ENGINEERSI AND MiiECHIANICS POCKET-PBOOK, CONTAINING UNITEI) STATES AND FOREIGN WEIGHTS AND MREASURES;'.I*AIIOS OF AREAS AND CIRCUMFERENCES OF CIRCLES, CIRCULAR BE'GMENTS AND ZONES OF A CIRCLE; SQUARES AND CUBES, SQUARE AND CUBE ROOTS; L,ENGTLIS OF CIRCULAR AND SEMI-ELLIPTIC ARCS; AND RULES OF ARITHMI'.ETIC. I:fENSURAT IOH O1r SURrFACOS ANID SOLIDS; TIHE MECIIANICAL POWERS: GEOSIETRY, TRIGONOMETRY, GRAVITV, STRENGTIH (i SIATERIALS, WATER WHEELS, IIYDRAULICS, HIYDROSTATICS, PNEUNIATICS, STATICS, DYNAMICS, GUNNERY, IHEAT, WVINGING ENGINES, TONNAGE, SI-OT, S &HEJLLf, CC. STEAAM AiND T! S TEAi~3- 1 R - TI j E; CO MBUSTION, VWATER, GUNPIOWI)ER, CABLES AND) ANCiHORS, FUEL,, AITR GUNS, &C%, &C. TABLES OF THE WEIGIITS OF METALS, PIPES, &c *:ISCELLANEOUS NOTES, DIMENSIONS OF STEAASIERS, SMILLS, MOTION OF BODIES IN FLUIDS, ORTHOGRAPHY OF TECHNICAL TERMS, &c., &c. E IGH II T EDITION. BY CHAS. H. -ASWEL L. MARINE ENGINEER. An examination of facts is the foundation of science. NEW YORK: A RP E R & B RO T H E RS, P UBLIS ItH ERS, 329 & 331 PEARL STREE T, FRANKLIN SQUARE. 1853. HON. WILLIABM S. MILLER, OF NEW YORK, IN TOKEN OF HIS EMINENT SERVICES, AS A MEMBER OF THE HOUSE OF REPRESENTATIVES OF THE UNITED STATES, IN OBTAINING THE PASSAGE OF THE ACT PROVIDING FOR THE CONSTRUCTION OF UNITED STATES MAIL STEAMERS, AND OF THE AUTHOR'S APPRECIATION OF HIS GREAT PUBLIC AND PRIVATE WORTH. NEW YonRK, February 10, 1853. Entered, according to Act of Congress, in the year 1853, by HARPERI & BROTHERS, In the Clerk's Office of the Southern District of New York PREFACE. THE -following work is submitted to the Engineers and Mechanics of the United States by one of their number, who trusts that it will be found a convenient summary, for reference to Tables, Results.and Rules connected with the discharge of their various duties. The Tables comprising the Weights of Metals, Iron and Leaden Balls, Cast Iron and Copper Pipes, &c., were calculated expressly for this work, and from specific gravities of the different materials taken for the purpose. The want of a work of this description in this country has long been felt, and this is peculiarly fitted to supply that want, in consequence of the adaptation of its rules to the metals, woods, and manufactures of the United States. Having for many years experienced inconvenience for the want of a compilation of tables and rules by a Practical Mechanic, together with the total absence of units for the weights and strengths of American materials, I was induced to attempt the labour and the experiments necessary to furnish this work. The proportions of the parts of the steam-engine and boilers will be found to differ in most instances materially from the English authorities; but as they are based upon the actual results of the most successful experience, I do not hesitate to put them forth, being well assured that an adherence to them will ensure both success and satisfaction. The principal sources of information from which I have compiled are Adcock, Grier, Gregory, the Library of UseA2 iv PREFACE. ful Knowledge, and the Ordnance Manual; and to the labours of the authors of these valuable works I freely acknowledge my indebtedness. In my own efforts, I have been materially assisted by the officers of the WTest Point Foundry Association, who liberally furnished me with the means of making such experiments as were considered necessary; and to the Engineer of that establishment, Mr. B. H. Bartol, and also to Mr. Chas. W. Copeland, I am indebted for much valuable information and assistance. To the Young Engineer I would say, cultivate a knowledge of physical laws, without which, eminence in his profession can never be securely attained; and if this volume should assist him in the attainment of so desirable a result, the object of the author will be fully accomplished WE have seen a proof copy of HASWELL'S Engineers' and Methanics' Pocket Book, and approve of its design, which in extent exceeds that of any of its class: a work of this description has long been wanted, apd we are satisfied of its usefulness and application. GOUVERNEUR KEIMBLE,) WILLIAM KEMBLE, West Point Foundry ssociation, ROBERT P. PARROTT, N. Y. B. H. BARTOL, JAMES P. ALLAIRE, X1lare Works, N. Y B. R. M'ILVAINE,. E. DETIo ALLEND, Civil Engineer's,. I. CHARLES W. H~ACLEYi, Professor of JMathematics, Columrnbia College, N. Y. HOGG & DELAMATER, Phcenix Foundry, NJV. Y. STILLMAN & CO., NJVovelty Works, NV. Y. T. F. SECOR & Co., Steam Engine.Manufacturers BROWN & BELL, Ship Builders, X Y SMITH & DIMON, MERRICK & TOWNE, Southwlark Foundry, Phila. December, 1843. We concur in the above, having used a copy of Haswell's Engineers' and Mechanics' Pocket Book. WILLIAM MAYNADIER, Captain Ordnance Corps, U. S. J. WILLIAM J. TOTTEN, Steam Engine Mianufacturers, JOSEPH TOMLINSON, Pittsburg, Penn. August, 1844. We concur in the above. JOHN ERICSSON, Civil Engineer, XJ. Y. C. W. COPELAND, Principal Engineer, U. S. J SAMUEL HUMPHREYS, Chief Naval Constructor, U. S. JN. J Foundry and Steam Engine.Manu. J. Rt. ANDERSON, 2 factory, Richmond, Va. March, 1846. We concur in the above. FRANCIS GRICE, Chief Naval Constructor, U. S. N. FREDERICK E. SICKELS, Steann Egincer, N'. Y. September, 1847. JOHaN FARON, JR., Engineer J. Y. and Liv'l. Line of JMail Steamers. A. MEHAFFEY, 0Gospoert Iron WVorks, Va. JADeZ CONEY, Steam Eng'ine Mfanufacturer, Bos. ton,.Mass. Vi RECOiM\IENDATIONS. We take pleasure in adding our names to the list of recommendations of Haswell's Engineers' and Mechanics' Pocket Book, which we consider the best of the kind. 1. P. MoRRIS & Co., Steam Engine J.Manufactulrers, 1 Philadelphia, Penn. With much pleasure we add our testimony to thb value of Has, well's Engineers' and Mechanics' Pocket Book, which we are gratified to learn is about to be published in a fifth edition, as each succeeding issue contains valuable additions to the preceding. There are several valuable English works of this description, but the peculiarities and differences in this country render his a far more useful book; we consider it, in fact, an almost indispensable pocket companion. MURRAY & HAZLEHURST, Vulcan Works, Baltimore, Md. We concur in recommending Haswell's Engineers' and Mechanics' Pocket Book, and consider it a very useful and valuable work. A. & C. REEDER, Steam En, ine oMeanufacturers, Baltimore, Old. In the daily use of Haswell's Engineers' and Mechanics' Pocket Book, we concur in the recommendation of it as one of very general reference and great utility; it is an indispensable pocket companion. REANEY, NEAFIE & CO., Penn Works, rtensington, Phia. August, 1848. CONTENT S. Page CONIC SECTIONS. Pa 11 Construction of Figures......... 48 EXPLANATIONS OF CHARACTERS... 12 Definitions.......-............. 54 To construct a Parabola. 54 U. S. WEIGHTS AND MEASURES. To construct a Hyperbola 54 jMeasures of Length................. 13 Ellipse. 55 13rabolas. — ~~~~....~~~~.~~~~~~~~ 56.Jeasures of Surface.............. Parabolas.56 MIeasures of Capacity.......14....... 14 MJeasures of Solidity.............. 15 Measures of Weight.............. 15 MENSURATION OF SURFACES. Miscellaneous................... 16 Triangles, Trapeziums, and TrapeMeasures of Value............. 16 zoids. 59 Mint Value of Foreign Coins... 16 Regular Polygtons................. 60 Re'gular Bodies, Irregular Figures, FOREIGN WEIGHTS AND MEASURES. and Circles.............. 61.Jrcs of a Circle.. 62 JMeasures of Length........... 17 Sectors, Segments, and Zones... 63 Measures of Surface.............. 18 Ungulas and Ellipses........... 64 Measures of Capacity............. 18 Parabolas aned Hyperbolas.... 65 JMeasurcs of Solidity.............. 19 Cylindrical Rings and Cycloids 66 MJeasures of Weight...20..... I Cylinders, Cones, Pyramids, Spheres, SCRIPTURE AND ANCIENT MEAS- acnd Circular Spiendles.....67 URES....... 21 By JM1athematical Formulza. Table for finding the Distances of Lines of Circle, Ellipse, and ParaObjects at Sea.................. 21 bola...........6. 6 Reduction of Longitude........... 22 Areas of QuLadrilaterals, Circle, Gylintder, Spherical Zonte or Segment, VULHAR FRACTIONS.............. 23 Circular Spindle, Spherical TriDECIMAL F RACTIONS 26 angle, or any Surface of Revoleution.68 D1uodecimals..................... 30 Capillary Tube........... 69 RULE OF THREE.31 Useful Factors....... 69 ompoused Propo:rtion..... -.-. 32 Exaemples in JMIensuration......... 70 Involutrion...................... 32 Evolutiocn.........-e- -.......... 32 AREAS OF THE SEGMENTS OF A Arithmetical Progression....... 34 CIRCLE....................... 72 Geometrical Progression.......... 35 LENGTHS OF CIRCULAR ARCS S 75 Perwmut ation................... 36 ColnbicLetion..36 To filnd the Length of an Elliptic Position 36.. Curv..........36 Fellooship....................... 37 LENGTHS OF SEMI-ELLIPTICA RCS. AC 78 Aqlligation............. 38 AREAS OF THE ZONES OF A CIRCLE. 8 Comnpoune d Interest................ 38 Discount....................MENSURATION 39 Equation of Payments. MENSURATION OF SOLIDS. ASnnunities.... *.. 40 Of Cubes and Parallelopipedons.... 81 Perpetuities 41........ReglarBodie............ 81 Chronologicel Problems. 41 " Cylicnders, Prisms, acnd Ungulas 81 To find tlie Moon's Age 42 "Cones and Pyramids........... 82 Table of Epacts, Dominical Letters, Wedges and Prismoids 83 &c.....................;; 42 "Sp4eres 83 Promiscuous Queestions 43 Spheroids..... 84 " Spher~oids ~~~~.~..84 " Circular Spindles............. 85 GEOMETRY.;' Elliptic Spindles...... 85 Definitions................ 46 "Parabolic Conoids and Spindles. 86 V111 CONTENTS. Page HYDROSTATICS. Of Hyperboloids and Hyperbolic Co- Pago noids..................6 Of Pressure...................... 167 "Cylindrical Rings............. 87 Construction of Banks............ 168 By Jlatlhematical F'ormule......... 87 Flood Gates............... 168 Cask Gtaugingr................... 88 Pipes.168 Examples in Mensuration.......... 89 -lydrostatic Press............. 169 AREAS OF CIRCLES............... 91 HYDRAULICS AND HYDRODYNAMICS. CIRCUMFERENCES OF CIRCLES..... 95 Of Sluices..................... 170 SQUARES, CUBES, AND ROOTS..~.... Of Vertical.Jpertures or Slits... 170 Of Strcams and Jets...............171 To find the Square of a Number Velocity of Streams...... 171 above 1000........-... 116 To find the Velocity of Water run. To finld the Cube or SquLare Root of I ning through Pipes.172 a highpr XNamnber than is contain- Waves...-1.................... [72 ed icn the Table..... - 118 Table showing the Head necessary To find the Cube of a NJVumber above to overcomle the Friction of Water 1000. 118 il Horizontal Pipes..........173 7'o find the Sixth Root of a X.NAmber 118 General Rules........... 174 To find tihe Cube or Square Root of Table of the Rise of Water ine Rivers l75 a JVNueber consistisng of lnLtegers WTER WHEELS.... 76 anld Decimals.................. 119 SIDES OF EQUAL SQUARES... 120 To find the Power of a Stream..... 176 Barkner's JMill..................178 PLANE TRIGONOMETRY............ 123 Tobfind the Centre of Gyration of a Oblique-as7gled Triangles....... 124 Water Wheel............-... 179.Motes........................ 179 NATURAL SINES, COSINES, AND TANGENTS.......................... 125 PNEUMATICS. Sines and Secants................. 127 IWeirht, Elasticity, and Rarity of MECHANICAL POWERS. ir........................ 180.Jleasurement of Heights by means Wheel and Axle........ of the Baroeter. 181,.cline d Plane.....V... 131 Velocity anld Force of Wind........ 181 inclined Plane.......... -....... 13l Wedge........................ -. 133 STATICS. Screw...... 133 Pzul4ley y...... - ~.... ~~~ ~~..~~~~~ 135 Pressure of Earth against Walls.. 182 DYNAMICS.........,............. 183 CENTRES OF GRAVITY. PENDULUMS....................... 185 See:faces.-....... 137 Solids............................ 138 CENTRE OF GYRATION 186........ 1 GRAVITATION.................. 139 CENTRES OF PERCUSSION AND OSPromiscuous Exanples............ 142 CILLATION... 1 Gravities of Bodies........143 CENTRAL FORCES. 190 Specific Gravities................. 143 Fly Wheels.... 192 Proof of Spirituous Liquors....... 144 Governors....... - 192 TABLE OF SPECIFIC GRAVITIES. 145 GUNNERY....... 193 STRENGTH OF MATERIALS. FRICTION194 TI'esile StclFRgtIO..........................148 Tensile Strength 148 Transverse Strength............. 149 HEAT. Deflexion..................... 156 Communication of Caloric......... 196 Torsional Streongth-Shafis....... 158 Radiation of Caloric.............. -- 197 Gudgeons and Shafts-.. 160 Specific Caloric.......... 197 TEETH OF WHEELS 161 Evaporation......... 200 Congelation and Liquefaction...... 200 1Felocitpj'of Wheels.............-. 162 Distillation......... *. — 200 Strength of Wheels *... 163 JMiscellaneous....... 201 General Explanations concerning Alelting Points of Alloys......... 202 Wheels.................... 164 Warming Apartments...........202 HORSE POWER.. 165 AIR.................... 2....... 203 ANIMAL STRENGTH. LI ANIMAL STRENGTH. LIGHT........................... 204 Mecn......-........-............. 165 Horses16 TONNAG........................ 204 CONTENTS..X ~STE ~AM. ~Pae S3TEAM. Page Composition Sheathing JV ails..... 24 Roiling Points, &c................ 206 Cement........................... 242 Volume, Gravity, &c.............. 207 Brown.Mortar................... 242 Elastic Force, Temperature, &c., & c. 208 Bricks, Laths, &c................. 243 Hyperbolic Lougarithms, dSc......... 269 Hay.......... 243 Wyarming.partments............. 209 Hills in an Jdcre of Ground...... 243 STCEAiM-E NGIN E. lDisplacement of Vessels........... 243 Wieight of Lead Pipe.............. 244 Condensing EEngine............. 210 Tin.......................... 244.Nonr-condensing Engine......... 212 Relative Prices of Wrougfht Iron 245 Boilers................. 213 Power required to Punch Iron and Boilers of.Non-condensing Engine. 214 Copper Plates.............. GENERAL RULES................ 215 IRON, COPPER, ETC. Engines........................ 215 Weight of Square Rolled Iron...... 248 Boilers............... 217 fWeight of Round Rolled Iron...... 247 ~Materials for Boilers............. 219 Weight of Flat Rolled Iron........ 248 Belts................... 219 Values of the Birmingham Gauges 251 Saturation in.Mlarine Boilers.....2 19 CSat in oilers...... AST IRON....................... 251 BLOWING ENGINES................. 220 WeigAht of Cast Iron Pipes........ 252 Horse Power of Engines...... 221-222 Streligth and Stiffness of.Metals... 253 To find the Quanitity of Water Evap- Wl:eight of a Square Foot.......... 254 orated in an Engine........2.... 222 Weight of Cast Iron and Lead Balls 255 To find the Power of an E7gine ne- tWeight of Copper Rods or Bolts... 256 cessary to raise Water.......... 2L22 Weight of Riveted Copper Pipes... 256 Tofind the Velocity necessary to Dis- C E.......................... 257 charge a Given Quantity of Water 223 Loss by Radiation................ 23 Braziers and Sheathing........... 267 Friction-Blowing off........ 223 LEAD............................ 257 BRASS.......................... 257 COFMBBUTSTION. CABLES AND ANCHORS............ 258 Fuels................ 224 CABLES.......................... 259 Evaporative Power of Coals and Pine Wood................ 225 Tables of Hemp Cables and Ropes.. 259 Analysis of Fiuels, c., 4-c...... 226 To ascetain the Strezgth of Cables. 259 f el,.......... 2 To ascertain the Weight of V.Ianilla WATER...................... 227 Ropes and Haiwsers............. GUNPOWDER................ 229 To ascertain the Stren.gth of Ropes. 260 Tables of variois Exzperiments, with To ascetain the Weight of CableBellistic Pen dulum-Proof- Effect laid Ropes........2 60..e.g.. o of Wads, 4-c......... 230-233 To ascertain the Weight of Tarred Penetratione of Shot and Shells. 23 Ropes and Cables.............. 260 Penetration oif Shells........... 235 e Rope... lWeight and Dimensions of Shot.... 235 PILING or BALLS AND SHELLS..... 261 Weight and Dimensions of Leaden [Veights and Dimenlsions of Balls Balls..........2.......... 236 and Shells...................... 262 DISIENSIONSAND WEIGHT OF GUNS, MISCELLANEOUS RULES. SHOT, AND SHELLS, U. S. ARMY. 237 Winding Engines................ 263 DInIENSIONS AND WEIGHT OF GUNS, Windile BaEancg s..............263 SHOT, AND SHELLS, U. S. NAVY.. 238 Frasidielesit Balnics.264 SHOT D SHELLS, S NAY. 238 Measuring of Timber............. 265 MDisplacement of Vessels........... 216 Externli Surface of Hulls of Vessels 2W6 Recapitulation of Weights and Variou s Substaices.. 2}30 i MISCELLANEOUS NOTES. Weights of a Cubic Foot of Various On Materials..................... 267 Substalzces............. 239 Solders, Cements, and Pai ts...... 268 Slating.............. 239 Paints, Lackers, and Stainiing..... 269 Capacity of Cisterns.............. 240 Tracing Paper................... 270 Conipositions................. 240.Zlloys........................... 270 Sizes of J.uts..................... 240 i Electric Conductors............... 270 Screws.................... 241 Illumination- Waves-Ice......... 271 Strength of Copper.............. 241 Babbitt's Aletal-Fall of Rain, 4-. 272 Digging......................... 241 Liquid Solder.................... 272 Coal Gas......................... 241 Angles of Equilibriumn of va-rious lecohol........................ 242 S.bstances..7........... 2 .Xi CONTENTS. Page RIVER ENGINES. Diameter and Lengths of Gas Pipes 273 Page Limits of Vegetation in the Temper- CONDENSING, with Side Wheels.... 288 ate Zone...27......4....... 274 heempen Cords..... 274 CONDENSING, with Side Wheels and Lap-welded Iront Boiler Tutbes.-.... 274 Iron Hll................... 290 Formsule for thicknesses of various NON-CONDENSING, with Side Wheels 291 Metal Pipes.........275 27" " Stern Wheel 292 Alloys of Copper, Tin, Zinc, atzd Lead........................ 275 IRON VESSELS. Longitudilal Compression.-... — 276 STEAMERS............ 293 Absorption of Moisture of Various Materials. -.... -.-....-.-..276. SHIP, SCHOONER, AND CANAL BOAT 294 Table of Recipr-ocals.............. 279 STEAM VESSELS, ENGINES, &C., STEAM ACTING EXPANSIVELY2. 8.. 280 Miscellaneouts lotes............ 295 SUGAR MILLS.................... 296 MARINE ENGINES. SAW MILL..-.............. 297 COTTON FACTORY............... 298 NAVAL STEAIIERS, with Side Wheels 281 "' " t with Side Wheels COTTON PRESS..........-....... 299 and Iron hulls................. 282 BLAST ENGINES................. 300 NAVAL STEAMERS, with Screw Pro- STEAM DREDGING MACHINE..... 301 pellers......................... 283 PILE DRIVING --------- 302 MERCHANT STEAMERS, with Side Wheels............... 284............. 303 MERCHANT STEAMERS, aetd Iron FLOUR MILLS -. 304 Hulls....................... 286 TUBULAR BOILERS -............... 305 MERCHANT STEAMERS, with Screw MOTION OF BOILERS IN FLUIDS..- 306 Propellers and Iront Hulls....... 289 ORTHOGRAPHY OF TECHNICAL TERMS........................ 309 EXPLANATION OF CHARACTERS. I EXPLANATIONS OF THE CHARACTERS USED IN THE FOLLOWING TABLES AND CALCULATIONS. -Equal to, as, 12 inches = 1 foot, or X8 X8 = 16 X4. + Plus, or more, signifies addition; as, 4+6+-5 15. - Minus, or less, signifies subtraction; as, 15-5 = 10. X Multiplied by, or into, signifies multiplication; as, 8 X 9 = 72. -Divided by, signifies division; as, 72 —9 = 8. Is to s t Proportion; as, 2:4:: 8: 16; that is, as 2 is to 4 so is 8::o0is to 16. to -/ Prefixed to any number signifies that the square root of that number is required; as, V/16 = 4; that is, 44 -= 16. / Signifies that the cube root of that number is required; as, Z/64=4; that is, 4X4X4=64. 2 added to a number signifies that that number is to be squared; thus, 42,means that 4 is to be multiplied by 4. added to a number signifies that that number is to be cubed; thus, 43,is = 44 X 4= 64. The power, or number of times a number is to be multiplied by itself is shown by the number added; as, 2 3 4 5, &c. — The bar signifies that the numbers are to be taken together; as, 8-2+6 = 12, or 3X5+3=- 24. Decimal point, signifies when prefixed to a number, that that number has a unit (1) for its denominator; as,.1 is -L,.155 is — 55 &C. ir Signifies difference, and is placed between two quantities when it is not evident which of them is the greater. o Degrees,'minutes, "seconds, "'thirds. < Signifies angle. a —1, a-2, a-_, &c., denote inverse powers of a, and are equal to12' 3, &c. a' a 7 Is put between two quantities to express that the Iormer is greater than the latter; as, a7 b, reads a greater than b. L Signifies the reverse; as, a L b, reads a less than b. Signifies therefore. Signifies because. () Parentheses are used to show that all the figures within them are to be operated upon as if they were only one; thus, (3+2) x 5=25. p is used to express the ratio of the circumference of a circle to its diameter = 3.1415926, &c. A A' A" A"' signifies A, A prime, A second, A third, &c. aXd, a.d, or ad, signifies that a is to be multiplied by d. Note. Thle degrees of temperature used in this work are those of Fahrenheit. NOTATION. 1-I. 2 _ II. t As often as a character is repeated, 3 =111. 1 so many times is its value repeated. 4- IV. 5 - V. A less character before a greater 6 - VI. diminishes its value, as IV _ I from 7- VII. V, or 1 subtracted from 5 - 4. 8 - VIII. 9= IX. A less character after a greater in10 = X. creases its value, as XI X+I, or 20- XX. 10+1 11. 30 - XXX. 40 = XL. 50 = L. 60 = LX. 70 -= LXX. 80 = LXXX. 90 = XC. 100 = C. 00 D, or 1 For every 0 annexed, this be500 = D, or IC. ~ comes 10 times as many. For every C and 0, placed one at 1,000 =M, or CID. Reach end, it becomes 10 times as many. 2,000 = MM. 5,000 =V, or IO0. 6,000 -= VI. 10,000 = X, or CCIO. 60,000- =YL, or L TXO. i A bar, thus -, over any 60,01)00 LX. times. 100,000 C, or CCCIOO0. 1,000,000 =_ M, or CCCCIDOD. 0 2,000,000 -= MM. EXAMPLEs.-1840, MDCCCXL. 18560, XVIIIDLX. WEIGHTS AND MEASURES. 13 UNITED STATES' WEIGHTS AND MEASURES. Measures of Length. Inches. Feet. Yards. Rods. Furl. 12 inches = 1 foot. 3 feet 1 yard. 36 = 3. 5~ yards -- 1 rod. 198 = 16i= 54. 40 rods = 1 furlong. 7920= 660 = 220 - 40. 8 furlongs-= 1 mile. 63360 = 5280 = 1760 = 320 = 8. The inch is sometimes divided into 3 barley corns, or 12 lines. A hair's breadth is the 48th of an inch. Gunter's Chain. 7.92 inches - 1 link. 100 links - 4 rods, or 22 yards. Ropes and Cables. 6 feet = I fathom. 120 fathoms = 1 cable's length. Geographical and Nautical Measure. 1 degree of a great circle of the earth = 69.77 Statute miles. 1 mile...... 2046.58 yards. Log Lines. 1 knot = 51.1625 feet, or 51 feet 14+ inches. I fathom = 5.11625 feet, or 5 feet 14+ inches. Estimating a mile at 6139~ feet, and using a 30" glass. If a 28' glass is used, and eight divisions, then 1 knot = 47 feet 9 + inches. 1 fathom - 5 feet 114 inches. The line should be about 150 fathoms long, having 10 fathoms between the chip and first knot for stray line. NOTE. —Bowditch gives 6120 feet in a sea mile, which, if taken as the length, will make the divisions 51 feet and 5 1-10 feet. Cloth. 1 nail = 24 inches = th of a yard. 1 quarter = 4 nails. 5 quarters - I ell English Pendulums. 6 points = 1 line. 12 lines -= 1 ich. Shoemakers'. No. 1 is 44 inches in length, and every succeeding number is ~ of an inch. There are 28 divisions, in two series of numbers, viz., from 1 to 13, and I to 15. B ~14 WEIGHTS AND MEASURESo Circles. 60 seconds - 1 minute. 60 minutes = I degree. 3600 = 60 360 degrees = 1 circle. 1296000 - 21600. 1 day is.... 002739 of a year, 1 minute is.....000694 of a day. Miscellaneous. I palm = 3 inches. 1 span = 9 inches. 1 hand = 4 inches. 1 metre = 3.281'74 feet. The standard of measure is a brass rod, which, at the temperature of 320 Fahrenheit, is the standard yard. The standard yard of the State of New- York bears, to a pendulum vibrating seconds in vacuo, at Columbia College, the relation of 1.000000 to 1.086141 at a temperature of 32~ Fahrenheit. 1 yard is.... 000568 of a mile. 1 inch is..0000158 of a mile. Measures of Surface. 144 square inches = 1 square foot. Inchs. 9 square feet - 1 square yard. 1296 Land. 304 square yards = 1 square rod. Yards. Ras. WrAdf. 40 square rods = 1 square rood. 1210. 4 square roods 1 acre. 4840-160. 10 square chains acre. 640 acres = 1 square mile. 3097600 = 102400 = 2560. NOTE.-208.710321 feet, 69.5701 yards, or 220 by 198feet squzare = 1 acre. Paper. 24 sheets - 1 quire. Sheets. 20 quires = 1 ream. 480. Drawing Paper. Cap.. 13 X16 inches. Columbier. 33iX 23 inclleh.. Demy.. 19~X15~ " Atlas.. 33 X 26 " Medium. 22 X18 " Theorem. 34 X 28 " Royal.. 24 X19 " Doub. Elephant, 40 X 26 Super-royal. 27 X19 Antiquarian. 52 X 31 " Imperial. 29 X21~ " Emperor. 40 X 60' Elephant. 27 X 224 " Uncle Sam. 48 X120 " Measures of Capacity. Liquid. 4 gills - 1 pint. Gills. Pints. 2 pints = I quart. 8. 4 quarts = 1 gallon. 32 -- 8. The standard gallon measures 231 cubic inches and containe WEIGHTS AND MEASURES. 15 8.3388822 avoirdupois pounds, or 58372.1754 troy grains of distilled water at 390 83 Fahrenheit; the barometer at 30 inches. The gallon of the State of New- York contains 221.184 cubic inches, or 8 pounds of pure water at its maximum density. The Imperial gallon (British) contains 277.274 cubic inches. Dry. 2 pints 1 quart. Pits. Quarts. Gallons. 4 quarts - 1 gallon. 8. 2 gallons = 1 peck. 16 0 8. 4 pecks - 1 bushel. 64 = 32 =-8. U; S. Standard Bushel. The standard bushel is the Winchester, which contains 2150.42 cubic inches, or 77.627413 lbs. avoirdupois of distilled water at its maximum density. Its dimensions are 18- inches diameter inside, 19-i inches outside, and 8 inches deep; and when heaped, the cone must not be less than 6 inches high, equal 2747.70 cubic inches for a true cone. The bushel of the State of New- York contains 80 lbs. of pure water at its maximum density, or 2211.84 cubic inches. 1728 cubic inchles - 1 foot. Inches. 27 cubic feet = I yard. 46656. Miscellaneous. 1 chaldron = 36 bushels, or. 57.25 cubic feet. I cord of wood.... 128 cubic feet. 1 perch of stone.... 24.75 cubic feet. Measures of Weight. Avoirdupois. 16 drachms = 1 ounce. Drachms. Ounces. Pounds 16 ounces = 1 pound. 256. 112 pounds = 1 cwt. 28672 _ 1792. 20 cwt. = I ton. 573440 - 35840 = 2240. 1 lb. -- 14 oz. 11 dwt. 16 gr. troy. The standard avoirdupois pound is the weight of 27.7015 cubic inches of distilled water weighed in air, at the temperature of tht:, maximum density (390.83), the barometer being at 30 inches. Troy. 24 grains = 1 dwt. orains. Dwt. 20 dwt. - 1 ounce. 480. 12 ounces = 1 pound. 5760 = 240. Apothecaries. 20 grains = 1 scruple. Grains. Scruples. Drachms. 3 scruples = 1 drachm. 60. 8 drachms = 1 ounce. 480 = 24. 12 ounces = 1 pound. 5760 = 288 =- 96. 16 WEIGHTS AND MEASURES. Diamond. 16 parts 1 grain = 0.8 troy grains. 4 grains - 1 carat = 3.2 " 7000 troy grains 1 lb. avoirdupois. 175 troy pounds 144 lbs. " 175 troy ounces = 192 oz. " 437~ troy grains - 1 oz. " 1 troy pound -.8228+lb. " Miscellaneous. 1 cubic foot of anthracite coal from 50 to 55 lbs. I cubic foot of bituminous coal from 45 to 55 lbs. i cubic foot Cumberland coal= 53 lbs. 1 cubic foot charcoal. 18.5 (hard wood). I cubic foot charcoal. _ 18. " (pine wood). 1 cord Virginia pine. -2700 " 1 cord Southern pine. = 3300' stone.... 14 " Coals are usually purchased at the conventional rate of 28 bushels (5 pecks) to a ton.=43.56 cubic feet. Measures of Value. 1 eagle = 258 troy grains. 1 dollar = 412.5 " 1 cent - 168 " The standard of gold an'd silver is 900 parts of pure metal, and 100 of alloy, in 1000 parts of coin. Relative Mint Value of Foreign Gold Coins, By Law of Congress, August, 1834. Coin. Weight. Value. Dwt. Cr. BRAZIL. 1 Johannes. 18.. $17.068 1 Dobraon..... 34 12.. 32.714 1 Dobra.... 18 06.. 17.305 1 Moidore..... 6 22.. 6.560 1 Crusado..... 164...638 ENGLAND. 1 Guinea... 5.116 1 Sovereign.. 5 34.. 4.875FRANCE. 1 Double Louis (1786) 10 11.. 9.694 1 Double Napoleon... 8 7. 7.713 COLOMBIA. 1 Doubloon. 17 84.. 15.538. MExico. 1 Doubloon. 17 84.. 15.538 PORTUGAL. 1 Dobraon. 34 12.. 32.714 1 Dobra..... 18 6.. 17.305 1 Johannes. 18.. 17.068 1 Moidore..... 6 22.. 6.560 1 Milrea...1...780 SPAIN. 1 Doubloon (1772).,. 17 81.. 16.030 1 Doubloon (1801). 17 9. 15.538 1 Pistole... 4 3.. 3.883 23.2 grains of pure gold = $1.00. Gold coin of the United States prior to 1834, like to that of England, 88.8 cents per dwt. By Act of 1834, its vwlte its 94.8 cents per dwt. Old U. S. kagle preceding 18:34. $it).1i68. FOREIGN WEIGHTS AND MEASURES. 17 Mint Value of Foreign Coins. ENGLAND. 1 Shilling..... $0.244 FRANCE. 5 Francs....... 0.935 1 Sous.. 0.0093 AUSTRIA. 1 Crown, or rix dollar.. 0.97 1 Ducat...... 2.22 PRUssts. 1 Ducat 2.202 RUSSIA. I Ducat - 10 roubles.. 7.724 1 Rouble.....-.. 0.748 SWEDEN. I Ducat...... 2.19 1 Rix dollar...... 1.08 The relative value of gold and silver is as 1 to 15 —. Mleasures of Length. BRITISH. Yard is referred to a natural standard, which is the length of a pendulum vibrating seconds in vacuo in London, at the level of the sea; measured on a brass rod, at the temperature of 620 Fahrenheit, =39.1393 Imperial inches. FRENCH. Old System. — Line - 12 points. 0.08884 U. S. inches. 1 Inch = 12 lines. = 1.0660 " 1 Foot - 12 inches. - 12.7925 1 Toise - 6 feet. _ 76.755 1 League = 2280 toises. (common). I League = 2000 toises. (post). 1 Fathom = 5 feet. A.=rew System. — Millimetre...03938 U. S. inches. 1 Centimetre.39380 1 Decimetre... - 3.93809 1.Metre.... - 39.38091 " 1 Decametre... -- 393.80917 " 1 Hecatometre.. - 3938.09171 " AUSTRIAN. 1 Foot.. 12.448 " PRUssIAN.. 1 Foot.... 12.361 A' SWEDISI.. I Foot.... = 11.690 PANISH.. 1 Foot... 11.034 1 League (common). = 3.448 U. S. miles. TABLE showing the relative length of Foreign Mleasures compared with BRITISH. Places. lMeasurs. Inches. Planches. Measures. Inches. Amsterdam. Foot 11.14 Malta... Foot 11.17 Antwerp.. 11.24 Moscow... " 13.17 Bavaria., " 11.42 Naples... Palmn 10.38 Berlin... I 12.19 Prussia.. Foot 12.36 Bremen... " 11.38 Persia... Arish 38.27 Brussels.. " 11.45 Rhineland. Foot 12.35 China... " Mathematic. 13.12 Riga.... " 10.79 Builder's 12.71 Rome... " 11.60 " Tradesman's 13.32 Russia.. " 13.75.. urveyor's 12.58 Sardinia Palmno 9.78 Copenhagen. 12.35 Sicily..., 9.53 Dresden.. " 11.14 Spain... Foot 11.03 England.. 12.00.".. Toesas 66.72 Florence.. Braccio 21.60 ".... Palmo 8.34 France... Pied de Roi 12.79 Strasburgh Foot 11.39. Metre 39.381 Ssweden. 11.69 Geneva... Foot 19.20 Turin... 12.72 Genoa... Palmo 9.72 Venice... 13.40 Hanmburgh. Foot 11.29 Vienna... 12.45 Hanover.. 11.45 Zurich... 11.81 Leipsic... 11.11 Utrecht. 10.74 Lisbon... " 12.96 Warsaw 14.03... Palmo 8.64 B2 18 FOREIGN WEIGHTS AND MEASURES. TABLE showing the relative length of Foreign Road Measures compared with BRITISH. Places. Measures. Yards. Places. Measures. Yards. Arabia... Mile 2148 Hungary.. Mile 9113 Bohemia.. 10137 Ireland...3038 China.... Li 629 Netherlands. " 1093 Denmark.. Mile 8244 Persia... Parasang 6086 England... " Statute 1760 Poland... iMile, long1 8101 ".. "G e(l eigrallh. 2025 Portugal.. League 6760 Flanders.. " 6869 Prussia... Mile 8468 France... League, mlarine 6075 Rome... " 2025 "..,.'" comrmlion 4861 Russia... Verst 1167 "..... " post 4264 Scotland.. Mile 1984 Germany Mile, long 10126 Spain... League, cornm. 7416 Hlamburgh.. " 8244 Sweden.. Mile 11700 Hanover.. " 11559 Switzerland. " 9153 Holland... " 6395 Turkey... Belri 1826 leasures of Surface. FRENCH. Old System. — Square Inch.. 1.1364 U. S. inches. 1 Arpent (Paris).. = 900 square toises. 1 Arpent (woodland). = 100 square royal perches..MNew System. —I.re... 100 square metres. i Decare... = 10 ares. 1 Hecatare... = 100 ares. 1 Square Metre.. S 1550.85 square inches or 10.7698 square feet. 1 Are.... 107G.98 TABLE showing the relation of Foreign Measures of Surface compared with BRITIsH. Places. Measures. yards. Places. Measures. Sq. yards. Sq. yards, Amsterdam. Morgen. 9722 Portugal.. Geira 6970 Berlin... great 6786 Prussia... Morgen 3053 CR~ry"... " small 3054 Rome... Pezza 3158 Canary Isles. Fanegada 2422 Russia... Dessetina 13066.6 England... Acre 4840 Scotland.. Acre 6150 Geneva... Arpent 6179 Spain... Fanegalda 5500 Hamburgh. Morgen 11545 Sweden.. Tunneland 59(10 Hanover.. " 3100 Switzerland Faux 7855 Ireland... Acre 7840 Vienna... Joch 6889 Naples... Moggia 3998 Zurich... Common acre 3875.6 Measures of Capacity. BRITISH. The Imaperial gallon measures 277.274 cubic inches, containing 10 lbs. avoirdupois of distilled water, weighed in air, at the temperature ol 620, the barometer at 30 inches. Faor Grain. 8 bushels - 1 quarter. 1 quarter = 10.2694 cubic feet. Coal, or heaped measure. 3 bushels = saclr. 12 sacks = 1 chaldron. Izmperial bushel = 2218.192 cubic inches. *Heaped bIushel, 19~ ins. diam., cone 6 ins. high = 281-5.4872 cub. ins. 1 chaldron = 58.658 cubic feet, and weighs 3136 pounds. 1 chaldron (Newcastle) = 5936 pounds. FRENCIH. JNews System.-I Litre = 1 cub. decimetre, or 61.074 U. S. cub. ins. Old System. - 1 Boisseau = 13 litses _ 793.964 cub. ins., or 3.43 galls 1 Pinte - 0.931 litres, or 56.817 cubic inches. SPANsIIs. 1 Wine Arroba = 4.2455 gallons. 1 Fanega conummon measure) -- 1.593 bushels.';Vlir hba,,e l ln thf frmn of a trilf a (iine FOREIGN WVEIGHTS AND MEASURES. 19 TABLE showing the relative Capacity of Foreign Liquid Measures compared with BRITISH. Places. Measures. Cub. Inch. Plae. Measures. Cub.lnch. Amsterdam. Anker 2331 Naples... Wine Barille 2544 Stoop 146.. Oil Stajo 1133 Antwerp. " 194 Oporto... Almude 1555 Bordeaux. Barricque 14033 Rome... Wine Barille 2560 Bremen... Stubgens 194.5... Oil 2240 Canaries.. Arrobas 949 ".. Boccali 80 (onstantinople Almud 319 Russia... Weddras 752 Copenhagen. Anker 2355,.. Kunkas 94 Florence. Oil Barille 1946 Scotland. Pint 103.5 Wine 2427 Sicily... Oil Caffiri 662 France... Litre 61.0'7 Spain... Azumbres 22.5 Geneva... Setier 2760... Quartillos 30.5 Genoa... Wine Barille 4530 Sweden... Einer 4794 Pinte 90.5 Trieste... Orne 4007 Hamburgh..' Stubgen 221 Tripoli... Mattari 1376 Hanover " 231 Tunis... Oil" 1157 Hungary.. Eimer 4474 Venice... Secchio 628'Leghorn.. Oil Barille 1942 Vienna... Elmer 3452 Lisbon... Almude 1040 "... Maas 86.33 Malta.. Caffiri 1270 TABLE showing the relative Capacity of Foreign Dry Measures compared with BRITIsH. Places. Measures. CubIrch. Places. Measures. Cub.Inch. Alexandria. Rebele 9587 Malta... Salme 16930 R Kislos 10418 Marseilles.. Charge 9411 Algiers... Tarrie 1219 Milan... Moggi 8444 Amsterdam. Mudde 6596 Naples...Tomoli 3122 " Sack 4947 Opqrto... Alquiere 1051 Antwerp.. Viertel 4705 Persia... Artaba 4013 Azores... Alquiere 731 Poland... Zorzec 3120 Berlin... Scheffel 3180 Rtiga... Loop 3978 Bremen... 4339 Rome... Rubbio 16904 Candia... Charge 9288... Quarti 4226 Constantinople Kislos 2023 Rotterdam. Sach 6361 Copenhagen. Toende 8489 Russia.. Chetwert 12448 Corsica... Stajo 6014 Sardinia. Starelli 2988 Florence.. Stari 1449 Scotland. Firlot 2197 Geneva. Coupes 4739 Sicily... Salme gros 21014 Genoa... Mina 7382 " " generale 16886 Greece... Medimni 2390 Smyrna... Kislos 2141 Hamburgh.. Scheffel 6426 Spain... Catrize 41269 Hanover.. Malter 6868 Sweden... Tunnar 8940 Leghorn... Stajo 1501. rieste. Stari 4521... Sacco 4503 ripoli... Caffiri 19780 Lisbon... Alquiere 817'unis... 21855. Fanega 3268 Venice... Stajo 4945 Madeira.. Alquiere 684 Vienna... Metzen 3753 Malaga... Fanega 3783 Measures of Solidity. FRENCH. 1 Cubic Foot. 2093.470 U.S. inches. Decistre.... 3.5375 cubic feet. Stere (a cubic metre). 35.375 Decastere... = 353.75 1 Stere..... 61074.664 cubic inches. For the Square and Cubic.Measures of other countries, take the length of the measure in table, page 17, and square or cube it as required. 2 - FOREIGN WEIGTS AND MEASURES. Measures of Weight. BRITISH. 1 troy Grain -.003961 cubic inches of distilled water.,1 troy Pound - 22.815689 cubic inches of water. FgENCII. Old System. — Grain... = 0.8188 grains troy. 1 Gros... 58.9548 " 1 Once... - 1.0780 oz. avoirdupois. 1 Livre... -.0780 lbs. " Newe Systein.-Milligramine...01543 troy grains. CentigaranInme...15433 Decigraintle.. 1.54331 Gramnme.. 15.43315 Decagramme.. - 154.33159 Hlecatogranmne. 1. 1543.3159 " 1 Millier = 1000 Kilograrinues = I1 ton sea weight 1 Kilogranme. - 2.204737 lbs. avoirdupois. 1 Pound avoirdupois -- 0.4535685 Kilogrammne. 1 Pound troy. = 0.3732.223 SPANISH.. 1 ". 1.0152 lbs. avoirdupois. SWEDISI.. 1 ". 0.9376 " AUSTRIAN... ". - 1.2351 PRUSSIAN... 1 ". = 1.0333 NOTE.-In the new French systemn, the valiues of the base of each measure, viz,, Mfetre, Litre, Stere, Are, anzd Grnamme, are decreased or increased by the following words preiczed to them. TiZus, Milli expresses the 1000th part. I-Iecato expresses 100 times the value Centi " 100th Chilio i" 1000. Deci " 10th " Myrio " 10000 I)eca 10 times the value. TABnLE showing the relative value of lForeign VWeights compared with BRITISH. Number Number equal to equal to Places. Weigt. 100 avoir- Pla. Weights. 100 avoir. dupois dupois pou ds. pounds. A leppo.. Rottoli - 20.46 Hanover.. Pound 93.20 c... Oke 35.80. Japan... Catty 76.92 Alexandria. ottoli 107. Leghorn.. Pound 133.56 Algiers 84. Leipsic... "(common) 97.14 Amsterdam. Pound 91.8 Lyons... (silk) 98.81 Antwerp.. 96.75 Madeira.. 143.20 Barcelona. 112.6 Mocha... Maund 33.33 Batavia... Catty 76.78 Morea... Pound 90.79 Bengal... Seer 53.57 Naples... Rottoli 50.91 Berlinl.. Pound 96.8 Rome... Pound 133.69 Bologna... 125.3 Rotterdam. 91.80 BJremen. 9. 90.93 Russia. 110.86 Brunswick.. 97.14 Sicily... 142.85 Cairo.... Rottoll 105. Smyrna.. Oke 36.51 Candia... i 85.9 Sumnatra.. Catty 35.56 China... Catty 75.45 Sweden Pound 106.67 Constantinople Oke 35.55.. 120.68 Copenhagen. Pound 90.80 Tangiers.. (miner's) 94.27 Corsica. 131.72 Tripoli... Rottoli 89.28 Cyprus. Rottoli 19.07 Tusnis... " 90.09 Damascus.. 25.28 Venice... Pound (heavy) 94.74 Florence. Pound 133.56 "... " (light) 150. Geneva. " (heavy) 82.35 Vienna... 81. Genoa. 9.. 92.86 Warsaw.. 112.25 IIamburgh.. 93.63 SCRIPTURE AND ANCIENT MEASURES. 21 Scripture Long Measures. Feet. Inches. Feet. Inches. A digit.. -0 0.912 A cubit. =1 9.888 A palm.. = 0 3.648 A fathom.. -7 3.552 A span.. 0 10.944 Grecian Long Measures. Feet. Inches. 11 Feet. Inches. A digit.. 0 0.7554y- i A stadium. = 604 4.5 A pous (foot). 1 0.08753 A mile.. — 4835 A cubit.. =1 1.5984 A Greek or Olympic foot = 12.108 inches. A Pythic or natural foot = 9.768 " Jewish Long Measures. Feet. Feet. A cubit 1.824 A mile... 7296 A Sabbath day's A day's journey. _ 175104 journey.. =3648. (or 33 miles 864 feet). Roman Long Measures. Feet. Inches. Feet. Inches. A digit.. 0.72575 A cubit.. 1 5.406 An uncia (inch) 0.967 A passus 4 10.02 A pes (foot). = 0 11.604 A mile..-4835 Miscellaneous. Feet. Feet. Arabian foot.. =1.095 Hebrew foot.. -1.212 Babylonian foot. 1.140 " cubit.. -1.817 Egyptian... -1.421 " sacred cubit 2.002 NOTE.-The above dimensions are British. TABLE forfinding the Distance of Objects at Sea, in Statute Miles. Height in Dlstance in Height in Distance in Height in Distance in Height in Distance in feet. miles. feet. miles. feet. miles. feet. miles. *.582 1. 11 4.39 30 7.25 200 18.72 1 1.31 12 458 35 7.83 300 22.91 2 1.87 13 4.77 40 8.37 400 26.46 3 2.29 14 4.95 45 8.87 500 29.58 4 2.63 15 5.12 50 9.35 1000 32.41 5 2.96 16 5.29 60 10.25 2000 59.20 6 3.24 17 5.45 70 11.07 3000 72.50 7 3.49 18 5.61 80 11.83 4000 83.7 8 3.73 19 5.77 90 12.55 5000 93.5 9 3.96 20 5.92 100 13.23 1 mile. 96.1 10 4.18 25 6.61 150 16.20 6.99 inehes. The difference in two levels is as the square of the distance Thus, if the height is required for 2 miles, 12:22:: 6.99: 27.96 inches; and if for 100 miles, 12: 1002:: 6.99: 1.103+ miles. For Geographical miles, the distance for one mile is 7.962 inches 22 DISTANCES. EXAMPLE.-If a man at the foretop-gallant mast-head of a ship, 100 feet from the water, sees another and a large ship (hull to), how far are the ships apart? A large ship's bulwarks are, say 20 feet from the water. Then, by table, 100 feet... - 13.23 20 ".. - 5.92 Distance.. 19.15 miles. NOTE.- 1 should be added for horizontal refraction. 1 3 To Reduce Longitude into Time. Multiply the number of degrees, minutes, and seconds by 4, and the product is the time. EXAMPLE. — Required the time corresponding to 500 31'. 500 31' 4 h.3 22' 4" Ans. If time is to be reduced, then h. m. 8. 4) 3 22 4 50 31 Ans. Or, multiplying by 15: thus, h. rn. S. 4 27 13X15-660 48' 15". Degrees of longitude are to each other in length, as the cosines of their latitudes. For every 50 they are as follows: Miles. Miles. Equator. 60. 500.. 38.57 50 59.77 550. 34.41 10 ~.. 59.09 600... 30. 150... 57.96 650,.. 25.36 200.. 56.38 700... 20.52 250... 54.38 750. 15.53 300... 51.96 800... 10.42 350. 49.15 850.. 5.23 400 45.96 900.. 0.00 456. 42.43 VULGAR FRACTIONS. 23 VULGAR FRACTIONS. A FRAcTION, Or broken number, is one or more parts of a UNIT. ExAMPLE.-12 inches are 1 foot. HIere, 1 foot is the unit, and 12 inches its parts; 3 inches, therefore, are the sAe fourth of a foot, for 3 is the quarter or fourth of 12. A Vulgar Fraction is a fraction expressed by two numbers placed one above the other, with a line between them; as, 50 cents is the i of a dollar. The upper number is called the sumerator, because it shows the number of parts used.. ~ -~s_-11':;ria r is called the Denominator, because it denominates, or gives name to the fraction. The Terms of a fraction express both numerator and denominator; as, 6 and 9 are the terms of. A Proper fraction has the numerator equal to, or less than the denominator; as, &, &c. An Improper fraction is the reverse of a proper one; as, 2, &c. A JMixed fraction is a comnpound of a whole number and a fraction; as, 5-, &C. A Compound fraction is the fraction of a fraction; as, x of A, &c. A Complex fraction is one that has a fraction for its numerator or denominator, or _1 3~ 1 both; as,, 0or, or or &c.. Flirartion denotes division, and its value is equal to the quotient obtained by dividing the numnerator by the denominator; thus, 12 is equal to 3. and 21 is eqnal to 41. REDUCTION OF VULGAR FRACTIONS. To find the greatest Number that will divide Two or more Numbers without a Remainder. RULE.-Divide the greater number by the less; then divide the divisor by tile remainder; and so on, dividing always the last divisor by the last remainder, until nothing remains. EXAMPLE.-What is the greatest common measure of 1908 and 936? 936) 1908 (2 1872 36) 936 (26 72 216 - So 36 is the greatest common measure. To find the least Cor mmon Multiple of Two or more Numbers. RuLE.-Divide by any numnber that will divide two or more of the given numbers without a remainder, and set the quotients with the undivided numbers in a line beneath. Divide the second line as before, and so on, until there are no two numbers that san be divided; then the continued product of the divisors and quotients will give the multiple required. ExAmPLE. —What is the least common multiple of 40, 50, and 25 1 5) 40.50.25 5) 8.10. 5 2) 8. 2. 1 4. 1. 1 Then 5X5X2X4 —200.Anas. 241 VULGAR FRACTIONS. To reduce Fractions to their lowest Terms. RULE.-Divide the terms by any number that will divide them without a re mainder, or by their greatest common measure at once. EXAMPLE. —Reduce A20 of a foot to its lowest terms. 72o. 10= -.8 *' 3 =4,or9 inches. To reduce a Mlixed Fraction to its equivalent, an Improper Fraction. NOTE. —Jixed and improper fractions are the same; thuss, 5~ -- 1 For illustration, see following examples: RULE. —Multiply the whole number by the denominator of the fraction, and to the product add the numerator; then set that sum above the denominator. EXAZMPLE-. Reduce 23-9 to a fraction. 23X6+2-= 140 6 6 ExAMPLE.-Reduce 1 23inches to its value in feet. 123+-6 = 203-; that is, 20 feet and f or i of a foot. To reduce a Whole Number to an equivalent Fraction having a given Denominator. RULE.-Multiply the whole number by the given denominator, and set the product over the said denominator. ExAMPLE.-Reduce 8 to a fraction whose denominator shall be 9. 8X9 = 72; then;72 the answer. To reduce a Compound Fraction to an equivalent Simple one. RULE.-Multiply all the numerators together for a numerator, and all the denominators together for a denominator. NOTE.-When there are terms that are common, they may be omitted. ExAMPi,E.-Reduce i of 3 of z to a simple fraction. 1 3 2 6 1 2 4 3 24-4 Or, X Xi= X, by cancelling the 2's and 3's. ExAMrPLE.-Reduce I of X of a pound to a simple fraction. iX = 8 ens. To reldce Fractions of different Denominations to equivalent ones having a common Denominator. RULE. —Multiply each numerator by all the denominators except its own for the new numerators; and multiply all the denominators together for a common denominator NOTE.-In this, as in all other operations, whole numbers, mixed, or compound fractions, mLst Jirst be reduced to the form of simple fractions. ExAMrPLE.-Reduce?, 2, and ~ to a common denominator. 1X3X4n- 121 2X2X4 —16 -", 6 2 By s. 3X2X3 = 18 2X3X4 = 24 The operation may be performed mentally; thus, Reduce, 3 X and 5 to a common denominator. 312 1 -1 66 5 a-'ff, 9 g7 -A. = f. VULGAR FRACTIONS. 25 To reduce Complex Fractions to Simple ones. RULE.-Reduce the two parts both to simple fractions; then multiply the numerator of each by the denominator of the other. ExAMPLE. —Simplify the complex fraction 43' 22 — 8 8X 5=40 5 n 425 -_4 3X24 = 72 9 ADDITION OF VULGAR FRACTIONS. RULE.-If the fractions have a common denominator, add all the numerators together, and then place the sum over the denominators. NOTE.-If the prepared fractions have not a common denominator, they must be reduced, to one. Also, compound and complez must be reduced to simple fractions. EXAMPLE.-Add 4 and 4 together. 1+- 3_= 4 Has. EXAMXPLE.-Add I of i of _6 to 2 of J. iX 41X 1-o = 0 2 of = -1 4= —5. Then, +S51=. Ans. SUBTRACTION OF VULGAR FRACTIONS. RULE.-Prepare the fractions the same as for other operations, when necessary; then subtract the one numerator from the other, and set the remainder over the common denominator. EXAMhPLE.-What is the difference between 4 and 4? 5-If = 4 dns. ExrMPLE.-Subtract 6 from 03. 6X9 = 54 3X8=24 =- 54 _ 2 43 A s. 8X9 - 72 MULTIPLICATION OF VULGAR FRACTIONS. RULE.-Prepare the fractions as previously required; multiply all the numerators together for a new numerator, and all the denominators together for a new denominator. EXA&PLC.-What is the product of 3 and? 4X- =,9~ = 4 JSn. XsaitLE. —What is the product of 6 and ~ of 5? X 2 of 5= - XtO 6 = 20 dns. C 26 APPLICATION OF RIEDUCTION OF VULGAR FRACTIONS, DIVISION OF VULGAR FRACTIONS. RULE.-Prepare the fractions as before; then divide the nunerator by the nlumerator, and the denominator by the denominator, if they wvill exactly divide; but if not, invert the terms of the divisor, and multiply the dividend by it, as in multiplication. ExAMrPL.-Divide, by 5-. 5 ~ 5 5 - s. To find the Value of a Fraction in Parts of a whole Numsber. RULE. —Multiply the whole number by the nulnerator, and divide by the denorn. hiallor; then, if atnything rematillns, mnultiply it by the parts in the next inferior deonomination, and divide by the dlenotiinator, as before, and so on as far as necessa ry, so shall the quotients placed in order be the value of the fraction requiredL ExAMPLE.-What is the value of ~ of of 9? I of 2 -18.. 3 9s. EAtMPLz.-Reduce { of a pound to avoirdupois ounces. 3 4) 3 (t lbs. 16 ounces in a lb. 4) 48 12 ounces,.dns. ExAMPLE.-R.educe o-% of a day to hours. x 24- = 73 2 hours, 2 ns. To reduce a Fraction from one Denomination to another. RULE.-Multiply the number of parts in the next less denominator by the numerator if the reduction is to be to a less name, but multiply by the denominator if to a greater. ExAMPLE-.Reduce ~ of a dollar to the fraction of a cent. 3-X 1 - ~ ~5, the answer. ExAMPLE.-Rteduce -} of an avoirdupois pound to the fraction of an ous'e X' = 61 = 8, the answer. ExAMpLLE.-Reduce. 2 of a cwt. to the fraction of a lb. 2XX2 224 _ 32 - XI f 1 7 iS -, the answer. ExAMPLE.-Reduce } of - of a mile to the fraction of a foot. _a of; 6 X 5 2 3s 0 3 1 80 2 6I4 0, the answer. 3 4 12 1 i- 2 EXAMPLz.-Reduce ~ of a square foot to the fraction of an inch. 144 4 ns. For Rude of Three in yulgar Fractions, see page 29. DECIMAL FRACTIONS. A D)CIMAL FRACTION is that which has for its denominator a UNIT (1), with as many ciphers annexed as the nlumerator has places; it is usually expressed by set ting down the numerator olly, with a point on the left of it. Thus, D is.4, J-o i is.85, - ~ ~3'765b Is.0075, and 1 0- 5 is.00125. When there is a deficiency of figdres in the nlumeerator, prefix ciphers to make up as many places as there are ciphers in the denomninator. DECIMI\AL FRACTIONS. 27 Mixed numbers consist of a whole number and a fraction; as, 3.25, which is the same as 3.25o, or 32 5, Ciphers on the right hand make no alteration in their value; for.4,.40,.400 are decimals of the same value, each being -A, or -. ADDITION OF DECIMALS. RU:LE.-Set the numbers under each other according to the value of their places as in whole numbers, in which state the decimal points will stand directly under each other. Then, beginning at the right hand, add up all the columns of numbers as in integers, and place the point directly below all the other points. ExMPrLE.-Add together 25.125, 56.19, 1.875, and 293.7325. 25.125 56.19 1.875 293.7325 376.9225 the sum. SUBTRACTION OF DECIMAL FRACTIONS. RULE.-Place the numbers under each other as in addition; then subtract as In whole numbers, and point off the decimals as in the last rule. EXAMPrLEs —Subtract 15.150 from 89.1759. 89.1759 15.150 74.0259 Renm. MULTIPLICATION OF DECIMALS. RuLE.-Place the factors, and multiply them together the same as if they were whole numbers; then point off in the product just as many places of decimals as there are decimals in both the factors. But if there be not so many figures in the product, supply the deficiency by prefixing ciphers. EXAMPLE. —Multiply 1.56 by.75. 1.56.75 780 1092 1.1700 Prod. BY CONTRACTION. To contract the Operation so as to retain only as many Decimal places in the Product as may be thought necessary. RULE.-Set the unit's place of the multiplier under the figure of the multiplicand whose place is the same as is to be retained for the last in the product, and dispose of the rest of the figures in the contrary order to what they are usually placed in. Then, in multiplying, reject all the figures that are more to the right hand than each multiplying figure, and set down the products, so that their righthiand figures may fall in a column straight below each other; and observe to increase the first figure in every line with what would arise from the figures omitted; thus, add 1 for every result from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, 4 from 35 to 44, &c., &c., and the sum of all the lines will be the product as required. ExAMPLrE.-MUltiply 13.57493 by 46.20517, and retain only four places of deci. um.als in the product. 13.574 93 71 502.64 54 299 72 8 144 96+2 for 18 271 50+2 " 18 679+4 " 35 14+1 5 9+2" 21 627.23 20 28 DECIIMAL FRACTIONS. ExAMPLE.-Multiply 27.14986 by 92.41035, and retain only five places of decimnals. A.ns. 2508.9206. DIVISION OF DECIMALS. RULE. —Divide as in whole numbers, and point off in the quotient as many places for decimals as the decimal places in the dividend exceed those in the divisor; but if there are not so many places, supply the deficiency by prefixing ciphers. ExAMPLE.-Divide 53.00 by 6.75. 6.75):,53.00 (= — 7.851+. Here 3 ciphers were annexed to carry out the division. BY CONTRACTION. RULE.-Take only as many figures of the divisor as will be equal to the number of figures, both integers and decimals, to be in the quotient, and find how many times they may be contained in the first figures of the dividend, as usual. Let each remainder be a new dividend; and for every such dividend leave out one figure more on the right-hand side of the divisor, carrying for the figures cut off as in Contraction of Multiplication. NOTE.- When there are not so many figures in the divisor as are required to be in the quotient, continue the first operation till the number of figures in the divisor be equal to those remaining to be found in the quotient, after which begin the contraction. ExAMPLE.-Divide 2508.92806 by 92.41035, so as to have only four places of decimals in the quotient. 92.410315) 2508.928106 (27.1498 1848 207+ 1 660 721 646 872+ 2 13849 9 241 4 608 3 696 912 832+4 80 74+2 6 ExAMPLE..-Divide 4109.2351 by 230.409, retaining only four decimals in the quoioent. A.ns. 17.8345. REDUCTION OF DECIMALS. To reduce a Vulgar Fraction to its equivalent Decimal. RvLE.-Divide the numerator by the denominator, annexing ciphers to the numerator as far as necessary. ExAMPLE.-Reduce 4 to a decimal. 5) 4.0.8 dns. To find the Value of a Decimal in Terms of an Inferior Denomination. RULE.-Multiply the decimal by the number of parts in the next lower denomination, and cut off as many places for a remainder, to the right hand, as there are places in the given decimal. Multiply that remainder by the parts in the next lower denomination, again cut. ting off for a remainder, and so on through all the parts of the integer. EXAMPLE.-What is the value of.875 dollars 3.875 100 Cents, 87,500 10 Mills, 5.000 Ans. 87 cents 5 mills. DECIMAL FRACTIONS. 29 EXAMPLE.-What is the content of.140 cubic feet in inches 3.140 1728 cubic inches in a cubic foot. 241.920 e.1ns. 241.o 2 0 cubic inches. ExAMPLE.-What is the value of.00129 of a foot.,A.ns..01548 inches. EXAMPLE.-What is the value of 1.075 tons in pounds..Ans. 2408. To reduce Decimals to equivalent Decimals of higher Denominations. RULE.-Divide by the number of parts in the next higher denomination, contin. aing the operation as far as required. ExAMPLE.-Reduce 1 inch to the decimal of a foot. 1211.00000.08333, &c., Rns. EXAMPLE,-Reduce 14 minutes to the decimal of a day. 60114.00000 241.23333.00972, &c., JAns. EXAMPLE.-.Reduce 14" 12"' to the decimal of a minute. 14" 12"' 60 60'852."' 60 14.2".23666', &c.,.,aRs. NOTE. —When there are several numbers, to be reduced all to the decimal of the highest. Reduce them all to the lowest denomination, and proceed as for one denornination. ExAMPLE,-Reduce 5 feet 10 inches and 3 barleycorns to the decimal of a yard. Feet. Inches. Be. 5 10 3' 12 70 3 3 213. 12 71. 3 5.9166 1.9722, &c., yards, dins. RULE OF THREE IN DECIMALS. RuLE.-Prepare the terms by reducing the vulgar fractions to decimals, cormpound numbers to decimals of the highest denomination, the first and third terms to the samie name; then proceed as in whole numbers. See Rule, page 31. EXAMPLE. —If i a ton of iron cost i of a dollar, what will.625 of a ton cost I i =.5.f 1.5:.75::.625 —.75.6%.5).46875'.9375 dollars, A.ns. 02 30 DUODECIMI[ALS. DUODECIMALS. In Duodecimals, or Cross Multiplication, the dimensions are taken in feet, inches, and twelfths of an inch. RuLE. —Set down the dimensions to be multiplied together, one under the other, so that feet may stand under feet, inches under inches, &c. Multiply each term of the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each immediately under its corresponding term, carrying I for every 12, from one term to the other. In like manner, multiply all the multiplicand by the inches of the multiplier, and then by the twelfth parts, setting the result of each term one place farther to the right hand for every multiplier. The sum of the products is the answer. EXAM'PLE.-Multiply 1 foot 3 inches by 1 foot one inch. Feet. Inches. 1 3 1 1 1 3 1 3 1 4 3 PROOF.-1 foot 3 inches is 15 inches, and 1 foot 1 inch is 13 inches; and 15X13 = 195 square inches. Now the above product reads 1 foot 4 inches and 3 twelfths of an inch, and 1 foot -- 144 square inches. 4 inches - 48 " 3 twelfths - 3 " 195 EXAMPLE.-HOW many square inches are there in a board 35 feet 4j inches long and 12 feet 31 inches wide 1 Feet. Inches. Twelfths. 35 4 6 12 3 4 424 6 8 10 1 6 11 9 6 0 434 3 11 0 0 EXAMPLE.-MUltiply 20 feet 61 inches by 40 feet 6 inches. By duodecimals, dfns. 831 feet 11 inches 3 twelfths equal 831 square feet and 135 square inches. By decimals.. 40 feet 6 inches - 40.5 20 " 6 " - 20.541666, &c. Feet. 831.937499 144 Square inches. 134.999856 fable showing the value of Duodecimals in Square Feet, and Decimals of an Inch. Sq. feet. Sq. inches. 1 Foot...... = 1 or 144. 1 Inch.. "12. 1 Twelfth...= 1. 1 of 1 twelfth.- I.083333, &c. 12 of 1tfh I of - of do.= ".006944, &c. Application qf this Table. What number of square inches are there in a floor 100-2 feet broad and 25 feet (i Inches and 6 twelfths long? fns. 2566 feet 11 inches 3 twelfths equal 2566 feet 135.inches. RULE OF THREE. 31 RULE OF THREE. The RULE OF THREE teaches how to find a fourth proportional to three given numbers. It is either DIRECT or INVERSE. It is Direct when more requires more, or less requires less. Thus, if 3 barrels of flour cost $18, what will 10 barrels cost? Or, if 300 lbs. of lead cost $25.50, what will 10 lbs. cost In both of these cases the Proportion is Direct, and the stating must be, As 3:18: 10: fns. 60,. 300: 25.50:: 10: d ins..85. It is Inverse when more requires less, or less requires more. Thus, if 6 men build a certain quantity of wall in 10 days, in how many days will 8 men build the like quantity? Or, if 3 men dig 100 feet of trench in 7 days, in how many days will 2 men perform the same work? Here the Proportion is Inverse, and the stating must be, As 8:10::6: dins. 7-i. 2: 7::3:'ins. 10~. The fourth term is always found by multiplying the 2d and 3d terms together, and dividing the product by the 1st term. Of the three given numbers necessary for the stating, two of them contain the supposition, and the third a demand. RULE.-STATE the question by setting down in a straight line the three necessary numbers in the following manner: Let the 2d term be that number of supposition which is of the same denomination as that the answer, or 4th term, is to be, making the demanding number the 3d term, and the other number the 1st term when the question is in Direct Proportion, but contrariwise if in Inverse Proportion, that is, let the demanding number be the 1st term. Then multiply the 2d and 3d terms togethelr, and divide by the 1st, and the prodact will be the answer, or 4th term sought, of the same denomination as the 2d term. NOTE.-If the first and third terms are of different denominations, r-educe them to the same. If, after division, there be any remainder, reduce it to the next lower denomination, and divide by the same divisor as before, and the quotient will be of this last denomination. Sometimes two or more statings are necessary, which may always be known by the nature of the question. EXAMPLE 1.-If 20 tons of iron cost $225, what will 500 tons cost? Tons. DnIls. Tons. 20: 225:: 500 500 210) 1125010 5625 dollars, Ains. EXAMPLE 2.-If 15 men raise 100 tons of iron ore in 12 days, how many men will raise a like quantity in 5 days? Days. Men. Days. As 5:15::12 12 5) 180 36 men, Ans. EXAMPLE 3.-A wall that is to be built to the height of 36 feet was raised 9 feel high by 16 men in 6 days: how many men could finish it in 4 days at the same rate of working I Days. Men. Days. Men. 4: 16: 6: 24 uAns. Then, if 9 feet require 24 men, what will 27 feet require? 9: 24:: 27: 72 Ans. EXAMPLE 4.-If the third of six be three, what will the fourth of twenty be! inas. 7& INVOLUTION-EVOLUTION COMPOUND PROPORTION. COMPOUND PROPORTION is the rule by means of which such. questions as would require two or more statings in simple proportion (Rule of Three) can be resolved in one. As the rulle, however, is but little used, and not easily acquired, it is deemed preferable to omit it here, and to show the operation by two or more statings. EXAMPLE.-How many men can dig a trench 135 feet long in 8 days, whets 16 tmen can dig 54 feet in 6 days? Feet. Men. Feet. Men. First... As 54: 16:: 135: 40 Days. Men. Days. Men. Second. ~ As 8: 40:: 6: 30 ns. EXAMPLE.-If a man travel 130 miles in 3 days of twelve hours each, in how many days of 10 hours each would he require to travel 360 miles? Miles. Days. Miles. Days. First.. As 130: 3:: 360: 8.307 Hours. Days. Hours. Days. Second.. As 10: 8.307:: 12: 9.9684 2nts. EXAMPLE.-If 12 men in 15 days of 12 hours build a wall 30 feet long, 6 wide, and 3 deep, in how many days of 8 hours will 60 men build a wall 300 feet long, 8 wide, and 6 deep.?.?ns. 120 days, INVOLUTION. INVOLUTION is the multiplying any number into itself a certain number of times. The products obtained are called POWERS. The number is called the Roor, or first power. When a number is multiplied by itself once, the product is the square of thatt number; twice, the cube; three times, the biquadrate, &c. Thus, of the number 5. 5 is the Root, or 1st power. 5X5= 25 " Square, or 2d power, and is expressed 52. 5X5X —125 " Cube, or 3d power, and is expressed 53. 5X 5X5X= 625 " Biquadrate, or 4th power, and is expressed 54. The little figure denoting the power is called the INDEX or EXPONENT. EXAMPLE.-What is the cube of 9 1. Sns. 729. ~EiXAMPLE.-What is the cube of? s ins. 2T. EXAMPLE.-What is the 4th power of 1.5? Ans. 5.0625, EVOLUTION. EVOLUTION is finding the RooT of any number. The sign / placed mefore any number, indicates the square root of that ntllbl;r is required or shown. The same character expresses any other root by placing the index above it. Thus, /25 = 5, and 4+2 = -,/36. And, /27= 3, and /64= 4. Roots which only approximate are called Surd Roots. TO EXTRACT THE SQUARE ROOT. RULE. —Point off the given number from units' place into periods of two figures each. Find the greatest square in the left-hand period, and place its root in the quotient; subtract the square number from the ieft-hand period, and to the remainder bring down the next period for a dividend. Double the root already found for a divisor; find how many times the divisor is contained in the dividend, exclusive of the right-hand figure, place the result in the qluotient, and at the right hand of the divisor. EVOLUTION. 33 Multiply the divisor by the last quotient figure, and subtract the product from the lividend; bring down the next period, and proceed as before. NOTE. —Mied decimals must be pointed off both ways from units. ExAMPLE.-What is the square root of 2? 11 2.000000 (1.414, &c. I 1 241100 41 96 2811 400 1 281 2824 11900 4 11296 28_8 604 EXAMPLE.-What is the square root of 144. 1I 144 (12 Ans. 221044 44 00 EXAMPLE.-What is the square root of 121. ns. 3.464101. SQUARE ROOTS OF VULGAR FRACTIONS. RULE.-Reduce the fractions to their lowest terms, and that fraction to a decimal, and proceed as in whole numbers and decimals. NOTE.- When the terms of the fractions are squares, take the root of each and set one above the other; as, % is the square root of 3X. EXAMPLE.-What is the square root of -9-2 sIns. 0.86602540. 12 To find the 4th root of a number, extract the square root twice, and for the 8th root thrice, &c., &c. TO EXTRACT THE CUBE ROOT. RULE.-From the table of Roots (page 99) take the nearest cube to the given number, and call it the assumed cube. Then say, as the given number added to twice the assumed cube is to the assumed cube added to twice the given number, so is the root of the assumed cube to the required root, nearly. And, by using in like manner the root thus found as an assumed cube, and proceeding as above, another root will be found still nearer; and in the same manner as far as may be deemed necessary. EXAMPLE.-What is the cube root of 10517.9 Nearest cube, page 99 i 10648, root 22. 10648. 10517.9 2 2 21296 21035.8 10517.9 10648. 31813.9: 31683.8:: 22: 21.9+ Alns. To extract any Root whatever. Let P represent the number, n " the index of the power, A " the assumed power, r its root, R " the required root of P. Then say, as the sum of n+lXA and n-1XP is to the sum of -n +lXP and at-1XA, so is the assumed root r to the requiled root R. EXAMPLE.-What is the cube root of 1500 The nearest cube, page 99, is 1331, root 11. ARITHMiIETICAL PROGRIESSION. P - 1500, n - 3, A = 1331, r _- 11; then, n+ X A = 5324, n+1XP =6000 n —1 X P = 3000, n-1 X A = 2662 8324: 8662:: 11: 11.446 —f Jn&s ARITHMETICAL PROGRESSION. ARITHMETICAL PROGRESSION is a series of numbers increasing or decreasing by a constant number or difference; as, 1, 3, 5, 7, 9, 15, 12, 9, 6, 3. The numbers which form the series are called Terms; tile first and last are called the Ez,tremes, and the others the JMeans. When any three of the following parts are given, the remaining two can be foiund, viz.: The First term, the Last term, the.N'JLmbcr of: terms, the COMMON DIFFERENCE, and the SUM of all the terms. When the First Term, the Common Difference, and the Number of Terms are given, to find the Last Term. RULE.-Multiply the number of terms less one, by-the common difference, and to the product add the first term. EXAMPLE.-A man travelled for 12 days, going 3 miles the first day, 8 the second, antd so on; how far did he travel the last day? 1.2 —1 5 +3 = 58.dns. When the Number of Terms and the Extrernes are given: tojfind the Common Difference. RULE.-Divide the difference of the extremes, by one less than the number of terms. EXAMPLE.-The extremes are 3 and 15, and the number of terms 7; what is the common difference? 15-3 + (7-1) = 2.ins. When the Extremes and Number of Terms are given, to find the Sum of all the Terms. RULE.-Multiply the number of terms by half the sum of the extremes. EXAMPLE. —fIow many times does the hammer of a clock strike in 12 hours 3 12X (13- 2) = 78 Ens. When the Common Difference and the Extremes are given, to find the Number of Terms. RULE.-Divide the difference of the extremes by the common difference, and add soe to the quotient. EXAMPLE. —A man travelled 3 miles the first day, 5 the second, 7 the third, and so on, till he went 57 miles in one day. How many days had he travelled at the close of the last day? 57 —3 -2+1 = 28 vAns. To find two Arithmetical Means between two g iven Extremes. RULE.-Subtract the less extreme from the greater, and divide the difference by three, the quotient will be the common difference, which, being added to the less extreme, or taken from the greater, will give the means. EXAMPLE.-Find two arithmetical means between 4 and 16. f — 4-3 = 4 com. dif. 4+4 = 8 one mean. 16-4 = 12 second mean. To find any Number of Arithmetical Means between two Extremes. RULE -Subtract the less extreme from the greater, and divide the difference by see more than the number of means required to be found, and then proceed as in the foregoing rule. GEOME:TRICAL PROGRESSION. S5 GEOMETRICAL PROGRESSION. GEOMETRICAL PROGRESSION is any series of numbers continually increasing by a constant multiplier, or decreasing by a constant divisor. As, 1, 2, 4, 8, 16, and 15, 72, 34. The constant multiplier or divisor is the RATIO. When any three of the following parts are given, the remaining two can bt tbund, viz.: The FIRST term, the LAST term, thie NuMtBEr of terms, tilhe RATIO, and the SUM OF ALL THE TERMS. ViVhen the Ratio, Number of Terms, and the First Term are given, to find the Last Term. RULE.-Write a few of the leading terms of the series, and place their indices. over them, beginning with a cipher. Add togetherthe Inost convenient indices, to tmake an index less by one than the number of the term sought. Multiply together the terms of the series or powers belonging to those indices, tind the product, multiplied by the first term, will be the answer. NoTE.- W-Ien the first term is equal to tle ratio, the indices must begin with a untit. EXArmPLE. —The first term is I, the ratio 2, and the number of terms 23; what is the last term? Indices. 0 1 2 3 4.5 6 7 Terms. 1, 2, 4, 8, 16, 32, 64, 128. 1+-'3+4+5+7 = 22. 1l28X32X16X8X4X2 a- 4194304X 1 4194:304 Jna. EXAMPLE.-If one cent had been put out at intcrest in 1630, whaent would it have amounted to in 1834 if it had doubled itself every 12 years? 1834 —1630 = 204 12 = 17.17+1 - 18. 0 1 2 3 4 7 1, 2, 4, 8, 16, 128, 1 —2+3 —4+7 = 17. 2X4XSX16X128- 131072.X 1 -$1.310.72 Ans. W;hen the First Term, the Ratio, and the Number of Terms are given, tofind the Sum of the Series. RULE.-Raise the ratio to a power whose index is equal to tlhe number of termss, from which subtract 1; then divide the remainder by the ratio less 1, and multiply the quotient by the first term. EXAMPLE.-If a man were to buy 12 horses, givinsg 2 cents for the first horse, 6 cents for the second, and so on, what would they cost him? 312 - 531441-1 = 531440 -(3-1) - 2 - 265720X2 = $5.314.40 Ans. By another Method, the greater Extreme bein, knownw (Greater extremeXratio)-less extreme - Sum of the Serieg. Ratio -1 354294X 3-2 106020 Thus - - = $5.314.40, 2ns., as above. A TABLE OF GEOMETRICAL PROGRESSION, Wher'eby anty q stions of Geomtetrical Progression proceeding frna 1, and of d ule ratio, may be solved by itnspection, if the 2oumtre of teins e.-< ceed not 50. 1 I 8 1281 15 16384 1 21097152 2 2 9 25 1 1 6 32768 23 41 194304 3 4 10 512 1.7 655365 24 8388Sh q 4 8 111 1024s 18 131072 25 16777216 5 16 12.2048 19 262144 26 233554432 6 32 13 4096 20 V3)088 27 67108864 7 64 34 8192 21 134857 28 134217728 36 PERBIUTATION- CO MBINATION —POSITION, TABLE — (Continued.) 29 268435456 37 68719476736 144 8796093022208 30 536870912 1 38 137438953472 45 17592186044416 31 1073741824139 274877906944 46 351] 84372088832 312 2147483648 40 54975581.3888 47 70368744177664 4294967296 41 1099511627776 48 140737488355328 34 8589934592 42 2199023255552 49 281474976710656 17179869184 43 1 4398046511104 50 562949953421312 36 34359738368 100 1 633825300114114700748351602688 PERMUTATION. PERMUTATION is a rule for finding how many different ways, any given number of things may be varied in their position. RULE.-Multiply all the terms continually together, and the last product will be the answer. EXAMPLE.-How many variations will the nine digits admit of 1 1X2X3X4X5X6X7X8X9 362880 Ans. COMBINATION. COMBINATION is a rule for finding how often a less number of thlings, can be chosen from a greater. RULE.-Multiply together the natural series, 1, 2, 3, &c., up to the number to be t3.ken at a time. Take a series of as many terms, decreasing by 1, from the number out of which the choice is to be made, and find their continued product. Di vide this last product by the former, and the quotient is the answer. EXAMPLE.. —How many combinations may le made of 7 letters out of 12? IX 2X 3X4X5X6X7=-5040. 12XllXlO10X9X8X7X6 =-3991680. —5040 = 792.fAns. ExAsrPLs. —How many combinations can be made of 5 letters out of 10 10X9X8X7X6 1X2X3X4X5 POSITION. POSITION is of two kinds, SINGLE and DOUBLE, and is determined by the number of SUPPOSITIONS. SINGLE POSITION. RULE.-Take any number, and proceed with it as though it were the correct one; then say, as the result is to the given sum, so is the supposed number to the number required. EXAMPLE.-A commander of a vessel, after sending away in boats 1, i, and j of his crew, had left 300; what number had he in command 1 Suppose he had 600. k of 600 is 200 of 600 is 100 of of 600 is 150 450 150: 300:: 600: 1200 dAns. IEXAMIPLE. —A person being asked his age, replied, if: of my age be multiplied by 2, and that product added to half the years I have lived, the sam will be 75. How odi was he? Anas. 37J years. DOUBLE POSITION. RU&zE.-Take any two numbers, and proceed with each according to the conda FELLOWSHIP-DOUBLE FELLOWSHIP. 37 tions of the question; multiply the results or erro~s by the contrary supposition thlat is, the first position by the last error, and the last position by the first error. If the errors be too great, mark them +; and if too little, -. Then, if the errors are alike, divide the differenLce of the products by the differeMlce of the errors; but if they are unlike, divide the sum of the products by the suu of the errorls. EXAMPLE.-F asked G how much his boat cost; he replied that if it cost him {6 times as much as it did, and $30 more, it would stand him in $300. What eras the price of the boat? Suppose it cost.. 60 or 30 6 times. 6 times. 360 180 and 30 more, and 30 more. 390 210 300 300 90+ 9030 2d position. 60 1st position. 90 2700 5400 90 5400 180) 8100 (45 Ens, 720 900 900 ExAMIPLE. —What is the length of a fish when the head is 9 inches long, the tail as long as its head and half its body, and the body as long as both the head and tail 3 Bns. 6 feet. FELLOWSHIP. FLLOWSSHIP is a method of ascertaining gains or losses of individuals engaged in joint operations. RULE.-As the whole stock is to the whole gain or loss, so is each share to the gain or loss on that share. ExAMPLE.- -Two men drew a prize in a lottery, of $9.500. A paid $3, and B paid $ for the ticket; how much is each one's share? 5 9.500:: 3: 5.700, A's share. 5: 9.500:: 2: 3.800, B's share. DOUBLE FELLOWSHIP, Or Fellowship with Time. RULE.-Multiply each share by the time of its interest in the Fellowship; then, is the sum of the products is to the product of each interest, so is the whole gain or loss to each share of the gain or loss. EXAIPI, E.-A shlip's company take a prize of $10,000, whlich they divide accordlng to their rate of pay and tine of service on board. The officers have been on board 6 months, and the men 3 months; the pay of the lieutenants is $100; midshipmen $50, and men $10 per month; andt there are 2 lieutenants, 4 midshipmen. and 50 muen. What is each one's share? 2 lieutenants.. 100 = 200X6 =1200 4 midshipmen. 50 = 200X 6= 1200 50 men. 10 500X3= 1500 3900 Lieutenants. 3900: 1200:: 10.000: 3.076.92. 2-$1.538.46 Midshipmen. 3900: 1200: 10.000: 3.076.92- 4= $769.23 Men.. 3900 1500::10.000 3.8,16.16 - 50- $76.900 I) 38 nALLIG ATION —-CO0IPOUND INTEREST. ALLIGATION. ALLIGATION is a method of finding the mean rate or quality of different materialse when mixed together. [Yhen it is required to find the imean price of the mixture, observe the following RULE.-Multiply each quantity by its rate, then divide the sum of these products by the sum of the quantities, and the quotient will be the rate of the composition. EXAMPLE.-if 10 lbs. of copper at 20 cents per lb., i lb. of tin at 5 cents, and I lb. (if lead at 4 cents, be mlixed together, what is the:value of the composition'! 10X20 n 200 1X 5= 5 lx 4- 4 12 ) 209 (17.- - ns. When the Prices and Ml~ean Price are given, to.find what Qeuantity of each Article must be taken. RULE 1.-Connect with a line each price that is less than the mean rate with one or more that is greater. Write the difference between the mixture rate and that of each of the simnples opposite the price with which it is connected; then the sum of the differences against any price will express the quantity to be taken of that price. EXAMPLE. —IHO much gunpowder, at 72, 54, and 48 cents per pound, will coumpose a mixture worth 60 cents a pound? (48 A 12, at 48 cents) 60< 54 )) 12, at 54 cents> iAns. (72J 112+6 = 18, at 72 cents) PROOF. —1X 48+12X 54+18X 72 2520.-12+12+12-6 = 60. Should it be required to mix a defisnite quantity of any one article, the quantities of each, determined by the above rule, must be increased or decreased in the proportionr they bear to the defined quantity. Thus, had it been required to mix 18 pounds at 48 cents, the result would be 18 at 48, 18 at 54, and 27 at 72 cents per pound..-igail, e/when the zwhole composition is limited, say, As the sum of the relative quantities, as found by the above rule, is to the whole quantity required, so is each quantity so found to the required quantity of each. EXAMPLE.-Were 100 pounds of the above mixture wanting, the result wotild be outained thus: As 42: 100:: 12: 281 42: 100:: 12: 284-. 42: 100:: 18: 42q. COMPOUND INTEREST. If any principal be multiplied by the alnount (in the following table) opposite the years, and under the rate per cent., the sum will be the amount of that principal at compound interest for the time and rate taken. EXAMPLE.-What is the amount of $500 for 10 years, at 6 per cent.?'Tabular numliber. 1.79084X 500 = $895.4.2,,ns. DISCOUINT —EQUATION OF PAYMSENTS. 39 TABLE: showving the amrounWt of ~1 or $1,.c., for a ~ny nuwmber of years inwt exceeding 24, at the rates of 5 and 6 per cent. compound interest. Years. 5 per cent. 6 per cent. Years. 5 per cent. }6 per cent. 1 1.05 1.06 13 1.88564 2.13292.2 1.1025 1.1236 14 1.97993 2.26090 3 1.15762 1.19101 15 2.07892 2.39655 4 1.21550 1.26247 16 - 2.18287 2.54035 5 1.27628 1.33822 17 2.29201 2.69277 6 1.34009 1.41851 18 2.40661 2.85433 7 1.40710 1.50363 19 2.52695 3.02559 8 1.47745 1.59384 20 2.65329 3.20713 9 1.55132 1.68947 21 2.78596 3.39956 10 1.62889 1.79084 22 2.92526 3.60353 11 1.71033 1.89829 23 3.07152 3.81974 1-2 1.79585 2.01219 24 3.22509 4.04893 DISCOUNT. The Time, Rate per Cent., and Interest being given, to find the Principal. RULE. —Divide the given interest, by the interest of $1, for the given rate and timne. EXAMxPLE.-What sum of money at 6 per cent. will in 14 months gain $14? $14 -.07 = $200 Ans. The Principal, Interest, and Time being given, to find the Rate per Cent. RULE. —Divide the given interest, by the interest of the given sum, for the time, at 1 per cent. EXAMPLE.-A broker received $32.66 interest for the use of $400, 14 months; what was that per cent.? The interest on $32.66 for 14 months, is 4.66. 32.66 — 4.66-=7 per cent. Ans. The Principal, Rate per Cent., and Interest being given, to find the Time. RULE.-Divide the given interest, by the interest of the sum at the rate per seunt. for one year. EXAMPLE.-In what time will $108 gain 11.34, at 7 per cent.? The interest on $108 for one year is 7.56. 11.34.-7.56 = 1.5 years, Ans. EQUATION OF PAYMENTS. Multiply each sum by its time of payment in days, and divide the sum of the products by the sum of the payments. EXAhIPLE.-A owes B $300 in 15 days, $60 in 12 days. and $350 in 20 days; whet is the whole due? 300X 15 = 4500 60X12= 720 350X20 = 7000 710 ) 12220 (17+ days,.dns. 40 ANNUITIES. ANNUITIES. The Annuity, Time, and Rate of Interest given, to find the Amount. RULE.-Raise the ratio to a power denoted by the time, from which subtract 1 divide the remainder by the ratio less 1, and the quotient, multiplied by the annuil ty, will give the amount. NOTE.-$1 or ~1 added to the given rate per cent. is the ratio, and the preceding table in Compound Interest is a table of ratios. EXAMPLE.-What is the amount of an annual pension of I$100, interest 5 per cent., which has remained unpaid for four years? 1.05 ratio; then 1.054 = 1.21550625-1 =.21550625'-(1.05-1).05 5= 4.310125X100= 431.0125 dollars. The Annuity, Time, and Rate given, to find the Present Worth. RULE.-Find the value of it for the whole tinme; and this amount divided by ths ratio, involved to the time, will give the answer. EXAMPLE.-What is the present worth of a pension or salary of $500, to continue 10 years at 6 per cent. compound interest? $500, by the last rule, is worth $6590.3975, which, divided by 1.0610 (by table, page 39, is 1.79084) = $3680.05 Ins. Or, by the following table, multiply the tabular number by the given annuity, and the product will be the present worth: TABLE showaing the prescott wort/c of $1 or 1i annuity, at 5 and 6 per cent. compound interest, for any number of years under 34. Years. 15 per cent. 6 per cent. Years. 5 per cent. 6 per cent. 1 0.95238 0.94339 18 11.68958 10.8276 2 1.85941 1.83339 19 12.08532 11.15811 3 2.72325 2.67301 20 12.46221 11.46992 4 3.54595 3.4651 21 12.82115 11.76407 5 4.32948 4.21236 22 13.163 12.041.58 6 5.07569 4.91732 23 13.48807 12.30338 7 5.78637 5.58238 24 13.79864 12.55035 8 6.46321 6.20979 25 14.09394 12.78335 9 7.10782 6.80169 26 14.37518 13.00316 10 7.72173 7.36008 27 14.64303 13.21053 11 8.30641 7.88687 28 14.89813 13.40616 12 8.86325 8.38384 29 15.14107 13.59072 13 9.39357 8.85268 30 15.37245 13.76483 14 9.89864 9.29498 31 15.59281 13.92908 15 10.37966 9.71225 32 15.80268 14.08398 16 10.83777 10.10589 33 16.00255 14.22917 17 10.27407 10.47726 34 16.1929 14.36613 EXAMPLE. —Same as above; 10 years at 6 per cent. gives 7.36008X500 = $3680.04 lns. When annuities do not commence till a certain period of time, they are said to be in REVERSION. To find the Present Worth of an Annuity in Reversion. RULE.-Take two numbers under the rate in the above table, viz., that opposite the sum of the two given times and that of the time of reversion, and multiply their difference by the annuity, and the product is the present worth. EXAMrPLE.-What is the present worth of a reversion of a lease of $40 per annum, to continue for six years, but not to commence until the end of 2 years, allowing 6 per cent. to the purchaser 1 By table, 8 years. 6.20979 " 2... 1.83339 4.37640X40= $175.05 7ons. PERPETUITIES-CHRONOLOGICAL PROBLEMIS. 41 For half yearly and quarterly payments, the amount for the given tine, multiplied by the number in the following table, will be the true amount: Rate per ct. Half yearly. Quarterly. Rate per ct. Half yearly. Quarterly. 3 1.007445 1.011181 5A 1.013567 1.020395 3J 1.008675 1.013031 6 1.014781 1.022257 4 1.009902 1.014877 6- 1.015993 1.024055 41 1.011126 1.016720 7 1.017204 1.025880 5 1.012348 1.018559 EXAMPLE.-What will an annuity of $50, payable yearly, amount to in 4 years, at 5 per cent., and what if payable half yearly? By table, page 39, 1.21550-1.-(1.05-1) = 4.310 X 50 $215.50 Ans., for yearly payment, and.. 215.50X1.012348 - V218.16 "half yearly do. PERPETUITIES. PERPETUITIES are such annuities as continue forever. RULE.-Divide the annuity by the rate per cent., and the quotient will be the answer. EXAMPLE.-What is the present worth of a $100 annuity, payable semi-annually, at 5 per cent.? 100..05 = 2000X1.012348 (from preceding table) = $2.024.70 Ans. For Perpetuities in Reversion, subtract the present worth of the annuity for the lime of reversion from the worth of the annuity, to commence immediately. EXAMPLE.-What is the present worth of an estate of $50 per annum, at 5 per cent., to commence in'4 years 1 50 -.05. = 1000. $50, for 4 years, at 5 per'cent.-= 3.54595 (from table) X50 = 177.29 $822.71 Aus., which in 4 years, at 5 per cent. compound interest, would produce $1000. CHRONOLOGICAL PROBLEMS. The Golden Number is a period of 19 years, in which the changes of the moon fall on the same days of the month as before. To find the Golden Number, or Lunar Cycle. RULE.-Add one to the given year; divide the sum by 19, and the remainder is the golden number. NOTE.-If 0 remain, it will be 19. EXAMPLE. —What is the golden number for 1830 1 1830+1-.19 = 96 rem. 7 Ans. To find the'Epact. RULE.-Divide the centuries of the given year by 4; multiply the remainder by 17, and to this product add the quotient, multiplied by 43; divide this sum plus 86 by 25, multiplying the golden number by 11, from which subtract the last quotient, and, rejecting the 30's, the remainder will be the answer. EXAMPLE.-Required the epact fos 1830. Centuries. 18-4 =-4 4.2 2X17 = 34. 4X43 = 172+34 = 206+86 =292+-25= 11, last quotient. qolden number, as ascertained above, 7XJl = 77-11 (last quotient)= 66, rejecting 30's = 6 Ans. EXAtPLE.-What is the epact for 1839?.dss. 15. D2 42 TABLE OF EPACTS. DOIMINICAL LETTERS, ETC. TO FIND THE AMOO1N'S AGE ON, ANY GIVEN DAY. RUlE.-To the day of the month add the epact and number of the month, lhon reject the 30's, and the answer will be the moon's age. Numbers of the Month. January 0, April 2, July 5 October 8, February 2, May 3, August 6, November 10, March 1, June 4, September 8, December 10. EXAMPLE.-For 5th February, 1841. Given day.. Epact... 714, age of the moon. Number of month. 2 Tihe CYCLE OF THE SUN is the 28 years before the days of the week return to the same days of the month. TABLE of Epacts, Dominical Letters, and an Almanac, fronm 1776 to 1875. February, ehbuarv,' *'January. January,. ISeptember, Marcl, May. April, June. Novemnber. August. October. July. December. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 N. B. —In leap-year, January and February must be taken in the columns nmarked *. Dom. [Dom. Dom. Dom.Years Days. Let- Years Days. Let-. Years Das. Let. Years Days Let-. ters.. ters.. ters.- ters. 1776 Friday0 GF 9! 1801 Sunday. D 15 1826 Wedn'y. A 22 1851 Sat'y. E 28 1777 Saturd'y E 20 1802 Monday. C 26 1827 Thursd. G 3 1852 Mon.* DC 9 1778 Sunday. D 1 1803 Tuesd'y. B 7 1828 Saturd.* FE 14 1853 Tues. B 20 1779 Monday. C 12 1804 Thurs.* AG 18 1829 Sunday. D 25 1854 Wedn. A 1 1780 Wedn.4* BA 23 1805 Friday. F 29 1830 Monday. C G6 1855 Thur. G 12 1781 Thursd. G 4 1806 Saturd'y E 11 1831 Tuesd'y. B 17 1856 Stat'y* FE 23 1782 Friday. F 15 1807 Sunday. D 22 1832 Thurs.* AG 28 1857 Sund. D 4 1783 Saturd'y E 26 1808 Tuesd.* CB 3 1833 Fridav F 9 1858 Mond. C 15 1784 Mond.* DC 7 1809 Wedn'y. A 14 1834 Saturd'y E 20 1859 Tues. B /26 1785; Tnesd'y B 18 1810 Thursd. G 25 1835 Sunday. D 1 1860 Thu.* AG 7 1786 Wedn'y. A 29 1811 Friday. F 6 1836 Tuesdd * CB 12 181 Friday F 18 1787 Thursd-. G 11 1812 Sunday* ED 17 1837 Wedn'y. A 23 1862 Satur. E 29 1788 Saturd.* FE 22 1813 Monday. C 28 1838 Thursd. G 4 1863 Sund. D 11 1789 Sunday. D 3 1814 Tuesd'y. B 9 1839 Friday. F 15 1864 Tue.* CB 22 1790 Monday. C 14 1815 Wedn'y. A 20 1840 Sund'y.* ED 26 1865 Wedn [ A 3 1791 Tuesd'y. B 25 1816 Friday.* GF I 1841 Monday. C 7 1866 Thur. G 14 1792 Thurs.* AG 6 1817 Saturd'y E 12 1842 Tuesd'y. B 18 1867 Friday F 25 1793 Friday. F.17 1818 Sunday. D 23 1843 Wedn'y. A 29 1868 Sun.* ED 6 1794 Saturd'y E 28 1819 Monday. C 4 1844 Friday.* GF 11 1869 Mond. C 17 1795 Sunday. D 9 1820 Wedn.* BA 15 1845 Saturd'y E 22 1870 Tues. B 28 1796 Tuesd.* CB 20 1821 Thursd. G 26 1846 Sunday. D 3 1871 Wedn. A 9 1797 Wedn'y. A 1 1822 Friday. F 7 1847 Monday. C 14 1872 Frid.* GF 20 t798 Thursd. G 12 1823 Saturd'y E 18 1848 Wed'y.* BA 25 1873 Satur. E lt2'9 Friday. F 2311824 Mond'y*0 DC 29 1849 Thurd. G 6 1874 Sund. D -12 1800 Saturd'y E 411825 Tuesd'y.| B 11 1850 Friday. F 17 1875 Mond. C 23 * Distinguishes the lyeasp- ers. 1t'ROMISOTIOUS QUESTIONS. 43 Use of the Table- To find the day of the week on which any given day of the month falls in any year from 1776 to 1875. EXAMPLE.-The great fire occurred in New-York on the 16th December, 1835; what was the day of the week? Against 1835 we find Sunday, and at top, under December, we find that the 13th Yms, Sunday; consequently, the 16th was Wednesday. PROMISCUOUS QUESTIONS. 1. If $100 principal gain $5 interest in one year, what amount A'ill gain $20 in 8 months? As 12 months: 5:: 8 months: 3.33, the interest for 8 months. And, as 3.33::100::20: 600 the answer. 2. A reservoir has two cocks, through which it is supplied; by one of thent it will fill in 40 minutes, and by the other in 50 minlites; it has also a discharging cock, by which, when full, it may be emptied in 25 minutes. If the three cocks are left. open, in what time would the cistern b.e filled, assuming the velocity of the watei to be uniform? The least common multiple of 40, 50, and 25 is 200. Then the 1st cock will fill it 5 times in 200 minutes, the 2d " 4 " 200 " or both 9 times in 200 minutes; and, as the discharge-cock will empty it 8 times in 200 minutes, then 9-8 = 1, or once in 3.20 hours, Ans. 3. Out of a pipe of wine, containing 84 gallons, 10 gallons were drawn off, and the vessel replenished with water; after which 10 gallons of the mixture was likewise drawn off, and then 10 gallons more of water were poured in, and so on for a third and fourth time it is required to find how much pure wine remained in the vessel,.ui)posing the two fluids to have been thoroughly mixed? 84-10 -- 74 As 84: 10:: 74: 8.80952 84: 10:: 65.19048: 7.76077 84: 10:: 57.42971: 6.83687 6.83687 50.59284 Ans. 4. A traveller leaves New-York at 8 o'clock in the morning, and walks towards New-London at the rate of 3 miles an hour, without intermission; another traveller sets out fromn New-London at 4 o'clock the same evening, and walks for New-York at the rate -of 4 miles an hour, constantly; now, supposing the distance between the two cities to be 130 miles, whereabout on the road will they meet? From 8 o'clock till 4 o'clock is 8 hours; therefore, 8 X3 = 24 miles, performed by A before B set out from New-London; and, con. sequently, 130-24:= 106 are the miles to travel between them after 44 PROMIISCUOUS QUESTIONS. that. Hence, as 7 = 3-+4: 3:: 106: 3 - = 453 more miles tiav. elled by A at the meeting; consequently, 24+45 -- 693 miles fiom New-York is the place of their meeting. 5. What part of $3 is a third part of $2 1 I of 2 of 3 = X 1 x.-2 Ans. 6. The hour and minute hand of a clock are exactly together at 12; when are they next together. As the minute hand runs 1 I times as fast as the hour hand; then, 11: 60:: 1: 5 min. 5-5- sec. The time, then, is 5 min. 5&5L sec. past 1 o'clock. 7. The time of the day is between 4 and 5, and the hour and minute hands are exactly together; what is the time! The speed of the hands is as 1 to 11. 4 hours X 60 = 240, which -.-11= —21 - min. added to 4 hours, Ans. 8. A can do a piece of work in 3 weeks, B can do thrice as much in 8 weeks, and C five times as much in 12 weeks; in what time can they finish it jointly 1 Week. Week. Week. As 3: 1: 1: work done by A in one week. 8 3: 3:: 13: B " B 12: 5. 1 " C 1 2 Then, by addition, +_+i-A5- will be the work done by them all in one week; these, reduced to a common denominator, become {-2 +2-`+3T --' —-; whence, 9: 6:: 8: 5- Ans. 9. A cistern, containing 60 gallons of water, has 3 unequal cocks for discharging it; one cock will empty it in 1 hour, a second in 2 hours, and a third in 3 hours; in what time will it be emptied if they all run together 1. First, i would run out in 1 hour by the second cock, and 1 by the third; consequently, by the 3 was the reservoir supplied one hour. _+1+1 - 32+6 being reduced to a common denominator, the sum of these 3_ -; whence the proportion, 11: 60: 6: 32,-1 minutes, the time required. 10. Suppose a cubic inch of common: glass weighs 1.49 ounces troy, the same of sea water.59, and of brandy.53. A gallon of this liquor in a glass bottle, which weighs 3.84 lbs., is thrown into th( water. It is proposed to determine if it will sink; and if so, host much force will just buoy it up 1 3.84X 12- 1.49 = 30.92 cubic inches of glass in the bottle..231 cubic inches in a gallon X.53 -122.43 ounces of brandy. Then, bottle and brandy weigh 3.84 X 12 4- 122.43 = 168.51 ounces, and contain 261.92 cubic inches, which, X.59 154.53 ounces, the weight of an equal bulk of salt water. And, 168.51-154.53 = 13.98 ounces, the weight necessary to sup. port it in the water. 11. How many fifteens can be counted with four fives? Ans. 4. PROMISCUOUS QUESTIONS. 45 12. What is the radius of a circular acre. (Side of a square X 1.128 = diameter of an equal circle.) 208.710321, the side of a square acre, X1.128 = 235.50 — 2 (for radius) =117.75 feet, Ans. 13. From Caldwell's to Newburg is 18 miles; the current of the river is such as to accelerate a boat descending, or retard one ascending 1- miles per hour. Suppose two boats, driven uniformly at the rate of 15 miles per hour through the water, were to start one from each place at the same time, where will they meet 3 Call x the distance from N to the place of meeting; its distance from C, then, will be 18-x. Speed of descending boat, 15+1.5 = 16.5 miles per hour. Speed of ascending boat, 15-1.5 = 13.5 miles per hour. = time of boat descending to point of meeting. 16.5 13.5 -time of boat ascending to point of meeting. These times are, of course, equal; therefore, 5-= 18-x 16.5 13.5 Then, 13.5x - 297-16.5x, and 13.5x+ 16.5x --- 297, or 30x = 297. 297 Hence, x = —- 9.9 miles, the distance from Newburg, Ans. 14. A steamboat, going at the rate of 10 miles per hour through the'water, descends a river, the velocity of which is 4 miles per hour, and returns in 10 hours; how far did she proceed? Let a = distance required, ---- time of going, = time of returning. 10+4 10-4 i,4- 10; 6x+14x = 840; 20x = 840; 840-. 20 = 42, Ans. 15. If a steamboat, going uniformly at the rate of 15 miles in an hour through the water, were to run for 1 hour with a current of 5 miles per hour; then, to return against that current, what length of time would she require to reach the place from whence she started 1 15+5 = 20 miles, the distance gone during the hour. Then 15-5= 10 miles, is her effective velocity per hour when returning, and 0.-10 = 2 hours, the time of returning, and 2+1 = 3 hours, or the whole time occupied, Ans. Or, let d represent the distance in one direction, t and t' the greater and less times of running (in hours), and cthe current or tide. tt'Then, - = velocity of boat through the water, and c = d16. The flood tide wave of a river runs 20 miles per hour, the current of it is 3 miles per hour. Assume the air to be quiescent, and a floating body set in motion at the commencement of the flow of the tide; how long will the body drift in one direction, the tide flowing six hoturs from each point of the river? Let x be the time required; 20x = distance the tide has run up, together with the distance which the floating body has moved; 3x = whole distance which tehe body has floated. Then 20x;-3 = 6X20, or the length in miles of a tide. x= 20 —Xx6- =7 hours, 3 minutes, 31 7 5 2 seconds, Ans. 20-_3 I0 9 GEOMIIETRY. GE O ME T R Y. Definitions. A Point has position, but not magnitude. A Line is length without breadth, and is either Right, Curved, or Mixed. A Right Line is the shortest distance between-two points. A Jlixed Line is composed of a right and a curved line. A Superficies has length and breadth only, and is plane or curved. A Solid has length, breadth, and thickness. An Angle is the opening of two lines having different directions, and is either Righ. Icute, or Obtuse. A Right Angle is made by a line perpendicular to another, falling upon it An Acute Angle is less than a right angle. An Obtuse Angle is greater than a right angle. A Triangle is a figure of three sides. An Equilateral Triangle has all its sides equal. An Isosceles Triangle has two of its sides equal A Scalene Triangle has all its sides unequal. A Right-angled Triangle has one right angle. An Obtuse-angled Triangle has one obtuse angle. An Acute-angled Triangle has all its angles acute. A Quadrangle or Quadrilateral is a figure of four sides, and has the following par, ticular names, viz.: A Parallelogram, having its opposite sides parallel. A Square, having length and breadth equal. A. Rectangle, a parallelogram having a right angle. A Rhombus (or Lozenge), having equal sides, but its angles not right angles. A Rhomboid, a parallelogram, its angles not being right angles. A Trapezium, having unequal sides. A Trapezoid, having only one pair of opposite sides parallel. NOTE.-A Triangle is sometimes called a Trigon, and a Square a Tetragon. POLYGONS are plane figures having more than four sides, and are either Regular or Irregular, according as their sides and angles are equal or unequal, and they ail aamed from the number of their sides or angles. Thus: A Pentagon has five sides. A Hexagon " six A Heptagon " seven An Octagon " eight " A Nonagon " nine " A Decagon " ten An Undecagos " eleven" A Dodecagon " twelve " A CIRCLE is a plane figure bounded by a curve line, called the Circumference (eo Periphery). An Arc is any part of the circumference:of a circle. A Chord is a right line joining the extremities of an arc. A Segment of a circle is any part bounded by an arc and its chord. The Radius of a circle is a line drawn from the centre to the circumference. A Sector is any part of a circle bounded by an arc and its two radii. A Semicircle is half a circle. A Quadrant is a quarter of a circle. A Zone is a part of a circle include, between two parallel chords. A Lune is the space between the intersecting arcs of two eccentric circles. A Gnomon is the space included between the lines forming two similar paralIelograms, of which the smaller is inscribed within the larger, so as to have one angle is each common to both. A Secant is a line that cuts a circle, lying partly within and partly without it. A Cosecant is the secant of the complement of an arc. A Sine of an arc is a line running from one extremity of an arc perpendicular to a diameter passing through the other extremity, and the sine of an angle is the sine of the arc that measures that angle. The Versed Sine-of an arc or angle is the part of the diameter intercepted( between the sine and the arc. GEOM-ETRY. 47 -The Cosine of an arc or angle is the part of the diameter intercepted between the line and the ce-stre. A Tangent is a right line that touches a circle without cutting it. A Cotangent is the tangent of the' complement of the arc. The Circumference of every circle is supposed to be divided into 360 equal parts called Degrees; each degree into 60 Minutes, and each minute into 60 Seconds, and so on. The Complement of an angle is what remains after subtracting the angle from 90 degrees. The Supplement of an angle is what remains after subtracting the angle from 18( degrees. To exemplify these definitions, let A c b, in the following diagram, be an assumned tre of a circle described with the radius A B. D A c b, an AI',' of the circle A C E D. B k, the Cosine of the arc A c o. A b, the Chord of that arc. A a, the Tangent of do. e D d, a Segmnent of the circle. C B b, the Complement, and b B E, the A B? the Radius. Supplement of the arc A c b. A B b, a Sector, C g, the Cotangent of the arc, written A D E B, a Semicircle. cot. C B E, a Quadrant. B g, tne Cosecant of the arc, written A e d E, a Zone. cosec n o h, a Lune. m m, the Coversed sine of the arc, or, by B g, the Secant of the arc A c bi. convention, of the angle A B b; written b k, the Sine of do. coversin. A k, the Versed Sine of do. A Prism is a solid of which the sides are parallelogranms, and are of three, four, sire, or more sides, and are upright or oblique. A Parnllelopipedon is a solid terminated by six parallelograms: thus, a four-sided prism is a parallelopipedon. A Pyramid is a solid bounded by a number of planes, its base being a rectilinear figure, and its faces triangles, terminating in one point, called the summit or vertex. It is regular or irregular, upright or oblique, and triangular, quadrangular, and so) on, from its equality of sides, inclination, or number of sides. A Cylinder is a solid formed by the rotation of a rectangle about one of its sides, at rest; this side is called the axis of the cylinder. It is right or oblique as the axis is perpendicular or inclined. An Ellipse is a section of a cylinder oblique to the axis. (See CONIC SECTIONS, page 54.) A S'phere is a solid bounded by one continued surface, every point of which is equally distant from a point within the sphere, called the coXltre. 8 GEIOMETRY. lThe Altitude, or }eight of a figure, is a perpendicular- let tall from its vertex to the opposite side, called the base. The Measure of an angle is an arc of a circle contained between the two lines that form the angle. and is estimated by the number of degrees in the arc. A Prismoid has its two ends as any unlike parallel plane figures of the same num. ber of sides, the upright sides being trapezoids. A Spheroid is a solid resembling the figure of a sphere, but not exactly round, one Of its diameters being longer than the other. n. Spindle is a solid formed by tile revolution of some curve round its base. A Segment is a part cut off by a plane, parallel to the base. A Frustum is the part remaining after the segment is cut off. A Cycloid is a curve formed by a point in the circumference of a circle, revolving on a right line the length of that circumference. An Epicycloid is a curve generated by a point in one circle which revolves about slnother circle, either on the concavity or convexity of its circumference. An Ungula is the bottom part cut off by a plane passing obliquely through the base of a cone or cylinder. The Perimeter of a figure is the sum of all its sides. A Problem is something proposed to be done. A Postulate is something required. A Theorem is something proposed to be demonstrated. A Lemma is something premised, to render what follows more easy. A Corollary is a truth consequent upon a preceding demonstration. A Scho7ium is a remark upon something going before it. abc G o A'B u4 F I p To construct a Diagonal Scale upon any Line, as A B-fig. 1. Divide the line into as many divisions as there are hundreds of feet, spaces of tel. feet, feet, or inches required. Drawv perpendiculars from every division to a parallel line C D. Divide these perpelndiculars and one-of the divisions A E, C F, into spaces of ten if for feet and hundredths, and into twelve if for inches; draw the lines A 1, a 2, b 3, &c., and they wvill complete the scale required. Thus: The line A B representing tell feet; A to E, E to G, &c., will measure one foot; A to a, C to 1, 1 to 2, &c., will measure 1-10th of a foot; and the several lines A 1, a 2, &c., will measure upon the lines k k, l 1, &,., 1-100th of a foot; ani o p will measire upon k k, 11, &c., 1-10th of a foot. D E c c D To circumscribe a Pentagon about a gonen Circle-fig. 2. Rvu,E.-Iliscrilbe a pentagon in the circle, defining the points s r v m m. GEOMETRY. 49 F'rom the centre o, draw o r,o v, &e. Through n, m, &c., draw A B, B C, &c., perpendicular to o n, o m, and complete. the figure. NoTE.-Any other polygon may be made in a similar manner, by drawing tangents to the points, first defining them in the circle. Upon a given Line A B, to form an Octagon-fig. 3. RULE.-On the extremities of A B, erect indefinite perpendiculars A F, B E, pioduce A B to m and n, and bisect the angles m A e and n Bf with A H and B C. Make A H and B C equal to A B, and draw H G, C D, parallel to A F, and equal to A B. From G and D, as centres with a radius equal to A B, describe arcs cutting A F, B F, in F and E. Join G F, F E, and E D, and the figure is made. Circles and Squares. 4. C 0 A B To describe a Circle that shall pass through any three given Points, as A B C-fig. 4. RULE. —Upon the points A and B, with any opening of the dividers, describe two arcs to intersect each other, as at e e; on the points A C describe two more to intersect each other in the points c c; draw the lines e e and c c, and where these two idues intersect o, place one foot of the dividers, and extend the other to A, B, or C, and it will pass through the three given points as required. To make a Square equal to a given Triangle —fig. 5. Ilet B d E be the triangle given. RULE.-Extend the side of the triangle B E to 0, making E O equal to half the:ength of the perpendicular of the triangle A d. Divide B 0 into two equal parts in K, and with the distance K B describe the semicircle B H O. Upon E erect the perpendicular E H, which will be the side of a square, equal to the triangle B d E. Triangles and Squares. ~~6. ~7. o A ~E X s1 \\<:^) F To make an Equilateral Triangle equal to two given Equilateral Triangles-fig. 6. Let the given equilateral triangles be A and B. RULE. —Draw a right line C D equal in length to one side of the triangle B. Erect the perpendicular D E, equal in length to one side of the triangle A. Draw C E, and complete the equilateral triangle C E F, and it will be equal in quantity to the two given equilateral triangles A and B. To make a Square equal to two given Squares-fig. 7. Iet the two squares given be A and B. E 50 GEOMETRY. RULE.-Draw the line C F, equal in length to one side of the largest square A. Raise the perpendicular E F, equal in length to one side of the smallest square B.;Draw C E, and C E is the side of the square C E G 0, which is equal in quantity lto;the two given squares A and B. Circles and Ellipses. S. 9. /1 0 11 B A Two Circles, F and G, being given, to make another of equal quantity —fig. 8. RULE.-Upon the diameter of the largest of the two circles at the point D, erect the perpendicular D E, equal in length to the diameter A B of the least circle. Draw B E, and divide it into two equal parts in O; take the distance B O or O E, and describe a circle. This circle will be equal in quantity to the two given circles F and G. To describe an Ellipse of any given length, without regard to breadth-fig. 9. Let A B be the given length. RuLE.-Divide it into three equal parts, as A s i B. Then, with the radius A s, describe A F o i n C; and from i, the circle B D n s o E; then with n F and o C do scribe F E and C D, and you have the ellipse required. 10. 11. -GIG.. d a E........... ging.. 10. To describe an Ellipse to any length and breadth given-fig. 10. Let the longest diameter given be the line F, and the shortest G. RULE.-Make A B equal to F, and C D to G, dividing A B equally at right angles in a. Make A o equal to D C, and dividing o B into three equal parts, set off two of those parts from a to b and from a to c, then with the distance c b make the two,equilateral triangles c d b and c e b, these angles are the centres, and the sides being continued are the lines of direction for the several arcs of the oval A C B D. NOrTE.-Carpenters, Bricklayers, and Masons are oftentimes obliged to work an -architrave, 4c., about windowts, of this form: they may, by the help of the four centres c, d, b, e, and the lines of direction h d, e f, d g, e i, describe another line around th' former, and at any distance required. as h if g. GEOMETRY. 5.1 To describe an Ellipse to any length and breadth required, another way —fig. 11. Let the longest diameter be A, and the shortest B. RULE.-Draw the line C D equal in length to A; also E F equal in length to B, and at right angles with C D. Take the distance C O or O D, and with it, from the point E and F, describe the arcs h and f upon the diameter C D. Strike in a nail or pin at h and atf, and put a string around them, of such a length that the two ends may just reach to E or F. Introduce a pencil, and bearing upon the string, carry it around the centre 0, and,t will describe the ellipse required. 12. 13. To find the Centre and two diameters of an Ellipse —fig. 12. Let A B C D be the ellipse. RULE.-Draw at pleasure two lines, Q G, M O, parallel to each other; bisect them in the points H N, and draw the line P E; bisect it in K, and upon K, as a centre, describe a circle at pleasure, as F L R, cutting, the figure in the points F L; draw the right line F L, bisect it in I, and through the points I K draw the greatest diameter A C, and through the centre K draw the least diameter B D, parallel to the line F L. To draw a Spiral Line about a given Point-fig. 13. Let B be the centre. RULE.-Draw A C, and divide it into twice the number of parts that there are to be revolutions of the line. Upon B describe IK I, G F, H E, &c., and upon I describe K F, G E, &c. Polygons. It \ 1 RUL-Daw t leaur tw lnes Q,M0 aale oec teriette 52. GEOMETRY. Upon a given line, to describe any Polygon beyond a Pentagon-fig. 14. Let A B be the given line. RULE.-Bisect the, line A B in Q, and erect the perpendicular Q P. lProm the point A describe the arc B H, and from B the arc A H, and divide B H into equal parts, as H 1, 2, 3, 4, 5, B. Let a pentagon be required. From the point H, with the interval H 1, describe the arc I 7, and the point I will be the centre of a circle containing the given line A B five times, the interval I B being the radius thereof. Take the point H for the centre of another circle, and H B for the radius; this circle will contain the line A B six times. From the point 7, with the radius 7 B, a circle drawn will contain A B seven. times. From the point H, with the interval H 2, describe the arc 2 8; and from the oint 8, with the radius 8 B, draw a circle, and A B shall be the side of an octagon. From 9, with the radius 9 B, you form a nine-sided figure; from 10 a ten sided figure; and so on to 12. Arches. 17. 15. * o0 16. d e A B To describe an Elliptic Arch on the Conjugate Diameter-fig, 15. RI ULE.-Draw the diameter A B, and in the middle at k, erect the perpendicular k o, equal to the height of the arch; divide the perpendicular k o into two equal parts tat e; continue the line A B on both sides at pleasure, and from the point k, with the distance k o, define c and d; through c e, d e, draw c ef and d e g at pleasure; d arud c are centres for the arcs A g and Bf, and e the cantre for the arc g o f, which will form the arch required. To draw a Gothic Arch-fig. 16. RULE 1.-Take the length of the line A B, and on the points A and B describe th, arcs A c and B d, and it will complete the arch required. RULE 2,-fig. 17.-Divide the line A B into three equal parts, at c and d; take A d or B c, and describe B e or A e, and it will give an arch of another form. 18. 19. A A RULE 3 18.-Divide the line A B also into three eual parts, e from the RUlE 3,fig. 18 —Divide the line A B also into three equal parts, e f; from the GEOMETRY. 53 points A and B let fall the perpendiculars A c and B d, equal in length to two of the divisions of the line A B; draw the lines c h and d g: from the points e f, with the length off B, describe the arcs A g and B h, and from the points c d describe the arcs g i and i h, and it will complete another Gothic arch. RULE 4,fig. 19.-As before, divide the line A B into three equal parts at a and b. and on the points A a, b B, with the distance of two divisions, make four arcs intersecting at c d. Through the points c d and the divisions a b, draw the lines c f and d e, and on the points a and b describe the arcs A e and B f, and on the poiPts c d the arcs e g and f g, and it will complete another Gothic arch. E 2 54 CONIC SECTIONS. CONIC SECTIONS. IDfinitions.. A Cone: is a solid figure having a circle for its base, and terminated' in a vertex. Conic Sections are the figures made by a plane cutting a cone An Ellipse is the section of a cone when cut by a plane obliquel) through both sides. A Parabola is the section of a cone when cut by a plane parallel to its side. A Hyperbola is the section of a cone when cut by a plane, makiwn a greater angle with the base than the side of the cone makes. The Transverse Axis is the longest straight line that can be drawn in an ellipse. The Conjugate Axis is a line drawn through the centre, at right angles to the transverse axis. An Ordinate is a right line drawn from any point of the curve perpendicular to either of the diameters. An Absczssa is a part of any diameter contained between its vertex and an ordinate. The Parameter of any diameter is a third proportional to that diameter and its conjugate. The Focus is the point in the axis where the ordinate is equal to half the parameter. A Conoid is a solid generated by the revolving of a parabola or hyperbola around its axis. A Spheroid is a solid generated in like manner to a conoid by an ellipse. To construct a Parabola —fig. 1. B A, 4 D_- 1 4. CONIC SECTIONS. Draw an isosceles triangle, A B D, whose base shall be equal to that of the proposed parabola, and its altitude, C B, twice that of it. Divide each side, A B, D B, into 8, or any number of equal parts; tlhen draw lines 1 1, 2 2, 3 3, &c., and their intersection will define the curve of a parabola. NOTE.-The following figures are drawn to a scale of 100 parts to an inch. 7o construct an Ilyperbola,* the Transverse and Conjugate Diameters' being given-fig. 2. a/ A Js Make A B the transverse diameter, and C D, perpendicular to. it, the conjugate. Bisect A B in 0, and from 0, with the radius O C or O D, describe the circle D f c F, cutting A B produced in F and f, which points will be the foci. In A B produced take any number of points, n n, &c., and from F and f, as centres, with B n, A n, as radii, describe arcs, cutting each other in s s, &c. Through s s, &c., draw the curve s B s, and it will be the hyperbola required. ELLIPSE. * To describe hyperbolas by another method, see Gregory's Mathematics, p. 160. 556 CONIC SECTIONS, To find the length of the Ordinate, E F, of an Ellipse, the Transverse, A B, Conjugate, C D, and Abscissce, A F and F B, being known —fig. 3. RULE.-As the transverse diameter is to the conjugate, so is the square root of the product of the abscissae to the ordinate which divides them. EXAMPLE.-The transverse axis, A B, is 100; the conjugate, C D, is 60; one abscissa, B.F, is 20; the other, A F, is (100-20)= 80. 100: 60:: /20X80: 24 Ans. The Transverse and Conjugate diameters, and an Ordinate being known, to find the Abscissoe —fig. 3. RULE.-As the conjugate diameter is to the transverse, so is the square root of the difference of the squares of the ordinate and semiconjugate to the distance between the ordinate and centre; and this distance, being added to and subtracted from the semi-transverse, will give the abscissas required. EXAMPLE.-The transverse diameter, A B, is 100; the conjugate, C D, is 60; and the ordinate, F E, is 24. 60:100:: /242-302: 30, distance between the ordinate and centre; then 100 — 2-30 — 20, one abscissa; 100 2+30 -- 80, the other abscissa. When the Conjugate, Ordinate, and Abscisscs are known, to find the Transverse-fig. 3. RULE. —TO or from the semi-conjugate, according as the greater or less abscissa is used, add, or subtract the square root of the difference of the squares of the ordinate and semi-conjugate. Then, as this sum or difference is to the abscissa, so is the conjugate to the transverse. EXAMPLE.-The ordinate, F E, is 24; the less abscissa, F B, is 20; and the conjugate, C D, is 60. 30 —/242_302 - 12; then 12: 20:: 60: 100 Ans. The Transverse, Ordinate, and Abscissce being given, to find the Conjugate-fig. 3. RULE. —AS the square root of the product of the abscissa is to the ordinate, so is the transverse diameter to the conjugate. EXAMPLE.-The transverse is 100, the ordinate 24, one abscissa:20., the other 80. 8(80X20: 24:: 100 * 60 Ans. PARABOLAS. Any three of the four following terms being given, viz., any two Ordinates and their Abscissae, to find the fourth-fig. 4. RULE. —AS any abscissa is to the square of its ordinate, so is any other abscissa to the square of its ordinate. CONIC SECTIONS. 57 4. e A.__ EXAMPLE.-The abscissa, e g, is 50, its ordinate, c g, 35.35, required the ordinate A F, whose abscissa, e F, is 100. 50: 35.352:: 100: V/2500 = 50 Ans. HYPERBOLAS. 5. B Wlhen the Transverse, the Conjugate, and the less Abscissa, B a, are given, to find an Ordinate, e n —fig. 5. NOTE.-In hyperbolas, the less abscissa, added to the axis, gives the greater. RULE.-As the transverse diameter is to the conjugate, so is the square root of the product of the abscissae to the ordinate required. When the Transverse, the Conjugate, and an Ordinate are given, to find the Abscissce-fig. 5. RULE.-To the square of half the conjugate add the square of the ordinate, and extract the square root of that sum. Then, as the conjugate diameter is to the transverse, so is the square root to half the sum of the abscissae. To this half sum add half the transverse diameter for the greater abscissa, and subtract it for the less. When the Transverse, the Abscissa, and Ordinate are given, to find the Conjugate-fig. 5. RULE. —-AS the square root of the product of the abscissae is 1t the ordinate, so is the transverse diameter to the conjugate. When the Conjugate, the Ordinate, and the Abscissa, are given, to find the Transverse-fig. 5. Ru[r. —Add the square of the ordinate to the square of half the conjugate, and extract the square root of that sum. To this root add half the conjugate when the less abcissa is used, and subtract it when the greater is used, reserving the difference or sum. Then, as the square of the ordinate is to the product of the abscissa and conjugate, so is the sum, or difference above found, to the transverse diameter. CONIC SECTIONS. EXAMPLES. —In the hyperbola, figs. 2 and 5, the transverse diameter is 100, the conjugate 60, and the abscissa, B n, is 40; required the ordinate e n. 100: 60:: V(40+100X40) —=74.8: 44.8 Ans. The transverse is 100, the conjugate 60, and ordinate e n, 44.8; what are the abscissae. Ans. 40 and 140. The transverse is 100, the ordinate 44.8, the absciss.a 140 and 40; what is the conjugate? Ans. 60. The conjugate is 60, the ordinate 44.8, and the less abscissa 40 ~ what is the transverse? Ans. 100. .IIENSURATION OF SURFACES. 59 MENSURATION OF SURFACES. OF FOUR-SIDED FIGURES. itt_ 2. a 4..z.... r_7o 7' find the Area of a four-sided Figure, whether it be a Square, Parallelogram, Rhombus, or a Rhomboid. RULE.-Multiply the length by the breadth or perpendicular height, and the product will be the area. OF TRIANGLES. To find the Area of a Triangle-figs. 5 and 6. iift RULE.-Multiply the base a b by the perpendicular height c d. and half the product will be the area. To find the Area of a Triangle by the length of its sides. RULE.-From half the sum of the three sides subtract each side separately; then multiply the half sum and the three remainders continually together, and the square root of the product will be the area. To find the Length of one side of a Right-angled Triangle, when the Length of the other two sides are given-fig. 7. RULE. —TO find the hypothenuse a c. Add together the square of the two legs a b and b c, and extract the square root of that sum. To find one of the legs. Subtract the' square of the leg, of which the length is known, from the square of the hypothenuse, and the square root of the difference will be the answer. NOTE. —For Spherical Triangles, see page 68..OF TRAPEZIUMS AND TRAPEZOIDS. To find the Area of a Trapezium-fig. 8. RULE.-Multiply the diagonal a c by the sum of the two perpendiculars falling upon it from the opposite angles, and half the product will be the area. MENSURATION OF SURFACES. To find the Area of a Trapezoid-fig. 9. RULE. —Multiply the sum of the parallel sides a b, d c, by a h, the perpendicular distance between them, and half the product will be the area. OF REGULAR POLYGONS. RULE.-Multiply half the perimeter of the figure by the perpendicular, falling from its centre upon one of the sides, and the product will be the area. To find the Area of a Regular Polygon, when the side only is given. RULE. —Multiply the square of the side by the multiplier opposite to the name of the polygon in the following table, and the product will be the area. Sides. NameofPolygon. Angle. AnPolygon. Area. A C Sid~noe. o _ Polygon. B _3 Trigon 1200 60~ 0.433012 2. 1.732.5773 4 Tetragon 90 90 1.000000 1.41 1.414.7071 S Pentagon 72 108 1.720477 1.238 1.175.8506 6 Hexagon 60 120'2.598076 1.156 -Radius ofsidel'gth 7 Heptagon 51 1281 3.633912 1.11.8677 1.152 8 Octagon 45 135 4.828427 1.08.7653 1.3065 9 Nonagon 40 140 6.181824 1.06.6840 1.4619 10 Decagon 36 144 7.694208 1.05.6180 1.6180 11 Undecagon 32-L~ 147A-1 9.365640 1.04.56i34 1.7747 12 Dodecagon 30 150 11.19615211.037.5176 1.9318 Additional uses of theforegoing Table. The third and fourth columns of the table will greatly facilitate the construction of these figures, with the aid of the sector. Thus, if it is required to describe an octagon, opposite to it, in column third, is 45; then, with the chord of 60 on the sector as radius, describe a circle, taking the length 45 on the same line of the sector; mark this distance off on the circumference, which, being repeated around the circle, will give the points of the sides. The fourth column gives the angle which any two adjoining sides of the respective fitures make with each other. Take the length of a perpendicular drawn from the centre to one of the sides of a polygon, and multiply this by the numbers in column A, the product will be the radius of the circle that contains the figure. The radius of a circle multiplied by the number in column B, will give the length of the side of the corresponding figure which that circle will contain. The length of the side of a polygon multiplied by the corresponding number in the column C, will give the radius of the circumscribing circle. 'I7ENSURA'rION OF SURFACES. OF REGULAR BODIES.'o find the Superficies of any Regular Body. RULE.-Multiply the tabular surface in the following table by the square of the linear edge, and the product will be the superficies. Number of Sides. Names. Surfaces. 4 Tetrahedron 1.73205 6 Hexahedron. 6.00000 8 Octahedron. 3.46410 12 Dodecahedron. 20.64573 20 Icosahedron. 8.66025 OF IRREGULAR FIGURES. 10. e e5,d G I To find the Area of an Irregular Polygon, ab c d efgfJig. 10. RULE. —Draw diagonals to divide the figure into trapeziums and triangles; find the area of each separately, and the sum of the whole will give the area required. To find the Area of a Long Irregular Figure, b d c a-fig. 11. RULE.-Take the breadth in several places, and at equal distances apart; add them together, and divide the sum by the number of breadths for the mean breadth; then multiply that by the length of the figure, and the product will be the area. OF CIRCLES. 12. 13, 14. u.\\ d ~d c4L~1I~3~L~~d e6... \ I,! Ic To find the Diameter and Circumference of any Circle. RULE 1.-Multiply the diameter by 3.1416, and the product will be the circumference. F '0 2 MENSURATION OF SURFACES. RULE 2. —Divide the circumference by 3.1416, and the quotienl will be the diameter. RULE 3.-Or, as 7 is to 22, so is the diameter to the circumfer. ence. Or, as 22 is to 7, so is the circumference to the diameter. Or, as 113 is to 355, so is the diameter to the circumference, &c To find t/he Area of a Circle. RULE 1.-Multiply the square of the diameter by.7854, or the square of the circumference by.07958, and the product will be the area. RULE 2. —Multiply half the circumference by half the diameter. RULE 3.-As 14 is to 11, so is the square of the diameter to the area, or, as 88 is to 7, so is the square of the circumference to the area. To find the Length of any Arc of a Circle-fig. 12. (See Table of Areas, page 75.) RULE 1.-From 8 times the chord of half the are a c, subtract the chora a o of the whole are; one third of the remainder will be the length nearly. RULE 2.-Multiply the radius a o of the circle by.0174533, and that product by the degrees in the arc. RULE 3.-As 180 is to the number of degrees in the arc, so is 3.1416 times the radius to its length. When the Chord of the Arc and the Versed Sine of half the Arc are given. RULE. - TO 15 times the square of the chord a b, add 33 times the square of the versed sine c d, and reserve the number. To the square of the chord add 4 times the square of the versed sine, and the square root of the sum will be twice the chord of half the arc. Multiply twice the chord of half the arc by 10 times the square of the versed sine, divide the product by the reserved number, and add the quotient to twice the chord of half the arce: the sum will be the length of the arc very nearly. NOTE.-1. diameter X.8862= side of an equal square. 2. circumference X.2821- " =' " 3. diameter X.7071 = " of the inscribed square. 4. circumference X.2251- " " 5. area X.9003 = " " 6. side of a square X 1.4142 = diam. of its circums. circle. 7. " " X4.443 -circum. " " 8. " " X ].128 = diam. of an equal circle 0. " " X3.545 =circum. " " 10. square inches X1.273 =round inches. When the Chord of the Arc, and the Chord of half the Arc are given. RULE. - From the square of the chord of half the are subtract NOTE.-If the length for any number of degrees, minutes, &c., is required (see page 67 for the units, radius being 1), multiply them by the number of degrees, &c. (n the are, and the answer is the length. MENSURATION OF SURFACES. 63 the square of half the chord of the arc, and the remainder will be the square of the versed sine: then proceed as above. NOTE.-The chord of half the arc is equal to the square root of the sum of the squares of the versed sine or height, and half the chord of the entire arc. When the Diameter and the Versed Sine of half the Arc are given. RULE. - From 60 times the diameter c o, subtract 27 times e d the versed sine, and reserve the number. Multiply the diameter by the versed sine, and the square root oi the product will be the chord of half the arc. Multiply twice the chord of half the arc by 10 times the versed sine, divide the product by the reserved number, and add the quotient to twice the chord of half the arc: the sum will be the length of the arc very nearly. NOTE. - When the diameter and chord of the arc are given, the versed sine may be found thus: From the square of the diameter,subtract the square of the chord, and extract the square root of tile remainder. Subtract this root from the diameter, and half the remainder will give the versed sine of half the arc. The square of the chord of half the arc being divided by the diameter, will give the versed sine; or, being divided by the versed sine, will give the diameter. To find the Area of a Sector of a Circle-fig. 13. RULE 1.-Multiply the length of the arc ad b by half the length of the radius a o. RULE 2.-As 360 is to the degrees in the arc of the sector, so is the area of the circle to the area of the sector. NOTE.-If the diameter or radius is not given, add the square of half the chord of the arc to the square of the versed sine of half the arc; this sum being divided by the versed sine, will give the diameter. To find the Area of a Segment of a Circle —fig. 12. (See table of Areas, page 72.) W'Vhen the Segment is less than a Semicircle. RULE 1.-Find the area of the sector having the same arc with the segment, then find the area of the triangle formed by the chord of the segment and the radii of the sector, and the difference of these areas will be the area required. RULE 2.-To the chord a b of the whole arc, add the chord a c of half the arc, and 3 of it more; then multiply the sum by the versed sine c d, and.40426 of the product will be the area. RULE 3.-Multiply the chord of the segment by the versed sine, divide the prodluct by 3, and multiply the quotient by 2. Cube the height or versed sine, divide it by twice the length of the chord, and add the quotient to the former product; it will give the area nearly. When the Segment is greater than a Semicirle. RULE. —Find, by a preceding rule, the area of the lesser portion of the circle, subtract it from the area of the whole circle, and the remainder is the area required. To find the.Area of a Circular Zone-fig. 14. (See table of Areas, page 80.) RULE 1.-When the zone is less than a semicircle. To the area of the trapezoid a h c d add the area of the segments a b, c d; their sutn is the area. 645 MENSURATION OF SURFACES. RULE 2. —When the zone is gr-eater than a semicircle. To the area of the parallelogram b g d h, add the area of the segments b ig, d k h; their sum is the area, T'ofind the Convex Susface of any Zone or Segznent-figs. 38 and 39. RuLE.-Multiply the height c b, or b d, of the zone or segment by the circumference of the sphere, and the product is the surface. NOTE. —WhIen the diameter of the circle is not given, multiply the mean length ot the two chords by half their difference, divide this product by the breadth of the zone, and to the quotient add the breadth. To the square of this sum add the square of the lesser chord, and the square root of their sum will be the diameter of the circle. OF UNGULAS. To find the Convex Surface of the Usgulas-figs. 27, 28, 29, ansd 30. RULEs. — For fig. 27, multiply the length of the arc line a b c of the base, by the height a d. For fig. 28, multiply the circumference of the base of the cylinder efg by half the sum of the greater and less lengths a e, cf. For fig. 29, multiply the sine ad, of half the arc ag, of the base a c g, by the diameter eg of the cylinder, and from this product subtract the product* of the arc ag c and cosine df. Multiply the dif ference thus found by the quotient of the height g b, divided by the versed sine e d. For fig. 30 (conceive the section to be continued till it meets the side of the cylinder produced), then find the surface of each of the ungulas thus formed, and their difference is the surface required. NOTE.-For rules to ascertain the surface of conical ungulas, see Ryan's Bonnycastle's Mensusratton, page 136 (1839). To find the Area of a Circular Ring or Space included between two Concelitric Circles-fig'. 54. RULE. —Find the areas of the two circles a d, b c separately, and their difference will be the area of the ring. OF ELLIPSES. ~~~~15.~~ 16. i? 16. c a -6 6 -..... (i d J 7'o find the Circumference of an Ellipse-fig. 15. RULE.-Square the axes a b and c d, and multiply the square root of half their sum by 3.1416; the product will be the circumference. To find the Area of an Ellipse-fig. 15. RULE.-Multiply the dianeters together, and their product.by.7854. When this product exceeds the other, add them together, and when the cosinfe is 0, the product is 0. MENSURATION OF SURFACES. 65 To find the Area of an Elliptic Segment, a eg-fig. 16. RvLE. —Divide the height of the segment ap by the axis a b, oi which it is a part, and find in the table of circular segments, page i2, a segment having the same versed sine as this quotient; then multiply the segment thus found and the two axes of the ellipse togethel, and the product will give the area. OF PARABOLAS. 17. / 18. 18. C,,~L-~ —n ~ /~ j;,, I —- - C d- e' a 6 b d To find the Area of a Parabola —fig. 17. RULE.-Multiply the base df by the height g C, and 3 of the product will be the area. To find the Area of a Frustrum of a Parabola-fig. 17. RULE.-Multiply the difference of the cubes of the two ends of the frustrum a c df by twice its altitude b e, and divide the product by three times the difference of the squares of the ends. To find the Length of a Parabolic Curve cut off by a Double Ordinatefig. 18. RULE. —TO the square of the ordinate a b add 4 of the square o f the abscissa c b; the square root of that sum, multiplied by 2, will give the length of the curve nearly. OF HYPERBOLAS. 20. 19 M a Z —--—.................. To find the Area of a Hyperbola —fig. 19. RULE. —TO the product of the transverse and abscissa add 5 of the square of the abscissa a b, and multiply the square root of the sum by 21. Add 4 times the square root of the product of the transverse and abscissa to the product last found, and divide the sum by 75. Divide 4 times the product of the conjugate and abscissa by the transverse, and th;s last quotient, multiplied by the former, will give the area nearly. F2 6 IENSURATION OF SURFACES. To find the Length of a Hyperbolic Curve-fig. 20. RULE.-As the transverse is to the conjugate, so is the conjugate to the parameter. To 21 times the parameter of the axis add 19 times the transverse, and to 21 times the parameter add 9 times the transverse, and multiply each of these sums by the quotient of the abscissa b a, divided by the transverse. To each of these two products add 15 times the parameter, and divide the former by the latter; multiply this quotient by the ordinate, and the product is the length of half the curve nearly. OF CYLINDRICAL RINGS. To find the Convex Surface of a Cylindrical Ring-fig. 54. RULE.-To the thickness of the ring a b add the inner diameter b c; multiply this sum by the thickness, and the product by 9.8696, and it will give the surface required. Tofind the Area of a Circular Ring-fig. 54. RULE. -The difference of the areas of the two circles will be the area of the ring. OF LUNES. 21. ee a -. To find the Area of a Lune-fig. 21. RULE -Find the areas of the two segments a d c b, a b e.from which the lune is formed, and their difference will be the area required. * OF CYCLOIDS. 22. s n / d To find the Area of a Cycloid-fig. 22. RULE.-Multiply area of generating circle a b c by 3, and the product is the area. * If semicircles be described on the three sides of a right-angled triangle as diameters, two lunes will be formed, their united areas being equal to the area of the'ri allgle. dam b4mn. MENSURATION OF SURFACES. 67 OF CYLINDERS. To find the Convex Sutface of a Cylinder-fig. 25. RULE.-Multiply the circumference by the length, and the prod. uct will be the surface. OF CONES OR PYRAMIDS.'o find the Convex Sutface of a Right Cone or Pyramid-figs. 31 and 33. RULE.-Multiply the perimeter or circumference of the base by the slant height, and half the product added to the area of the base will be the surface. To find the Convex Surface of a Frustrum of a Right Cone or Pyramidfigs. 32 and 34. RULE.-Multiply the sum of the perimeters of the two ends by the slant height or side, and half the product will be the surface. OF SPHERES. To find the Convex Surface of a Sphere or Globe-fig. 37. RULE -Multiply the diameter of the sphere by its circumference, and the product is the surface. OF CIRCULAR SPINDLES. To find the Convex Surface of a Circular Spindle —Jg. 45. RULE.-Multiply the length fc of the spindle by the radius o c of the revolving arc; multiply the said arc fa c by the central distance o e, or distanQe between the centre of the spindle and centre of the revolving arc. Subtract this last product from the former, double the remainder, multiply it by 3.1416, and the product is the surface. NOTE.-The same rule will serve for any zone or segment, cut off perpendicularly to the chord of the revolving are; in this case, then, the particular length of the part, and the part of the arc which describes it, must be taken, in lieu of the whole length and whole arc. BY MATHEMATICAL FORMULAE. LINES. CIRCLE. Ratio of circumference to diameter, p - 3.1416. 8c'-c Length of an arc = nearly; c the chord of the are, and c the chord of half the are. Length of 1 degree, radius being 1, =.0174533 " 1 minute, -=.0002909 " 1 second, =.0000048 ELLIPSE. Circumference a -90p2-(a2+b2) nearly, a and b being the axes. PARABOLA. Length of an arc, commencing at the vertex, = 2 -+be ) nearly, a being the abscissa, and b the ordinate. 68 MENSURATION OF SURFACES. QUADRILATERALS. Half the product of the diagonals X the sine of their angle. CIRCLE. pr'2; or diam. 2 X.78539816; or circum. 2 X.0795774. CYLINDER. Curved surfalce - height X perimeter of base. SPHERICAL ZONE OR SEGMENT. 2prh; or, the height of the zone or segment X the circumferenct of the sphere. CIRCULAR SPINDLE. 2p(rc- a,/ I -.c 2); a being the length of the arc, and c its chord, or the length of' the spindle. SPHERICAL TRIANGLE. s-1800 pr', 1-80 —; s being the sum of the three angles. ANY SURFACE OF REVOLUTION. 2prXI; or, the length of the generating eleln il x th-, (r-it-(1,1~1ference described by its centre of gravity. ILLUSTRATIONS. —Let a b c be the side of a cylinder, b r the radius; then a b c is the generating element, b the centre of gravity (of the line), and b r the radius of the circle described by a b c. a~ ----------—. g Then, if abc-= 10, br-5: 10X(5+SX i 3.1416) = 314.16. Parabola. vjr \x acX(2brXp), p being in this and all other'. instances = 3.1416, b the centre of gravity, / and b r the radius of its circumference. / MENSURATION OF SURFACES. 69 Or, take a uniform piece of board or thick pasteboard, and cut out the figure of which the area is required weigh both pieces together, and then the figure separately, and say, as the gross weight is to the entire surface, so is the weight of the figure to its surface. CAPILLARY TUBE. Let the tube be weighed when empty, and again when filled with alercury; let w be the difference of those weights in troy grains, and I the length of the tube in inches. Diameter =.019252V l. USEFUL FACTORS, in wihic] p represents the Circtlmference of a Circle whose Dianetko is 1. Then p - 3.1415926535897932384626+-1 4 - = 1.273239 p -- 6.283185307179-+ - =.079577 4p - 12.566370614359+ 4p /p = 1.772453 p -= 1.570796326794+',/p=.886226 _P. = 0.785398163397+ 2V/p= 3.544907 4p - 4.188790 V- =.797884 1. P 1p =.523598 =p.392699.= 564189 112p -.261799 360, 0_ 114.591559 pTP =.008726 P 1 —' = 2.094395 -.318309 i = 1.909859 0 p -.636619 = 3.0735 70 MENSURATION OF SURFACES. Examples in Illustration of the foregoing Rules. Required the area, 1. Of the rhombus, fig. 3, a c 12 feet 6 inches, and its height a b, 9 feet 3 inches. Aans. 115.625 feet. 2. Of the triangle a b c, fig. 5, a b being 10 feet, and c d 5 feet..Ins. 25 feet. 3. Of the triangle a b c, fig. 7, its three sides measuring respectively 24, 36, and 48 feet..lns. 418.282. 4. In the right-angled triangle a be, fig. 7, the base is 56, and the height 33; whAlt is the hypothenuse? Ans. 65. 5. If the hypothenuse of a triangle be 53, and the base 45, what is the perpendicular?.- ns. 28. 6. Required the area of the trapezium, fig. 8, the diagonal a c 84, the perpendiculars 21 and 28. ens. 2058. 7. Of the trapezoid, fig. 9, a b 10, d c 12, and a h 6 feet..ans. 66. 8. Of an octagon, the side being 5. 52 - 25X4.828427 = 120.710675./nss. 9. The length of a perpendicular from the centre to one of the sides of an octagon is 12; what is the radius of the circumscribing circle? 12X1.08 (table, page 60) = 12.96 Ans. 10. The radius of a circle being 12.96, what will be the length of one side of an inscribed octagon? 12.96X.765 (page 60) = 9.914./ns. 11. The length of the side of a decagon is 10; what is the radius of the circumnscribing circle 10OX1.618 (page 60) = 16.18.Atns. 12. The chord a b, fig. 12, is 48, and the versed sine c d 18; what is the length of the arc? By Rule 4, twice the chord of half the arc is 60, then 60.2 = 30, chord of half the arc, and 30X8 = 240 —48 = 192 -3 = 64.ns. 13. The diameter c C, fig. 12, is 50, and the versed sine c d 18; what is the length of the arc It By rule 6... 50X60+18X27 = 2514V/50X 18-= 900 - 30. Then 30X2 = 60X10OX18 = 10800+ 2514 = 4.2959+30X2 = 64.2959 Ans. 14. The diameter of a circle is 50, and the chord of half the arc 30; what is the length of the arc?.ns. 64.2959. 15. What is the area of a sector, the chord of the arc being 40, and the versed sine 151 Acts. 558.125. 16. The radius of a sector o b, fig. 13, is 20, and the degrees in its arc 22; what Is the area? Ins. 76.7947. 17. The radius o c is 10, and the chord a c 10; what is the area of the segment acb d, fig. 12 3 AIns. 59.36. 18. The greater chord, b d, fig. 14, is 96, the lesser, a c, 60, and the breadth 26; what is the area of the zone. AIns. 2136.75. 19. The sine of half the arc, fig. 29, is 7, the diameter of the cylinder 15, the cosine on eg, at the intersection of a c, 2.7, the versed sine 4.8, and the height, bg, 12; what is the convex surface? Ans. 196. 20. The height, ap, of an elliptic segment, fig. 16, is 10, and the axes 25 and 35 respectively; what is the area? 10 -35=.2857 tabular versed sine, and segment =.185153X35X25 = 162.0088 ens. 21. In the parabolic frustrum, a c df, fig. 17, the ends a c and df are 6 and 10, and the height b e is 5; what is the area? 103 —63 784 102-6 -64 = 12.25Xi of 5 = 40.8.Ans. 22. The abscissa c b, fig. 18, is 12, and its ordinate ab 6; what is the length o. a d? ains. 30.198. 23. The transverse and coljugate diameters of a hyperbola, fig. 19, are 100 ant 60, and the abscissa a b 60; what is the area..Ins. 4320. 24 What is the curve a c d of the hyperbola, fig. 20, the abscissa a b 40? Ins. 59.85. MENSURATION OF SURFACES. 71 25. The chord a c, fig. 21, is 19, the heights ed 6.9, and eb 2.4; what is the area of the lune?.ins. 65.3. 26. The generating circle ab c, fig. 22, is 4 inches diameter; what is the area of the cycloid bc d?.dns. 37.6992. 27. The base of a cone, fig. 31, is 3 feet, and the slant height 15 feet; what is the convex surface? Rnts. 70.686. 28. The thickness of a cylindric ring, fig. 54, is 3 inches, and the inner diameter 12 inches; what is the convex surface?./dns. 444.132. 29. What is the convex surface of a globe, fig. 37, 17 inches in diameter? -dns. 6.305 square feet. 30. WVhat is the surface of the circular spindle, fig. 45, the length fc 14.142, the radius o c 10, and the central distance o e 7.071 inches?.lns. 190.82 inches. 31. What is the surface of an octahedron, the linear side being 2 inches? 22X3.46410 (tabular surface) = 13.85640.Ans. 32. What is the convex surface of a cylinder, fig. 25, the diameter of the base a b 10, and the height b c 10 inches?.ins. 314.16. 724 AREAS OF THE SEGMENTS OF A CIRCLE. rTABLE of the Areas of the Segments of a Circle, the diameter of w2hith is Unity, and supposed to be divided into 1000 equal Parts. Versed Versed Se r ea I Versed Se -re. Versed Seg Area. Sine. Seg r. Sie. Seg. Aa. Sine. Se. rea. Sie. Seg...001.00004.055.01691.109.04638.163.08332.002.00011.056.01736.110.04700.164.0840f.003.00021.057.01783.111.04763.165.08480.004.00033.058.01829 D.112.04826.166.08554.005.00047.059.01876.113.04889 [.167.08628.006.00061.060.01923. 114.04952. 168.08703.007.00077.061.01971.115.05016.169.08778.008.00095.062.02019.116.05080..170.08853.009.00113.063.02068.117.05144.171.08928.010.00132.064.02116.118.05209.172.09004.011.00153.065.02165.119.05273.173.0907s.012.00174.066.02215 [.120.05338.174.09155.013.00196.067.02265.121.05403.175.09231.014.00219.068.02315.122.05468.176.09307.015.00243.069.02365.123.05534.177.09383.016.00268.070.02416.1241.05600.178.09460.017.00294.071.02468.125.05666.179.09536.018.00320.072.02519.126.05732.180.09613.019.00347.073.02571.127.05799.181.09690.020.00374.074.02623.128.05865.182.09767.021.00403.075.02676.129.05932.183.09844.022.00432.076.02728.130.05999.184.09922.023.00461.077.02782.131.06067.185.09999.024.00492.078.02835.132.06134.186.10077.025.00523.079.02889.133.06202.18'7.10155.026.00554.080.02943.134.06270.188.10233.027.00586.081.02997.135.06338.189.10311.028.00619.082.03052 t.136.06407.190.10390.029.00652.083.03107.137.06476.191.10468.030.00686.084.03162.138.06544.192.10547.031.00720.085.03218.139.06614.193.10626.032.00755.086.03274.140.06683.194.10705.033.00791.087.03330.141.06752.195.10784.034.00827.088.0338'7.142 [.06822.196.10863.035.00863.089).03444.143.06892.197.10943.036.00900.090.03501.144.06962.198.11022.037.00938.091.03558. 145.07032. 199.11102.038.00976.092.03616.146.07103.200.11182.039.01014.093.03674.147.07174.201.11262.040.0)1053.094.03732.148.07245.202.11342.041.01093.095.03790.149.07316.203.11423.042.01133.096.03849.150.07387.204.11503.043.01173.097.03908.151.07458.205.11584.044.01214.098.03968.152.07530. 206.11665.045.01255.099.04027.153.07602.207.11746.046.01297.100.04087.154-.07674.208.11827.047.01339.101.04147.155.07746.209.11908 048.01381.102.04208.156.07819.210.11989.049.01424.103.04268.157.07892.211.12071.050.01468.104.04329.1.58.07964.212.12152.051.01511.105.04390.159.08038.213.12234.052.01556.106.04452.160.08111.214.12316.053.01600.107.04513.161.08184.215.12398.054.01645.108.014575..162.08258.I 216 i.12481 AREAS OF THE SEGMENTS OF A CIRCLE. 73 TABLE-(Continued.) Versed Seg Aea.| Versed Seg. Area. Versed Seg. Aea g.Versed SArea. _ Sine. Sine. Sine Sine..217.12563.272.17286.327 j.22321.382.27580.218.12645.273.17375.328.22415.383.27677.219.12728.274.17464.329.22509.384.27774.220.12811.275.17554.330.22603.385.27872.221.12894.276.17643.331.22697.386.27969.222.12977.277.17733.332.22791.387.28066.223.13060.278.17822.333.22885.388.28164.224.13143.279.17912.334.22980.389.28261.225.13227.280.18001.335.23074.390.28359.226.13310.281.18091.336.23168.391.28456.227 *13394.282.18181.337.23263.392.28554.228.13478.283.18271.338.23358.393.28652.229.13562.284.18361.339.23452.394.28749.230.13646.285.18452.340.23547.395.28847.231.13730.286.18542.341.23642.396.28945.232.13815.287.18632.342.23736.397.29043.233.13899.288.18723.343.23831.398.29141.234.13984.289.18814.344.23926.399.29239.235.14068.290.18904.345.24021.400.29336.236.14153.291.18995.346.24116.401.29434.237.14238.292.19086.347.24212.402.29533.238.14323.293.19177.348.24307.403.29631.239.14409.294.19268.349.24402.404.29729.240.14494.295.19359.350.24498.405.29827.241.14579.296.19450.351.24593.406.29925.242.14665.297.19542.352.24688.407.30023.243.14751.298.19633.353 I.24784.408.30122.244.14837.299.19725.354.24880.409.30220.245.14923.300.19816.355.24975.410.30318.246.15009.301.19908.356.25071.411.30417.247.15095.302.20000.357.25167.412.30515.248.15181.303.20092.358.25263.413.30614.249.15268.304.20184.359.25359.414.30712.250.15354.305.20276.360.25455.415.30811.251.15441.306.20368.361.25551.416.30909.252.15528.307.20460.362.25647.417.31008.253.15614.308.20552.363.25743.418.31106.254.15701.309.20645.364.25839.419.31205.255.15789.310.20737.365.25935.420.31304.256.15876 311.20830.366.26032.421.31402.257.15963.312.20922.367.26128.422.31501.258.16051.313.21015.368.26224.423.31600,259 16138.314.21108.369.26321.424.31699.260.16226 315.21201.370.26417.425.31798.261.16314.316.21294.371.26514.426.31897.262.16401.317.21387.372.26611.427 1.31995.263.16489.318.21480.373.26707.428.32094.264.16578.319.21573.374.26804.429.32193.265 1.16666.320.21666.375.26901.430.32292.266.16754.321.21759.376 1.26998.431.32391.267.16843.322.21853.377.27095.432.32490.268.16931.323.21946.378.27192.433.32590.269.17020.324.22040.379.27289.434.32689.270.17108.325 1.22134.380.27386.435.32788.271.17197.326.22227.381.27483.436.32887 G 74 AREAS OF THE SEGMENTS OF A CIRCLE. TABLE —(Continued). Versed rea. Versed eg..Irea. Versed S eg rea. e Seg. Sine. e. Sine. Seg. re. Sine. Seg. Sine. rea..437.32986.453.34576.469.36171.485.37770.438.33085.454.34676.470.36271.486.37870.439.33185.455.34775.471.36371.487.37970.440.33284.456.34875.472.36471.488.38070.441.33383.457.34975.473.36571.489.38169.442.33482.458.35074.474.36671.490.38269.443.33582.459.35174.475.36770.491.38369.444.33681.460.35274.476.36870.492.38469.445.33781.461.35373.477.36970.493.38569.446.33880.462.35473.478.37070 494.38669.447.33979.463.35573.479.37170.495.38769.448.34079.464.35673.480.37276.496.38869.449.34178.465.35772.481.37370.497.38969.450. 34278.466.35872.482.37470.498.39069.451.343'77.467.35972.483.37570.499.39169-.452.34477.468.36072.484.37670 o.500.39269 USE OF THE ABOVE TABLE. To find the Area of a Segment of a Circle. RULE.-Divide the height or versed sine by the diameter of the circle, and find the quotient in the column of versed sines. Take the area noted in the next column, and multiply it by the square of the diameter, and it will give the area required. EXAMPLE.-Required the area of a segment; its height being 10, and the diameter of the circle 50 feet. 10+50=.2, and.2, per table,-.11182; then.11182X502 - 279.55.Ans. LENGTHS OF CIRCULAR ARCS. 75 TABLE of the Lengths of Circular Arcs. eight. Length. Height. Length. Height. Length. Height. Lengh..100 1.0265.1-56 1.0637.212 1.1158.268 1.1816 101 1.0270.157 1.0645.213 1.1169.269 1.1829.102 1.0275.158 1.0653.214 1.1180.270 1.1843.103 1.0281.159 1.0661.215 1.1190.271 1.1856.104 1.0286.160 1.0669.216 1.1201.272 1.1869.105 1.0291.161 1.0678.217 1.1212.273 1.1882.106 1 1.0297.162 1.0686.218 1.1223.274 1.1897 107 1.0303.163 1.0694.219 1.1233.275 1.1908.108 1.0308.164 1.0703.220 1.1245.276 1.1921.109 1.0314.165 1.0711.221 1.1256.277 1.1934.110 1.0320.166 1.0719.222 1.1266.278 1.1948.111 1.0325.167 1.0728.223 1.1277.279 1.1961.112 1.0331.168 1.0737.224 1.1289.280 1.1974.113 1.0337.169 1.0745.225 1.1300.281 1.1989.114 1.0343.170 1.0754.226 1.1311.282 1.2001.115 1.0349.171 1.0762.227 1.1322.283 1.2015.116.1.0355.172 1.0771.228 1.1333.284 1.2028 117 1.0361.173 1.0780.229 1.1344.285 1.2042.118 1.0367.174 1.0789.230 1.1356.286 1.2056.119 1.0373.175 1.0798.231 1.1367.287 1.2070.120 1.0380.176 1.080'7.232 1.1379.288 1.2083.121 1.0386.177 1.0816.233 1.1390.289 1.2097.122 1.0392.178 1.0825.234 1.1402.290 1.2120.123 1.0399.179 1.0834.235 1.1414.291 1.2124.124 1.0405.180 1.0843-.236 1.1425.292 1.2138.125 1.0412.181 1.0852.237 1.1436.293 1.2152.126 1.0418.182 1.0861.238 1.1448.294 1.2166.127 1.0425.183 1.0870.239 1.1460.295 1.2179.128 1.0431.184 1.0880.240 1.1471.296 1.2193.129 1.0438.185 1.0889.241 1.1483.297 1.2206.130 1.0445.186 1.0898.242 1.1495.298 1.2220.131 1.04-52.18'7 1.0908.243 1.1507.299 1.2235.132 1.0458.188 1.0917.244 1.1519.300 1.2250.133 1.0465.189 1.0927.24-5 1.1531.301 1.2264.134 1.0472.190 1.0936.246 1.1543.302 1.2278.135 1.0479.191 1.0946.247 1.1555.303 1.2292.136 [1.0486.192 1.0956.248 1.1567.304 1.2306.137 1.0493.193 1.0965.249 1.1579.305 1.2321.138 1.0500.194 1.0975.250 1.1591.306 1.2335.139 1.0508.195 1.0985.251 1.1603.307 1.2349.140 1.0515.196 1.0995.252 1.1616.308 1.2364.141 1.0522.197 1.1005.253 1.1628.309 1.2378.142 1.0529.198 1.1015.254 1.1640.310 1.2393,143 1.0537.199 1.1025.255 1.1653.311 1.2407 144 1.0544.2()0 1.1035.256 1.1665.312 1.2422 145 1.0552.201 1.1045.257 1.1677.313 1.2436.146 1.0559.202 1.1055.258 1.1690.314 1.2451.147 1.0567.203 1.1065.259 1.1702.315 1.2465.148 1.0574.204 1.1075.260 1.1715.316 1.2480.149 1.0582.205 1.1085.261 1.1728.317 1.2495.150 1.0590.206 1.1096.262 1.1740.318 1.2510.151 1.0597.207 1.1006.263'1.1753.319 1.2524.152 1.0605.208 1.1117.264 1.1766.320 1.2539.153 1.0613.209 1.1127.265 1.1778.321 1.2554.154 1.0621.210 1.1137.266 1.1791.322 1.2569.155 1.0629.211 1.1148.267 1.1804.323 1.2584 76 LENGTHS OF CIRCULAR ARCS. TABLE-(Continued). Height. Length. Height. Length. Height. Length. Height. Lengh..324 1.2599.369. 1.3307.413 1.4061.457 1.4870.325 1.2614.370[ 1.3323.414 1.4079 [.458 1.4889.326 1.2629.371 1.3340.415 1.4097.459 1.4908 327 1.2644.372 1.3356.416 1.4115.460 1.4927 328 1.2659.373 1.3373.417 1.4132.461 1.4946.329 1.2674.374 1.3390.418 1.4150.46,2 1.4965.330 1.2689.375. 1.3406.419 1.4168.463 1.4984.331 1.2704.376 1.3423.420 1.4186.464 1.5003.332 1.2720.377 1.3440.421 1.4204.465 1.5022.333 1.2735.378 1.3456.422 1.4222.466 1.5042.334 1.27'50.379 1.3473.423 1.4240.467 1.5061.335 1.2766.380 1.3490.424 1.4258.468 1.5080.336 1.2781.381 1.3507.425 1.4276.469 1.5099.337 1.2786.382 1.3524.426 1.4295.470 1.5119.338 1.2812.383 1.3541.427 1.4313.471 1.5138.339 1.282','.384 1.3558.428 1.4331.472 1.5157.340 1.2843.385 1.3574.429 1.4349.473 1.5176.341 1.2858.386 1.3591.430 1.4367.474 1.5196.342 1.2874.387 1.3608.431 1.4386.475 1.5215.343 1.2890.388 1.3625.432'1.4404.476 1.5235.344 1.2905.389 1.3643.433 1.4422.477 1.5254.345 1.2921.390 1.3660.434 1.4441.478 1.5274.346 1.2937.391 1.3677.435 1.4459.479 1.5293.347 1.2952.392 1.3694.436 1.4477.480 1.5313.348 1.2968.393- 1.3711.437 1.4496.481 1.5332.349 1.2984.394 1.3728.438 1.4514.482 1.5352.35 0 1.3000.395 1.3746.439 1.4533.483 1.5371.351 1.3016.396 1.3763.440 1.4551.484 1.5391.352 1.3032.397 1.3780.441 1.4570.485 1.5411.353 1.3047.398 1.3797.442 1.4588.486 1.5430.354 1.3063.399 1.3815 ].443 1.4607.487 1.5450.355 1.3079.400 1.3832.444/ 1.4626.488 1.5470.356 1.3095.401 1.3850.4456 -1.4644.489 1.5489,357 1.3112.402 1.3867.446. 1.4663.490 1.5509.358 1.3128.403 1.3885.447 1.4682.491 1.5529.359 1.3144.404 1.3902.448/ 1.4700.492 1.5549.360 1.3160.405 i 1.3920.449 1.4719.493 1.5569.361 1.3176.406 i 1.3937.450] 1.4738.494 1.5585.362 1.3192.407 1.3955.451[ 1.4757.495 1.5608.363 1.3209.408 1.3972.452. 1.4775.496 1.5628.364 1.3225.409 1.3990.453 1.4794.497 1.5648.365 1.3241.410 1.4008.454 1.4813.498 1.5668.366 1.3258.411 1.4025.455 1.4832.499 1.5688.367 1.3274.412 1.4043.456 1.4851.500 1.5708.368 1.3291 To find the Length of an Arc of a Circle by the foregoing Table RULE.-Divide the height by the base, find the quotient in the column of heights, and take the length of that height from the next right-hand column. Multiply the length thus obtained by the base of the arc, and the product wvill be the length of the arc required. ExAMrPIE. —What is the length of an arc of a circle, the span or base being 100 feet, and the height 25 feet? 25- 100 -.25, and.25, per table, gives 1.1591; which, being multiplied by 100 -115.9100, the length. LENGTH OF AN ELLIPTIC ARC. 77 NOTE. —When great accuracy is required, iJ; in the division of a height by the base, there should be a renzainder. Find the lengths of the curves fiom the two nearest tabular heights, and subtract the one length from the other. Then, as the base of the arc of which the length is required is to the remainder in the operation of division, so is the difference of the lengths of the curves to the complement required, to be added to the length. EXAMPLE.-What is the length of an arce of a circle, the base of which is 35 feet, aInd.the height or versed sine 8 feet? 8 —.35u.28 — ~,.28, 8=1.1333,.229 1.1344, 1.1333x35 = 39.6655, 1.1344X35 39.7040, 39.7040-39.6655 =.0385, difference of lengths. Hence, as 35: 20::.0385:.0220, the length for the remainder, and.0220+ 39.6655 _ 39.6875, and..6875X 12, for inches = 84, making the length of the are 39 feet 84 inches. To find the length of an Elliptic Curve which is less than half of the entire Figuzre. CGEOMETRICALLY.-Let the curve of?which the length is?-equired be a b c. Extend the versed sine b d to meet the centre of the curve in e. Draw the line c e, and from e, with the distance e b, describe b h; bisect c h mll i, and from e, with the radius e i, describe k i, and it is equal half the arc a b c. To find the 7emngth whezs the Curve is greater than half the entire Figure. RvLE:.-Find by the above problem the ourve of the less portion of the figure, and subtract it from the circumference of the ellipse, and the remainder will be the length of the curve required. G 2 G 2 78 LENGTHS OF SEMI-ELLIPTIC ARCS. TABLE of the Lengths of Semi-elliptic Arcs. Ieight. Length. Height. Length. Height. Length. Height. Length..100 1.0416. 315 1.2960. 545 1.6409.775 2.0187.101 1.042>6.320 1.3038.550 1.6488.2780 2.0 73.102 1.0436.325 1.3106.555 1. 6567. ~785 2.0360.103 1.0446.330 1,3175. 560 1.6646.790 2.0446.104 1.0456.335 1.3244.565 1.6725.795 2.0533.105 1.0466.340 1.3313!.570 1.6804 c.800 2.0620.110 1.0516.345 1.3383. 575 1.6883.805 2.0708.115 1.0567.350 1.3454.580 1.6963.o810 2.0795.120 1.0618 355 1.3525..585 1.7042.815 2.0883.125 1.0569.360 1. o3597.590 1.7123. 820 2.0971.130 1.0720.365 1.3659.595 1.7203. 825 2.1060.135 1.0773.370 1.3741.600 1.7283 ].830 2.1148.140 1.0825.375 1.381 i.605 1.7364.835 2.123',145 1.0879.380 1.3888.610 1.7444.840 2.1326.150 1.0933.385 1.3961.615 1.7525.845 2.1416.155 1.0989.390 1.4034.620 1.7606.850 2.1505.160 1.1045.395 1.4107.625 1.7687.855 2.1595.165 1.1106.400 1.4180.630 1.7768.860 2.1685 170 1.1157.405 1.4253.635 1.7850.865 2.1775.175 1.1213.410 1.4327.640 1.7931. 0. 2.1866.180 1.1270.4.15 1.4402.645 1.8013.875 2.1956.185 1.1327.420 1.4476 650 1.8094.880 2.2047,190 1.1384.425 1,4552.655 1.8176 ~.885 2.2139.195 1. 1442.430 1.4627.660 1.8258.890 2o9230,200 1.1501.435 1.4702.665 1.8340.895 2.2322,205 1.1.560.440 1.4778.670 1.8423. 900 2.2414.210 1.1620.445 1.4854 6'75 1a8505 ].905 2.2506.215 1.1680.450 1,4931.680 1.8587.910 2.2597 e220 1.1741.455 1.5008 o685 1.8670.915 2.2689 225 1 o1802.460 1.5084.690 1.8753 o 920 2.2780.230 1.18654.465 1.5161 695 1.8836. 925 2.2872.235 1.1926.470 1.5238.700 1.8919.930 2.2964.240 1.1989.475 1.531i o705 1.9002.935 2.3056.245 1.)2051.480 1.5394 710 109085.940 2,3148.250 1.2114.485 1.5472.715, 1.9169.945 2.3241.255 1.217 7.490 1.5550 720 1.9253.950 2.3335.260 1.2241.495 1.5629.725 1.9337.955 2.3429.265 1.2306.500 1.5709.730 1.9422.960 2.3524.270 1. 2371.505 1.5785.735 1.9506.o965 2.3619.275 1. 2436 o510 1.5863.740 1.9599. 9'70 2.3714.280 1.2t501.515 1.5941.745 1.9675 o 975 2.3810.285 }1,2567.520 1o6019.750 1.9760.980 2, 3906t:.290 1.2634.525 1.6097.755 1.9845.9835 24002'.295 1.2700 0530 1.6175 760 1.9931 o.990 2.4098.300 1.2I767.535 1,6253.765 2,0016.995 2.4194.305 1,2834.540 1.6331.770 2.0102.1000 2.4291.310 1.2901 To find tIe Lengtih of the Curve of a Right Seemi-f'lipzse. Proceed with the foregoing table by the rule for ascertaining the lengths of cirtular arcs, page 76. ExAniPLE.-What is the length of the curve of the arch of a bridge, the sptae being 70 feet, and the height 30.10 feet? 30.10 +70.430 - per table, 1.4627, and 1.4627X70 = 102.3890, the length rte quired. SEMI-ELLIPTIC ARCS. 79 When the Curve is not that of a Right Semi-Ellipse, the height being half of the Transverse Diameter. RULE.-Divide half the base by twice the height; then proceed as in the foregoing example, and multiply the tabular length by twice the height, and the product will be the length required. EXXAIPLE.-What is the length of the profile of arch (it being that of a semi-ed 4ipse), the height measuring 35 feet and the base 60 feet? 60.-2 = 30 —35X2=.428, the tabular length of which is 1.4597. Then, 1.4597X35X2=-102.1790, the length required. NOTE.- When the quotient is not given in the column of heigrhts, divide the difference between the two nearest heights by.5; multiply the quotient by the excess of the height given and the height in the table first above it, and add this susm to tHeh tabular area of the least height. Thus, if the height is 118,.115, per table, = 1.0567.120, " -1.0618.0051.-.5 =.00102 X (118 - 115) =.00, which, added to 1.0567 = 1.05976, the length for 118. 80 AREAS OF THE ZONES OF A CIRCLE. TABLE of the Areas of the Zones of a Circle. Height. Area. Height. Area. Height. Area. Height. Area..001.00100.115.11397.245.23480.375.33604 002.Q0300.120.11883.250.23915.380.33931.003.00300.125.12368.255.24346.385.34253.004.00400.130.12852.260.24775.390.34569.005:.00500.135.13334.265.25201.395.34879.010.01000.140.13814.270.25624.400.35182.015.01499.145.14294.275.26043.405.35479.020.01999.150.14772.280.26459.410.35769.025.02499.155.15248.285.26871.415.36051.030.02998.160.15722.290.27280.420.36326.035.03497.165.16195.295.27686.425.36594.040.03995.170.16667.300.28088.430.36853.045.04494.175.17136.305.28486.435.37104.050.04992.180.17603.310.28880.440 *37346.055.05489.185.18069.315.29270.445.37579.060.05985.190.18532.320.29657.450.37805.065.06482.195.18994.325.30039.4-55.38015.070.06977.200.19453.330.30416.460.38216.075.07472.205.19910.335.30790.465.38466.080.07965.210.20365.340.31159.470.38853.085.08458.215.20818.345.31523.475.38747.090.08951.220.21268.350.31883.480.38895.095.09442.225.21715.355.32237.485.39026.100.09933.230.22161.360.32587.490.39137.105.10422.235.22603.365.32931.495.39223.110.10910.240.23043 [.370.33270.500.39270 To find the Area of a Zone by the above Table. RULE 1.- Yhern the zone is greater than a part of a semicircle, take the height on each side of the diameter of the circle, of which it is a part; divide the heights by the diameter; find the respective quotients in the column of heights, and take out the areas opposite to them, multiplying the areas thus found by the square of the diameter or chord, and the products, added together, twill be the area required. NOTE.- When the quoetient is not given in the column of heights, divide the difference between the two nearest heights by 5, anld multiply the quotient by the ezcess between the height givene and the height in the table first above it, and add this surr to the tabular area of the least height. Thus, if the height is.333,.30416-.30790 =.00374 — 5 =.000748 X 3 (excess of 333 over 330) =.002244+.30416 —.306404, the area for 333. EXAMPLE.-What is the area of zone, the diameter of the circle being 100, and the heights respectively 20 and 10, upon each side of it? 20- 100 =.200, and 200, per table, -.19453X><1002 = 1945.3. 10-100 =.100, and 100, per table, -=.09933X 1002 - 993.3. Hence, 1945.3+993.3 = 2938.6 dns. RULE.-YWhlen the zonze is less than a semicircle, proceed as in rule 1 for one height. EXAMPLE.-What is the area of a zone, the longest chord being 10, and the height 4? 4- 10 -.400 =.35182X 102 = 35.182.Sns. MENSURATION OF SOLIDS 81 MENSURATION OF SOLIDS. 23. OF CUBES AND PARALLELOPIPEDONS. 24. _ _ C: 6c To find the Solidity of a Cube-fig. 23. RULE.-Multiply the side of the cube by itself, and that product again by the side, and this last product will be the solidity. To find the Solidily of a Parallelopipedon-fig,. 24. RULE.-Multiply the length by the breadth, and that product by the depth, and this product is the solidity. OF REGULAR BODIES. To find the Solidity of any Regqniar e Bo(d. RULE.-Multiply the tabular solidity in the following table by the cube of the linear edge, and the product is the solidity. TABLE of the Solidities of the Regqlar Bodies when the Li -ear' Edge is Number of Sides. Names. Solidities. 4 Tetrahedrion. 0 11785 6 Hexahedron. 1.00000 8 Octahedro n. 0.47140 12 Dodecahedron. 7.66312 20 Icosahedron. 2.18169 OF CYLINDERS, PRISMS, AND UNGULAS. 25. 26. a 27. ~~~ ~e. -628. 30 29. Tofind the Solidity of Cylinders, Prisms, and UngelOas-figs. 25, 26, and 27. RuLE.-Multiply the area of the base by the height, and the prod. uct is the solidity. 8MENSURATION OF SOLIDS. VTo find the Solidity of an Ungula, fig. 28, when the section passes obliquely through the cylinder, a b cd. RULE.-Multiply the area of the base of the cylinder by half the sum of the greater and less heights a e, cf of the ungula, and the product is the solidity. Whe/n t/he Section passes througlh the base of the Cylinder and one of its sides-fig. 29, abc. RULE.-From -l2 of the cube of the right sine a d, of half the are ag of the base, subtract the product of the area of the base, and the cosine df of said half arc.* Multiply the difference thus found by the quotient of the height, divided by the versed sine, and the product is the solidity. When the Section passes obliquely through both ends of the Cylindey, ad c e —fig. 30. RULE.-Find the solidities of the ungulas a d c e and d b c, and the difference is the solidity required (conceiving the section to be continued till it meets the side of the cylinder). NOTE. —For rules to ascertain the solidity of conical ungulas, see Ryan's BonnyS castle's Mensuration, page 136 (1839). OF CONES AND PYRAMIDS. 3i. 32. Tofind the Solid'ity of a Cone or Py-anzid-figs. 31 and 33. RULE. —Mulltiply the area of the base by the height c d, and ~ the product will be the content. To find the Solidity of the F'rustrum of a Cone-fig. 32. RuisE. —Divide the difference of the cubes of the diameters a b, c d of the two ends by the difference of the diameters; this quotient, multiplied by.7854, and again by I of the height, will give the solidity. To find the Solidity of the FrPustrum of a Pyramid-fig. 34. RuLE. —Add to the areas of the two ends of the frustrum the square root ofbeyhetd, and their prouct, and this sm, multiplied by of the height a b, will give the solidity. If the height of the base e less than radius, otherwise. add thems * If the height of the base be less than radius, otherwise. add them. MENSURATION OF SOLIDS. 83 OF WEDGES AND PRISTMOIDS. 35. 36. To find the Solidity of a Wedge-fig. 35. RULE.-TO the length of the edge of the wedge d e add twice the length of the back a b; multiply this sum by the height of the wedge if, and then by the breadth of the back c a, and 1 of the product will be the solid content. To find the Solidity of a Prisnoid-fig. 36. RULE.-Add the areas of the two ends a b c, d ef, and four times the middle section g h, parallel to them, together; multiply this sum by - of the height, and it will give the solidity. 37. 38. OF SPHERES. 39 6 \ To find the Solidity of a Sphere-fig. 37. RuL.E.-Multiply the cube of the diameter by.5236, and the product is the solidity. To find the Solidity of a Spherical Segment-fig. 38. RULE.-TO three times the square of the radius of its base a b, add the square of its height c b; then multiply this sum by the height. and the product by.5236. To find the Solidity of a Spherical Zone or iFrustrzum-fig. 39. RULE. —TO the sum of the squares of the radius of each end ab, cd, add 1 of the square of the height b d of the zone; and this sum, multiplied by the height, and the product by 1.5708, will give the solidity. 84 MENSURATION OF SOLIDS. OF SPIIEROIDS.* 40. eC__ 41,fc 204 e 43. 42. e 44. To find the Solidity of a Spheroid-fig. 40. RULE.-Multiply the square of the revolving axis c d by the fixed axis a b; the product, multiplied by.5236, will give the solidity. To find the Solidity of the Segment of a Spheroid-figs. 41 and 42. RULE.-When the base ef is circular, or parallel to the revolving axis c d, fig. 41. Multiply the fixed axis a b by 3, the height of the segment a g by 2, and subtract the one product from the other; then multiply the remainder by the square of the height of the segment, and the product by.5236. Then, as the square of the fixed axis is to the square of the revolving axis, so is the last product to the content of the segment. RULE.- ThCn the base ef is perpendicular to the revolving axis c d, fig. 42. Multiply the revolving axis by 3, and the height of the segment cg by 2, and subtract the one from the other; then multiply the remainder by the square of the height bf the segment, and the product by.5236. Then, as the revolving axis is to the fixed axis, so is the last product to the content. To find the Solidity of the MSoiddle Festr of a Spheroid-figs. 40. and 44. RULE.- Vhen the ends ef and g h are circular, or parallel to the revol. —ingaxiscpyd,fig.43. the square of the revolving axis c d by the fixed add the square of the diameter of either end, ef or g h; then multiply this sum by the length a b of the frustrum, and the product again by.26152368, and this will give the solidity. RULE.- When the ends ef and g h are elliptical, er perpendicular to the revolving axis c d, fig. 44. To twice the product of the transverse and conjugate diametersact the middle section a b, add the product of the transverse and conjugate of either end; multiply theis sum by the length Ik of the frustrum, and the product by.2618, and this will give the solidity. * Spheroids are either Prolate or Oblate. They are prolate when produced by the revolution of a semi-ellipse about its transverse diameter, and oblate whee produced product.5an ellipse revolving about its conugate diameter..:, an ellipse revolving about its conlugate diameter. MENSURATION OF SOLIDS.: 85 OF CIRCULAR SPINDLES. 46. -- To find the Solidity of a Circular Spindle —fig. 45 RULE.-Multiply the central distance o e by half the area of the e-evolving segment a c ef. Subtract the product from I of the cube fe of half the length; then multiply the remainder by 12.5664 (or four times 3.1416), and the product is the solidity. To find the Solidity of the Frtstrn'm, or Zone of a Circular Spindle — fig. 46. RIULE.-From the square of half the length hi of the whole spindle, take 3 of the square of half the length n i of the frustrum, and multiply the remainder by the remainder by the said half-length of the frustrum n; multiply the central distance o i by the revolving area* which generates the frustrum; subtract the last product from the former, and the remainder, multiplied by 6.2832 (or twice 3.1416), will give the solidity. 47. OF ELLIPTIC SPINDLES. To find the Solidity of an Elliptic Spindle-fig. 47. RULE. —To the square of the greatest diameter a b, add the square of twice the diameter ef at i of its length; multiply the sum by the length, and the product by.1309, and it will give the solidity nearly. To find the Solidity of, FPrustrum or Segment of an Elliptic Spindlefig. 48. RULE.-Proceed as in the last rule for this or any other solid formed by the revolution of a conic section about an axis, viz.: Add together the squares of the greatest and least diameters, a b, c d, and the square of double the'diameter in the middle, between the two multiply the sum by the length ef, and the product by.1309, and it will give the solidity. NoTE.-For all such solids, this rule is exact when the body is formed by the conic section, or a part of it, revolving about the axis of the section, and will always be very near when the figure revolves about another line. The area of the frustrum can be obtained by dividing its central plane into segments of a circle, and triangles or parallelograms. H 86 MIENSURATION OF SOLIDS. OF PARABOLIC CONOIDS AND SPINDLES. 49. 9 50. 51 e To find the Solidity of a Parabolic Conoid* —fig. 49. RvLE. —Multiply the area of the base d by half the altitude fg and the product will be the solidity. NOTE.-This rule will hold for any segment of the paraboloid, whether the base be perpendicular or oblique to the axis of the solid.'To find the Solidity of a Prmust'ruet of a Paraboloid-fig. 49. RULE.-Multiply the sum of the squares of the diameters a b and d c by the height ef, and the product by.3927. To find the Solidity of a Parabolic Spindle-fig. 50. RULE.-Multiply the square of the diameter a b by the length d ct and the product by.4188, and it will give the solidity. To find the Solidity of the Middle FPr'strnm of a Parabolic Spiillefig. 51. RULE.-Add together 8 times the square of the greatest diameter o d, 3 times the square of the least diameter ef, and 4 times the product of these two diameters; multiply the sum by the length a b, and the product by.05236, and it will give the solidity. OF HIYPERBf)LOIDS AND HYPERBOLIC CONOIDS. C 52. 53. To find the Solidity of a Hyperboloid-fig. 52. RULE.-To the square of the radius of the base a b, add the square of the middle diameter n m; multiply this sum by the height c r, and the product again by.5236, and it will give the solidity. * The parabolic conoid is = i its circumscribing cylinder. MENSURATION OF SOLIDS. 87 To.find the Solidity of the FruLstrum of a Hyperbolic Conoid-fig. 53. RULE.-Add together the squares of the greatest and least semidiameters as and d r, and the square of the whole diameter n m in the middle of the two; multiply this sum by the height r s, and the product by.5236, and it will give the solidity. OF CYLINDRICAL RINGS. 54. a dd To find the Solidity of a Cylindrical Ring —fig. 54. RULE.-To the thickness of the ring a b, add the inner diameter be; then multiply the sum by the square of the thickness, and the product by 2.4674, and it will give the solidity. BY MATHEMATICAL FORMULE. FRUSTRUM OF A RIGHT TRIANGULAR PRISM. The base X~3(h+h'+h"), h being the heights. FRUSTRUM OF ANY RIGHT PRISM. The base X its distance from the centre of gravity of the section. CYLINDRICAL SEGMENT. Contained between the base and an oblique plane passing through a diameter of the base, twice the height X the quotient of the square of the radius — 3; or 2h Ar2, r being the radius and h the height SPHERICAL SEGMENT. 6 (3r2' -h2), r being the radius of the base, and h' the height of the segment. SPHERICAL ZONE. p(h3R"+3r 2+h"), Rr being the radii of the bases. SPHERICAL SECTOR. Jr x the surface of the segment or zone. ELLIPSOID. -, a being the revolving diameter, and b the axis of revolution. 88 nMIENSURATION OF SOLIDS. PARABOLOID. 4 area of the base X the height. CIRCULAR SPINDLE. p(.c3-2sVr2-i-C2 ), S being the area of the revolving segment, and c its chord. ANY SOLID OF REVOLUTION. 2prs; or, the area of the generating surface X the circumference described by its centre of gravity. NOTE.-If bounded by a curved surface, find the area by the rule for irregular plane figures. To find a Cylinder of a Given Solidity (b) with the Least Surface. Let a - altitude, and d = diameter of base. pd~a pd2 Then - b, and p- sum of bases. 4' 2 4b Convex surface = pda, or d. pd2 4b Therefore Pd + - a minimum. CASK GAUGING. Casks are usually comprised under the following figures, viz.: 1. The middle frustrum of a spheroid. 2. The middle frustrum of a parabolic spindle. 3. The two equal frustrums of a paraboloid. 4. The two equal frustrums of a cone, and their contents can be computed by the preceding rules for these figures. To find the Content of a Cask by four Dimensions. PRULE.-Add together the squares of the bung and head diameters, and the square of double the diameter taken in the middle between the bung and head; multiply the sum by the length of the cask, and the product by.1309. Or, 1.1309 (d2+-D2~+2M2), I being the length, dD the head and bung diameters, and M a diameter midway between them. Or, 1(d+ M3 ), d D M being the areas of the diameters. 2D + d Or, I X area of 2D ULLAGE CASKS. Wheen the Cask is standing. I X (- + ), 1 being the height of the fluid. MENSURATION OF SOLIDS. 89 W7hen the Cask is on its bilge. RULE. —Divide the wet inches by the bung diameter, and opposite to the quotient in the column of versed sines, page 72, take the area of the segment; multiply this area by the content of the cask in inches or gallons, and the product by 1.25 for the ullage required. EXAMPLE.-What is the ullage of a cask that contains 25689 cubic inches, and has 8 inches of liquor on its bilge, the bung diameter 32 inches 1 Ans. 4930. Examnples in Illustration of the foregoing Rules. 1. The side of a cube a b c, fig. 23, is 5 inches; what is the solidity?./ins. 125. 2. The length of a parallelopipedon a b, fig. 24, is 8 inches, its depth be and breadth c d 4 inches; what is the solidity?.Ains. 128. 3. The base of a cylinder a b, fig. 25, is 30 inches, and the height b c 50 inches; what is the solidity n? J/s. 20.4531 cubic feet. 4. The sides of a prism a b, b c, fig. 26, are each 5 inches, and the height b d 10 inches; what is the solidity! s-1-5-1-5 2 — 7.5; then 7.5-5.7.5-5. 7.5-5 = 2.5. /7.5X2.5X2.5X2.5X0 - 108.6./nss. 5. The base of an ungula, fig. 27, is a semicircle, the radius 5 inches, and the height of the figure 25 inches; what is the solidity? / ans. 981.75. 6. The base of an ungula, fig. 28, is a circle, the radius 8 inches, and the heights of the sides 2 and 4 inches; what is the solidity. ains. 603.180. 7. The base of a cone, fig. 31, is 20 inches, and the height c d 24; what is the solidity? ns. 2513.28. 8. The diameter of the greater end c d of the frustrum of a cone, fig. 32, is 5 feet, the less end a b 3 feet, and the perpendicular height 9 feet; what is the solidity?./lss. 115.4538. 9. What is the solidity of the frustrum of a hexagonal pyramid, fig. 34, a side c d of the greater end being 4 feet, and one of the lesser end 3 feet, and the height a b 9 feet?.ns. 288.3864. 10. How many solid feet are there in a wedge, fig. 35, the base a b is 64 inches long, c a 9 inches, d e 42 inches, and the height df 28 inches? ins. 4.1319. 11. What is the solidity of a rectangular prismoid, fig. 36, the base being 12 by 14 inches, the top 4 by 6 inches, and the perpendicular height 18 feet? lns. 10.666. 12. What is the solidity of the sphere, fig. 37, the diameter being 17 inches? wJns. 2572.4468. 13. The segment of a sphere, fig. 38, has a radius a b of 7 inches for its base, and its height b c is 4 inches; what is the solidity? Aqns. 341.3872. 14. The greater and less diameters of a spherical zone, fig. 39, are each 3 feet, and the height d b 4 feet; what is the solidity in cubic feet? lAns. 61.7843. 15. In the prolate spheroid, fig. 40, the fixed axis ab is 100, and the revolving axis c d is 60; what is the solidity?.ns. 188496. 16. The segment of a spheroid, fig. 41, is cut from a spheroid, the fixed axis of which, ab, is 100 inches, the height of the segment ag is 10, and the revolving axis 60; what is the solidity of it? lns. 14660.8. 17. The fixed axis c d, fig. 42, is 60, the revolving axis 100, and the height of the segment cg 8 inches; what is its solidity? - nes. 9159.509. 18. What is the solidity of the middle frustrum of a spheroid, fig. 43, the middle diameter c d being 60, ef and gh each 36, and the length a b 80? 1cns. 177940,224. H2 90 MENSURATION OF SOLIDS. 19. In the middle frustrum efgh, fig. 44, an oblate spheroid, the diameters of the ends are 40 by 24 inches, the middle diameters are 50 and 30, and the height Ik is 18 inches; what is the solidity? fans. ]8661.104. 20. What is the solidity of a circular spindle, fig. 45, the distance o e 7.07 inches, the length fe 7.07 inches, and the radius o c 10 inches? danS. 210.96. 21. The length of an elliptic spindle a b, fig. 47, is 85, its greatest diameter c d 25, and the diameter ef, at ~ of Its length, 20 inches; what is its solidity? ans. 24756.4625. 22. What is the solidity of the parabolic conoid, fig. 49; its height gf is 60, and the diameter d c of its base 100 inches? odns. 235620. 23. What is the solidity of the parabolic frustrum a b c d, fig. 49, its diameters de 58, a b 30, and height ef 18 inches? ofnZs. 30140.5104. 24. What is the solidity of the parabolic spindle, fig. 50, a b being 40, and c d 100 inches Ao ns. 67008.8. 25. The middle frustrum of a parabolic spindle, fig. 51, has a b 60, c d 40, and ef 30 inches; what is its solidity? o nss. 63774.48. 26. In the hyperboloid, fig. 52, the height c r is 10, the radius as 12, and the middle diameter n m 15.8745; what is the solidity-? o?s. 2073.454691. 27. The frustrum of the hyperbolic conoid, fig. 53, has r s 12, a b 10, d e 6, anc n m 8.5 inches; what is the solidity? ofns. 667.59. 28. The thickness a b, fig. 54, of a cylindric ring is 3 inches, and the inner diame. ter c b 8 inches; what is the solidity? dans. 244.2726. 29. Required the solidity of an icosahedron, its linear edge being 2? sAns. 17.45352. 30. Required the content of a cask, the length being 40, the bung and head dli. ameters 24 and 32, and the middle diameter 28.75 inches..Ans. 25689.1'25. AREAS OF CIRCLES~ 91 AREAS OF CIRCLES, from 1 to 100. Diameter Area. Diameter. Area. Diameter. Area. Diameter. Area..00019 5. 19.635 12. 113.09 19. 283.62 1. 20.-629.D l115.46 2,87.27 3 0006 21.647 e 117.85.4 291.03 -.~00306 4 22.690. 120.27. 294.83 01227 1 23.758 1 122.71 Y298.64 02 24.850 4 125.18 4 302.48 16.02761.* 25.967 3 127.67 4 306.35.04908. 27.108. 130.19 7 310.24 5.07669 6. 28.274 13. 1332.73 2. 314.16 69. 4 29.464 135.29 318809 4 1. 4.1104. 30.679.4 137.88 4 32206 7.150 8 31.919 34 140.50 3 326.05 2 3.141.~ 33.183.~ 143.13. 330.06.1963. 34.471 2.a 145.80. 334.10 49 3.46 4 65.784 4 148.48 4 338.16 4.2485 47 4 424. - 4 9085.8 37. 122 2 151.20 e 342.25 3.3067 738.484 14. 153.93 21. 346.36.3712 ], 39.871 ( 156.69 350.49 3.76 4 41.581 152.48 4 3545.5 1.9 42.718.3 162.29 358.84 -I.5184. 44.178 1 165.13. 363.05 6013 3 45.663, 167.98. 367.28 47.173 43 170.87 371.54 11.6902 7 48.707 4 173.78 47 375.82..7854 8. 50.265 15. 1 76.71 22. 380.13 ~.9940. 4 51.848. 179.67. 5 384.46 [ 1.227.9 53.456 182.65. 388.82 1.484. 55.088 6 185.66.3 393.20 1.767.~ 56.745. 188.69. 397.60 8 2.073. 581.426 91.74.1 402.03.[ 2.405. 60.132.4 194.82 4 406.49 7 2.761 * 61.862.7 197.93 7 410.97 2 3.141 9. 63.617 16. 201.06 23. 415.47 13.546 I 65.396.} 204.21 420.00 I 3.976 I 67.200 4. 207.39 424.55 4.430 [ 69.029.3 210.59. 429.13 I 4.908. 70.882 213.82 433.73 5.411. 72.759 217.07. 438.30 4 5.939.[ 74.662. 220.35. 443.01 j 6.491 7! 76.588.7 223.65 447.69 3. 7.068 10. 78.539 17. 226.98 24. 452.39. 7.669 I. 80.515.8 230.33. 457.11 8.295.~ 82.516 4 233.70 461.86. s8.946 84.540. 237.10. 466.63 9.621 86,590 240.52 471.43.5 1*0.320 88.664. 243.97 476 25 i 11.044 / 90.762.4 247.45 481.10 r 11.793. 7 92.885. 250.94.7 485.97 8 12.566 11. 95.033 18. 254.46 25. 490.87 * 1 3.364 I 97.,05. 8 258.o1. 495.79. 14.186 8 99.402. $261.58. 500.74 115.033 * 101.62. 265.18. 505.71. 15.904.i 103.86. 268.80. 510.70 16.800 106.13.8 272.44. 515.72 17.720 8 108.43.P 276.11.~ 520.70 18.665 87 110.75.8 279.81. 8 525.83 92 AREAS OF CIRCLES. TAB LE-(Continued). Diameter Area. iDiameter. Area. Diametr. Area. Diameter Area.'6. 530.93 33. I 855.30 40. 1256.6 47. 1734.9,. 536.04'. 861.79 1264.5 1744. 1 541.18 A 1 868.30 8 1272.3.4 1753.4 546.35 874.84 1280.3 4 1762.7 1 551,54 8 881.41 2 1288.2 1772.0 * 556 76 888. 00 1296.2 1781.3 562.000 894.61 1304.2 1790.7 567.26.1 901.25 1312.2 100. 27. 572.55 34. 907.92 41. 1320.2 48. 1809.5.1 577.87 1 914.61 1328.3. 1818.9. 83.20 / 921.32. 1336.4. 1828.4 3 588.57 928.06 1344.5 1837.9 4 593.95 I 934.82. 1352. I 1847.4 - 599.37 941.60 1360.8 1856.9 a 604.80 948.41 3 1369.0 1866.5 4 610.26 [ 7 955.25 4 1377.2 4 1876.1 248. 615.75 35, 962.11 42. 1385.4 49. 1885.7 621.26 968.99 1393. 7 1895.3 626.79 975.90 1401.9 4 1905.0 632.35 982.84. 1410.2 1914.7 637.94 I 989.80 4 1418.6 4 1924.4 a 643 54 8 996.78 1426.9 1934.1!4 649.18 1003.7 8 1435. 3 1943. 9 7 654.83 1010.8 1443.7 4 1953.6 9. 660.52 36. 1017.8 43. 1452.2 50. 1963.5 4 666.22 1.024.9 8 1460.6 4 1973.3 4 671.95 8 1032.0 4 1469.1 8 1983.1:. 677.71. 1039.1 1477.6 1993.0 4 683.49.2 1046.3 4 1486.1 I 2002.9 689 29 1053.5 6 1494. 7 5 2012..4 695.12 1060.7 1503.3 4 2022.8 7 700.98. 1067.9 1511.9 4 2032.8 30. 706.86 37 1075.2 44. 1520.5 51. 2042. S 4 712 76 t 1 1082.4 2 1529.1 4 2052.8 t 718.69 8 1089.7 4 1537.8 4 2062.9 724 64 1097.1 4 1546.5 2072.9 4 730.61 1104.4 1555.2 2083.0 4 736.61.5 1111.8 4 1564.0 2093.2 4, 742.64 1119.2. 15'72.8. 2103.3 748.69 8 1126.6.7 1581.6 2113.5 31. 754.76 38 1134.1 45. 1590.4 52. 2123.7 760.86.8 1141.5.4 1599.2 4 2133.9 766.99. 1149.0 4 1608.1 4 2144.1. 4 773.14 1156.6. 1617.0 4 2154.4 4 779.31 1164.1 4 1625.9 4 2164.7 4 785.51 1171.7 1634.9 * 2175.0 4 791.73 1179.3. 1643.8 i 2185.4 8 797.97 1186.9 4 1652.8 2195.7 32. 804.24 39. 1194.5 46. 1661.9 53. 2206.1. 810.54. 1202.2.4 1670.9 4 2216.6 4 816.86 1209.9.* 1680.0 4 2227.0 823.21 e 1217.6 16839.1 2237.5 4 829 57 j 1225.4 i 1698.2 4 2248.0 4 835.97 1233.1 4 1707.3 2258.5 e; 842.39 s 1240.9 4 1716.5 4 2269.0 4 848 83 a 1248.7 7 1725.7 7 2279.6 AREAS OF CIRCLES 93 TABLE —(Continued ). Diameterl Area. Diameter. j Area. Diameter. Area. Diameter. Area. 54 2290.2 61. 2922.4 68. 3631.6 75. 4417.8 l 2300.8.4 2934.4 3645.0. 4432.6 1 2311.4. 2946.4 36584 4 4447.3 2322.1 I 2958.5 3671.8. 4462.1 i 2332.8. 2970.5. 3685.2 2 4476.9 2343.5 5 2982.6 3698.7 8 4491.8 2354.2.8 2994.7. 3712.2 4506.6 7 2365.0 7 3006.9 8 3725.7.7 4521.5 5S. 2375.8 62. 3019.0 69. 3739.2 76. 4536.4 1 2386.6. 3031.2 1 3752.8.2 4551 4 2397.4 8 3043.4 4 3766.4. 4566.3 4 2408.3 4 3055.7 a 3780.0 4581.3 1 2419.2 1 3067.9 4 3793.6 1 4596 3. 2430.1 3080.2 3807.3 4611.3 2441.0 3092.5 3821.0 4626.4 7 2452.0 7 3104.8 7 3834.7 4641 5 56. /2463.0 63. 3117.2 70:. 3848. 4 77. 4656 6 4 2474.0 [. 3129.6 [. 3862.2 4671.7 4 2485. 0 I 3142.0 8 3875.9 4686.9 2496 1 a 3154.4 3889.8 4702.1 1 2507.1' 3166.9 1 3903.6 3 4717.3 4 2518 2 3179.4 2 3917.4 4732.5 4 2529'4 3 3191.9 39.31 3 4747 7 7 25405 7 3204.4.7 3945.2 7 4763.0 57 2551 7 648 3216.9 71. 3959.2 78. 4778.3 4g 2562 9 3229.5 8 3973.1. 4793.7. 2574.1 * 3242.1 43987.1. 4809.0 4 2585.4 [. 3254.8. 4001.1. 4824.4 2596.7 1 3267.4 4015.1 4839.8.- 2608.0 3280.1 4029.2. 4855.2.2 2619.3 2 3292.8 / 4043.2 4870.7. 2630.7 7 3305.5,7 4067.3. 4886.1 58. 2642 0 65. 83318.3 72. 4071.5 79. 4901.6 4 26534 1 3331.0 a. 4085.6. 4917.2 4 2664.9 8 3343.8 4 4099.8 4 4932.7 2676.3 8 3356.7. 4114.0 4948.3 8 2687.8. 3369.5. 4128.2.2 4963.9. 2699.3. 3382.4 a 4142.5 4979.5 2. 2710.8 3395.3 8 4156.7. 4995.1 ~ 2722.4. 3408.2 7 41710 2 5010.8 59. 2733.9 66. 3421.2 73. 4185 3 80. 5026.5 4 2745.5.2 3434.1 4 4199.7.2 5042.2 4 2757.1.4 3447.1.~ 4214.1. 5058.0 * 2768.8 | 3460.1 a 4228.5 5073.7 2 2780.5. 3473.2.~ 4242.9. 5089.5 8 28792.2. 3486.3 2 4257.3 5105.4./ 2803.9.4 3499.3 1 4271.8 5121.2 7 2815.6.2 3512.5.7 4286.3.7 5137.1 60. 2827.4 67. 3525.6 74. 4300.8 81. 5153.0.2 2839.2. 3538.8 28 4315.3. 5168.9. 2851.0 4. 3552.0. 4329.9 5184.8 2 2862.8. 3565.2.4 4344.5. 5200.8. 2874.7. 3578.4. 4359.1. 5216.8 * 82886.6.| 3591.7. 4373.8 * 5232 8.2 2898.5. 3605.0 4 4388.4. 5248. 8.2 2910.5 [.1 3618.3 7 4403.1. 5264.9 O94 AREAS OF CIRCLES. TABLE-(Continued). ~eiameter. Area. Diameter. Area. Diameter re. Diameter._ Area. 82. 5281.0 87. 5944. 6 92. 6647.6 97. 7389.t -1 5297 1 8 5961.7, - 6665.7 9. 7408.8.4 5313!2 5978.9. 6683.8.4 7427 9 3. 5329.4. 5996.0 (. 6701.9. 7447.0. 5345.6! 6013.2. 6720.0. 7466.2.4 5361.8 ~ 6030.4. 6738.2 7485.3 ~. 5378.0 Q 6047.6 3 6756.4 7504.5.7 5394 3. 6064.8 7 6776.4.7 7523.7 83. 5410.6 88. 6082.1 93. 6792.9 98. 7542.9 5426.9 6. 099.4. A 6811.1. 7562.2.4 5443.2. 6116.7.4, 6829.4. 7581.5 4 5459.6 4 6134.0Q 6847.8 *e 7600.8 - 54-76 0 4 6151.4. 6866.1 -7620.1.4 5492.4 A 6168.8 2 6884.5 5 7639.4 55008.8 6186.2. 6902.9.9 7658.9 7 5525.3 * a 6203.6 6921.3. 7678.2 4.8 5541.7 89. 6221.1 94. 6939.7 99. 7697.7. 5558.2 6238.6 6958.2. 7717.1. 5574.8 4 6256.1.~ 6976 7 4 73fi6.6 5591.3 6273.6 * 6995.2 T 7756.1, 5607.9 6291.2 - 7013.8. 7775.6 5624.5 6308.8. 7032.3 7795.2 5641.1 3 6326.4.4 7050.9 4 7814.7 5657.8 6344.0 7 7069.5 s t834.3 85. 5674.5 90. 6361.7 95. 7088.2 100..785 3.9 5691. 2 6 379).4 7106.9. 5707.9 6397.1 4 7125.5 5 72416 6 414.8 71441.3 5741. 4 6432.6 [ 7163.0.4 5758 2 6450.4. 7181.8 5775 0 3 6468.2 3 7200.5. 5'91.9 7 6486.0 7 7219.4 8. 5808.8 91 6503.8 96. 7238.2. s75825.7 6521.7 * 7257.1 1 t58412.6 1 $ 6539.6 ] 7275.9 5859.5 3 6557.6.8 7294.9 4 5876.5 s 6575.5 I 7313.8 e 5993.5 6. fi593.5. 7332. 8. 5910.5 3 6611.5.4 7351.7 7. 5927.6 7 6629.5. 7370.7 CIRCUMFERENCES OF CIRCLES. CIRCUMFERENCES OF CIRCLES, from 1 to 100. Diarmeter Circumference. Diameter. Circumference.l Diameter. Circumference. Diameter. Circum.ferrex 4.0490 5. 15.70 12. 37.69 19. 59.69 __i. ].16.10. 38.09. 60.08 12.0981.1 1649 38.48.~ 60.47.1963. 16.88. 38.87 60.86.3926 ~ 17.27 4 39.27 4 61.2 5 8'90 i 17.67 4./ 39.66. 61.6;. 5890 18.06 4 40.05 62.04 ].7854.4 18'.45 * 40.44. 62.4': s5.9817 6. 18.84 13. 40.84 20. 62.83 [....7. 19.24 4 41.23 4 63.2A 9 1.178. 19.63 4 41.62 4 63.61 16 1.374 4 20.02! 42.01 I 64.01 1.7.~ ]20.42. 42.41.4 64.40 1.570 20.81 4 42.80 4 64.79 1. 767! 21.20./ 43.1.9 4 65.18 s5 7. 21.57 4 43.58 7. 65.58 g 1.963 7. 21.99 14. 43. 98 21. 65.97 f2.159. 22.38. 44.37 66.36 3 2.356 S.. 22.77 4. 44.76 4 66.75 313 56. 23.16.;3 45.16. 67.15 2.552: 23.56.~ 45.55. 67.54 7 2.748 23.95 4 45.94 67.93. 24.34 I. 46.33 6 68.32 i 2.945 24.74 7 46.73 4 68.72 3.141 8. 25.13 15. 47.12 22. 69.11. 3.534. 25.52 4. 47.51.. 69.50 3.927.4 25.91.4 47.90.4 69.90. 4.319 43 26.31.4 48.30. 70.29 4.712 4~ 26.70. 48.69.4 70.68. 5.105. 27.09! 49.08. 71.07 5.497.A 27.48 I. 49.48.4 71.47. 5.890. 27.88 / 49.87 7 71.86. 2. 6.283 9. 28.27 16./ 50.26 23. 72.25 4 6.675. 28.66. 50.65. 72.64 7.068 4 29.05.4 51.05 4 73.04. 7.461 29.45.A 51.44. 73.43.4 7.854, 29.84.4 51.83.4 73.82. 8.246 30.23.4 52.22. 74.21 8.639 3 30.63./ 52.62 /. 74.61.4 9.032. 31.02. 53.01.4 75.. 9.424 10. 31.41 17. 53.40 24. 75.394.] 9.817.~ 31.80. 53.79. 75.79. 10.21.$ 32.20.~ 54.19.~ 76.18. 10.60 8 32.59 54.58. 76.57. 10.99. 32.98.~ 54.97. 76.96 11.38 33.37. 55.37.* 77.36. 11.78 4 33.77. 55.76.4 77.75 12.17 34.16. 56.16 W 78.14 12.56 11. 34.55 18. 56.54 25. 78.54 4 12.95. 34.95.8 56.94.4 78.93 i 13.35 4 35.34.4 57.33.4 79.32. 13.74. 35.73. -57.72.4 79.71 i 14.13 36.12.4 58.11.4 80.10 ] 14.52. 36.52. 58.51.4 80.50. 14.92 4 36.91.4 58.90. 80.89.4 15.31 7 37.30.1 59.29. 81.28 isW 96 CIRCUMFERENCES OF CIRCLES. TABI,E-(Continued). Diamete Circumference Diameter. Circumference. Diameter. Circumforenee. Diame'er. Circuml r-c 26. 81.68 33. 103.6 40. 125.6 47 147.6 1 82.07.4 104. * 126.. 148. y 82.46. 104.4. 126.4 4 148.4,4 82.85 4 104.8.4 126.8.* 148.8.' 83.25.1 105.2.4 127.2 149.2 5 83.64! 105.6 *8 127.6 149.6 8- 84.03! 106. 3. 128..3 150. 7 84.43 7 106.4 7 128.4 7 150.4 27.8 84.82 34. 106.8 41. 128.8 48. 150.7.e 85.21. 107.2.! 129.1. 151.1.8 85.60 107.5 129.5.4 151.5 86..8 107.9.4 129.9 151.9, 86.39 4 108.3.8 130.3 e 152.3. s 86.78 * 108.7. 130.7 152.7 o. 87.17 3 109.1 8 131.1 3 153.1.7 87.57 109.5.7 131.5 153.5 28. 87 96 35. 109.9 42 131.9 49.~ 153.9 4 88.35 1 110.3 132.3 154.3 4 88s75 8. 110.7.4 132.7.4 154.7 3 89.14 111.1. 133.1 155.1 89.53 1. 111.5. 133.5 155.5 89.92 5 111.9.5 133.9 /4 155.9 90.32 4 112.3. 134.3 4 156.2 90.71 7 112.7 [ 134.6. 156.6 o9 8 91 10 36. 113. 43. 135. 50. 157. 91.49.8 113.4.1 135.4 I. 157.4 91.89. 113.8 1 135.8 157.8 4 92.28 114..2 8. 136.2 / 158.2 92.67. 114.6.4 136.6.4 158.6 93.06.8 115.. 137. 159. 93.46 3 115.4. 137.4 4 159.4 93.85 7 115.8 4 137.8.7 159.8 30') 94.24 37. 116.2 44. 138.2 151. 160.2 94.64 4 116.6!. 138.6.1 160.6.] 95.03 117.. 139. 161.. 95.42 117.4.- 139.4 *. 161.3 1 95.81 1 117.8.4 139.8.4 161.7 96.21 2 118.2.8 140.1.4 8 162.1 96.60 T. 118.6.3 140.5 4 162.5 7 96.99 7 118.9 7. 140.9 7. 162.9 0,8 9'7. 38 38. 119.3 45. 141.3 52. 163.3 97.78. 119.7. 141.7. 163.7 98.17 8 120.1.4 142.1. 164.1 98.56 120.5 - 142.5.8 164.5 98.96. 120.9.4 142.9 164.C 5 99.35,' 121.3.5 143.3 8 165.3 99.74 121.7.- 143.7. 165.7. 100.1.7 122.1.4 144.1 7. 166 1 3** 100.5 39, 122.5 46. 144.5 53. 166 5 o 1009 122.9 I. 144.9 * 166.8, 101.3. 123. 3. 145.2.4 167.2 101.7. 123.7 I.4 145.6 i.3 167.6 3 102.1 t 124.. 146.. 168.. 102.4 124.4.5 146.4.4 168.4 4 102.8. 124.8.4 146.8. 168.8 103.2.7 125.2. 147.2 7. 169.2 ~i~~~1 ~ CIRCTUMFERENCES OF CIRCLES. 97 TABLE —(Continued). 01ameter Circumference. Diameter. ircumference. Diameter. Circumference. Diameter. jCircumfer'ee 54, 169.6 61. 191.6 68. 213.6 I75. 235.6 - 170.. 192. I 214. * 236. ~ 170.4 8 192.4.4 214.4.4 236.4.4 170.8 4 192.8. 214.8 4 236.7 171.2 1 193.2 4 215.1. 237.1. 171.6 4 193.6 215.5. 237.5. 172. 8 193.9 215.9. 237.9.4 172.3. 194.3 4 216.3 7 238.3 55. 172.7 62. 194.7 69. 216.7 76. 238.7 ~ 173.1, 195.1 1 217.1 I 239.1 4 173.5 4 195.5 4. 217.5. 239.5 4 173.9 4. 195.9 4. 217.9, 239.9 ~ 174.3.~ 196.3.~ 218.3. 240.3 174.7.5 196.7. 218.7,4 240.7 175.1 4 197.1 4 219.1 8 241.1. 175.5 197.5 219.5 241.5 66'. 175.9 63. 197.9 70. 219.9 77. 241.9 176.3.1 198.3. 220.3. 242.2 4 176.7 4. 198.7 220.6.4 242.6 177.1.4 199.. 221..8 243. 177.5.~ 199.4. 221.4.4 243.4 177.8 2 199.8. 221.8. 243.8 178.2 / 200.2 8. 222.2.4 244.2 178.6 7 200.6.7 222.6 e4 244.6 67. 179. 64. 201. 71. 223. 78. 245. 4 179.4. 201.4. 223.4.] 245.4 179.8.4 201.8 4 223.8! 4 245.8 4 180.2 4 202.2 224.2 /. 246.2 L. 180.6 4. 202.6 224.6. 246.6 4 181.. 203. I 8 225. 8 247. i 181.4 a 203.4. 225.4 I 247.4 181.8 7 203.8. 225.8.I 247.7 58. 182.2 65. 204.2 72. 226.1 79. 248.1. 182.6. 204.5. 226.5 4. 248.5 4 182.9,4 204.9 s 226.9 l 4 248.9 183.3 4 205.3. 227.3. 249.3 183.7 1. 205.7.1 227.7.' 249.7 184.1 8 206.1 2. 228.1.85 250.1 184.5.] 206.5! 228.5'4 250.5 4 184.9. 206.9. I 228.9.8 250.9 59. 185.3 66. 207.3 73. 229.3 80. I251.3 4g 185.7 4. 207.7 4. 229.7.. 251.7 4 186.1. 208.1. 230.1 I 252.1 4 186.65.4 208.5.3 230.5. 252. " 4 186.9. 208.9. 230.9. 252.S * 187.3 209,3 4 231.3 R 253 2 187.7.4 209.7.i 231.6. 253.6 ~ 188.1. 210. 4 232.. 254. 60. 188.4 67. 210.4 74. 232.4 81. s 254.4. 188.8. 210.8 4 232.8. 254.8 189.2 4 211.2. 233.2 ].~ 255.2. 189.6. 211.6 4 233.6. 255.6.4 190.. 212.. 1 234.. 256. 190.4 212.4 234.4 256.4 4 190.8 4. 212.8 4 234.8 256.8 191.2. 213.2 4 235.2. 1 257.2 J -98 6CIRCUM.IFEi:ENCES OF CIRCLES. TABLE-(Continued). i lalmeter Circurnfreuce. Diameter. Circumference. Diameter. Circumference. Diameter. Circumfer'lc 82. 257 6 87 273.3 92. 289. 97. 304.7 258.'.1 273.7 1 289.4. 305.1 258 3 41 274.1.1 289.8. 305.5 258 7 * 274.4. 290.2.8 305.9.2 259.1'. 274.8.i 290.5 J 306.3 259.5 275.2 5 290.9 306. 6 259.9 275.6.8 291.3.i 307.. 260.3. 276..1 291.7 307.4 83. 260.7 88, 276.4 93. 292.1 98. 307.8 261 1. 276.8 I, 292.5 8 308.2 261. 5 277.2 8. 292.9.' 308.6 261 9 277.6 3 293.3 4 309.0 i 262. 3. 278. 293. 7. 309.4 262.7 8 278.4 I 294.1 5 309.8. 263.1 278.8. 294.5. 310.2' 263.5 a 279.2. 294.9 8 310.6 84 263 8 89 279.6 94. 295.3 99. 311.0 264~ 2. 279.9.- 295.7 311.4.i 26'41G,. ] 280.3 4 296. 4 311.8. 265. a 80.7 4 296.4 8 312.1 265. 4 281.1 * 296.8 4 312.5 265.2 8 8 281.5 297.2 312.9.s 266.2 281.9.-I 297.6 313.3 266.6 u8 282.3 4 298. 4 313.7 85 267. 90) 282.7 95. 298.4 100. 314.1 267.4 1 283.1 * 298.8. 267.8 283.5. 299.2 268.2 283.9. 299.6 2369 6.7 284.3 4 300. 2683.9 8 284.7 2 300.4 i 269.3 285.1 4 300.8:7 269.7 7 285.4. 301.2 86-. 270.1 91. 285.8 96. 301.5 * 270.5 286.2. 301.9. 270.9.1 286.6.1 302.3 271.3 287... 302. 7.+ 271.7 1. 27.4 4. 303.1 272.1 5 287.8.8 303.5 4. 272.s 5 288.2 4/ 303.9 272.9. 288.6 304.3 SQUARES, CUBES, AND ROOTS. 99 TABLE of Squares, Cubes, and Square and Cube Roots, of all Numbers from 1 to 1000. Number. Square. Cube. Square Root. Cube Root. 1 1 1 1. 1. 2 4 8 1.414213 1.259921 3- 9 27 1.732050 1.442250 4 16 64 2. 1.587401 5 25 125 2.236068 1.709976 6 36 216 2.449489 1.817121 7 49 343 2.645751 1.912933 8 64 512 2.828427 2. 9 81 729 3. 2.080084 10 100 1000 3.162277 2.154435 11 121 1331 3.316624 2.223980 12 144 1728 3.464101 2.289428 13 169 2197 3.605551 2.351335 14 196 2744 3.741657 2.410142 15 225 3375 3.872983 2.466212 16 256 4096 4. 2.519842 17 289 4913 4.123105 2.571282 18 324 5832 4.242640 1 2.62041 19 361 6859 4.358898 2.668402 20 400 8000 4.472136 2.714418 21 441 9261 4.582575 2.758923 22 484 10648 4.690415 2.802039 23 529 12167 4.795831 2.843867 24 576 13824 4.898979 I 2.884499 25 625 15625 5. 2.924018 26 676 17576 5.099019 2.962496 27 729 19683 5.196152 3. 28 784 21952 5.291502 3.036589 29 841 24389 5.385164 1 3.072317 30 900 27000 5.477225 3.107232 31 961 29791 5.567764 3.141381 32 1024 32768 5.656854 3.174802 33 1089 35937 5.744562 3.207534 34 1156 39304 5.830951 3.239612 35 1225 42875 5.916079 3.271066 36 1296 46656 6. 3.301927 37 1369 50653 - 6.082762 3.332222 38 1444 54872 6.164414 3.361975 39 1521 59319 6.244998 3.391211 40 1600 64000 6.324555 3.419952 41 1681 68921 6.403124 3,.448217 42 1764 74088 6.480740 3.476027 43 1849 79507 6.557438 3.503398 44 1936 85184 6.633249 3.530348 45 2025 91125 6.708203 3.556893 46 2116 97336 6.782330 3.583048 47 2209 103823 6.855654 3.608826 48 2304 110592 6.928203 3.634241 49 2401 117649 7. 3.659306 50 2500 125000 7.071067 3.684031 61 2601 132651 7.141428 3.708430 52 2704 140608 7.211102 3.732511 53 2809 148877 7.280109 3.756286 54 2916 157464 7.348469 3.779763 65 3025 166375 7.416198 3.802953 100 SQUARES, CUBES, AND ROOTS. TABLE-(Continued). Number. Square. Cube. Square Root. Cube Root. 56 3136 175616 7.483314 3.825862 57 3249 185193 7.5498334 3.84850] 58 3364 195112 7.615773 3.870877 59 3481 205379 7.681145 3.892996 60 3600 216000 7.745966 3.914867 61 3721 226981 7.810249 3.936497 62 3844 238328 7.874007 3.957892 63 3969 250047 7.937253 3.979057 64 4096 262144 8. 4. 65 4225 274626 8.062257 4.020726 66 4356 287496 8.124038 4.041240 67 4489 300763 8.185352 4.061548 68 4624 314432 8.246211 4.081656 69 4761 328509 8.306623 4.101566 70 4900 343000 8.366600 4.121285 71 5041 357911 8.426149 4.140818 72 5184 373248 8.485281 4.160168 73 5329 389017 8.544003 4.179339 74 5476 405224 8.602325 4.198336 75 5625 421875 8.660254 4.2171.63 76 5776 438976 8.717797 4.235824 77 5929 456533 8.774964 4.254321 78 6084 474552 8.831760 4.272659 79 6241 493039 8.888194 4.290841 80 6400 512000 8.944271 4.308870 81 6561 531441 9. 4.3267419 82 6724 551368 9.055385 4.344481 83 6889 571787 9.110433 4.362071 84 7056 592704 9.165151 4.379519 85 7225 614125 9.219544 4.396830 86 7396 636056 9.273618 4.414005 87 7569 658503 9.327379 4.431047 88 7744 681472 9.380831 4.447960 89 7921 704969 9.433981 4.464745 90 8100 729000 9.486833 4.481405 91 8281 753571 9.539392 4.497942 92 8464 778688 9.591663 4.514357 93 8649 804357 9.643650 4.530655 94 8836 830584 9.695359 4.546836 95 9025 857375 9.746794 4.562903 96 9216 884736 9.797959 4.577857 97 9409 912673 9.848857 4.594701 98 9604 941192 9.899494 4.610436 99 9801 970299 9.949874 4.626065 100 10000 1000000 10. 4.641589 101 10201 1030301 10.049875 4.657010 102 10404 1061208 10.099504 4.672330 103 10609 1092727 10.148891 4.687548 104 10816 1124864 10.198039 4.702669 105 11025 1157625 10.246950 4.717694 106 11236 1191016 10.295630 4.732624 107 11449 1225043 10 344080 4.747459 108 11664 1259712 10.392304 4.762203 109 11881 1295029 10.440306 4.776856 110 12100 1331000 10.488088 4.791420 111 12321 1367631 10.535653 4.805896 SQUARES, CUBES, AND ROOTS. 101 TABLE-(Continued). Number. Square. i Cube. Square Root. Cube Root. 112 12544 1404928 10.583005 4.820284 113 12769 1442897 10.630145 4.834588 114 12996 1481544 10.677078 4.848808 115 13225 1520875 10.723805 4.862944 116 13456 1560896 10.770329 4.876999 117 13689 1601613 10.816653 4.890973 118 13924.1643032 10.862780 4.904868 119 14161 1685159 10.908712 4.918685 120 14400 1728000 10.954451 4.932424 121 146411 1771561 11. 4.946088 122 14884 1815848 11.045361 4.959675 123 15129 1860867 11.090536 4.973190'24 15376 1906624 11.135528 4.986631 125 15625 1953125 11.180339 5. 126 15876 2000376 11.224972 5.013298 127 16129 2048383 11.269427 6.026526 128 16384 2097152 11.313708 5.039684 129 16641 2146689 11.357816 5.052774 130 16900 2197000 11.401754 5.065797 131 17161 2248091 11.445523 5.078753 132 17424 2299968 11.489125 5.091643 133 17689 2352637 11.532562 5.104469 134 17956 2406104 11.575836 5.117230 135 18225 2460373 11.618950 5.129928 136 18496 2515456 11.661903 5.142563 137 18769 2571353 11 704699 5.155137 138 19044 2628072 11.747344 5.167649 139 19321 2685619 11.789826 5.180101 140 19600 2744000 11.832159 5.192494 141 19881 2803221 11.874342 5.204828 142 20164 2863288 11.916375 5.217103 143 20449 2924207 11.958260 5.229321 144 20736 2985984 12. 5.241482 145 21025 3048625 12.041594 5.253588 146 21316 3112136 12.083046 5.265637 147 21609 3176523 12.124355 5.277632 148 21904 3241792 12.165525 5.289572 149 22201 3307949 12.206555 5.301459 150 22500 3375000 12.247448 5.313293 151 22801 3442951 12.288205 5.325074 152 23104 3511808 12.328828 5.336803 153 23409 3581577 12 369316 5.348481 154 23716 3652264 12.409673 5.360108 155 24025 3723875 12.449899 5.371685 156 24336 3796416 12.489996 5.383231 157 24649 3869893 12.529964 5.394690 158 24964 3944312 12.569805 5.406120 159 25281 4019679 12.609520 5.417501 160 25600 4096000 12.649110 5.428835 161 25921 4173281 12.688577 5.440122 162 26244 4251528 12.727922 5.451362 163 26569 4330747 12.767145 5.462556 164 26896 4410944 12.806248 5.473703 165 27225 4492125 12.845232 5.484806 166 27556 4574296 12.884098 5.495865 167 27889 4657463 12.922848 5.506879 12 U)i2 SQUARES, CUBES, AND ROOTS. TABLE-(Continued). Number. Square. Cube. Square Root. Cube Root. 168 28224 4741632 12.961481 5,.517848 169 28561 4826809 13. 5.528775 170 28900 4913000 13.038404 5.539658 171 29241 5000211 13.076696 5.550499 172 29584 5088448 13.114877 5.561298 173 29929 5177717 13.152946 5.572054 174 30276 5268024 13.190906 5.582770 175 30625 5359375 13.228756 5.593445 176 30976 5451776 13.266499 5.604079 177 31329 5545233 13.304134 5.614673 178 31684 5639752 13.341664 5.625226 179 32041 5735339 13.379088 5.635741 180 32400 5832000 13.416407 5.646216 181 32761 5929741 13.453624 5.656652 182 33124 6028568 13.490737 5.667051 183 33489 6128487 13.527749 5.677411 184 33856 6229504 13.564660 5.687734 185 34225 6331625 13.601470 5.698019 186 34596 6434856 13.638181 5.708267 187 34969 6539203 13.674794 5.718479 188 35344 6644672 13.711309 5.728654 189 35721 6751269 13.747727 5.738794 190 36100 6859000 13.784048 5.748897 191 36481 6967871 13.820275 5.758965 192 36864 7077888 13.856406 5.768998 193 37249 7189057 13.892444 5.778996 194 37636 7301384 13.928388 5.788960 195 38025 7414875 13.964240 5.798890 196 38416 7529536 14. 5.808786 197. 38809 7645373 14.035668 5.818648 198 39204 7762392 14.071247 5.828476 199 39601 7880599 14.106736 5.838272 200 40000 8000000 14.142135 5.848035 201 40401 8120601 14.177446 5.857765 202 40804 8242408 14.212670 5.867464 203 41209 8365427 14.247806 5.877130 204 41616 8489664 14.282856 5.886765 205 42025 8615125 14.317821 5.896368 206 42436 8741816 14.352700 5.905941 207 42849 8869743 14.387494 5.915481 208 43264 8998912 14.422205 5.924991 209 43681 9123329 14.456832 5.934473 210 44100 9261000 14.491376 5.943911 211 44521 9393931 14.525839 5.953341 212 44944 9528128 14.560219 5.962731 213 45369 9663597 14.594519 5.972091 214 45796 9800344 14.628738 5.981426 215 46225 9938375 14.662878 5.990727 216 46656 10077696 14.696938 6. 217 47089 10218313 14.730919 6.009244 218 47524 10360232 14.764823 6.018463 219 47961 10503459 14.798648 6.027650 220 48400 10648000 14.832397 6.036811 221 48841 10793861 14.866068 6.045943 222 49284 10941048 14.899664 6.055048 223 49729 11089567 14.933184 6.064126 SQUARES, CUBES, AND ROOTS. 103 TABLE-(Continued). tiouber. 1 Square. Cube. Square Root. Cube Root. 224 50176 11239424 14.966629 6.073177 225 50625 11390625 15. 6.082201 226 51076 11543176 15.033296 6.091199 227 51529 11697083 15.066519 6.100170 228 51984 11852352 15.099668 6.109115 229 52441 12008989 15.132746 6.118032 230 52900 12167000 15.165750 6.126925 231 53361 12326391 15.198684 6.135792 232 53824 12487168 15.231546 6.114634 233 54289 12649337 15.264337 6.153449 234 54756 12812904 15.297058 6.162239 235 55225 12977875 15.329709 6.171005 236 55696 13144256 15.362291 6.179747 237 56169 13312053 15.394804 6.188463 238 56644 13481272 15.427248 6.197154 239 57121 13651919 15.459624 6.205821 240 57600 13824000 15.491933 6.214464 241 58081 13997521 15.524174 6.223083 242 58564 14172488 15.556349 6.231678 243 59049 14348907 15.588457 6.240251 244 59536 14526784 15.620499 6.248800 245 60025 14706125 15.652475 6.257324 246 60516 14886936 15.684387 6.265826 247 61009 15069223 15.716233 6.274304 248 61504 15252992 1.5.748015 6.282760 249 62001 15438249 15.779733 6.291194 250 62500 15625000 15.811388 6.299604 251 63001 15813251 15.842979 6.307992 252 63504 16003008 15.874507 6.316359 253 64009 16194277 15.905973 6.324704 254 64516 16387064 15.937377 6.333025 255 65025 16581375 15.968719 6.341325 256 65536 16777216 16. 6.349602 257 66049 16974593 16.031219 6.357859 258 66564 17173512 16.062378 6.366095 259 67081 17373979 16.093476 6.374310 260 67600 17576000 16.124515 6.382504 261 68121 17779581 16.155494 6.390676 262 68644 17984728 16.186414 6.398827 263 69169 18191447 16.217274 6.406958 264 69696 18399744 16.248076 6.415068 265 70225 18609625 16.278820 6.423157 266 70756 18821096 16.309506 6.431226 267 71289 19034163 16.340134 6.439275 268 71824 19248832 16.370705 6.447305 269 72361 19465109 16.401219 6.455314 270 72900 19683000 16.431676 6.463304 271 73441 19902511 16.462077 6.4'71274 272 73984 20123648 16.492422 6.479224 273 74529 20346417 16.522711 6.487153 274 75076 20570824 16.6552945 6.495064 275 75625 20796875 16.583124 6.502956 276 76176 21024576 16.613247 6.510829 277 76729 21253933 16.643317 6.518684 278 77284 21484952 16.673332 6.526519 279 77841 21717639 16.703293 6.534335 104 SQUARES, CUBES, AND ROOTS, TABLE —(Continued). Number. Square. Cube. - Square Root. Cube Root. 280 78400 21952000 16.733200 6.542132 281 78961 22188041 16,763051 6.549911 282 79524 22425768 16.792855 6.557672 283 80089 22665187 16.822603 6.565415 284 80656 22906304 16.852299 1 6.573139, 285 81225 23149125 16.881943 6.580844 286 81796 23393656 16.911534 6.588531 287 82369 23639903 16.941074 6.596202 288 82944 23887872 16.970562 6.603854 289 83521 24137569 17. 6.611488 290 84100 24389000 17.029386 6.619106 291 84681 24642171 17.058722 6.626705 292 85264 24897088 17.088007 6.634287 293 85849 25153757 17.117242 6.641851 294 86436 25412184 17.146428 6.649399 295 87025 25672375 17.175564 6.656930 296 87616 25934336 17.204650 6,664443 297 88209 26198073 17.233687 6.671940 298 88804 26463592 1.7.262676 6.679419 299 89401 26730899 17.291616 6.686882 300 90001) 27000000 17.320508 6.694328 301 90601 27270901 17.349351 6.701758 302 91204 27543608 17.378147 6.709172 303 91809 27818127 17.406895 6.716569 304 92416 28094464 17.435595 6.723950 305 93025 28372625 1.7.4642,19 6.731316 306 93636 28652616 17.492855 6.738665 307 94249 28934443 17.521415 6.745997 308 94864 29218112 17.549928 6.753313 309 95481 29503629 17.578395 6.760614 310 96100 2979100G 17.606816 6.767899 311 96721 3008023i 1'7.635192 6.775168 312 97344 30371328 17.663521 6.782422 313 97969 30664297 17.691806 6.789661 314 98596 30959144 17.720045 6.796884 315 99225 31255875 17.748239 6.804091 316 99856 31554496 17.776388 6.811284 31'7 100489 31855013 17.804493 6.818461 318 101124 32157432 17.832554 6.825624 319 101761 324617a59 17.860571 6.8327'71 320 102400 32768000 17.888543 6.839903 321 103041 33076161 17.916472 6.847021 322 103684 33386248 17.944358 6.854124 323 104329 33698267 17.972200 6.861211 324 104976 34012224 18..6.868284 325 1051;25 34328125 18.027756 6.875343 326 106276 34645976 18.055470 6.882388 327 106929 34965783 18, 083141 6.889419 328 107584 35287552 18. 110770 6.896435 329 108241 35611289 18,138357 6.90:3436 330 108900 35937000 18.165902 6.910423 331 109561 36264-691 18.193405 6.917396 332 110224 36594368 1R.220867 6.924355 333 110889 36926037 14.248287 6.931300 334 111556 37259704 18.275666 6.938232 335 112225 37595375 18.303005 6.945149 SQUARES, CUBES, AND ROOTS. 105 TABLE-(Continued). Number. Square. Cube. Square Root. Cube Root. 336 112896 37933056 18.330302 6.952053 337 113569 38272753 18.357559 6.958943 338 114244 38614472 18.384776 6.965819 339 114921 38958219 18.411952 6.972682 340 115600 39304000 18.439088 6.979532 341 116281 39651821 18.466185 6.986369 342 116964 40001688 18.493242 6.993491 343 117649 40353607 18.520259 7. 344 118336 40707584 18.547237 7.006796 345 119025 41063625 18.574175 7.013579 346 119716 41421736 18.601075 7.020349 347 120409 41781923 18.627936 7.027106 348 121104 42144192 18.654758 7.033850 349 121801 42508549 18.681541 7.040581 350 122500 42875000 18.708286 7.047208 351 123201 43243551 18.734994 7.054003 352 1'23904 43614208 18.761663 7.060696 353 124609 43986977 18.788294 7.067376 354 125316 44361864 18.814887 7.074043 355 126025 44738875 18.841443 7.080698 356 126736 45118016 18.867962 7.087341 357 127449 45499293 18.894443 7.093970 358 128164 45882712 18.920887 7.100588 359 128881 46268279 18.947295 7.107193 360 129600 46656000 18.973666 7.113786 361 130321 47045881 19. 7.120367 362 131044 47437928 19.026297 7.126935 363 131769 47832147 19.052558 7.133492 364 132496 48228544 19.078784 7.140037 365 133225 48627125 19.104973 7.146569 366 133956 49027896 19.131126 7.153090 367 134689 49430863 19.157244 7.159599 368 135424 49836032 19.183326 7.166095 369 136161 50243409 19.209372 7.172580 370 136900 50653000 19.235384 7.179054 371 137641 51064811 19.261360 7.185516 372 138384 51478848 19.287301 7.191966 373 139129 51895117 19.313207 7.198405 374 139876 52313624 19.339079 7.204832 375 140625 52734375 19.364916 7.211247 376 141376 53157376 19.390719 7.217652 377 142129 53582633 19.416487 7.224045 378 142884 54010152 19.442222 7.230427 379 143641 54439939 19.467922 7.236797 380 144400 54872000 19.493588 7.243156 381 145161 55306341 19.519221 7.249504 382 145924 55742968 19.544820 7.255841 383 146689 56181887 19.570385 7.262167 384 147456 56623104 19.595917 7.268482 385 148225 57066625 19.621416 7.274786 386 148996 57512456 19.646882 7.281079 387 149769 57960603 19.672315 7.287362 388 150544 58411072 19.697715 7.293633 389 151321 58863869 19.723082 7.299893 390 152100 59319000 19.748417 7.306143 391 152881 59776471 19.773719 7.312383 X106 SQUARES, CUBES, AND ROOTS. TABLE-(Continued). Number. Square. Cube. Square Root. Cube RooL 392 153664 60236288 19.798989 7.318611 393 154449 60698457 19.824227 7.324829 394 155236 61162984 19.849432 7.331037 395 156025 61629875 19.874606 7.337234 396 156816 62099136 19.899748 7.343420 397 157609 62570773 19.924858 7.349596 398 158404 63044792 19.949937 7.355762 399 159201 63521199 19.974984 7.361917 400 160000 64000000 20. 7.368063 401 160801 64481201 20.024984 7.374198 402 161604 64964808 20.049937 7.380322 403 162409 65450827 20.074859 7.386437 404 163216 65939264 20.099751 7.392542 405 164025 66430125 20.124611 7.398636 406 164836 66923416 20.149441 7.404720 407 16564.9 67419143 20.174241 7.410794 408 166464 67911312 20.199009 7.416859 409 167281 68417929 20.223748 7.422914 410 168100 68921000 20.248456 7.428958 411 168921 69426531 20.273134 7.434993 412 ]69744 69934528 20.297783 7.441018 413 170569 70444997 20.322401 7.447033 414 171396 70957944 20.346989 7.453039 415 172225 71473375 20.371548 7.459036 416 173056 71991296 20.396078 7.465022 417 173889 72511713 20.420577 7.470999 418 174724 73034632 20.445048 7.476966 419 175561 73560059 20.469489 7.482924 420 176400 74088000 20.493901 7.488872 421 177241 74618461 20.518284 7.494810 422 178084 75151448 20.542638 7.500740 423 178929 75686967 20.566963 7.506660 424 179776 76225024 20.591260 7.512571 425 180625 76765625 20.615528 7.518473 426 181476 77308776 20.639767 7.524365 427 182329 77854483 20.663978 7.530248 428 183184 78402752 20.688160 7.536121 129 184041 78953589 20.712315 7.541986 430 184900 79507000 20.736441 7.547841 431 185761 80062991 20.760539 7.553688 432 186624 80621568 20.784609 7.559525 433 187489 81182737 20.808652 7.565353 434 188356 81746504 20.832666 7.571173 435 189225 82312875 20.856653 7.576984 436 190096 82881856 20.880613 7.582786 437 190969 83453453 20.904545 7.588579 438 191844 84027672 20.928449 7.594363 439 192721 84604519 20.952326 7.600138 440 1 93600 85184000 20.976177 7.605905 441 194481 85766121 21. 7.611662 442 195364 86350388 21.023796 7.617411 443 196249 86938307 21.047565 7.623151 444 197136 87528384 21.071307 7.628883 445 198025 88121125 21.095023 7.634606 446 198916 88716536 21.118712 7.640321 447 199809 89314623 21.142374 7.646027 SQUARES, CUBES, AND ROOTS. l07 TABLE —(Continued). Num'aer. Square. Cube. Square Root. Cube Root. 448 200704 89915392 21.166010 7.651725 449 201601 90518849 21.189620 7.657.414 450 202500 91125000 21.213203 7.663094 451 203401 91733851 21.236760 7.668766 452 204304 92345408 21.260291 7.674430 453 205209 92959677 21.283796 7.680085 454 206106 93576664 21.307275 7.685732 455 207025 94196375 21.330729 7.691371 456 207936 94818816 21.354156 7.6i97002 457 208849 95443993 21.377558 7.702624 458 209764 96071912 21.400934 7.708238 459 210681 96702579 21.424285 7.713844 460 211600 97336000 21.447610 7.719442 461 212521 979'72181 21.470910 7.725032 462 213144 98611128 21.494185 7.730614 463 214369 99252847 %1.517434 7.736187 464 I215296 99897344 21.540659 7.'741753 465 216225 100544625 21.563858 7.747310 466 217156 101194696 21.587033 7.752860 467 218089 101847563 21.610182 7.758402 468 219024 102503232 21.633307 7.763936 469 219961 103161709 21.656407 7.769462 470 220900 103823000 21.679483 7.774980 471 221841 104487111 21.702534 7.780490 472 222784 105154048 21.725561 7.785992 473 223729 105823817 21.748563 7.791487 474 224676 106496424 21.771541 7.796974 475 225625 107171875 21.794494 7.802453 476 226576 107850176 21.817424 7.807925 477 227529 108531333 21.840329 7.813389 478 228484 109215352 21.863211 7.818845 479 229441 109902239 21.886068 7.824294 480 230400 110592000 21.908902 7.829735 481 231361 111284641 21.931712 7.835168 482 232324 111980168 21.954498 7.840594 483 233289 112678587 21.977261 7.846013 484 234256 113379904 22. 7.85'1424 485 235225 114084125 22.022715 7.856828 486 236196W 11-4791256 22.045407 7.862224 487 237169 115501303 22.068076 7.867613 488 238144 116214272 22.090722 7.872994 489 239121 116930169 22.113344 7.878368 490 240100 117649000 22.135943 7.883734 491 241081 118370771 22.158519 7.889094 492 242064 119095488 22.181073 7.894446 493 243049 119823157 22.203603 7.899791 494 244036 120553784 22.226110 7.905129 495 245025 121287375 22.248595 7.910460 496 246016 122023936 22.271057 7.915784 497 247009 122763473 22.293496 7.921100 498 248004 123505992 22.315913 7.926408 499 249001 124251499 22.338307 7.931710 500 250000 125000000 22.360679 7.937005 501 251001 125751501 22.383029 7.942293 602 252004 126506008 22.405356 7.947573 503 253009 127263527 22.427601 7.952847 108 SQUARES, CUBES, AND ROOTS. TABLE-(Continued). Number. Square. Cube. Square Root. Cube Root. 504 254016 128024064 22.449944 7.958114 505 255025 128787625 22.472205 7.963374 506 256036 129554216 22.494443 7.968627 507 257049 130323843 22.516660 7.973873 508 258064 131096512 22.538855 7.979112 509 259081 131872229 22.561028 7.984344 510 260100 132651000 22.583179 7.989569.511 261121 133432831 22.605309 7.994788 512 262144 1]34217728 22.627417 8. 513 263169 135005697 22.649503 8.005205 514 264196 135796744 22.671568 8.010403 515 265225 136590875 22.693611 8.015595 516 266256 137388096 22.715633 8.020779 517 26'7289 138188413 22.737634 8.025957 518 268324 138991832 22.759613 8.031129 519 269361 139798359 22.781571 8.036293 520 270400 140608000. 22.803508 8.041451 521 271441 141420761 22.8254:24 8.046603 522 272484 142236648 22.847319 8.051748 523 273529 143055667 22.869193 8.056886 524 274576 143877824 22.891046 8.062018 525 275625 144703125 22.912878 8.067143 526 276676 145531.576 22.934689 8.072262 527 277729 146363183 22.956480 8.077374 528 278784 147197952 22.978250 8.082480 529 279841 148035889 23. 8.087579 530 280900 148877000 23.021728 8.092672 531 281961 149721291 23.043437 8.097758 532 283024 150568768 23.065125 8.102838 533 284089 151419437 23.086792 8.107912 534 285156 152273304 23.108440 8.112980 535 286225 153130375 23.130067 8.118041 536 287296 153990656 23.151673 8.123096 537 288369 154854153 23.173260 8.128144 538 289444 1557208'72 23.194827 8.133186 539 290521 156590819 23.216373 8.138223 540 291600 157464000 23.237900.8.143253 541 292681 158340421 23.259406 8.148276 542 293764 159220088 23.280893 8.153293 543 294849 160103007 23.302360 8.158304 544 295936 160989184 23.323807 8.163309 545 297025 161878625 23.345235 8.168308 546 298116 162771336 23.366642 8.173302 547 299209 163667323 23.388031 8.178289 548 300304 164566592 23.409399 8.183269 549 301401 165469149 23.430749 8.188244 550 302500 166375000 23.452078 8.193212 551 303601 167284151 22.473389 8.198175 552 304704 168196608 23.494680 8.203131 553 305809 169112377 23.515952 8.208082 554 306916 170031464 23.537204 8.213027 555 308025 170953875 23.558438 8.217965 556 309136 171879616 23.579652 8.222898 557 310249 172808693 23.600847 8.227825 558 311364 173741112 23.622023 8.232746 559 312481 174676879 23.643180! 8.237661 SQUARES, CUBES, AND ROOTS. 109 TABLE-(Continued). Numbp. Square. Cube. Square Root. Cube Ioot. 560 313600 175616000 23.664319 8.242570 561 314721 176558481 23.685438 8.247474 562 315844 177504328 23.706539 8.252371 563 316969 178453547 23.727621 8.257263 564 318096 179406144 23.748684 8.262149 565 319225 180362125 23.769728 8.267029 566 320356 181321496 23.790754 8.271903 567 321489 182284263 23.811761 8.276772 568 322624 183250432 23.832750 8,281635 569 323761 184220009 23.853720 8.286493 570 324900 185193000 23.874672 8.291344 571 326041 186169411 23.895606 8.296190 572 327184 187149248 23.916521 8.301030 573 328329 188132517 23.93 7418 8.305865 574 329476 189119224 23.958297 8.310694 575 330625 190109375 23.979157 8.315517 576 331776 191102976 24. 8.320335 577 332929 192100033 24.020824 8.325147 578 334084 ]193100552 24.041630 8.329954 579 335241 194104539 24.062418 8.334755 580 336400 195112000 24.083189 8.339551 581 337561 196122941 24.103941 S.344341 582 338724 197137368 24.124676 8.349125 583 339889 198155287 24.145392 8.353904 584 341056 199176704 24.166091 8.358678 585 342225 200201625 24.186773 8.363446 586 343396 201230056 24.207436 8.368209 587 344569 202262003 24.228082 8.372966 588 345744 203297472 24.248711 8.37'7718 589 346921 204336469 24.269322 8.382465 590 348100 205379000 24.289915 8.387206 591 349281 206425071 24.310491 8.391942 592 350464 207474688 24.331050 8.396673 593 351649 208527857 24.351591 8.401398 594 352836 209584584 24.372115 8.406118 595 354025 210644875 24.392621 8.410832 596 355216 211708736 24.413111 8.415541 597 356409 212776173 24.433583 8.420245 598 357604 213847192 24.454038 8.424944 599 358801 214921799 24.474476 8.429638 600 360000 216000000 24.494897 8.434327 601 361201 217081801 24.515301 8.439009 602 362404 218167208 24.535688 8.443687 603 363609 219256227 24.556058 8.44836) 604 364816 220348864 24.576411 8.453027 605 366025 221445125 24.596747 8.457689 606 367236 222545016 24.617067 8.462347 607 368449 223648543 24.637370 8.466999 608 369664 224755712 24.657656 8.471647 609 370881 225866529 24.677925 8.476289 610 372100 226981000 24.698178 8.480926 611 373321 228099131 24.718414 8.485557 612 374544 229220)928 24.738633 8.490184 613 375769 230346397 24.758836 8.494806 614 376996 23 1475544 24.779023 8.499423 615 378225 232608375 I24.799193 8.504034 110 sqUAREs, CUBES, AND ROOTS. TABLE —(Continuted). Number. Square. Cube. Square Root. Cube Root. 616 379456 233744896 i24.81947 8.508641 617 380689 234885113 24.839484 8.513243 618 381924 236029032 24859605 8.517840 619 383161 237176659 1 24.879710 8.522432 620 384700 238328000 24.899799 8.527018 621 385641 239483061 24.919871 8.531600 622 386884 240641848 24.939927 8.536177 623 388129 241804367 24.959967 8.540749 624 389376 242970624 24.979992 8.545317 625 390625 244140625 25. 8.549879 626 391876 245314376 25.019992 8.554437 627 393129 246491883 I 25.039968 8.558990 628 394384 247673152 25.059928 8.563537 629 395641 248858189 25.079872 8.568080 630 396900 250047000 25.099800 8.572618 631 398161 251239591 25.119713 8.577152 632 399424 252435968 25.139610 8.581680 633 400689 253636137 25.159491 8.586204 634 401956 254840104 25.179356 8.590723 635 403225 256047875 25.199206 8.595238 636 404496 257259456 25.219040 8.599747 637 405769 258474853 25.238858 8.604252 638 407044 259694072 25.258661 8.608752 639 408321 260917119 25.2'78449 8.613248 640 409600 262144000 25.298221 8.617738 641 410881 263374721 25.317977 8.622224 642 4112164 264609288 25.337718 8.626706 643 413449 265847707 25.357444 8.631183 644 414736 267089984 25.377155 8.635655 645 416025 268336125 25.396850 8.640122 646 417316 269586136 25.416530 8.644585 647 418609 270840023 25.436194 8.649043 648 419904 272097792 25.455844 8.653497 649 * 421201 2'73359449 25.475478 8.657946 650 422500 274625000 25.495007 8.662301 651 423801 275894451 25.514701 8.666831 652 425104 2'77167808 25.534290 8.671266 653 426409 278445077 25.553864 8.675697 654 427716 279726264 25.573423 8.680123 (55 429025 281011375 25.592967 8.684545 656 430336 282300416 25.612496 8.688963 657 431649 283593393 25.632011 8.693376 658 432964 284890312 25.651510 8.697784 659 434281 286191179 25.670995 8.702188 660 435600 287496000 25.690465 8.706587 661 436921 288804781 25.709920 8.710982 662 438244 290117528 25.720360 8.715373 663 439569 291434247 25.748786 8.719759 664 440896 292754944 25.768197 8.724141 665 442225 294079625 25.787593 8.728518 666 443556 295408296 25.806975 8.732891 667 444889 296740963 25.826343 8.737260 668 446224 298077632 25.845696 8.741624 669 447561 299418309 25.865034 8.745984 670 448900 300763000 25.884358 8.750340 671 450241 302111711 25.903667 8.754691 SQUARES, CUBES, AND ROOTS. 111 TABLE —(Continued). Number. Square. Cube. Square Root. Cube Root. 672 451584 303464448 25.922962 8.759038 673 452929 304821217 25.942243 8.763380 674 454276 306182024 25.961510 8.767719 675 455625 307546875 25.980762 8.772053 676 456976 308915776 26. 8.776382 677 458329 310288733 26.019223 8.780708 678 459684 311665752 26.038433 8.785029 679 461041 313046839 26.057628 8.789346 680 462400 314432000 26.076809 8.793659 681 J463761 315821241 26.095976 8.797967 682 465124 317214568 26.115129 8.802272 683 466489 318611987 26.134268 8.806572 684 467856 320013504 26.153393 8.810868 685 469225 321419125 26.172504 8.815159 686 470596 322828856 26.191601 8.819447 687 |471969 3242~125703 26.210684 8.'823730 688 473344 325660672 26.229754 8.828009 689 474721 327082769 26.248809 8.832285 690 476100 328509000 26.267851 8.836556 691 477481 329939371 26.286878 8.840822 692 478864 331373888 26.305892 8.845085 693 480249 332812557 26.324893 8.849344 694 481636 334255384 26.343879 8.853598 695 483025 335702375 26.362852 8.857849 696 484416 337153536 26.381811 8.862095 697 485809 338608873 26.400757 8.866337 698 487204 340068392 26.419689 8.870575 699 488601 341532099 26.438608 8.874809 700 490000 343000000 26.457513 8.879040 701 491401 344472101 26.476404 8.883266 702 492804 345948088 26.495282 8.887488 703 494209 347428927 26.514147 8.891706 704 495616 348913664 26.532998 8.895920 705 497025 350402625 26.551836 8.900130 706! 498436 351895816 26.570660 8.904336 707 499849 353393243 26.589471 8.908538 708 501264 0354894912 26.608269 8.912736 709 502681 356400829 26.627053 8.916931 710 504100 357911000 26.645825 8.921121 711 505521 359425431 26.664583 8.925307 712 506944 360944128 26.683328 8.929490 713 508369 362467097 26.702059 8.933668 714 509796 363994344 26.720778 8.937843 715 511225 365525875 26.739483 8.942014 716 512656 367061696 26.758,176 8.'946180 717 514089 368601813 26.776855 8.950343 718 515524 370146232 26.795522 8.954502 719 516961 371694959 26.814175 8.958658 720 518400 373248000 26.832815 8.962809 721 519841 374805361 26.851443 8.966957 722 521284 376367048 26.870057 8.971100 723 522729 377933067 26.888659 8.975240 724 524176 379503424 26.907248 8.979376 725 525625 381078125 26.925824 8.983508 726 527076 382657176 26.944387 8.987637 727 528529 384240583 26.962937 8.991762 112 SQUARES, CUBES) AND ROOTS. TABL.E —(Coniinued). Number Square. Cube. Square Root. Cube Root. 728 529984 385828352 26.981475 8.995883 729 531441 387420489 27. 9. 730 532900 389017000 27.018512 9.004113 731 534361 390617891 27.037011 9.008222 732 535824 392223168 27.055498 9.012328 733 537289 393832837 27.073972 9.016430 734 538756 395446904 27.092434 9.020529 735 540225 397065375 27.110883 9.024623 736 541696 398688256 27.129319 9.028714 737 543169 400315553 27.147743 9.032802 738 544644 401947272 27.166155 9.036885 739 546121 403583419 27.184554 9.040965 740 547600 405224000 27.202941 9.045041 741 549081 406869021 27.221315 9.049114 742 550564 408518488 27.239676 9.053183 743 [ 552049 4.10172407 27.258026 9.057248 744 553536 411830784 27.276363 9.061309 745 555025 413493625 27.294688 9.065367 746 556516 415160936 27.313000 9.069422 747 558009 416832723 27.331300 9.073472 748 559504 418508992 27.349588 9.077519 749 56 001 420189749 27.367864 9.081563 750 562500 421875000 27.386127 9.085603 751 564001 423564751 27.404379 9.089639 752 565504 425259008 27.422618 9.093672 753 567009 426957777 27.440845 9.097701 754 568516 428661064 27.459060 9.101726 755 570025 430368875 27.477263 9.105748 756 571536 432081216 27.495454 9.109766 757 573049 433798093 27.513633 9.113781 758 574564 435519512 27.531799 9.117793 759 576081 437245479 27.549954 C.121801 760 577600 438976000 27.568097 9.1.25805 761 579121 440711081 27.586228 9.129806 762 580644 442450728 27.604347 9.13;3803 763 582169 444194947 27.622454 9.13778)7 764 583696 44594-3744 27.640549 9.141788 765 585225 447697125 27.658633 9.145774 766 586756 449455096 27.676705 9.149757 767 588289 451217663 27.694764 9.153737 768 589824 452984832 27.712812 9.157713 769 591361 454756609 27.730849 9.161686 770 592900 456533000 27.748873 9.165656 771 594441 458314011 27.766886 9.169622 772 595984 460099648 27.784888 9.173585 773 597529 461889917 27.802877 9.177544 774 599076 463684824 27.820855 9.181500 775 600625 465484375 27.838821 9.185452 776 602176 467288576 27.856776 9.189401 777 603729 469097433 27.874719 9.193347 778 605284 470910952 27.892651 9.197289 779 606841 472729139 27.910571 9.201228 780 608400 474552000 27.928480 9.205164 781 609961 476379541 27.946377 9.209096 782 611524 478211768 27.964262 9.213025 783 613089 480048687 27.982137 9.216950 SQUARES, CUBES, AND ROOTS. 113 TAB LE-(Continued). Number. Square. Cube. Square Root. Cube Root. 784 614656 481890304 28. 9.220872 785 616225 483736025 28.017851 9.224791 786 617796 485587656 28.035691 9.228706 787 619369 487443403 28.053520 9.232'J18 788 620944 489303872 28.071337 9.237527 789 622521 491169069 28.089143 9.240433 790 624100 493039000 28.106938 9.244335 791 625681 494913671 28.124722 9.248234'792 627264 496793088 28.142494 9.252130 793 628849 498677257 28.160255 9.256022 794 630436 500566184 28.178005 9.259911 795 632025 502459875 28.195744 9.263797 796 633616 504358336 28.213472 9.267679 797 635209 506261573 28.231188 9.271559'798 636804 508169592 28.248893 9.275435 799 638401 510082399 28.266588 9.279308 800 640000 512000000 28.284271 9.283177 801 641601 513922401 28.301943 9.287044 802 643204 515849608 28.319604 9.290907 803 644809 517781627 28.337254 9.294767 804 646416 519718464 28.354893 9.298623 805 648025 521660125 28.372521 9.302477 806 649636 523606616 28.390139 9.306327 807 651249 525557943 28.407745 9.310175 808 652864 527514112 28.425340 9.314019 809 654481 529475129 28.442925 9.317859 810 656100 531441000 28.460498 9.321697 811 657721 533411731 28.478061 9.325532 812 659344 535387328 28.495613 9.329363 813 660969 537366797 28.513154 9.333191 814 662596 539353144 28.530685 9.337016 815 664225 541343375 28.548204 9.340838 816 665856 543338496 28.565713 9.344657 817 667489 545338513 28.583211 9.348473 818 669124 547343432 28.600699 9.352285 819 670761 549353259 28.618176 9.356095 820 672400 551368000 28.635642 9.359901 821 674041 553387661 28.653097 9.363704 822 675684 555412248 28.670542 9.367.505 823 677329 557441767 28.687976 9.371302 824 678976 559476224 28.705400 9.375096 825 680625 561515625 28.722813 9.378887 826 682276 563559976 28.740215 9.372675 827 683929 565609283 28.757607 9.386460 828 685584 567663552 28.774989 9.390241 829 687241 569722789 28.792360 9.394020 830 688900 571787000 28.809720 9.397796 831 690561 573856191 28.827070 9.401569 832: 692224 575930368 28.844410 9.405338 833 693889 578009537 28.861739 9.409105 834 695556 580093704 28.879058 9.412869 835 697225 582182875 28.896366 9.416630 836 698896 584277056 28.913664 9.420387 837 700569 586376253 28.930952 9.424141 838 702244 588480472 28.948229 9.427893 839 703921 590589719 28.965496 9.431642 K2 114 SQUARES, CUBES, AND ROOTS. TABLE-(Continued). Number. Square. Cube. Square Root. Cube Root. 840 705600 592704000 28.982753 9.435388 841 707281 594823321 29. 9.439130 842 708964 596947688 29.017236 9.442870 843 710649 59907710'7 29.034462 9.446607 844 712336 601211584 29.051678 9.450341 84'5 714025 603351125 29.06888:3 9.454071 846 715716 605495736 29.086079 9.457799 847 717409 607645423 29.103264 9.461524 848 719104 609800192 29.120439 9.465247 849 720801 611960049 29.137604 9.468966 850 722500 614125000 29.154759 9.472682 851 724201 616295051 29.171904 9.476395 852 725904 618470208 29.189039 9.480106 853 727609 620650477 29.206163 9.483813 854 729316 622835864 29.223278 9.487518 855 731025 625026375 29.240383 9.491219 856 732736 627222016 29.257477 9.494918 857 734449 629422793 29.274562 9.498614 858 736164 631628712 29.291637 9.502307 859 737881 633839779 29.308701 9.505998 860 739600 636056000 29.325756 9.509685 861 741321 638277381 29.342801 9.513369 862 743044 640503928 29.359836 9.517051 863 744769 642735647 29.376861 9.520730 864 746496 644972544 29.393876 9.524406 865 748225 647214625 29.410882 9.528079 866 749956 649461896 29.427877 9.531749 867'751689 651714363 29.444863 9.535417 868 753424 653972032 29.461839 9.539081 869 755161 656234909 29.478805 9.542743 870 756900 658503000 29.495762 9.546402 871 758641 660776311 29.512709 9.550058 872 760384 663054848 29.529646 9.553712 873 762129 665338617 29.546573 9.557363 874 763876 667627624 29.563491 9.561010 875 765625 669921875 29.580398 9.564655 876 767376 672221376 29.597297 9.568297 877 769129 674526133 29.614185 9.571937 878 770884 676836152 29.631064 9.575574 879 772641 679151439 29.647932 9.579208 80 774400 681472000 29.664793 9.582839 881 776161 683797841 29.681644 9.586468 882 777924 686128968 29.698484 9.590093 883 779689 688465387 29.715315 9.593716 884 781456 690807104 29.732137 9.597337 885 783225 693154125 29.7489,1-9 9.600954 886 784996 695506456 29.765752 9.604569 887 786769 697864103 29.782545 9.608181 888 788544 700227072 29.799328 9.611791 889 790321 702595369 29.816103 9.615397 890 792100 704969000 29.832867 90.619001 891 793881 707347971 29.849623 9.622603 892 795664 709732288 29.866369 9.626201 893 797449 712121957 29.883105 9.629797 894 799236 714516984 29.899832 9.633390 895 801025 716917375 1 29.91655(0 9.636981 SQUARES, CUBES, AND ROOTS. 115 TAsrE-(Continued). Number. Square. Cube. Square Root. Cube Root. 896 802816 719323136 29.933259 9.640569 897 804609 721734273 29.949958 9.644154 898 806404 724150792 29.966648 9.647736 899 808201 726572699 29.983328 9.651316 900 810000 729000000 30. 9.654893 901 811804 731432701 30.016662 9.658468 902 813604 733870808 30.033314 9.662040 903 815409 736314327 30.049958 9.665609 904 817216 738763264 30.066592 9.669176 905 819025 741217625 30.083217 9.672740 906 820836 743677416 30.099833 9.676301 907 822649 746142643 30.116440 9.679860 908 824464 748613312 30.133038 9.683416 909 826281 751089429 30.149626 9.686970 910 828100 753571000 30.166206 9.690521 911 829921 756058031 30.182776 9.694069 912 831744 758550528 30.199337 9.697615 913 833569 761048497 30.215889 9.701158 914 835396 763551944 30.232432 9.704698 915 837225 766060875 30.248966 9.708236 916 839056 768575296 30.265491 9.711772 917 840889 771095213 30.282007 9.715305 918 842724 773620632 30.298514 9.718835 919 844561 776151559 30.315012 9.722363 920 846400 778688000 30.331501 9.725888 921 848241 781229961 30.347981 9.729410 922 850084 783777448 30.364452 9.732930 923 851929 786330467 30.380915 9.736448 924 853776 788889024 30.397368 9.739963 925 855625 791453125 30.413812 9.743475 926 857476 794022776 30.430248 9.746985 927 859329 796597983 30.4466'74 9.750493 928 861184 799178752 30.463092 9.753998 929 863041 801765089 30.479501 9.757500 930 864900 804357000 30.495901 9.761000 931 866761 806954491 30.512292 9.764497 932 868624 809557568 30.528675 9.767992 933 870489 812166237 30.545048 9. 771484 934 872356 814780504 30.561413 9.774974 935 874225 817400375 30.577769 9.778461 936 876096 820025856 30.594117 9.782946 937 877969 822656953 30.610455 9.785428 938 879844 825293672 30.626785 9.788908 939 881721 827936019 30.643106 9.792386 940 883600 830584000 30.659419 9.795861 941 885481 833237621 30.675723 9.799333 942 887364 835896888 30.692018 9.802803 943 889249 838561807 30.708305 9.806271 944 891136 841232384 30.724583. 9.809736 945 893025 843908625 30.740852 9.813198 946 894916 846590536 30.'757113 9.816659 947 896809 849278123 30.773365 9.820117 948 898704 851971392 30.789608 9.823572 949 900601 854670349 30.805843 9.827025 950 902500 857375000 30.822070 9.830475 951 904401 860085351 30.838287 9.833923 116 SQUARES, CUBES, AND ROOTS. TABL —(Continued). Number. Sqaure. Cube. Square Root. Cube Root. 952 906304 862801408 30.854497 9.837369 953 908209 865523177 30.870698 9.840812 954 910116 868250664 30.886890 9.844253 955 912025 870983875 30.903074 9.847692 956 913936 873722816 30.919249 9.851128 957 915849 87646'7493 30.935416 9.854561 958 917764 879217912 30.951575 9.857992 959 919681 881974079 30.967725 9.861421 960 921600 884736000 30.983866 9.864848 961 923521 887503681 31. 9.868272 962 925444 890277128 31.016124 9.871694 963 927369 893056347 31.032241 9.875113 964 929296 895841344 31.048349 9.878530 965 931225 898632125 31.064449 9.881945 966 933156 901428696 31.080540 9.885357 967 935089 904231063 31.096623 9.888767 968 937024 907039232 31.112698 9.892174 969 938961 909853209 31.128764 9.895580 970 940900 912673000 31.144823 9.898983 971 942841 915498611 31.160872 9.902383 972 944784 918330048 31.176914. 9.905781 973 946729 921167317 31.192947 9.909177 974 948676 924010424 31.208973 9.912571 975 950625 926859375 31.224990 9.915962 976 952576 929714176 31.240998 9.919351 977 954529 932574833 31.256999 9.922738 978 956484 935441352 31.272991 9.926122 979 958441 938313739 31.288975 9.929504 980 960400 941192000 31.304951 9.932883 981 962361 944076141 31.320919 9.936261 982 964324 946966168 31.336879 9.939636 983 966289 949862087 31.352830 9.943009 984 968256 952763904 31.368774 9.946379 985 970225 955671625 31.384709 9.949747 986 972196 958585256 31.400636 9.953113 987 974169 961504803 31.416556 9.956477 988 976144 964430272 31.432467 9.959839 989 978121 967361669 31.448370 9.963198 990 980100 970299000 61.464265 9.966554 991 982081 973242271 31.480152 9.969909 992 984064 976191488 31.496031 9.973262 993 986049 979146657 31.511902 9.976612 994 988036 982107784 31.527765 9.979959 995 990025 985074875 31.542620 9.983304 996 992016 988047936 31.559467 9.986648 997 994009 991026973 31.575306 9.989990 998 996004 994011992 31.591138 9.993328 999 998001 997002999 31.606961 9.996665 1000 1000000 1000000000 31.622776 10. Additional use of this table can be made by the aid of the following Rules: To find the Square of a Number above 1000 - when the Number is divisible by any Number without leaving a Remainder. RULE. —If the number exceed by 2, 3, or any other number of times, any SQUARES, CUBES, AND ROOTS. 117 number contained in the table, let the square affixed to that number in the table be multiplied by the square of 2, 3, 4, 5, or 6, &c., and the product will be the answer. EXAMPLE. —Required the square of 1550. 1550 is 10 times 155, and the square of 155 in the table is 24025. Then 24025X 102 = 2402500 Jns. When the Number is an Odd Number. RULE.-Find the two numbers nearest to each other, which, added together, make that sum; then the sum of the squares of these two numbers, as per table, multiplied by 2, will give the answer, exceeded by 1, which is to be subtracted, and the remainder is the answer required. ExAMPLE.-What is the square of 1345. The nearest two numbers are 6732 1345. ~ 672 - Then, per table, 6732 = 459291584 904513X2 = 1809026-1 = 1809025 4ns. To find the Cube of a Number greater than is contained in the Table. RULE. —Proceed as in squares to find how many times the number exceeds one tf the tabular numbers. Multiply the cube of that number by the cube of the number of times the number sought exceeds the number in the table, and the prod. act will be the answer. ExAmPLE.-What is the cube of 1200? 1200 is 3 times 400, and the cube of 400 is 64000000. Then 64000000X33 = 1728000000 Ans. To find the Squares of Numbers following each other in Arithmetical Progression. RULE.-Find the squares of the two first numbers in the usual way, and subtract the less from the greater. Add the difference to the greatest square, with the adiitiori of 2 as a constant quantity; the sum will be the square of the next:lumber. EXAMPLE.-What are the squares of 1001, 1002, 1003, 1004, and 1005 1 10002 = 1000000 9992 = 998001 1999 Add 2 Add 10002 = 1000000 1002001 Square of 1001. Difference, 2001+2 = 2003 1004004 Square of 1002. Difference, 2003+2 = 2005 1006009 Square of 1003. Difference, 2005+2 = 2007 1008016 Square of 1004. Difference, 2007+2 = 2009 1010025 Square of 1005. To find the Cubes of Numbers following each other in Arithmetical Progression. RULE.-Find the cubes of the two first numbers, and subtract the less from the treater; then multiply the least of the two numbers cubed, by 6; add the product. 118 SQUARES, CUBES, AIND ROOTS. with the addition of 6, to the difference, and continue this the first series of differences. For the second series of differences, add the cube of the highest of the above numbers to the difference, and the sum will be the cube of the next number. EXAMPLE.-What are the cubes of 1001, 1002, and 1003? First Series. Cube of 1000 = 1000000000 Cube of 999 = 997002999 2997001 Difference. 999X6+6 = 6000 3003001 Difference of 1000. 6000 +6 - 6006 3009007 Difference of 1001. 6006 +6 = 6012 3015019 Difference of 1002. Second Series. Cube of 1000. = 1000000000 Difference for 1000, 3003001 1003003001 = Cube of 1001. Difference for 1001, 3009007 1006012008 = Cube of 1002. Difference for 1002, 3015019 1009027027 -= Cube of 1003. To find the Cube or Square Root of a higlher Number than i' contained in the Table. RULE.-Find in the column of Squares or Cubes the number nearest to that number whose root is required, and the number from which that square or cube is derived will be the answer wihen decirrals are not of importance. EX ~AMPLE.-What is the square root of 562500? In the table of Squares, this number is the square of 750; therefore 750 is the square root required. EXAMPLE.-What is the cube root of 2248090 1 In the table of Cubes, 2248091 is the cube of 131; therefore 131- is the cube root required, nearly. To find the Cube Root of any Number over 1000. RULE.-Find by the table the nearest cube to the number given, and call it the assumed cube. Multiply the assumed cube and the given number respectively by 2; to the product of the assumed cube add the given number, and to the product of the given number add the assumed cube. Then, as the sum of the assumed cube is to the sum of the given number, so is the root of the assumed cube to the root of the given number. EXAMPLE.-What is the cube root of 224809? By table, the nearest cube is 216000, and its root is 60. 216000 X 2+224809 - 656809, And 224809X2+216000- 665618. Then, as 656809: 665618:: 60: 60.804+ To find the Sixth Root of a _Numder. RULE.-Take the cube root'of its square root. ExANiPLE.-What is the / of 441? /441 --- 21, and /21 = 2.7589 Jdns. SQUARES, CUBES, AND ROOTS. 119 TO FIND THE CUBE OR SQUARE ROOT OF A NUMSBER CONSISTING OF INTEGERS AND DECIMALS. RULE.-Multiply thile difference between the root of the integer part and the root of the next higher integer by the decimal, and add the product to the root of the integer given; the sum will be the root of the nulnber required. This is correct for the square root to three places of decimals, and in the cube s oot to seven. ExAMPIu,.-What is the square root of 53.75, and the cube root uf 843.75? / 54 = 7.3484 /844 - 9.4503 53 = 7.2801 Q/843 = 9.4466.0683.0037.75.75.051225.002775 J 53 7.2801 Z/843 = 9.4466,/53.75 - 7.331325 3843.75 - 9.449375 When the Number is a Root of any of the Numbers from 1 to 1000, as in the preceding Table. Find in the column of Square Root or Cube Root the number for which the Square or Cube is required, and opposite, in their respective columns, is the square or cube required. TO' FIND THE ROOT OF A NUMBER EXCEEDING 1000. By anc Inversion of the preceding' Table. Look in the column of Square or Cube, according to the root required, for the number for which the root is required, and opposite, in the column of number, is the root. Or, when the number is divisible by 4, twice the square root o! bihe quotient is its square root. 120 SIDES OF EQUAL SQUARES.'TABLE of the Sides of Squares-equal in Area to a Circle of any Diameter, fr'om 1 to 100. Side of equal, mee Side of equal ameter Side of equal ide of equal I~ianetrSqae imtr Sdofqul reter. SrDiameter. qa r.. Da ter. Sq uare. 1 Square. ]_ Squ.re. Square. 1.00 0.886 15. 13.293 29. 25.7001 43. 38.107.25 1.107.25 13.514.25 25.922.25 38.329.5 1.329.5 13.736.5 26.143.5 38.550. 75 1.550.75 13.958.75 26.3658.75 38.772 2. 1.772 16. 14.179 30. 26.586 44. 38.993.25 1.994.25 14.401.25 26.808.25 39.215.5 2.215.5 14.622.5 27.029.5 39.437.75 2.437.75 14.844.75 27.251.75 39.658 3. 2.658 17. 15.065 31. 27.473 45. 39.880.25 2.880.25 15.287.25 27.694.25 40.101.5 3.101.5 15.508.5 27.916.5 40.323.75 3.323.75 15.730.75 28.137.75 40.544 4. 3.544 18. 15.952 32. 28.359 46. 40.766.25 3.766.25 16.173.25 28.580.25 40.987.5 3.988.5 16.395.5 28.802.5 41.209.75 4.209.75 16.616.75 29.023.75 41.431.5 4,431 19. 16.838 33. 29.245 47. 41.652.25 4.652.25 17.059.25 29.467.25 41.874.5 4.8741.5 17.281.5 29.688.5 42.095.75 5.095.75 17.502.75 29.910.75 42.317 6. 5.317 20. 17.724 34. 30.131 48. 42.538.25 5.538.25 17.946.25 30.353.25 42.760.5 5.760.5 18.167.5 30.574.5 42.982.75 5.982.75 18.389.75 30.796.75 43.203 7. 6.203 21. 18.610 35. 31.017 49. 43.425.25 6.425.25] 18.832.25 31.239].25 43.646.5 6.646.5 19.053.5 31.461.5 43.868.75 6.868.75 19.275.75 31.682.75 44.089 8. 7.089 22. 19.496 36. 31.904 50. 44.311.25 7.311.25 19.718.25 32.125.25 44.532.5 7.532.5 19.940.5 32.347.5 44.754.75 7.754.75 20.161.75 32.568.75 44.976 9. 7.976 23. 20.383 37. 32.790 51. 45.197.25 8.197.25 20.604.25 33.011,.25 45.419.5 8.419.5 20.826.5 33.233.5 45.640.75 8.640.75 21.047.75 33.455.75 45.862 10. 8.862 24. 21.269 38. 33.676I 52. 46.083.25 9.083.25 21.491.25 33.898.25 46.305.5 9.305.5 21.712.5 34.119.5 46.526.75 9.526.75 21.934.75 34.341t.75 46.748 11. 9.748 25. 22.155 39. 34.562 53. 46.970.25 9.970.25 22.377.25 34.784~.25 47.191.5 10.191.5 22.598.5 35.005.5 47.413.75 10.413.75 22.820.75 35.227.75 47.634 12. 10.634 26. 23.041 40. 35.449 54. 47.856.25 10.856.25 23.263.25 35.670.25 48.077.5 11.077.5 23.485.5 35.892..5 48.299.75 11.299.75 23.706.75 36.113.75 48.520 13. 11.520 27. 23 928 41. 36.335 55. 48.742.25 11. 742.25 24.149.25 36.556.25 48.964.5 11.964. 5 24.371.5 36.778.5 49.185.75 12.185.75 24.592.75 36.999.75 49.407 14. 12.407 28. 24.814 42. 37.221 56. 49.628.25 12.628.25 25.035.25 37.443.25 49 850.5 12.850.5 25.257.5 37.664.5 50.071.75 13.071.75 25.479.75 37.886.75 50.2J3 SIDES OF EQUAL SQUARES. 121 TABLE- (Continued). Side of equal Side of e qual. Side of equal Side of equal Diameter.. Diameter. quar. ameer. Square. Diameter. Side of equal _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ S q u a re. 57. 50; 514 68.- 60.263 79. 70.011 90. -79.760.25 50.736.25 60.484.25 70.233.25 79.981.5 50.958.5 60.706.5 70.455.5 80.203.75 51.179.75 60.928.75 70.676.75 80.425 58. 51.401 69. 61.149 80. 70.898 91. 80.646.25 51.622.25 61.371.25 71.119.25 80.868.5 51.844.5 61.592.5 71.341.5 81.089.75 52.065.75 61.814.75 71.562.75 81.311 59. 52.287 70. 62.035 81. 71.784 92. 81.532.25 52.508.25 62.257.25 72.005.25 81.754.5 52.730.5 62.478.5 72.227.5 81.975.75 52.952.75 62.700.75 72.449.7582.197 60. 53.173 71. 62.922 82. 72.670 93. 82.419.25 53.395.25 613.143.25 72.892.25 82.640.5 53.616.5 63.365.5 73.113.5 82.862.75 53.838.75 63.586.75 73.335.75 83.083 61 54.059 72. 63.808 83. 73.556 94. 83.305.25 54.281.25 64.029.25 73.778.25 83.526.5 54.502.5 64.251.5 73.999.5 83.748.75 54.724.75 64.473.75 74.221.75 83.970 62. 54.946 73. 64.694 84. 74.443 95. 84.191.25 55.167.25 64.916.25 74.664.25 84.413.5 55.389.5 65.137.5 74.886.5 84.634.75 55.610.75 65.359.75 75.107.75 84.856 63. 55.832 74. 65.580 85. 75.329 96. 85.077.25 56.053.25 65.802.25 75.550.25 85.299.5 56.275.5 66.023.5 75.772.5 85.520.75 56.496.75 66.245.75 75.993.75 85.742 64. 56.718 75. 66.467 86. 76.215 97. 8.5.964.25 56.940.25 66.688.25 76.437.25 86.185.5 157.161.5 66.910.5 76.658.5 86.407.75 57.383.75 67.191.75 76.880.75 86.628 65. 57.604 76. 67.353 87. 77.101 98. 86.850.25 57.826.25 67.574.25 77.323.25 87.071.5 58.047.5 67.796.5 77.544.5 87.293.75 58.269.75 68.017.75 77.766.75 87.514 66. 58.490'77. 68.239 88. 77.987 99. 87.736.25 58.712.25 68.461.25 78.209.25 87.958.5 58.934.5 68.682.5 78.431.5 88.179.75 59.155.75 68.904.75 78.652.75 88.401 67. 59.377 78. 69.125 89. 78.874 100. 88.622.25 59.598.25 69.347.25 79.095.25 88.844.5 59.820.5 69.568.5 79.317.5 89.065.75 60.041.75 69.790.75 79.538.75 89.287 USE OF THIS TABLE. To find a Square that shall have the same Area as a Given Circle. EXAMPLE.-What is the side of a square that has the same area as a circle of 734 inches? By table of Areas, page 93, opposite to 73.25 is its area, 4214.1; and in the above t;ble, page 121, is 64.916, the side of a square that has the same area as a circle of 734 inches in diameter. EXAMPLE.-What should be the side of a square that wotld give the same area as a board that is 18 inches wide and 10 feet long? 1, 122 SIDES OF EQUAL SQUARES. 18 inches is 1.5 feet. 10 15.0 feet. 14 4 square inches in a foot. 60 0 600 150 2160.0 inches area. By table, page 92, 2164.75 inches area have a diameter of 52.5 inches, which i1 Tbe above table gives an equal side of 46.526, which is the answer very nearly. PLANE TR1GONOhMETRY. 12 3 PLANE TRIGONOMETRY. A B C the three angles (A the right angle); a b c the three sides respectively opposite to them; R the tabular radius (1 or 1000000); S the area of the triangle, and p half its perimeter = ( a+2 ). RIGHT-ANGLED TRIANGLES. B B B COosiwe. e -' " C Ba. etA CTa.ne A OCosivA E: a....' u A - t tang. B sin. B a=-/b2+c2, b-,(aZ-c2), c= /(a2-b2), b = c --, also-= a GIVEN. To fnd A C and B A. Hyp. B C, (R BC::: sin. B: A C, legBA, 4 = 10.9, Ans. The examples above given are deduced froml instances in successful practice; where the diameter has been less, fracture has almost universally taken place, the strain being increased beyond the ordinary limit. Relative Values of Cast and Wrought Iron. When cast iron shafts of less diameter than 12 inches are required, the Values here given must be reduced friom J' to -j0, according to the quality of the iron and the diamett r of the shaft to be used; but when they exceed this diameter, the Values must be increased in a like manner. The necessity of this arises in consequence of the strength of a wrought iron shaft decreasing as its diameter increases. Grier makes the difillerene between cast and wrought iron for all diameters as.963 to 1.000. Relative Values of Cast Iron, Oak, and Pine. The Value for cast iron being 125, that for oak is 280, and that of white pine 257. Grier, in his Mechanics' Calculator, gives the following rule: For Cast Iron Shafts of second and third MIovers. The diameters for second and third movers are found by multiplying the diam. eter for a prime mover by.8 and.793 respectively. Torsional Strength of Hollow Cylinders. Relative Torsional Strength of Cast Itron Shafts of different Forms, having equal Areas of Cross Sections. From Major Wade's experiments, on shafts having sections of 1, 2, and 3 square inches. Hollow Cylinders, where interior and exterior diameters were in the Solid Cylin- Solid Square. proportion of der. 4 to 10. 1 5 to to 10. 7 to 10. 8 to 10. 1.000 1.8750 1.2656 l 1.4433 1.7000.1 2.0864 2.7377 Formula of Lieutenant Rodman, U. S. A., deduced from the above experiments, modified to introduce the element V, and the length in feet in preference to inches. V D5 —d4 _W. D1 W - the weight which a shaft will sustain in pounds (independent f its own weight), D d the exterior and interior diameters in inches, And I the distance from the axis in feet at which W is applied. 160 GUDGEONS AND SHAFTS. GUDGEONS AND SHAFTS. To find the Dimensions of a Gudgeon. 0.30V(wl) = d, w representing the stress in 100 lbs., I the length in inches, and d the diameter in inches. If a Cylindrical Shaft has no other lateral stress to sustain than its own weight, and is Fixed at one End, d =.00002412. Let the stress supposed to be in the middle be n times the weight of the shaft; then, When supported at both Ends, If the weight of the shaft be not taken into account, d = /.00012 n2. If the weight of the shaft is taken into account, d = V/.00012 (n+1)12. When a Hollow Shaft is supported at each End, d _ 0 —- 48 tD3, w representing the stress in lbs., I the length in inches, D the interior diameter, and d the diameter in inches. When a Hollow Shaft is Fixed at each End, and Loaded in the Middle, d 00048 wl For hollow Cylindrical Shafts, when Supported at one End, d =,Y/.00048 awl-+D3. If the hollow shaft support the weights at distances m and n from each end, and is supported at each end, d = Y/.00048-w+D3. The last four formulas do not take into account the weight of the shaft. The above is for Cast Iron. For Cylindrical Shafts of Cast Iron to resist Torsion. (Buchanan.) Let P be the number of horses' power, and R the revolutions of the shaft in a minute; then 240 P V/~ = d. For Wrought Iron, multiply this result by.963; for Oak, by 2. 238; for Pine, by 2.06. If a shaft has to sustain both lateral stress and torsion, then, For cast iron, s(240 P+1) =d R/ 2 TEETHI OF WHEELS. 161 TEETH OF WHEELS. To Construct a Tooth. DIVIDE the pitch into 10 parts. Let 3.5 of these parts be below the pitch line, and 3.0 of them above. The thickness should be.47 of the pitch. The length should be.65 of the pitch. The Diameter of a wheel is measured from the pitch line. The wood used for teeth is about ~ the strength of cast iron, therefore they should be twice the depth to be of equal strength. To find the Diameter of a Wheel, the Pitch and Number bf Teeth being given. Pitch X number of teeth 3.1-416 -= diameter. NoTE.-The pitch, as found by this rule, is the are of a circle; the true pitch required is a straight line,-and must be measured from the centres of two contiguoils teeth. To find the Pitch, the Diameter and Number of Teeth being given. Diameter X 3.1416 number of teeth pitch. NOTE. —The pitch, as found by this rule, is the arc of a circle; the true pitch required is a straight line, and must be measured from the centres of two contiguous teeth. To find the Number of Teeth, the Diameter and Pitch being given. Diameter X 3.1416 = number of teeth. pitch Tofind the Diameter when the true Pitch is used. Number of teeth X pitch X.32 = diameter. Dimensions of 7Wheels in operation. Diameter. Breadthl. Pitch. Length of Tlhickness of elocit. per Pressure. teeth. teeth. _eco_ d. Feet. Ins. Inches. Inches. ISclies. Inches. Feet. Lbs. 10 7. 2.8 1.625 1.3 3. 11000 6 12. 4.2 2.25 1.9 6.6 20000 7 10 4.5 1.9 1.125.875 1.1 3300 14 4 8. i 3. i 1.75 1.4 [ 1.87 9000 02 162 VELOCITY OF WHEELS. VELOCITY OF WHEELS. THE relative velocity of wheels is as the number of their teeth. To find the Velocity or Number of Turns of the last Wheel to one of the first. RvLE.-Divide the product of the teeth of the wheels that act as drivers by the product of the driven, and the quotient is the number. EXAMPLE.-If a wheel of 32 teeth drive a pinicn of 10, on the axis of which there is one of 30 teeth, acting on a pinion of 8, what is the number of turns of the last? 32 30 960 _ x - = - - 12, Ans. To find the Proportion that the Velocities of the Wheels in a train should bear to one another. RULE.-Subtract the less velocity from the greater, and divide the remainder by one less than the number of wheels in the train; the quotient is the number, rising in arithmetical progression from the less to the greater velocity. EXAMPLE.-What are the velocities of three wheels to produce 18 revolutions per minute, the driver making 3 revolutions per minute 1 18 —3 - 15 3183 --- = 7.5; then 3+7.5 = 10.5, 3-1 =and 10.5+7.5 = 18; thus, 3, 10.5, and 18 are the velocities of the three wheels. To find the Number of Teeth required in a Train of Wheels to produce a certain Velocity. RULE.-As the velocity required is to the number of teeth in the driver, so is the velocity of the driver to the number of teeth in the driven. EXAMPLE.-If the driver has 90 teeth, makes 2 revolutions, and the velocities required are 2, 10, and 18, what are the number of teeth in each of the other two 1 2d wheel, 10: 90 2: 18 teeth. 3d wheel, 18 90:: 2: 10 teeth. STRENGTH OF WHEELS. 163 STRENGTH OF WHEELS. THE strength of the teeth of wheels is directly as their breadth and as the square of their thickness, and inversely as their length. The stress is as the pressure. To find the Thickness of a Tooth, the Strain at the Pitch Line being given. RULE.-Divide the pressure in pounds at the pitch line by 3000, and the square root of the quotient is the thickness of the tooth in inches. EXAMPLE. —The pressure is 9000 lbs., what is the thickness of the tooth required? 9000 /30- -- 1.732 inches, Ans. The Breadth should be 2.5 times the pitch. Therefore, as the thickness should be 0.47 of the pitch, the pitch for the above example will be 3.685 inches, and the breadth 3.685 X2.5 - 9.2125 inches. To find the Horses' Power of a Tooth, the Dimensions and Velocity being given. Thickness 2 X 3000 =- pressure. Pressure X velocity in feet per minute 33000 = horses' power. 33000 Thickness X 2.1277+ = the pitch. Thickness X 1.5384+ = the length. Tofind the Dimensions of the Arms of a Wheel. RULE.-Multiply the power at the pitch line by the cube of the length of the arms, and divide this product by the product of the number of arms and 280; the quotient will be the breadth and cube of the depth. EXAMPLE.-If the power be 1600, the diameter of the wheel 10 feet, and the number of arms 6, what will be the dimensions of each arm? 1600 X ] 0 —23 200000 60X02' 200 = 119; if the breadth be 5 inches, then 6x280 1680 119 = 23.8, and -/ of 23.8 = 2.87, the depth. 5K 161 GENERAL EXPLANATIONS CONCERNING WHEELS. GENERAL EXPLANATIONS CONCERNING WHEELS. Pitch Lines. —TntE touching circumferences of two or more wheels, which act upon each other. Pitch of a Wheel.-The distance of two contiguous teeth, measured upon their pitch line. Length of a Tooth.-The distance from its base to its extremity. Breadth of a Tooth. —The length of the face of the wheel. Spur Wheels.-Wheels that have their teeth perpendicular to their axis. Bevel Wheels.-Wheels having their teeth at an angle with their axis. Grown fWheels.-Wheels which have their teeth at a right angle with their axis. JMitre Wheels.-Wheels having their teeth at an angle of 450 with their axis. opur Gear.-Wheels acting upon each other in the same plane. Bevel Gear.-Wheels acting upon each other at an angle. When two wheels act upon one another, the greater is called the spur or driver, and the lesser the pinion or driven. When the teeth of a wheel are made of a different material fiom the wheel, they are called cogs. TABLE of the Strength of Teeth and Arms. Teeth. With 6 Arms. Iorses' power hPs.ssure in lbs. at 3 test per Thickness in Breadth in Dptlfor I Breadth of second. Pitch in inches. nes. es oot adius i nnhe _.cllt~s. inch ice cii in inches. 22.25.25.119.75 0.87.25 85.5.50.238 1.25 1.24.42 191 1..75.357 1.75 1.67.60 337 2. 1..475 2.50 1.76.80 520 3. 1.25.590 3. 2. 1. 800 4. 1.50.730 4. 2.20 1.30 1040 5. 1.75.835 4.25 2.40 1.40 1370 7. 2..955 5. 2.50 1.70 1720 9. 2.25 1.070 5.50 2.70 1.80 2100 10.5 2.50 1.190 6. 2.85 2. 2560 13. 2.75 1.310 6.75 3. 2.20 3000 15. 3. 1.430 7.25 3.20 2.40 3600 18. 3.25 1.550 8. 3.30 2.60 4150 21. 3.50 1.670 8.50 3.40 2.80 4800 24. 3.75 1.790 9.25 3.50 2.90 5700 27.5 4. 1.910 10.25 3.60 3.40 6300 31.5 4.25 2.025 10.50 3.70 3.50 6900 34.5 4.50 2.150 11. 3.80 3.70 7700 38.5 4.75 2.270 11.75 3.90 3.90 8500 42.5 5. 2.390 12.25 4. 4. Tredgold. HORSE POWER-ANIBIAL STRENGTH. 165 HORSE POWER. As this is the universal term used to express the capability of first movers of magnitude, it is very essential that the estimate of this power should be uniform; and as it is customary, in Europe, to estimate the power of a horse equivalent to the raising of 33000 lb.~. one foot high in a minute, there can be no objection to such an estimate here. The estimate, then, of a horse's power in the calculations in this work, is 33000 pounds avoirdupois, raised through thb space of one foot in height in one minute, and in this I am supported by the practice of a majority of the manufacturers of steam-engines in this country. ANIMAL STRENGTH. MIEN. The mean effect of the power of a man, unaided by a machine, working to the best possible advantage, and at a moderate estimation, is the raising of 70 lbs. 1 foot high in a second, for 10 hours in a day. Two men, working at a windlass at right angles to each other, can raise 70 lbs. more easily than one man can 30 lbs.;Mr. Bevan's results with experiments upon human strength are, for a short peWith a drawing-knife. a force of 100 lbs. an auger, both hands... " 100 " a screw-driver, one hand.. 84 a bench vice, handle... 72 " 72 a chisel, vertical pressure... 72 a windlass... " 60 " pincers, compression. " 60 " a hand-plane... " 50" a hand-saw.... " 36" a thumb-vice... " 45" a brace-bit, revolving. " 16 Twisting by the thumb and fingers only, 14 and with small screw-drivers. By Mr. Field's experiments in 1838, the maximum power of a strong man, exerted for 21 minutes, is = 18000 lbs. raised one foot in a minute. A man of ordinary strength exerts a force of 30 lbs. for 10 hours in a day, with a velocity of 21 feet in a second, = 4500 lbs. raised one foot in a minute, =- of the work of a horse. A foot-soldier travels in 1 minute, in common time, 90 steps, = 70 yards. in quick time, 110 " = 86 " in double quick-time, 140 " -109 " He occupies in the ranks, a front of 20 inches, and a depth of 13, without a knapsack; the interval between the ranks is 13 inches. Average weight of men, 150 lbs. each. 5 men can stand in a space of 1 square yard. A man travels, without a load, on level ground, during 8% hours a day, at the rate of 3.7 miles an hour, or 31~ miles a day. He can carry 111 lbs. 11 miles in a day.. Daily allowance of' water for a man, 1 gallon for all purposes. 166 ANIMAL STRENGTIH. A porter going short distances, and returning unloaded, carries 135 lbs. 7 miles a day. He can carry, in a wheelbarrow, 150 lbs. 10 miles a day. The muscles of the human jaw exert a force of 534 lbs. HORSES. A horse travels 400 yards, at a walk, in 4- minutes; at a trot, in 1 minutes; at a gallop, in 1 minute. HIe occupies in the ranks a front of 40 inches, and a depth of 10 feet; in a stall, from 3-l to 45 feet front; and at picket, 3 feet by 9. Average weight = 1000 lbs. each. A horse, carrying a soldier and his equipments (say 225 lbs.), travels 25 miles in a day (8 hours). A drasught horse can draw 1600 lbs. 23 miles a day, weight of carriage included. The ordinary work of a horse may he stated at 22.500 lbs., raised I foot in a minute, for 8 hours a day. in a horse mill, a horse moves at the rate of 3 feet in a second. The diameter of the track should not be less than 25 feet. A horsepower in machinery is estimated at 33.000 lbs., raised 1 foot in a minute; hut as a horse can exert that force but 6 hours a day, one machinery horse power is equivalent to that of 4.4 horses. The expense of conveying goods at 3 miles per hour per horse teams being 1, the expense at 44 miles will be 1.33, and so on, the expense being doubled when the speed is 51 miles per hour. The strength of a horse is equivalent to that of 5 men. TABLE, of 1te AleoutLnt of Labour a Horse of avereage Strength is capable of persfornMting. at (Jrifer'ent Velocities, o7b CaZlals, Rail'roads, and Turtvikes. Force of traction estimated at 83.3 lbs. Velocity i nmiles Duration of t I Useful effect for one day in tons, drawn one nile. per hour. day's work. On a Canall On a Railroad. On a Turnpike. Miles. Hours. Tons. Tons. Tons. -2 — 11 520 115 14 3 8 243 92 12 3'- 5~- 153 82 10 4 4, 102 72 9. 5 2 52 57 7.2 6 2 30 48 6. 7 1~ 19 4l 5.1 8 1 12.8 36 4.5 9 9.0 32 4.0 10 6.6 28,8 3.6 The actual labour performed by horses is greater, but they are injured by it. HYDROSTATICS. 167 HYDROSTATICS. HYDROSTATICS treat of the pressure, weight, and equilibrium of non-elastic fluids. The pressure of a fluid at any depth is as the depth of the fluid. The pressure of a fluid upon the bottom of the containing vessel is as the base and perpendicular height, whatever may be the figure if the containing vessel. Fluids press equally in all directions. The Centre of Pressure is that point of a surface against which any fluid presses, to which, if a force equal to the whole pressure,were applied, it would keep the surface at rest. The centre of pressure of a parallelogram is at a of the line (measuring downward) that joins the middles of the two horizontal sides. In a triangular plane, when the base is uppermost, the centre of pressure is at the middle of the line, raised perpendicularly from the vertex; and when the vertex is uppermost, the centre of pressure is at i of a line let fall perpendicularly fi'om the vertex. OF PRESSURE. The pressure of a fluid on any surface, whether vertical, oblhque,.r horizontal, is equal to the weight of a column of the fluid, whose case is equal to the surface pressed, and height equal to the distance:f the centre of gravity of the surface pressed, below the surface of:he fluid. To find the Pressure of a Fluid upon the Bottom of the Containing Vessel. RULE. —Multiply area of base in feet by height of fluid in feet, and:heir sum by the weight of a cubic foot of the fluid. EXAMPLE.-What is the pressure upon a surface 10 feet square,;he water (fresh) being 20 feet deep? 102 x 20 X 62.5 = 125000 lbs., Ans. The side of any vessel sustains a pressure equal to the area of the 3ide, multiplied by half the depth. The pressure upon an inclined, curved, or any surface, is as the area )f the surface, and the depth of its centre of gravity below thefluid. EXAMPLE..-What is the pressure upon the sloping side of a pond 10 feet square, the depth of the pond being 8 feet? 102 X62.5 - 25000 lbs., Ans. Or, on a hemisphere just covered with water, and 36 inches in liameter, trea of 36 = 1017.8, centre of gravity (page 138) 36 2+2 = 9. 9: ns..75 foot; then,.75X 101 7.8-, - 144 _ 5 3, which, X62.5 - 131.25 lbs., Ans. 1 68 HYDROSTATICS. The pressure upon a number of surfaces is-found by multiplying the sum of the surfaces into the depth of their common centre of gravity, below the surface of the fluid. CONSTRUCTION OF BANKS. A bank, constructed of a given quantity of materials, will just resist the pressure of the water when the square of its thickness at the base is to the square of its perpendicular height, as the weight of a given bulk of water is to the weight of the same bulk of the material the bank is made of, increased by twice the aforesaid weight of the given bulk of water. Thus, if the bank is made of a stone 2 times heavier than water, the thickness of the base should be to the height, as 3 to 6. If the height, compared to the thickness of the base, be as 10 to 7, stability is always ensured, whatever the specific gravity of the material may be. The bottom of a conical, pyramidal, or cylindrical vessel, or of one the section of which is that of an inverted frustrum of a cone or pyramid, sustains a pressure equal to the area of the bottom and the depth of the fluid. FLOOD GATES. To find the Strain which a Fluid will exert to make it turn upon its Hinges, or open. RULE.-Multiply i of the square of the height by the square of the breadth, and take a bulk of water equal to the product. EXAMPLE.-If the gate is 6 feet square, 62 x- 62 -324 cubic feet, or 20250 lbs.'o find the Strain the Water exerts upon its Hinges. RULE.-Multiply - of the breadth by the cube of the height, and take a bulk of water equal to the product. EXAMPLE.-With the same gate, iX63 -=216 cubic feet, or 13500 lbs. PIPES. To find the Thickness of a Pipe. RULE.-Multiply the height of the head of the fluid in feet by the diameter of the pipe in inches, and divide their product by the co. liesion of one square inch of the material of which the pipe is composed. By experiment it has been found that a cast iron pipe, 15 inches in diameter, and { of an inch thick, will support a head of water of 600 feet; and that one of oak, of the same diameter, and 2 inches thick, will support a head of 180 feet. The cohesive power of cast iron, then, would be 12,000 lbs.; of oak, 1350 lbs. That of lead is 750 lbs.; sand wrought iron boiler plates, riveted together, is from 5S to 30,000 lbs. HYDROSTATICS. 169 In conduit pipes, lying horizontal, and made of lead, their thickness, compared to their diameter, should be, As 2j, 3, 4, 5, 6, 7, 8 lines, To 1, 1,, 2, 3, 41, 6, 7 inches. And when made of iron, As 1, 2, 3, 4, 5, &c., lines, To 1, 2, 4, 6, 8, &c., inches. T!le tenacity of lead is increased to 3000 by the addition of 1 part of zinc in 8. HYDROSTATIC PRESS. To find the Thickness of the Metal to resist a Given Pressure. Let p pressure per square inch in pounds, r=radius of cylinder, and c = cohesion of the metal per square inch. Then -- r =thickness of metal. c —p The cohesive force of a square inch of cast iron is frequently estimated at 18000 lbs., but 16000 is preferable. A cylindrical ring, the diameters of which were 5.3 arid 10 8 nlch. es, burst at a pressure of 9000 lbs. per square inch. These dimensions by the above rule would give) 9000 X 5.3 — 2 3.4 ins. 16000-900 4 ins. 10.8 —-5.3'rhe thickness was'2 =- 2.75 ins. P 170 HYDRAULICS AND HYDRODYNAMICS. HYDRAULICS AND HYDRODYNAMICS. HYDRAULICS treats of the motion of non-elastic fluids, and Hydrodynamics of the force with which they act. Descending water is actuated by the same laws as falling bodies. Water will fall through 1 foot in i of a second, 4 feet in ~ of a second, and through 9 feet in J of a second, and so on. The velocity of a fluid, spouting through an opening in the side of a vessel, reservoir, or bulkhead, is the same that a body would acquire by falling through a perpendicular space equal to that be-,:ween the top of the water and the middle of the aperture. Then, by rule 4 in Gravitation, / height X 64.33 = velocity. EXAMPLE.-What is the velocity of a stream issuing from a head of 10 feet. V/10 X 64.33 = 25.36 feet, A Or, /10 X 8.02 = 25.36 feet, Ans. If the velocity be 50.72 feet per second, what is the head? 50.722. f64.33 40 feet, Or, 50.722 *82 - 40.2 feet, This would be true were it not for the effect of friction, which in pipes and canals increases as the square of the velocity. The mean velocity of a number of experiments gives 5.4 feet for a height of one foot. The theoretical velocity is (x/64~) 8. OF SLUICES. To find the Quantity of Water which will flow out of an Opening. RULE.-Multiply the square root of the depth of the water by 5.4; the product is the velocity in feet per second. This, multiplied by the area of the orifice in feet, will give the number of cubic feet per second. EXAMPLE.-If the centre of a sluice is 10 feet below the surface of a pond, and its area 2 feet, what quantity of water will run out in one second? /10 X 5.4 X2 -= 34.1496 feet, Ans. NOTE.-If the area of the opening is large compared with the head of the water take z of this velocity for the actual velocity. OF VERTICAL APERTURES OR SLITS. The quantity of water that will flow out of one that reaches as high as the surface is 2 of that which would flow out of the same aperture if it were horizontal at the depth of the base. velocity at bottom X depth X 2 Or, velocity at bottom X breadth of slit = number of cubic feet per second. HYDRAULICS AND HYDRODYNAMICS. 17] OF STREAIMS OR JETS. To find the Distance a Jet will be projected from a Vessel. through an opening in the Side. RULE. —B C will always be equal to twice the square root of A 0 X A OB. a If o is 4 times as deep below A, as a is, o will discharge twice the quan- - tity of water that will flow from a in the same time, because 2 is the / square root of A. o, and 1 is the square root of A a. c(: NOTE.-The water will spout the farthest when o is equidistant from A and B; and if the vessel is raised above a plane, B must be taken upon the plane. The quantities of water passing through equal holes in the same time are as the square roots of their depths. ExAMPLE. -A vessel 20 feet deep is raised 5 feet above a plane; how far will a jet reach that is 5 feet from the bottom? * /15 X 10 X 2 = 24.48 feet, Ans. When a prismatic vessel empties itself by a small orifice, in the time of emptying itself, twice the quantity would be discharged if it were kept fiull by a new supply. To find the Vertical Hei,,ht of a Stream projected from a Pipe. RULE.-Ascertain the velocity of the stream by computing the quantity of water running or forced through the opening; then, by rule 5 in Gravitation, page 140, find the required height. EXAMPLE.-If a fire-engine discharges 16.8 cubic feet of water through a J inch pipe in one minute, how high will the water be projected, the pipe being directed vertically? 16.8 X 1728. area of —' inches in a foot - seconds in a minute =91.6, or velocity of stream in feet per second; then, by rule, page 140, 91.6-8 = 11.45, and i1.452 = 131.10 feet, Ans. NOTE.-This rule gives a theoretical result; the result in practice is somewhat less. VELOCITY OF STREAMIS. In a stream, the velocity is greatest at the surface and in the middle of the current. To find the Velocity of a River or Brook. RULE.-Take the number of inches that a floating body passes over in one second in the middle of the current, and extract its square root; double this root, subtract it from the velocity at top, and add 1; the result will be the velocity of the stream at the bottom; and the mean velocity of the stream is equal the velocity at the surface -/ velocity at the surface +.5. 172 HYDRAULICS AND HYDRODYNAMICS. EXAMPLE.-If the velocity at the surface and in the middle of a stream be 36 inches per second, what is the mean velocity?,/36 x 2-36+1 = 25, the velocity at bottom. 36- /36+.5 = 30.5, Ans. To find the Velocity of Water running through Pipes. RULE.-Multiply the height in feet by 2500, divide this product by the product of the length in feet into 13.88 divided by the number of inches in the diameter of the pipe, and the square root of the quotient is the velocity in feet per second. Thus, V2500 X h (l X138)= velocity, when h represents the height, I the length, and d the diameter. EXAMPLE.-The head of a reservoir is 1 foot, the diameter of the pipe 5 inches, and the length of it 100 feet; what is the velocity in feet per second? 2500x (l 00X3.88 ) =27769, and /9 —3, Ans. If the height is required, 13.88 v2X d X -- 2500 = height. In this, as in the annexed table, the friction is assumed to be in the ratio of the diameters. Late and extended experiments show that this is not strictly accurate. See table, page 175. Quantities of Water discharged from Orifices of varzous forms, the Altitude being constant, at 34.642 Inches. Cubic incre' Nature and dimensions of the tubes and orifices. discharged in a minute. i. A circular orifice in a thin plate, the diameter being 1.7 inches 10783 2. A cylindrical tube 1.7 inches in diameter, and 5.117 inches long. 14261 3. A short conical adjutage, 1.7 inches in diameter 10526 4.'The same, with a cylindei 3.41 inches long added to it. 10409 5. The same, the length of the cylinder being 13.65 inches long.. 9830 6. The same, the length of the cylinder being 27.30 inches long. 9216 Results prove that the discharge of water through a straight cylindrical pipe of an unlimited length may be increased only by altering the form of the terminations of the pipe, by making the inner end of the pipe of the same form as the vena contracta, and the ex. tremity a truncated cone, having its length about 9 times the diameter of the cylinder or pipe attached, and the aperture at the outlet to the diameter of the cylinder as 18 is to 10. By giving this form, the discharge is over what it would be by the cylinder alone as 24 is to 10. rABLE showing the Head necessary to overcome the Friction of Water in dtorizontal Pipes, BY MR. SMEATON. Velocity of Water in Pipe per Second. Feet. Incthes. Feet. Inches. Feet. Inches. Feet. Inches. Feet. Inches. Feet. Inches. Feet. Inches. Feet. Inches. Feet. Inches. Feet. Inches. Bore of the n 6 1 0 1 6 2 0 2 6 3 0 6 4 0 4 6 5 0 Pipes. 4.5 1 4.7 2 11.10 4 9.7 7 1.7 10 1.0 13 8.0 17 10. 22 6.7 28 0.2 1 inch. 2 3.0 11.1 1 11.3 3 2.5 4 9.2 6 8.6 9 1.3 11 10.6 15 0.5 18 8 1, Z 2.2 8.4 1 5.5 2 4.9 3 6.9 5 0.5 6 10. 8 11. 11 3.4 14 1 1.8 6.7 1 2. 1 11.1 2 10.3 4 0.4 5 5.6 7 1.6 9 0.3 11 2.5 1I 4 o 1.5 5.6 11.7 1 7.2 2 4.6 3 4.3 4 6.7 5 11.3 7 6.2 9 41 1 1.3 4.8 10. 1 4.5 2 0.5 2 10.6 3 10.9 5 1.1 6 5.4 8 0.1 12 1.1 4.2 8.7 1 2.4 1 9.4 2 6.2 3 5. 4 5.5 5 7.7 7 2 1.0 3.7 7.8 1 0.8 1 7. 2 9.9 3 0.4 3 11.6 5 0.1 6 27 2.9 3.3 7.0 11.5 1 5.1 2 0.2 2 8.8 3 6.8 4 6.1 5 72 2.7 2.8 5.0 9.6 1 2.3 1 8.2 2 3.3 2 11.7 3 9.1 4 8 3 q.6.6 2:4 5.0 8.2 1 0.2 1 5.3 111.4 2 6.6 3 2.7 4 3 Gc'.6 2.1 4.4 7.2 10.7 1 3.1 1 8.5 2 2.7 2 9.8 3 61 4.5 1.9 3.9 6.4 9.5 1 1.4 1 6.2 1 11.8 2 6.1 3 1.4 4j t3.4 1.7 3.5 5.8 8.6 1 0.1 1 4.4 1 9.4 2 3.1 2 9.6 5 4 1.4 2.9 4.8 7.1 10.1 1 1.7 1 5.8 1 10.6 2 4. 6.3 1.2 2.5 4.1 6.1 8.6 11.7 1 3.3 1 7.3 2 7 Ps.3 1.0 2.2 3.6 5.4 7.6 10.2 1 1.4 1 4.9 19. 8 m.25.9 1.9 3.2 4.8 6.7 9.1 11.9 1 3. 1 6.7 9.2.8 1.7 2.9 4.3 6.0 8.2 10.7 1 1.5 1 4.8 10.2.8 1.6 2.6 3.9 5.5 7.5 9.7 1 0.3 1 3.3 11.19.7 1.5 2.4 3.6 5.0 6.8 8.9 11.3 1 2.0 12 Look for the velocity of water in the pipe in the upper line, and in the column heneath it, and opposite to the given diameter, is thle height of the column or head requisite to overcome the friction of such pipe for 100 feet in length, and obtain the required velocity. 174U HYDRAULICS AND HYDRODYNAMICS. GENERAL RULES. Discharge by Horizontal Pipes. 1. THE less the diameter of the pipe, the less is the proportional discharge of the fluid. 2. The greater the length of the discharging pipe, the greater the diminution of the discharge. 3. The friction of a fluid is proportionally greater in small than in large pipes. 4. The velocity of water flowing out of an aperture is as the square root of the height of the head of the water. Theoretically the velocity would be / height X 8. In practice it is V height X 5.4 = velocity in feet per second. Discharge by Vertical Pipes. The discharge of fluids by vertical pipes is augmented, on the principle of the gravitation of falling bodies; consequently, the greater the length of the pipe, the greater the discharge of the fluid. Discharge by Inclined Pipes. A pipe which is inclined will discharge in a given time a greater quantity of water than a horizontal pipe of the same dimensions. Decdlctions from various Experiments. 1. The areas of orifices being equal, that which has the smallest perimeter will discharge the most water under equal heads; hence circular apertures are the nost advantageous. 2. That in consequence of the additional contraction of the fluid vein, as the head of the fluid increases the discharge is a little diminished. 3. That the discharge of a fluid through a cylindrical horizontal tube, the diameter and length of which are equal to one another, is the same as through a simple orifice. 4. That the above tube may be increased to four times the diameter of the orifice with advantage. 5. The velocity of motion that would result from the direct, unretarded action of the column of a fluid which produces it, being a constant, or. 8 The velocity through an aperture in a thin plate, with the same pressure, is 5. Through a tube from two to three diameters in length, projecting outward, 6.5 Through a tube of the same length, projecting inward.5.45 Through a conical tube of the form of the contracted vein.... 7.9 0survilineal and rectasngular pipes discharge less of a fuid than rectilineal pipes. Dischargefroom Reservoirs receiving no Suepply of Water. For prismatic vessels the general law applies, that twice as much would be dis. charged from the same orifice if the vessel were kept full during the time which is required for emptying itself. Discharges'front Comzpound or Divided Reservoirs. The velocity in each may be considered as generated by the difference of the heights in the two contiguous reservoirs; consequently, the square root of the difference will represent the velocity, which, if there are several orifices, must be inversely as their respective areas. Discharge by Weirs and Rectanzgular Notches. The quantity of water discharged is found by talking 2 of the velocity due to the mean height, using 5.1 for the coefficient of the velocity. EXAMPLE.-What quantity of water svill flow from a pond, over a weir 102 inchos in length by 12 inches deep? s/1l foot X 5.1X 8.5 area of weir-= 28.9 cubic feet in one second. HYDRAULICS AND HYIDR0DYNAMIICS. 175 TABLE of the Rise of Water in Rivers, occasioned by the erection of Piers, (4c. _,. " _ Amount of obstruction compared with area of section of the river. 560~1 2 -3 -4 5 6 1 8 >So [ 1et Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. Feet. 1.0157.0377.0698.1192.2012.3521.6780 1.609 6.639 2.0277.0665.1231..2102.3548.6208 1.196 2.838 11.71 3.0477.1144.2118.3618.6107 1.069 2.058 4.885 20.15 4.0760.1822.3372.5759.9719 1.701 3.276 7.775 32.07 5.1165.2793.5168.8782 1.490 2.607 5.020 11.92 49.15 6.1558.3736.6912 1.181 1.993 3.487 6.715 15.94 65.75 7.2078.4983.9221 1.575 2.658 4.651 8.958 21.26i 87.71 8.2678.6423 1.188 2.030 3.426 5.995 11.54 27.40 113.0 9.3359.8054 1.490 2.557 4.296. 7.517 14.48 34.36 141.7 10.4119.9877 1.827 3.122 5.268 9.219 17.75 42.14 173.8 Velocity of Water in Pipes and Sewers. By the experiments of Mr. Rofe, of the Birmingham Waterworks, the following valuable tables are obtained: Table of the heads of water necessary to maintain different velocities oJ water in 100 feet of pipe. V represents the velocities in feet per minute, and C the con. stant number for these velocities. V. C.'V. C. V. C. 60.. 8.62 90.. 17.95 140.. 38.90 70. 11.40 100.. 21.56 150.. 44. 80.. 14.58 120.. 29.70 180.. 62.13 Table of the constant numbers for different diameters. D represents the diameter of the pipe in inches, and C the constant number for their diameters. D. c. D. c. D. c. 4....028 6....078 8...134 5...053 7...104 Then, when h represents the head of water, C-D+= H. ExAMPLE.-It is required to determine what head of water would be necessary to send water through 1500 feet of six-inch pipes to an elevation of 80 feet, and at a velocity of 180 feet per minute: 62.13 62-+1 = 10.22 inches, which X 15 (the number of 100 feet) — 153.3 6+.078 inches (12 feet 9 inches). This, added to 80 feet, gives 92 feet 9.3 inches for the answer. The time occupied in an equal quantity of water through a pipe or sewer of equal lengths, and with equal falls, is proportionally as follows: Tn a right line as 90, in a true curve as 100, and in passing a right angle as 140. 176 WATER WHEELS. WATER WHEELS. THIS subject belongs properly to Hydrodynamics, but a separate classification is here deemed preferable. WATER WHEELS are of three kinds, viz., the Overshot, Undershot, and Breast. The Overshot Wheel is the most advantageous, as it gives the greatest power with the least quantity-of water. The next in order, in point of efficiency, is the Breast Wheel, which may be considered a mean between the Overshot and the Undershot. For a small supply of water with a high fall, the first should be employed; where the quantity of water and height of fall are both moderate, the second form should be used. For a large supply of water with a low fall, the third form must be resorted to. Before proceeding to erect a water wheel, the area of the stream and the head that can be used must be measured. Find the velocity acquired by the water in falling through that height by the rule, viz.: Extract the square root of the height of the head of the water (from the surface to the middle of the gate), and multiply it by 8. NoTE.-Where the opening is small, and the head of water is great, or propor tionally so, use from 5.5 to 8 for the multiplier. EXAMPLE.-The dimensions of a stream are 2 by 80 inches, from a head of 2 feet to the upper surface of the stream; what is the velocity of the water per minute, and what is its weight? 2 feet and 1 of 2 inches = 25 inches = 2.08 feet, /2.08 x *6.5 X 60 = 561.60 feet velocity per minute. And 80 X2 X561.6 feet X 12 inches, +1728 = 624 cubic feet, X 62~ lbs. = 39000 lbs. of water discharged in one minute. To find the Power of an Overshot Wheel. RULE.-Multiply the weight of water in lbs. discharged upon the wheel in one minute by the height or distance in feet from the lower edge of the wheel to the centre of the opening in the gate; divide the product by 50000, and the quotient is the number of horses' power. EXAMPLE.-In the preceding example, the weight of the water discharged per minute is 39000 lbs. If the height of the fall is 23 feet, the diameter of the wheel being 22, what is the power of the wheel I 23 feet - 8 inches clearance below - 22.4 -= 22.33. 39000 x22.33 -50000 = 17.41 horses' power, Ans. To find the Power of a Stream. RULE.-Multiply the weight of the water in lbs. discharged in one minute by the height of the fall in feet; divide by 33000, and the quotient is the answer. * Estimate of velocity. WATER'WHEELS. 17' EXAMPLE. —What power is a stream of water equal to of the following dimensions, viz.: 1 foot deep by 22 inches broad, velocity 350 feet per minute, and fall 60 feet; and what should be the size of the wheel applied to it? 12X22X350X 12 -. 1728 X 62 —X60 feet.]33000 = 72.9, Ans. Height of fall 60 feet, from which deduct for admission of water, and clearance below, 15 inches, which gives 58.9 feet for the diamreter of the wheel. Clearance above 3 15 inches. " below 12 15 inches. The power of a stream, applied to an overshot wheel, produces effect as 10 to 6.6. Then, as 10: 6.6:: 72.9: 48 horses' power equal that of an overshot wheel of 60 feet applied to this stream. When the fall exceeds 10 feet, the overshot wheel should be applied. The higher the wheel is in proportion to the whole descent, the greater will be the effect. The effect is as the quantity of water and its perpendicular height multiplied together. The weight of the arch of loaded buckets in pounds, is found by multiplying 4 of their number, X thenumber of cubic feet in each, and that product by 40. To find the Power of an Undershot Wheel when the Stream is confined to the Wheel. RULE.-Ascertain the weight of the water discharged against the floats of the wheel in one minute by the preceding rules, and divide it by 100000; the quotient is the number of horses' power. NOTE.-The 100000 is obtained thus: The power of a stream, applied to an undershot wheel, produces eftect as 10 to 3.3; then 3.3: 10:: 33000: 100000. When the opening is above the centre of the floats, multiply the weight of the water by the height, as in the rule for an overshot wheel. EXAMPLE.-What is the power of an undershot wheel, applied to a stream 2 by 80 inches, from a head of 25 feet 1 V/25X6.5X60= 1950 feet velocity of water per minute, and 2 X80 - 160 inches X 1950 X 12- -1728 - 2166.6 cubic feet X 62.5 - *135412 lbs. of water discharged in one minute; then 135412-+ 100000 = 1.35 horses' power. NoTE. —The maximum work is always obtained when the velocity of the wheel Is half that of the stream. Let V represent velocity of float boards, and v velocity of water; then -V X force of the water, will be the force of the effective stroke. The effect of an undershot wheel to the power expended is, at a medium, one half that of an overshot wheel. The virtual or effective head being the same, the effect will be very nearly aS the quantity of water expended. When the fall is below 4 feet, an undershot wheel should be applied. Tofind the Powver of a Breast Wheel. RuLE.-Find the effect of an undershot wheel, the head of water of which is the difference of level between the surface and where it strikes the wheel (breast), and add to it the effect of that of an overshot wheel, the height of the head of which is equal to the differ*Equal 160X 12-.1728X62.5X1950 — momentum of water and its velocity. P178 WATER WHEELS. ence between where the water strikes the wheel, and the tail water; the sum is the effective power. EXAMPLE.-What would be the power of a breast wheel applied to a stream 2 x 80 inches, 14 feet from the surface, the rest of the fall being 11 feet?,/14 X6.5 X 60 = 1458.6 feet velocity of water per minute. And 2X80X 1458 X12. 1728 - 1620 cubic feet X62.5 = 101250 lbs. of water discharged in one minute. Then 101250.' 100000 = 1.012 horses' power as an undershot. /11 X 6.5 X 60 = 1290 feet velocity of water per minute. And 2 X 80 X 1290 X 12-.1728 = 1433 cubic feet X 62.5 = 89562 lbs. of water discharged in one minute. XI height of fall — 50000 =19.703 horses, which, added to the above, =20.715, Ans. When the fall exceeds 10 feet, it may be divided into two, and two breast wheels applied to it. When the fall is between 4 and 10 feet, a breast wheel should bs applied. The power of a water wheel ought to be taken off opposite to the point where the water is producing its greatest action upon the wheel. BARKER'S MILL. The effect of this mill is considerably greater than that which the same quantity of water would produce if applied to an undershot wheel, but less than that which it would produce if properly applied to an overshot wheel. For a description of it, see Grier's Mechanics' Calculator, page 234. Make each arm of the horizontal tube, from the centre of motion to the centre of the aperture of any convenient length, not less than - of the perpendicular height of the water's surface above these centres. Multiply the length of the arm in feet by.61365, and the square root of the product will be the proper time for a revolution in seconds; then adapt the gearing to this velocity. Or, if the time of a revolution be given, multiply the square of it by 1.6296 for the proportional length of the arm in feet. Divide the continued product of the breadth, depth, and velocity of the stream in feet by 14.27; multiply the quotient by the square root of the height, and the result is the area of either aperture. Multiply the area of either aperture by the height of the head of water, and this product by 56; the result is the moving force in lbs. at the centre of the apertures. EXAMPLE.-If the fall be 18 feet from the head to the centre of the apertures, then the arm must not be less than 2 feet (as 8 of 182), \/2 X.61365 = 1.107, the time of a revolution in seconds; the breadth of the race 17 inches, the depth 9, and the velocity 6 feet per second; what is the moving force? 17 inches-= 1.41 feet, 9 inches =.75 feet; then 1.41X.75X6 — 14.27 XV /8 X 18 X 56 = 1895 lbs., Ans. WATER WHEELS. 179 To find the Centre of Gyration of a WaVter Wheel. RULE.-Take the radius of the wheel, the weight of its arms, and the weight of its rim, as composed of floats, shrouding, &c. Let R represent the weight of rim, " r " the radius of the wheel, " A " the weight of arms,'W," the weight of the water in action when the buckets are filled, as in operation. Then V(RX rx2 X2+A X r2 X 2+W X r2-R+A+W X 2)- centre of gyration. EXAMPLE.-In a wheel 20 feet diameter, the weight of the rim is 3 tons, the weight of the arms 2 tons, and the weight of the water 1 ton; what is the distance of the centre of gyration from the centre of the wheel? R - 3 tons X102X2 - 6)0 A — 2 " X102X2-400 W — 1 X102. -- 100 1100 3+2+ = 6 X 2 12 - 91.6, the square root of which is 9.5, or 9~ feet, Ans. NoTEs.-At the mill of Mr. Samuel Newlin, at Fishkill Creek, N. Y., 5 barrels of flour canll be ground, and 400 bushels of grain elevated 36 feet per hour wilh a stream and ovelshot wheel of the following dimensions, viz.: Height of head to centre of opening, 241 inches; opening, 14 by 80 inches; wheel 22 feet diameter by 8 feet face; 52 buclkets, each I foot in depth. The wheel making 31 revolutions, driving 3 run of 41 feet stones 130 turns in a minute, with all the attendant machinery. This is a case of maximum effect, in consequence of the gearing being well set up, and kept in good order. At the furnace of Mr. Peter Townsend, Monroe Works, N. J., 30 to 34 tons of No 1 Iron are made per week, with the blast from two 5 feet by 5 feet 1 inch blowing cylinders. The wheel (overshot) being 24 feet diameter, by 6 feet in width, having 70 buckets of 14 inches in depth. The stream is i by 51 inches, having a head 61 feet; the wheel and cylinders each making 44 revolutions pet minute. Rocky Glen Factory, Fishkill, N. Y., containing 6144 self-acting mule spindles, 160 looms, weaving printing cloths 27 inches wide of No. 33 yarn (33 hanks to a pound), and producing 24,000 hanks in a day of 11 hours, is driven by a breast wheel and stream of the following dimensions, viz.: Stream 18 feet by 2 inches, head 20 feet, height of water upon wheel 16 feet, diameter of wheel 26 feet 4 inches, face of wheel 20 feet 9 inches, depth of buckets 154 inches, number of buckets 70. Revolutions, 4~ — per minute. 180 PNEUMATICS. PNEUMATICS. WEIGHT, ELASTICITY, AND RARITY OF AIR. THE pressure of the air at the surface of the earth is, at a mean rate, equal to the support of 29.5 inches of mercury, or 33.18 feet of fresh water. It is usually estimated in round numbers at 30 inches of mercury and 34 feet of water, or 15 lbs. pressure upon the square inch. The Elasticity of air is inversely as the space it occupies, and directly as its density. When the altitude of the air is taken in arithmetic proportion, its Rarity will be in geometric proportion. Thus, at 7 miles above the surface of the earth, the air is 4 times rarer or lighter than at the earth's surface; at 14 miles, 16 times at 21 miles, 64 times, and so on. The weight of a cubic foot of air is 527.04 grains, or 1.205 ounces avoirdupois. At the temperature of 330, the mean velocity of sound is 1100 feet per second. It is increased or diminished half a foot for each degree of temperature above or below 330~. To compute Distances by Sound. RULE.-Multiply the time in seconds by 1100, and the product is the distance in feet. EX.AMPLE.-After observing a flash of lightning, air at 500, it was 5 seconds before I heard the thunder; what was the distance of the cloud 1 50-33 1100+ 2 x5 -5280 - 1.049 miles, Ans. To compute what Degree of Rarefaction may be effected in a Vessel. Let the quantity of air in the vessel, tube, and pump be represented by 1, and the proportion of the capacity of the pump to the vessel and tube by.33; consequently, it contains 4 of the air in the united apparatus. Upon the first stroke of the piston this fourth will be expelled, and iF of the original quantity will remain: 4 of this will be expelled upon the second stroke, which is equal to. ~F of the original quantity; and, consequently, there remains in the ap paratus T of the original quantity. Calculating in this way, the following table is easily made: No. of Strokes. Air expelled at each stroke. Air remaining in the vessel. 1 4 =4 i =4 3 3 9 3X3 16 - 4X4 16 - 4X4 9 3X3 27 3X3X3 64 4X4X4 64 -4X4X4 27 3X3X3 81 3X3X3X3 256- 4X4X4X4 9256 4X4X4X4 81 3x33X3 243 3X3X3X3X3 1024 1 4X4X4X4X4 1024 - 4X4X4X4X4 PNEUMATICS. 18 And so on, continually multiplying the air expelled at the preceding stroke by 3, and dividing it by 4; and the air remaining atter each stroke is found by multiplying the air remaining after the preceding stroke by 3, and dividing it by'4. Measurement of Heights by Means of the Barometer. ipproximate Rule. For a mean temperature of 550, = required difference in height in feet, h = the height of the mercury at the lower station, h' = the height of the mercury at the upper station, h-h' X = 55000 X. Add 1- of this result for each degree which the swean temperature of the air at the two stations exceeds 550, and deduct as much for each degree below 550. Velocity and Force of Wind. Miles in an Feet in a Pressure on a square hour. minute. foot in pounds avoir- Description. dupois. 1 88.005 Barely observable. 2: 176.020 2 264.020 } Just perceptible. 3 264.045 4 352.080 Light breeze. 5 440.125 6 528.180 Gentle, pleasant wind. 8 704.320 10 880.s5004 Brisk blow. 15 1320 1.125 Brisk blow. 20o 1760 2.000 I Very brisk. 25 2200 3.125 30 2640 4.500 35 3080 6.125 High wind. 40 3520 8.000 Very high. 45 3960 10.125 50 4400 12.500 Storm. 60 5280 18.000 Great storm. 80 7040 32.000 Hurricane. 100 8800 50.000 Tornado, tearing up trees, &c. To find the Force of Wind acting perpendicularly upon a Surface. RULE. —Multiply the surface in feet by the square of the velocity in feet, and the product by.00228; the result is the force in avoirdupois pounds. Q 182 STATICS. STATICS. PRESSURE OF EARTH AGAINST WALLS. A B IL D c LET A B C D be the verticsal section of a wall, behind which is d bank of earth, A Dfe; I-t DG be the line of rupture, or natural slope which the earth would assume but for the resistance of the wall. In sandy or loose earth, the angle G D H is generally 300; in firmer earth it is 360, and in some instances it is 45~. The angle formed with the vertical by the earth, A D G, that exerts the greatest horizontal stress against a wall, is half the angle which the natural slope makes with the vertical. If the upper surface of the earth and the wall which supports it are both in one horizontal plane, Then the resultant I n of the pressure of the bank, behind a vertic'al wall, is at a distance D n of I A D. In vegetable earths, the friction is i the pressure; in sands, T.4 The line of rupture A G in a bank of vegetable earth is =.618 of A D. When the bank is of sand, it is.677 of A D. If of rubble, it is.414 of A D. Tlicknless of Walls, both Faces Vertical. Brick. Weight of a cubic foot, 109 lbs. avoirdupois, bank of vegetable earth behind it, A B =-. 6 A D. Unshewn stones. 135 lbs. per cubic foot, bank as before, AB =.15 A D. Brick. Bank clay, well rammed, A B -.17 A D. Hewn freestone. 170 lbs. per cubic foot, bank of vegetable earth, A B —.13 A D; If the bank is of clay, A B =.14 A D. Bricks. Bank of sand, A B =.33 A D. Unhewn stone. Bank of sand, A B =.30 A D. Hewn freestone. Bank of sand, A B =.26 A D. When the bank is liable to be saturated with water, the thickness of the wat must be doubled. For farther notes, and for the Equilibrium of Piers, see Gregory's XJathemati, pages 220 to 2t2. DYNAMICS. 183 DYNAMICS. DYNAMICS is the investigation of body, force, velocity, space, and time. Let them be represented by their initial letters bfv s t, gravity by g, and momentum or quantity of motion by m; this is the effect pro-:duced by a body in motion. Force is motive, and accelerative or retardative. Motive force, or momentum, is the absolute force of a body in motion, and is the product of the weight or mass of matter in the body, multiplied by its velocity. Accelerative or retardative force is that which respects the velocity of the motion only, accelerating or retarding it, and is found by the force being divided by the mass or weight of the body. Thus, if a body of 4 lbs. be acted upon by a fcace of 40 lbs., the accelerating force is 10 lbs.; but if the same force of 40 act upon another body of 8 lbs., the accelerating force then is 5 lbs., only half the former, and will produce only half the velocity. Uniform?, Motion. The space described by a body moving uniformly is represented by the product of the velocity into the time. With momenta, m varies as b v. EXAtPLrE.-Two bodies, one of 20, the other of 10 lbs., are impelled by the same momentum, say 60. They move uniformly, the first for 8 seconds, the second for 6; what are the spaces described by both? b 60 60 - v, or- = 3, and = 6. v 2 u iThen tv - 3X8 = 24 = s, and 6X6 = 36 - s. Thus the spaces are 24 and 36 respectively. Motion Uniformly Accelerated. In this motion, the velocity acquired at the end of any time whatever, is equal to the product of the accelerating force into the time, and the space described is equal to the product of half the accelerating force into the square of the time. The spaces described in successive seconds of time are as the odd numbers, 1, 3, 5, 7, 9, &c. Gravity is a constant force, and its effect upon a body falling freely is represented by g. The following theorems are applicable to all cases of motion uniformly accelerated by any constant force: t s = t= =gFt2 F - -F = -/-2gfs. ts v s v gF lgF' v 2s v2 gt gtz2 2gs When gravity acts alone, as when a body falls in a vertical line, F is omitted, and we have, v2 8 = Agt - - = - ttv. 2g s=gt = /2 gs. * See page 142 for notice of experiments on bodies falling freely, made by the author in 1852. 184 DYNAMICS. v 2s 2s g v g v 2s v2 g -- t't t2 2s NOTE.-g is obviously 32.166 from what has been given in rules for Gravitation, aid is. the force of gravity. If, instead of a heavy body falling freely, it be propelled vertically upward or downward with a given velocity, v, then s- tv gt2; an expression in which - must be taken when the projection is upward, and + when it is downward. Motion over a Asixed Pulley. Let the two weights which are connected by the cord that goes over the pulley be represented by xV and w; then W-+w F in the fornnulm where F is used; so that W-+w s x it. Or, if the resistance of the friction and inertia of the pulley be represented by r, then W-w gD EXAMPLE.-If by experiment it is ascertained that two weights of 5 and 3 lbs. over a pulley, the heavier weight descended only 50 feet in 4 seconds, what is the measure of r? If r is not considered, the heavier weight would fall 64& feet. Then Ww gt2 = 50 feet. And, as 5-3+rr: 5+3:: 64: 50; That is r: 5+-3:: 144-: 50. Whence... r 8X140= 2.293 lbs., ofns. 50 TABLE of the Effects of a Force of Traction of 100 lbs. at different Velocities, on Canals, Railroads, and Turnpikes. Velocity. On a Canal. On a Railroad. On a Turnpike. Miles Feet per Moass Useful Nass Useful lass Usenfu per hr. second. moved. effect, moved. effect. mooed, effect. lbs. Ibs. Ibs. Ibs. Ibs. lbos. 2- 3.66 55.500 39.400 14.400 10.800 1.800 1.350 3 4.40 38.542 27.361 14.400 10.800 1.800 1.350 3-. 5.13 X28.316 20.100 14.400 10.800 1.800 1.350 4-1 5.86 21.680 15.390 14.400 10.800 1.800 1.350 5 7.33 13.875 9.850 14.400 10.800 1.800 1.350 6 8.80 9.635 6.840 14.400 10.800 1.800 1.350 7 10.26 7.080 5.026 14.400 10.800 1.800 1.350 8 11.73 5.420 3.848 14.400 10.800 1.800 1.350 9 13. 20 4.282 3.040 14.400 10.800 1.800 1.350 10 14.66 3.468 2.462 14.400 10.800 1.800 1.3 50 13.5 19.9 1.900 1.350 14.400 10.800 1.800 1.350 The load carried, added to the weight of the vessel or carriage which contains It. forms the total mass moved, and the useful effect is the load. The force of traction on a canal vtlries as the square of the velocity; on a railroad or turnpike the force of traction is constant, but the mechanical power neceossary to move the carriage increases as the velocity. PENDULUMS. 185 PENDULUMS. THE Vibrations of Pendulums are as the square roots of their lengths. The length of one vibrating seconds in New-York at the level of the sea is 39.1013 inches. To find the Length of a Pendulum for any Given Number of Vibrations in a Minute. RULE.-As the number of vibrations given is to 60, so is the square root of 39.1013 (the length of the pendulum that vibrates seconds) to the square root of the length of the pendulum required. EXAMPLE.-What is the length of a pendulum that will make 80 vibrations in a minute! As,39.1013 x 60 = 375, a constant number, 375 Then -- = 4.6875, and 4.6875 --— = 21.97 inches, Ans. 80 The lengths of pendulums for less or greater times is as the square of the times; thus, for ~ a second it would be the square of 39.1013 i, or - -_ 9.7753 inches, the length of a i second pendulum at New-York. To find the Number of Vibrations in a Minute, the Length of the Pendulum being given. RULE.-As the square root of the length of the pendulum is to the square root of 39.1013, so is 60 to the number of vibrations required. EXAMPLE.-HOW many vibrations will a pendulum of 49 inches long make in a minute? /49: ~/39.1013:: 60: number of vibrations. 375 Or, ~ = 53.57 vibrations, Ans. To find the Length of a Pendulum, the Vibrations of which will be -the same Number as the Inches in its Length. RULE.-Square the cube root of *375, and the product is the answer. EXAMPLE. — /375 = 7.211247, and 7.2112472 = 52.002, Ans. The Length of a Pendulum being given, to find the Space through which a Body will fall in the Time that the Pendulum makes one Vibration. RULE. —Multiply the length of the pendulum by 4.93482528, and it will give the answer. * 375 is the constant for the latitude of New-York; in any other place, multiply the square root of the length of the pendulum at that place by 60. Q2 186 CENTRE OF GYRATION. EXAMPLE.-The length of the pendulum is 39.1013 inches; what it. the distance a body will fall in one vibration of it 1 39.1013X4.9348 = 192.9578 inches, or 16.8298 feet, Ans. All vibrations of the same pendulum, whether great or small, are performed very nearly in the same time. In a Simple Pendulum, which is, as a ball, suspended by a rod or line, supposed to be inflexible, and without weight, the length of the pendulum is the distance from its centre of gravity to its point of suspension. Otherwise, the length of the pendulum is the distance from the point of suspension to the Centre of Oscillation,* which does not coincide with the centre of gravity of the ball or bob. CENTRE OF GYRATION. THE Centre of Gyration is the point in any revolving body, or system of bodies, that, if the whole quantity of matter were collected in it, the angular velocityt would be the same; that is, the momentum of the body or system of bodies is centred at this point. If a straight bar, equally thick, was struck at this point, the stroke would communicate the same angular velocity to the bar as if the whole bar was collected at that point. To find the Centre of Gyration. RULE 1.-Multiply the weight of the several particles by the squares of their distances in feet from the centre of motion, and divide the sum of the products by the weight of the entire mass; the square root of the quotient will be the distance of the centre of gyration from the centre of motion. EXAMPLE.-If two weights of 3 and 4 lbs. respectively be laid upon a lever (which is here assumed to be without weight) at the respective distances of 1 and 2 feet, what is the distance of the centre of gyration from the centre of motion (the fulcrum)? 3X1 =- 3. 4X22 = 16. 3-~-16 19 3+46 = 19 - 2.71, and /2.71 = 1.64 feet, Ans. 3+4 7 That is, a single weight of 7 lbs., placed at 1.64 feet from the fulcrum, and revolving in the same time, would have the same impetus as the. two weights in their respective places. * See Centre of Oscillation. t The angrular velocity of a body or system of bodies is the motion of a line c pln necting any point with the axis of motion, and is the same in all parts of the same revolving system. CENTRE OF GYRATION. 187 RULE 2. —Multiply the distance of the centre of oscillation, from the centre or point of suspension, by the distance of the centre of gravity from the same point, and the s4uare root of the product will be the answer. ExAMPLE. —The centre of oscillation is 9 feet, and that of gravity is 4 feet from the centre of the system, or point of suspension; at what distance from this point is the centre of gyration? 9 X4 - 36, and /36 = 6 feet, Ans. The following are the distances of the centres of gyration from the centre of motion in various revolving bodies, as given by Mr. Farey: In a straight, uniform Rod, revolving about one end; length of rod X.5773. In a circular Plate, revolving on its centre; the radius of the circle X.7071. In a circular Plate, revolving about one of its diameters as an axis; the radius X.5. In a Wheel of uniform thickness, or in a Cylinder revolving about the axis; the radius X.7071. In a solid Sphere, revolving about one of its diameters as an axis; the radius X.6325. In a thin, hollow Sphere, revolving about one of its diameters as an axis; the radius X.8154. In a Cone, revolving about its axis; the radius of the circular base X.5477. In a right-angled Cone, revolving about its vertex; the height of the cone X.866. In a Paraboloid, revolving about its axis; the radius of the circular base X.5773. In a straight Lever, the arms being R and r, the distance of the centre of gyralion from the centre of motion -= /R —+r 3(R —r)' NOTE.-The weight of the revolving body, multiplied into the height due to the velocity with which the centre of gyration moves in its circle, is the energy of the body, or the mechanical pow20er which must be commaunicated to it to give it that motion. EXAMPLE.-In a solid sphere revolving about its diameter, the diameter being 2, feet, the distance of the centre of gyration ir 12X.6325 = 7.59 inches. 188 CENTRES OF PERCUSSION AND OSCILLATION. CENTRES OF PERCUSSION AND OSCILLATION. THE Centres of Percussion and Oscillation are in the same point when their bodies are symmetrical with regard to the plane of their motion; or when they are solids of revolution, which is commonly the case, their properties are similar, and their point is, that in a body revolving around a fixed axis, which, when stopped by any force, the whole motion, and tendency to motion, of the revolving blody is stopped at the same time. It is also that point of a revolving body which would strike any obstacle with the greatest effect, and from this property it has received the name of percussion. As in bodies at rest, the whole weight may be considered as col. lected in the centre of gravity; so in bodies in motion, the whole force may be considered as concentrated in the centre of percussion: therefore, the weight of a bar or rod, multiplied by the distance of the centre of gravity from the point of suspension, will be equal to the force of the rod, divided by the distance of the centre of percussion front the same point. EXAMPLE. —rhe length of a rod being 20 feet, and the weight of a foot in length equal 100 oz., having a ball attached at the under end weighing 1000 oz., at what point of the rod from the point of suspension will be the centre of percussion?* The weight of the rod is 20X 100 = 2000 oz., which, multiplied by half its length, 2000X10 = 20000, gives the momentum of the rod. The weight of the ball = 1000 oz., multiplied by the length of rod, - 1000X20, gives the momentum of the ball Now the weight of the rod multiplied by the square of the length, and divided by 3, - 200 0 266666, the force of the rod, and the weight of the ball multi3 plied by the square of the length of the rod, 1000X202 = 400000, is the force of the ball: therefore, the certre of percussion = 266660+400000 = 16.66 feet. 20000+20000 EXAM.PLE. —SUuppose a rod 12 feet long, and 2 lbs. each foot in length, with 2 balts of 3 lbs. each, one fixed 6 feet fiom the point of suspension, and the other at the end of the rod; what is the distance between the points of suspension and percussion? 12X 2X6 - 144, momentum of rod, 3X 6 = 18 " of ist ball, 3X 12 - 36 " of 2d 198 24X144 1152, force of rod, 3X 36= 108 " of 1st ball, 3X144= 432 " of 2d ball, 1692 1692 therefore the centre of percussion -= 1 = 8.545 feet from the point of suspension 198 As the centre of percussion is the same with the centre of escillation in the non-application to practical purposes, the following is the easiest and simplest mode of finding it in any beam, bar, &c.: Suspend the body very freely by a fixed point, and make it vibrate in small arcs, counting the nunber of vibrations it makes in any time, as a minute, and let the number of vibrations made in a minute be called n; then shall the distance of the 140850 centre of oscillation from the point of suspension be =50 inches. For the length of the pendulum vibrating seconds, or 60 times in a minute, being 39k inch* an 20 feet long, 2+ca b = 100 oz. weight of a foot in length, abX = centre of percussion c = 1000 " fixed at end, i abXa-Fac CENTRES OF PERCUSSION AND OSCILLATION. 189 es, and the lengths of the pendulums being reciprocally as the square of the number of vibrations made in the same time, therefore n2 60: 394: 602X9 140850 2-,being the length of the pendulum which vibrates n times in a minute, or the distance of the centre of oscillation below the axis of motion. There are many situations in which bodies are placed that prevent the application of the above rule, and for this reason the following data are given, which will be found useful when the bodies and the forms here given correspond: 1. If the body is a heavy, straight line of uniform density, and is suspended by one extremity, the distance of its centre of percussion is 2 of its length. 2. In a slender rod of a cylindrical or prismatic shape, the breadth of which is very small compared with its length, the distance of its centre of percussion is nearly 2 of its length from the axis of suspension. If these rods were formed so that all the points of their transverse sections were equidistant from the axis of suspension, the distance of the centre of percussion would be exactly A of their length. 3. In an Isosceles triangle, suspended by its apex, and vibrating in a plane perpendicular to itself, the distance of the centre of percussion is 4- of its altitude. A line or rod, whose density varies as the distance from its extremnity, or the point of suspension; also Fly-wheels, or wheels in general, have the same relation as the isosceles triangle, the centre of percussion being distant from the centre of suspen sion' of its length. 4. In a very slender cone or pyramid, vibrating about its apex, the distance of its centre of percussion is nearly 4 of its length.'The distance of either of these centres from the axis of motion is found thus: If the Axis of Mtlotion be in the vertex of the figure, and the motion be flatwise; then, in a right line, it is A of its length. In an Isosceles Triangle =; of its height. In a Circle - 4 of its radius. In a Parabola= -- of its height. But if the bodies move sidewise, it is, In a Circle = 4 of its diameter. In a Rectangle, suspended by one angle, =.A of the diagonal. In a Parabola, suspended by its vertex, = - axis + parameter; but if suspended by the middle of its base,- 4 axis + I parameter. In the Sector of a Circle 3 X ar X radius 4X chord radius of base 2 In a Cone = 4 axis - X axis 9 5 X axis 2X radius 2 In a Sphere X radius + radius q+ t, t representing the length of the thread 5(t X radius) by which it is suspended. EXAMPLE.-What must be the length of a rod without a weight, so that when hung by one end it shall vibrate seconds? To vibrate seconds, the centre of oscillation must be 39.1013 inches ftom that of suspension; and as this must be 2 of the rod, Then 2: 3:: 39.1013: 58.6519, dns. EXAMPLE.-What is the centre of percussion of a rod 23 inches long? a of 23 — 15.3 inches from the point of suspension or motion. EXAMPLE.-In a sphere of 10 inches diameter, the thread by which it is suspended being 20 inches, where is the centre of percussion or oscillation? 2X52 50 5(20+ - 5 - 20 + - 0=- + 25 = 25.4, J2ns. 190 CENTRAL FORCES. CENTRAL FORCES. ALL bodies moving around a centre or fixed point have a tendency to fly off in a straight line: this is called the Centrifugal Force; it is opposed to the Centripetal Force, or that power which maintains the body in its curvilineal path. The centrifugal force of a body, moving with different velocities in the same circle, is proportional to the square of the velocity. Thus, the centrifugal force of a body making 10 revolutions in a minute is four times as great as the centrifugal force of the same body making 5 revolutions in a minute. To find the Centrzfjugal Force of any Body. RUiE 1.-Divide the velocity in feet per second by 4.01, also the square of the quotient by the diameter of the circle; the quotient is the centrifugal force, assuming the weight of the body as 1. Then this, multiplied by the weight of the body, is the centrifugal force. ExAMPLE.-What is the centrifugal force of the rim of a flywheel 10 feet in diameter. running with a velocity of 30 feet in a second. 30-: 4.01X 7.48 — I0 = 5.59 times the weight of the rim, Ans. NOTE.-When great accuracy is required, find the centre of gyration of the body and take twice the distance of it from the axis for the diameter. RuLE 2.-Multiply the square of the number of revolutions in a minute by the diameter of the circle in feet, and divide the product by the constant number 5870; the quotient is the centrifugal force when the weight of the body is 1. Then, as in the previous rule, this quotient, multiplied by the weight of the body, is the centrifugal force. EXAMPLE.-What is the centrifugal force of a grindstone, weighing 1200 lbs., 42 inches in diameter, and turning with a velocity of 400 revolutions in a minute 1 4002 x 35 1200 - 114480 lbs., Ans. 5870 The central forces are as the radii of the circles directly, and the squares of the times inversely; also, the squares of the times are as the cubes of the distances Hence, let v represent velocity of body in feet per second, w " weight of body, r " radius of circle of revolution, c " centrifugal force. Then r2 c, and 2 r; rX32 surface. The mean weight of a column of air a foot square, and of an altitude equal to the height of the atmosphere, is equal to 2116.8 lbs. avoirdupois. it consists of oxygen 20, and nitrogen 80 parts; and in 10.000 parts there are 4.9 parts of carbonic acid gas. The mean pressure of the atmosphere is usually estimated at 14.7 lbs. per square inch. 13.29 cubic feet of air weigh a lb. avoirdupois, hence 1 ton of air will occupy 29769.6 cubic feet. The rate of expansion of air, and all other Elastic Fluids, for aU temperatures, is uniform. From 320 to 2120 they expand from 1000 to 1376, equal to i of their bulk for every degree of heat. Weight and Volume of Air required for the Combustion of several 1Materials. Matenals. Weight in lbs. Volumes in cubic feet at 320. at 320. at 60_. Virginia pine.. 4.47 55.8 59.4 Turf... 4.6 57.5 61.1 Bituminous coal.. 9.26 115.8 123. Anthracite coal.- - Coke l 11.46 143.4 152.3 Oil. Tallow.... 15. 187.6 199.3 Wax.... 19 to 20 tons of air with coke or anthracite coal, and 18 tons with charcoal, are required to make 1 ton of pig iron, the ore yielding 33 per cent. of iron. See Heat, page 201. *-1 6 equal.002087 for each degree. 204A LIGHT-TONNAGE. LIGHT. LIGHT is similar to caloric in many of its qualities, being emitted in the form ol rays, and subject to the same laws of reflection. It is of two kinds, NI'atural and.trtificial; the one proceeding from the sun and stars, the other from heated bodies. Solids shine in the dark only when heated from 6000 to 7000, and in daylight when the temperature reaches 10000. Relative intensity of light fiom the burning of various bodies is, for wax, 101 parts; tallow, 100; oil in an Argand lamp, 110; in a common lamp, 129; and an ill-snuffed candle, 229. By experiments on coal gas, it appears that above 20 cubic feet are required to produce light equal in duration and in illuminating powers to a pound of tallow candles, six to a pound, set up and burned out one after the other. In distilling 56 lbs. coal, the quantity of gas produced in cubic feet when the distillation was effected in 3 hours was 41.3, in 7 hours 37.5. in 20 hours 33.5, and in 25 hours 31.7. TONNAGE. BY a law of Congress, the tonnage of vessels is found as follows: FOR A DOUBLE-DECKED. Take the length from the fore part of the stem to the after side of the sternpost above the upper deck; the breadth at the broadest part above the main wales; half of this breadth must be taken as the depth of the vessel; then deduct from the length 3 of the breadth, multiply the remainder by the breadth, and the product by the depth; divide this last product by 95, and the quotient is the tonnage. EXAMPLE.-What is the tonnage of a ship of the line, measuring, as above, 210 feet on deck, and 59 feet in breadth 59- 2_= 29.5, depth. 210 — of 59 = 174.6X59X29.5 —95 = 3198.8 tons. FOR A SINGLE-DECKED. Take the length and breadth as above directed for a double-deckod, and deduct from the length W of the breadth; take the depth from the under side of the deck-plank to the ceiling of the hold; then proceed as before. EXAMPLE.-The length of a vessel is (as above) 223 feet, the breadth 39~ feet, and the depth of hold 236 feet; what is the tonnage 223- 3 of 39.5 =- 199.3X 39.5X23.5 95 -1947.3 tons. A ton will stow 3- bales cotton. NoTE. —The burden of similar ships are to each other as the cubes of their like dimensions. TONNAGE. 205 CARPENTERS' MEASUREMENT. FOR A SINGLE-DECKED. Multiply the length of keel, the breadth of beam, and the depth of the hold together, and divide by 95. FOR A DOUBLE-DECKED. Multiply as above, taking half the breadth of beam for the depth of the hold, and divide by 95. To find the Tonnage of English Vessels. IvULE. —Divide the length of the upper deck between the afterpart of the stein and the forepart of the sternpost into 6 equal parts, and note the foremost, middle, and aftermost points of division Measure the depths at these three points in feet. and tenths of a foot, also the depths tfom the under side of the upper deckl to the ceiling at the limbel strake; or, in case of a break in the upper deck, from a line stretched in continuation of the deck. For the breadths, divide each depth into 5 equal parts, and measure the inside breadths at the following points, viz.: at 1 and at A from the upper deck of the foremost and aftermost depths, and at ~ and, from the upper deck of the midship depth. Take the length, at half the midship depth. from the afterpart of the stem to the forepart of the sternpost. Then, to twice the midship depth, add the foremost and aftermost depths for the sum of the depths; and add together the foremost upper and lower breadths, 3 times the upper breadth with the lower breadth at the midship, and the upper and twice the lower breadth at the after division for the sum of the breadths. Multiply together the sum of the depths, the sulm of the breadths, and the length, and divide the product by 3500, which will give the number of tons, or register. If the vessel have a poop or half-deck, or a break in the upper deck, measure the inside mean length, breadth, and height of such part thereof as may be included within the bulkhead; multiply these three measurements together, and divide the product by 92.4. The quotient will be the number of tons to be added to the result as above found. For Open Vessels. The depths are to be taken from the upper edge of the upper strake. For Steam rVessels. The tonnage due to the engine-room is deducted from the total tonnage calculated by the above rule. To determine this, measure the inside length of the engine-room from the foremost to the afternnost bulkhead; then multiply this length by the midship depth of the vessel, and the product by the inside midship breadth at 0.40 of the depth from the deck, and divide the final product by 92.4. S 206 STEAM. STEAM. STEAM, arising from water at the boiling point, is equal to the pressure of the atmosphere, which is 14.706 lbs. on the square inch. Under the pressure of the atmosphere alone, water cannot be heated above the boiling point. A cubic inch of water, evaporated under the ordinary atmospheric pressure, is converted into 1700 cubic inches of steam, or, in round numbers, 1 cubic foot, and gives a mechanical force equal to the raising of 2200 lbs. 1 foot high. The force of steam is the same at the boiling point of every fluid. The elasticity of the vapour of spirit of wine, at all temperatures, is equal to 2.125 times that of steam. It has already been stated (see Heat) that the sum of sensible and latent heats is 12020, and that 1400 of sensible heat becomes latent upon the liquefaction of ice; also, that 1 lb. of water converted into steam at 2120 will heat 51 lbs. of water at 320 to 2120, and that the sum is 6- lbs. of water. The practical estimate of the velocity of steam, when flowing into a vacuum, is about 1400 feet in a second when at an expansive power equal to the atmosphere; and when at 20 atmospheres, the velocity is increased but to 1600 feet. When flowing into the air under a similar power, about 650 feet per second, increasing to 1600 feet for a pressure of 20 atmospheres. Specific gravity of steam at the pressure of the atmosphere.488, air being 1. 27.206 cubic feet of steam at the pressure of the atmosphere, equal 1 lb. avoirdupois. A pressure of 1 lb. on a square inch will raise a mercurial steam gauge (syphon) 1.01995 inches. A column of mercury 2 inches in height will counterbalance a pressure of.9804 lbs. on a square inch. TABLE of the Boiling Points corresponding to Altitudes of the Barome. ter between 26 and 31 Inches. Barometer. Boiling point. I Baromneter. i Boilingpoint. arometer. Bailing point. — ~t~126. 204.910 28. 208.43~ 30. 21.0o 26.5 205.790 28.5 1 209.310 30.5 212.8' 27. 206.670 29. 210.19~ 31. 213.760 27.5 i07.550 29.5 211.0 70 TABLE of the Expansive Force of Steam, fromt 212~ to 352~~, in atmospheres of 30 inches of MlIercury. (From experiments of Committee of Franklin Institute.) Atmo- Degrees of Atmo. Degrees of Atmo- Degrees of At.mo- Degrees of ppheres. heat. spheres. heat. spheres. heat. spheres. heat. 1. 212.0 3.5 284.0 6. 315.50 8.5 340.50 1.5 235.0 4. 201.50 6.5 321.0 9. 345.0 2. 250.0 4.5 298.50 7. f 326.0 9.5 349.0 2.5 264.~ 5. 304.50 7.5 331.0 10. 352.50 3. 275._ 5.5 310.0 _8. _ 336.0 A.~ ~ ~~~ I 3~0 I _ I a._ - __ __..... STEAM. 207 To ascertain the Number of Cubic Inches of Water, at any Given Temperature, that must be mixed with a Cubic Foot of Steam to reduce the Mixture to any Required Temperature. RuLE.-Front the required temperature subtract the temperature of the water; then find how often the remainder is contained in the required temperature, subtracted from 1202~, and the-quotient is the answer. ExAMPLE.-The temperature of the condensing water of an engine is 800, and the required temperature 1000; what is the proportion of condensing water to that evaporated? 1202-100 100-80=20. Then 1 -55.1, ns. 20 Or, let w represent temperature of condensing water, t the required temperature, and h the sum of sensible and latent heats. h-t Then — = water required. t-w'T'o ascertain the Quantity of Steam required to raise a Given Quantity of /Vater to any Given Temperature. RULE.-Multiply the water to be warmed by the difference of temperature between the cold water and that to which it is to be raised, for a dividend; then to the temperature of the steam add 9900, and from that sum take the required temperature of the water foir a divisor; the quotient is the quantity of steam in the same terms as the water. EXAMPLE. — What quantity of steam at 2120 will raise 100 cubic feet of water at 800 to 2120? 100X2120 —80'>12o __990~x2O- -11 3.3 cubic feet of water formed into steam, occupying (13.3X 2120+9900-2120 1700) 22610.0 cubic feet of space. To ascertain the Specific Gravity of Steam. RULE.-Divide the constant number 829.6 (1700X).488) by the volume of the steam at the temperature or pressure at which the gravity is required. To ascertain the Volume ofJ Steam under a given Pressure, the Temperature answering to that Pressure, in Steam at its Maximum Densily for its Temperature being given. 1+.00202 x(t —32) 18329=V. p When p represents the pressure of the steam in pounds per square inch, t the temperature in degrees (Fah.), and V the volume required. To ascertain the Temperature of Steam. From the pressure in inches of mercury subtract the constant 0.1, divide the logarithm of the remainder by 5.13, and to the quotient add the logarithm 2.132794; find the natural number of the sum of the logarithms, from it subtract the constant 51.3, and the remainder will be the temperature. ExAMPLE. —When the pressure is = 397.8 inches of mercury, what is the temperature of the steam? 397.8- 0.1 = 397.7. log. 397.7 =2.59956. Then 2.59956 - 5.13=0.506737 log. 2.132794 Natural number, 436. = log. 2.639531 Constant temperature, 51.30 384.70 = temperature required. 208 STEAM, TABLE of the Elastic Force, Temperature, and Volume of Steam. From a Temperature of 32~0 to 387.30, and from a Pressure of.200 to 408. inches of IMercury. Elastic Force in [ Elastic Force in Inches of Pounds per I Temper- ol- Inches of Pounds per Temper- Vol. mercury. square incli. ature. ume. ioercury. square inch. ature. ume..200.098 32 187407 49.98 24.5 239.9 1064.221.108 35 170267 51. 25 241. 1044.263.129 40 144529 53.04 26 243.3 1007.316.155 45 121483 55.08 27 245.5 973.375.184 50 103350 57.12 28 247.6 941.443.217 55 88388 59.16 29 249.6 911.524.257 60 75421 61.2. 30 251.6 883.616.302 65 64762 63.24 31 253.6 857.721.353 70, 55862 65.28 32 255.5 833.851.417 75 47771 67.32 33 257.3 810 1..49 80 41031 69.36 34 259.1 788 1.17.573 85 35393 71.4 35 260.9 767 1.36.666 90 30425 71.44 36 262.6 748 1.58.774 95 26686 75.48 37 264.3 729 1.86.911 100 22873 77.52 38 265.9 712 2.04 1. 103 20958 7 9.56 39 267.5 695 2.18 1.068 105 1 9693 81.6 40 269.1 679 2.53 1.24 110 16667 83.64 41 270.6 664 2.92 1.431 115 14942 85.68 42 272.1 649 3.33 1.632 120 13215 87.72 43. 273.6 635 3.79 1.857 125 11723 89.76 44 275. 622 4.34 2.129 130 10328 91.8 45 276.4 610 5. 2.45 135 9036 93.84 46 277.8 598 5.74 2.813 140 7938 95.88 47 279.2 586 6.53 3.1 145 7040 97.92 48 280.5 575 7.42 3.636 150 6243 99.96' 49 281.9 564 8.4 4.116 155 5559 102. 50 283.2 554 9.46 4.635 160 4976 104.04 51 284.4 544 10.68 5.23 165 4443 1 06.08 52 285.7 534 12.13 5.94 170 3943 108.12 53 286.9 525 13.62 6.67 175 3538 110.16 54 288.1 516 15.15 7.42 180 3208 112.2 55 289.3 508 17. 8.33 185 2879 114.24 56 290.5 500 19. 9.31 190 2595 116.28 57 291.7 492 21.22 10.4 195 2342 118.32 58 292.9 484 23.64 11.58 200 2118 120.36 59 294.2 477 26.13 12.8 205 1932 122.4 60. 295.6 470 28.84 14.13 210 1763 124.44 61 296.9 463 29.41 14.41 211 1730 126.48 62 298.1 456 30. 14.7 212 1700 128.52 63 I299.2 449 30.6 15. 212.8 1669 130.56 64 300.3 443 31.62 15.5 214.5 1618 132.6 65 301.3 437 32.64 16. 216.3 1573 134.64 66 302.4 431 33.66 16.5 218. 1530 136.68 67 303.4 425 34.68 17. 219.6 1488 138.72 68. 304.4 419 35.7 17.5 221.2 1440 140.76 69 305.4 414 36.72 18. 222.7 1411 142.8 70 306.4 408 37.74 18.5 224.2 1377 144.84 71 307.4 403 38.76 19. 225.6 1343 146.88 72 308.4 398 39.78 19.5 227.1 1312 148.92 73 309.3 393 40.80 20. 228.5 1281 150.96 74 310.3 388 41.82 20.5 229.9 1253 153.02 75 311.2 383 42.84 21. 231.2 1225 155.06 76 312.2 379 43.86 21.5 232.5 1199 157.1 77 313.1 374 44.88 22. 233.8 1174 159.14, 78 314. 370 45.90 22.5 235.1 1150 161.18 79 314.9 366 46.92 23. 236.3 1127 163.22 80. 315.8 362 46.94 23.5 237.5 1105 16i5.26 81 316.7 358 48.96 24. 238.7 1084 167.3 82 317.6 354 STEAM. 209 TABLE —(Continued). Elastic Force in Elastic Force in Inches of Pounds per Temper- Vol- IIlches of Pounds per Temper- Volmercury. square inch. ature. ume. mercury. square inch. ature. ulne. 169.34 83 318.4 350 285.59 140 357.9 218 171.38 84 319.3 346 295.79 145 (60.6 210 173.42 85 320.1 342 306. 150 363.4 205 183.62 90 324.3 325 316.19 155 366. 198 193.82 95 328.2 310 326.39 160 368.7 193 203.99 100 332. 295 336.59 165 371.1 187 214.19 105 335.8 282 346.79 170 373.6 183 224.39 110 339.2 271 357. 175 376. 178 234.59 115 342.7 259 367.2 180 378.4 174 244.79 120 345.8 251 377.1 185 380.6 169 254.99 125 349.1 240 387.6 190 382.9 166 265.19 130 352.1 233 397.8 195 384.1 1601 275.39 135 355. 224 408. 200 387.3 158 HYPERBOLIC LOGARITHMS. ToJind the mean Pressure by Hyperbolic Logarithms. RULE.-Divide the length of the stroke by the length of the space into which the steam is admitted; find in the table the logarlithm of the number nearest to that of the quotient, to which add 1. The sum is the ratio of the gain. EXAMPLE.-Suppose the steam to enter the cylinder at the press-:tre of 40 lbs. per square inch, and to be cut off at - of the length of the stroke; what is the mean pressure, the stroke being 10 feet? 10-' 2.5 =4. Hyp. log. of 4 = 1.38629-+1 = —2.38629. Then, as 4: 2.38629::40: 23.8629 lbs. TABLE. No. Logarithm. No. Logarithm. No. Logarithm. No. Logarithm. 1.25.22314 4. 1.38629 6.75 1.90954 12. 2.48490 1.5.40546 4.25 1.44691 7. 1.94591 13. 2.56494 1.75.55961 4.5 1.50507 7.25 1.98100 14. 2.63905 2..69314 4.75 1.55814 7.5 2.01490 15. 2.70805 2.25.81093 15. 1.60943 7.75 2.04769 16. 2.77258 2.5.91629 5.25 1.65822 8. 2.07944 17. 2.83321 2.75 1.01160 5.5 1.70474 8.5 2.14006 18. 2.89037 3. 1.09861 5.75 1.74919 9. 2.19722 19. 2.94443 3.25 1.17865 6. 1.79175 9.5 2.25129 20. 2.99573 3.5 1.25276 6.25 1.83258 10. 2.30258 21. 3.04452 3.75 1.32175 6.5 1.87180 11. 2.39789 22. 3.09104 TABLE of the Density of Steam under different Pressures. Aphere.tmo- Density. Volume - Vle.Density. Volume. phere. phere. phere. 1.00059 1694 5.00258 387 12.00581 172 2.00110 909 6.00306 326 14.00670 149 3.00160 625 8.00399 250 16.00760 131 4,00210 476 10.00492 203 18.00849 117 The volumes are not direct, in consequence of the increase of heat. For table showing the ratio of the expansion of steam, see page 275. S2 210 STEAM-ENGINE, STEAM-ENGINE. It is inconsistent with the design of this work to treat of the operation of the steam-engine, and this article will be confined to the exhibition of some rules of construction, the utility of which have been fully tested in the varied purposes of Land, River, and Marine practice. The extremes of proportions here given are for the particular requirements of variations in speed, differences in draughts of water, pressures of steam, &c., &c. CONDENSING ENGINE, For a range of pressures (indicated by a mercurial gauge) of from 10 to 60 pounds per square inch. Condenser. The capacity of it should be from 3 to A- that of the steam cylinder. Air-Pump. The capacity of it, exclusive of piston space, should be from 4 to ~ that of the steam cylinder. Steam and Exhaust Valves. Let a represent area of steam cylinder in inches, s stroke of piston in inches, and r number of revolutions aXsXr per minute, then the area of the valve a X s X r 24000 By experiment, when there were 325 square inches of valve for 290.400 cubic inches in the cylinder, with 26 revolutions, the proportion was found to be a proper one. And 318 inches of valve for 371.000 inches in the cylinder, with 20 revolutions. has been used, and the operation, as shown by an Inditator, was held satisfactory. Foot Valve. The dimensions of it should give an area of fronm -' to 6: of the capacity of the steam cylinder in inches. Delivery Valve. When a solid piston is used in the air-pump, its dimensions should correspond with that-of the foot valve; but whet an open piston alone is used, this proportion may not be obtained. Out-board Delivery Valve. The area of it should be from - to ~ that of the foot valve. Feed Pumnps. Their capacity should be Tl m to l that of the steam cylinder. Injection Cocks. There should be two to each condenser, th( area of each sufficient to supply 70 times the quantity of wate: evaporated when the engine is working at its maximum; and in ma rine engines there should be three, viz., a Side, Bottom, and Bilge STEAM-ENGINE. 211 The Side and Bottom injections will require (for 20 revolutions) an area in square inches of from ~-l- to?-E the number of cubic feet in the steam cylinder for the for mer, and from I to - for the latter.* The Bilge injection is properly a branch of the bottom injection pipe, and may be of less capacity. Piston Rods. Their diameter (if of wrought iron) should be - that of the cylinder or air-pump. Cross Heads. Of wrought iron. (Steam cylinders.) For the section at their centre, let a represent area of cylinder in inches, p extreme pressure in pounds per square inch that they may be subjected to, and 1 their length between the centres of journals in feet; then, axpXl d2Xw d2 Xw =_ d2 2< w, and = d, or --- w, 700 w d" where d represents the depth and w the width of the section. Air-pumps, - de'Xw. If the sections are cylindrical, for 18 I2 Xw read /d2 w X 1.7. SURFACE CONDENSERS. The area of their surfaces should be from to a of that of the heating surface of the boiler, when a natural [raught is employed: with a blower, or forced draught, these proporions should be increased to I and -. With the double vacuum condenser of I. P. Pirsson, the lowest proportions bove given have been found sufficient. * The proportions here given will admit of a sufficient quantity of water when he engine is in operation in the Gulf Stream, where the water is at times at the emperature of 84~, and the quantity of water (when the steam is at 10 lbs. presstre) required to give it and the water of condensation a temperature of 1000, is 70 imes that of the quantity evaporated. 212 STEAM-ENGINE. NON-CONDENSING) OR HIGH-PRESSURE ENGINE, For a range of pressures of from 50 to 130 pounds per square inch. Steam and Exhaust Valves. Let a represent area of steam cylinder in inches, s stroke of piston in inches, and r number of revaXsXr ulutions per minute, then the area of the valve -= X 30000 A decrease in the capacity of the cylinder is not attended with a corresponding diminution of their area. Thus, a 24 inch cylinder by 3 feet stroke has valves that contain 20 square inches, which is a proportion of 1 inch in every 814 inches capacity of cylinder. A 12 inch cylinder by 4 feet stroke has 9 inches area of valve, which is i inch in every 600 inches capacity, and a 6 inch cylinder by 1 foot stroke has 1 inch in every I25 inches capacity. Piston Rod. Its diameter should be from ~ to that of the cylinder. Feed Pumps. Their capacity, when the pressure of the steam is not to exceed 60 lbs., should be _-'- the contents of the cylinder; when not to exceed 130 lbs.,, and in a similar ratio for higher pressures. Cross Hfeads. See rule under head of condensing engines, page 211. STEAM-ENGINE. 213 BOILERS. INTERNAL FURNACES AND FLUES. BITUMINOUS COAL, WITH A NATURAL DRAUGHT. With a Pressure of 20 lbs. Steam per square inch, cut off at " the Stroke of the Piston, and with 20 Revolutions of the Engine. Fire and Flue SuSfa1ce.* For every cubic foot in the steam cyliloder, the length of the flues, including steam chimney, not exceeding 45 feet, there should be from 16 to 18 square feet. Or, surface for one revolution for each cubic foot, 5o0 of a square foot. Grates. Their area in square feet should be from 70 to - of the,ubic feet in the steam cylinder. ANTHRACITE COAL, WITH A BLAST. With a Pressure of 30 lbs. per square inch, cut off at ~ the Stroke, and with 20 Revolutions. Fire and Flue Surface.* For every cubic foot in the steam cylin-.ler, the length of the flues not exceeding 50 feet, there should he from 12 to 14 square feet. Or, surface for one revolution for each cubic foot, 1io of a square foot. Grates. Their area should be from 1-5L5 to -5L that of the cubic feet in the cylinder. Steam Room. There should be at least 5 times the space in the Ateam room (independent of steam pipes, &c.), that there is in the ~ylinder. iNTIIRACITE COAL, WITH AN EXHAUST DRAUGHT, TP5E FLAME RETURNED THROUGH OR PASSING BETWEEN SMALL TUBES. With a Pressure of 50 lbs. per square inch, cut off at i the Stroke, aend with 50 Revolutions. Fire and Flue Surface.* For every cubic foot in the steam cylinler, the passage of the flame not exceeding 30 feet, there should hei from 65 to 70 square feet. Or, surface for one revolution for each cubic foot, 1.35 square feet. Grates. Their area in square feet should be from 2.75 to 3.25,imes that of the cubic feet in the cylinder. Steam Room. There should be at least 7 times the space that;here is in the cylinder. * Estimated from above the grate bars, including steam chimney, and for salt vater. 214 STEAM-ENGINE. EXTERNAL FURNACE, WITH INTERNAL RETURN FLUES. WOOD OR COAL, WITH A NATURAL DRAUGHT. For a Pressure of 100 lbs. per square inch, cut off at ~ the Stroke, ant with 25 Revolutions. Fire and Flue Surface. For every cubic foot in the steam cylin der, the length of the flues not exceeding 50 feet, there should be from 65 to 75 feet.* Or, surface for one revolution for each cubic foot, 2.8 square feet. Grates. Their area in square feet should be from 3. to 3.25 time, that of the cubic feet in the cylinder.* PLAIN CYLINDRICAL, WITH EXTERNAL FURNACE AND FLUE WOOD OR COAL, WITH A NATURAL DRAUGHT. For a Pressure of 100 lbs. per square inch, cut of at A the Stroke, ants with 25 Revolutions. Fire and Flue Surface. For every cubic foot in the steam cylinder, their length not exceeding 30 feet, there should be from 60 to 7( feet.* Surface for one revolution for each cubic foot, 2.6 square feet. Grates. Their area in square feet should be from 4 to 5 tirnes that of the cubic feet in the cylinder.* In the last case the range of surface is extended to meet the application of small and large engines, as the proportion of surface varies materially with cylinders of 1( and 30 inches diameter, being inversely as their capacities. Steam Room. There should be at least 5 times the space in the boiler that there is in the cylinder. NoTEs.-Four copper boilers, with a natural draught and bituminous coal, flues 46 feet in length, including steam chimney, with 14 square feet of fire and flue surface, and -A6 of a square foot of grate surface for every cubic foot in the cylinders, furnished steam at 20 lbs. pressure, cut off at i of the stroke of the piston, for 18.5 revolutions. The mean of four cases, with iron boilers, and anthracite coal, with a blast; flues 50 feet in length, gave, with 11.5 square feet of fire and flue surface, and 15,- of a square foot of grate surface for every cubic foot in the cylinders, steam at 35 lbs. pressure, cut off at - of the stroke of the piston for 22 revolutions. The space in the steam room of the boilers and chimney was about 5 times that of the cylinders in the preceding cases. The ranges here given, of from 16 to 18, 12 to 14, &c., are for the purpose of covering the ordinary differences of construction, thickness of metal, &;c. When there are two engines, or an increased number of revolutions, these proportions of steam room must be increased. When a heater is used, and the temperature of the feed water is raised above that obtained in a condensing engine, the proportions of surfaces may be correspondingly decreased. For further notes of practical results, and for the dimensions and proportions of earious Engines, Boilers, JIills, &-c., 4c., see pages 272-280. * These proportions are for the evaporation of fresh water; if sea water is used the surfaces must be increased. STEAM-ENGINE. 215 GENERAL RULES. ENGINES. Steam Cylinders. When Vertical, multiply their diameter by the extreme pressure of steam in pounds per square inch that they may be subjected to, and divide by 2400; the result will give the thickness in inches. When Horizontal, divide by 2000; and when Inclined, divide as above, in a ratio inversely as the sine of the angle of inclination. Shafts. To resist torsion. For wrought iron, multiply the extreme of pressure upon the steam piston that they may be subjected to, or the resistance to be overcome, in pounds, by the length of the crank or arm in feet; divide this product by 125, and the cube root of the quotient is the diameter of the journal in inches. If for Cast Iron, add I to the diameter thus found. When two shafts are to be used, as in steamers with one engine? Take the cube root of 3 of' the cube of the diameter found as above, and the result is the diameter for each shaft. See pages 158, 159 for other rules and notes of experiments, &c. Journals. Their length should exceed the diameter not less than n the proportion of 10 to 9, and in some cases the proportion should be increased in the ratio of 3 to 2. Steam Pipes. Their area should exceed that of the steam valve. Connecting Rods. Their length should be 2~ times the stroke of lie piston when it is at all practicable to afford the space. When, iowever, it is imperative to reduce this proportion, it may be a little ess than twice the stroke. The diameter of the neck should be the same as that of the piston od. The diameter of the centre of the body is found in the following nanner: Multiply the length of the body (between the necks) in feet, by he area of a neck, divide the product by 3 the stroke of the piston r the throw of the rod, and the quotient is the area of the centre, rom which the diameter may be determined. The length of connecting links should be half of that of the stroke,r of the throw of their attachments. Where a pair of connecting rods is used, as in some descriptions f engines, and with links, their necks should have an area of' of that of the attached rod. When a second set of connecting rods or links are used, as with ir-pump connections, &c., their necks should have a diameter in ratio inversely as their throw to the stroke of the piston. 21 6 STEAM-ENGIN5E. Straps of Connecting Rods, Links, &c. The area of the Strap at its least section should be "5- that of the neck of the attached rod. The Key should be in thickness A- the diameter of the neck, the width of Gib and Key combined should be 1.25 times, and the Slot should be 1 35 times that of the same diameter. The draft of keys should be from 6 to A of an inch per foot. Pins. For Cranks, their area should be 1.6 times that of the attached rod, and for Air Pumps, 1.5 times. Cranks. Of Wrought Iron. The Hub, compared with the nelU of the shaft, should be 1.75 the diameter and 1.05 the depth. T;he Eye, compared with the pin, should be twice its diameter and 1 5 the depth. The Web, at the periphery of the hub, should be, in widthl-. 1 the width, and in depth, one half the depth of that of the hub. At the periphery of the eye it should be, in width, iPL the width, and, in depth, -1 the depth of the eye. The Radii for the fillets of the sides of the web should be one half of the width of the web at the end for which the fillet is designed; for the fillets at the back of the web, they should be one half of that at the sides of their respective ends. When of Cast Iron. The diameters of the Hub and Eye should be, respectively, twice the diameter of the neck of the shaft, and 2.25 times that of the crank pin. Beams. Their length from centres should be 1.8 the stroke of the piston, and their depth one half of their length. If strapped, the strap at its smallest dimensions should have at least - the area of the piston rod, its depth being equal to half of its breadth. The end centres should have each one, and the main centre two and a hall times the area of the piston or driving rod. This proportion for the strap is when the depth of the beam is-as above; consequently, when the depth is less, the area must be increased. Cast Iron Beams, when of uniform thickness, should have a thickness of not less than Ig of their depth. To ascertain the vibration of the end centres, let L represent lengtt of beam, and S stroke of piston. L — /((L — 2)2-(S-. 2)=-vi-bration at each end. Cocks. The angle of the sides of their plug should be from 7: to 80 from the plane of it. Piston Rods of different materials should have their diameters it the following ratios: Wrought Iron, 1. Tempered Steel,.8. Cast Iron, 1.6. Safety Valves. 10 inches area of valve for every 250 square fee of fire surface is sufficient. Fly Wheels and Governors. See rules, page 192. STE AMI-ENGINE. 217 BOILERS Boilers-When Sea Water is used. The heating and grate sur faces should be increased about I over that for fresh water. Furnaces —With Coal. The space over the grate bars in height should be from 2~ to 3 feet. With Wood. The capacity of them above the grate bars, compared with one where coal is used, should be as 5 to 3. The area at the bridge wall should be from - to -To that of the lower flues. Flues. The area of them should decrease with their length, and nearly in proportion with the reduction of the temperature of the heated air; their area at the base of the smoke pipe being from YiP to -7 that of immediately behind the bridge wall. Large flues absorb more heat than small, as both the volume and intensity of the heat is greater with equal surfaces. Grates. The spaces between the bars should be as near or frequent as possible. NOTE.-The bulk of heated air at the bridge wall is about 460 cubic feet for each pound of coal consumed on the grates. With a blast, there will be required about 20 square inches of area at the bridge wall for every square foot of grate, and with a natural draught about 30 inches. Smoke Pipes, or Chimneys. Their area at the base should always exceed that of the extremity of the flue or flues. The intensity of the draught is as the square root of the height. BLOWING ENGINES. Exhausting into a Condenser. ONE STEAM CYLINDER, WITH ONE BLOWING ENGINE. Cylinder. The diameter of it should be.283 that of the steam cylinder, and the stroke of it I-o ONE STEAM CYLINDER, WITII TWO BLOWING ENGINES. Cylinders. The diameter of each should be.2 that of the steam cylinder, and their stroke b-. When the stroke by this rule would be less than 10 inches, the diameter should be decreased, and the stroke of 10 incIhes retained. BLOWERS. When driven with a Belt. Their Width should be equal to their radius; the Opening, in their sides, ~ of their diameter; the Nozzle, not to exceed g and not less than 1 of it, and the Depth of the Fan - less than tho diameter of the opening in the side. Pulley. Its Diameter should not be less than -1 of that of the blower. When Direct Acting. Their Width should not exceed ~ of their diameter; the Opening, in their sides, -, and the depth of the Nozzle, ~ of it. To ascertain the Pressure of a Fan. Let r -radius of blower in fee't (taken from the centre of the blade of the fan), and t= rtime of one revolution in seconds. Then 5r... the height in feet of a column of air that the fan will support T 218 STEAIM-ENGINE, NoTs. —The pressures given are intended for those indicated by the steam gauge or safety valve. When other pressures, revolutions, and points of cutting off than those given are required, the areas of the fire and flue and grate surfaces must be proportionally increased or diminished. The proportion, however, will not be exactly in the ratio of expansion, pressures, and revolutions, as many of the sources of losses, as radiation and leaks, may be uniform and alike in extent. To ascertain the Areas of F;ire anld Flue and Grate Surfaces for a required Capacity oj Cylinder, when the Pressure, Revolutions, and Point of Cutting off differ from the Units given under the several descriptions of Boilers, pages 213 and 214. Reduce the given capacity of cylinder to conform to the appropriate units of pressures, revolutions, and point of cutting off; then multiply this capacity by the units for surfaces given, and the products wiEl be the required areas. EXAMPLE.-The capacity of a cylinder is 12.5 cubic feet, the pressure of steam 40 lbs. per square inch, the revolutions of the engine 50 per minute, and the point of cutting off J of the stroke of the piston. Boiler to have internal furnaces and flues, and to operate with a natural draught. Given pressure 40 lbs. mercurial gauge = — 54.7 lbs. Unit pressure 20 lbs. mercurial gauge 34.7 lbs. 54.7 12.5 X 34 = 19.7 capacity at 54.7 lbs. pressure, 34.7 50 19.7x-=49.25 capacity at 50 revolutions, 20 and 49.25 X 3- 73.875 capacity at half stroke. Then 73.875 X 17. -= 1255.875 square feet of fire and flue surlace. 73 R75 X.75 = 55.406 square feet of grate surface. STEA1II-ENGINE. 219 MATERIALS FOR BOILERS. IRON. The tensile strength of iron boiler plates ranges from 60,000 lbs. to 60,000 lbs. per square inch of section. This strength is increased by their exposure to a moderate temperature, and re. duced when the tension is crosswise to the direction of the fibres. COPPER. The tensile strength of copper boiler plates is about 30,000 lbs. per square inch of section. At a temperature of 5500, this strength is reduced to 22,500 lbs. and at 817~, to 15,000 lbs. Construction of Boilers. The necessary allowances for the spaces between the rivets, oxidation and the loss of elasticity by extreme tensions, reduces the measure of this strength for practical purposes as follows: Iron. 10,000 to 12,000 lbs. Copper....... 7,000 lbs. RIVETING. A double riveted joint is equal to -17, and a single riveted joint is equal to ~5%t of the strength of the plate. BELTS. Two 15 inch belts over a driver of six feet in diameter, running wvith a velocity of 2128 feet in a minute, transmit the power from.he water-wheel at Rocky Glen Factory, the dimensions of which ire given in page 179. An 11 inch belt over a driver of 4 feet in diameter, running from L200 to 2100 feet in a minute, will transmit the power from two 6 nch cylinders having 11 inches stroke, and averaging 125 revolu-.ions per minute, with a pressure of 60 lbs. per square inch. Two 6 inch belts over a driver of 5.9 feet in diameter, running 5700 feet in a minute, will transmit the power from two 9 inch cylnders having 8 inches stroke, and averaging 150 revolutions per ninute, with a pressure of 60 lbs. per square inch. SATURATION IN MARINE BOILERS. 100 parts of sea water contain 3 parts of its weight in saline natter, and is saturated when it contains 36 parts; then, if the uantity in the boiler be taken as 100 parts of water, and s parts e used for steam, b parts blowed out; to fix on the degree of satuation to contain x parts of saline matter, the quantity of salt enering and the quantity leaving in the same time, will be equal 3s when 3 (s+b) = xb; hence b -- - If x = 30, the water in the boiler will not reach to a higher degree f saturation when v of the quantity used for steam is allowed to scape.- And as it requires but about - of the quantity of fuel to oil water that is required to convert it into steam, the loss of fuel 4ill be I X -= I part. — Tredgold. 220 STEAM-ENGINE. BLOWING ENGINES. For Smelting, 4.c. The quantity of oxygen in the same bulk of air is different at dit ferent temperatures. Thus, dry air at 850 contains 10 per cent. less oxygen than when at the temperature of 32~; when saturated with vapour, it contains 12 per cent. less. Hence, if an average supply of 1500 cubic feet per minute is required in winter, 1650 feet will be required in summer. Manufacture of Pig Iron. Coke or Anthraczte Coal. 18 to 20 tons of air are required toI each ton of pig iron. Charcoal. 17 to 18 tons of air are required for each ton of pig iron. (1 ton of air at 600 = 29750 cubic feet.) Pressure. The pressure ordinarily required for smelting purposes is equal to a column of mercury from 3 to 9 inches. Reservoir. The capacity of it, if dry, should be 15 timesthat of the cylinder or cylinders. Pipes. Their area, leading to the reservoir, should be ~ that of the cylinder (blast). To ascertain the Pouter of an Engine.,et P represent pressure of blast in lbs. per squoare inch, v velocity of the piston in feet per minute, a area of cylinder in square inches, f friction of pistons, and from curvatures, &c., estimated at 1.25 per square inch of piston. Poower. P Xa Xv Xf- power in lbs. to be raised 1 foot in a minute, and 3-000f- horses' power required. If the cylinder is single acting, divide the results by 2. To ascertain the Dimensions of Driving Engine. Let'*p represent mean effective pressure upon piston of steam cylinder in lbs. per square inch, *1' the lbs. pressure upon the piston necessary to overcome the friction of the engine, r revolutions of engine per minute. Cylinder. Divide the power in pounds by p-fxv, and the quo. tient is the area. Then - stroke of piston. 2 rThe quantity of air at atmospheric density delivered into the reservoir, in consequence of escapes through the valves, and the partial vacuum necessary to produce a current, will be about 2 less than the capacity of the cylinder. See page 288 for dimeinsions of Furnace, Engines, &c. * See page 221 for these values. STEAM-ENGINE. 221 To find the Power of a Condensing Engine. Let S represent velocity of cylinder piston in feet per minute, n velocity of air-pump piston in feet per minute, *P mean effective pressure upon cylinder piston in lbs., m pressure upon cylinder piston necessary to overcome the friction of the air-pump and its gearing, *b the lbs. pressure upon the air-pump piston, f the lbs. pressure upon the piston necessary to overcome the friction of the engine. The value of m is about 2 lbs. per square inch, that of b 9.5 lbs., and f, at a fair estimate, is ~ of the pressure per steam gauge. P-fr-m X S-nb Then 300'- horses' power. EXAMPLE. —The diameter of a cylinder is 60 inches, the stroke ot the piston 10 feet, the revolutions 20 per minute, the diameter of the air-pump 46 inches, and the stroke 4 feet; the pressure of the steam 20 lbs. per square inch, cut off at ~ the length of the stroke. Then S - 10 X 2 X20- 400, n- 4X2X20: 160, P =(per rule and example, page 209) 20.855 X 2827.4 = 60235.4, m = 2827.4X 2 = 5654.8, b - area of 46 X 9.5 - 15788, f= 20 X.2X2827.4 = 11309.6. 60235.4 — 11309.6+5654.8 X400 -- 160 X 15788 33000=447.9 horses' 33000 power. To find the Power of a Non-condensing Engine. S (*P-f+ 14.7) -horses' power; 33000 f of the pressure per steam gauge. EXAMPLE 1.-What is the power of an engine, the diameter of the cylinder being 10 inches, the stroke 4 feet, the pressure of the steam, per gauge, 60 lbs., making 45 revolutions? 360 X 74.7-(7.5+14.7) 22.2 X 78.54- -33000 = 44.98, Ans. 2. The same with 30 lbs., cut off at ~ the stroke, and making 25 revolutions 1 200 X31.61-(3.75+ 14.7) 18.45 X 78.54-~-33000 = 6.26, Ans. * These values are best obtained by an Indicator: when one is not used, refer to rule and table, page 209. In estimating the value of P, add 14.7 lbs., for the, tmospheric pressure, to that indicated by the steam gauge or safety valve. T2 22Q STEAM-ENGINE. To find the Power of an Engine in the usual Methods. aXPXS 33000 = horses' power. (d-1 )2xS 5640 = nominal horses' power. d2 Xe/s = nominal horses' power. When a represents area of piston, d diameter of cylinder, s stroke of piston, and P and S as in preceding page. To find the Quantity of Water required to be Evaporated in an Engine, the Capacity of the Cylinder, Pressure of Steam, and the Number of Revolutions being given. C x 2 When C represents cubic feet of steam expended in V the cylinder, r the number of revolutions, and V the volume of the steam at the pressure as shown by an indicator. EXAMPLE.-What quantity of water will an engine require per revolution, the diameter of the cylinder being 70 inches, the stroke of the piston 10 feet, and the pressure 30 lbs. per square inch, cut off at one third of the stroke 3 120 Area of cylinder = 3848.4 inches. 1 of stroke = — 40 inches. 3 Then 3848.4 X 40 + 1728...... 89.08 cubic feet. Volume of steam at the above pressure, per table, page 208, 883 cubic feet. Hence 89.082.. -.2017 cubic feet. 883 NOTE. —This refers to the expenditure of steam alone; in practice, howevexr, a large quantity of water (differtng in different cases) is carried into the cylinder in mechanical combination wit/l the steam. It is also independent of any losses by waste, clearance of piston, capacity of steam chests, nozzles, &c., 4'c. To find the Power of an Engine necessary to raise Water to anla Given Height. RULE.-Multiply the. weight of the column by the velocity in fee; per minute, and divide by 33000 EXAMPLr.-It is required to raise a column of fresh water, It inches in diameter by 86 feet high, with a velocity of 128 feet pe minute; what power is necessary 1 86 feet' 2.31 feet, the height equal to 1 lb. per square inch37.2lbs. Area of 16 inches =201.X37.2 lbs. X128=957081.6-i 33000 = 29, horses' power. To which must be added an allowanc for friction and waste, say -. STEAM-ENGINE. 223 To find the Velocity necessary to Discharge a Given Quantity of Water in any Given Time. RULE. —Multiply the number of cubic feet by 144, and divide the product by the area of the pipe or opening. EXAMPLE.-The diameter of the pipe is 16 inches, and the quantity. of water 179 cubic feet; what is the velocity! 179 X 144- -201 — = 128.2 feet, Ans. To find the Area, the Velocity and Quantity being given. RULE.-Proceed as above, and divide the product by the velocity. LOSS BY RADIATION. To ascertain the Loss of Heat per Square Foot in a Second. Let T = temperature of pipe, which is, say — L less than that of the steam t - temperature of the air, I = length of the pipe in feet, d = diameter in inches, v = velocity in feet per second, R = radiation in degrees of heat. 1.71 (T-t) dv- =aR. Tredgold. FRICTION OF ENGINES. The side-beam Engine taken as the Standard. The vibrating engine (oscillating) has a gain of 1.1 per cent. The direct-action engine, with slides, has a loss of 1.8 per cent. dii; " " " rollers," "gain" 0.8 "' " 9" " a " ". a parallel motion, has a gain of 1.3 per cent. By experiments made in London with a non-condensing engine, it appeared that where the pressure on the piston was 12 lbs. per square inch, the friction was about 1 lbs. per square inch. BLOWING OFF. To ascertain the Loss of Heat by the blowing off of saturated Water from a Stean-boiler. STX E-t = proportion of the heat lost; and, to obtain the loss per cent., invert this formula. S representing the sum of the sensible and latent heats. T the temperature of the feed water. t the difference in the temperatures of the water blowed off and supplied to tile boiler. E the quantity of water evaporated, proportionate to that blowed oftf the latter being a constant quantity, and represented by 1. Values of E at the following de- l. 1.5_ _ 75 2 2.5_153, &. grees of saturation, viz.: 32 32 32 32 3 The loss by blowing off at -L is 11.98 per cent. 224F COMIoBUSTION. COMBUSTION. CoMsBusTIoN is one of the many sources of heat, and denotes the combination of a body with any of the substances termed Supporters of Combustion: with refer ence to the generation of steam, we are restricted to but one of these combinations and that is Oxygen. All bodies, when intensely heated, become luminous. When this heat is produ ced by combination with oxygen, they are said to be ignited; and when the body heated is in a gaseous state, it forms what is called Flame. No bodies appear visible, even in a faint light, below about 8700. Carbon exists in nearly a pure state in charcoal and in soot. It combines with no more than 2- of its weight of oxygen. In its combustion, 1 lb. of it produces sufficient heat to increase the temperature of 13000 lbs. of water 10. Hydrogen exists in a gaseous state, and combines with 8 times its weight of oxygen, and 1 lb. of it, in burning, raises the heat of 42000 1Ls. of water 10. The quantity of air chemically required for the combustion of 1 lb. of bituminous coal is 150.35 cubic feet. The bulk of the products of combustion in a furnace is about 465 cubic feet for each pound of bituminous coal burned. An. increase in the rapidity of combustion is accompanied by a diminution in the evaporative efficiency of the combustible. FUEL. With equal weights, that which contains most hydrogen, ought in its combustion, to produce the greatest quantity of heat where each kind is exposed under the most advanlageous circumstances. Thus, pine wood is preferable to hard wood, and bituminous to anthracite coal. When wood is employed as a fuel, it ought to be as dry as possible. To produce the greatest quantity of heat, it should be dried by the direct application of heat. As usually employed, it has about 25 per cent. of water mechanically combined, the heat necessary for the evaporating of which is lost. Different fuels require different quantities of oxygen; for the different kinds of coal, it varies from 1.87 to 3 lbs. for each lb. of coal. 60 cubic feet of air is necessary to afford 1. lb. of oxygen; and making a due allowance for loss, nearly 90 cubic feet of air will be required in the furnace of a boiler for each lb. of oxygen. BITUMINOUS COAL. Accurate experiments upon the practical burning of this description of coal in a steam boiler gave an evaporation of from 6 to 9 lbs. of fresh water, under a pressure of 30 lbs. per square inch, for one pound of coal. Cumnberland coal being the most, and Scotch the least effective. Average Weight per Cubic Foot in Pounds. Cumberland.52.844 Western...... 47.23 Virginia. 49.276 Foreign (Eng.)..... 49.845 ANTHRACITE COAL. Experiments similar to those referred to above, gave an evaporation, with the aid of a blast, of fiom r7 to 9O lbs. of fresh water for 1 lb. of coal. Average weight per cubic foot..... 53.5 lbs. CHARCOAL. The best quality is made from oak, maple, beech, and chestnut. Wood will fmirnish, when properly burned, about 16 per cent. of coal. A bushel of coal from hard wood weighs about 30 lbs., and from pine 29 lb& COI\IBUSTION, 22,5 COKE. Sixty' bushels Newcastle coals (lumps) will make 92 bushels good coke, and 60 bushels (fine) will make 85 bushels of a similar quality. 60 bushels Newcastle and Pictou coal (one half of each) make 84 bushels Inferior; 60 bushels Pictou, or Virginia coal, make 75 bushels of bad. A bushel of the best coke weighs 32 lbs. The production of coke by weight is about i that of the coal. Coal furnishes 60 to 70 per cent. of coke by weight. 1 lb. in a common locomotive boiler will evaporate 7- lbs. water at 2120 Intmr steam. WOOD. Virginia Pine. One cord of the best description weighs about 2700 lbs.: 106.6 cubic feet being required to stow a ton. In practical evaporating power, 2' to 24- lbs. is an equivalent to 1 lb. of the best bituminous or anthracite coals. Western Wood. One cord of the description used by the river steamboats, is equal in evaporating qualities to 12 bushels (960 lbs.) of Pittsburgh coal. MISCELLANEOUS. One pound of anthracite coal in a cupola furnace will melt from 5 to 10 lbs. of cast iron; 80 bushels bituminous coal in an air furnace will melt 10 tons cast iron. Small coal produces about 4 the effect of large coal of the same species. 10 lbs. fresh water have been evaporated it a tubular boiler by 1 lb. of anthra. cite coal. fABLE showing the WVeights, Evaporative Powers per Weight, and Bulk and Character of Fuels, frotm Report of Professor Walter R. Johnson, 1844. Weight Lbs. of steam Lbs. ofsteam Weight of Numbe r of Specific from water at rrom water at Clinker cubic feet ignationof Fuel. gravity. i2120 by t lb. 212 by 1 cub. from 00 required to loot, of fuel. foot of fuel. lbs. of coal. stow a ton BITUMINOUS. Cumberland, maximu7n 1.313 52.92 10.7 573.3 2.13 42.3 4 minimum 1.337 54.2!) 9.44 532.3 4.53 41.2 Blossburgh.. 1.324 53.05 9.72 522.6 3.40 42.2 Midlothian, screenmed 1.283 45.72 8.94 438.4 3.33 49. " average. 1.2'394 54.04 8.29 461.6 8.82 41.4 Newcastle.... 1.257 50.82 8.66 453.9 3.14 44. Pictou...... 1.318 49.25 8.41 478.7 6.13 45. Piltsburgh.. 1.252 46.81 8.20 384.1,94 47.8 Sydney.... 1.338 47.44 7.99. 3861 2.25 47.2 Liverpool..... 1.262 47.88 7.84 411.2 1.86 46.7 Clover Hill.... 1.2. 5 45.49 7.67 359.3 3.86 49.2 Cannelton, Ia.... 1.273 47.65 7.34 360. 1.64 47. Scotch...... 1.519 51.09 6.95 369.1 5.63 43.8 ANTHRACITE. Peach Mountain. 1.464 53.79 10.11 581.3 3.03 41.6 Forest Improvemlent 1.477 53.66 10.06 577.3.81 41.7 Beaver Meadow, No. 5 1.554 56.19 9.88 572.9.60 39.8 Lackawanna.... 1.421 48.89 9.79 493. 1.24 45.8 Beaver Meadow, No. 3 1.610 54.93 9.21 526.5 1.01 40.7 Lehigh....... 1.590 55.32 8.93 515.4 1.08 40.5 Natural Vigini 1.323 46.64 847 407.9 Natural Virginia.. 1.323 46.64 8 47 407.9 5.31 48.3 Midlothian.... 32.70 8.63- 282.5 10.51 68.5 Cumnberland.... 31.57 8.99 284. 3.55 70.9 WooD. Dry Pine Wood... 21.01 4.69 98.6 106.6 The above table exhibits the ultimate effects. As a safe estimate for prantical values, a deduction (for the coals) of 1-4t should be made. * Winchester bushel = 2150.4~2 cubic inciles. 226 COMBUSTION. TABLE showing the Results of Mr. Bull's Experiments. CompasaWoods. Weight of tiC. s WWe gihtof Corpara tive vWoods. o car. cod tive yal Wood. |per cord. per cord. Lbs. Lbs. Shell-bark Hickory 4469 100 Hard Maple.. 2878 60 Red-heart Hickory 3705 81 Jersey Pine.. 2137 54 White Oak... 3821 81 Yellow Pine.. 1904 43 Red Oak.... 3254 69 White Pine.. 1868 42 Relative Value of the following Fuels by Weight: Seasoned oak... 125 Charcoal. 285 " " artificially. 140 Peat. 115 Hickory... 137 Welsh coal.. 312. White pine... 137 Newcastle... 309 Yellow pine... 145 Anthracite 250 Good coke.... 285 ANALYSIS OF FUELS. Newcastle Coal, Cannel Coal Cumberland Coa, Antracite. caking kind. American. Carbon... 75.28 64.72 80. Carbon... 89.458 Hydrogen,. 4.18 21.56 Bitumen. 18.40 Earthy matter 7.112 Nitrogen.. 15.96 13.72 Ash. 1.60 Volatile" 3.430 Oxygen.. 4.58 0.00 100. 100. 100. 100. Oak. Ash. Maple. Chestnut. Norway Pine. Volatile matter... 76.9 81.3 79.3 76.3 80.4 Charcoal.... 22.7 17.9 20. 23.3 19.2 Ashes.......4.7.7.4.4 Weight of sundry Fuels to form a Cubic Foot of Water at 52e inte Steam at 220~. Lbs. Lbs. Newcastle coal... 8. Peat.. 30.5 Pine wood (dry).. 20.2 Olive oil 5.9 Oak wood (dry).. 12. Coke... 9. TABLE showing the Heating Power of different Substances Weight of Lbs. of Weight of Lbs. of water in steam by I water in steam by I Name. lbs.,heated lb. of com-. Ibs.,heated lb. of com 1~ by 1 lb. bustible, Na l by 1 lb.1 bustilde. ofthe com-2 from 52~ to1 o fte com- from 5 to bustible.'.20~. bustible. 22Q~. Alcohlol..... 11000 Pine, seasoned. 546e6 4.66 Olive oil.... 14500 12. Coal, Newcastle 9230 7.90 Beeswax, yellow. 14000 11. —, Welsh... 11840 10.1 Tallow.... 15000 12., Anthracite 9560 8. Oak wood, seasoned 4600 3.90 - Cannel. 9000 7.7 --— dried on Coke..... 9110 7.7 a stove.... 5960 5.12 Peat..... 3250 2.8 Experiments made upon the Baltimore and Ohio Rail-road determined the relative values of the following fuels in a.locomotive boiler to be, 1 ton Cumberland coal (bituminous) =1.25 tons anthracite coal, and 2.12 cords pine wood. 1 ton anthracite = 1.75 cords pine wood. WATER. 227 WATER. FRESH WATER. The constitution of it by weight and measure is, By weight. By measure. Oxygen.88.9 1 Hydrogen. 11.1 2 One cubic inch at 620, the barometer at 30 inches, weighs 252.458 grains, and it is 830.1 times heavier than atmospheric air. A cubic foot weighs 1000 ounces, or 62~ lbs. avoirdupois; a column 1 inch square and 1 foot high weighs.434028 lbs. It expands v of its bulk in freezing, and averages.0002517, or for'every degree of heat from 400 to 2120. Maximum density, 39.38~. TABLE of Expansion at different Temperatures. Temperature. Expansion. Temperature. Expansion. 120.00236 640.00159 220.00090 1020.00791 320.00022 2120.04330 400.00000 0. 0330 Showing an increase in bulk from 400 to 2120 of.00000-, equal to 1 cubic foot in every 23.09 feet. The height Of a column of 1 lb. per square inch, is 2.30 feet, The height of a column ~of 1 "d A" circular " " 2.93 " water at 60~, equivalent to the 1 inch of mercury, 1.129" the atmosphere. "33.86" River or canal water contains Io 1 of its volume of gaseous matSpring or well water " 1 ter. A cubic inch weighs.0361 of a lb., and at 212~ has a force of 29.56 inches mercury. SEA WATER, according to the analysis of Dr. Murray, at the specific gravity of 1.029, contains, Muriate of soda.... 220.01 = Sulphate of soda 33.16 = Muriate of magnesia 42.08 = Muriate of lime 7.84 = T 303.09 = TABLE showing the Deposites that take place at different Degrees of Saturation and Temperature. When'1000 parts were reduced by evaporation. Quantity of sea water. Boiling point. Salt in 100 parts. Nature of deposite. 1000 214~ 3.. None. 299 217~ 10. Sulphate of lime. 102 2280 29.5 Common salt. '22;z8 WATER. Boiling Point at different Degrees of Saturation. Proportion of salt in 100 parts by Boiling point. Pr100 oportion of salt in Boiling point. weight. 100 parts by weight. Saturated 36.37 226.0 18.18 219.0 solution 15.15 217.90 33.34 224.90 12.12 216.7e 30.30 223.7W 9.09 215.50 27.28 222.5~ 6.06 214.40 24.25 221.40 3.03 213.2 21.22 220.20 Seawater 2 SALT WATER. A cubic foot of it weighs 64.3 lbs.; a cubic inch, 03721 lbs. The height of a column of 1 lb. per square inch, is 2.26 feet, water at 600, equivalent to the 1.". circular' " 2.87 " pressure of.... 1 inch of mercury, " 1.09 " (Specific gravity, 1029). the atmosphere. "32.92 " Saline Coantents of Sea Waterfol'om differcnt Localities. Baltic. 6.60 Equator.. 39.42 Black Sea... 21.60 South Atlantic.. 41.20 Arctic.28.30 Sea of Marmora..42.00 Irish Sea... 33.76 North Atlantic.. 42.60 British Channel.. 35.50 Dead Sea... 385.00 Mediterranean.. 39.40 There are 62 volumes of carbonic acid in 1000 of sea water. IO"' TABLE of Results and Properties of Gunpowder, determined by the.Experiments and Report of Capt. A. MORDtECAI, U. S. Army. N 24-POUNDER GUN. Weight of ball and wad...... 24.25 lbs. MUSKET PENDULUx. Weight of ball.. 397.5 grains. " powder....... 6. powder.. 120. Windage of ball......... 135 inch. Windage of ball...... 05 inch..q o' o Relative Force. Standard== i ~ Composition. Manufacture. 1000.:' 5 24-Pounder Musket 0 ~~~~~~~~~~~~~~~~;.S. ________________________ ______.__ ~o ~ o o Gu. Peodn'um. Kind of Grain. -...Gun. [Pendulum. i. -V.0-. 5 o InitiaI s Initl -' Where from. Gluzing. it n CD UU 1=: ivelo,:ity velocity 0 c:~ inuFeet. 0 iu eet. is Per ct Per ec. Per ec. Oz. Cannon, large Glazed. 77 275 2.77 1710 964.. - 677 916 A small... 569 314 3.35 1659 935 -- 720 927 3 Musket...... 1134 214.....- 1484 808 896 4 Rifle.76{4 10 Dupont's Mills, Wilming 6174 142-.....907 g Rifle.......ton, Delaware.. 5344 282 3.55 1668 940.. 728 1044 E Musket 1642.....834 937 X p 4 Rifle..... 13152.....943 955 p5 Sporting.... Esquerades, French..... 935 9 PFrench. Cannon, uneven... Rough. 66 183 2.09... 7- 780 F large~~~~~~~~~~~ 103 18 1o91.... 8..15.... 756 775 v zlarge.. 7 252 medium 71 12.5 2.5 Dupont's Mills, Delaware 182 1.91 552 8715 756 75 1 ~~~~~~~~~~~~~~].1 ]5528715]... 163 186 2.95 1527 861 7681 751 2 ~. very large, even I211 200 1504 848 740' 762 0 Sporting.. 77 13 10 Dupont's Mills, Delaware Hiihly glazed 72808 100 4.4'2 1774 1000' 1888 11000 1047 G 6 Blasting, uneven... 70 1.5 15 Glazed. 295 212 1660 936. ---- 917 S Cannon, medium... 76 13.710.3 arec, Delware 19 216 2.69754 879 B 2 91 262009....... 88.... ]7072 Rifle.. 76 15.9.. Loomises, Hazard, & Co., 2378 204.... 820 934 C 5 Sporting...... 7 5 Connecticut......o..... 8881. 6 ~~~~~7Cannon, uneven ---... 731 872 English. Musket... 10 WalthamAbbeyEgland 2832.841 8 Rifle 11600...... 865 820 ~~~~Sporting.. 1 —- l. Hall & Sons, Dartford, I, 0 Sporting It I... -......... netiu.... 9" CnoueeEngland.. -. 3.. 8.... 80 Cannon, uneven. ~75 12.5 12.5 B uchet, France.. Rough 316 835 804 French. M~usket......2410.........7 830' Musket......Stockholm, Sweden I lazed. 2.... Sedis ___________________..72.....Swedish. NOTE.- The report of Captain M1ordecai embraced a greoter number of specimens, and furniahed other elements. A selection onlsy has been made of such (qualities of ponderr, results, h., as are considered essential for general pfrposer 230 GUNPOWDER. TABLE showing the dIcrease of Weight in 1 lb. of various Kinds of Gunpowder, exposed to nMoisture for a Period of 17 Days. Experiments of Captain A. Mordecai, U.S.A. Kiod of Increase of Kind of Increase of Kind of Increase of Powder. VWeight. Powder. Weight. Pl'oder. Weight. Per cent. Per cent. Al1 2.77 E 5 3.55 F 2 2.95 3 3.35 F 2.09 G 6 4.42 B 2 2.6851 F 1 1.91 The powders A and B were slightly caked by this exposure; E much more so; the F's were not at all caked; and the G became hard caked on the surface after 6 days' exposure. TAsLE showing the Relative Force, by the Musket Pendulum, of various Kinds of Gunpowder in good order, and of the same Powders dried, after Exposure to Air saturated with Moisture. Experiments of Captain A. 5Mordecai, U.S.A. Kind Powder in gooed Powder dried after Exposure to Moisture. Loss ofForce Gain of of er.tof Fxorc Force by Pow- Initial Velocity Moisture ab- Absorbed Moist- Initial Velocity yte o the Exder. of Ball. sorbed. ure retained. of Ball. sposur. Feet. Per cent. Per cent. Feet. Per cent. Per cent. A 1 1256 20.88.83 891 29.06.. — A 4 1499 23.25.29 1480 1.27 A 5 1684 27.05.31 1516 9.98 E 5 1351 23.11 0. 1472. 8.96 F 1 1404 20.83 0. 1315 6.34 F 0 1373 22.59.20.1143 16.75 C 6 1856 19.53 0. 1472 20.69 The least dense powders return nearest to their original strength. The gain by the powder E 5 arose from the circumstance that the density of it in its original state was so great as to impede its comoustion; hence, the grain becoming swollen by exposure to moisture, and its density correspondingly diminished, its combustibility was increased in a greater degree than its strength was impaired by the moisture. In a large charge, like to that of a 24-pound gun, Captain Mordecai thinks the result would not be the same. TABLE showing the Loss of Force by Windage. Experiments of Captain A. Mordecai, U.S.A. 24-POUNDER GUN. Initial Velocity of Ball in Feet per second. Powder. Ball. Without Windage, Windase, Windage, VWindage..135 Int:es..24.5 Incllhes..35.t Inche. Lbs. Lbs. Feet. F ret. Feet. Feet. 4 24.25 1631 1450 1332 1197 6 24.25 1963 1702 1596 1465 A comparison of these results shows that 4 lbs. of powder give to GUNPOWDER. 231 a ball without windage nearly as great a velocity as is given by 6 lbs. to a ball having 0.14 inch windage, which is the true windage of a 24-pound ball; or, in other words, this windage causes a loss of nearlv one third of the force of the charge. TABLE showing' the Loss of Force by the Vent of a Gun. Experiments of Captain A. Mordecai, U.S.A. 24-POUNDER GUN. Powder. Vent. Velocity of Ball. I'owder. Vent. Velocity of Ball Lbs. ntslles. Feet. Lbs. Incshes. Feet. 3.175 1251 6 closed 1705 3 closed 1259 6.25 1625 6.2.5 1696 6.175 1627 6 175 1702 j 6 closed 1612 These experiments show that the loss of force, by the escape of gas from the vent is altogether inconsiderable, when compared with the whole force of the charge. TABLE showing' the Effect of different Descriptions of Wadding. Experiments of Captain A. Mordecai, U.S.A. Powder. Velocity Kind. Weighlt. Kind of sVad. Per Second. Grains. Feet. r Ball wrapped in the cartridge paper, 1308 G 6 77 the whole crumpled into a wad 1377 i and inserted over the powder " 77 1 felt wad on powder and 1 on ball. 1346 c 77 2 felt wads on powder and 1 on ball 1482 French 1 77. 1 elastic wad on powder and 1 on I 1096 sporting 1 ball 1169 77.17 2 pasteboard wads on powder.. 1200 " 7.17 2 felt wads on powder. 1088 G 6 77.17 2 elastic wads on powder. 1079 4 77.17 2 pasteboard wads on powder. 1193 " 77.17 2 felt wads on powder.. 1106 " 77.17 2 felt wads on powder and 1 on ball. 1190 c; 77.17 I Ball wrapped in the cartridge paper, 1237 as in the first experiment The felt wads were cut from the body of a hat, weight 3 grains. The pasteboard wcads were y-jth of an inch thick, weight 8 grains. The cartridge paper was 3X 4.5 inches, weight 12.82 grains. The elastic wads were', Baldwin's indented," a little more than l oth of an inch thick, weight 5.127 grains. The most advantageous wads are those made of thick pasteboard, or of the ordinary cartridge paper. 232 GUNPOWDER. Comparison of the Faorce of a Charge in various Arms, as determined by Experiments in Small Arms, of Rifle Calibre. By Captain A. Mordecai, U.S.A. Kind of Arm. Lock. Powder, Windage., Ball' Velocit Grains. Inches. Grains Feet. Common rifle... Percussion 100.015 219 2018 " "f...70 ].015 219 1755 * Hall's rifle.. Flint 70.000 219 1490 * Hall's carbine.. Percussion 70.000 219 1240 * Jenks's carbine.. 70.000 219 1687 Cadet's musket Flint 70.045 219 1690 Pistol...... Percussion 35.015 218.5 947 * Loaded at the breach. PRACTICAL DEDUCTTONS of Captain Mordecai, suggested by his Experi ments. Proof of Powder. The common eprouvettes are of no value as instruments for determining the relative force of different kinds of gunpowder. In the proof of gunpowder a cannon pendulum should be used. In a 24-pounder gun, new cannon powder should give, with a charge of 6 lbs., an initial velocity ef not less than. 1600 feet to a ball of medium weight and windage. For the proof of powder for small arms, a small ballistic pendulum is best adapted. The initial velocity of a musket ball (18 to the pound), of 0.05 inches windage, with a charge of 120 grains, should be: With new musket powder, not less than 1500 feet. With new rifle powder, not less than 1600 feet. With fine sporting powder, not less than 1800 feet. Manusfacture of Powder. The powder of greatest force, whether for cannon or small arms, is produced by incorporation in the "' cylinder mills." Efect of Wads. In the service of cannon, heavy wads over the ball are in all respects injurious. For the purpose of retaining the ball in its place, light grommets should be used. On the other hand, it is of great importance, and especially so in the use of small arms, that there should be a good wad over the powder for developing the full force of the charge, unless, as in the rifle, the ball has but very little windage. Effect of the Size of the Grain. Within the limits of the difference in the size of grain, which occurs in ordinary cannon powder, the granulation appears to exercise but little influence on the force of it, unless the grain be exceedingly dense and hard. Effect of Glazing. Glazing is favorable to the production of the greatest force, and to the quick combustion of the grains, by affording a rapid transmission of the flame through the mass of the powder. Effect of Using Percussion Primers. The increase of force by the use of primers, which nearly closes the vent, is constant and appreciable in amount, yet not of sufficient value to authorize a reduction of the charge. GUNPOWDER. 233 BORE, WEIGHT OF CHARGES AND RANGES FOR U. S. SMALL ARMS. Kind of Arm. Bore. Windage of Ball. Powder. Ball. Incles. inches. Grainls. Musket... 69 *.05 120 397.5 grains. Wad, 10.2 grains. "....69 i'.04 110 410.2 " " 10.2' Rifle....54.525 75 218.5 " " 8.4 " Pistol..54.525 35 218.5 " " 5.5'.....54.525 30 218.5 " 5.5 " *.05679 lbs., or 18 to the pound t.05861 lbs., or 17 to the pound Rangesfor Small Arms. Musket. With the ball of 17 to the pound, and a charge of 110 grains of powder tcc., an elevation of 36' is required for a range of 200 yards; and for a range of 506 yards, an elevation of 30:s0' is necessary, and at this distance the ball will pass;hrough a pine board 1 inch in thickness. Rifle. With the chlrge of 70 grains, an effective range of from 300 to 350 is obtained; bht as 75 grains can be used without stripping the ball, it is deemed better to use it; to allow for accidental loss, deterioration of powder, &c. Pistol. With the charge of 30 grains, the ball is projected through a pine board 1 inch thick at a distance of 80 yards. PROOF OF POWDER. Ordilzary Proofof f Powder.-One oz. with a 24 lb. ball. The mean range of new, proved at any one time, must not be less than 250 yards; but none ranging below 225 yards is received. Powder in magazines that does not range over 180 yards is considered unserviceable. Good powder averages from 280 to 300 yards; small grain, from 300 to 320 yards. DIMENSIONS OF POWDER BARRELS. Whole length. 20.5 inches. Length. interior in the clear.18. Interior diameter at the head.. 14. " " "4 at the bile.16. " Thickness of staves and heads.5 "B Weight of barrels about..... 25 lbs US 234 PENETRATION OF SHOT AND SHELLS. PENETRATION OF SHOT AND SHELLS. PENETRATION IN MASONRY. Experiments at Fort Monroe Arsenal in 1839. Mean penetration. Calibre. 1ICharge. Elevation. Distance. )Dressed I olonac Hard _ granite. freestone. brick. Shot. Lbs. Yards. Inches. Inches. Inches. 32 Pounder (Gun). 8 10 880 3.5 12. 15.25 Shell. 8 Inch Howitzer Sea8 Inch coast zt 6 10 35"' 880 1. 4.5 8.5 The solid shot broke against the granite. The shells broke into small fragments against each of the three materials. PENETRATION IN WHITE OAK. Experiments at New-York HarbouTr in 1814. Calibre. Charge. Distance. Penetration. Remarks. Lbs. Yards. Incies. Shot wrapped so as 32 Pounder. 11 100 60 to destroy the wind11 150 54 age. PENETRATION IN COMIPACT EARTHf (Half sand, half clay). Distances in yards. Calibre. Charge. 27 109 328 1094 Shot. Inches. Inches. Inches. Inches. 6.885 109.1 102.4 93.4 69.7 Shells. 8.782 4.4 lbs. *48.4 *45.3 38.6 23.2 Musket. 154 grains 9.85 8.6 4.3 The penetrations in other kinds of earth are found by multiplying the above by 0 63 for sand mixed with gravel; by 0.87 for earth mixed with sand and gravel, weighing 125 lbs. to a cubic foot; by 1.09 for compact mould and fresh earth mixed with sand, or hall clay; by 1.44 for wet potter's clay; by 1.5 for light earth, settled; and by 1.9 for light earth, fresh. * With this charge, and at these distances, the shells were often broken. PENETRATION OF SHELLS. 235 PENETRATION OF SHELLS. Eleva. D.stace. In Compact Earth, In Oak Wood, In Masonry, tiona.. ins. 10 ins. 102 ins. 8 ins. 10 ins. 12 ins. S ins. 10 ins. 1 2 ins. Yards. Inches. Inches. Inches. Inches. Inches. Inches. Inches. Inches. Inches. 30 656 7.8 17.7 19.6 3.9 7.8 8.6 1.9 3.5 3.9 1312 9.8 25.6 27.5 4.7 11.8 13.7 2.3 4.7 5. 1 450 656 11.8 19.6 21.6 5.9 9.8 10.6 3,1 3.9 4.3 1312 15.7 27.5 29.5 7.8 13.7 15.7 3.9 5.5 5.9 o 656 19.6 29.5 31.5 8.6 13. 14.5 4.3 5.9 6.3 1312 21.6 31.5 33.4 9.8 13.7 15.7 4.7 6.3 6.6 Falling with maximum 23.6 33.4 35.4 9.8 13.7 15.7 4.7 6.6 7. velocity. ( _.. The penetration in other kinds of earth and stone may be obtained by using the coefficients given for the other tables. For woods, use for beech and ash I, for elm 1.3, for white pine and birch 1.8, and for poplar 2. 144 grains of powder in a musket, at 5 yards' distance, will project a ball 3 inches into seasoned white oak, and 100 grains in a rifle, at the same distance, 2.05 inches. PENETRATION IN SEASONED WHITE OAK. Experiments at Washzngton Arsenal in 1839. Arms. Charge in ODistance in Penetration grains. yards. in incies. Remarks. Musket......t. 144 5 3. Common rifle., ~ 100 5 2.5 I Arms loaded with Hall's rifle...... 100 5 2. new musket pow. ( 70 5.6 der. Hall's carbine, musket cal- s 80 5.8 ibre........ 90 5 1.1 I Charge too great {k100 5 1.2 for service. Pistol....... 51 5 725 WEIGHT AND DIMENSIONS OF SHOT. GRAPE. CALIBRE OF 8 Inc. I42 32 24 1 8 14 iclmest. rncies Incises, Inc. es. Inches. Inches. Diameter of high guage. 3.60 3.17 2.90 2.64 2.40 2.06 low gauge 3.54 3.13 2.86 2.60 2.36 2.02 Mean weight in lbs. 6.24 4.25 3.25 2.45 1.83 1.19 CARCASSES. CALIBRE OF 113 incl. lOinc. 8inctl. 1 42 1 32 124 is8 12 Mean weight in lbs. l 194 87.63 43.62 29Y.5 21.60 15.84| 12.15 8 ~~~19.6]4. 6. _,!' [ WEIGHT AND DIMENSIONS OF SHOT. CANISTER. 24 and 9 and 12 lb. howitzer. Calibreof 42 32 8 inch 18 12 24lb. 6 howi how. Field. Mountair iizer. i izer. Ins. Ills. Ills. Ills Ins.. Ins. Diam. of high gauge, 2.26 2.06 1.87 1.70 1.49 1.35 1.17 1.08 Musket " low gauge, 2.22 2.02 1.84 1.67 1.46 1.32 1.14 1.05 ball. Mean. weight in lbs., 1.57 1.19.90.67.45 -.33.235.17.056 WEIGHT AND DIMENSIONS OF LEADEN BALLS. TABLE showing the Number of Balls in a Pound, from 1.- ths to 210-o0 of an Inch Bore. Dian. Diam. Number Diam. iaim. Number Diam. 2)iam. Number hf parts of in decimnals of balls in in parts of in decimnls of balls in a in parts of il decimals of balls in a an inch. of an inch pound. an inch. of all inch. pound. an inch. of an uch. pound. 1.670 1.570 25.301 170 a.1 5 1.326 2.537 30.295 180 1.157 3.510 35.290 190 1.051 4.505 36..285 200.977 5 2.488 40.281 210 a* l".919 6.469 45.276 220 * 7.873 7.453 50.272 230 X.835 8 * -17 ~.426 60.268 240 * 13.802 9.405 70.265 250 H.775 10.395 75.262 260 * 3.750 11.388 80.259 270 730 12 *.375 88.256 280.710 13.372 90.252 290 4 1 1.693 14.359 100 * 2.249 300 TO.677 15.348 110.247 310.662 16.338 120.244 320.650 17.329 130.242 330.637 18.321 140.239 340 *.625 19 *.314 151-.237 350 _.615 120.307 160 The exact decimals would be as follows: 1. 5 1.3125 13.8125 7.4375 1 5.9375 1 t.6875 l.3125 1.8750 1.5000 1.2500 EXPANSION OF SHOT heated to a White Heat. Calibres. 42 32 24 18 12 Inchses. Incles. Inches. Incnes. Inches. Experiment at Expansion. 0.11 0.10 0.08 0.06 0.04 Fort Monroe, 1839. BRONZE FOR CANNON. Copper 90, Tin 10. Specific gravity is greater than the mean of copper and tin, viz., 8,766, PRINCIPAL DIMENSIONS AND WEIGHTS OF GUNS, SHOT, AND SHELLS. 02Xi ~ 1 oi~.s~ Length Diameter Diameter Weigt Proport. Di-m Diameter PwFrom ]ih "tOf' O ]' ~Po. beteen r SEA-COAST o to Total. Of WeightdOfwshotr i Of de rieihts cham - iattert 4 SES COM.S o~fhbore os rnm. u t. ud high l *| Of shot. Of shells stein, of gon I. b c.1.0 X bzse ring. Ions. of Gun. shells. gouge. gauge. and`bol. ber IRLON. In-clses. Tsches. Inches. Inches. Inches. Ins. Lbs. Ins. Ins. Ins. In. lbs. Lbs. Lbs Lbs. Ins. Ins. Ins._ 10 Inch... 10. 96. 112. 124.25 8..1 75 9500 9.85 9.90 9.80 1.60 136. 89.39 5.12 69.8.1.5 7. 9.5'181 8... 8. 85.5 98. 109. 6.40.175 5800 7.85 7.90 7.80 1.25 69.5 44.62 2.66 83.4.15 6.4 7.5 184 42 Pounder. 7.018 109. 1 17. 129.4 7.018.2 8688 6.83 6.86 6.76 1.15 43.3 30.38 1.64 200..188 - - 183 42 42... 7. 110. 1 7. 129. 7..175 8300 6.83 6.86 6.76 1.15 43.3 30.38 1.64 191.7.17 - -!1843 M 32 " 6.41 107.59 114. t25.2 6.41.2 7531 6.24 6.27 6.18 1. 33.04 22.41 1.33 227.9.161 - - 1829 2 32.. 6.4 107.6 1 14. 125.2 6.4.175 7100 6.24 6.27 6.18 l. 33.04 22.41 1.33 214.8.16 l- - 1840 24 ". 5.823 108.17 114. 123.95 5.823.2 6500 5.68 5.70 5.61.85 24.92 16.22 1.09 220.7.14A3 - - 1819 Sieete. s 8 Inch Hower..8. 0238.5 52. 61.5 5.82.175 2650 7.85 7.90 7.80 1.25 69.5 44.62 2.66 38.1.15 4.62 8. 1840 ct F o Sea-coast. o Siege and Garri son.: 24 Pounder. 5.82 108. 114. 124. 5.82.175 5600 5.68 5.70 5.61.85 24.92116.22 1.09 224.7.14 - - 1840 Siege and Garrisen. 18 Pounder. 5.3 109. 114. 123.25 5.3.17.5 4750 5.17 5.18 5.10.80 18.79112.5.75 252.6.13 - 1840 R 12 4'. 4.62 103.4 108. 116. 4.62.1 75 3500 a4.52 4.53 4.46.70 12.56 8.3.53 280..10 - -1 840 B BRONZE. Field. I I 24 Pound.:owr. 5.82 56.25 65. 71.2 4.2.175 1320 5.68 5.70 5.6!1.85 24.92116.22 1.09 53..14 4.62 4.75 1840 o 12. 4.62 46.25 53. 58.6 3.67 1.75 785 4.52 4.53 4.46.70 12.56 8.3.53 62.81.10 3.67 4.25 1840 12' Gun 4.62 74.'78. 85. 4.62.175 1800 4.52 4.5314.46.70 12.56 8.3.53 144..10 - - 1840 Z 6( (( j 3.67 f57.05 60. 65.6 3.67.175 880 3.58 3.60 3.54 - 6.241 - -- 141..09 - - 1840 BRONZE. m 4ountain. I i 1 0 0 0 4 12Pound. How'r. | 4.62 3 7. 21 2.71.' 17537g2112.7g1&52 2i2l. 751 ]90f5 7 1 9. q r 17 i A Qo QA o, n PRINCIPAL DIMENSIONS IAND WEIGHTS OF GUNS, SHOT, AND SHELLS. 02 U. S. NAVY. DiaD Imetsr. Wg eig 1t 1 I W'indage. i Charges n! D i 5amet ~ ~eighto.f S Dipmet LDonle 1enblA to a', Gun corn- Eb 8 0 0:bes. incses. ns. ins. nsns. Ins. los. Lbs. Lbs. los. Description of Guns. te. fIR tero f par toPatt en. l0 inch, chamnber of a 42. 10. 106. 9.85 9.88 9.82 107 11 ito r 112.18.2 10 8.25 Navy Com.miss., 1841. bor ~ ~ ~ ~ ~ ~ i=ta ae. bor" ca, -~ IZ b weig. /o r e. 0l~~ ~~,~a toC $~; = 108 inch, to t chamber of a 42. 10. [106. 9.85 9.88 9.82 127 100.... 90 1 to 7t1100:.1 0 8.2 ay onis. ~1 11.tom 1O inch,[ chamber of a 32 8. 102. 7.85 7.887.82 64 47. 63 1 106.18.12 10 6.25 1841. to 150; 8 inch 8. 95.4 7.85 7.88 7.82 64 47. 53 1.5 ito 93 100.18.12 8 7.2 Bureau of Ordnance, 1845. o 42 Pounder... 7.018 101. 6.85 6.88 6.82 42 31. 71 1.2 1 to 189 109.198.138 10 8.25 Navy Corn., 1821, 1824. r 32 " (oldpattern) I6.41 102.7 6.25 6.28 6.22 32 25.51 61 1.2 1 to214 110.19.13 8 6.25 " " 1824. c 32 "(new "). 6.4 107.9 6.25 56.2816.22 32 25.5 57 1.2 1 to200 112.10.12 8 6.2 BureauofOrdnance, 1846. 32.6.4 103.95:6.25 6.28 6.22 32 25.5 51 1 1 to 180 108.18.12 7 5.2 " 1846.. 32 ".". 6.4 97.2 16.2516.28 16.22t32 25.5 46 1.2 1 to 161 102.18.12 6 4.2 " " 1846. 2 32... 6.4 92.75 6.251 6.286:2 32 25.5142 1.2 1 to 150 96.18.12 5 4.o Navy Commiss., 1842. 32 6...4 75.04 6.25 6.28 6.22 32 25.5132 1.2 1 to 112 79.18.12 4.5 4.2 BureauofOrdnance,1846. 32 " "..4 68.4 6.25 6.28 0.22 32 25.5 27 1' 1 to 95 72.18.12 4.2 1846. 24 Pounder. ". 5.82 100. 5.67 5.7 5.641 24 17. 49 1. Ito 29 119.18.12 8 6.25 Navy Commis., 18324. 24 " (med.). 5.82 75.5 5.6715.7 5.64 24 17. 32 1. to 150 92.5.18.12 5 4.25 " " 1826. Carronades. 42 Pounder... 7. 54. 6.85 6.88 6.82 42 31 24 1.5 1 to 69 71.18.12 3.8 3.2.25 1821. t4 32... 6.40 49. 6.25 6.28 6.221 32 224 21 1.5 ito 73.5 77.18.12 3 2.5.25 " 1821. Including chamber. t Solid shot. Hollow shot. MISCELLANEOUS. 239 MISCELLANEOUS. RECAPITULATION OF WEIGHTS OF VARIOUS SUBSTANCES. Cubic foot in pounds. Cubic inch in poundg: FCast iron.... 450.55.2607'Wrought Iron.. 486.65.2816 Steel..... 489.8.2834:Copper. 555..32118 Lead..... 708.75.41015 Brass..... 537.75.3112 Tin..... 456..263 White Pine.... 29.56.0171 Yellow Pine. 33.81.019 White Oak.. 45.2.026 Live Oak... 70..040 Salt Water (sea)... 64.3.03721 Fresh Water... 62.5.03616 Air......07529 Steam.03689 Weights of a Cubic Foot of various Substances in ordinary use. Lbs. Lbs, Loose earth or sand. 95 Clay and stones.. 160 Common soil... 124 Cork.... 15 Strong soil... 127 Tallow... 59 Clay. 135 Brick... 125 SLATING. Sizes of Slates. Doubles.... 14 by 6 inches, Ladies'...... 15 8 " Countess.... 22 "11 " Duchess... 26 "15 " Imperial and Patent... 32 "26 " Rags and Queens. 39 "2'7 ~ From the West Point Foundry Association at Cold Spring, N. Y. Other es riments have given.2613 as the weight of a cubic inch. t Ulster Iron Company, Saugerties, N. Y. t From Phelps, Dodge, & Co.'s Works, in Derby, Conn. 240 MISCELLANEOUS. CAPACITY OF CISTERNS IN U. S. GALLONS. For each 10 Inches in Depth. 2 feet diameter. 19.5 8 feet diameter 313.33 2~ ".. 30.6 8s ". 353.72 3 ".. 44.06 9 ".. 396.56 3t ".. 59.97 9 ".. 461.40 4 ".. 78.33 10 ".. 489.20 42 ".. 99.14 11 ".. 592.40 5.. 122.40 12 705. 5,.. 148.10 13 ". 827.4 6.. 176.25 14 ".. 959.6 6 ".. 206.85 15 ". 1101.6 7 ".. 239.88 20 ".. 1958.4 r7 ".. 275.40 25 ".. 3059.9 TABLE OF COMIPOSITIONS-BRASS, ETC. Copper. Tin. Zinc. 2 0 1 For Yellow Brass. 3 0 1 " Spelter. 4 1 " Lathe bushes. 6 1 0 " Shaft bearings. 5 1 i " " " (hard). 8 1 0 " Wheels, boxes, cocks, &c. 9 1 0 "Gun metal. 3 0 1 " Brass. 10 1 0 " Valves. 78 22 5. "Bells and Gongs. 80 10 5.6, and lead 4.3 SIZES OF NUTS, EQUAL IN STRENGTH TO THEIR BOLTS. Dimensions in Inches. Diameter Diameter Depth of Diameter Diameter Depth of [iamter Diameter Depth of of bolt. of nut. ut. of bolt. of nut. nu nut. [otlt. of nut. nut. *.23.14., 2.05 1.40 2. 4.05 2.75;y.45.28 l3 1. 83 2.25 1.50 2. *.4]45 3..55.42 1. 2.45 1.65 3. 4.80 3.30.i 85.55 1.- 2.55 1.70 3.( 5.20 3.60 o.95.62 1.9 2.85 1.95 3.j 5. 55 3.85 1.05.70 1 3.05 2.05 3 34 5.95 4.15 1.25.83 2. 3,25 2.20 4. 6.30 4.40 B 1.45.96 ( 2. 3.45 2.35 4.- 7.10 5. 1, 1.65 1.10 2. 3.55 2.50 5. 7.85 5.50 NOTE.-The depth of the head should equal the diameter of the bolt; the deptl of the nut should exceed it in the proportion of 9 or 10 to 8. MISCELLANEOUS. 241 SCREWVS. TABLE showing the Number of Threads to an Inch in V thread Screws. Diam. in inches, - 5 3 1 5 3 7 1 1 No. of threads, 20 18 16 14 12 11 10 9 8 7 7 6 Diam. in inches, 1-1 1 13 1 2 21 2- 23 3 31 3No. of threads, 6 5 5 4'4 4 4 4 3- 3- 31 Diarn. in inches, 33 4 41 Q1 43 5 51 5 53 6 No. of threads, 3 3 2~ 21 23 23 25- 2X- 2- 21 The depth of the threads should be half their pitch. The diameter of a screw, to work in the teeth of a wheel, should be such that the angle of the threads does not exceed 10~. T'ABLE of the Strength of Coppc?' at different Temperatures. Temperature. i Strength in lbs, Temperature. Strength in lbs. Temperature. Strength in Ibs. 1220 33079 1 4820 26981 8010 18854 2120 32187 5450 25420 9120 14789 3020 30872 6020 22302 10160 11054 3920 27154 Franklin Institute. DIGGING. 23 cubic feet of sand, or 18 cubic feet of earth, or 17 cubic feet of clay, make a ton. 18 cubic feet of gravel or earth before digging, make 27 cubic feet when dug. COAL GAS. A chaldron of bituminous coal yields about 10.000 cubic feet of gas. Gas pipes i inch in diameter supply a light equal to 20 candles. 1.43 cubic feet of gas per hour give a light equal to one good candle. 1.96 cubic feet equal four candles. 3. it" "d i ten " X 242 MIISCELLANEOUS. ALCOHOL Is obtained by distillation from fermented liquors. Proportion of Alcohol in 100 parts of the following Liquors: Scotch Whiskey. 54.32 Sherry... 19.17 Irish " 53.9 Claret.. 15.1 Rum. 53.68 Champagne.. 13.8 Brandy.. 53.39 Gooseberry. 11.84 Gin. 51.6 Elder... 8.79 Port..22.9 Ale. 6.87 Madeira... 22.27 Porter.. 4.2 Currant.. 20.55 Cider. 9.8 to 5.2 Teneriffe... 19.79 Prof. Brande. WEIGHT OF COMPOSITION SHEATHING NAILS. Number Le in Number i Number. ngth in Number i Nme Lengthin Number in Nme inches. a pound.'ubr inches. a pound. N be. inches. a pound. 1 4 290 6 1 190 10 1 101 2 ] 2601 7 1 184 1] 1 14 74 3 1 212 8 14 168 12 2 64 4 14 2011 9 P1 110 13 24 59 45 1 199 CEMENT. ClAshyes, 3 parts Mixed with oil, will resist the weather equal to Sand, I marble. HYDRAULIC CEMENT. A barrel contains 300 lbs., equal to 4 struck bushels. BROWN MORTAR. One third Thomaston lime, Two thirds sand, and a small quantity of hair. Lime and sand, and cement and sand, lessen about ~ in bulk when made into mortar. MISCELLANEOUS. 24'3 BRICKS, LATHS, ETC. Dimensions. Common brick... 8 to 7 X44 X2~ inches. Front brick. 8.. 84 4 X4X2 " 20 common bricks to a cubic foot, when laid; 15 " " " a foot of 8 inch wall, when laid. Laths are i to 1~ inches by four feet in length, are usually set of an inch apart, and a bundle contains 100. Stourbridge fire-brick, 94 X 44 X 24 inches. HAY. 1.0 cubic yards of meadow hay weigh a ton. When the hay is ta1ien out of large or old stacks, 8 and 9 yards will make a ton. 11 to 12 cubic yards of clover, when dry, weigh a ton. HILLS IN AN ACRE OF GROUND. 40 feet apart. 27 hills, 8 feet apart. 680 hills, 35 ".. 35 " 6 " ".. 1210 " 30' ".. 48 " 5 " ".. 1742 " 25 ".. 69 " 32" ".. 3556 " 20 ".. 108 1" 3 " "..4840 " 15 " ".. 193 " 2- " ".. 6969 " i2 ".. 302" 2 " ". 10890 " 10 ".. 435 " 1 " ".. 43560 " DISPLACEMENT OF ENGLISH VESSELS OF WAR, WHEN LAUNCHED AND WHEN READY FOR SEA. Razee. Corv. Brig. Rate of Guns. 120 80 74 52 46 28 50 18 18 Tons. Tons. Tons Tons. Tons. Tons. Tons. Tons. Tons. Weight of hull, launched. 2467 1882 1617 1448 1042 795 413 281 213 Weight received on board 2142 1723 1359 1044 1067 670 370 326 242 IW;gVit complete.... 4609 3605 2976 24921 2109 1465 783 607 455 Edye's.N'. C. :244 MISCELLANEOUS. WEIGHT OF LEAD PIPE PER YARDS From ~ to 4i Inches Diameter. Weight in Ihs. and oz Weight in 1bs. and ao. i inch medium. 3 - 14 inch extra light 9 - " strong. 4 - " lig5ht. 13 i inch light 3 --," medium. 15 8 " medium. 4 -- " strong. 19 " strong. 5 - 14 inch medium. 16 -- " extra strong 6 6 " strong. 20 i inch light. 5 - 2 inch light. 16 12 " medium 6 8 " medium. 20' strong. 7 8 " strong. 23 " extra strong 8 4 24 inch light. 25 - inch extra light 5 - " medium. 30 " light. 6 4 " strong. 35 -- " medium. 8 - 3 inch light. 30 " strong. 9 12 " medium. 35 " extra strong 10 8 " strong. 44 -- 1 inch extra light 6 14 34 inch medium. 45 " light 8 5 " strong. 54 " medium. 10 5 " extra strong 70 " strong. 12 4 4 inch waste, light, 15 14 lI inch extra light 8 5 " " medium, 21 -- " light. 9 12 " " strong, 26 - medium. 11 - 4I inch "light, 17 4 " strong. 12 8 " " medium, 24 -' extra strong 14 10 " " strong, 29 -- Very light Pipe. Diameter. Weight in lbs. and oz. Diameter. Weight in lbs. and cz 4 inch.. 1 - inch 3 6 ".. 1 - 1" _ 1 " 5 10.. 2 1 ".. 6 14 ~" 24 T I N. Mean thickness. Description. Size of sheet. n wire Thicknes of sheet. Mean weight No. on wire Thickness of sheet. of one sheet. gauge. Inches. Inches. Lbs. Single. 10 X 14 31.0125, (or 80 to 1 inch) 0.5 Double X 10 X 14 27.0181, (or 55 to 1 inch) 0.75 There are usually 225 sheets in a box. MIISCELLANEOUS. 245 RELATIVE PRICES OF AMERICAN WROUGHT IRON. Round. Square. 4: inches 27 4 inches.. 27 34 to 24 inches 26 to 21 3~ to 24 inches. 26 to 20 2 " i ". 19 2". 21 rl:3 is. 520 to29 " 4 " 19 to 26 Flat. Hoops. i and i inch to l 26 to 28 1. to inch 24 to 33 4 " 1 " " 1x4 19 14 " i " "..15 19 to 23 Band... 20 ILUSTRATION.-If 4 inch round iron is worth $135 per ton, then and iron is worth $100 per ton, for 27 is to 20'as 135 to 100. POWER REQUIRED TO PUNCII IRON AND COPPER PLATES. Through an Iron Plate, with a Punch.1 Inch in Diameter..08 inches thick.6025 lbs..17 " "..11950 ".24 " ".. 17100' Through a Copper Plate, with a Punch ~ Inch in Diameter..08 inches thick.3983 lbs..17. 7833 " The force necessary to punch holes of different diameters through metals of various thicknesses, is directly as the diameter of the hole and the thickness of the metal. To ascertain the Force necessary to Punch Iron or Copper Plates. RULE.-Multiply, if for iron, 150000, and if for copper, 96000, by the diameter of the punch and the thickness of the plate, each in inches; the product is the pressure in pounds. The use of oil reduces the above 8 per cent. X2 246 WEIGHT OF SQUARE ROLLED IRON. IRON. CAST IRON expands,0 of its length for one degree of heat; greatest change in the shade in this climate, lil —y of its length; exposed to the sun's rays, T~U; shrinks in cooling from I to, of its length; is crushed by a force of 93,000 lbs. upon a square inch; will bear, without permanent alteration, 15,300 lbs. upon a square inch, and an extension of' of its length. Weight of modulus of elasticity for a base of an inch square, 18,400,000 lbs.; height of modulus of elasticity, 5,750,000 feet. WROUGHT IRON expands y310I — of its length for one degree of heat; will bear on a square inch, without permanent alteration, 17,800 lbs., and an extension in length of ~-,I'; cohesive force is diminished 3- io by an increase of I degree of heat. Weight of modulus of elasticity for a base of an inch square, 24,920,000 lbs.; height of modulus of elasticity, 7,550,000 feet. Compared with cast iron, its strength is 1.12 times, its extens'bsiity 0.86 times, and its stiffness 1.3 times. WEIGHT OF SQUARE ROLLED IRON, From { Inch to 12 Inches, AND ONE FOOT IN LENGTH. Size in Weight in Size in Weight in Size in Weight in Size in Weight in inche. pounds. inches. pounds. inches. pounds. inches. pounds.,.013 2. 13.520 4.1 64.700 7. 190.136.03 2 15.263 4. 68.448 7. 203.024.0.53 2 17.112 442 72.305 8. 216.336 ~-3.118 2.3 19.066 4. 76.264 8. 4230.068 i.211 2.| 21.120 4.7 80.333 8'4 244.220 476 24e 23.292 F5. 84.480 8. 258.800 1.845 ~2.l 25.560 5.4 88.784 9. 273.792 1.32845 2 7 27.939 5.1 93.16fi8 9.4 289.220' 15.320 3. 30.416 54 97.657 9.j 305.056.3 1.901 3 33.010 5.a 102.240 94 321.332 4 2.588 3 1 35.704 5[ 106.953 10. 337.920 1. 3.380 31 38.503 5 111.756 10.4 355.136 1. 4.278 3 - 41.408 5 4 7 116.671 1 0. 372.672 1. 5.280 3 2 44.418 6. 121.664 10[. 390.628 1.4J 6.390 3.3i 47.534 6.4 132.040 11. 408.960 1. 7.604 31 50.756 6.4 142.816 11. 427.812 1. 8.926 4 54.084 6 154012 1 1. 447.024 1.1 10.352 4. 57.517 7. 165.6 11.4 466.684 1.] 11.883 4I. 61.055 7vX 177.672 1 2. 486.656 ExAMPLE.-What is the weight of a bar of rolied iron 1- inches square and 12 inches long? In column 1st find 1., and oppositt tc it is 7.604 pounds, which is 7 lbs. and 61,,4 WEIGHT OF ROUND ROLLED IRON. 24?7 of' a lb. If the lesser denomination of ounces is required, the result is obtained as tollows: Multiply the remainder by 16, pointing off the decimals as in multiplication of decimals, and the figures remaining on the left of the point indicate the number of ounces. Thus, T6 4 of a lb. =.604 16 9.664 ounces. The weight, then, is 7 lbs. 9.1- ounces. If the weight for less than a foot in length was required, the readiest operation is this: ExAMrtTC,. —What is the weight of a bar 64 inches square and 94 inches long? In column 5th, opposite to 64, is 132.040, which is the weight for a foot in length. 6~X12 inches 132.040 6. " is 66.020 3. is 4 of 6 33.010. "' is $ of 3 5.5016 " is of i = 2.7508 9.4 = 107. pounds. 1000 WEIGHT OF ROUND ROLLED IRON, From 4 Inch to 12 Inches Diameter, AND ONE FOOT IN LENGTH. Diameter Weight in Diameter Weight in Diameter Weightin Diamter Weight in a usehes. pounds.in inches. pounds. in inches. pounds. in inches. pounds..010 2.4 11.988 4.4 53.760 7. 4 159.456 041 2. 13.440 4. 56.788 8. 169.856 * 041 2. 14.975 4. 59.900 8.~ 180.696.- 1.1'19 2.1 16.688 4.4 63.094 8.~ 191.808.165 2.5 18.293 5. 66.752 8.4 203.260.373 2.4 20.076 5.4 69.731 9. 215.040 4,.663 2.2 21.944 5.4 73.172 9.1 227.152 1.043 3. 23.888 5.4 76.700 9.~ 239.600:i 1.493 3.1 25.926 5.4 80.304 9.4 252.376 W 2.032 3.~ 28.040 5. 84.001 10. 266.288 1. 2.654 3.* 30.240 5.3 87.776 10.4 278.924 1 3.360 3. 32-.512 5.4 91.634 10.- 292.688 1 * 4 4 4 1 7r2 3.4 34.886 6. 95.552 10.34 306.800 1.W 5.019 3.4 37.332 6.4 103.704 11. 321.216 1. 5.972 3. 39.864 6.4 112.160 11.4 336.004 1 s. 7.010 4. 42.464 6.3 120.960 11.4 351.104 Y1. i 8.128 4.4 45.174 7. 130.048 11.4 366.536 1. 9 333 1 4. 4 47.952 7.4 139.544 12. 382.208 2. 10.616 4.4 50.815 7.4 149.328 The application of this table is precisely similar to that of the preceding one. 248 WEIGHT OF FLAT ROLLED IRON. WEIGHT OF FLAT ROLLED IRON From 4XI Inch to 54-X6 Inches, AND ONE FOOT IN LENGTH. Breadth Thickness Weight in Breadth lThickness Weight in Breadth Thickness Weight in;.n inches. in inches. pounds. in inches in inches pounds. in incies. in inches. pounds..4.4 0.211 12 1., 5.808 2. 0. 845. 0.422 1 1. g 0.633 I 1.689.* 0.634.~ 1.2661 I 2.534..I 0.264! [ 1.900.i 3.379. 0. 528.3 2.535 I i 4.224. 0.792 I 5 3.168. 5.069. 1.056 I 3.802 5.914 ~..4 0.316. 4.435 1. 6.758.4 0 633 1 i 5 069 7.604. 0.950 1. 5.703 1. 8.448. 1.265 1.~ 6.337 1 1 9.294. 1.584 1. 6.970 1.4 10.138. 0.369.1 8 0.686 1 10.983 4 0.738. 1 1.372 1 11.828.* 1.108. 2.059 14 12.673.' 1.477. 2.746 2.4. 0.898. 1.846 3.432.4 1.795. 2.217 I 4.119. 2.693 3L.4 0.422 4.805. 3.591.8 0.845 1 5.492 4.488 1.267 1. 6.178.5.386 1.690 1 1. 6.8641 I 6.283 2.112 I1. 7.551 I / 7.181 2.534. 1 8.237 L. 8.079 2 956 1 0.739 1.4 8.977 1.4 8 0.475. 1.479 1. 9.874.I 0.950. 2.218. 1 0.772.4 1.425 I 2.957 1.4 11.670. 1.901.8 3.696 1.4 12.567. 2.375 [ 4.435 1. 13.465.I 2.850 5.178 2I 14.362.4 I 3.326 1. 5.914 2.i4..950 1. 3.802 1 B. 6.653 I. 1.900 14.' 0.528 1 4 7.393 1. 2.851.4 1.056 i! 8.132.4 3.802.4 1.584 1. 8.871 / 4.752. 2.112 9.610. 5.703.4 2.640 1. 0.792. 6.653.4 3.168.~ 1.584 1. 7.604. 3.696 8 2.376 1.4 8.554 1. 4.224. 3.168 1.4 9.505 1.4 4.752. 3.960 1.:10.455 1.4.4 0.580 [ 4.752 1.4 11,.406. 1.161. 5.544 1.4 12.356.4 1.742 1. 6.336 1.4 13.307 j 2.325 1. 7.129 1. 14.257.4 2.904 1.i 4.921 2. 15.208.1 3.484 1.8 8.713 2.4 16.158.4 4.065 1.4 9.505 2.4. 1.003 1 4.646 1., 10.297. 2.006 1i. 5.227 1.4 11.089.4 3.009 WEIGHT OF FLAT ROLLED IRON;. 2419 TABLE-(Continued). Breadth Thickness Weight in Breadth Thicknesa Weight in Breadth Thickness Weight in in inches. in inches. pounds. in inches. in inches. poindsi in inclles. in inches. polunds., 2.4 4 4.013 -2.4 2.323 3. 14 17.742 5.016 I3 3.485 1. 19 010' 6.019.4.647 2. 20.277. 7.022 5.808 2. 1 22.811 1. 8.025 4 6.970 2. 25.346 1.. 9.028 8 8.132 2.4 27.881 1. 10.032 1. 9.294 3..4 1.373 1. 11.035 1. 10.455 4 2.746 1.4 12.038 1. 1.1.617 4.119 1 13.042 1 * 3 12.779 4 5.492 1. 14.045 1. 13.940.5 6.865 1 15.048 [ 1.n 15.102' 8.237 2. 16.051 1.9 16.264 9.610 2.4 17.054 17 17.425 1. 10.983 2.- 18.057 2. 18.587 1. I 12.356 2. 1.056 2 8.1 19.749 1. 13.730 4 2.112 2.4 20.910 14 15.102. 3.168 2.: 22.072 1. 16.475 4 4.224 2. 23.234 I. 17.848 ~ 5.280 2. n 24.395 1. 19.221 6.336 2. 1215 17 20.594 7.392 8. 2.429 2. 21.967 1. 8 8.448 3.644 2.4 24.712 1l. 9.504 [. 4.858 2. 27.4,58 1.4 10.560 6.072 2]4 30.204 1.4 11.616.4 7.287 3. 32.950 1.4 12.672 i 8.502 3.4. 1.479 1. 13.728 1. 8 9.716 8 2.957 1.4 14.784 1. 1 -10.931. 4.436 14. 15.840 1.4 12.145 4 5.914 2. 16.896 1. 13.360. 7.393 2. 17.952 1. 14.574 4 8.871 2.4 19.008 ao 1 15.789 10.350 2.4 20.064 1 1.; 17.003 1. 11.828 20..# 4 1.109 1. 18.218 1.4 13.307.4 2.218 2. 19.432 1 14.785 3.327 2. 20.647 ]. 16.264.+ 4.436 2. 21.861 1. 17.742 5.545 2. 8 23.076 1. 19.221.4 6.654 2.i 24.290 1 20.699.~ 7.763 2. 8 25.505 1.7 22.178 I. 8.872 2. 26.719 2. 23.656 1.4 9.981 3..8 1.267 2.: 26.613.4 11.090.4 2.535 2.4 29.570 1.4 12 199. 3.802 2.i 32.527 14 13.308.+ 5.069 3. 35.485 1.4 14.417 6.337 3.4 38.441 1.4 115.526., 7.604 3.4.4 1.584 14 16.635. 8.871. 3.168 2. 8 17.744 1 10.138 4 4.752 2.4 18.853 1. 11.406. 6.336 2.4 19.962 1. 12.673 7.921 2.4 21.071 1.8 13.940. 9.505 2.4J 22.180 1t 15.208 7 11.080. 1.162 1' 16.475 1. 12.673 250 WEIGHT OF FLAT ROLLED IRON. TABLE —(Continued). Breadth Thickness Weight in Breadth Thickness Weight in Breadth Thickness Weight in in inches. in inches. pounds. ininches. in inches. pounds. in inches. in inches. pounds. 3. 1. 14.257 4. 2.4 34.217 5.4 2.- 44.355 1 15.841 2.4 38.019 2.4 48.791 1. 17.425 2.3 41.820 3. 53.226 1. 19.009 3. 45.623 3. 57.662 1 20.594 3.~4 49.425 3.4 62.097 1. 22.178 3.4 53.226 3[ 4 66.533 1.7 23.762 3.4 57.028 4. 70.968 2. 25.346 4. 60.830 4.1 75.404 2.4 28.514 4.4 64.632 4.4- 79.839 2.4 31.682 4.4 4 4.013 4. 84.275 2.4 34.851 8.026 5. 88.710 3. 38.019. 12.039 5..~ 4.647 3.4 41.187 1. 16.052 4 9.294 3.' 44.355 1. 20.066 4 13.940 4..4 1.690 1. 24.079 1. 18.587 4 3.380 1.4 28.092 1. 23.234 ~ 6.759 2. 32.105 1.- 27.881 4 10.138 2.1. 36.118 1. 32.527 1. 13.518 2 -. 40.131 2. 37.174 1. 16.897 2.4 44.144 2.4 41.821 1.4 20.277 3. 48.157 2.4 46.468 1.' 23.656 3.~ 52.170 2.3 51.114 2. 27.036 3.- 56.184 3. 55.761 2.4 30.415 [ / 3. 60.197 3.4 60.408 2.4 33.795 4. 64.210 3.4 65.055 2.4 37.174 I 4 68.223 3.4 69.701 3. 40.554 4 72.235 4. 74.348 3.4 43.933 5. I 4.224 4. I 78.995 3.4'47.313. I 8.449 4. 83.642 3.3 50.692 12.673 4.4 88.288 4.4.4 1.79 1. 16.897 5. 92.935.4 3.591 1.4 21.122 5. 97.582 4 7.181 1.2- 25.346 5.4. 4 858 4 10.772 1I 29.570. 9.716 1. 14.364 2. 33.795 4 14.574 1.4 17.953 2.- 138.019 1 1. 19.432 1. }- 21.544 2.- 42.243 1.4 24.290 14 25.135 2.1 46.468 1.. 29.148 2. 28.725 3. 50.692 14 34.006 2.4 32.316 3.~ 54.916 2. 38.864 2.~ 35.907 3.- 59.140 2.' 43.722 2*4 39.497 3 63.365 2. 48.580 3. 43.088 4. 67.589 2.4 53.437 3.4 46.679 4.~ 71.813 3. 58.296 3. - 50.269 4. -76.038 3. 63.154 34 53.860 4 4.- 80.262 3.9 68.012 4. 57.450 54. 5 4.436 3. 72.870 4,4. 4 3.802 ] 8.871 4. 77.728. 7.604 t. 13.307 4.~ 82.585 4 11.406 1. 17.742 4.~ 8'7.443 1. 15.208 1.4 22.178 4.3 92.301 1. 19.010 1.- 26.613 5. 97.159 14. 22.812 1 A 31.049 5.4 102.017 1.4 26.614- 2. 35.484 5.~ 106.876 2. 30.415 1 2. 39.920 6. 1116.592 WEIGHT OF FLAT ROLLED IRON. 2 51 ExkMPLEs.-What is the weight of a bar of iron 54 in. in breadth by } in. thick? In column 4, page 250, find 54, and below it, in column 5, i; and opposite to that,s 13.307, which is 13 lbs. and 3i 0~- of a lb. For parts of a lb. and of a foot, operate precisely similar to the rule laid down bor table, page 247. VALUES OF THE BIRMINGHAMI GAUGES. tFor Iron Wire and for Sheet Iron and Steel. h'o. 0 1 1 2 3 4 5 6 7 8 9 110 11 2 neh..340.300.284.259.238.290.203.180.165.148.134.1.0.109 qo. 13 114 15 1 16 17 1 18 1 19 1 1 21 1 122 123 12 1 25 nch..095.083.072 1.065 1.058.049.042 1.035.032.028.025.022.020 I o. 26 1 2728 29 130 131 132 133 134 135 36 nch..018.016.014.013.012.010.09.008.007 1.005.004 For Sheet Brass, Silver, Gold, 4.c. o. 1 1 2 1 3 4 1 5 16 17 18 19 110 111 1 2 1 13 Ich..004 1.005 |.008 I.010.012.013.015.016 1.019 |.024.029.034 036 1 14 1 15 116 1 17 118 119 120 1 21 122 1 23 1241 251 26 ich..041.047.051.057.061.064.067.072.074.077.082.095.103 o. 1 27 1 28 129 1 30 1 31 1321 33 34 1 35 1 36 ch. 1.113.120.124.126.133.143.145.148.158.167 CAST IRON. To ascertain the weight of a cast iron Bar or Rod, find the weight of a wrought in bar or rod of the same dimensions in the preceding tables, and from the weight duct the.-th part; or say, As 486.65. 450.55:: the weight in the table:to to the weight required. Thus, What is the weight of a piece of cast iron 4X3~X12 inches. In table, page 250, the weight of a piece of wrought iron of these dimensions is 692- lbs. Then 486.65: 450.55:: 50.692: 46.93 lbs. 9 find the Weight of a piece of CAST or WROUGHT IRON of atny size or shape. 3y the rules given in Mensuration of Solids (see page 81), ascertain the number cubic inches in the piece, multiply by the weight of a cubic inch, and the duct will be the weight in pounds. EXAMPLES. Vhat is the weight of a block of wrought iron 10 inches square by 1.5 inches in 10X 10X15 - 1500 cubic inches..2816 weight of a cubic inch. 422.4000 pounds. Vhat is the weight of a cast iron ball 15 inches in diameter. By table, page 255-= 176.7149 cubic inches..2607 weight of a cubic inch. 460.6957 pounds. ~2<02 WEIGHIT OF CASr IRON PIPES. WEIGHT OF CAST IRON PIPES OF DIFFERENT T HICKNESSE9, From 1 Inch to 36 Inches Bore, AND ONE FOOT IN LENGTH. P ore. Thicknesl Weight. Bore. Thllickness Weight. Bore. Thickness Wesght. [Mnches IncheE. Pounds. Inches. Inches. Pounds. Inches. Inches Pounds. 1.. 3.06 6. 49.60 1 1. -i 58.82 5.05! 58.96. I. 74.28 I,, I4 1 3.6,7 6.4 4 34.32. 90.06 6.. 43.68 4 106.14 a4 6.89.4 53.30 1. 122.62 9.801. 63.18 12.. 61.26 1S 4@ *8 7.80 7..4 36.66.4 77.36 1104 46.80 [ 93.7C 2, B 8.74 8 56.96 7. 110.48 12.23 [ 67.60 1. 127.42 * 9.65 1. 78.39 12.-. 63.70 * 13.48 7.~.- 39.22. 80.4C 2 10.57. 49.92. 97.46 4 14.66 [. 60.48.* 114.72 19,0.5 71.76 1. 132.37 2 4 88 11.54 1. 83.28 13..4 66.1 4, 15.91 8..~ 41.64.4 83.4( 20.59 52.68 I. 101.0 3, 12 28 [ 64.27. 118.97 1 17.15.7 76.12 1. 137.21 22.15 1 88.20 13.4.4 68.64 27.56 8.'.4 44.11 86.51 3.o ~ 18.40 56.1..6 104.70 23.72 3 68. 4 123.3( 29.64 80).50 1. 142.1( SW4 4 19.66 1. 93.28 14..4 71.0 25.27 9. 4 46.50 89.6 31.20 6 58.92 I 108.4( 3..4 20.90. 71.70. 127.6( 26.83 7 84.70. 147.0 33.07 1, 97.98 14. 73.7' t4. 22.05 9.4. ~ 48.98. 92.6( 28.28, 62.02.4 112.1 34.94.3 75.32 131.8t 4. 23.35 7 88.98 1. 151.9: 8 29.85 1 102.90 15..4 75.9 36.73 10.. 51.46., 95.7 4. 24.49.5 65.08.4 115.7 *85 31.40 4 78.99 4 136.1 38.58 7 93.24 1. 156.8 4 4.4 25.70 1. 108.84 15.-.~ 78.4 32.91 10 -. 53.88.g 98.7 4 40.43 5 68.14. 119.4 ~O y 26.94. 82.68 ] 140.4 34.34 [ 97.44 1. 161.8 4 42.28 1. 112.68 16.. 80.8 5,). 29.40 11.' /2 56.34.4 101.8 37.44!.5} 71.19. 123.1.4 45.94 3 86.40. 44.i 6 43 31.82 a 101.83 1. 166.( i4 40.56 1 1 117.60 16.. i 83. WEIGHIT OF CAST IRON PIPES. 253 TABLE —(Continued). Bor, Thickness Weight. Bore. Thickness Weight. Bore. Thickness Weight. Inches. Inches. Pounds. Inches. Inches. Pounds. Inches. Inches. Pounds. 16 1.* 104.82 22., 4 138.60 30. 1. 303.86 4 126.79. 167.24 1. 343.20 4 149.02. 196.46 31. - 233.40 1. 171.60 1. 225.38 I 273.40 17. /.4 / 85.73 23..5 144.77 1. 313.68 * 107.96 8 174.62 1. 354.24 4 130.48. 204.78 32. 8 240.76 4 153.30 1. 235.28 4 281.94 1. 176.58 24. 4 150.85 1. 323.49 L7.j.[ 88.23 [ 181.92 14. 365.29 4 111.06 7 213.28 33.. 248.10 -4 134.16 1. 245.08 290.50 4 157.59 25. * 8 156.97 I 333.24 1. 181.33. 189.28 1 376.26 18..4 114.10. 221.94 34..3 255.45. 137.84 1. 254.86.4 298.88 4 161'.90 26. 3 196.62 1. 342.88 1. 186.24 * 230.56 I.' 387.13 19..4 120.24. 264.66 1.4 431.76 4 145.20 27. 2 204.04 35. 4 262.70 4 170.47 7 239.08 s 307.62 1.1 195.92 1 X s274.56 I. 352.86 20. 6 126.33 28. 211.32 1. 398.10 4 152.53, 247.62 I1 443.96 4 179.02 1. 284.28 36..4 270.18 1. 205.80 29. 218.70 316.36 21. 4 132.50. 256.20 1. 362.86. 159.84 1. 294.02 1.4 409.34 7 187.60 30. 226.20 1. 456.46 1. 215.52 W 264.79 NoTE. —These weights do not include any allowance for spigot and faucet ends. STRENGTH AND STIFFNESS OF METALS. Cast Iron being 1. Wrought iron..... 1.12 1.3 Brass.......435.49 Zinc......365.76 Tin.......182.25 Lead..046.0385 Y 254 WEIGHT OF A SQUARE FOOT OF CAST IRON, ETC. WEIGHT OF A SQUARE FOOT OF CAST AND WROUGHT' 1RON, COPPER, AND LEAD, From I th to 2 Inches thick. Wrought Iron. Copper Lead. Thickness. Cast Iron. Hard rolled. Hard rolled. Lead. Hord rotl _. Hr r Pounds. Pounds. Pounds. Pounds. 1.~ 2.346 2.517 2.890 3.691.i 4.693 5.035 5.781 7.382 3 7.039 7.552 8.672 11.074 9.386 10.070 11.562 14.765 ~.A-% 11.733 12.588 14.453 18.456 14.079 15.106 17.344 22.148 716 16.426 17.623 20.234 25.839 i 18.773 20.141 23.125 29.530 9* 1 21.119 22.659 26.016 33.222 23.466 25.176 28.906 36.913.:f 25.812 27.694 31.797 40.604. 4 28.159 30.211 34.688 44.296 13-~ 30.505 32.729 37.578 47.987 v ~ 32.852 35.247 40.469 51.678 15 35.199 37.764 43.359 55.370 1 inch 37.545 40. 282 46.250 59.061 }4 42.238 45.317 52.031 66.444 1j ~ 46.931 50.352 57.813 73.826 I. ] 51.625 55.387 63.594 81.210 1.~ 56.317 60.422 69.375 88.592 1.1 61.011 65.458 75.156 95.975 1.1 65.704 70.493 80.938 103.358 1.7 ji 70.397 75.528 86.719 110.740 2. 75.090 80.563 92.500 118.128 NoTE. —The SPEcIFIC GRAVITY of the Wrought Iron is that of Pennsylvania plates, and of the Copper, that of plates from tlhe works of Messrs. Phelps,' Dodge. & Co., in Connecticurt. The Leqd. a meatn from several places. WEIGHT AND CAPACITY OF CAST IRON AND LEAD BALLS. 255 WEIGHT AND CAPACITY OF CAST IRON AND LEAD BALLS, Fromn 1 to 20 Inches in Diameter. Diameter in Capacity in cubic CAST IRON. LEAD. inches. inches. Pounds. Pounds. 1..5235.1365.2147 1.4 1.7671.4607.7248 2. 4.1887 1.0920 1.7180 2.4 8.1812 2.1328 3.3554 3. 14.1371 3.6855 5.7982 3.4 22.4492 5.8525 9.2073 4. 33.5103 8.7361 13.744 4.4 47.7129 12.4387 19.569 5. 65.4498 17.0628 26.843 5.~ 87.1137 22.7206 35.729 6. 113.0973 29.4845 46.385 6.4 143.7932 37.4528 58.976 7. 179.5943 46.8203 73.659 7.4 220.8932 57.5870 90.598 8. 268.0825 69.8892 109.952 8.4 321.5550 83.8396 131.883 9. 381.7034 99.5103 156.553 9.4 448.9204 117.0338 184.121 10. 523.5987 136.5025 214.749 11 696.9098 181.7648 285.832 12. 904.7784 235.8763 371.096 13. 1150.346 299.6230 471.806 14. 1436.754 374.5629 589.273 15. 1767.145 460.6959 724.781 16. 2144.660 559.1142 879.616 17. 2572.440 670.7168 1055.066 18. 3053.627 796.0825 1252.422 19. 3591.363 936.2708 1472.970 20. 4188.790 1092.0200 1717.995 WEIGHT OF A SQUARE FOOT OF WROUGHT IRON, AS PER BIRMINGHAM WIRE GAUGE. No. of No. of No. of Lbs. No. of Lbs. L bs. Lbs. Lbs. G~uge. LGauge. Gauge. Gauge. 1 12.55 8 6.64 15 2.97 21 1.32 2 11.25 9 6.29 16 2.62 22 1.15 3 10.45 10 5.5 17 2.19 23.99 4 9.55 11 4.73 18 1.92 24.95 5 8.66 12 4.3 19 1.7 25.84 6 8.34 13 3.64 20 1.41 26.78 7 7.5 14 3.23. _. 256 WEIGHT OF COPPER RODS AND PIPES. WEIGHT OF COPPER RODS OR BOLTS, From i to 4 Inches in Diameter, AND ONE FOOT IN LENGTH. Diameter. Pounds. Diameter. Pounds. Diameter. Pounds. X. s.1892 1 3.4170 2.A 13.6677.-76.2956 1,3g 3.8312 2. 15.3251 2.] 17.0750 6 {.4256 1. 4.2688 2.~ 18.9161 ~'~/.5794 1. 4.7298 2. 20.8562.7567 1. 5.2140 2.4 22.8913.IS.9578 1. 9 5.7228 2' 25.0188. 1.9982 1. X 7.9931 3 3 27.084308 *W 2.3176 1. 3 9.2702 3 39.77 4. 2.6605 1. 10.6420 3.4 45.4550 I. 3.0270. 12.10892 34. 48.4331 WEIGHT OF RIVETED COPPER PIPES, 13From 5 to 30 Inches in Diameter, on 3 to -ths thick, AND ONE FOOT IN LENGTH. Dtasn. Thickness Weightin Diam. Thickness Weight in Diaun. Thiehuem Weight in inins. in t6ths. pounds. in ins, in t6ths. pounds. in ins, in lIths. pounds. 5. 3 12.497 9.4i 4 -30.598 19. 4 60.142 5. 4 16.880 10. 4 32.208 19. 5 75.233 54 3 13.628 112. 4 35.200 20. 5 8. 208 5.4 4 18.395 12. 4 38.456 21. 5 82.5984 6. 3 1.765 13. 4 41.45 22. 5 86.455071 6. 4 19.908 14. 4 44.640 23. 5 90.3571 6. om 4 21.415 In15 5 59.588 fro 25. 9to8.ths t 12hick, 3Thickness ght in. Thicknes Weight in 50 ia.752 Thick26 5 10eight 897 7. 4 22.932 1 6. 5 63.470 27. 5 105.700 7.4 4 24.447 17. 4 53.856 28. 5 109.446 8. 4 25.96880 17. 5 67.3441 29. 5 11375.221 8.4 4 27.471 18. 4 57.037 30. 5 116.997 9. 4 2819.985 18.23. 5 71.258 The above weights include the laps on the sheets for riveting and caulking. The weights of the rivets are not added; the aeamber per lineal foot of pipe depends upon the distance they are placed apart, and their soze upon the diameter of the pipe. ter of the pipe. COPPER, LEAD, AND BRASS.:257 COPPER. To ascertain the Weight of Copper. RuLE.-Find by calculation the number of cubic inches in the piece, multiply them by.32118, and the product will be the weight in pounds. EXAMPLE.-What is the weight of a copper plate i an inch thick by 16 inches square? 162- 256.5 for ~ an inch. 128.0X.32118 = 41.111 pounds. BRAzIER'S SHEETS. 30X60 inches, and from 12 to 100 lbs. per square foot, SHEATHING COPPER. 14X48 inches, and from 14 to 34 oz. per square foot. LEAD. To ascertain the Weight of Lead. RULE.-Find by calculation the number of cubic inches in the piece, and multiply the sum by.41015, and the product will be the weight in pounds. EXAMPLE. —What is the weight of a leaden pipe 12 feet long, 3h inches in diameter, and 1 inch thick! By rule in Mfensuration of Sunfaces, to ascertain the area of cylindrical rings, Area of (3,- 1+1) = 25.967 " 3'" =. 11.044 Difference, 14.923, or area of ring. 144 = 12 feet. 2148.912X.41015 - 881.376 pounds. BRASS. To ascertain the Weight of ordinary Brass Castings. RULE.-Find the number of cubic inches in the piece, multiply by.3112, and the product will be the weight in pounds. Y2 258 CABLES AND ANCHORS. CABLES AND ANCHORS. TABLE showing the Size of Cables and Anchors proportioned to the Tonnage of Vessels. Tonnage f CrCables. Chain Cables.. oof' Weight of Weight of a Weight of a onnage o Circumference Diameter in ro oin Anchor in fathom of fathom of vese in inches. inches. ons. pounds. Chain. Cable. 5 3..-5.. 56 5. 2.1 8 4..9'1.4 84 8. 4.10 4. ~.1- 2.~ 112 11. 4.6 15 5.9 4. 168 14. 6.5 25 6..- 5. 224 17. 8.4 40 6. 6. 336 24. 9.8 60 7. IF 7. 392 27. 11.4 75 7. 43 9. 532 30. 13. 100 8. I-3 10. 616 36. 15. 130 9.. f 12. 700 42. 18.9 150 9. i 14. 840 50. 21. 180 10. 1. 16. 952 56. 25.7 200 11. 1._ 18. 1176 60. 28.2 240 12. 1. 20. 1400 70. 33.6 270 12.W 1.6 21. 1456 78. 36.4 320 13. 14 22.- 1680 86. 42.5 360 14. 1. 5 25. 1904 96. 45.7 400 14.1 1. 27. 2072 104. 49. 440 15.i 1. 30. 2240 115. 56. 480 16. 1. 33. 2408 125. 59.5 520 16. i. 36. 2800 136. 63.4 570 17. 14 39. 3360 144. 67.2 620 1.7 - 42. 3920 152. 71.1 680 18. 1. 4 45. 4200 161. 75.6 740 19. 1. 49. 4480 172. 84.2 820 20. 1. 52. 5600 184. 93.3 900 22. 1.o1 56. 6720 196. 112.9 1000 24. 2. 60. 7168 208. 134.6 The proof in the U. S. Naval Service is about 121 per cent. less than the above for the larger sizes, and from 25 to 30 per cent. for the smaller. The mean results of experiments at the U. S. Navy Yard, Washington, D. C., give for the cohesive force of chain iron, per square inch, 43500 lbs. The diameter of chain cables assigned to the tonnage of vessels as above is based upon the practice of the Merchant Service, where shorter and fewer chains are used than in the Naval Service; hence heavier chains and anchors are necessary. A first class ship of the line has three chains 24 inches in diameter and 180 fathoms in length, and anchors of 9000 lbs. weight. CABLES. 259 CABLES. TABLE showing what Weight a good Hemp Cable will bear with Safety. Circumference. Pounds. Circumference. Pounds. Circumference. Pounds. 6. 4320. 10.25 12607.5 14.50 25230. 6.25 4687.5 10.50 13230. 14.75 26107.5 6.50 5070. 10.75 13867.5 15. 27000. 6.75 5467.5 11. 14520. 15.25 27907.5 7. 5880. 11.25 15187.5 15.50 28830. 7.25 6307.5 11.50 15870. 15.75 29767.5 7.50 6750. 11.75 16567.5 16. 30720. 7.75 7207.5 12. 17280. 16.25 31687.5 8. 7680. 12.25 18007.5 16.50 32670. 8.25 8167.5 12.50 18750. 16.75 33667.5 8.50 8670. 12.75 19507.5 17. 34680. 8.75 9187.5 13. 20280. 17.25 35707.5 9. 9720. 13.25 21067.5 17.50 36750. 9.25 10267.5 13.50 21870. 17.75 37807.5 9.50 10830. 13.75 22687.5 18. 38880. 9.75 11407.5 14. 23520. 18.25 39967.5 10. 12000. 14.25 24367.5 To ascertain the Strength of Cables. MIultiply the square of the circumference in inches by 120, and the product Ze. the weight the cable will bear in pounds, with safety. To ascertain the Weight of Manilla Ropes and Hawsers. Multiply the square of the circumference in inches by.03, and the product is the weight in pounds of a foot in length. This is but an approximation, and yet it is sufficiently correct for many purposes. TABT.E showing what Weight a Hemp Rope will bear with Safety. Circumference. Pounds. Circumference. Pounds. Circumference. Pounds. 1. -200. 3. 2450. 6. 7200. 1.4 312.5 3. 2812.5 6.1 7812.5 1.~ 450. 4. 3200. 6.1 8450. 1.4 612.5 4.1 3612.5 6.4 9112.5 2. 800. 4.1 4050. 7. 9800. 2.1 1012.5 4.4 4512.5 7. 10512.5 2.1 1250. 5. 5000. 7.~ 11250. 24f 1512.5 5.I 5512.5 7.4 12012.5 3. 1800. 5.1 6050. 8. 12800. 3. 1 2112.5 5.; 6612.5 26}0 CABLES, To ascertain the Strength of Ropes. Multiply the square of the circumference in inches by 200, and it gives thl weight the rope will bear in pounds, with safety. To ascertain the Weight of Cable-laid Ropes. Multiply the square of the circumference in inches by.036, and the product it the weight in pounds of a foot in length. To ascertain the Weight of Tarred Ropes and Cables. Multiply the square of the circumference by 2.13, and divide by 9; the product is the weight of a fathom in pounds. Or, multiply the square of the circumference by.04, and the product is the weight of a foot. For the ultimate strength, divide the square of the circumference in inches by 5; the product is the weight in tons. WIRE ROPE. TABLE of Experiments made by the British Admiralty with Wire and Hempen Rope and Chain. COMPARATIVE STRENGTH. Circumference of. Circumference of Diameter of Breaking Wire Rope. Hempen Rope. Chain. Weight Inches. Inches. Inches. Lbs. 2 5 i 14224 3 8 13 26880 4 10 3 43232 WEIGHT PER FATHOM. Wire Rope. Hempen Rope. Ciain. Lbs. Lbs. Lbs. 3.875 6. 1.6 8.5 14. 36 14.6125 25. 53 Notes. A square inch of hemp fibres will support a weight of 9200 lbs. The utmost strength of a good hemp rope is 6400 lbs. to the square inch; in practice it should not be- subjected to more than half this strain. It stretches from 4 to and its diameter is diminished from 4 to before breaking. A difference in the quality of hemp may produce a difference of ~ in the strength of ropes of the same size. The strength of Manilla is about.i that of hemp. White ropes are one third more durable. PILING OF BALLS AND SHELLS. 26 PILING OF BALLS AND SHELLS. To find the Number of Balls in a Triangular Pile. RULE.-Multiply continually together the number of balls in one side of the bottom row, and that number increased by 1; also, the same number increased by 2; - of the product will be the answer. EXAMPLE. —What is the number of balls in a pile, each side of the. base containing 30 balls? 30X 31 X 32. 6 - 4960, Ans. Tofind the ANumber of Balls in a Square Pile. RULE.-Multiply continually together the number in one side o0f the bottom course, that number increased by 1, and double the same number increased by 1; - of the product will be the answer. EXAMPLE.-HOW many balls are there in a pile of 30 rows 1 30X31 X 61 -6 =- 9455, Ans. To find the Number of Balls in an Oblong Pile. RULE. —From 3 times the number in the length of the base row subtract one less than the breadth of the same; multiply the itmainder by the same breadth, and the product by one more than.o.t same, and divide by 6. ExAMPLE.-Required the number of balls in an oblong pile, ftbe numbers in the base row being 16 and 7 X 16 X3-7 —1 X7X7+ 1 —6 392, Ans. Tofind the Number of Balls in an Incomplete Pile. RULE.-From the number in the pile, considered as complete, subtract the number conceived to be in the upper pile which is wanting. 262 WEIGHT AND DIMENSIONS OF BALLS AND SHELLS. WEIGHT AND DIMENSIONS OF BALLS AND SHELLS. THE weights of these may be found by the rules in Mensuration, also, in the tables, pages 237, 238, and 255. To find the Weight of an Iron Ball from its Diameter. An iron ball of 4 inches diameter weighs 8.736 lbs. Therefore, 8.736 of the cube of the diameter is the weight, for the weight of spheres is as the cubes of the diameters. EXAMPLE. —What is the weight of a ball 10 inches in diameter T S-.66 of 103 = 136.5 lbs., Ans. To find the Diameter from the Weight. EXAMPLE.-What is the diameter of an iron ball, its weight being 99.5 lbs.? 8/364 X 99.5 = 9 inches, Ans. Or, multiply the cube of the diameter in inches by.1365, and the sum is the weight. And divide the weight in pounds by.1365, and the cube root of the product is the diameter. To find the Weight of a Leaden Ball. A leaden ball of 4 inches diameter weighs 13.744 lbs. Therefore, i.'4_ of the cube of the diameter. EXAMPLE. —What is the weight of a leaden ball 10 inches in di-:-aeter I 13.744 of 103 - 214.7 lbs., Ans. 664 Inversely, 63.-44 X weight = diameter. Or, multiply the cube of the diameter in inches by.2147, and the aum is the weight. And divide the weight in pounds by.2147, and the cube root of the product is the diameter. To find the Weight of a Cast Iron Shell. Multiply the difference of the cubes of the exterior and interior diameter in inches by.1365. EXAMPLE.-What is the weight of a shell having 10 and 8.50 inch..,s for its diameters? 103 —8.53 X.1365 = 52.6 lbs., Asm. MIXSCELLANEOUS IULES. 265 MEASURING OF TIMBER. SAWED 01r IEWN timber is measured by the cubic foot. The unit of board measure is a superficial foot 1 inch thick. To measure Round Timber. Multiply the length in inches by the square of ~ the mean girth in inches, and the product, divided by 1728, will give the contents in cubic feet. When the length is given in feet, and the girth in inches, divide by 144. When all the dimensions are in feet, the product is the content without a,ivislon LxC2 Or, 144, L the length in feet, and C half the sum of the circumferences of the two ends in inches. Or, ascertain the contents by the rules in Mensuration of Solids, page 82, and multiply by.75734. EXAMPLE.-The girths of a piece of timber are 31.416 and 62.832 inches, and its length 50 feet; required its contents. 31.416+62.832 2 -+ 11.781, and 11.7812 X50 —144-=48.1916 cubic feet, Ans. 50 X 47.1242 Or, -.144 —48.1916 cubic feet. 600 Or, 10s3-203-5-20-10X.7854X ---- 63.632X.75734 - 48.19106 cubic feet, Ans. To measure Square Timber. Multiply the length in inches by the breadth in inches, and the product by the depth in feet; divide by 144,_and the quotient is the content. NOTE.-When all the dimensions are in feet, omit the divisor of 144. BOARD MEASURE. Multiply the length by the breadth, and the productis the content. NOTE. —This rule only applies when all the dimensions are in feet. When either the length or breadth are given in inches, divide their product by 12; and when all the dimensions are in inches, divide it by 144. Pine spars, from 10 to 4b inches in diameter inclusive, and spruce spars,. are to be measured by taking the diameter, clear of bark, at ~ of their length from the large end. Spars are usually purchased by the inch diameter; all under 4 inches are considered poles. Spruce spars of 7 inches and less should have 5 feet:in length for every inch diameter. Those above 7 inches should have 4 feet in length for every inch diameter. z 266 MISCELLANEOUS RULES. DISPLACEMENT OF VESSELS. To ascertain the Displacement of a Vessel. RULE.-Multiply the mean of the lengths of the keel and between the perpendiculars, by the area of the immersed midship section in square feet, and divide this product, if for a Ship of the Line or a fill-built Mercha.ntman, by 42; if for a Frigate or an ordinary Merchantnman, by 43; if for a Sloop or a Naval Steamer, by from 44 to 47; if for a Merchant Sea Steamer, by from 45 to 50; and if for a River Steamer, by from 48 to 56. EXAMPLE. —The lengths of a Ist class Sea Steamer, now (1847' constructing at the Navy Yard, Gosport, are as follows: Between the perpendiculars. 250 feet Length of keel 236 feet And the area of the midship section below the load-line is 65( square feet; what should be the displacement by the above rule 1 250+236 — 2 = 243. 243 x 650 Then - = 3547+-tons. 44.5 Note.-The exact displacement is 3535 tons. Each man in a vessel requires for his own weight, clothes, &c,, and stores, water, &c., for one month, a displacement of 663 lbs. EXTERNAL SURFACE OF HULLS OF VESSELS. To ascertain the Bottom and Side Surface of a Vessel. RULE.-Multiply the length of the curve of the midship section, taken from the top of the upper or spar deck beams on one side, to the same point on the other (omitting the width of the keel), by the mean of the lengths of the keel and between the perpendiculars, and take from -8- to ~-0O of the product (according to the capacity of the vessel) for the surface required in square feet. EXAMPLE.-The lengths of a 2d class Steamer, now (1847) constructing at the Navy Yard, Brooklyn, are as follows: Between the perpendiculars. 210 feet Length of keel....201 feet And the curved surface of the midship section is 76 feet; what is the surface 1 210+201 2- 205.5. 205. x76 x 87 Then 205.5 x 76 x 87 13,587 square feet, o 100 AWte.-The exact surface is 13,630 square feet. MISCELLANEOUS NOTES. 26i MISCELLANEOUS NOTES. ON MIATERIALS, ETC. Wood is from 7 to 20 times stronger transversely than longitudinally. In Buffon's experiments, b, d, and I being the breadth, depth, and iength of a piece of oak in inches, the weight that broke it in pounds was bd2(54s 5 10). The hardness cf metals is as follows: Iron, Platina, Copper, Silver, Gold, Tin, Lead. A piece spliced on to strengthen a beam should be on its convex side. Springs are weakened by use, but recover their strength if laid by. A pipe of cast iron 15 inches in diameter and.75 inches thick will sustain a head of water of 600 feet. One of oak, 2 inches thick, and of the same diameter, will sustain a head of 180 feet. When the cohesion is the same, the thickness varies as the height X the diameter. When one beam is let in, at an inclination to the depth of another, so as to bear in the direction of the fibres of the beam that is cut; the depth of the cut at right angles to the fibres should not be more than. of the length of the piece, the fibres of which, by their cohesion, resist the pressure. Metals have five degrees of lustre-splendent, shining, glistening, glimmering, and dull. THE Vernier Scale is 44, divided into 10 equal parts; so that. it divides a scale of lOths into 100ths when the lines meet even in the two scales. A luminous point, to produce a visual circle, must have a velocity of 10 feet in a second, the diameter not exceeding 15 inches. Tides. The difference in time between high water averages ibout 49 minutes each day.'In Sandy soil, the greatest force of a pile-driver will not drive a pile over 15 feet. Afall of - of an inch in a mile will produce a current in rivers. Melted snow produces about 8 of its bulk of water. All solid bodies become lumittous at 800 degrees of heat. At the depth of 45 feet, the temperature of the earth is uniform throughout the year. A Sper7naceti candle.85 of an inch in diameter consumes an inch in length in I hour. Silica is the base of the mineral world, and Carbon of the organized. &sund passes in water at a velocity of 4708 feet per second. 268 MISCELLANEOUS NOTES. SOLDERS. For Lead, melt 1 part of Block tin; and when in a state of fusion, add 2 parts of Lead. Resin should be used with this solder. For Tin, Pewter 4 parts, Tin I, and Bismuth 1; melt them to. gether. Resin is also used with this solder. For Iron, tough Brass, with a small quantity of Borax. CEBMENTS. Glue. Powdered chalk added to common glue strengthens it. A glue which will resist the action of water is made by boiling I pound of glue in 2 quarts of skimmed milk. Soft Cement. For steam-boilers, steam-pipes, &ec. Red or white lead in oil, 4 parts; Iron borings, 2 to 3 parts. Hard Cement. Iron borings and salt water, and a small quantity of sal ammoniac with fresh water. PAINTS. White Paint. Inside work. Outside work. White-lead, ground in oil 80. 80 Boiled oil.. 14.5 9 Raw oil..- 9 Spirits turpentine. 8. 4 New wood work requires about 1 lb. to the square yard for 3 coats. Lead Colour. White-lead, ground in oil, 75 Litharge....5 Lampblack.. 1 Japan varnish...5 Boiled linseed oil. 23 Spirits turpentine. 2.5 The turpentine and varnish are added as the paint is required fot use or transportation. Gray, or Stone Colour. White-lead, in oil. 78. Spirits turpentine. 3. Boiled oil.. 9.5 Turkev umber...5 Raw oil... 9.5 Lampblack...25 1 square yard of new brick work requires, for 2 coats, 1.1 lb.; for 3 coats, 1.5 lbs. Cream Colour. Ist coat. 2d coat White-lead, in oil 66.6 70. French yellow. 3.3 3.3 Japan varnish.... 1.3 1.3 Raw oil.... 28. 24.5 Spirits turpentine 2.25 2.25 I square yard of new brick work requires, for 1st coat, 0.75; for 2d coat, 0.3 lbs. MISCELLANEOUS NOTES. 2.69 Black Paint (for Iron). Lampblack. 28 Linseed oil, boiled. 73 Litharge... 1 Spirits turpentine. 1 Japan varnish.. 1 The varnish and turpentine are added last. Liquid Olive Colour. Olive paste.. 61.5 Dryings... 3.5 Boiled oil.. 29.5 Japan varnish. 2. Spirits turpentine 5.5 Paint for Tarpaulins (Olive). Liquid olive colour. 100 ( Spirits turpentine. 6 Beeswax. 6 1 square yard requires 2 lbs. for 3 coats. Dissolve the beeswax in the turpentine, and mix the paint warm. LACKER for Iron Ordnance. Black-lead, pulverized 12 Red-lead... 12 Litharge... 5 Lampblack. 5 Linseed oil. 66 Boil it gently for about 20 minutes, stirring it constantly during that time. Lacker for Small Arms, or for Water Proof Paper. Beeswax.. 18. l Spirits turpentine. 80 Boiled linseed oil 3.5 Heat the ingredients in a copper or earthen vessel over a gentle fire, in a water bath, until they are well mixed. Lacker for Bright Iron Work. Linseed oil, boiled. 80.5 l Litharge... 5.5 White-lead, ground in oil,11.25 I Pulverized rosin. 2.75 Add the litharge to the oil; let it simmer over a slow fire for 3 hours; strain it, and add the rosin and white-lead; keep it gently warmed, and stir it until the rosin is dissolved. Staining Wv/ood and Ivory. Yellowt. Dilute nitric acid will produce it on wood. Red. An infusion of Brazil wood in stale urine, in the proportion of a lb. to a gallon for wood,', be laid on when boiling hot, and should be laid over with alum water before it dries. Or, a solution of dragon's blood, in spirits of wine, may be used. Black. Strong solution of nitric acid, for wood or ivory..Mahogany. Brazil, Madder, and Logwood, dissolved in water and put on hot. Blue. Ivory may be stained thus: Soak it in a solution of verdigris in nitric acid, which will tulrn it green; then dip it into a solution of pearlash boiling hot. Purple. Soak ivory in a solution of sal ammoniac into four times its weight of nitrous acid. Z2 270 MISCELLANEOUS NOTES. TRACING PAPER. By weight. Spirits of turpentine. Resin.. Boiled nut-oil..... Use with a soft brush or sponge. ALLOYS. W~hite argentan.. 8. 3. Pinchbeck.....5. Chinese white copper 40.4 25.4 2.6 31.6 Chinese silver.. 65.2 19.5 13. 2.5 12 White argentan. 8 3.5 3. Pinchbeck.... 5. 1. German silver.. 1. 1. Britannia metal. 4 4. Wlhen fused, add. 4. 4. Printing characters. 4. 1. Small type and stereotype plates.. 9. 2. 2. Telescopic mirrors. 100. 50. Bronze. Statuary.. 91.4 9 5.5.4 1.7 Large cannon 90. 10. Small cannon.. 93. 7. Medals.. 100. 8. Cymbals... 80 20. Tutenague copper. 8. 5. 3. Newton's fusible metal, melts at a temperature less than that of boiling water 3. 5. 8. Babbitt's metal. See page 272 for directions. A metal that expands in cooling... 9. 2. 1. NoTE. —The more infusible metals should be melted first. RELATIVE POWER OF ELECTRIC CONDUCTORS.'Copper... 1.000 Platina...188 Gold...936 Iron....158 Silver....736 Tin....155 Zinc...285 Lead....083 The conducting power with similar metals is inversely as their lengths; and with equal lengths it is proportional to the mass, and not to the surface. Increasing the temperature decreases the conducting power. MIISCELLANEOUS NOTES. 271 RELATIVE COSTS, PER HOUR, OF VARIOUS METHODS OF ILLUMINATION. With equal Light. Parker's hot oil lamp (sperm oil).902 Carcel, mechanical lamp (sperm oil) 1.280 French, common lamp (sperm oil) 1.707 Tallow, mould candles 2.520 Spermaceti candles. 5.352 Wax candles 5.892 Wax. The consumption of a wax candle is 125 grains per hour in still air; the intensity of its light is -~-th that of a Carcel lamp. Specific gravity..970 Three in a pound, 1 inch diameter, X12 inches long. Three in a pound, g "' X15 " Six in a pound, 8, " " X 9 " An inch of one, 1 inch diameter, will burn for 1 hours =.0277 lbs. Spermaceti. The consumption of a spermaceti candle is.0246 lbs. per hour, and its light is 1 —th of a Carcel lamp. Specific gravity.....943 Three in a pound, AP- inch diameter, X 15 inches long. Four in a pound, -A "... X13~ " " Tallow. The consumption of a tallow candle is about.025 lbs. per hour, the intensity of its light is from T-L to f- that of a Carcel lamp. Specific gravity..941 Three in a pound, 1 inch diameter, X 121 inches long. Three in a pound,,.. "s X15 " " Four in a pound, -,' " X 134 " " C. E. and 21. Journal WAVES. The undulations of waves are performed in the same time as the oscillations of a pendulum, the length of which is equal to the breadth jf a wave, or to the distance between two neighbouring cavities or eminences. STREDIGTH OF ICE. Ieb 2 inches thick will bear infantry. "4 " " cavalry or light guns. "6 f'" heavy field-guns. i' 8 " " 24-pounder guns on sledges; weight not over 1000 lbs. to a square foot. 272 MISCELLANEOUS NOTES, DIRECTIONS FOR PREPARING AND FITTING BABBITT'S ANT': ATTRITION METAL. Melt 4 pounds of copper; add, by degrees, 12 pounds best quality Banca tin, 8 pounds regulus of antimony, and 12 pounds more of tin, while the composition is in a melted state. After the copper is melted and 4 or 5 pounds of tin have been added, the heat should be reduced to a dull red, to prevent oxydation; then add the remainder of the metal as above. In melting the composition, it is better to keep a small quantity of powdered charcoal on the surface of the metal. The above composition is called hardening; for lining, take one pound of this bardening, and nielt with it 2 pounds Banca tin, Which produces the lining metal for use. Thus the proportions for lining metal are 4 pounds of copper, 8 of regulus of antimony, and 96 of tin. The article to be lined, having been cast with a recess for the lining, is to be nicely fitted to a former, which is made the same shape as the bearing. Drill a hole in the article for the reception of the metal, say half or three quarters of an inch, according to the size of it. Coat over the part not to be tinned with a\ clay wash; wet the part to be tinned with alcohol, and sprinkle on it powdered sal ammoniac; heat it till a fume arises from the sal ammoniac, and then immerse it in melted Banca tin, care being taken not to heat it so that it will oxydize. After the article is tinned, should it have a dark colour, sprinkle a little sal ammoniac on it, which will make it of a bright silver colour, and cool it gradually in water; then take the former, to which the article has been fitted, and coat it over with a thin clay wash, and warm it so that it will be perfectly dry, heat the article until the tin begins to melt, lay it on the former, and pour in the metal, which should not be so hot as to oxydize, through the drilled hole, giving it a head, so that as it shrinks it will fill up. After it is sufficiently cool, remove the former. P.S.-A shorter method may be adopted when the work is light enough to handle quickly, viz.: when the article is prepared for tinning, it may be immersed in the lining metal instead of the tin, brushed lightly in order to remove the sal ammoniac from the surface, placed immediately on the former, and lined at the same heating. SOLDERING FLUID FOR SOFT SOLDER. To 2 fluid ounces of muriatic acid add small pieces of zinc until bubbles cease to rise. Add half a tea-spoonful of sal ammoniac and 2 fluid ounces of water. By the application of this, Iron or Steel may be soldered without being previously tinned. FALL OF RAIN AND SNOW. The average quantity of water which falls in rain and snow at Philadelphia in one year is 36 inches. MISCELLANEOUS NOTES. 273 ANGLES OF EQUILIBRIUM AT WHICH VARIOUS SUBSTANCES STAND, AS TAKEN WITH A CLINOMETER. D egrees. Lime-dust as it falls from a spout... 45 Wheat flour as it falls from a spout. 44 Malt flour as it falls from a spout. 40 Saw-dust as it falls from a spout.. 44 Dry sand as it falls from a spout... 40 Sand less dry as it falls from a spout. 39.6 Wheat corn as it falls from a spout... 37 Malt corn as it falls from a spout. 37 Common mould as it falls from.a spout... 37 Pease as they fall from a spout. 35 Coarse gravel heaps. 35 to 38 Common gravel. 35 to 36 Large flints.40 to 45 Flints, half size....... 35 Flints approaching to sand..... 34 to 35 TABLE of the Diameter and Length of Gas-pipes to transmit given Quantities of Gas to Branch Pipes and Burners.-DR. URE. No. orcubic feet of gas per hour. Length of pipe in feet. Diameter of pipe in inche. 50 100 0.40 250 200 1. 500 600 1.97 700 1000 2.65 1000 1000 3.16 1500 1000 3.87 2000 1000 4.47 2000 2000 5.32 2000 4000 6.33 2000 6000 7. 6000 1000 7.75 6000 2000 9.21 8000 1000 8.95 8000 2000 16.65 These dimensions are applicable to the mains which conduct the gas to the places where it is to be used. If they send off branches for burners, the diameter may be reduced, or the length may be greiater. For example, if a pipe of 5.32 inches, which transmits 2000 cubic feet through a length of 2000 feet, gives off, in this space, 1000 cubic feet of gas, then the same diameter can continue to transmit the gas through a length of 2450 feet. 274 MISCELLANEOUS NOTES. LIMITS OF VEGETATION IN THE TEMPERATE ZONE. The vine ceases to grow at about 2300 feet above the level of the sea; Indian corn, at 2800; oak, at 3350; walnut, at 3600; ash, at 4800; yellow pine, at 6200; and fir, at 6700. Perpetual Snow. Under the equator, at 15,800 feet above the level of the sea; in latitude 45~, at 8400; and in latitude 650, at. 5000. Hempen Cords. When twisted, will support the following weights to the square inch of their section. Lbs. Lbs. i to I inch diameter.. 8.746 3 to 5 inches diameter 5.345 1 to 3 inches diameter. 6.800 5 to 7 inches diameter. 4.860 LAP-WELDED IRON BOILER TUBES (Prosse'~'s Patent). TABLE of the Diameter, Thickness, and WVeig'ht of Iron Boiler Tubes. External Thickness of Average Price External Average Thickness of Price Diameter. Wire Guage. Weight. per fdot. Diameter. Weight. Wire Guage. per foot Inches. No. Ibs. per ft. Cents, Inches. lbs. per ft. No. Cents. -1 16 1 20 -23 25 12- 48 15. 1 29 3 3 " 51 1- 13 1 31 3- 4 12 85 2 " 2 34 4 5 1 1 1 10 It s 2 (" 5 6 — 10 1 50 2 2 2 39 6 9 I 2 20 42. 142- 2,91 43 7 12 8 3 00 ~_..t14~i.14 2 46 8 1 16 7 4 00 MISCELLANEOUS NOTES. 276 Formula for ascertaining the thickness in Inches of several descriptions of Pipe of uniform Strength. (Not riveted.) Wrought iron...000059 x p d +-.12 Cast iron..000162 X p d +.33 Copper..000107 Xp d+.16 Lead..000165 X p d +.20 p representing the pressure in pounds per square inch, and d the diameter of the pipe. ALLOYS OF COPPER AND TIN. GUN METAL. Tin to be added to 1 pound of Copper. Soft gun metal... 1 oz Mathematical instruments 1.25 " Wheels.... 1.5 " Guns, large.. 1.75 i" small... 1.50 " Machinery bearings.2 to 2.50 Musical bells..... 3 " Gongs, cymbals, &c. 3.50 House bells, small.. 4 " " " large..... 4.50 Church bells. 5 " Speculum metal 7 to 8 " Temper metal for adding small quantities of copper, 32 oz. COPPER AND ZINC. Tin to be added to\l pound. Copper castings.1.125 to 1.50 oz. Gilding metal. 1 " 1.50 " Tombac (red brass). 2 Red sheet brass. Pinchbeck and Bath metal 3 to 4 " Bristol brass.... 6 " Ordinary brass..... 8 Muntz's metal for ship fastenings, 10.66' sheathing, &c. Soft spelter solder.. 16 " COPPER, ZINC, TIN, AND LEAD. Tin. Zinc. Copper. Brass Extremely tenacious.. 1.5 oz..5 oz. 16 oz. Wheels... 1.5 " 16 " 2 ox For turning... 2 " 1.5" Bearings, Nuts, &c. 2.5 " 1.5" 276 MISCELLANEBOUS NOTES. Longitudinal Compression. Results of Experiments upon thce Resistance to Compression of varaiou Substances of the Dimension of 1 Cubic Inch.Loss by laterisal. lXbeigllt in Materials. Weight Freezing lbtelts. Xegin lbs. in10.000 of an oz. METALS. STONES, &C. Cast iron, gun metal. 105.000 Granite, Patapsco.. 11.200 Wrought iron,. 116.000 *Sandstone, Aquia Creek. 5.340 begin to 40000 " Seneca.. 10.764 yield " Aquia Creek, stra- 8332?i332 Brass, fine.. 164.800 ta laid horizont'y Copper, cast. 117.000:Marble, Stockbridge. 10.382 8.8'Pin, cast... 15.550 [ " East Chester. 23.917 6.2 Lead.. 7.730 " Symington, large crys WOODS. " same, strata horizontal 10.124 Oak.. 4.500 " same, strata vertical. 9.124 Pine.. 1. 1.930 " same, fine crystal. 18.248 Elm... 1.290 Italia... 12.624 " Lee, Mass. 22.702 9.9 STONES, &C.! Hastings, N. Y.. 18.941 11.8 Gneiss. 19.68)0 Montgomery co., Penn. 8.950 8.8 Brick, hard. 4.368 Baltimore, large crystal 8.057 21.9 common. 4.000 " " small cystal 18.061 8.1 fire. *n.c.a. * Same as that of the Capitol, Treasury Department, and Patent Office, Washington, D. C. 1 Same as that of the Smithsonian Institute.,6" " " (: City Hall, New York; 4.. it General Post-office, Washington, D. C. National W~ashington Monument. Resistance of Stones, Afc., to the Effects of Freezing. Various experirrents show that the power of stones, &c., to resist the effects of freezing is a fair exponent of that to resist compression. Conducting and Cooling Powers Absorption of Moisture of various Buaildinlg iIaterials. Materia~l. ~Absorption of Moisture Conducting Cooli ng Specific by Weight. b.y Bulk.-lte as 100. Slate as 100. Gravty. White marble, Carrara, soft' 3.10 8.42........ 2.717 " " " 4 4 C hard 8.50 23.09........ 2.717 Slate..... 3.50 97.58 100. 100. 2.788 Asphaltum 5. 12.86 45.19 105.57 2.572 Stucco. 16. 35.56... 2.223 Fire-brick.. 32 - 70.43 61.70 103.13 2201 Stock brick. 109. 199.57 60.14 96.97 1.831 Hlair and lime.. 109.12 184.52 109.38 37.93 1.691 Chalk... 133;50 206.79 56.38 74.58 1.542 Roman cement... 133.56. 208.:35 20.88 72.63 1.560 Plaster and sand. 147. 192.27 I 18.70 63.31 1.308 Beech-wood.. 185.50; 138.04 22.44 84.71.744'laster of Paris. 187.50 220.50 20.26 60.81 1.176 Oak. 224.75 128.04 33.66 55.79.569 Fir-wood.. 622.75 265.41 27.61. 69.44.426 Lead......... 521.35 95.67 10.56 _ ~~~~~~~~~~~.... i4ISCELLANEOUS NOTES. 277 Practical Deductions from above and previously-given Results. Asphaltum is the best composition for resisting moisture, and, being a slow conductor of heat, it is best adapted where economy of heat and dryness are required. Slate is a very dry material, but, from its quick conducting power, it is ill adapted for the retention of heat. Cements. Plaster of Paris and Woods are well adapted for the lining of rooms, being the warmest substances, while Hair and lime. being a quick conductor, is one of the coldest compositions. Fire-brick absorbs much heat, and is therefore well adapted for the lining of fire-places, furnaces, &c.; while, on the contrary, Iron, being a high conductor of heat, is one of the worst of substances for this purpose. Common brick is not a very slow conductor of heat, it is 1.8 times higher in the scale than oak-wood, and about — lth lower than firebrick. Hard marble absorbs more moisture than soft. AA 278 MISCELLANEOUS NOTES. TABLE OF RECIPROCALS. 279 TABLE of RECIPROCALS, or Decimnal E.pressions, jromt I to 250. No. Recip. No. Recip. No. Recip. No. I Recip. No. Recip. 1 1.000000 51.019607 101.009900 151.006623 201.004975 2.500000 52.019231 102.009803 152.006579 202.004951 3 -.333333 53.018868 103.009709 153.006536 203.004927 4.25(000 54.018519 104.009616 154.006494 204.004901 5.200000 55.018182 105.009523 155.006451 205.004879 66.166667 56,017857 106.009433 156.006411 2061.004855 7.142857 57.017543 107. 009345 157.006370 207.004831 8.125000 58.017242 108.009260 158.006329 208.004807 9.111111 59.016949 109.009174 159.006290 209.004785 10. 100000 60.016667 110.009091 160.006250 210.004762 11.090901 61.016393 111.009010 161.006211 211.004740 12.083333 62.016129 112.008928 162.006172 212.004716 13.076923 63.015873 113.008850 163.006135 213.004695 14.071428 64.015625 114.008771 164.006097 214.004673 15i.066667 65.015385 115.008695 165.006061 215.004651 16.062500 66.015151 116.008620 166.006025 216.004629 17.058823 67.014925 117.008548 167.005988 217.004609 18.055556 68.014-705 118.008475 168.005952 218.004588 19.052632 69.014492 119.0084031169],005917 219.004566 20.050000 70.014285 120.008333 170.005882 220.004546 211.047620 7 1.0140851121.008264 171.005847 221.004525 22.045455 72.013889 122.008196 172.005813 222.004505 23 043078 73.013698 123.008130 173.005781 223.00448.5 24 *041667 74.013513 124.008065 174.005748 224.004465 25 *040000 751.013333 125.008000 175.005715 225.004444 26 *038462 76.013158 1261.007936 176.005682 226.004425 27'037038 77.012987 127'.007875 177.005650 227.004406 28 *035715 78.012820 128.007812 178.005618 228.004386 29 *034483 79.012659 1291.0077521179.005586 229.004366 30.033333 80.012500 130-.007693 180.005556 230.004348 3 1.032259 81.012346 131.007634 181.005524 231.004329 32.032250 82.012195 132.007576 182.005495 232.004311 33.030303 83.012048 133.007519 183.005464 233.004292 34.029412 84.011904 1341.007463 184.005434 234.004273 35.028572 85.011765 1351.007408 185.005406 235.004256 36.027778 86.011628 136i.007352 186.005376 236.004238 37.027028 87.011494 137i.007299 187.005347 2371.004220 38.026316 88.011364 1381.007247 188.005320 238.004201 39.025642 89.011235 139:.007195 189.005292 239.004184 10.025000 90.011111 140.007143 190.005264 240.004167 41.024390 91.010989 141'.007093 191.005235 241.04150 42.023809 92.010870 142;.007042 192.005208 242.004132 13.023255 93.010753 1143.006994 193.005182 243.004116 14.022727 94.010639 1144.006944 194.005155 244.004098 15.022222 95.010527 1145.006896 195.005129 245.004081 46,021739 96.010417 146:.006850 196.005102 246.004065 47.021276 97.010310 147i.006802 197,005076 247.004048 48.020833 98.010204 1481.006756 198.005051 248.004033 49.020408 99.010101 1149.006712 199.005026 249.004016 50.020000 100.0100001150!.006667 200.005000 250.004000 Use of the foregoing Table.-In the column of reciprocals is given the decimal expression for any number from 1 to 250, which number is the denominator of a fraction. Thus:.033333 is the decimal expression for -w. Hence for > it would be.033333 X 7-.233331. 280 STEAM ACTING EXPANSIVELY. To find the Mean Pressure of Steam on'a Piston. RULE. —Divide the length of the stroke, added to the clearance in the cylinder at one end, by the length of the stroke at which the steam is cut off, added to the clearance,* and the quotient will express the relative expansion it undergoes. Find in the following table, in the column of expansion, a number corresponding to this; take out the multiplier opposite to it, and multiply it into the full pressure of the steam per square inch as it enters the cylinder.t TABLE showing the Mean Pressure of Steam. Expansion. 5Multiplier. Expansion. Multiplier. Expansion. Mlultipllcm. 1.0 1.000 3;4.654 5.8 479 1.1.995 3.5.644 5.9.474 1.2.985 3.6.634 6..470 1.3.971 3.7.624 6.1.466 1.4.955 3.8.615 6.2.462 1.5.937 3.9.605 6.3.458 1.6.919 4..597 6.4.454 1.7.900 4.1.588 6.5.450 1.8.882 4.2.580 6.6.446 1.9.864 4.3.572 6 7.442 2..847 4.4.564 6.8.438 2.1.830 4.5.556 6.9.434 2.2.813 4.6.549 7..430 2.3.797 4.7.542 7.1.427 2.4.781 4.8.535 7.2.423 2.5.766 4.9.528 7.3.420 2.6.752 5..522 7.4.417 2.7.738 56.1.516 7.5.414 2.8.725 5.2.510 7. 6.411 2.9.712 5.3.504 7.7.408 3..700 5.4.499 7.8.405 3.1.688 5.5.494 7.9.402 3.2.676 5.6.489 8..399 3.3.665 5.7.484 EXAMPLE.-Suppose the steam to enter the cylinder at a pressure of 34.7 lbs. per square inch, and to be cut off at ~ the length of the stroke of the piston. The stroke being 10 feet, 10 feet - 120 inches +.5 for clearance = 120.5, stroke 4 = 30 inches +.5 " = 30.5. Then 120.5- -30.5 = 3.95, the relative expansion, which falls between 3.9 and 4. Referring to the table, the multiplier for 3.9 is.605, and the difference between that and the multiplier for 4 is.008. Hence, multiplying.008 by.5, and subtracting the product.004 from.605, the remainder,.601, is the multiplier for 3.95, Therefore,.601 x34.7 lbs. = 20.855 lbs. per square inch, the mean effective pressure of the piston required. * When great accuracy is required, the space between the cylinder and the steam valve must be added to the clearance. t.J pressure equal to that of the vacuum existing must be added to the pressure of the boiler, as indicated by the steam gauge. MARINE ENGINES. 281 MARINE ENGINES. NAVAL STEAMERS WITH SIDE WHEELS. "MISSOURI." Length between perpendiculars, 220 feet; beam, 40feet; depth of hold, 23.5 feet; with a displacement of 2800 tons, at a draught of water of 18.3 feet. Immersed Section, 550 square Feet. Cylinders. Two, each of 215 cubic feet in capacity. Condensers. 75 cubic feet in each. Air Pumps. 50 cubic feet in each. Force Pumps. 7' inches diameter by 46 inches stroke. Water Wheels. 28 feet in diameter by 11 feet in width, 21 arms in each; buck. ets (divided both in depth and length), 14 and 16 inches. Shafts ('Wrought iron). Diameters of journals, 17X18i, and 12X15 inches. Boilers. 6000 square feet of fire and flue surface; flues 40 feet in length, including steam chimney. Grates. 260 square feet. Steam Room. 1770 cubic feet. Pressure. Average, 10 lbs. per square inch, cut off at of the stroke of the piston; attainable, 20 lbs. Revolutions. From 12 to 183 per minute. Dip of Wheel. 5 feet at load-line Consumption of Fuel. 35 to 40 tons of bituminous coal per day. Weights. Engines, Boilers, Water Wheels, Water in boilers, Coal bunkers, Engineer's Stores, Tools, &c., &c., 490 tons of 2240 lbs.; viz.: Engines.... 211 tons. Water wheels... 47 tons. Boilers and smoke pipe 120 Engineers' stores, tools, &c. 7 Water in boilers. 82 " Coal bunkers... 23 " HULL. Of Live Oak and copper fastened. Launching weight, 1280 tons. Rig. Full barque. Armament. Two 10-inch chambered guns on pivots, and eight 8-inch chambered guns on carriages. NOTE.-The " Mississippi" and this vessel differ alone in the design of their elln gines; the "Missouri" having those of direct action, at an inclination of 260 30/,. and the former havina vertical side levers, with equal capacity of cylinders, the stroke of the piston being 7 feet. Their hulls should have had 20 feet additional length. "POWHATA/N." INCLINED ENGINES. Length between perpendiculars, 250 feet; beam, 45 feet; depth of hold, 26.5 feet; with a displacement of 3600 tons, at a draught of water of 18.5 feet. Immersed Section, 663 square Feet. Cylinders. Two, of 70 inches in diameter by 10 feet stroke of piston; capacity of piston space, 534.5 cubic feet. Air Pumps. 52.125 inches in diameter by 42 inches stroke of piston = 51.5 cubic feet in capacity. Condensers. 95 cubic feet in each. Feed Pumps. 8 inches in diameter = 1.2 cubic feet in each. Water Wheels. Diameter, 31 feet; breadth of buckets, 10 feet; depth of ditto, 26 inches. Arms, 23 in each wheel. Dip of wheels at load-line (18.5 feet), 5.5 feet. AA2 282 MARINE ENGINES. Water Wheel Shafts. Journals, 18.3 and 13 inches in diameter, and 20 and 15 inches in length. Boilers. Four of copper (double return ascending flues); length, 16 feet, breadth, 15.25 feet; height, 13 feet. Heating Surface. 8500 square feet. Grates. 354 square feet. Steam Room. 2300 cubic feet. Cross Area of lower Flutes. 75.7 square feet. Smoke Pipe. 63.6 square feet in area, and 65 feet in height above the grate level. Pressure of Steam. 10 lbs. per square inch, cut off at one half of the stroke of lhe piston, throttle one quarter open. Revolutions. 12.7 per minute. Fuel. Bituminous coal, with a natural draught. Consumption. At load line, moderate sea, and at above pressure, revolutions, &c., 3950 lbs. per hour = 42.3 tons per day. Speed. 10 knots per hour. Coal Bunkers. 800 tons capacity. Slip of wheels, 16.45 per cent. HULL. Launching draught, 10.62 feet; displacement, 1585 tons. Angles of en trance at 17 feet 5 inches and at 19 feet 5 inches are respectively 480 and 540 40'. Average Displacvment per Inch. From 18 to 19 feet draught, 22.09 tons. Rig. Full barque. Weights. Engines and frames, flooring, &c., complete...... 491,282 lbs. Boilers (copper and brass)......... 324,977" (iron, lead, &c.), in appurtenances....... 12,455 Coal bunkers, deck plates, &c............ 117,367 "' Water wheels................. 97,499 Extra pieces................. 5,135" Hoisting engine and boiler............. 4,569 Total.................1,208,304 lbs. Day's work (boilers, water wheels, and coal bunkers not included, as they were made by the pound), 56.222. Turnaing, bowring, and planing, 775.141 square inches. Armnament. Three 10-inch chambered guns on pivots, and six 8-inch chambered guns on carriages. "NIx" and " SALAMANDER." OSCILLATING ENGINES. IRON HULLS. Length onl deck, 186 feet; beano, 26 feet; depth of hold, 11 feet 6 inches; draught of water at load-line, 6 feet 9 inches. Immersed Section, 168 square Feet. Cylinders. 48 inches in diameter; stroke of piston, 4.5 feet. Boilers. Four horizontal tubular. Revolutions. 36 to 38 per minute. Speed. 134 knots per hour. Fuel. Consumption, 2450 lbs. of bituminous coal per hour. HULL. Frames, 3.SX3X% inches, and 2 feet from centres; thickness of plates, i and 4ths thick. Ceiled with teak 8 inches thick. Weights. Engines, boilers, and coal bunkers, 179,200 lbs. Armament. Four 68-pounder pivot guns, and two medium 32-pounders upon carriages. MARINE ENGINES. 283 NAVAL STEAMERS WITH SCREW PROPELLERS. (Ericsson). " PRINCETON." SEMI-CYLINDRICAL ENGINES. Length between per. pendiculars, 156.5 feet; beam, 30.5 feet; hold, 21.5 feet; tonnage, 673.5, with a displacement of 1046 tons. Immersed Section at load-line, 338 square Feet. Cylinders. Two, each of 54 cubic feet in capacity. Pressure. 25 lbs. per square Inch, cut off at A uf the stroke. Propeller. 120 square feet of surface. Pitch. 35 feet. Diameter. 14 feet. Revolutions. 36 per minute. Shafts (TYrought iron). Diameter of journals, 12X16 inches. Boilers. 2500 square feet of fire and flue surface. Flues 50 feet in length. Grates. 134 square feet. Steam Room. 1150 cubic feet. Fuel. 1 ton of anthracite coal per hour (blast). W4feights. Engines... 92 tons. i Water in boilers.... 34 tons. Boilers... 17" Coal bunkers. 6 HULL. Of White Oak and tree-nailed. Launching weight, 418 tons. Sail. Area in square feet, 11,762. Surface of sails in proportion to immersed section, M4.92 square feet to 1. 284 MARINE. ENGINES. MERCHANT STEAMERS, WITH SIDE WHEELS. NEW YORK AND LIVERPOOL, Collins's Line. "ARCTIC." SIDE-LEVER ENGINES. Length between perpendiculars, 280 feet; beam, 46 feet; depth of hold, 24 feet; depth of hold to spar deck, 32 feet; with a displacement of 3724 tons, at a draught of water of 19 feet. Immersed Section, 685 square Feet. Cylinders. Two, of 95 inches in diameter, with a stroke of piston of 10 feet; capacity of piston space, 984.48 cubic feet. Water Wheels. 34 feet 5 inches in diameter by 11.5 feet in depth; width of buckets, 21.5 inches. Arms, 36 in each wheel. Dip of wheels at load-line, 6.5 feet. Boilers. Four, of iron (vertical tubular, with double-storied furnaces); length, 20 feet 7 inches; breadth of two forward, 14 feet; of two after, 15 feet; height of each, 13 feet 8 inches. Heating Surface. 19.484 square feet. Grates. 587.8 square feet. Steam Room. 4350 cubic feet. Cross Area of Flues. 175 square feet. Smoke Pipe. 78.5 square feet in area, and 75 feet in height above the grate level. Water. 392,000 lbs. Pressure of Steam. 17 lbs. per square inch, cut off at one half of the stroke of the piston. Average Revolutions. 14.7 per minute. Fuel. Bituminous coal, with a natural draught, 7640 lbs. per hour. Average Speed.. 12.2 knots per hour for an entire passage. Slip of Wheels. Average of 11 days, 18 per cent. Weight of Hull. 1380 tons. Displacement per Inch. At 19 feet draught, 22 tons; at 17.6 feet, 20.3 tons. Rig. Brig. IARINE ENGINES. 285'"NORTH STAR." VERTICAL BEAM ENGINES. Lenlgth on deck, 269.5 feet; breadth of beam, 38 feet; depth of hold, 22 feet; depth of hold to spar deck, 29.5 feet; draught of water at load-line, 14 feet. Immersed Section, 496.5 square Feet. Cylinders. Two, of 60 inches in diameter, with a stroke of piston of 10 feet. Water Wheels. 33 feet in diameter by 8 feet in width; depth of buckets, I3S inches. Arms. 28 in each wheel. Dip of wheels at load-line, 7.75 feet. Boilers. Four, of iron (drop flued, with one tier of upper return flues); length, 23 feet 11 inches; breadth, 11 feet; height, 10.5 feet; height of steam chimney, 12 feet. Heating Surface. 4892 square feet. Gr,'aes. 266 square feet. Cross Area of Flues. 31.5 square feet. Smoke Pipes. Two, of 5 feet in diameter and 55 feet in height above the grate level. Pressure of Steam. 20 lbs. per square inch, cut off at one half the stroke of the piston. Revolutions. 16 per minute. Fuel. Anthracite coal, with a natural draught. Coal Bunkers. Capacity of 600 tons. Weights. Engines...... 510,000 lbs. I Boilers....... 176,000 lbs. Launching Draught. Engines, boilers, &c., all on board, 9 feet 6 inches. Rig. Brigantine. 286 MARINE ENGINES. IRON HULLS. "EUXINE.'" OSCILLATING ENGINES. PENS'R. & O. S. NAV'. CO. Length on deck, 222.7 feet; breadth of beam, 29.4 feet; depth of hold, 18.5 feet; length of engine room, 74 feet; length of quarter deck, 70.3 feet, depth, 3 feet; draught of water at load-line, 9.25 feet. Immersed Section, 252 square Feet. Tonnage of Hull. 1096 tons. Qu2arter deck, 68. Total, 1164. Contents ofs Engine Room, 435. Register, 729. 340 7 Cylinders.,74 inches in diameter; stroke of piston, 7 feet; least thickness, 1.5 inches. Openings, 5 by 23 inches. St7iffing boxes, 39 inches in depth. Cylinder Tr7'Znnions. 25.5 inches in diameter, and 9 inches in length. Thickhess of Metal, 2.5 inches. Packi?,.a, 6.5 inches in'depth when screwed down. Air Pumps. 42 inches in diameter. Water Wthecls. 28 feet in diameter by 8 feet 5 inches in width. Buckets, 27 inches in depth. Atnms,. three sets of 25 in each wheel. Revolutions. 19.5 per minute. Steam Pipe. 17 inches in diameter. Piston Rods. 9 inches in diameter. Shafts. Journals, 15X21 inches and 10.5X12 inches. Orank Pins. 9.25X 16.375 inches. Cranks. Hub, 25.75X16 inches. Eye, 18.X1 3.1 inches. Web, 9.5 and 9, X19, and 14.1- inches. Columns. Of Wr'ought Iron. 8 of 7 inches diameter. Tapered at upper end to 5.75 inches. Engines occupy 15 feet in length. Boilers. Four tubular: length, above, 10 feet 10.5 inches; below, 10 feet; breadth, 10.25 feet; height, 10 feet 10.5 inches. Tubes, 198 in each (792 in all); 3.25 inches internal diameter, and 6.5 feet in length. Furnaces, three in each boiler Smoke Pipes, two of 55 inches in diameter, 37 feet 7 inches in height. HIULL. Of Wrought Iron. Launching draught, 6.5 feet. Frames. 15 and 24 inches apart from centres. Plates. 15 strakes from keel to gunwale, to iths of an inch thick. MARINE - ENGINES. 287 CREW PROPELLERS. BRITISH AND N. A. R. MAIL STEAM-SHIP Co. " ANDES" and "ALPS." BEAM ENGINES. Length on deck, 236.5feet; breadth of beam, 34 feet l inch; depth of hold, 24 feet; draught of water at load-line, 16.75 feet. Immersed Section, 530 square Feet. Cylinders. Two, of 66 inches in diameter, with a stroke of piston of 4.5 feet. Boilers. Two tubular; length, 15.5 feet; breadth, 9.5 feet; height, 14 feet. Tubes, 832 of 3.25 inches in diameter by 6.5 feet in length. Furnaces, twelve, 6.75 feet in length.:Smoke Pipe. 5 feet 11 inches in diameter by 40 feet in height above boiler. Pressure of Steam. 10 lbs. per square inch. Revolutions. 25 per minute. Propeller. True screw, two blades. Pitch, 18 feet; diameter, 14 feet; geared, 1.5 revolutions to 1 of engine. Fuel. Bituminous coal. Cosnsumption. 2000 lbs. per hour. Coal Bunkers. Capacity of 400 tons. HULL. Frames, double angled iron, 5.25X5X3 inches; distance apart from centres, 15 inches. IKeel. 9X3 inches. Plates. 1 to iths of an inch in thickness, abuted at ends; clincher built at edges, and double riveted. Rivets. - iths of an inch in diameter, and 3.5 inches apart. Keelsons. Two, same dimensions as frames. Weights. Engines..... 336,000 lbs. I Boilers and water... 336,000 lbs, Rig. Barque. 288 RIVER ENGINES. RIVER ENGINES. With Side Wheels. " NIAGARA." CONDENSING. For 123 square Feet of Immersed Section. Length of vessel, 265 feet; beam, 28 feet 6 inches; depth of hold, 9 feet 3 inches; draught (loaded), 4 feet 9 inches. Cylinder. 216 cubic feet in capacity. Condenser. 88 cubic feet. Air Pump.:33.5 cubic feet. force Pumps. 51 inches diameter by 41 feet stroke. Pressure. 40 to 45 lbs. per square inch, cut off at i the stroke of the piston. Revolutions. 24 per minute. Water Wheels. 30 feet in diameter by 11 feet face..Arms. 24 in each flange. Bulckets. Two, of 15 inches deep. 1)ip (at load line). 30 inches. Shafts ( Wrought iron). Journal, 14 inches. Boilers. Two, of 27 feet in length by 10 feet front. Shlell. 8 feet six inches in diameter. Fire and Flue Surface. 3000 square feet. Grates. 108 square feet. Steam Room. 1200 cubic feet. Fuel. 3200 lbs. of anthracite coal per hour (maximum). Blowers. Two, of 9 feet in diameter. Fans. Ten, of 24 inches by 3 feet face. Blowing Engines. Two, of 10 inches diameter of cylinder by 12 inches stroke. Revolutions. 150 per minute. Weights. Engines... 186,000 lbs. Boilers 65,000" Wood, in engines and wheels 29,000" Water in boilers 76,000" Total.. 356,000 lbs. "SSOUTH AME RICA." CONDENSING. For 132 square Feet of Immersed Section. Length of vessel, 250 feet; beam, 27 feet; depth of hold, 9 feet; draught (loaded), 5 feet. Cylinder. 175 cubic feet in capacity. Sir Pump. 38.5 cubic feet. Condenser. 72 cubic feet. Force Pumps. 5 inches diameter by 4 feet stroke. Pressure. 35 to 40 lbs. per square inch, cut off at A the stroke of the piston. Revolutions. 23.5 to 24 per minute. Water Wheels. 29 feet in diameter by 11 feet face..Jrms. 24 in number. Buickets. Two divisions, and 30 inches deep. Dip (at load-line). 30 inches. Shafts ( Wrought iron). Diameter of journal, 13 inches. Boilers. Two, 27 feet in length by 9.5 in width. Fire and flue surface, 3000 square feet. Shell. 8 feet in diameter. Grates. 100 square feet. Steam Room. 1000 cubic feet. Consumption of Fuel. 3000lbs. of anthracite coal per hour (maximum). Blowers. Four, 4 feet diameter by 26 inches face. 4 arms. Fans, 13 incllie deep. Revolutions, 75 per minute. Blowing Engines. Cylinder, 8 inches in diameter by 12 inches stroke. Weight of Boilers. 62,000 lbs. The above engines and boilers were designed and constructed at the Phoanis Foundry, New-York. RIVER ENGINES. 289 CONDENSING. For 160 square Feet of Immersed Section. Length of vessel, 335 feet; beam, 35 feet; depth of hold, 11 feet 6 inches. Cylinder. 313 cubic feet in capacity. Air Pump.' 54 cubic feet. Pressure. 38 lbs. per square inch. Revolutions. 22 per minute. Water Wheels. 34 feet in diameter by 10 feet 8 inches face. Buckets. 36,nches deep. Boilers. Shell. 91 feet in diameter. Surface. 3660 square feet. Steam Room. 6570 cubic feet. Grates. 145 square feet. Blowing Engines. Two cylinders, of 14 inches diameter by 14 inches stroke. rwo blowers, 10 feet in diameter by 2 feet in width. Ten fans in each, of 26 inche n depth. Revolutions. 100 per minute. Fuel.: 3000 lbs. of anthracite coal per hour. "CLIFTON." VERTICAL BEAM ENGINES. CONDENSING. Length on deck, 180 feet; beam, 29 feet; depth of hold, 8.75 feet; draught of water at load-line, 3 feet. Immersed Section, 75 square Feet. Cylinder. 40 inches in diameter, with a stroke of piston of 8 feet. Air Pump. 23 inches in diameter, with a stroke of piston of 2 feet 7 inches. Condenser. 40 inches in diameter by 4 feet in height. Water Wheels. 26 feet in diameter by 7 feet in width; buckets, 20 inches in depth. Arms. 28 in number, 7X2.5 inches. Dip of Wheel. 22 inches. Boiler. One, drop-flued, 11 feet in width, 26 feet in length, and 8.75 feet in di ameter of shell, back of the furnaces; steam chimney, 9.5 feet high. Grates. 7 feet in length, having an area of 69 square feet. Frames. Moulded, 12 inches; sided, 4 inches, and 24 inches apart from centres. Keel. 4 inches deep. SHating Surface. 1217 square feet; upper flues, 5 of 16.5 inches in diameter. B B 290 RIVER ENGINES. IRON HULL. GLASGOW AND ROTHESAY. "OSPREY." STEEPLE ENGINE. Length on deck, 196.6 feet; beam, 18.5 feet; depth of hold, 8.8 feet; draught of water (light) 4 feet 7 inches. Immersed Section at light draught, 80.5 square Feet. Cylinder. 54 inches in diameter, with a stroke of piston of 4 feet 4 inches. Air Pump. 24 inches in diameter, with a stroke of piston of 4 feet 4 inches. Water Wheels. 18.5 feet in diameter by 6 feet 7 inches in width. Arms. 17 in number. Buckets. 15 inches deep. Boilers. Two tubular; length, 8 feet 3 inches; breadth, 11 feet 3 inches; depth, 7 feet 9 inches. Furnaces. Six, 3 feet in width; 6.5 feet in length; and 3 feet in depth. Tubes. 354 of 3 inches in diameter by 6 feet in length. Smoke Pipes. Two, of 3 feet 8 inches in diameter and 31 feet in height above grate level. Revolutions. 32 per minute. Pressure. 15 lbs. per square inch. Fuel. 1792 lbs. of bituminous coal per hour. Speed. 15 statute miles per hour. HULL. Frames. 2.5X2.5Xsths of an inch; and 27 inches apart from centre. Plates. to.5ths thick. Clincher built. Launlching Draught. 2 feet 7~ inches. RIVER ENGINES. 291 NON-CONDENSING.. "BUCKEYE STATE." HORIZONTAL ENGINES. Length of vessel, 260 feet; beam, 29feet 4 inches; depth of hold, 7 feet 6 inches; draught (light), 3 feet 6 inches; loaded, 4 feet. Immersed Section, 116 square Feet. Cylinders. Two, of 29.5 inches in diameter by 8 feet stroke of piston. Water Wheels. 31 feet 8 inches in diameter by 12 feet in width. Buckets.: (20) 2.5 feet in depth. Boilers. 5 of 42 inches in diameter and 30 feet in length, with two flues in each, 16 inches in diameter. Heating Surface. 2394 square feet. Grate bars. 4.1 feet in length;.08 inch thick; spaces between each,.08 inch. Area, 56 feet. Smoke Pipes. Two, of 66 inches in diameter, and 64 feet high. Water Wheel Shafts. 17 inches in diameter. Pressure. Average, 145 lbs. per square inch Cut off at 5 feet. Revolutions. 18 per minute. Fuel. 4281 lbs. of bituminous coal per hour. HULL. Keelson. 11X17inches; bilge do., 3X10.5inches; false do., 3X8 inches; bottom plank, 4 inches; deck beams, 3.5X6.25 inches; plankshear, 2.5X25 inches, and deck plank, 2 inches. Frames. Throats, 7inches; sides, 3.5 inches; distance friom centre to centre, 17 inches. Freight. 108 tonls at this draught of water. NOTES.-Areas of immersed section of hull at light draught (102.7 square feet) and of cross bucket surface, are as 1 to 1.71. Areas of grate and fire-flue surface as 1 to 18. Fuel consumed upon each square feet of grate, 50 lbs. per hour. Areas of smoke pipes, 47 feet; of flues, 16.7 feet; and of bridge wall, 27 feet. For 300 square Feet of Immersed Section. Vessel 260 feet in length, 38 feet beam, and 8 feet draught when loaded. Cylinders. Two, each 30 inches in diameter by 10 feet stroke of piston (98.'3 cubic feet). Force Pumps. 63 inches in diameter by 25 inches stroke. Water Wheels. 33 feet in diameter by 15 feet in width, 19 arms in each; buckets, 36 inches deep. Shafts (Cast iron). Diameter of journals, 16 and 14 inches. Connecting Rod. 35 feet in length. Piston Rod. Diameter, 6 inches..Steam Valves. 50 square inches each. Exhaust Valves. 63 square inches Boilers. 5 of 42 inches in diameter by 34 feet in length, with 2 return flues ilt each, 16 inches in diameter, having 2278 square feet of heating surface. Grates. R4 square feet. Boiler Plates. (Shells), quarter of an inch in thickness. Flues. Full quarter if an inch. Pressure. 75 to 100 lbs. per square inch, cut off at i the stroke of the piston. Revolutions. 16 to 21 per minute. Dip of wheel. 5 feet when loaded. Consumption of Fuel. 2.3 cords yellow pine per hour. Weights. Engines, 160 tons; Boilers, 9000 lbs. NoTE.-32.75 square feet of fire and flue surface is the proportion for each cubic hkot in the cylinders at the above given revolutions. 292 RIVER ENGINES. STERN WHEEL. HORIZONTAL ENGINES. Length on deck, 110 feet; beam, 14 feet (deck projecting over, 4 feet); depth of hold, 3.5 feet, draught of water at load-line, 12 inches. Immersed Section, 10.25 square Feet. Cylinders. Two, of 10 inches in diameter, with a stroke of piston of 3 feet. Water Wheels. 13 feet in diameter by 8.5 feet in width. Buckets. 13 in num her, and 8 inches in depth. Revolutions. 33 per minute. Boiler. One tubular, 100 2-inch tubes. Fuel. 4480 lbs. in 24 hours. HULL. Plates. Keel, No. 3; bilges, No. 4; bottom, No. 5; sides, Nos. 6 and 7 Frames. 2.5XI inch, and 20 inches apart from centres. IRON VESSELS. 293 STEAM VESSELS (Marine). "GREAT BRITAIN." Displacement at 16 Feet draught, 3000 Tons. Length of keel, 289 feet; beam, 51 feet; depth of hold, 32 feet 6 inches Plates. 7, 11,, and 3a thick. Frames. 18 and 24 inches apart. 5 thick, by 3.5X6 inches; by 2.5X6 inches. and by 3X4 inches, j.. WEIGHT OF HULL (in iron). 1040 tons.'MICHIGAN." Displacement at 8 Feet 8 Inches draught, 658 Tons. Length between perpendiculars, 162 feet 6 inches; beam, 27 feet; depth of hold, 12 feet. Plates. I,' X, 4' and 3 thick. 8, 2, 8,,-6 4- -IT Frames. 2 feet apart. 1 and 3 thick, by 4X4.5 inches, j; and l and X, by 4X2.25 inches, L. Deck Beams. 3, by 4.5X7 inches,.. WEIGHT OF HULL, including bulwarks and berth deck, 507,387 lbs. "SPENCER." Displacement at 9 Feet 3 Inches, 440 Tons. Length between perpendiculars, 143 feet; beam (average), 20 feet 3 inches; depth of hold, 11 feet 6 inches. Plates. B, -i,' and X of an inch. Frames. 20 inches apart. 5 thick, by 2.5X4.5 inches, L. Deck Beams. X, by 2.87X5.5 inches, L. WEIGHTS. Plating...... 125,922 lbs. Knees....... 13,845 lbs. Bulk-heads..... 11,663" Rivets. 18,005 " Kelson....... 3,093 " Stanchions..... 2,997 " Deck beams.. 34,463 " Sundries...... 20,918'" Frames.. 44,661 " Total..275,567 lbs. FOYLE." Length on deck, 196 feet; beam, 25 feet; depth of hold, 16 feet. Tonnage. 761 tons. Draft of Water. 8.5 feet. Plates. 13 strakes of plates from keel to gunwale, diminishing upward from i to ~ an inch in thickness. Frames. 18 inches anart amidships, and 22 inches forward and aft. l-'7X2.}X4 inches L. 294 IRON VESSELS. SHIP (Clipper). (English.) T*'I'lYPIOON." Length of keel and fore rake, 198.5feet; breadth of beam. 32 feet; depth of hold, 20 feet; length of poop, 76 feet. B',;rden. 2000 tons by measurement, or 1400 tolls by weight. lTo7,eage. o. M., 976. N. AT., 1100. FIralnes. 5X3X-ths of an inch. and 15inchesapartfrom centres. Plates. 13ths at keel, and 5ths at gunwale. S.em. 10X3 inches, tapering to 7X3. Stern Post. 1OX-4 inches. Keel. 9X3 lathes. SCHOONER. (English.) "SaAilMROCK." Length on deck, 82.5 feet; breadth ona deck, 20 feet; depth of hold, 11.1 feet.'onnage. English, JVNew Register, 133.~ —L tons. Frames. 18 inches apart. Plates. 10 strakes fiom keel to gunwale, of 5ths to iths thick. Draft of water. Launching, forward, 4 feet 9 inches; aft, 5 feet 8j inches; loaded, 10 feet forward and 10 feet 6 inches aft. Clincher built and schooner rigged. CANAL BOAT (very full built). Length between perpendiculars, 80 feet; beam, 14 feet; depth of boldi 7 feet. Plates. Nos. 3 and 4, wire gauge. Fraces. 18 inches apart, and 38 thick, by 3 inches wide, I Deck and beams of wood. WEICGHT OF IIUT.I, 42,300 lbs. STEAM-VESSELS, ENGINES, ETC. 295 IISCELLANEOUS NOTES. 296 SUGAR-MILLS SUGAR MILL. For Expressing 20,000 lbs. of Cane Ju;ce per Day. NON-CONDENSING ENGINE. Cylinder 15 inches in diameter by 4 feet stroke. Pressure. 50 lbs. per square inch, cut off at i the stroke of the piston. Revolutions. 36 per minute. Boiler. 1 of 62 inches in diameter by 30 feet in length, with 2 18 inch return flues. Grates. 36 square feet. Rolls. Two set of 3 each, of 24 inches in diameter by 5 feet in length; geared 2k to 36 of engine, giving a speed of pe'riphery of 15b feet per minute. Fly Wheel. 18 feet in diameter; weight, 5 tons. This arrangement of a second set of rolls is a late improvement; its object, that of distributing the cane over an increased surface of rolls, reducing their speed, and affording more time for the juice to run off; an increase of 20 per cent. is effected by it. For a Crop of 3000 boxes of Sugar of 500 lbs. each. A non-condensing engine with a cylinder of 11 inches in diameter by 4 feet stroke, making 48 revolutions, with a pressure of steam 60 lbs. per square inch, driving 1 set of rolls, 24 inches by 4 feet, at a speed of periphery of 36 feet per minute. Boiler. 52 inches by 24 feet, with 2 16 inch return flues. Grate Surface. 25 square feet. Fly Wheel. 16 feet diameter; weight, 4 tons SAW-MILL. 297 SAW-MILL. NON-CONDENSING ENGINE. Two Vertical Saws of 34 inches Stroke. Lathes,,4c. Cylinder. 10 inches in diameter by 4 feet stroke. Pressure. 90 to 100 lbs. per square inch, full stroke. Revolutions. 35 per minute. Boilers. 3, plain cylindrical, 30 inches in diameter by 20 feet in length. NoTE. —This engine has cut, of yellow pine, 30 feet by 18 inches in one minute. 298 COTTON FACTORY. COTTON FACTORY. CONDENSING ENGINE. Driving 13,000 Spindles (Mules and Tihrostles), with 256 Looms for i- cloth No. 30. CyliTnders. Two, 22 inches in diameter by 3 feet stroke of piston. Pressure. 15 to 45 lbs. per square inch, cut off at one third of the stroke. Revolutions. 50 per minute. tONDLuNslNG ENGINE (British). Driving 22,060 Hand-mule Spindles, with preparation, 260 Looms, and common Sizing. Engine. Cylinder, 37~ inches diameter by 7 feet stroke; indicated presrsure (average), 16.73 lbs. per square inchi; revolutions, 17 per minute. Friction of engine and shafting, indicated 4.75 lbs. per square inch of piston. Total power, = 1. Available, deducting friction, -.717. Estimated power of en gine, 134 horses. NoTEs.-Each indicated horse power will drive 305 hand-mule spindles, with preparation, or 230 self-acting " or 104 throstle " " or 101 looms, with common sizing. Including preparation: 1 throstle spindle - 3 hand-mule, or 24 self-acting spindles. 1 self-acting spindle = 14 hand-mule spindles. Exclusive of preparation, taking only the spindle: 1 throstle spindle = 34 hand-mule, or 25 self-acting spindles. 1 self-acting spindle = -1 hand-mule spindles. iThe tlJrostles are the common, spinning 34 twist for power-loom weaving: tie spintdles revolve 4000 per minute. The self-acting mules are, one half spinning 36's weft, spindles revolving 4800; the other half spinning 36's twist, spindles revolving 5200. The hand-mules spinning about equal quantities of 36's weft and twist. Weft spindles 4700, and twist spindles 5000 revolutions per minute. Average breadth of looms 37 in. (weaving 37 in. cloth), making 123 picks per minate. All common calicoes about 60 reed, Stockport count, and 68 picks to the inch. No power consumed by the sizing. When the yarn is dressed instead of sized, one horse power cannot drive so many looms, as the dressing machine will absorb from - to 4 of the power. COTTON-PRESS. 299 COTTON-PRESS. NON-CONDENSING ENGINE. For 1000 Bales in 12 Hozers. Cylinder. 14 inches in diameter by 4 feet stroke. Pressure. 40 lbs. per square inch, at fill stroke. Revolutions. 60 per minute. Boilers. 3, plain cylindrical, without flues, 30 inches in diameter by 26 feet iu length Grates. 32 square feet. Presses. 4, geared 6 to 1, with 2 screws each of 7-~ inches in diameter by 1t in 0itClh. Shaft (Wrought iron). Journal, 8j inches. Fly Wheel. 16 feet in diameter; weight, 4 tons, 30)0 BLAST ENGINES. BLOWING OR BLAST ENGINES. Dimensions of a Furnace, Engines, 4c. At Lonakoning (Md.). Furnace. Diameter at the boshes 14 feet, which fall in, 6.33 inches in every fobot rise. Engine (.Non-condensing). Diameterof cylinder 18 inches, length ofstroke 8 feet. Revolutions. 12 per minute, with a pressure of 50 lbs. per square inch. Boilers. Five: each 24 feet in length, and 36 inches in diameter. Blast Cylinders. 5 feet diameter and 8 feet stroke. At a pressure of from 2 to 2~ lbs. per square inch, the quantity of blast is 3770 cubic feet per minute, requiring a power of about 50 horses to supply it. At Mount Savage (Md.). For Blowing four Furnaces, 14feet in diameter, each making 100 tons of Pig Iron per week. Engine (Condensing). Diameter of cylinder 56 inches, length of stroke 10 feet. Revolutions. 15 per minute. Pressure. 60 lbs. per square inch, cut off at 4 of the stroke. Boilers. 6 of 60 inches in diameter, and 24 feet in length, with 1 22 inch flue in each, double returned. Grates. 198 square feet. Blast Cylinder. 126 inches in diameter by 10 feet stroke. Revolutions. 15 per minute. Pressure of Blast. 4 to 5 lbs. per square inch. Area of Pipes. 2300 square inches, or 1 that of the cylinder. I'or Blowing two Furnaces and two Fineries, making 240 tons of Forge Pig per week. Engine (JVNon-condensing). Diameter of cylinder 20 inches, length of stroke 8 feet. Revolutions. 28 per minute. Pressure. 50 to 60 lbs. per square inch (full stroke). Boilers. 6 of 36 inches in diameter, and 28* feet in length (without flues). Grates. 100 square feet. Blast Cylinders. 2 of 62 inches in diameter by 8 feet stroke. Revolutions. 22 per minute. Pressure of Blast. 24 lbs. per square inch..lrea of Pipes. 3 feet, or -4 that of the cylinders. One Blast furnace has 2 3 inch and 1 34 inch tuyeres; the other has 3 of 3 inches. One Finery has 6 tuyeres of 14 inches, and the other 4 of 1k inches. The ore yields from 40 to 45 per. cent of iron. The temperature of the blast is * 40 feet would have afforded much economy of fuel. STEAM DREDGING MACHINE. 30 STEAM DREDGING MVIACHINE. NON-CONDENSING ENGINE. For Dredging 30 feet from Water-line. Six full Buckets per Minute. Clylinder. 12 inches in diameter by 5 feet stroke of piston. Pressure. 60 to 70 lbs. per square inch, full stroke. Revolutions. 20 per minte, geared 30 of engine to 1 of buckets. Boilers. 2 of 20 feet in length by 36 inches in diameter, with 1, 15 inch return flue lat each. Ways. 55 feet in length by 6 feet in width. Buckets. 10 of 28 inches ill width by 58 in length, and 14 in depth. At a depth of 18 feet, 10 buckets full of mud are discharged per minute, the engineo making 30 revolutions. NOTE.-This engine is geared too slow, being but 2k to 1. fHulls. 2 of 50 feet in length, 12 feet in width, and 9 in depth each; connectid on deck, space between for the ways, 7A feet. C, c t302 PILE-DRIIVING. PILE-DRIVING. NON-CONDENSING ENGINE. Driving two Pics. q/iiders. Two of 6 inches in diameter by 18 inches stroke. Pressure. 60 poundls per square inch, full stroke. Revolutions.; From 60 to 80 per minute Boiler. Tubular. Shell 3A- feet in diameter by 6 feet in length. Furnmce end, 3 feet 9 inches in width, 3A feet in length, and 6 feet in height. Ress. 1000 lbs. each, lifted 5 times in a minute. F11raste. 8j feet In width by 26 feet in length. Leaders. 3 feet in width by 24 feet in height. HYDROSTATICC PRESS, 3 o HYDROSTATIC PRESS. NOM-CONDENSINGC ENGINE. 8:0 Ba:les of Co wtoot per Hour Cyinder. 10 inches illn diameter by 3 feet stroke. Presswure. 50 lbs. per square inch, fill stroke. Revolutions. 45 to 60 per mulute. Presses. Two, with 12 inch rams, having 4. feet stroka. Pumps, Two of 2 inches (dlinamlcr lby 6 inchbs stroker L OUR MILLS. FLOUR MILLS. At Richmond (Va.). OVERSHOT WATER WHEELS. For 30 Barrels of Flour per.Hour. Water Wheels. Five, of 18 feet in diameter by 14 feet 6 inches face. Buckets. 15 inches in depth. Head of Water. 2.5 feet; opening over each wheel, 2.5 inches by 14 feet. For further and very exact notes of a Flour 3Mill, Blast Engine, and Gotion MJill, refer to page 179. TUBULAR BOILERS t0' MARINE BOILERS. T.AB,, oif J)trensiiO s of' several Tubular Marine Boilers in use,n England. Y \ 34 Kit V _ | | m 01 "" } s w to su pply of _ _ _ _ _ _ _ _ _ _ [ i s tet:_'=E 2' __ ~_ 0 c,, _. I.s. Feet. Cub. ft. Sq. ft. 8q. ft. Sq ins. 9q. inq. Feet. Royt l George 61 5. 101.5 21. 144 6.76 3067 2818 Fair. IBragansg l. 62 5.5 115.3 27.4 98 3.7 5 4158 402)0 Almnda -e luidiee 5. 8 58 5.5 100.9 28.5 192 5 6i2 4914 4 3669 Too much. Infeinal 65 5. 115.2 8.a 165 7.85 30 2[ 3347 Scarce. T l'agus 6 2 5.75 120.5 15.4 112 6.6 2461 2637 Short. Royal Consorta 65 5.5 126.7 P'A.4 179 5.75 4479 3956 Abundane,a Tlhe Queen.,59 4.S 1760 1804 n uvincible.n. 40 4 16, 36.4 12. 5 73 11. 948 1040 Abundancte lPhnlix, IR. N. i 112 4.42 36410 i 3380 $Scarce. Cc2 d( 1(i ) NMOTION OF BODIES IN FLUIDS. MOTION OF BODIES IN FLUIDS. s ABLE of the Weights required to give different Velocities to several different Figures.'TuE diameter of all the figures but the small hemisphere is 6.375 inches, and the altitude of the cone 6.625 inches. The small hemisphere is 4.75 inches. The angle of the side of the cone and its axis is, consequently, 25~ 42' nearly. Velocity Cone. Whole C d Hemisphere. Small hemoff secot un- -o. isphere. second. Vertex. Base. globe. Flat. Round. feet. oz. oz. O. OZ oz. oz. o. OZ. 3.028.064.027.050 t.051.020.028 4.048.109.047.090.096.039.048 5.071.162.068.143.148.063.072 6.098.225.094.205.211.092.103 7.129.298.125.278.284.123 1.141 8.168.382.162.360.368.160 r.184 9.211.478.205.456.464.199 I.233 10.260.587.255.565.573.2421.287 12.376.850.370.826.836.347.418 15.589 1.346.581 1.327 1.336.552.661 16.675 1.546.663 1.526 1.538.634.754 20 1.069 2.540 1.057 2.528 2.542 1.033 1.196 pomber. 126 291 124 285 288 119 140 iiumber From this table several practical inferences may be drawn. 1. That the resistance is nearly as the surface, the resistance in creasing but a very little above that proportion in the greater surfaces. 2. The resistance to the same surface is nearly as the square of the velocity, but gradually increasing more and more above that proportion as the velocity increases. 3. When the hinder parts of bodies are of different forms, the resistances are different, though the fore parts be alike. 4. The resistance on the base of the hemisphere is to that on the convex side nearly as 2.4 to 1, instead of 2 to 1, as the theory assigns the proportion. 5. The resistance on the base* of the cone is to that on the vertex nearly as 2.3 to 1. And in the same ratio is radius to the sine of the angle of the inclination of the side of the cone to its path or axis. So that, in this instance, the resistance is directly as the sine of the * This is a complete refutation of the popular assertion, that a taoper spar will low in water easiest when the base is foremost. MOTION OF BODIES IN FLUIDS. 307 angle of incidence, the transverse section being the same, instead of the square of the sine. 6. Hence we can find the altitude of a column of air, the pressure of which shall be equal to the resistance of a body moving through it with any velocity. Thus, let a = the area of the section of the body, similar to any of those in the table, perpendicular to the direction of motion, R = the resistance to the velocity, in the- table, and z = the altitude sought, of a column of air whose base is a and its pressure R. Then a z = the contents of the columns in feet, and 1.2 a z, or 6 a x its weight in ounces. Therefore, ~ a z = R, and x -z- X- is the altitude sought in feet, namely, A of the quotient of the resistance of any body divided by its transverse section, which is a constant quantity for all similar bodies, however different in magnitude, since the resistance R is as the section a, as by article 1. When a= — of a foot, as in all the figures in the foregoing table except the small hemisphere, then x - X R, becomes x = 5R, where R is the resistance in the table, to the similar body. If, for example, we take the convex side of the large hemisphere, whose resist-;ance is.634, or at a velocity of 16 feet per second, then R =.634, and x = -R= 2.3775 feet, is the altitude of the column of air whose pressure is equal to the resistance on a spherical surface, with a velocity of 16 feet. And to compare the above altitude with that-which is due to the given velocity, It will be 322; 162:: 16: 4, the altitude due to the velocity 16, which is near double the altitude that is equal the pressure. And as the altitude is proportional to the square of the velocity; therefore, in small velocities the resistance to any spherical surface is equal to the pressure of a column of air on its great circle, whose al19 titude is 3 2' or.594 of the altitude due to its velocity. But if the cylinder be taken, where resistance R = 1.526, then xz= 1 5R= 5.72, which exceeds the height 4, due to the velocity, in the ratio of 23 to 16 nearly. And the difference would be still greater if the body were larger, and also if the velocity were more. If any body move through a fluid at rest, or the fluid move against the body at rest, the force or resistance of the fluid against the body will be as the square of the velocity and the density of the fluid; that is, R= - dv2. For the force or resistance is as the quantity of matter or particles struck, and the velocity with which they are struck. But the quantity or number of particles struck in any time are as the velocity and the density of the fluid. Therefore, the resistance or force of the fluid is as the density and square of the velocity. The resistance to any plane is also more or less, as the plane is greater or less, and therefore the resistance on any plane is as the area of the plane a, the density of the medium, and the square of the velocity; that is, R = adsv. If the motion be not perpendicular, but oblique to the plane, or to the face of the body, then the resistance in the direction:of the motion will be diminished in the 308 MBOTION OF BODIES IN FLUIDS. triplicate ratio of radius to the sine of the angle of inclination of the plane to th, direction of the motion, or as the cube of radius to the cube of the sine of that an gle. So thatit R advzs3, 1 -- radius, and s = sine of the angle of inclination. The real resistance to a plane, from a fluid acting in a direction perpendicular to its face, is equal to the weight of a column of the fluid, whose base is the plane and altitudtle equal to that which is due to the velocity of the motion, or througi which a heavy body must fall to acquire that velocity. Thle resistance to a plane running through a fluid is the same as the tforce of thl rluid in motion with the saine velocity on the plane at rest. But the force of the fifid in motion is equal to the weight or pressure which generates that motion:ind this is equal to the weight or pressure of a column of the fluid, the base li which is the area of the plane, and its altitude that which is due to tile velocity. I. If a be the area of a plane, v its velocity, n the density or specific gravity ol V2 the fluid, and i g 16.0833 feet; then, the altitude due to the velocity v being o,, V2?zV2t'leretbre aXnX- —, wvill be the whole resistance or force It. gI. If the direction of motion be not perpendicular to the face of the plane, bult anvv"s3 oiblique to it, itl an angle2e; then R 3. If W represent the weight of the body, a being resisted by the absolute force I): then the retarding force f, or -_, will be as EXAMPLE.-If a plane I foot square be moved througil water at the rate of 32.16 32.162 feet per secondl, then.- 16.08, the space a body would require to fall to ac — q'uire a velocity of 32.16 feet per second; therefore, 1X62.5 (weight of a cubic foot 32.162 of water) X 6-) - = 1005 lbs., the whole resistance of the plane. The resistance to a sphere moving through a fluid is but half the resistance te its great circle, or to the end of a cylinder of the same diameter, moving with an equal velocity. Rt - --- being the half of that of a cylinder of the same diameter, R repre senting radius. ILLUSTRATION.-A 9 lb. iron ball, the diameter being 4 inches, when projected at a velocity of 1600 feet per serond, will meet a resistance which is equal to a sweight of 132.66 lbs. over the pressure of the atmosphere. The resistance that a body sustains in moving through afluid is zn proportion tst the square of the velocity. And it is the same, whether the plane moves against the fluid or the fluid against tie plane. The following Table shows the results of e.perirnents with a plane one foot square, at an immersion of 3feet below the surface, and at different velocities per second. velocity. Resistance. Velocity. Resistance. e Resistance. 5 feet 29.5 lbs 8 feet 71.7 lbs. 11 feet 136.3 ibs. 6 " 40. " 9" 90.6" 12 " 162.1 " 7 54.6" 10 " 112. " 13L" 213. " ORTHOGRAPHY OF TECHNICAL TERMS. 309 ORTHOGRAPHY OF TECHNICAL TERMS. THE orthography and use of the following terms are so variedin general practice, that it is thought proper to treat of them, with a view to establishing an uniformity of expression. Abut. To meet, to adjoin to at the end, to border upon. Alignment. Setting to a line, a row. Arabesque. Applied to painted and sculptured ornaments of imaginary foliage, in which there are no figures of animals. Synonymnous with MIoresque. Ashlar. Rough stones from the quarry. When faced and squared, they are termed smooth. Bagass.e. Sugar-cane in its crushed state, as delivered from the rollers. Baluster. A small column or pilaster; a collection of them, joinsd by a rail, forms a balustrade. Bevel. A term for a plane having any other angle than 450 or 90oo Boomkin. A short spar projecting from the bow or quarter of a vessel, to extend a tack of the fore or main sail to windward. Buhr-stone. Mill-stone which is nearly pure silex, full of pores and cavities. Camboose. The cooking-room of a vessel, usually confined to merchant vessels; in vessels of war it is termed Galley. Cag. A small cask, differing from a barrel only in its size. Calk. To stop seams and pay them with pitch, &c. To point an iron shoe, so as to prevent slipping. Cam. An irregular curved instrument, having its axis eccentric to the shaft upon which it is fixed. Capstan. A vertical windlass. Caravel. A small vessel (of 25 or 30 tons burthen) used on the coast of France in herring fisheries. Chamfer. A slope, groove, or small gutter cut in wood, metals, or stones. To chamfer is to slope, to channel, or to groove. Chimney. The flue of a fire-place or furnace, constructed of masonry. Chinse. To chinse is to calk slightly with a knife or chisel. Clincher built. A term applied to the construction of vessels, when the lower edge of the bottom planks overlays the next unJcr it. 10 t) IRTHTOGRAPIIY OF TECHNICAL TERMIS. Coamings. Raised borders around the edges of hatches. Cog. In Mechanics, A short piece of wood or other material lei into the faces of two b)odies, to impart motion one to the other. A tecm applied to a tooth in a wheel when it is made of a different material than that of the wheel. Colter. The fore iron of a plow, that cuts the earth or sod. Compass. In Geometry, An instrument for describing circles, measuring figures, &c. Corridor. A gallery or passage in or around a building, connected with various departments, sometimes running within a quadrangle: it may be open or enclosed. In fortifications, a covert..way. Damasquineriec. Inlaying in metal. Davit. A short boom fitted to hoist an anchor or boat. Dilatation. Expanding; opposed to contraction. It differs front extension; thus, a line may be extended, but a body is dilated. Doctel. To fasten two boards or pieces together, by pins inserted'n their edges. This is very similar to cogging, but is used in a diminutive sense. An illustration of it is had in the manner a cooper secures two or more pieces in tile head of a cask. Evaporation. The conversion of a fluid into vapor, specifically lighter than the atrnospheric air. Felloe. The pieces of wood forming the circumference of a carriage wheel, into which the spokes are inserted. Flange. A projection from an end or from the body of an instrument, or any part composing it, for the purposes of receiving, confining, or of securing it to a support or to a second piece. Frap. To bind togetiher with a rope, as to frap a fall or vessel. Furring. In Carpentry, Strips of board or pieces of joist to sup.. ply deficiencies of timber, or to prepare a fair surface for lathing. Gallery. In Archiltcttree, A covered part of a building used for walking. Graving. Burning off grass, shells, dc., from a ship's bottom Synonymous with Breani'ng. Ciornomet. A wreath or ring of rope. Jib. The projecting beam of a crane from which the pulleys and weight are suspended. A sail in a vessel. Keelsan. The timber within a vessel laid upon the middle of the floor timbers, and exactly over the keel. Lacquer. A spirituous solution of lac. Lapsided. A term expressive of the condition of a vessel or any Itbdy when it will not float or set upright. ORtTHOGRAPHY OF TECHNICAL TERiMS 3 ] Mitered. In Me.han-ics, Cut to an angle of 450, or two pieces joined so as to make a right angle. JMold. In Mechanics, A matrix in which a casting is formed. A number of pieces of vellum or like substance, between which gold and silver are laid for the purpose of being beaten. Thin pieces of materials cut to curves or any required figure. In Naval Architecture, pieces of thin board cut to the lines of a vessel's timber, &c., &c. Fine earth, such as constitutes soil. A substance which forms on bodies in warm and confined damp air. This orthography is by analogy, as gold. sold, old, bold, cold, fold, &c. Molding. In Architecture, A projection beyond a wall, fronl a column, wainscot, &c., &c. Mortise. A hole cut in any material to receive the end or te3oln of another piece. Ogee. A molding with a concave and convex outline, like to an S. Peek. The upper or pointed corner of a sail extended by a gaff;'r a yard set obliquely to a mast. To peek a yard is to point it perpendicularly to a mast. Pipe. In Mechanics, A metallic tube. The flue of a fire-place or furnace, constructed of metal; usually of a cylindrical form. Piragua. A small vessel with two masts and boom sails. Pluimber block. A' bearing to receive and support the journal of a shaft. Porch. An arched vestibule at the entrance of a building. A ves. tibule supported by columns. A portico. Portico. A gallery near to the ground, the sides being open. A piazza encompassed with arches supported by columns, where persons may walk: the roof may be flat or vaulted. Rabbet. In Mechanics, To pare down an edge of a board or a plate for the purpose of receiving another board or plate by lapping.'To lap and unite edges of boards and plates. The groove in the side of a keel for receiving the garboard strake of plank. Rarefaction. The act or process of distending bodies, by separating their parts and rendering them more rare or porous. It' is opposed to condensation. Recta. In Mechanics, To bevel out a hole. Reeming. The opening of the seams between the planks of a vessel, for the purpose of calking them. Rotary. Turning on. an axis, as a wheel. Shammy. Leather prepared: from the skin of a chamois goat. Sheer. In Naval Architecture, The curve or bend of a ship's deck or sides. To sheer, to slip or move aside. 3 1P2 ORTHOGRAPHY OF TECHNICAL TERMS. Shoar. An oblique brac":, the upper end resting against the sub stance to be supported. Signalled. Communicated by signals. Slue. To slue is to turn a substance on an axis within its figure. Sponson. An addition to the outer side of the hull of a steam vessel, commencing near the light water-line and running up to the guard. Used for the purpose of shielding the guard from the shock of a sea. Sponson sided. The hull of a vessel may be so termed when her frames have the outline of a sponson, and the space afforded by the curvature is included in the hold. Stack. In Masonry, A number of chimneys or pipes standing together. The chimney of a blast furnace is so called. Strut. An oblique brace to support a rafter. Swage. To bear or force down. An instrument having a groove on its under side, for the purpose of giving shape to any piece subjected to it when receiving a blow from a hammer. Template. A mould cut to an exact section of any piece or structure. Templet. In Masonry, A wooden bearing to receive the end of a girder. Tompion. The stopper of a piece of ordnance. The iron bottom to which grape-shot are secured. Treenails. Wooden pins employed to secure the planking of a ve,.sel to the frames. Supplementary. Burden. A load. The quantity that a ship will carry. The vease repeated in a song-the chorus. Hence burdensome. Carvel built. A term applied to the manner of construction of small boats, to signify that the edges of their bott6m planks are laid to each other like to the manner of planking vessels. Opposed to the term clincher. Coccoon. The case which certain insects make for a covering during the period of their metamorphosis to the pupa state. Contrariwise. Conversely, opposite. Crossways is a corruption of this word. Draught. A representation by delineation. The depth which a vessel or any floating body sinks into water. The act of drawing. A detachment of men from the main body, &c., &c., &c. Draft is a corruption of this word. ORTHOGRAPHY OF TECHNICAL TEiRM'. 3113 Edgewise. An edge being put into a particular direction. Hence, endwise and side-wise have similar significations with reference to an end and a side. Gearing. In Mechanics, a seriev of teethed or cogged wheels for transmitting motion. To gear is to prepare. To connect, as by an articulation Newel. Anl upright piece of timber, around which winding stairs ttlurn. Planle. i n Geometry and Milech.airics, at surface, which coincides witl a right line. Resin. The residuum of the distillation of turpentine. Sidewise. See Edgewvise. Slantwise. Oblique; not perpendicular. Strake. A breadth of plank. Syphered. Oveilapping the chamfered edge of one plankl upon the chamfered edge of another, in such a manner t.hat the joint shall be a plane surface. T ND t THE UNgD