep pool, it will gra- 
 dually accelerate its motion, till the increasing re- 
 sistance it encounters becomes equal to its descend- 
 ing force, or the excess of its weight above that of 
 the volume of water displaced ; and it then conti- 
 nues to fall with an uniform velocity. This veloci- 
 ty, which serves as a limit, is hence called terminal ; 
 and it evidently depends on the form, the size, and 
 the density of the plunging mass. To simplify the 
 consideration, suppose the body were globular, its dia- 
 meter in feet being denoted byt/, its specific gravity by 
 g, and the velocity by v . The excess of its weight 
 
 above that of water will therefore be j..d 3 (cj 1) ; 
 
 x v * 
 while the resistance of a circular section is - . d* . T-, 
 
 and that of the sphere itself the half of this, or 
 
 v *._. Whence equating these expressions, we 
 8 04 
 
 obtain v =2 9% \/ (d (g 1 ) ). The terminal velocity 
 is therefore in the subduplicate ratio of the diameter 
 of the ball, and its preponderating density. Thus, 
 an eighty pound iron shot, ^plunging into a fresh-wa- 
 ter lake, would acquire a celerity of 20.4 feet in a 
 second, or would descend through the space of 204 
 fathoms every minute ; for reckoning the specific 
 gravity at 7.5, the diameter of the ball would be 
 about 8J- inches, or .7 14 parts of a foot; whence 
 .714x6.5). In the ocean, the celerity 
 
NATURAL PHILOSOPHY. 
 
 must, owing to the greater density, be somewhat less, 
 or 201 fathoms. But a ball of a pound and a quar- 
 ter, or 64t times lighter, and consequently having a 
 diameter very little more than 2 inches, would sink 
 twice as slow, or at the rate of 100 fathoms every 
 minute. Whether the balls drop softly into the wa- 
 ter, or enter it with vehement impulsion, they will, 
 in a very short time, acquire their terminal velo- 
 city. The same principle regulates the ascent of 
 lighter bodies, the formula being only modified into 
 vzz9^v/(^(l #) ) Thus, reckoning the specific 
 gravity of cork to be .20, a ball of that substance, two 
 inches in diameter, if not affected by compression, 
 would rise from the bottom of a lake with a velocity 
 of 34< fathoms in a minute. 
 
 Jn other fluids, it is only requisite to substitute the 
 relative value of g. Thus, for atmospheric air, the 
 formula is changed into v 9$\/(d(84<Qg 1) ). 
 An eighty pound iron-shot, if fired upwards, would 
 hence acquire, in falling back to the ground, the ce- 
 lerity of 608 feet in a second, which is sufficient to 
 produce the most destructive effects. 
 
 In the case of drops of rain, the formula will be 
 modified into v=9%'%9\/d ; and for hailstones, we 
 may assume 9^ 30^/d. Thus, the largest drop of 
 rain being only the fifth part of an inch diameter, the 
 celerity of its descent cannot exceed 9'29.A=34 
 feet in a second, or 2040 every minute ; but the or- 
 dinary drops in this climate will seldom fall half as 
 
440 ELEMENTS OF 
 
 fast. On the other hand, hailstones in the south oi 
 Europe have sometimes the enormous diameter of 
 
 20 
 two inches. Here v zz 9% 30 . zz 113^ feet, or 
 
 exceeding a mile and a quarter in a minute ; a ra- 
 pidity of stroke which destroys corn-fields, and ra- 
 vages vineyards. 
 
 That the resistance which fluids oppose to pene- 
 tration, increases with the square of jhe celerity, is 
 easily shown by a series of glass-beads, of different 
 densities, dropped into a very tall jar full of water ; 
 for these balls in a short space acquire a certain ter- 
 minal velocity, with which, thenceforth, they pur- 
 sue their uniform descent. While the standard of 
 water is 1000, let the beads be adjusted to the spe- 
 cific gravities of 1001, 1004, 1009, 1016, proceed- 
 ing by the square numbers as far, perhaps, as 1100, 
 and have all of them an inch in diameter. Then, 
 
 for the first bead v=9 - = or an 
 
 inch ; so that reckoning from the limit, or less than 
 a foot below the surface, the velocities which they 
 attain are expressed in inches each second, by the 
 consecutive digits, 1, 2, 3, 4, &c. to 10. With the 
 heavier beads, such as those marked 1009, 1016, 
 1025, &c. the correspondence between these num- 
 bers and the results of actual experiment is very sa- 
 tisfactory. But those beads near the beginning of 
 the series descend more slowly than the theory in- 
 dicates : Thus, the bead 1004, instead of falling 
 
NATURAL PHILOSOPHY. 44*1 
 
 through two inches, scarcely describes one, and the 
 first bead, instead of a single second, takes more 
 than three seconds to pass through an inch. This 
 sluggishness proceeds evidently from the imperfect 
 fluidity of water, the effect of the tenacity, or mu- 
 tual adhesion of its particles, which cramps their in- 
 ternal mobility. Hence the slow subsidence of 
 earthy matters discharged into lakes ; and hence, 
 likewise, the suspension, in our atmosphere, of those 
 very minute aqueous globules which compose the 
 clouds. 
 
 If, instead of the tall jar, there be substituted 
 tubes of the same height, but only 2, 2, and 1 in- 
 ches in diameter, all the beads will descend much slow- 
 er than before, especially in the narrowest one. The 
 retardation is evidently caused by the difficulty with 
 which the water in this case recedes, and gives way to 
 the passage of the ball. This experiment illustrates 
 the peculiar obstruction found in drawing loaded boats 
 through the narrow and shallow parts of a canal. 
 
 Since the resistance of water to the progress of a 
 vessel is proportional to the square of the velocity, 
 the most advantageous mode of applying force in na- 
 vigation is to support a slow motion. If a steam en- 
 gine of what is called a 20 horse power, give an impul- 
 sion of four miles an hour, it would require one 80 
 horse power to double this rate, and one of 180 to in- 
 crease the celerity to twelve miles an hour. The same 
 large engine would therefore be capable of drawing 
 nine such boats only three times slower. But if ani- 
 
442 ELEMENTS OF 
 
 mal exertion were employed, the effect would appear 
 still more diminished, since the excess of muscular ac- 
 tion would be proportionally reduced. Thus, if a horse 
 were set to drag a boat, which required the force of 
 9 Ib. to pull it along a canal at the rate of a mile in an 
 hour, he would continually accelerate his pace till he 
 reached the limit of three miles an hour, when the 
 resistance, augmented to 81 Ibs., becomes equal to his 
 corresponding power of draught. If the force of 
 4 Ibs. were sufficient to produce the motion of one 
 mile in an hour, the horse would advance always 
 faster, till he attained the pace of four miles, when 
 his power of traction sinks to 64 Ibs., the increased 
 measure of resistance. The same horse would, there- 
 fore, be capable of dragging a train of 30 similar 
 boats only four times slower, and might hence pro- 
 duce more than seven times as much effect in trans- 
 porting goods. 
 
 In general, let P denote the force required to pull 
 a boat along a canal, at the rate of a mile in an hour, 
 and p the traction exerted at any other velocity, v 
 when it becomes just equal to the accumulated re- 
 sistance : Then, resuming a former expression, 
 
 (12 v y> p = Pv z ; whence v /p , and 
 
 v J 
 
 P = | 1 . Thus, on the Leeds and Liver- 
 \ v J 
 
 pool canal, a horse, walking at the rate of l miles 
 in an hour, draws a boat of 45 tons burthen, whence 
 
NATURAL PHILOSOPHY. 443 
 
 the actual draught is 115 Ibs., and P = 74 Ibs. 
 If the pace were quickened only to two miles, the 
 traction would be increased to 296 Ibs., and require 
 3 horses : were it accelerated to three miles, the 
 force would amount to 666 Ibs., requiring 8 horses : 
 a traction of 1184 Ibs., or the labour of 18 horses, 
 would be necessary to bring the rate of travelling to 
 four miles ; and no fewer than 38 horses, pulling 
 with the force of 1850 Ibs., would have been requi- 
 red to bring the speed up to the rate of five miles in 
 an hour. 
 
 The advantages resulting from the present mode 
 of interior navigation are hence very conspicuous. 
 The impulsion of locomotive engines along the track 
 of a railway seems, to a certain extent, well calcu- 
 lated for the rapid conveyance of passengers and 
 light goods ; but the system of canals, with their 
 slow regulated movements, which, at so little ex- 
 pense of labour, distribute or transport the ponde- 
 rous materials and the staple productions of the 
 country, must always be regarded as the grand feed- 
 er of our extended commerce. 
 
 An impulse may be conveyed through any fluid, 
 as along a solid substance, by a chain of tremulous 
 commotion, without any actual transfer of the parti- 
 cles which are effected in succession. A system of 
 alternate contraction and dilatation pervades the 
 whole extent. The celerity with which those vi- 
 
444* ELEMENTS OF 
 
 brations are transmitted, it may be shown, is that 
 due to half the altitude of the modulus of elasticity. 
 The rapidity with which an impulse will shoot 
 through water is hence equal to 8^350,000, or 
 4*730 feet every second. Suppose a train of pipes of 
 a mile in length were filled with water, a violent 
 blow, struck with a hammer at one end, would be 
 transmitted through the fluid, and felt at the other 
 end in little more than a second. 
 
 But if the water should extend along an open canal, 
 or spread itself within a wide basin, the succession of 
 alternating contractions and dilatations will produce 
 a corresponding series of swells and concavities over 
 the whole surface. These connected elevations and 
 depressions must likewise modify and retard the tide 
 of vibratory commotion which would rush impetu- 
 ously below. The rise of the water above its level 
 in any place will create a pressure acting on all sides, 
 but more immediately exerted against the nearest 
 portions of the fluid. Every swell over the surface 
 must hence tend to subside, and every hollow again 
 to mount upwards. A system of oscillations having 
 been once excited will therefore continue its opera- 
 tion, till the tenacity of the fluid particles extinguish 
 the disturbing force. 
 
 Sir Isaac Newton, who first examined this intri- 
 cate subject, was satisfied with comparing the motion 
 of waves on the surface of a pool to the undulation 
 of water in an inverted syphon. Suppose a hori- 
 
NATURAL PHILOSOPHY. 445 
 
 /ontal glass tube AB (fig. 176.) turned up at each 
 end, were filled with any liquid, whether alcohol, or 
 water, or mercury ; disturbed by raising the column 
 on the one side to E, it will sink equally in the other 
 to F. Abandoned in this situation, the fluid would 
 again retreat towards its former position, urged by 
 the pressure of EG on the double of CE. Bisect 
 AB in O, and the inciting force will be expressed by 
 
 CE 
 
 and since CAO remains constant, this force 
 
 will be proportional to EG, the space of derange- 
 ment. Hence the fluid, after it has gained the level 
 CD, will yet shoot just as far beyond it to G and H. 
 In this new position, it will be pressed back by an 
 inverted force, and made to return to E and F. 
 But these vibratory motions, whether wide or nar- 
 row, must all be performed in the same time. They 
 correspond exactly with the oscillation of a pendu- 
 lum, whose length is OC, or half the extent of the 
 recurved tube from C to D. If the undulations of 
 water be therefore assimilated to these movements, 
 the interval between each swell and the consequent 
 subsidence will be equal to the time of the vibration 
 of a pendulum, which has for its length the distance 
 of the top of a wave from the middle of the subja- 
 cent hollow. 
 
 But waves never emerge and sink again in the 
 same place. They seem to take their origin from 
 some agitated spot, and appear thence to advance in 
 
446 ELEMENTS OF 
 
 expanding concentric circles. The subaqueous pro- 
 pulsion accompanying them decides no doubt the 
 direction of their progress, and prevents them from 
 remaining in a pendulous state. If we examine at- 
 tentively the motion of a wave, we shall find that 
 the fore-part is always in the act of rising, while the 
 hinder-part is constantly sinking. The whole sys- 
 tem hence appears to roll onwards, though there is 
 actually no translocation of any portion of the mass, 
 and each particle in succession merely oscillates with 
 nearly a vertical ascent and descent, 
 
 The motions of waves are perfectly imitated on 
 the stage, by turning slowly an open helical screw 
 applied round an horizontal axis. This effect is 
 produced by the varying rate of the vertical eleva- 
 tion and depression, which must be as the versed 
 sine of the distance of each point from the summit. 
 Its celerity is greatest in emerging from the level of 
 the water ; it becomes stationary when it has gain- 
 ed its utmost ascent ; but it again acquires an equal 
 and opposite celerity as it sinks under that level, till 
 it comes to pause at the limit of depression. The 
 figure and apparent motion of a wave hence result 
 from this unequal and reciprocating vertical play of 
 each particle, combined with the continued and uni- 
 form advance of the inciting energy. 
 
 The vertical motions of the particles which com- 
 pose a wave are thus exactly similar to the oscilla- 
 tions of a pendulum. Let a OA (fig. 177.) repre- 
 
NATURAL PHILOSOPHY. 
 
 sent a level surface of the water, describe a circle 
 a FAQ, erect the perpendicular POQ, and having 
 divided each quadrant into equal arcs at the points 
 A, B, C, &c. draw the parallels BE, CD, &c. If 
 those arcs AB, AC, AP, AD, AE, &c., denote the 
 times elapsed, the corresponding altitudes OF, OG, 
 OP, and again OG, OF, &c. will express the suc- 
 cessive positions of a point which emerged from O. 
 In like manner, f, g> Q, and again g, f will indicate 
 the series of positions of O, in its subsequent de- 
 scent, answering to the times elapsed, AP&, APc, 
 APQ, APQd, and APQe. The particles which 
 succeed to this will gain the same positions at equal 
 intervals of time, and must consequently, by their 
 combination, mark the outline of a curve, which has 
 the arcs of a circle for its abscissa?, and the corre- 
 sponding series for its ordinates. This is called the 
 Curne of Sines. (See Geometry of Curve Lines, 
 pp. 406-8.) When its ordinates are all diminished 
 or augmented in a given ratio, it becomes changed 
 into the Hat-manic Curve. Such then is the sinu- 
 ous curve which represents the concatenation of a 
 system of waves. Let the horizontal line AN 
 (fig. 178.) be distinguished into successive portions 
 at the points B, C, D, E, equal to the arcs of the 
 circle APQ, (fig. 17?.) which measure the intervals 
 of time, and consequently the progress of the swell ; 
 and from those points, erect perpendiculars equal to 
 the corresponding sines OF, OG, OP, &c. ; the 
 
448 ELEMENTS OF 
 
 summits of these being joined, will indicate the con- 
 tour of the undulating surface of the water. The 
 front of the wave AVG rises in the first instant to 
 R, and the whole appears to advance into the posi- 
 tion BXRH. In the next instant, the same portion 
 mounts to S, and the wave comes into the situation 
 CYSI. In another instant, the fore-part of the 
 wave attains the elevation T, and the outline changes 
 into DTK. Again, the hinder-part of this wave, 
 by an inverted or reciprocating motion, descends 
 first from V to X, then to Y, and next sinks to D. 
 Each wave thus appears to roll onward, although 
 none of its component particles really change their 
 places. They merely vibrate in the same vertical 
 lines, rising or falling with a variable celerity. 
 
 It is evident that the relative elevations of waves 
 may differ in every proportion. Fig. 179. exhibits 
 the contour of a very high wave or billow ; and 
 fig. 180. represents the outline of a gentle swell or 
 undulation. 
 
 Waves are always seen rolling towards the shore ; 
 but an obstacle opposed to them becomes the centre 
 of a new series, which spread in circles. One set of 
 waves, however, does not interfere with the motion 
 of another, and they will cross each other without 
 occasioning the smallest interruption. Sometimes 
 the ordinary undulations are combined with a distant 
 swell, called the bore, which rises impetuous after 
 certain considerable intervals. 
 
NOTES. 
 
 NOTE I. P. 25. 
 
 THIS capital experiment was first devised and performed on a 
 small scale by Mr Canton in 1760. It established incontestably 
 the compression of water, but seems to have been generally over- 
 looked by succeeding popular writers, many of whom still con- 
 tinue to repeat the erroneous conclusion of the Academicians del 
 Cimento, which represents that fluid as absolutely incompressible. 
 The instrument alluded to in the text holds about 12 pounds of 
 water, which was introduced with great care and patience : The 
 contraction and subsequent dilatation in the stem, from abstracting 
 and restoring the pressure of nine-tenths of an atmosphere, amounts 
 to 3 or 4 inches, and is rendered visible at the distance of several 
 benches in a large class-room, by help of a drop of quicksilver rest- 
 ing on the top of the aqueous column. 
 
 I have likewise had constructed, by our ingenious young optician 
 Mr John Adie, a large and delicate instrument, suggested by the 
 plan of Oerstedt, and capable of extensive application. It bears 
 safely a pressure of 12 or 15 atmospheres, and not only measures 
 easily the contraction of different fluids, but serves to indicate the 
 various compressibility of solid substances. From a series of ex- 
 periments which I have instituted, 1 may venture to anticipate the 
 detection of some interesting and important facts in the economy 
 of Nature. 
 
 The theory of the compression of bodies, carried to its full extent, 
 might give rise to several bold but striking speculations regarding 
 
 VOL. i. 2 F 
 
450 NOTES. 
 
 the internal constiltttion of our globe. Let the density of any sub- 
 stance, at a depth corresponding to the distance x from the centre in 
 miles, be denoted by e?, (that at the surface being assumed the unit,) 
 and the radius and the modulus of elasticity expressed by r and 
 m. Since the power of internal gravitation is directly as the dis- 
 tance from the centre, it will be demonstrated in the Second Vo- 
 
 lume of this Work, that Hyp. Log. d = ~ , or, adopting com- 
 
 mon logarithms, and inserting the numerical values, 
 395G 2 
 
 For Atmospheric Air, this formula becomes, 
 3956 2 # 2 
 
 9090 
 For pure Water, it passes into, 
 
 3956 2 
 
 41570 
 
 And for white Marble, the formula is, 
 3956*- ^ 
 Log. d- 728 7 2 oO ' 
 
 Hence it may be computed, that if the same law of condensation 
 continued, Aii- would become as dense as Water at the depth of 33| 
 miles ; it would even acquire the density of Quicksilver at a far- 
 ther depth of 163 miles. 
 
 The idea which I formerly threw out in the article Meteorology, 
 of the Supplement to the Encyclopedia Britannica, that the ocean 
 may rest on a subaqueous bed of compressed air, is therefore not 
 devoid of probability. Supposing the rate of contraction were to 
 proceed more slowly than at first, still the required measure of 
 condensation would be attained at a depth which forms a very 
 small part of the radius of the globe. 
 
 But Water, under the weight of an enormous column, must like- 
 wise largely contract. At the depth of 93 miles, it would be com- 
 pressed into half its former bulk ; and at the depth of 362J miles, 
 it would acquire the ordinary density of quicksilver. Even Mar- 
 
NOTES. 451 
 
 ble itself, subjected to its own pressure, would become twice as 
 dense as before at tbe enormous depth of 28 7 j miles. 
 
 It is curious to remark, tbat, from its rapid compressibility, Air 
 would sooner acquire the same density with Water, than this fluid 
 would reach the condensation of Marble. For the coincidence of 
 Air and Water, the formula becomes 
 9441^707 Q1OQO 
 
 a* = 15649936 ''I Log. 840 ; whence the depth 
 
 is 35J miles. 
 
 For equal densities of Water and Marble, the formula is 
 
 * = 15649936-^2: Log . 2 . 3 4 ; and the depth 
 
 descends to 172 T 9 n miles. 
 
 If we calculate for a depth of 395 f miles, which is only the tenth 
 part of the radius of the earth, we shall find that Air would attain 
 the enormous density of 101960 billions ; while, at the same depth, 
 Water would acquire but a density of 4.3492, and Marble only 
 3.8095. 
 
 At the centre of the earth, the several formula will become 
 simpler. The logarithm of the final augmented density would be 
 . 15649936 15649936 Af ^ .. 15649936 
 
 forAir -9T690-' f rWater 2415707' apdf rMarble ^287200- 
 Air would hence reach the inconceivable density expressed by 764 
 with 166 ciphers annexed, while Water would be condensed 
 3009000 times, and Marble acquire the density of 119. 
 
 Such are the prodigious results deduced from the law of gravi- 
 tation, even supposing the structure of the globe were uniform. 
 But if we take into the estimate the augmented power from con- 
 densation, the numbers would become still more stupendous. It 
 follows, therefore, that if the great body of our earth consisted of 
 any such materials as we are acquainted with, its mean density 
 w.ould very far surpass the limits assigned by the most accurate 
 investigations. The astronomical observation by Dr Maskelyne 
 on the deflection of a pendulum, caused by the attraction of the 
 sides of Mount Schehallien, and the nice experiments made with 
 the Balance of Torsion by Mr Cavendish, on the mutual action 
 
452 NOTES. 
 
 of ponderous balls of lead, nearly concur in representing the mean 
 density of the globe as only about jive times greater than that of 
 water. It seems, therefore, to follow conclusively, that our planet 
 must have a very widely cavernous structure, and that we tread 
 on a crust or shell whose thickness bears but a very small propor- 
 tion to the diameter of its sphere. Physical Science can extend her 
 prospects to the farthest verge of possibilities ; but Chemistry, even 
 in its present advanced stage, fails altogether in aiding inquiry ; 
 and the various hypotheses framed by Geologists are built with 
 such scanty and slender materials, as to furnish no safe guidance 
 through those boundless speculations. 
 
 It is evident, that immense compression would totally derange 
 the powers of elective attraction, and change the whole form and 
 constitution of bodies. When air becomes denser than gold, it is 
 hard to conjecture what transmutations this plastic fluid must un- 
 dergo. The bowels of the earth may contain substances thus trans- 
 formed, bearing no longer any resemblance to their aspect on its 
 surface. 
 
 But since an absolute void is inadmissible, the vast subterranean 
 cavity must be filled with some very diffusive medium, of asto- 
 nishing elasticity or internal repulsion among its molecules. The 
 only fluid we know possessing that character is LIGHT itself, 
 which, when embodied, constitutes Elemental Heat or Fire. It 
 is elicited from every substance by percussion or compression, by 
 electrical agency or chemical affinity. With every species of light 
 our vision is equally perfect, and, consequently, the luminous parti- 
 cles must, in all cases, dart forward with the same celerity, or travel 
 at the rate of about 200,000 miles in a second. But since atmos- 
 pheric air is projected into a vacuum with the velocity of only a 
 quarter of a mile each second, the motion of light is thus 800,000 
 times more rapid. Wherefore, the propulsive force of light com- 
 pared with that of air is expressed by the square of this number, 
 or 640 billions; and hence its Modulus of Elasticity must amount 
 to the stupendous column of 3200 billions of miles, an extension 
 which is 889 times greater than the diameter of the orbit of 
 
NOTES. 453 
 
 Uranus, the most distant of the planets yet discovered. Only 
 such surpassing powers of repulsion would appear at all adequate 
 to balance the cumulative mass of compression, and restrain the con- 
 densation of our globe within moderate limits. 
 
 We are thus led, by a close train of induction, to the most im- 
 portant and striking conclusion. The great central concavity is 
 not that dark and dreary abyss which the fancy of Poets had pic- 
 tured. On the contrary, this spacious internal vault must contain 
 the purest ethereal essence, Light in its most concentrated state, 
 shining with intense refulgence and overpowering splendour. 
 
 NOTE II. P. 96. 
 
 The problem of the Swiftest Descent, or the investigation of the 
 Brachystochronous Curve, was solved only by Mathematicians of 
 the first order about the close of the seventeenth and beginning of 
 the eighteenth century. It particularly exercised the ingenuity, 
 but it likewise unhappily excited the contentious spirit, of the 
 Bernoullis. The last solution which James the elder brother pro- 
 duced in 1718 seems uncommonly elegant, and deserves the more 
 attention, as it may be considered as the germ of the Method of 
 Variations with which Lagrange has enriched the Higher Calcu- 
 lus. I have endeavoured in the text to give the substance of that 
 solution without involving it in symbolical notation ; but perhaps 
 the reader will not be averse to see the demonstration remoulded. 
 
 If a body descend by its own gravity in the shortest time possi- 
 ble from one point to another in the same vertical plane, it must 
 describe every element of its cur vilineal path likewise in the shortest 
 time. Conceive this element to be composed of two minute portions 
 AC and BC, viewed as straight lines ; or let the extreme points 
 A and B stand equidistant on both sides of the horizontal line CD, 
 and a body descend with a certain velocity from A towards this 
 line at C, and next with a different velocity from C to B, so as to 
 complete the track ACB in the shortest time. As this is only a 
 
NOTES. 
 
 limiting position, assume the proximate path AcB, and from A and 
 B describe the minute arcs Ca and cb, the track AC is thus length- 
 ened by ca, while BC is shortened by Cb ; and consequently ca 
 must be described in the same instant of time as C6, or ca is to 
 Cb as the velocity of description above the line CD is to the velo- 
 city below it. But the elemental triangles being viewed as right 
 angled, ca and Cb are the sines of the angles cCa and cC6, which 
 are equal to the angles which AC and CB make with a vertical 
 line. Wherefore the Brachystochronous Curve must be such, that 
 the sine of the declination of every portion of it from the perpen- 
 dicular shall be proportional to the corresponding celerity of de- 
 scent. Now this property belongs to the Cycloid. For in fig. 57, 
 the tangent at E being parallel to the cord GD of the generating 
 circle, makes with a vertical the angle GDC, of which the sine 
 CG is proportional to the velocity acquired in falling directly from 
 C to H, or along the cydoidal arc from A to E. 
 
 NOTE III. P. 101. 
 
 Huygens strongly recommended the adoption of the length of 
 a second's pendulum as an universal standard measure, it being 
 divided into three equal parts, which he proposed to call horary 
 feet. The project was afterwards frequently resumed and aban- 
 doned. The great difficulty consists in determining experimen- 
 tally the position of the centre of oscillation. In 1778, Mr Hatton 
 proposed to avoid this, by applying a moveable point of suspension, 
 and taking for the standard the difference between the lengths of 
 the two pendulums thus formed. The ingenious Mr Whitehurst, 
 in 1787, improved the plan, by making his pendulums to vibrate 
 84 and 42 times in a minute, their lengths being 20 and 80 inches, 
 and leaving, consequently, a difference of 5 feet as the standard. 
 In performing the experiment, he found the second's pendulum in 
 London to have the length of 39| inches. 
 
 After a long interval of time, the question of a standard pendu- 
 
NOTES. 455 
 
 him was again revived. The very skilful artist, Mr Troughton, ' 
 proposed a cylinder of brass, suspended alternately from either end, 
 and two- thirds of the intermediate distance to be assumed, accord- 
 ing to theory, for the length of the pendulum. It was soon found, 
 however, liable to much inaccuracy from the unequal density of 
 the metal. Captain Kater, in 1818, happily availed himself of the 
 beautiful property demonstrated by Huygeus, that the point of 
 suspension and the centre of oscillation are interchangeable. He 
 employed a bar of plate brass, with two moveable knife edges for 
 suspension, and a heavy weight attached. A small intermediate sli- 
 der was annexed, and shifted along, till the vibrations from both 
 points of suspension became isochronous. The number of those 
 vibrations was reckoned from their successive coincidences with the 
 oscillations of the pendulum of a chronometer. In this way was 
 formed a standard pendulum, the interval between the two points 
 of suspension marking its true length. After applying several de- 
 licate corrections, Captain Kater fixed the length of a pendulum vi- 
 brating seconds in London, at the temperature of 62 in vacuo, at 
 39.1386 inches. In 1819, this scientific observer rectified his con- 
 clusion, and determined the length of the pendulum at different re- 
 mote stations. For London, he finally gives 39.13929, and for 
 Leith Fort 39.15554 inches. 
 
 NOTE IV. P. 110. 
 
 On the principle of Centrifugal Force, might be constructed a sort 
 of Revolving Battery. Suppose a wide vertical cylinder holding 
 a supply of balls, and carrying a very long horizontal branch, to be 
 whirled rapidly about its axis : it would evidently discharge from 
 the mouth of the pipe a torrent of shot with considerable force. 
 Each ball, driven along the tube by centrifugal action, and descri- 
 bing at the same time a circle, must be projected at an oblique an- 
 gle. To find this deviation from the tangent, or a perpendicular to 
 the tube, is a curious problem. Resuming the notation of the text, 
 and putting x for the distance of the ball from the centre of mo- 
 
456 NOTES. 
 
 tion^rr ^, and consequently, vdv = * , and the integral 
 v z = 7 ^- t or yg . = v t the radiating velocity. But the velo- 
 
 city of revolution being , is evidently to v as 2 : or ^/32 : 1, 
 * 
 
 which is the ratio of radius to the tangent of declination, or 
 10 1' 35". Hence the obliquity of discharge remains in all cases 
 precisely the same. 
 
 In Holland and Flanders, the bleachers use a very long scoop 
 for watering their linen. The stream must therefore issue from 
 the groove always nearly at the same angle. 
 
 NOTE V. P. ]68. 
 
 If the Concentrator of Force receive its impulsion from a cord 
 wound about the circumference of a cylinder, the moving power 
 will evidently travel with an accelerated velocity. Such a mode 
 of action is easily obtained from a descending weight or an expand- 
 ing spring. But where animal power is employed, or the pressure 
 of a regular engine, the primary motion must necessarily be uni- 
 form. This condition is very nearly, if not absolutely, obtained, 
 by substituting a sort of tapered fusee instead of the cylinder, the 
 pressure beginning at a considerable distance from the axis, and 
 then gradually approaching it. It is a curious, though not a diffi- 
 cult problem, to determine the nature of the curve, which is a 
 species of hyperbola. The reversed incurvation serves to soften 
 and equalize the discharge of the momentum acquired by the Con- 
 centrator. 
 
 NOTE VI. P. 209. 
 
 The common atmospheric engine, having a counterpoise at tlje 
 farther end of the beam, acts merely by pulling. The perpendi- 
 
NOTES. 457 
 
 cular motion of its pump rod is therefore produced by a flexible or 
 jointed chain (like that of a watch) bending on a sector. But in 
 Mr Watt's engine, the power, being communicated by alternately 
 drawing and pushing, it became necessary to direct the rod of 
 the piston in a vertical and rectilineal path. This is effected, at 
 least approximately, by a simple and ingenious contrivance called 
 the Parallel Motion. Fig. 113. represents it : AC is one half of 
 the beam turning on its centre C, ADEF is a parallelogram jointed 
 at its four angles, and carrying the rod from F, which is guided by 
 the bar BE, fastened to a firm beam at B, and working on the joint 
 E. It is easy to see that E must describe a small arc, and that 
 the deviation from the tangent and towards B will be inversely as 
 the radius BE. But, for the same reason, the point D deviates 
 
 towards C by a quantity as -r^ ; and EF being kept parallel to 
 L/JJ 
 
 CD, the point F will decline proportionally from the vertical to- 
 wards C. Thus, the deviation of the top of the piston-rod to the 
 
 I 1 P* F* 
 
 right and to the left will be denoted by |jp, and by ^^ . ^- or 
 
 T7P 1 
 
 I ; and consequently, to correct those opposite effects, it is 
 i^iJ 
 
 1 FF 
 
 requisite that ^ = - , or BE . EF = CD 8 . Wherefore the 
 
 beam CA must be so divided, that the part CD shall be a mean 
 proportional between the remainder DA and the length of the stay 
 BE. 
 
 This mode of rectification is sufficient for practice, though 
 strictly applicable only to small arcs of vibration. The extended 
 path of E is a complex curve line. 
 
 NOTE VII. P. 259. 
 
 This property may be demonstrated geometrically. For let the 
 parabola BAC (fig. 134.) have its parameter arid its axis AB equal 
 to the diameter of the circle ; if AD be taken the third part of AB, 
 
458 NOTES. 
 
 then DE, in the parabola, will be equal to the corresponding chord 
 of the circle, DB will be proportional to the square of the chord 
 AE. The strength of the beam will hence be represented by the 
 rectangle BDEF. But this space is a maximum ; for the tangent 
 EF being drawn, AD= AT, and consequently DT=DB. But the 
 elementary rectangle KD is equal to KG or EL. The rectangle 
 BDET has thus its increments balanced by the decrement, and is 
 therefore the greatest that can be inscribed in the given parabola. 
 
 NOTE VIII. P. 281. 
 
 The power of traction exerted by a horse, being evidently com- 
 pounded of the velocity and the strain, is hence denoted by 
 v (12 v) 2 , which becomes a maximum when v is equal to four 
 miles, or the third part of 12. But the same conclusion may be 
 derived geometrically. Let OAC (fig. 133.), be a parabola, of 
 which AO is the axis, and AB a vertical tangent : If the part DB 
 denote the actual velocity, the perpendicular DE, being as the 
 square of AD, will express the strain. Wherefore the exterior 
 rectangle DEF will represent the compound effect. Contiguous 
 to E assume the point K, and draw the secant EKT ; then AT 
 will approximate to an equality with AG. The elementary rect- 
 angle GK, being equal to KH, must therefore be double of KD. 
 But the rectangle EB in the state of a maximum, will have its in- 
 crement KD on the one side equal to its decrement EL on the 
 other. Consequently GH is double of EL, and the part AD dou- 
 ble of DB, which is therefore the third part of the whole AB. 
 
 NOTE IX. P. 399. 
 
 
 When air is projected from a small orifice, either into free space 
 or into a close vessel with a wider exit, it will evidently spread out 
 in diverging streamlets, and hence suffer rarefaction. The radia- 
 ting discharge of a fluid thus involves a principle, as I noticed in 
 
NOTES. 4-59 
 
 tbe article Meteorology of the Supplement to the Encyclopaedia 
 Britannica, which explains a number of curious facts. Hence tin- 
 suspension and play of a little ball above a jet of air from a con- 
 densing engine ; and hence, too, the lowered temperature of such a 
 stream from a condensing engine, or of a current of the High Pres- 
 sure Steam. 
 
 The same principle elucidates the seeming paradox in the action 
 of fluids, lately considered by the ingenious M. Hachette. If a 
 pipe bent directly downwards expand at its lower extremity into 
 a cone, the base of which is horizontal, encircled by a narrow brim, 
 and pierced with a little central hole, but having a circular plate 
 fitting loosely with a weight appended : a strong current of air is- 
 suing through that narrow orifice, so far from blowing away the 
 cover, will draw it forcibly and support the load. The sheet of 
 air between the opposite surfaces being kept rarefied by its diver- 
 ging streamlets, the external atmosphere presses strongly upwards. 
 Hence arises the uncertainty in some cases of safety valves. 
 
 A similar effect is produced in the expanding discharge of water 
 from a narrow outlet. The streamlets dividing from the centre, 
 draw in air and keep it rarefied by their rapid divergence. The 
 play of a ball over a. jet d'eau depends on a modification of the same 
 principle. A like explanation may be given of another fact, men- 
 tioned to me by a friend, that the common bellows work with 
 great difficulty when the valve is much larger than the hole which 
 it covers. 
 
 NOTE X. P. 407. 
 
 The latest and best set of experiments on the motion of fluids, 
 was performed between the years 1811 and 1815 at Fahlun, by 
 Lagerhielm, Forselles, and Kalistenius, at the expense of the Min- 
 ing Society, and published in 1818 and 1822 at Stockholm in the 
 Swedish language, under the title of Hydrauliska Fvrsok. The 
 profound mathematical knowledge displayed in that work is highly 
 creditable to the practicable engineers of Sweden. I regret that I 
 cannot afford space for any detail of their experiments. It may 
 
460 NOTES. 
 
 suffice to remark, that they found the discharge of water, from 
 pressure through a circular hole, to be almost exactly 3-5ths, and 
 the shock of a current against a plane, 9-10ths of the quanti- 
 ties assigned by theory ; and that the resistance which a wedge 
 encounters in water, consists of two distinct portions, as already 
 stated in the text, the one depending on the square of sine of the 
 ngle, and the other on the cosine of that angle. 
 
 NOTE XL P. 432. 
 
 I find that, adopting round numbers, what is called a single 
 horse power in mechanical performance may be estimated at one 
 thousand cubic feet of water, raised to the height of one foot in a 
 minute. This might be exemplified in the small river Leven, which 
 issues from a Lake of that name, and runs mostly through Fife- 
 shire about twelve miles, till it discharges itself into the Firth 
 of Forth. The mean flow being reckoned 4000 cubic feet for 
 every minute, the descent of 306 feet would certainly leave a fall of 
 200 available for the purpose of driving Machinery. The whole im- 
 pulsion of the stream of the Leven is hence represented by 4 X 200, 
 or 800 horse power ; or what is equivalent to the operation of 40 
 steam engines, each of 20 horse power. 
 
 NOTE XII. P. 437. 
 
 A ship must evidently sail faster in proportion to the extent of 
 canvas exposed to the wind, and the smallness of its resistance in 
 ploughing the water. The rate of going will hence depend as 
 much on the stability of the vessel as on its tapered form. The late 
 Admiral Chapman of the Swedish Navy, the ablest and most scien- 
 tific shipbuilder of his day, gave, as the result of very long expe- 
 rience, that the square of the effectual velocity of a ship may be 
 
 B*L 3 
 
 expressed by ~ 3 , where B denotes the breadth, and D the 
 
NOTES. 461 
 
 depth at the bilge, and L the length of the hull. Fast sailing is 
 thus the least affected by the breadth, but depends chiefly on the 
 length and on the smallness of the draught of water. It hence 
 follows, that the absolute efficiency of navigation, or the quantity 
 of goods transported in a given time, will be represented by 
 
 D 5 
 
 Our merchant ships, in their construction, are notoriously infe- 
 rior to those of most other nations, and particularly the Swedes 
 and Americans. They sail worse, have less stability, and suffer 
 more from stress of weather. These defects have been chiefly 
 caused by the operation of the absurd rule for the admeasurement 
 of tonnage, which takes into account only the length and breadth 
 of the ship, assuming the depth as always the same portion of the 
 latter. The builders are therefore tempted to increase the depth at 
 the expense of the breadth, and the vessel is thus made to draw 
 more water and carry less sail. Another evil arises from this con- 
 struction, the necessity of deepening our harbours. Let us hope, 
 that so gross a mode of ascertaining the tonnage will be speedily 
 abolished. 
 
 NOTE XIII. P. 448. 
 
 I have endeavoured, in a popular way, to reconcile the New- 
 tonian theory of waves with the actual appearances. But to inves- 
 tigate the subject thoroughly would prove a most arduous task. 
 The profound researches of Lagrange and Poisson on the vibra- 
 tions of fluids, are only fine speculations, which yet seem to bring 
 out no definite or practical results. 
 
 END OF VOLUME FIRST. 
 
 EDINBURGH : 
 PRINTED BY JAMES WALKEB. 
 
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ERRATA. 
 
 age. Line. 
 
 43, 2, from hot, for AG read AC 
 53, 2, from do. for PR read CQ 
 3, 11, from top, for GSR razd GSP 
 58, 6, from do. for A'O read A'O' 
 61, 2, from do. for cylinder read cone 
 63, 1 & 2, from do. for APB is to BPC 
 
 read BPC is to APB 
 6 3, 3, from do.>r CB read DB 
 68, 2, from do. for fig. 31 read fig. 10 
 
 6, from do. 'after ACB read fig. 31 
 
 14, from do. for AB read at B 
 
 76, 14, from do. for fig. 35 read fig. 36 
 86, 10, from bot. for CDB read BCD 
 91, 5 & 6, from do. invert the ivords small 
 arc 4' chord 
 
 10, from do. for EBFB read FB 
 
 94, 10, from do. for CD read KD 
 
 95, 14, from top, for 95 raze? 95 * 
 
 96, 17, from do. for IKN read IKL 
 
 \o.fc 
 y.for 
 
 99, 3, from do.for + B. AB 2 razc?-B. AB 9 
 
 Page. Line. 
 
 268, 3, from hot. transfer and thickness /o 
 
 after breadth 
 304, 2, from top,/or Ee read E<p 
 
 317, 4, from do.for I foot read 1 ft. 8 in. 
 
 318, 8, from do. for 875 read 925 
 
 319, 11, from do. for but read be 
 
 323, 10, from do. for AF read NF 
 
 10, from bot. for former read latter 
 
 324, 6, from do. for AB' raw* BB' 
 
 325, top line, for BNB read BMB 
 
 330, bot. line,^- 6 inches read 4 inches 
 
 331, 8 & 9, from top, for l T 7 f read 1 T ' B 
 
 5, from bot./or NI 2 raw* NE e 
 
 333, 8, from do. for hy read y 
 
 4, from do. for h z y* read if 
 
 336, 2, from top, for BNB read BM h 
 
 337, 8, from do. /or 152 read 151 
 
 16, from do. for NL read Ni ' 
 
 339, 3, from bot. for reduced two-third* 
 
 read reduced to two-thirds 
 bot. line, for breadth read depth 
 
 340, top line,>r (21.6)- raM?(21.5) 8 
 
 341, bot. \ine,for LM read LN 
 345, 3, from 'bot. for ba 3 read b 3 a> 
 355, 11, from bot. for column read plate 
 
 375, 14, from top,/or JK read IM 
 
 376, 3, from bot. for GE read CE' 
 from do.for CF read CF' 
 
 2, from do.>r ECF read E'CF' 
 
 - from do.for BCE read BCE' 
 
 bottom linear FH read FH' * 
 
 same do. for it rearf HC 
 
 377, 11, from top,/or GH read EG 
 
 12, from do.fot GC read E 
 
 380, 9, from do./or 156 read 158 
 
 382, 6, from do. for AC and CGH read 
 
 BC and'CG 
 
 383, bot. line,/or 158 read 159 
 
 385, 14, from top,/or 159 read 160 
 
 12, from bot. for AM readQM 
 
 2, from do.for 160 read 161 
 
 386, 1 1, from top, for t read t 
 from do.^r uead ir 
 
 3, from bot. before an read half 
 399, 10, from top, for affecting read effect- 
 ing ; and in one or two other 
 places, the like oversight ocrurs 
 403, 9, from bot. for 166 read 170 
 406, 14, from top,' for 167 read 171 
 424, top Vine, for inches read miles 
 
 426, 2, from bot. and] for 3 (\ g)ix* 
 
 427, 1, from top, J read a 3 (g])w 
 
I 
 
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