THE MECHANIC'S, MACH4INIST'S, AND EN(INEER'S PRACTICAL BOO1 OF REFERENCE: C~NTAINING TABLES AND FORMULLA FOR USE IN SUPERFICIAL AND SOLID MENSURATION; STRENGTH AND WEIGHT OF MATERIALS; MECHANICS; MACIIINERY; HYDRAULICS, HYDRODYNAMICS; MARINE ENGINES, CHEMISTRY; AND MISCELLANEOUS RECIPES. ADAPTED TO AND FOR THE USE OF ALL CLASSES OF PRACTICAL MECHANICS. TOGETHER WITH THE ENGINEER'S FIELD BOOK: CONTAINING FORMULzE FOR TIIE VARIOUS METHODS OF RUNNING AND CIHANGi'NG LINES, LOCATING SIDE' TRACKS AND SWITCHES, &C., &C. TABLES OF RADII ANTD THEIR LOGARITHMS, NATURAL AND LOGARITHMIC VERSED SINES AND EXTERNAL SECANTS, NATURAL SINES AND TANGENTS TO EVERY -DEGREE AND MINUTE OF THE QUADRANT, AND LOGARITHMS OF NATURAL NUMBERS FROM 1 TO 10,000. BY CHARLES HASLETT, Civil Engineer. EDITED BY CHARLES W. HACKLEY, Professor of Mathematics in Columbia College, N. Y. NEW YORK: STRINGER & TOWNSEND, 222 BR-OAD'WAY. 1856. Enterediaccording to the Act of Congress, in the year 1855, by STRINGER & TOWNSEN D, In the Clerk's Office of the District Court of the United States for the Southern District of New York. It. CRAIGHEAD, Stereotyper and Electrotyper,.3: FVsey,'tret, y. i' P R E FAC E. No more usefiul little works have ever been presented to the public than lthe various pocket companions of: i character analogous to that here offered. These have been a good deal, though not yet too much, multiplied of late; and where the:formulas, rules, and: tables which they contain have been skilfully framed under the guidance of scientific men, they have afforded to the Practical Engineer, Architect, and Mechanic, the most welcome aid in the constructions and computations which make part of their daily occupation, and'which, without the ever-at-hand suggestions and directions of these unpretending little servants, might consume hours and: days in the turning over of large volumes, or in painful investigations based on general principles of science where the individual happened to be competent to conduct them. The wants to be supplied in such a work are discovered by experience and observation in the different callings for which they are more especially intended. That these wants have not all been met in the works of a similar kind which have already appeared will be made evident by a simple inspection of the amount and variety of new matter:contained in the present volume. It is not every one, however practically expert he may be in his own pursuit, that is capable of arranging and digesting in the best manner the: knowledge necessary for his own use which he may have been years in acquiring, so as to render it available for the use vi PREFACE. ofi others. Such a task, to be well performed, requires a combination of mental qualities not always, perhaps not often, found in! the same individual. A happy concurrence of cireumstances. his by accident secured for the composition of the present work the labors of several skilful hands, both as compilers from the best foreign sources, and as original producers of valuable material never before in print. The result of so much well directed industry is the rich collection, not a line of'which is not invaluable, which, in the aptest form for imrmediate use, has been crowded into. the space of a single small volume. Steam and its application play so important'a palt in the economy. of life at the present day,:that., the most useful practicall rules and. formulas for all the ordinary cases occurring,c'annot with propriety be oi-itted in a. wvork of this kind.. A'.due. attention will be found to have been'paid to the' matter, and some of the newest modes of managing in steam supplied with the means of the requisite computation. The laying out of Railroad curves is one of' the: most important j anld at the same time'laborious and troublesome duties.which the t Oivil Engineer has to perform'.:. So much of this occurring on every i li'ne of Railroad, any,. however slight, improvement'of methods which may serve:to facilitate or lessen the labor of this.process is a real boon' to thlat large anid eminently Uiseful arid "accomplished body | of men to whom the' supervision of such bperations'is.comnmitted.. The use of the more common. trigonometric functions? to wit, sines, cosines, tangents, and cotangents, which ordinary tables- firnishi isl not well adapted to the' peeifliar problems whipch are presented in the construction of Railroad'curves.'The additional columns. of secants and cosecants in the.: tables of Dr. Bowditcll-h sometimes afford a slight'additional facility, which wotild be amuch increased had we also columns of natural secants as well as logarithmic. PREFACE. vii Still there would be much labor of computation which may be saved by the use of tables of external secants and versed sines, which have been employed with great success recently by the Engineers on the Ohio and Mississippi Railroad, and which, with the formulas and rules necessary for their application to the laying down of curves, drawn up by MR. HASLETT, one of the Engineers of that Road, I are now for the first time given to the public. This portion of the volume alone, by the great abridgment of labor for which it provides the means, and the simplicity and convenience of the matter which it furnishes, will give it an extensive circulation among Practical Engineers. But besides this, the Architect, the Shipbuilder, the Mason, the Carpenter, the Joiner, the Manufacturer and Artisan in iron and every species of 7material, will find rules and recipes for all kinds, of estimates, computations, constructions, compositions, mixtures, et cetera, which will excite surprise at their number, novelty, and value to every one. The contents of this volume are of so varied a nature that it was inot deemed necessary to make any strenuous efforts to arrange them systematically. Being solely intended for a book of reference, the relative order of the subjects is immaterial; and the copious Table of Contents and Index afford all the assistance that can be desired by those who wish to consult its panges. T timj EDroIto, | CotUBIsA COLLEGE, Sept. 18b5a C O N TEN T S. PAGE ARITHIMETIC.AL SIGNS,.... 13'ALGEBRAIC SYMTBOLS,.... 15 PRACTrCAL GEOMrRY,... 6 1. From any given point. in a straight line, to erect a perpendicular 16 2, 8. When a pelpendictlar is to be made at or near thle end of a given line, 16 4. To bisect any given angle, 16 5. To find the centre of a circle, or radius, that shall cut any three | iven points, not in a direct line,...... 6. To find the ccntre of a given circle, 1.... 7T. To find the length of alny given arc of a circle,.1 8. -'Through any given point, to draw a tangent to a circle,. 1 9. To draw from or to the circumference of a circle lines tending towatrds the centre, when tlhe centre is inaccessible,. 1 10, 11. To describe an arc, or segnient of a circle, of large ralii,.. 1 12. To describe an ellipse, hving the two diameters given,.. I 13, 14. To describe an elliptic arch, the width and rise of span being! iven,. 1S, 19 15. T6) describe a parabola, tile dimensions being given,... 19 16. To obtain by measurement the length of any direct line, though intercepted by some mIaterial object,. 19 17. A round piece of timber being gi'ven, out of w)hich to cut a beam of strongest section,.. 2 18. To mneasure the distance between two objects, both being inaccessible,.. 2 19, 2:). To ascertain the distance, geometrically, of any inaccessib'e object on an equal plane,.... 21. 2 21. To lind the proper piosiiton for an eccentric, in relation to the C'Rank int a steam —en-ine, tile angle of eccentlic rod, and travel of the valve. bein given,. 21 22. Tlle throw of'in eccentlie, and the travel ()f the valve in a steanmengine, also the length of one lever for communicating motion tfo the valve, being given, to determine the propei lcgtll tbr bhe othe,. 21 23. To inscribe any regular polygon in' a aiven circle,. 24. To construct a squnare upon a given right line,..'5. -To form a square equal in;iea to a given triangle,..'2i. To form a square equal in a'rean to a given rcctanigle 22 21. To find the lefigth for a rectanhle whose area shiall be equtal to that of a given square; thei bl adth of the rectangle being also given,. 22n 23. To bi-ect any given tianrale. 23 29 To describe a circle of reatiest diameter in a given trian'', 23 - ).'To form a rec'tangle of gre'Vtest sulifacc, in a given triagle.. 23 I DECT-\L AIITHMETIC,.... 2 Reduction,. 24 XC CONTENTS. _____-|~~~~~~~~ ~PAGGE MENSURATION,.. 27 MENSURATION OF SUPERFICIES,. 27 1. To find tile area of any parallelogram, 28 2. To find the area of a trapezoid, 28 3. To find the area of a trianle.. 28 4. Any two sides of a right-angled triangle being given, to find the third side,..... 29 5. To find the area of any regular polygon,. 30 Table of multipliers for polygonls fr.om-three to twelve sides,. 31 1. The breadth offa polygon given; to fln'd the radius of a circle to contain that polygon, 31 2. The radius of a circle given, to find the length of side,. 1 3. The length of side given, to finid the radius, 31 6. Having the diameter of a circle given, to find the circumference or the circumference given, to find the diameter,.. 7. To filnd the length of any arc of a circle,. 3 8. To find the diameter of a circle, by having the chord and versed sine givenll....... 32 9. To find the area of an ellipsis, or oval.... 33 10. To find the area of a parabola, or its segment,... 88 Some of the properties of a~circle,... 38 11. To'find the area Iofa sector of a circle... 34 12. To find the area of a segment of a circle,... 34 13. To find the area of a circular ring or space included between two colicenltric circles-... 14.,To find the area of an ellipsis,... 3 MENSURATION OF SOLIDS,..... 1. To find tll convex surface and solid content of a cylinder,. 3I 2. T'o dcltermine the dimensiolns of any cylindrical vessel, whereby to conltain the greatest cubical contents, bounded by the least superficial surface,..6 3. To find the surface and solid content of a cone or pyrnmid, 37 4. To find the surface of the frustum of a cone or pyramid, 37 5. To find the solid content of the frustum of a cone,. 7.. 6. To find the solid content of the fi'usturn of a pyramid,. 38 7. To find the solidity of a wedge,... 38 8. To find the convex surface aznd solid content of a sphere or globe, 38 9. To find the convexisurface and solid content of the segment of a sphere.....39 10. To find the soli(litv of a spheroid,.89 11. To find the solidity of the segrient of a spheroid when the base is circular or. parallel to the. revolving axis,... 40 12. To find the convex surface and solid content of a cylindric ring, 40 DECIMAL AiPPROXIMATIONS, for facilitating Calculations in Mensuration,..... 41 INSTRUMEINTAL ARIrTIMETIC, or Utility of the Slide Rule,.. 42 Numeration..... 42 To inmltiply numbers by the rule,... 42 To divide numbers upon the rule,... 43 Proportion,;or rule of threc direct,. 48 Rule of three inverse;.... 43 Square and cube roots of nmbers,... 43 The mean proportion between two numbers,.. 43 Mensuration of surface,.. 44 Tables of gauge-points for the engincer's rule and theo common slide rule,..... 45 Mensuration of solidity and capacity, 46 1 Land surveying,......... 46 Power of bteam-engines,.. 46 Of engine boilers,. 47 CONTENTS. xi PAGE TIiE Lxcws OF MOTTON,..... 47 To find the velocity of a falling body at the end of any number of seconds, 47 To find the space passed ovcr by a falling body in any number of - seconds,. 43 To find the space passed over by a falling body when, the velocity is given, 48 Rules and formule when -a body is acted on by any force,. 43 Rule for finding the accelerating force.of a body,. 9 Time of a body falling down an inclined plane,.. 49 The time of oscillation of a simple pendulum,.. 49 Centrifugal force,.50 The centre -of gyration,. 53 The centre of percussion,........ 3:ON WRouK,.54 To find the units of work in raising a given weight a given height,. 54 1Iorse'lpower.55 Maln-power,. 55 Work of animals,. 55 Traction of horses at various rates 6ftravelling. 56 Accumulated work,. 561 Work'done by machines,. 56 The llorse-power of an; engine,... 57 On the strength of animals, 57 To Calculate tle Diffelent Parts of a Crane as respects Ilechani'cal Advantage,.. 5 Equilibrium and Pressure of Beams-thle Parallelogralla of Forces,.. 5. SPECIFTC G RAVTTY,.. 1 Table of specific gravities,......... C2 Weiglt of given bulks of water and air,... 6G THE MT ECII.ASNICAL POWERS, AND TrHE It A:-itLIcATION',.... (;I The Lever,.. 63 The Pulley......... 65 The Wheel and -Axle,... 65 The Ilclined Plane,. 66 The Wedge,. 6 The Screw,.67:TOOTHED. WHEELS,.... 6 G ON TIJE VELOCITY OF WHEELS, DRUMS, PULLEYS, &C.. 68'STEAAM POWER AND TI-E ST EAM- EoNGINE,.... 71 To estimate the amount of advantage gained by using iteaml expanslvely;n a steam-enine.. 71 To calculate the effect of a lever and weight upon the safety-valve of a steam boiler, &c.,... 73 To find the-proper diameter for a safety-valve,... 7 General proportions of locomotive engines,. 76 Steam-engine boilers, and their proportions,... 7 Of the Ipressure of steam, in inches of mercury, at different temperatures,... Temperature of steam at different iiressures in atmosIbheres,.. 9 The elastic folce of steam, and corresponding temperature of the water with Which'it is in contact.. i8 The force and temperature of steam in atmospheres, 81 Table of the heating. power of various substances, exhibiting the atmost - quantity of water evaporated by given weights, &c., 3. Nominal horse-ipowei of low pressure entines, Nomifial horse-power of high pressure engines,..84 xii CON TENTS. PAGE Proportions of condensing engines. 85 Revolutions per mile of driving wheels, and consumption of steam and water for each sized wheel,..... 86 Pressure ofsteam excliisive of that of the atmosphere,. 87 Pressure exerted by the elastic force of steam at various degrees of temperatare.b, 8.. 8 AMALGAfMS, ~...89. S9 VARNISHES,..89 PRACTICAL TABLES: WEIGUT OF METALS — WROUGHT IRON; SQUARE, ROUND, AND FLAT,...97 TABLE OF GRADIENTS, and Resistance periTon for each,.. 118 Ultimatel Breaking Weight in Tons of Cast-Iron. Pillars,. 119 Table of Strengths of Cast-Iron Shafts,... 123 Strength of the Teeth of Cast-Iron Wheels.. 127 Method of Ascertaining the Weight of Pipes of Various Metals, and any Diameter,...12:Weight of Cast-Iron Balls,. 129 Table of the Weight of Flat and Rolled Iron...... 130 IWeight of Cast-Iron Pipes,....... 131 Table of the Weight of Malleable Iron,..... 1 IDimensions and Weight of Coppers,.. 33.Weight of Cast-Iron Plates,.. 133:Bore and Weight of Cocks; Weight of Lead, Lead Pipe, and Copper Tubing,..134 STRENGTHI OF MATERIALS,......... 135 Table of tenacities, resistances to compression, &c., of the common materials of construction,.135 Strength and weight of ropes and chains,...'. 186 Resistance to lateral pressure, or transverse action,.. 186 Elasticity and strength of val i us species of timber,. 136 Table showing tfle weight a beam of cast-iron one inch in breadth will sustain, 137 Resistance of bodies to flexure by vertical pressure,.. 141 Dimensions of cylindrical columns of cast-iron to sustain given loads, 142 i Elasticity of torsion..143 Table of the weight of-a superficial foot of plate or sheet iron, copper, and brass,.... 144 Comparative weights of different bodies,. 145 The mensuration of timber-flat or broad measure, cubic or solid measure,................ 146 Cast metal cylinders,......... 148 Table of the weight of cast-iron pipes,.. 148 Strength ofjournals of shafts,.. 149 Table of the diameters of shafts,. 151 Strength of wheels,... 152 Proportions of wheels,... 154 ALLOYS, OR MISCELLANEOUS METALS,.154 MENSURATION OF CIRCLES: Table of the Diameters, Circumferences and Areas of Circles,..160 Table of the circumferences and areas of circles, from 1 to 50 feet, advancing by an inch... 16T IVORY,............. 174 i CONTENTS. xiii PAGE Centre of Attraction, Equilibrium, &ec.,... 175 Cohesion,.. 17 MECIIANICAL LAWS OrF ELASTIC FLUIDS,.. 177 Foyle's or Mariotte's Law,.. 177 Dalton's and Gay-Lussac's Law,. 177 sAmonton's Law,. 177 iDaltonls Experiment,. 178 Illuminating gases,.......... 178 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMIBERS, 180 Blacking Recipes,.202 STRAIN AND STRESS OF MATERTALS,.. 204 Tensile strength of east iron,.207 Tensile and compressive strength of wrought iron,. 208 Compression of cast iron,. 209 Transverse strenglth of beams,. 209 Transverse strength of cast-iron bars,.211 1Deflection of beams,.212 Transverse flexure of a wrought-iron bar by horizontal pressure,.. 218 Hollow rectangular beanls,.. 214 Experimnents on the transverse strength of rectangular tubes of wrought I iron, 215 Strength of cast-iron pillars,.216 Effects of temperature on the strength of cast iron,.. 217 Comparative strength of long pillars; resistance to torsion,. 217 Strength of ropes,... 218 PROCESSES FOR STAINING WVOODs,.. 218 LocARITna Is; Application and Utility of Common Logarithmic Tables,..219 Table to determine the Distances of the Movable Points in a Parallel Motion,. 223 CAPILLARY ATTRACTION,. 224 Woods. to Polish, Preserve, &c. 226 STEAMs ENGINE: To Estimate the Amount of Effective Poer,l. 227 To determine the proper velocity for the piston,.227 Approximate velocities for pistons,. 228 Parallel motion,.. 228 I PROPERTIES AND MISCELLANEOUS EFFECTS OF HEAT: with Tables 6f Expansion and Dilatation of Liquids and Solids,. 228 Degrees of the three thermometrical scales,. 232 Table of the Weight of Substances of Construction,. 23 Conducting Power of Materials used in the Construction of Houses,........... 235 Capacity of Bodies for transmitting Heat,.... 235 Solders,.. 235 Table of Proportions for Shafting with Half-lap Couplings,. 236 Gradations of Temperature,.. 23;6 PROPERTIES OF NUMBERS,... 38 Table of useful numbers,. 249 Surface of Boiler Tubes of different Lengths and Diameters,. 241 xiv CONTENTS. PAGE Recipes for Making different Kinds of Glass,... 242 TABLE OF PRIME AUutBERS to 1000,.243 Table of Solid Inches and Solid Feet,. 245 Weight of Cast-Iron Plates,..... 245 Horse-power that a Cast-Iron Wheel is capable of transmitting, 246 Dimensions of Wheels in actual use. 246 Table of Comparative Dimensions and Strength of Ropes and Chains..... 247 Equivalents and Specific Gravities of Sixty-two Simple Substances.248 Feeding Properties of different Vegetables,. 249 PENDULUMS,.249 SYIMBOLS AND EQUIVALENTS OF BINARY COMJPOUNcDS. 250 Recipe for Dyeing Hats,.. 256 LIQUEFACTION OF GASES,........ 257! Proportionate Strength of Wheels in IIorse-power,... 258 Knot Table,......... 259 CEMENTS,...262'Analy sis of certain Organic Substances, from the best Authorities,...269 Table to calculate the Pitch of Toothed Wheels,.. 270 i Strength of Cables,. 271 Size of Cables and Anchors in proportion to Tonnage,... 272 Strain of Hemlpen Ropes,.. 273 Weight of Copper Rods or Bolts,.273 I Weight of Riveted Copper Pipes,. 274 Capacities of Cisterns in gallons,.2. 274 Table of Weights of Square Feet of Copper, Lead, and Iron,. 275 Table of the quantity of Water per Linear Foot in pumps,.. 276 Properties of Atmospheric Air, 277 Screws, 277i Component Parts of Several Varieties of Wood,.. 278 Specific Gravity of Gases,. 278 Table of Change Wheels for Screw-cutting,.279 Weight of Atmosphere at different Heights,.. 279,Table of the Quantity and Weight of Water in Pipes of any given size,.280 Specific Gravities and other Properties of Bodies, 281 Properties of woods,.... 282 The fusing. point of metals,...252. Properties of liquids,. 283 Table of Fusibilitv of Metals.. 84 Table of the Discharge of Water over Weirs,... 285 CONTENT3. XV PAGE Composition of different Gunpowders,.. 28,5 Alloys,. -.286 Man and Horse Power,... 286. Speed and Force of Wind at different Velocities,... 287i The Boiling Point,..287 Table of the Sizes of Nuts equal ill Strength to their Bolts,. 288 The Powers of various Species of Fuel,..... 288 Table of Ratios of the Successive Hardnesses of Bodies,,, 289 Ductility and Malleability of Metals,.. 289 Conducting and Radiating Powers of various Substances,,. 290 Reflection and Transmission of Heat,. 291 Table of Scale of Proofs for Chain-rigging,.2. 291 MASONRY. 292 Different kinds of,.292 The methods of joining stone,.'294 BRiCKLLATING,.......... 297 Foundations,.297 PLASTEI'TINGG,.... 300 Mortar and cement,....... 300 Digging, &c.,...01 Coarse and fine stff,... 301 Stucco for inside of walls, 3. 01 Gauge stuff, 32.. H!iggins' stucco,..... 02 Parker's and Hamelein's cements,. 302 Maltha, or Greek mastic,.... 303 Plaster in imitation of marble,.... 303 Composition...... 3,04 To make glass and stone paper,..... 04 84 Woon)VOPul, CARPENTRY, &e,........305 Causes of the decay of timber,..305 Circumstances favorable to vegetable decomposition,. 05 ]Means of prevelting decay,. 807 Seasoning timber,.. 07 Framing- of timber,... 12 Composition and resolution of forces,.12 Construction of roofs,. 314 Dimensions of timbers used in a roof, 816: Examples of roofs,... 317 Floors,............ 31... 8 Trusses,. 3.20 Of connecting timbers,..... 321 Timber partitions,........322 Gluing joints,........ 323 Dovetailing,....... 324 Mortice and tenon,........ 826 Grooving and lapping.... 82 Bending and gluin-up, 328. Scribing, 329 Finishing of joiners' work,..... 880 TERMS USED IN BUILDING,.. 331 GLUES,..342 xvi CONTENTS. PAGE i PAPERS,..........342I BRONZING.. 3.47 ENAMELS,.. 34 MARBLE STAINING,....349 COMPOUND COLORS IN DYEING,.. 350 I POLISHES,............. 351 BOOKBINDERS' RECIP,....... 3-5-2'CRAYONS,......3-58 GILDING,....... - 59 GLASS STAINSS,.. 361 FACTITIOUS STONES,........ 3621 INKS,......... 365 WAXES,.. ~.3691 THE ENGINEER'S FIELD-BOOK,...... 371 FORAIULE FOR RUNNING LINES, LOCATING SIDE TRACKS, &C.,.. 375 TABLE OF CHORDS CORRESPONDING TO EVERY 100 FEET ON CURVE FRONIr 200 TO 1000 FEET, CALCULATED TO EVERY 15'MINUTES' RATE OF CURVATURE, FROM 15 MINUTES TO 8 DEGREES, RADIUS OF l BElING 5730 FEET,.......... 4081 TABLE OF RADII AND THEIR LOGARITIIHMS,. 409 TABLE OF CHORDS,...... 414 TABLES OrF NATURAL AND LOGARITIIMIC VERSED SINES AND EXTERNAL SECANTS,... 4.. NATURAL SINES AND TANGENTS TO EVERY DEGREE AND AMINUTE OF TI1E QUADRANT,..4571 LOGARITHaMS OF NUMBERS FROAM 1 TO 10,000,. 483 TABLE OF TIlE LENGTHS OF CIRCULAR ARCS, RADIUS BEING UNITY,. 499 EXPLANATION OF THE USES AND APPLICATIONS OF THE TABLE OF LONG CHORDS,.....501 TABLE OF MIDDLE ORDINATES,... 503 TABLE FOR LOCATING TURNOUTS,. 505 MISCELLANEOUS NOTES AND IEXAMPLES...... 510 1 INDEX,... 5.13 i_ __ MIECHtANIC'S, MACHINIST'S, AND EINGINEER'S PRACTICAL BOOK OF REFERENCE: CoNTAININTABLES AND FORiMULXE FOR tTSE IS SUPERFICIAL AND SOLID MENSURATION; STRENGTH AND WEIGHT OF MIATERPALS; MECHANICS; MACHNERPY: HYDRAULICS; HYDRPODYNAMICS; MARINE ENGINES; CHEMISTRY; AND bMISCELLANEOUS RECIPES. ADAPTED TO AND FO.t TIlE 1sE Of ALL CLASSES OF PRACTICAL MECHANICS, EDITED BY CHARLES NV. HACKLEY, prfoasgx of ftatltTraitS i~t Columbia Coalgt, X. A THE PRACTICAL BOOK OF REFERENCE. ARITHMETICAL SIGNS. TnE following definitions of arithmetical signs which are employed in mechanical calculations, will be found of great value to those who do not yet understand them, and of some interest to; those who are already familiar with their meanings. - This is the sign of equality, and signifies equal to. For example: 12 inches = 1 foot (12 inches is equal to 1 foot). + This is the sign of addition, and signifies plus, or more. For example: 5 + 3 = 8 (5 added to 3 is equal to 8). - This is the sign of subtraction, and signifies minus, or less. For example: 10 - 8 = 2 (10 minus 8 leaves or is equal to 2). x This is the sign of multiplication, and signifies multiplied by, or into. For example: 10 x 3 = 30 (10 mnultiplied by 3 is equal to 30). i- This is the sign of- division, and signifies divided by. For example: 156 ~- 6 = 26 (156 divided by 6 is equal to 26); or, 24. 4 = 6 24 (24 divided by 4 is equal to 6); or - = 6 (24 fourths are equal to 6 wholes).:::: ~This is the sign of proportion, and signifies proportion. For example: 4:6:: 8: 12(as 4 is to 6, so is 8 to 12); or 3: 5 3 9 9: 15 (that is, as 3 is to 5, so is 9 to 15); I - 1-. / This is the sign of the SQUARE root. When it is placed before a number (as thus, 4/ 5 = 25) it means that the square root of that number is required. For example: V 25 = 5, because 5 x 5 = 25; or, 9 = 3because 3 x 3=-i9; or, *f 64 = 8, because 8 x 8 64. V Tlis is the sign of eh: CUBE root. When it is placed before a number, it means that the cube root of that number is required. For example: / 64 = 4 (that is, 4 x 4 = 16 and 4 x 16 = 64); or, V 216 = 6 (that is, 6 x 6 = 36, and 6 x N6 = 216). 2 14 AR-ITHMAETICAL SIGNS. 2 When this mark is added to a number (thus, 62), it means that i that number is to be squared. For example: 52-= 2. (that is,: 5 x 5 = 25); or 62 = 36 (that is, 6 x 6 = 36). s When this mark is added to a number, it means that that num-l ber is to be cubed. For example; 59-= 5 x 5 x 5 = 125 (that is, 5 x.5 =25, and 5 x 25 = 125; or, 7- 343 (that is, 7 x 7 _ 49, and 7 x 49 = 243). The index or power (as the small figure annexed is called) shows how many times a number is to be multi-:l plied by itself. This is called the bar. It signifies that all the numbers or quantities under it are to be taken together. For example: 3 + 5 x 4 = 32 (3 plus 5 are equal to 8, and that, multiplied by 4, is, equal to 32); or, 7 - 3 + 8 = 12 (7 less S is equal to 4, and th/at, if added to 8, is equal to 12); or, 5 x 4 + 3 = 35 (that is, 4 and 3 are 7, which, if multiplied by 5, is equal to 35); or, 5 x 6 + 4 = — I (that is, 6 and 4 are 10, and ten times 5 are 50). The parenthesis, ( ) is sometimes used in place of the bar, thus: (6 + 4) x 5 = -50. The meaning of this sign is therefore. This sign signifies because. ~ The meaning of this sign is perpendicular. Z This sign signifies an angle. This sign denotes difference, and is placed between two quan- | tities (as x - y) when it is not known which of them is the greater. > or 72 The meaning of these signs is GREATER than. For example: A B > C D (that is, A B is greater than C D). < or c- The meaning of these signs is LESS than. For example: AB < C D (that is, AB is less than CD). This is a decimal point. When placed before a number (thus,.1), it means that that number has a unit (1) for its denominator. For example,:.1 is the same as Il;.125 is the same as j!o20;.01 is the same as.y-;.001 is the same is;.0001 is the same as ~f-ijr; 42.85 is the same as 42~-; 57.217 is the same as 57~6. ~ This is a degree mGark. It is written and printed as follows: 25~ (that is, 25 degrees). This is a miznute sign,. " These two accents signify seconds. "' These three accents signify thirds. They read thus: 57' 17':43" 39"' (that s,. 57, degrees, 17 minutes, 43 seconds, and 39 thirds). ALGEBRAIC SYMBOLS. 16 ALGEBRAIC SYMBOLS. The advantage of these, in a work like the'present, may be thus illustrated: Let I denote the length, b the breadth, and d the depth of an iron beam. If it be desired to express the product of the length and ibreadth, divided by the depth, it is done as follows: lb That is to say, multiplication is expressed by simply writing the letters which represent numbers one after the other; division, by drawing a line under the dividend, and writing the divisor below. The sum of the length and breadth, divided by the depth, would i be expressed briefly thus: I +b d The square of the length, multiplied by the cube of the breadth, thus: P b)3 The square root of the length, divided by the fourth root of the' breadth, thus: The square root of the difference- of the length and breadth, divided by the depth, thus: V/-b d The square root of the quotient of the sum and difference of the length and breadth, thus: / I+ b Any other letters-as a, b;, &c.-mas stand for the given dimensions. These explanations will serve to give the sense of the symbols which will be met with throughout the work. 1~T PhtAOT~eA-L VGxoMwItRY. PRACTICAL GEOMETRY. 1. From any given point, in a straight line, to erect a perpendicular; or, to make a line at right angles with a given: line. On each side of the point A, from which the line is to be made, take equal distances, as A b, 7 A c; and from b and c as centres, with any distance greater than b A or c A, describe arcs cut,. ting each other at d; then will the line Ad be the perpendicular required. 2. Wh7en a perpendicular is to be made at or near e the end of a given line. With any convenient radius, and with any dietance from the given line A b, describe a portion of a circle, as b A c, cutting the given point in A; — ~ -," draw, through the centre of the circle n, the line b n c; and a line from the point A, cutting the intersections at c, is the perpendicular required. 3. To do the same otherwise. t From the given point A, with any conve. nient radius, describe the are d c b; from d cut the arc in c, and from c cut the are in b; also from c and b as centres, describe arcs cutting c/ each other in t; then will the line A t be the perpendicular as required. Note. —When the threa sides of a triangle are in the proportion of 3 4, and 5 equal parts, respectively, two of the sides form a right angle; and observe that in each of these or the preceding problems, the perpendiculars may be continued below the given lines, if negessarily required. \ ~ / 4. To bisect any given angle. From the point A as a centre, with any radius less than the extent of the angle, describe an arc, c.v ] g as c d; and from c and d as centres, describe: arcs cutting each other at b; then will the line A-b bisect the angle as required. PRACTICAL GEOMETRY. 17 5. To find the centre of a circle, or radius, that slhall cut any th]ree. given points, not in a direct line. d From the middle point b as a centre, with any radius, as b c, b d, describe a portion of a circle, r' as c s d; and from r and t as centres, with an equal radius, cut the portion of the circle in c s and d s; draw lines through where the arcs cut eaeh other; and the intersection of the lines at s is: the centre of the circle as required. 6. To find the centre of a given circle. D Bisect any chord in the circle, as A B, by a perpendicular, C D; bisect also the diameter E D in!'; and the intersection of the lines atf is the centre of the circle required. 7. To dfind the length of any given are of a circle. With the radius A C, equal to, ith the length i of: the chord of the arc A B, and from A as a centre, cut the are inmc; also from B as a centre,.->l with -equal radius, cut the chord in b6; draw the c line- C b; and twice: the length of the C line is the length of the arc nearly. 8. Through any given point, to draw a tangent to a circle. X, B Let the given point be at A; draw the line A C, on which describe the semicircle AD C; draw the line ADB, cutting the circumference in D, which is the tangent as required. 9., To draw from or to the circumference of a circle lines tending tow grds the centre, when the centre is inaccessible. Divide the whole or any given 3. portion of the circumference into the desired number of equal parts; t then, with any radius less than r B D the distance of two divisions, de- c scribe arcs cutting each other, as A A 1, B 1, C 2, D 2, &c.; draw the lines C 1, B 2, D 3, &c., which lead to the centre, as required. To draw the end lines. As A r, F r, from C describe the are r, and with the radius C 1, 2* -8 PIRAaOrIAL GXOMZTRY; from A or F as centres, cut the former arcs at r, or r, and the lines A r, F r, will tend to the centre as reqpuired 10. To describe an are, or segment of a circle, of large radii. Of any suitable material, construct a triangle, as A B C; make lB A B, BC, each equal in length to the chord of the are D E, and height, twice that of the arc B b. At each end of the chord D E fix a pin, and at B, in the triangle. fix a tracer (as a pencil), move the triangle along the pins as guides; and the tracer will describe the arc required. 11. Or otherwise. Draw the chord A C B; also draw the line. H D L parallel with the chord, and equal to the height of the segment; bisect the chord in C, and erect the perpen32 3 i, 2 3 1 dicular CD; join AD, DB; \t 3'' 3 draw A HE perpendicular to A D, and B I perpendicular a1 2to B D; erect also the per-A I c2 $ 2 1 -B pendiculars An, B; divide A B and H I into any number of equal parts; draw the lines 1 1, 2 2, 3 3, &c.; likewise divide the lines A a, B n, each into half the number of equal parts; draw lines to D from each division in the lines A n, B n, and, through where they intersect the former lines, describe a curve, which will be the arc or segment required. 12. To describe an ellipse, having the two diameters given. On the intersection of the two diameters as a centre, with a -t radius equal to the difference of the semidiameters, describe the are a b; and from S/',/a s b as a centre, with half the chord be a, deL?& cR %,0 scribe the arc c d; from. o, as a centre, with V,... o 6 "".. the distance o d cut the diameters in d r, d t; draw the lines r, s, s, and t, s, s; then: from r and t describe the arcs s, s, s, s; also 5, r,~ will from d and d, describe the smaller ares s,,,, which will complete the ellipse as required.. 13. To describe an elliptic arch, the width and rise of span being given. Bisect with a line at-right angles the- chord or spanl- A B; erect PxtAIEcAL GxEoYETa.t:tM perpendicular A q, and draw the line q D equal and parallel to AC; bi- _ seet AC and A/ in rand n; make C I z equal to CD, and draw the line Ir q; draw also the line n s D; bisect s D with A a line at right angles, and meeting the line CD in g; draw the line g q, make C P equal to C k, and draw the line g P i; then from g as a centre, with the radius g D, describe the are s D i; and from k and P as centres, with the ran- dins A k, describe the arcs A s and B i, which completes the arch as required. Or, 14. Bisect the chord A B, and fix at right angles any straight guide, as b c; prepare, of any suitable material, a rod or staff, equal to half the chord's length, as def; from the end of the staff, equal to the height A of the arch, fix a pin e, and at the A extremity a tracer f; move the staff, keeping its end to the guide and the fixed pin to the chord, and the tracer will describe one half the are required. 15. To describe a parabola, the dimensions being given. Let AB equal the length, and CD the breadth of the required parabola; divide CA, C B into any number of equal parts; also divide the perpen- a D 6 diculars Aa and B13b: 3 into the same num- i1 ber of equal parts; A i 2 then from a and b draw lines meeting each division on the line A C B; and a curve line drawn through each intersection will form the parabola required. 16. To obtain by measurement the length of any direct line, though intercepted by some material object. Suppose the distance between A and B is required, but the right line is intercepted by the object C. On the point d, with any convenient radius, describe the are c c, make the are twice the radius in length, through which draw the line d c e; and on e describe another -20 PRACTICAL GEOMETRY. are equal in length to once the radius,: as e ff; draw - A- c r B the line efr equal to efd; c /j on r describe the are jj, in length twice the radius; }C\ /j continue the line through,}ef ~ rj, which will be a right line, and de, or e r, equall the distance between dr, by which the distance between A and B is obtained as required. 17. A round piece of timber being given, out of whlich to cut a beamt JoD C of strongest section. Divide into three equal parts any diameter in e/ \ the circle, as A d, e C; from d or e, erect a perpendicular meeting the circumference of the circle, as dB; draw AB and BC, also AD equal to B C, and D C equal to AB, and the rectangle will be a section of the beam as required. 18. To measure the distance between two objects, both being inaccessible. From any point C draw any line C c, I -tic/ / and bisect it in D; take any point E in the prolongation of A C, and draw the line E e, making D e equal to D E;! in like manner take any point F in the e D —- a: C~ —-G ]prolongation of B C, and make Df J' Xx2Je equal to F D. Produce A D and ec till they meet in d, and also B D and fcl C /~\\ till they meet in b; then a b equal A B, or the distance between the objects as b B required. l 19. To ascertain the distance, geometrically, of any inaccessible object on an equal plane. _/__ —-_~ Let it be required to find the distance between A and B, A being inaccessible; produce the line in the direction of A B to any \ /. -x point, as D;. draw the line. D d at any angle to the line A B; bisect the line D d, through which draw the line B b, making cb equal to B c; draw the line d b a; also through c, in the direction c A, draw the line a cA, intersecting the line d b a; then b a equal B A, ja /the digtance required. PRACt1CAL GOM"lX tr. 2 la. Og6SWje. A, Prolong A B to any point D, making B C equal to C D; draw the line D a at' any angle with DA, and the line C b simnilar to B c; draw also the line D E.F, -which intersects the line B a; then a b B equal B A, or the distance required. 21. To find the proper position for an 6 eccentric, inl relation to the crank in a steam engine, the angle of eccentric rod, and travel of the valve, being given. Draw the right line A B, as the situation of the crank: at commencement of the stroke; draw also the line Cd, as the proper given angle of eccentric rod with: the crank; then from C as centre, describe a circle equal to the travel of the valve;'draw' the line ef at light angles to the line Ca, draws also the lines I 1, and 2 2, parallel to the line ef; and at a distance / s from ef on each side, equal to the c lap and lead of the valve, draw the angular lines C 1, C 2, which are the e < angles of eccentric with the crank, for forward or backward motion, as may be required. 22. The throw of an eccentric, and the travel of the valve in a steamengine, also the length qf one lever for communicatinzq motion to tlre valve, being given, to determine the proper length for tihe other. On any right line, as A B, describe a circle AD, equal to the throw of eccentric and travel of valve; then from C as a centre, with a radius equal Ito the length of lever given, i cut tile line A B, as at d, on - which describe a circle, equal:to the throw of eccentric or travel of valve, as may be required; draw the tangents B a, B a, cutting each other in the line A B, and d B is the leigth of'the lever as required. Note.-The throw of an eccentric is equal to the sum of twice the,distance between the centres of formation and revolution, as ab, Or to the degree of eccentrieity it is made to describe, as cd., And The travel of a valve is equal to the sum of the widths of the two steam openings, and the vulve's excess of length monre than just sUfficient to covet the openings. W22 PxrACTICAL G OMETraY. 23. To inscribe any regular polygon in a given circle. Divide any diameter, as A]B, into so many equal parts as the polygon is required to have sides; from A arind B -as centres, with a radius A e \\ equal to the diameter, describe arcs cuttingi B each other in C; draw the line CD through the second point of division on the diameter e, and the line D B is one side of the polygon required. 24. To construct a square upon a given Ae i right line. /is.,); From A and B as centres, with the radius A B, describe the arcs Ac b, B c d, 05 b and from c, with an equal radius, describe the circle or portion of a circle e d, A B, be; from bd cut the circle at e and c; draw the lines A e, B c, also the line st, which completes the square as required. 25. To form a square equal in area to a given triangle. Let ABC be the given triangle; let ~n fall the perpendicular B d, and make A e Lt 5 a \ half the height dB; bisect e C, and describe the semicircle e in C; erect the perpendicular A s, or side of the square, then A s t x is the square of equal area as rea e -a i C quired. 26. To form a square equal in area to a given rectangle. Let the line A B equal the length and breadth of the given rectangle; bisect the line in e, and describe the semicircle AI) B; then from A with the breadth, or from B with the length, of the rectangle, cut the line A B at C, and erect the perpendicular C D, meeting A C e.B the curve at D, and CD equal a side of the square required..27. To find the length for a rectangle whose area shall be equal to that of a given square, the breadth of the rectangle being also given.,Let A B C D be the given square and D E the given breadth of PRACTICAL GEOIETRY. 23 rectangle; continue the line B C E F d to F, and draw the line D; also - d continue the line D C to g, and draw the line Ag parallel to D F; D C from the intersection of the lines -- at g, draw the line g d parallel to - D E, and E d parallel to D g; then E D d g is the rectangle as required. A B 28. To bisect any given triangle. Suppose A B C the given triangle; bisect one of its sides, as A B in e, from which describe the semicircle A r B; bisect the same in r, and from B, with the distance B r, cut the diameter AB in v; draw the line v y parallel to A C, which will bisect the triangle as required. 29. b' describe a circle of greatest a e B diameter in a given triangle. Bisect the angles A and B, and draw e the intersecting lines A D, B D, cutting each other in D; then from D as centre, with the distance or radius D C, describe the circle C ef, as required. c c C 30. To form a rectangle of greatest surface, in a given triangle.!P Let A B C be the given triangle; bisect any two of its sides, as A B, B C, in e and d; draw the liae e d; also, at right angles with the line e d, a draw the lines e p, dp, and e p p d is the rectangle required, A e 3 RATIO OF THE HA4RDNESS OF METALS. 1. Iron, 4. Silver, 6. Tin, 2. Platina,.. Gold. 7. Lead. 3, Copper. STRENGTh OF WOOD. All waods are from 7 to 20 times stronger transyersely than longitudinally. They become stronger both ways wlien dry. 24 DECIMAL AuRITHMEIIC. DECIMAL ARITH-METIC.. Decimal Arithmetice is the most- simple and explicit mode of performing practical' calculations, on account -of its doing away with the necessity of fractional parts in the fractional. form, thereby reducing long and tedious operations to a few figures arranged and worked in all respects according:to the: usual rules of common arithmetic. Decimals simply signify tenths; thus, the decimal of a foot is the tenth part of a foot, the decimal of that tenth is the hundredth of a foot, the decimalof that hundredth is the thousandth of-a foot, and i so might the divisions be: carried on and lessened: to infinity: but in I practice it is seldom necessary to take into account any degree of: less measure than a one-hundreedth part of the integer or whole. number. And, as the entire: system consists in.supposing. the whole. number divided into tenfths, hundredths, thousandths, &c., no:peculiar notation is required,.otherwise than placing a mark or dot to. distinguish between the whole and any part of the whole, thus.. 34.25 gallons signify 34 gallons, 2 tenths, and 6 hundredths of a.gallon; 11.04 yards signify 1 1 yartds and: 4 hundredths'of a: yard;, I 16.008 shillings signify 16 shillings anid. 8 thousandth. parts of a. shilling; from which it must appear plain that ciphers on the right.hand of decimals are of no value wv-hatever,'but, placed! on the left:l hand they diminish the decimal value in a tenfold prop(ortion: for.6 signify 6 tenths;.06 signify 6: hundredths; and.006. signify 6, thousandths. of the integer or whole number. Reduction. Reduction means the converting or changing of vulgr fractions to deci'mals of equal value; also finding the fractional'value' of any decimal given. Rule 1. Add to the numerator of the fraction any number "of ciphers at pleasure, divide the sum by the denominat-tor, and'the quotient is the decimat:- f::equivalent value.. Rule 2, Multiply the given decimal by the various fractional denominations of the integer, or whole number, cutting off from the right hand of each product, for decidrmals, a number of figures equal to the given number of decimals, and thus proceed until the lowest degree, or required value,. is obtained. Ex. 1. Required the decimal equivalent, or decimal of equal value, to 3 of a foot. 3.00 2i -= 25, the decimal required. Ex. 2. Reduce the fraction. of: an inch, to a decimal of equal Yalue. - 1.. o08o _ ~ 125, the decimal reqtuirde. DIcIMAL ARITHMETIC. 25 Ex. 3. What is the decimal equivalent to " of a gallon 7.00~- 875, the decimal equivalent. Ex. 4. Required the fractional value of the decimal.40625 of an inch.'40625 Multiply by 8 8 3-25000 x=2 2'50000 x 52=lg 2 1 00000 ~ and 2 of an inch, the value required. EX. 5. What is the fractional value of'625 of a cwt.? 625 Multiply by 4 qrs 4 2'500 x 28 lbs. 28 14000 =;2 quarters and 14 lbs., the value required. Ex. 6. Ascertain the fractional value of'875 of an imperial gallon.'875 Multiply by 4 quarts 4 3'500 x 2 pints. 2 1-000 = 3. quarts and 1 pint, the value rerequired. Ex. 7. What is the fractional value of'525 of a ~. sterlingS'525 Multiply by 20 sh. 20 10-500 x 12 pence 12, 6-000 = 10 shillings and 6 pence, the value quired. Independent of the mark or dot which distinguishes between integers and decimals, the fundamental rules-viz. Addition, Subtraction, Multiplication, and Division —are in all respects the same as in Simple Arithmetic; and an example. in each, illustrative of placing the separating. point, will no doubt render the whole system Sufficiently intelligible, even to the dullest capacity. 91~~~~ 26 D)ECIMAL ARITPHMETIC. Ex. 1. Add into one sum the following integers and decimals: 16'625; 11'4; 20'7831; 12'125; 8'04; and 7'002. 16-625 11-4 20'7831 12-125 8-04 7'002 75'9751 = the sum required. Ex. 2. Subtract 119'80764 from 234'98276. 234'98276 119'80764 115 17512 = the remainder required. Ex. 3. Multiply 62'10372 by 16'732. 62'10372 16-732 12420744 18631116 43472604 37262232 6210372 1039'11944304 = the product required. Observe that the number of figures in the product from the right hand, accounted as decimals, are equal to the number of decimals in the multiplier and multiplicand taken together. Ex. 4. Divide 39'375 by 9'25. 9'25) 39-375 (4'256 = the quotient required. 3700 2375 1850 Observe that the number of deci5250 mals, in the divisor and quotient 4625 together, must be equal to the 6250 number in the dividend. 6250 700 Note.-The operation might be still continued, so as to reduce the quotient to a degree of greater exactitude; but in practice it is quite unnecessary, being even now reduced to a measure of greater nicety than is commonly required. XIlENSURATIO0I. 27 M1ENSURATION. ml[ensuration is the method of calculating the comparative magnitudes of figures, and it is divided into two parts-Mensuration of Superficies or Surfaces, and Mensuration of Solids. The magnitude of a surface is called its area, and is the space inclosed between its boundary lines. The magnitude of a body is called its solid contents, and is expressed in cubic feet, inches, &c. Mensuration of Superficies. Square. Rectangle. B3 B AA D C D C A SQUARE is a quadrilateral figure, which has all its sides equal, and all its angles right angles. A RECTANGLE is a four-sided figure, which has its angles, right! angles, and its opposite sides parallel. Rhombus. Rhomboid. ID BA - D a'C a C A RHOMBUS is a parallelogram, whose sides are equal, but whose angles are not right angles. A RuHOMBOID is a parallelogram, whose adjacent sides are unequal, and whose angles are not right angles. A TRAPEZOID is a four-sided figure, which has but two of its sides parallel. A CIRcLE is a figure bounded by one line, called the circumference, and is such that all lines drawn to the c A circumference from a certain point within the figure, called the centre, are equal to each other. Any of these lines is called a radius; - and a line drawn through the centre, termi- E nating both ways in the circumference, is called a diameter. The portion of circle cut off by a diameter is called a semicircle. An ARC of a circle is any portion of the circumference. 28, 3MENSIt-RATION. A SEGMENT of a circle is a figure contained by an arc and its chord. A VERSED SINE is a line drawn from the zmiddle of a chord perpendicular to the circumference. A SECTOR of a circle is a figure contained by two radii and an are, as A C B E. PROBLEM L To find the area of any parallelogram. RULE. Multiply the length by the perpendicular height, and the product will be the area. EXAMPLE. Required the area of a rhomboid whose length AB= 20 5, and perpendicular height a A = 1-1'5. 205 x 11.75 = 240-875, the area. Note.- In a square, or rectangle, the perpendicular height is the breadth: therefore, to find the areas of a square and rectangle, multiply the length by the breadth. PROBLEM II. To find the area of a trapezoid RULE Add together the two'paiallel sides, multiply their sum by the breadth or height, and half the product is the area. EXAMPLE. Required the area of a trapezoid whose sides A B and CD are 14'5 and 10'25, and breadth, a A ='1-25. B A14-5 + 10'25 x 7'25 2 - = 89-71875,: the area. D C PROBLEM IIL To find the area of a triangle. RULE. Multiply one of its sides as a base by a perpendicular let fall from the opposite angle, and take half the product for the aream Or, from half the sum of the three sides subtract each side separa4tely, and multiply the three remainders so obtained and the half sum together, and the square root of the product will be the area. EXAMPLE 1. Required the area of a triangle AB C, whose base. A B -— 16-5, and perpendicular D C = 10-25. C 165 x 10-25 \2-?_- - 84-6625, the area. ]B I < DA TMENSURATION. 29 ExAMPrL 2. What is the area of that triangle whose three sides are 8, 12, and 16 respectively? 8 + 12 + 16 + 2 + - 18, the half sum of the sides; then, 18 18 18 8 12 16 10 6 2 and /18 x 10 x 6 x 2=46 47, the area. PROBLEM IV. If any two sides of a right-angled triangle be given, the third side may! be found by the following rules. 1.-To the square of the base add the square of the perpendicular; and the square root of the sum will be the hypothenuse or longest side. 2.-Multiply the sum of the hypothenuse, and one side by their difference; and the square root of the product will be the other side. EXAMPLE 1. Given the base A B = 16, and perpendicular B C = 12; required the length of the hypothenuse A C. 4/162 + 122 = 20, the length of the hypothenuse A C. A EXAMPLE 2. Given the base A B = 16, and hypothenuse A C = 20; required the length of the perpendicular B C. 4'20+ 16 x 4 = 12, length of the perpendicular B C. Note. —The diagonal line, or hypothenuse in a square, is equal to the square root of twice the square ot' the side. And the side of a square is equal to the square root of half the square of its diagonal. Thus suppose each side of a square equal 12 feet: 122 X 2 —- 288 - 16'9705 bet, the diagonal. Or, -2- - ----- 12 feet, the length of each side. Similar triangles, or those which are D equi-angular to each other, have the sides about their equal angles proportional; thus, in the annexed figure the triangles A B C and C D E are similar, B and therefore have the sides about the equal angles proportional:' C 3 _ __ 30 MEN-SURATION. AC: BC:: CE: DE; AB: BC:: CD'DE, &c. The utility, then, of the above triangles for practical purposes, as, for instance, ascertaining the heights of buildings, &c., will be seen from the following: Suppose D E to be an eminence, of which it is required to find the height, and E C the length of the shadow cast:-by the sun; then, in order to find D E, we may erect perpendicularly at C a pole of any known. length, as B C, and after measuring the length of its shadow A C, state-as the length of the pole's shadow is to the height of the pole itself, so is the length of the shadow of D E to the height of D E; or, AsAC: CB:: CE: ED; and supposing A C = 6 feet, B C = 4 feet, and C E = 30 feet, then.A... E D would be 20 feet. Again, supposing we wished to find the distance between two objects A and B; draw D B of any length at right angles to A B, and in D B take any point C,,through which draw A E; also, at D, at right angles to D B, draw D E, making Bv D the triangle D E C, and state, AsDC: DE:: BC:BA. PROBLEM V. To find the area of any regular polygon. RULE. Multiply the sum of its sides by a perpendicular drawn from its centre to one of its sides, and take half the product for the area. Or, multiply the square of the side of a polygon (from three to twelve sides)' by the numbers in the fourth column of the table for polygons, opposite the number of sides required, and the product will be the area nearly. EXAMPLE 1. Required-the area of the regular - pentagon A B C D E, each side being 7-5, and perpendicular FG = 6'4.! B 7'5 x 5 x 6 120, the area. EXAMPLE 2. What is the area of a regular hexagon, each side being 8'75 in length? 8-752 x 2!598 = 199'009375, the area. MENSURATION. 31 TABLE of multipliers for polygons from three to twelve sides. Names. Sides. Multipliers. Multipliers. Multipliers. Areas. Trigon.... 3 2 1 73 *579.433 Tetragon.. 4 141 1'412'705 1000 Pentagon... 5 1 238 1174 852 172 Hexagon... 6 1-156 = Radius. = Length 2-598 of side. Heptagon.. 7 1'11'867 1-16 3'634 Octagon... 8 108 765 1 307 4'828 Nonagon... 9 1'062'681 1-47 6-1818 Decagon.... 10 1'05'616 1-625 7-694 Undecagon. 11 1'04 *561 1-777 9'365 Dodecagon.. 12 1-037'515625 1-94 11'196 1. The breadth of a polygon given, to find the radius of a circle to contain that polygon. RULE. Multiply half the breadth of the polygon by the numbers in the first column opposite to its name, or number of sides, and the product will be the radius of a circle to contain that polygon. And if the polygon have an unequal number of sides, the half breadth is accounted from its centre to one of its sides. 2. The radius of a circle given, to find the length of side. RULE. Multiply the radius of any circle by the numbers in the second column opposite the polygon required, and the product will be the length of side nearly that will divide that circle into the proposed number of sides. And, 3. The length of side given, to find the radius. RULE. Multiply the given length of side by the numbers in the third column opposite the polygon required, and the product will be the radius of a circle to contain that polygon. EXAMPLE 1. Required the radius of a circle to contain an octagon, whose breadth AB = C 18S5 inches. Half of 18'5 = 9-25, and 9'25 x 1'08 = 9'99 -- or ten inches nearly, the radius of the circle O D. EXAMPLE 2. Given the~ radius O D =- 999 inches, required the length of side D C. 9'99 x'765 ='7'64235, the length of side. 32 MENSURATION. EXAMPLE 3. Given the length of side D C = 7'64285; required the radius D O. 7-64235 x 1'307 = 9'98855145, or 9'99 in. nearly. PROBLEM VI. Having the diameter of a circle given, to find the circumference; or the circurnference given, to find the diameter. RULE 1. As 7 is to 22, so is the diameter to the circumference. Or, as 22 is to 7, so is the circumference to the diameter. 2. As 1 is to 3'1416, so is the diameter to the circumference. Or, as 3-1416 is to 1, so is the circumference to the diameter. EXAMPLE 1. Required the circumference of a circle when the diameter is 23'5. 23'5 x 22 - = 73 the circumference. EXAMPLE 2. The circumference of a circle is 73', required the diameter. 73c' x 7 x x 23-5, the diameter. 22 EXAMPLE 3. Required the circumference of a circle whose diameter is 30. 3-1416 x 30 = 94'248, the circumference. EXAMPLE 4. What is the diameter of a circle when the circumference is 94"248? 94'248.- 3'1416 = 30,.the diameter. PROBLEM VII. To find the length of any arc of a circle. RULE. Subtract the chord of the whole arc from eight times the chord of half the. arc; and.1 of the remainder is the length of the are nearly. EXAMPLE. Required the length of the are A B C; the chord of half the arc A B = 19-8, and chord of the whole arc A C = 34-4. x 19-8 x 8 = 158-4, and 1584- 34- 4_41-33, the length of C -A the are. PROBLEM VIII. To find the diameter of a circle, by having the chord and versed sine given. RULE. Divide the square of half the chord by the versed sine, to MENSURAT1O.2. 33 the quotient of which add the versed sine, and the sum will be the diameter.:Or, if the sum of the squares of the semichord and versed sine be divided by the versed sine, the quotient will be the diameter of the circle to which that segment corresponds. EXAMPLE. Given the chord AB = 24, and c versed sine CD = 8; required the diameter of the circle CE. B Half the chord= 12, and 122 8 = 18 + 8= D 26, the diameter. 12"+ 8' Or, -= 26, as before. 8 PROBLEM IX. To find the area of an ellipsis, or oval. RULE. Multiply the longest diameter by the shortest, and the product by'7854; the result is the area. An oval is 25-inches by 16'5: what are its superficial contents 25 x 16'0 = 412-5 x'7854 = 323'9775 inches, the area..Note.-Multiply half the sum of the two diameters by 3'1416, and the product is the circumference of the oval or ellipsis. PROBLEM X. To find the area of a parabola, or its segment. RULE. Multiply the base by the perpendicular height, and two- thirds of the product is the area. ~Whlat is, the area of a parabola whose base is 20 feet and height 12. 240 x 2 20 x 12- — = 160 feet, the area. Some of the properties of a circle. 1. It is the most capacious of all' plane figures, or contains the greatest area within the same perimeter or outline. 2. The areas of circles are to each other as the squares of their diameters, or of their radii. 3. Any circle whose diameter is double that of another, contains four times the area of the other. 4, The?area of a, circle is equal to the area of a triangle whose base is equal to the circumference, and perpendicular equal to the radius. 341 ATENSUII AT:ON. 5. The area of a circle is equal to the rectangle of its radius, and a right line equal to half its circumference. 6. The area of a circle is found by squaring the diameter, and multiplying by the decimal *7854; or by multiplying the circumference by the radius, and dividing the product by 2? EXAMPLE 1. Required the area of a circle, the diameter being 30-5. 30'52 x* 7854 = 730'618350, the area required. EXAMPLE 2. What is the area of a circle when the diameter is 1? In this case the circumference is.31416, half of which is 1-5708, and half of 1 = -5; then 1'5708 x'5 = 7854, the area. Having the area of a circle given, togind the diameter. RULE. As 355 is to 452, so is the area to the square of the diameter. Or, multiply the square root of the area by 1'12837, and the product will be the diameter. Or, divide the area by the decimal'7854, and extract the square root. EXAMPLE. Required the diameter of that circle whose area is 122-71875. 122'71875 x 452 ~i7351 452-= 12-5, diameter. 355 Or, /122-71875 = 11-077; and 11-077 x 1-12837 = 12'49895, or| 12a5, diameter. PROBLEM XI. To find the area of a sector of a circle. RULE. Multiply the length of the are by the radius of the circle, and half the product will be the area. I B ZBEXAMPLE. Required the area of a sector of a circle whose arc A B C = 26'666, and raC ~ A ~~dius B O = 16'9. 26-666 x 16,9 = 225-3277, the area. 2 PROBLEM XIL To find the area of a segment of a circle. RULE. Multiply the versed sine by the decimal'626, to the square of the product add the square of half the chord; multiply twice the square root of the sum by * of the versed sine, and the product will be the area. AILNSUr. ATNO..5 EXAMPLE. Required the area'of a segment of a circle whose chord AB = 48, and versed sine CD = 18. 18 x'626 =11.2682= — 126'967824; which D add to 576, being the square of half the chord = 702967824, twice the square root of which is 53'026 X 12; being i of the | versed sine = 636'312, the area. The following is a near approximate to the preceding rule: To the cube of the versed sine, divided by twice the length of the chord, add j of the product of the chord, multiplied by the versed sine; and the sum will be the area of the segment nearly. Take the last example: 188 Versed sine = 18, and chord 48, then, 8 60-7; and 48x2 48 x 18 x 2 = 576 + 60'7 = 636'7, the area nearly. Or, the area of a segment may be found by finding the area of a sector having the same radius as the segment; then deducting the area of the triangle, leaves the area of the segment. PROBLEM XTII. To find the area of a circular ring or space included between two concentric circles. RULE. Add the inside and outside diameters together, multiply the sum by their difference, and by'7854, and the product will be the area. EXAMPLE. The diameters of two concentric circles, AB and CD, are 10 and 6; required the area of the ring or space contained between them. 10 + 6 x 4 x'7854 = 50'2656, the area. PROBLEM XIV. To find the area of an ellipsis. RULE. Multiply the transverse or longer diameter by the conjugate or shorter diameter, and by'7854, and D the product will be the area. EXAMPLE. Required the area of an ellip-:B sis whose longer diameter A B = 12, and shorter diameter C D - 9. 13 x 9 x'7854 = 84'8232, the area. C Note.-If half the sum of the two diameters be multiplied by 3'1416, the product will be the circumference of the ellipsis. 3'1416 x 21 Thus 12 + 9 = 21, and 2:= 361384, the circumference. 36 IMENSURATION. Mensuration; of Solids. By solids are meant all bodies, whether solid, -fluid, or bounded space, that can be comprehended within length, breadth, and thickness. PROBLEMf I. To find the convex surface and solid coistent of a cylinder. RULE 1. Multiplv the circumference of the base by the height of the cylinder, and the product is the convex surface.:-RULE 2.' Multiply the area of the base by the height of the cylindbr,~anhd the product is the solid content. EXAfMPLE 1. Required the convex surface of the cylinder A B C D, whose base A B = 32 inches, anid perpendicular height B C = 6 feet. C l)fe- 38q416 x 32 x 72 inches = 7-238'2464 square or su-r perficial inches, and 7238-2464 -- 144 = 50'2658 superficial- feet. L.J I EXAPILE 2. Rcquired the solid content, in cubic inches and cubic feet, of the cylinder as above. 322 x'7854 x 72 =.57905'9712 cubic inches, and 57905-9712 -. -172'8 - 33'5104 cubic feet. EXAMPLE 8. Suppose the cylinder A BC D be intended to contain F a fluid, and that the sides and bottom are each one inch in thick-iless, how manyj imperial gallons would it contain? 32- 2 = 30 inches diameter; and 72- 1= 71 inches deep; 302 x'7854 X 71 then 277274 181 gallons. Or, 50187'06 X 003607 = 181, as before. PROBLEM IL To determine the dimensions of any cylindrical vessel, whereby to -contain the greatest cubical contents, bounded by the least superficial surface. RULE. Multiply the given cubical contents by 2-56, and the cube root of~ the product equal the diameter, and half the diameter equal the depth. EXArMPLE. Suppose a cylindrical vessel is to be made so as to contain 600 cubic feet, and of such dimensions as to require the *least possible materials by which it is constructed, what must be its depth and diameterl? 600 X 2-56 = V/1536 = 11-5379 feet diameter, and 11'5379 -2 = 5-76895 feet in-depth.'Note.-If the vessel is to be constructed with two ends, then the cube root of four times the solidity divided by 3'1416 equal both the lengthatiid diameter, so yS toexpose the least possible surface, or be composed of the least possible materials, of which to be X constructed. M5.Z kS U it 1;1N. I S PROBLEM IIL. To find the surface and solid content of a cone or pyramid. RULE 1. Multiply the circumference of the base by the slant height, and half the product will be the slant surface; to which add the area of the base, and the product will be the whole surface. RULE 2. Multiply the area of the base by the perpendicular height, and ~ of the product will be the solid content. EXAMPLE 1. Required the convex surface of a D cone whose base A B = 20 inches, and slant height B D = 29'5. 3'1416 x 20 x 292 = 926'772 square inches, and divided by 144 = 6-435 superficial feet. EXAMPLE 2. Required tile solidity of the coneB as above, the perpendicular C D being 28 inches. 202 x'7854 _ = 2932-16 cubic inches. and divided by 1728 = 1'697 cubic feet. PROBLEM IV.! To find the surface of the frustum of a copne or pyramzid. RULE. Multiply the sum of the perimeters of the two ends by the slant height, and half the product will be the slant surface; to which add the areas of the two ends, and the product will be the t whole surface. EXAMPLE. Required the convex surface of the frustum of a cone ABCD, whose base C,<-: -D AB = 20 inches, the slant height BC = 19, t and top end CD = 1 1. 31416 x 20 + 8-1416 x 11 x 19 = 925'2012 square inches, and divided by 144 B - a = 6'425 feet nearly. - PROBLEM V. To find the solid content of thelfresthm of a cone. RULE. To the product of the diameters of the two ends add the sum of their squares; multiply this sum by the perpendicular height and by'2618; the product is the solid content. EXAMPLE 1. Required the solid content of the frustum in Problem IV., whose perpendicular E F = 18 inches. 20 x 11 = 220, and.220 + 202 + 112 x 18 x'2618 = 3491-8884 cubic inches, and divided by 1728' ='2208 cubic feet nearly. 4 38 MENSURATION. EXAMPLE 2. Required the content, in imperial gallons, of the inverted frustum of a cone A B C D, whose inner dimensions are 34[ feet deep, 18 inches diameter at bottom, and 22 inches diameter at top. C- 22 x 18 = 396, and 396 + 222 + 182 X 42 13238'7024 x'2618 = 277 27 = 47'745 galls. nearly. Or, 13238'7024 x 0-00360654 47-75 galIons nearly, as before. PROBLEM VI. To find the solid content, of the frustum of a pyramid. RULE. To the sum of the areas of the two ends add the square root of their product; multiply this sum by the C perpendicular height, and i of the product is the solid content. \/ } \ EXAMPLE. Required the solid content of the frustum of a pyramid A B C D, whose perpendicular height = 24 inches, the area of the base \ = 144 inches, and area of the top end = 64. -i i \ 144 + 64 = 208, and 4/144 x 64 = 96; then 208 + 96 x 24 2432 cubic inches, and - 1728 A1~ —~a 3 = 1-4074 cubic feet nearly. PROBLEM VIL To find the solidity of a wedge. RULE. To the length of the wedge add twice the length of the base; multiply that sum by the height, and by the breadth of the base, and one-sixth of the product will be the solidity. EXAMPLE. Required the content in cubic inches of the wedge AB CDE, whose base ABC = 12 inches long and 4 inches broad, the I: \\\length of the edge D E = 10 inches, and perpenA dicular height r E = 20 inches. 2'r 10+24 x 20 x 4 C iB -6 = 452-33 cubic inches. PROBLEM VIII. To find the convex surface and solid content of a sphere or globe. RULE 1. Multiply the square of the diameter by 3'1416; the product will be the convex superficies. MENSURATION. 39 RULE 2. Multiply the cube of the diameter by'5236, and the product is the solid content. EXAMPLE 1. Required the convex surface of a sphere, whose diameter AB = 25J - inches. 25-52 x 3'1416 = 2042'8254 square inches, B A -144 -= 14-1862 square or superficial feet. EXAMPLE 2. Required the solid content of a sphere whose diameter A B = 251 inches. 25'5Sx'5236 = 8682-00795 cubic inches; -- 1728 - 50243 cubic feet. PROBLEM IX. Tofind the convex surface and solid content of the segment of a sphere. RULE 1. Multiply the height of the segment by the whole circumference of the sphere, and the product is the curved surface. RULE 2. Add the square of the height to three times the square of the radius of the base; multiply that: sum by.the height, and by'5236, and the product is the solid content. EXAMPLE 1. The diameter A B of the sphere ----- -— A A B C D = 20 inches; what is the convex surface of that segment of it whose height E D = 8 inches? 3-1416 x 20 x 8 = 502'656 square inches; - 144 = 3'49 superficial feet. EXAMPLE 2. The base F G of the segment F D G = 18 inches, and perpendicular ED = 8; what is the solid content? 82 = 64, and 92 x 3 = 243; then 243+64 x 8 x'5236 = 1285-9616 cubic inches, - 1728 ='7441 cubic feet. EXAMPLE 3. Sappose A B C D to be a sugar-pan, and that the diameter of the'mouth A B is 4 feet, the depth D C being 25 inches, how many imperial gallons will it contain? 25- 625, and 242 x 3 = 1728; then B AC a 80800'77 1728 + 625 x 25 x'5236 = 2787' 2 - 277-274 111-084 gallons. PROBLEM X. To find the solidity of a spheroid. RULE. Multiply the square of the revolving axis by the fixed axis, and by'5236, and the product will be the solidity. 40 MENSURATION. EXAMPLE 1. Required the solid content of the -i. < prolate spheroid A B C D, whose fixed axis A C is C A 50, and revolving axis B D 30. 302 x 50 x'5236 23562, the solidity. EXAMPLE 2. What is the solid content of an oblate spheroid, the fixed axis being 30, and revolving axis 50? 502 x 30 x'5236 =39270, the solid content. PROBLEM XL To find the solidity of the segment of a spheroid vwhen the base is circular or parallel to the revolving axis. RULE. From triple the fixed axis take double the height of the segment; multiply the difference by the square of the height, and by'5236; then say, as the square of the fixed axis is to the square of the revolving axis, so is the former product to the solidity. EXAMPLE 1. Required the solid content of the segment AB C, whose height B r is 10; the revolving axis E F being 40, and fixed axis B D 25. lB 25 x 3- 10 x 2 = 55, and 55 x 10 x c.. A'5236 = 2879'8. Then, as 252: 402 2879'8 ~7372'3 nearly. EXAMPLE 2. What is the solid content of the segment of a spheroid I whose height = 20 inches, the revolving axis being 25, and fixed axis 50? 50 x 3- 20 x 2 = 110, and 110 x 202 x'5236 23038'4; then, as 502: 252:: 230384: 5759'6 inches, the solid content. PROBLEM XII. To find the convex surface and solid content of a rylindric ring. RULE i. Multiply the thickness of the ring added to the inner diameter by the thickness and by 9-8698, and the product will be the convex surface. RULE 2. To the thickness of the ring add the inner diameter; multiply that sum by the square of the thickness and by 2-4674, and the product will be the solid content. EXAMPLE 1. The thickness of a cylindric ring A C or D B = 2 inches, and inner diameR 4D C A ter = 18, required the convex superficies. 18 + 2 x 2 x 9'8698 = 394-792 square inches, and 144 = 2741 superficial feet nearly. MENSURATION. 41 EXAMPLE 2. Required the solid content of the ring as above. 18 + 2 x 22x 2'4674 = 197 392 cubic inches, and -- 1728 ='114 cubic feet. Note.-A cubic foot is equal to 1728 cubic inches, or 2200 cylindrical inches, or 33t0 spherical inches, or 6600 conical inches. Also, the cubic foot being considered unity, or 1, A cylinder I foot in diameter and 1 foot In length.............................. = 7854 A sphere 1 foot in diameter................................................... ='5236 And a cone 1 foot in diameter at the base and 1 foot in height.................. ='2619 Decimal Approximations, FOR FACILITATING CALCULATIONS IN MENSURATION. Lineal feet multiplied by'00019 = miles. " yards, "' 000568 = " Square inches, "' 007 = square feet. " yards, "' 0002067 = acres. Circular inches, " -.00546 = square feet. Cylindrical inches, "' 0004546 = cubic feet. 4" feet, " 02909 = cubic yards. Cubic inches, "'00058- = cubic feet. " feet, " 03704 = cubic yards. ". " 6'232 = imperial gallons. " inches, "' 003607 =.. Cylindrical feet, " 4-895 = " " It inches, "'002832 =.. Cubic inches, "' 263 = lbs. avs. of cast iron. " at ". 281 = " wrought do. " " "'283 = " steel. i" " " *3225 = " copper.'" ". 3037 = " brass. t " " -"26 = " zinc.." " " *4103 " lead. it " " "' 2636 =' tin. t it it" *4908 = " mercury. Cylindrical inches, "'2065 = " cast iron..... 2168 = " wrought iron. 44 " "' 2223 = " steel...." *2533 - " copper. " "'2385 " brass. cc" "' 2042 = " zinc. ad " "'3223 = " lead. s.." *207 = " tin. s " " 3854 = a mercury. Avoirdupois lbs., "' 009 = cwts. 64 " "'00045 = tons. P' INSTRUMENTAL AtHMETIC. INSTRUMENTAL ARITHMETIC; OR, UTILITY OF THE SLIDE RULE. The slide rule is an instrument by which the greater portion of operations in arithmetic and mensuration may be advantageously performed, provided the lines of division and gauge points be made properly correct, and their. several values familiarly understood. The lines of division are distinguished by the letters A B CD, A B and C being each divided alike, and containing what is termed a double radius, or double series of logarithmic numbers, each series being supposed to be divided into 1000 -eqas parts- and distributed along the radius in the following manner: From 1 to 2 contains 301 of those parts, being the log. of 2. 3 " 477 3. " 4 " 602 4. 5 699 a. " 6 " 778 " 7 " 845 7. 8 " 903 S. 8 9 954 9. low0 being the whole number. The line D, on the improved rules, consists of only a single radius; and although of larger radius, the logarithmic series is the same, and disposed of along the line in a similar proportion, forming exactly a line of square roots to the numbers on the lines B C. Numeration. Numeration teaches us to estimate or properly value the numbers and divisions on the rule in an arithmetical form. Their values are all entirely governed by the value set upon the first figure, and, being decimally reckoned, advance tenfold from the commencement to the termination of each radius: thus, suppose 1 at the joint be one, the 1 in the middle of the rule is ten, and 1 at the end one hundred. Again, suppose 1 at the joint ten, 1 in the middle is 100, and 1 or 10 at the end is 1000, &c., the intermediate divisions on which complete the whole system of its notation. To Multiply Numbers by the Rule. Set 1 on B opposite to the multiplier on A; and against the number to be multiplied on B is the product on A. Multiply 6 by 4. Set I on B to 4 on A: and against 6 on B is 24 on A. The slide thus set, against 7 on B is 28 on A 8 "32 9 " 36 " 10 " 40 11 44 " 12 " 48 15 60 2I ", WO, &c., &c. IN'STRUM;NTA L ARmTHM1OETIC. ~3 To divide Numnibers upon the Rtue. Set the divisor on B to 1 on A, and against the number to be divided on B is the quotient on A. Divide 63 by 3. Set 3 on B to 1 on A, and against 63 on B is 21 on- A. 1Proportion, or Ruile of Three Direct..Rule. Set the first term on B to the second on A, and against the third upon B is the fourth upon A. 1. If 4 yards of cloth cost 38 shillings, what will 30 yards cost at -the same rate - Set 4 on B to 38 on A, and against 30 on B is 285 shillings on A. 2. Suppose I pay 31s. 6d. for 3 cwt. of iron, at what rate is that per ton? I ton - 20 cwt. Set 3 upon B to 31'5 upon A, and against 20 upon B is 210 upon A. Rlule of- Tlhree Inverse. Retle. Invert the slide, and the operation is the same as direct proportion. 1. I know that six men are capable of performing a certain given portion of work in eight days, but I want the same performned in three: how many men must there be employed? Set 6 upon C to 8 upon A, and against 3 upon C is 16 upon A. 2. The lever of a safety valve is 20 inches in length, and 5 inches between the fixed end and centre of the valve: what weight mustl there be placed on the end of the lever to equipoise a force or pressure of 40 lbs. tending to raise the valve..2 Set 5 upon C to 40 upon A, and against 20. on C is 10 on A. 3. If 8Q yards of cloth, 1i yards in width, be a sufficient quantity, how much will be required of that which is only Iths in width, to effect the same purpose? Set 1'5 on C to 8'75 on A, and against 8'75 upon C is 15 yards upon A. Square and Cube Roots of Nurmbers. On the engineer's rule, when the lines C and D are equal at both ends, C is a:table of squares, and D a table of roots, asSquares, 1 4 9 16 25 36 49 64 81 on C. Roots, 1 2 3 4 5 6 7 8 9 on D. To find the geometrical mean proportion between two numbers. Set one of the numbers upon C to the same number upon D, and against the- other number upon C is the mean number or side of an equal square upon D. Required the mean proportion between 20 and 45. Set 20 upon:C to 20 upon D, and against 45 upon C is 30- on D. 44 INSTRUMEiTAL AmtTntEgric. To cube, any nu-mber, set the number upon C to 1 or- 10 upon D, and against the same number upon D is the cube number upon C. Required the cube of 4. Set 4 upon C to I or 10 upon D, and against 4 upon D is 64 upon C. To extract the cube-root of any number, invert the slide, and set the number upon B to 1 or 10 upon D, and where two numbers of equal value coincide, on the lines B D, is the, root of the given number. Required the cube root of 64. Set 64 upon B to I or 10 upon D, and against 4 upon B is 64 upon D, or root of the given number. On the common rule, when 1 in the middle of the line C is set opposite to 10 on D, then C is a table of squares, and D a table of roots. To cube any number by this rule, set the number upon C to 10 upon D, and against the same number upon D is the cube upon C. MITensuratiou. of Surface. 1. Squares,.Rectangles, &c. Rule. When the length is given in feet, and the breadth in incihes, set: the breadth on B to 12 on A; and against the length on A is the content in square feet on B. If the dimensions are all inches, set the breadth on B'to 144 upon A; and against the length upon A is the number of square feet on B. Required the content of a- board 15 inches broad and 14 feet long. Set 15 upon B to 12 upon A; and against 14 upon A is 17'5 square feet on B. 2. Circles, Polygons. &c. Rule. Set'7854 upon C to 1 or 10 upon D then will the lines C and D be a table of areas and diameters. Areas, 3'14 706 12'56 19'63 2827 38'48 50'26 63'61 upon. Diameters, 2 3 4 5 6 7 8 9 upon D. In the common rule, set'7854 on C to 10 on D; then C is a line or table of areas, and D of diameters, as before. Set 7 upon B1 to 22 upon A; then B and A form or become a table of diameters and circumferences of circles. Circumferences, 3'14 6'28 9'42 12'56 15'7 18'85 22 25'13 28'27 upon A. Diameters, 1 2 3 4 5 6 7 8 9 upon B. Polygons fromz 3 to 12 sides. Set the gauge-point upon C to 1 or 10 upon D; and against the length of one side upon D is the area upon C. Sides, 3 5 6 7 8 9 10 11 12. Gauge-points,'433 1'7 2'6 3'63 4'82 6'18 7'69 9'37 1117. Required the area of an equilateral triangle, each side 12 inches in length. Set'433 upon C to 1 upon D;'and against 12 6upon D are 62'5 square inches upon C. TllSTRUMEiNTAL- A M-ETI. 45 TA;BLE -OF G&AUGE-POINTS FO'I- THE'. ENGINEER'S RULE., Names. FF P. F F 1. Il F, 1 | F. 1. Cubic inches,.. 578 83 1728 106 l273 10 121 Cubic feet,... 1 144 1 1833 22 121 33 Imperial gallons,. 163 8 231 277 294 353 306 529 Water in lbs.,. 16 23 276 293 352 305 528 Gold ".. 814 1175.141 149 178 155 269 Silver ".. 15 216 261 276 334 286' 5 Mercury..! 118 169 203 216 258 225 389 Brass ".. 193 177 333 354 424 369 637 Copper ".. 18 26:319 331 397 345 596 Lead ".. 141 203 243 258 21 27 465 Wrio't iron " ~. ~. 207 297- 357 338 453. 394 682 Cast ".. 222 32 384 407 489 424 7-33 Tin.. 219. 315- 37,8 401 481 ]i419 728 Steel ".. 202 292 352 372 448 385 671 Coal.. 127 183 22 33 28 242 42 Marble ".. 591 85 102 116 13 11-3 195 reestone ".. 632 915 11 1162 14 141 21 FOR THE COMMON SLIDE RULE. Names. F, F, F| F, I, I. 111,I. F,1. 1, 1. F. I. Cubic inches,.. 36 518 624 660- 799 62-5 113:Cubic feet;.. 625 9 108 114 138 119 206 Water in lbs;.. 10 144 174 184 22 191 329 Gold ".. 5071'735 88 96 118/ 939 180 Silver ". 938- 136 157 178 208 173 354 Mercury ".. 738 122 127 1382 162 141 242 Brass.-. 12:174: 207. 221 265 23 397 Copper ".. 112 163 - 196 207 247 214 371 Lead " 880-1 126 1.52 162 194, 169 / 289 Wro't iron ".. 129 186 222 235 283 247 423 Cast' 1 39 2 2411 254 304 265 458 Tin 1 1387:1 i135": 235': 25: 300 201 454 Steel ", 136 183 22 233 278. 239 418 Coal. 95 -114 138 146 176 151- l: 262 Marble ". [. 370 53 637 725: 81; 72 121' Freestone ".. 394 57 69 728 873- 7-55 1132 46 TINSTRUMENTAL ARITnMETIC. Mensurationa of Solidity and Capacity. Genleral Rule. Set the length upon B to the gauge-point upon A; and against the side of the square, or diameter on D, are the cubic contents, or weight in lbs. on C. 1. Required the cubic contents of a tree 30 feet in length, and 10 inches quarter girt. Set 20 upon B to 144 (the gauge-point) upon A; and against 10 upon D is 20'75 feet upon C. 2. In a cylinder 9 inches in length and 7 inches diameter, how many cubic inches? Set 9 upon B to 1273 (the gauge-point) upon A; and against 7 on D is 346 inches on C. 3. What is the weight of a bar of cast iron 3 inches square, and 6 feet long? Set 6 upon B to 32 (the gauge-point) upon A; and against 3 upon D is 168 lbs, upon: C. By the common rule. 4. Required the weight of a cylinder of wrought iron 10 inches long, and 51 diameter. Set 10 upon B to 23 (the gauge-point) upon A; and against 5Y2 upon D is 66'65 lbs. on C. 5. What is the weight of a dry rope 25 yards long, and 4 inches circumference? Set 23 upon B to 47 (the gauge-point) upon A; and against 4 on D is 53'16 lbs. on C. 6. What is the weight of a short linked chain 30 yards in length, and 6iths of an inch in diameter? Set 30 upon B to 52 (the gauge-point) upon A; and against 6 on D is 129'5 lbs. on C. Land Surveying. If the dimensions taken are in chains, the gauge-point is 1 or 10; if in perches, 160; and if in yards, 4840. Rule. Set the length upon B to the gauge-point on A; and against the breadth upon A is the content in acres upon B. 1. Required the number of acres or contents of a field 20 chains 50 links in length, and 4 chains 40 links in breadth. Set 20'5 on B to I on A; and against 4'4 on A is 9 acres on B. 2. In a piece of ground 440 yards long, and 44 broad, how many acres? Set 440 upon B to 4840 on A; and against 44 on A is 4 acres on B. Power of Steam-Engines. | Condensing Engines -Rule. Set 3-5 on C to 10 on D; then D is a line of diameters for cylinders, and C the corresponding number of horse power; thus, Horse power,. 3S 4 5 6 8 10 12 16 2025 30 40 50 on C. C. D.. 10 in. 10V 12 13Y1 153 17 18M 21M 24 26% 29% 33% 37% on D. THE LAwS oF' MOTIoN. 4T The same is effected on the common rule by setting 5 on C to 12 on D. Non-condensing Engines.-Rule. Set the pressure of steam in lbs.:per square inch on B to 4 upon -A; and against the cylinder's diameter on D is the number of horse power upon C. Required the power of an —engine, when the cylinder is 20 inches diameter, and steam 30 lbs. per square inch. Set 30 on B to 4 on A; and against 20 on D is 30 horse power on C. The same is effected on the common rule by setting the force of the steam on B to 250 on A. Of Engine Boilers. How many superficial feet are contained in a boiler 23 feet in length and 5i in width? Set 1 upon B to 23 upon A; and against 5'5 upon B is 126'5 square feet upon A. If 5 square feet of boiler surface be sufficient for each horse power, how many horse power of engine is the boiler equal to? Set 5 upon B to 126'5 upon A; and against 1 upon B is 25'5 upon A. The Laws of Iotion. If M - mass of a material body, And W= the weight of it..'. W- M x 32'19; Or the mass of a body is equal to its weight divided by 32'19. EXAMPLE. Find the weight of a body whose mass is 3~:.. = 35 x 3219,= 112'66 lbs. The gravity of a material body is its weight. Falling bodies fall through the same space in the same time, whatever may be their weight. A body one ton will fall to the ground no faster than a body one pound. The velocity of a body is the number of feet passed over in one second. Put v = the velocity of a falling body, at the end of t seconds,.. v =- 32'19 x t. The quantity 32'19 is the velocity of a falling body at the end of one second. Rule, tofind the Velocity of a Falling Body at the end of any Number of Seconds. Multiply the number of seconds by 32"19, the product will be the velocity. *8. TaE. LAws. oF MoTIONs EXAMPLE. Find the velocity of a body falling from a height in nine seconds: Velocity =32'19 x 9 =-289:71. Put s for the number of feet a falling body falls through in t seconds: 32'19 t2 2 Rule: to find te Space passed over by,a Falling Body in any Number of Seconds. Square the number of seconds, and multiply the result by 16'09, the product will be the distance passed over-inufeet. EXAMPLE. A stone fell from the top of a- chimney to the bottom in four seconds; find the height of the chimney: Height of chimney = 1609 x 16 257'44 feet. s 6439,.where v is the velocity. Rule to find the Space passed over by a Falling Body when the Velocity is given. Square the velocity, and divide by 64'39.; the quotient will be the number of feet passed over. The quantity 32'19 is frequently called.the accelerating force of gravity, and is denoted byf. The following formulae include all cases that can occur in falling bodies. s = space passed over = — - r2 _t2v 2. v = velocity at the end of (t)'seconds -ft t 2fs; t= time = v- -; f 2 / -v- vs' 2s The above formulae and rules are applicable only to:the ease when the body is acted upon by the force of gravity. Rules aed Formulce?when a body is acted on by any:force. Put M-= mass acted on by a force of F pounds..a.,velocity at;the end of. a second, which is called accelerating -force..:-= space passed. over.in (t) seconds,. producing a velocity (v). THE;LAws oF MoTIov, 49 p' Anid 2 s = _ Rule for finding the~ accelerating force of a body. Divide the force by the mass (remembering that mass is equal to weight divided by 32'19) or the velocity, by the time, either quotient will give the accelerating-force.:EXA-MPLE. A force of 25 lbs. acts on a body whose weiglit is 84 lbs. Find the accelerating force. 84 The mass- = 2q19- = 2'6 nearly; 25 |.'-. a- 2....-= 9'62 nearly. The velocity at the end of 10 seconds = 9'62 x 10 = 96'2. Time of a Body falling down an Ilnclined Plane. Let A B C be an inclined plane, B C perpendicular, and A B'parallel to the horizon. The velocity at A in falling down A C is the:same as it wiuld'be in falling perpendicularly down the heighlt B C. Put t = time in falling from C to A. I = A C the length of the inclined plane. h = B C the height of ditto. Let A D EB be a circle whose diameter A B is perpendicular to the horizon. The times of a body falling down aniy chords A D, A E are equal, and equal to the time in falling vertically through A B. _ - Th7e Time of Oscillation of a Simple Penduluzm. Let A B the length of the pendulum = I, And 7r = 3-14159, &c.; g = 32919 T = time in seconds oscillating from the point B toD. The are B C = CD is small. 6CF q~r~~TmaiA'w-ss OIF' MOTI&N; Rule to find the Time of one Oscillation of a Simple Pendulum. Divide tile length of the pendulum by 3219; extract the square root of this quotient, and multiply the result by 3'1416, and the product wvill be the time of oscillation in seconds. If L be the length: of -a penduluni whichi oscillates in one second,,Thie value of L, for the latitude of:London: is-, 39'1386: inches. A pendulum 9"-, 47,, 2572 inches long, wil-l oscillate, in a, li-lf - thllird, a quarter seconds respectively. If n be the iu:fiuber of oscillations mi-ade- by a- pendulum in one hlour, then L I = 36.00 x The time'of oscillation is not dependent on the weight of the bob. C'eutrifuqal Force. Let the weightW, placed at B, be connected with a cord, or wire, with,_ the fixedcl point A - round which: it revolves;with a: unifomnz ve- - locity. Put V = velocity of rotation. r = A B, the length of the cord in feet. F =centrifugal force, or the: forcea which1 is exerted to break:the cord in the direction of its-lengtli 32'19 x rIf it be the number of revolutions in one mlinte, 331.'.F= - x Wr t2. 1000000 If' t1 be measured, in tons, then F will be in tons also. If to be the angular: v-elocity, If T be the time of the wveight making ari complete' revolution,.. v —:-anguflar velocity T- -- If there be several bodies at B, C, D, and Iqvolving round the axis passing through A, and perpendicular to the plane, A-D B C, Tiu: LANWSS OF NOT OWN. 51 Where,- = angular velocity, VW3; 1'2; W3, &c.: the weights at B C D, &c., and r', r2, r3, &c, the distances A B, A C,,D,.&c. EXAMPLE. Let the weiglhts at B and C be 80 and 90 lbs. respectively, revolving at a distance A B = 8 feet, A C= 12 feet, with a velocity making 40 revolutions per minute. Find the centrifugal force, or the pressure on the axis passing through A. 27r x -40 4r w 60 3 F..= — 8 x 80 + 12 x 90 -30178 lbs. The moment of inertia. If.( W1 + W + W3 + &c.) k2 =:I rl + W1'-22 + W3 r32 + &c. Each side of this equation is called the monment of inertia, and the distance k is called the radius of gyration of the revolving system. Let a constant force P' act at a distance A f= a from the axis of motion. The angular velocity at the end of a second - gey E'a (Wl + 1V1 + W':1 + &c.) k2' The angular velocity at the end of one revolution'2 4/1P'ar 4/1'V, + W., + 1'/3 + &c. x k If a point 0 be determinled fiom tthe equation A O= -- A 6U;' where G is the centre of gravity;of the system, then 0 is called the centre of oscillation. lThe values of k in. Geometrical Solids. A rectangular parallelopipedon revolving about an axis passing through its centre of gravity, and parallel to either of its edges. hb + c~ 12 where 1 c are the length and breadth at right angles to the axis of revolution. An uprighttriangular prism about a vertical axis passing through its centre of gravity. =2 42 - C 48 36 52 THE:LAWS OF I'MOTION, The section of the prism perpendicular to the revolving axis is an isosceles triangle, the base being denoted by (a), and the perpendicular upon it from the angle contained by the equal sides by (c). In a cylinder,; whose radius is (r), revolving about its axis, k'2 Ini a hollow cylinder, whose internal and external radii are a and b respectively, revolving about its axis, k. a2 + b2 2 In a cylinder;, whose radius is r and length 1, revolving round a line at right angles to its axis,' and passing through its middle, 2 j2 22 12 4 In a sphere, whose radius is r, revolving about its diameter, 2 rIn a hollow'spllere, whose internal and external radii are (a) and (b) respectively, revolving about its diameter, k- _ 2 (b' — a') 5(6V- a2) In a cone, whose base is a circle, radius r, 3 r2 10 In a cone, whose radius of base is r and height h, revolving about a lilei at right angles to its axis, and passing throughl its centre of glavity, k2 = S (4 r2 + 2) 80 The square of the radius of gyration about any line in a revolving system, is equal to the square of the radius of gyration about a I line parallel to it passilng thlroughl tile centire of gravity and the square of the distance fiom the centre of gravity to the line about lwhich the system revolves. TuIE CEINTRE 0F' PIEIilCUISS1ON'. 53 *Let CG be;tihe centre of,gravity iof,any ibod-y;: draw A B any line a-bout -wlthich: the system revolves. Let C) "be parallel to A:B,'-and draw G H perpendicular to A B. Let ~ - radius of gyration'when revolving > aboutA B.'.k —radius of gyration when revolving' about C,D. ~ =2 k= +GH2'.,This important: fieorem will readily extend fle theorems hllich are given above to.most lpractical cases. The Centre of -.yratieni is that part of a body revolving aboutt an axis, into w.hich, if the q1whole qruantity of mattler er-e collected, the saime moviny force would genr.eate the samte angular velocity. Tobfind the centre of Gyration, multiply the weight of the se-veral particles by the;sqcuares of their:distanees from the:centre of motion, and divide the sum of the products by the weight of the whole mass; the square root' of the quotient will be the distance of the centre of gyration, from the centre of motion. The distances of the centre:of gyration firom tile centre of motion, of different revolving bodies, are as follows: Inl a straight rod re:evoling about one:end, Athe length:x'573. In a circular plate, revolving on its:ce:tre, thie radius x'7071. 1n;a ciircutlar plate, revol.ving:about one diameter, the radius.-x *5. In a thin circular ring, revolving about one diameter, radius x'7071. In a solid sphere, revolving about one diameter, the radius x 6325. In a thin hollow spheIe, revolving about one diameter, the radius x:8164. In a cone, revolving about its axis, the radius-of the base x.5477. In a riglit-angled cone, revdlving about its vertex, the height x 856. In a paraboloid, revolving about its axis, the radius -of the.base x'5773. Tlhe -Centre o.f.Percussio Jis that point in,a, body revolJvig about a fixedacl is, tLto::wichltt e wthuole of t/ev force or moti, s so.lected. It is, therefore, tfhait point of!a revolving body whliehl wtVild sti'i;ke any obstacle wit~h the -greatest -effect;- and, fro this pro-i perty, it has received the name of the centre of percussion. 5*] 54 ON WRORK. The centres of oscillation and percussion are in the same point. If a heavy straight bar, of uniform density, be suspended at one extremity, the distance of its centre of percussion is two-thirds of its length. In a long slender rod of a cylindrical or prismatic shape, the centre of percussion is nearly two-thirds of the length from the axis of suspension. In an isosceles triangle, suspended by its apex, the distance of' the centre of percussion is three-fourths of its altitude. In a line or i rod whose density varies as the distance from the point of suspension, also in a fly-wheel, and in wheels in general, the centre of{ percussion is distant from. the centre of suspension tlhree-fourths of the length. In a very slender cone or pyramid, vibrating about its apex, the distance of its centre of percussion is nearly four-fifths of its length. On Work. A unit of work is one pound avoirdupois raised vertically one foot. If U denotes the units of work in raising W lbs. A feet-. U Wh. Rule tofind the U'nits of Work itn Raising a given }Vcight a given Height..Multiply the height in feet by the weight in pounds, the product I will be the units of work done. ~ EXAMPLE. Find the units of work in raising half;a ton 30 feet high,.: U= 1120 x 30 33600 units of work. It is important to observe, in the application of the above formula to practical cases, that the height (ih) is the vertical distance tlirough which the centre of gravity of the body whose weight is ( W) is raised. ExaMPrLE. Find the units of work in lowering the surface of| water in a well one yard; the depth to the surface of water being 40, and diameter 3 feet. Theweight of a cubic foot of water is 62} lbs. The weight'of water= 9 x'7854 x 3 x 62'5=1325'36 lbs. The height through which the centre of gravity is raised = 415 feet..: U= 1325 36 x 41'5 = 55002 units of work. The work done in raising a body up an inclined plane, or any curved surface, is equal to the work done in raising the body vertically through the height of the inclined plane. There are 29000 units of work done in sawing a square foot of green oak. ON WOORK. 55 Hor.se Power. A horse power is 33000 units of work done in one minute. Put H, equal to the horse power, and U, the units of work done, in T hours:.. 3'3000' U. 60 T' The following results are taken from MboRIOIN A Man laboring Eight Hours per Day will perform the following Units of Work. Raising his own body;. 4250 Drawing, or pushing horizontally,. 3120 Pushing and drawing alternately in a vertical direction,. 2380 Turning a handle,... 2600 Working with his arms and legs, as in rowing,... 4000 A Man laboring Six Hours per Day. Raising material with a pulley,. 1560 Raising material with the hands,.. 1470 Raising material upon the back, and returning empty,.. 1126 A Man laboring Ten Hours per Day. Raising material with a wheelbarrow on ramps,..:.. 720 Throwing earth to the height of five feet,.... 470 Useful Work of a Man raising Water-Duration of Labor, Eight Hours per Day. With a windlass from deep wells,.. 2560 With an upright chain pump. 1730 With a Chinese wheel,.2167 With an Archimedean screw. 1505 Raising water from a well with a pail and rope,... 1054 Work of Animals. A horse, in a common pumping engine,. 17550 A mule, ditto,....11700 An ass, ditto,....3510 EXAMPLE. Required the horse power of an engine that will saw 368 planks, each being 30 feet by 2 feet 6 inches, in twelve hours. There are 29000 units of work done in sawing one square foot; Then 30 x 2'5 x 368 x 29000= —units of work done in sawing the planks. Put x = -the horse power of the engine; Then 60 x 12 x 33000 x x = units of work done by the engine In twelve hours. 30 x 2%5 x.368 x 29000 Hence, x = x 12 x 33000 33 horse power. 56,ON WORk. EXAMPLE. How many tons of coals would two men raise, working with a wheel and axle, from a pit whose depth is 20 yards, in 12 hours-? From the LTable, a man working with -a w-heel and axle,will do 2600 units of work in one minute. Then, 2600 x 60 x 12:X 2 = work done by the two men. Put x = the tons of coals raised. Then, 2240 x 20.x 3 x x = -work done by the two men. 2600:x 60 x 12 x 2.2240 x.. x 3. 27:85;tons;raised. 2240'.x,20 x.3 The Traction of Horses at various rates of Travelling. It i's a w'ell.known' fact,' that the traction or force -which a'horse can exert decreases with the increase of speed.'Rate in miles per.hour, 2 3.3 4 4, 5 Force exerted by the horse, 166 lbs. 125 104, 83, 62j, 41j. Accumulated'Work. If at force be applied to move a body subject to no resistance whatever, it.vwill'be wholly occupied in increasing the speed -of the body. In this case the work which is done by the action of the force applied is accumulated in the body, therefore it is called accuiulated work. Put VF the velocity of the body or feet per second. Alnd IF= -the veight.of -the.body in pounds. Accumulated work = W. 64 If WT be measured in tons, and V be measured in miles.-per hour, Accumulated work =. 4 —5 WF 45 A railway train 80 tons moves uniformly at the rate of 30 miles per hour, find the accumulated work. 3388 x 80 x 900 Accumulated \vork = — - 5420800.'45 420800 The horse power of the engine 5 00 164 nearly..38000 Generallyy the:horse power of the engine whereq W is in tons and VY in miles per hour. - 33750 The friction of a railway train is from 8 to 10 lbs. per ton.!Work doine by Machines. The moving power, which is applied to any machine -moving unliformly, is employed in overcoming the resistance of friction, and useful work done at the working points,of the machine. Hence, ON Wou;. 57 the aggregate number.of units of useful work yielded by any machine at its working point is less than the number received upol the machine directly from the moving power, by the number ofi units expended upon the resistance of friction. (The machine moving uniformly.) General Rule to fied the ICorkl done by any Machine. Find the distance through which the power (P) applied to the machine has travelled in one minute, and let this distance be called (a). Find the distance through which the weight (W), producing' useful work, has travelled in one minute. and let this distancee be (1). Then a P- b T — work done by friction per minute. And a P _ work: applied per minute. b W= useful work done per minute. The Horse Power of an Engine. Let P be the mean effective pressure of the steam on the piston. I be the length of the stroke in -feet. nb be the number of strokes per minute. IP.Horse power of the engine 33000 The nominal horse power = 00- as adopted by the Admiralty. On the Strength of Animals. Let P be the force in lbs. that any animal can exert when moving at (v) miles per hour. Put Kf= the greatest effort the animal can exert when standing. And c = the greatest number of miles per hour the animal can give itself when unimpeded by any weight. According to Bouguer, P = (1 — ). K Euler, P ( — = Euler, P =(! — K. It is readily seen that (v) miles per hour is equal to (88 v) feet per nminute. Put U the units of work done by the animal per mninute, Then, according to Bouguer, U = 88 (v -. K 58 TL.E CRANE...Accrding to Euler, U =88.(v — 4). 2 r" Euler, U= 88v(1 - v ) The values of U will be the greatest when v According:to Bouguer. C Eu ler..-4/ v = —. - " uler. Substitute these values in the formula for P and U, then there will result: K - the load of the animal -iwhen producing.,the greatest effect. 2K 3= " "..4K. yr. r 22 c K-= thegreatest effect,.by first frlmula. 17.6,cX:, P7.6 =K ".by second.formula..35 2 c-k "'= by -third formula. To Calcnlate the Different Parts of a Crane as respects lechanical Advantage. (1.) The numlber of revolutions of the pinion to one of the wheel, the length of the handle, and the force applied being given, to find the diameter of the barrel. RULE. Multiply the diameter of the circle described by the winch, or handle, in inches, by the power applied in,lbs, and by I the number. of:revolutions of the pinion to oneof -he.heeli.divide this product by the weight to be raised in lbs., and the quotient is the diameter of the barrel.in inches. EQLHUIBIURIA AND IRU.SSRE OT.1oF BEAMS 59 (2.) The:diam'zeter of the barrel, the length: of the aandle, andi the forree applied given, to find the number of revolutions of the pinio'n to one of the uwheel. RULE. Multiply the: weight. to; be raised in lbs. by the diameter of the barrel in inches, and divide the product by the diamneter' of the: circle described by the. handle in inches, multiplied by the po-wer applied in lbs., and the quotient:is. the revolutions of the pinion to one of the wheel. (3.) Thlie diameter of the barrel, the number of revolutions of the pinion to one of the wheel, axnd the power applied cqiven,, to find the length of the handles. RULE:. Multiply the weight to be raised in lbs. by the barrel's diameter in inches, and' divide the product by the power applied in lbs., multiplied by the number of revolutions of the pinion to one of the wheel, and: half the quotient is the length of the handles.(4.)- Thie diameter of the barrel, th'e: evolutions of th.e pinion to one of the' wheel, and length of,' handles- given, to find' the potwer requiredRULE. Multiply the weight; to be raised in lbs. by the diameter of the barrel in inches, and divide: the product by, theI diameter of the circle described by the handle multiplied by the revolutions of the pinion to one of the wheel, and the quotient is the power applied. The handles of a crane& should tot: be: less than 2 feet 11 inches or 3 feet from th.e: grou'nd,. and the jib to stanid at an angle of about 46 degrees.; Equnilibrinm and EPressure of lBeams' The Parallelogram of Forces. It has been proved by experiment that three forces, proportional to the tlwo sides of a parallelogrant and its diagonal, are ill a state of equilibrium when tleir directions; are in the direction of these lines. Let_ two forces, represented in' direc-:' tion and magnittde by the- lines A2 O and B O; act: at thee point 0,; then a. third force CO' in direction and magniu tude can be found; so that that the three forces, are in a state of equilibrium. 0 Draw A D, B D, parallel to 0 B, O A, respectively; join D 0, and produce it / to C, making C-0 equal to 0 D, then d 2 o a is the force required... —.. o' The two'forces A: 0, B O are called components, and C 0 the resultant of the (60 EQUELIBRIEUM ANI) PRESSURE OF' BEAuMS. components. The components and resultant are called the parallelogram of forces. p Any resultant force can be readily decomposed into two components, which will be the sides of a parallelogram whose diagonal is the resultant. Let the beams O E, 0 B sustain a weight )7P ((W) tons at the point 0; draw OD verti- cal, and make it equal to (W ) inches,then draw 1) A, 1) B parallel to O F and O E respectively; measure D A, D B in inches which will be the pressure in tons in the 20 S F directions OF and OE. In this case EFis a tie beam to prevent the lower ends of the beams OE, OF from spreading. Draw 0 D vertically equal to (W) - _D'P - inches, then draw DA, DB parallel to OF, O_ and aA, bB, parallel to EF, then AD will be the thrust in 0 F, and D B in O0E, and -Aa equal to bB will be the thrust in the direction of the tie beam E F. Draw O)D vertically equal to (Vr) p =//w D inches,. and draw D)A parallel to OF, and D B parallel to OF, then OBj OA A 1 //t ( )+ will represent the pressures in the directions OF, 0 E. c -~ Let A B be a beam whose centre of gravity 0 is 0,:and resting against an upright wall BE, the lower end resting on an abutment cut in the'beam A E at A. Through -the centre of gravity O draw X Z t the line CD vertically equal to thie weight of the beam, draw B 6C, D F parallel to EA, join CA; then CF represen:,s the' thrust at A in the direction CF and _FD represents the thrust at B, and also the horizontal thrust at -A. To Conmpute the Tension of the'guisc' ind Shear-leg of a pair of Shearcs. Let B C be the shear-leg:and A C the guise, and (IV) weight in tons supported at C. ,SPECLFC GRAVITY. 61 Make C as: many inches as(-V) contains tons, draw D E parallel to A C, then DlE/' mimeasured il in ches wlill bethe tension in tons of the/ D g1uise A C, and CE measured in inches will be the pressure / in the direction of the shear- / leg CE. To Compute the Tension on the guise arithmietically. Put AB c, B C= a, and A C= b. b (ba —'aa — CW) ] Then, tension in A C - _-_ ) W az(ba'-ca — a2) H;W And tlie pressure in C:B =- a(~aTi. ) - 2c (a+ b6-+c) a+ b -c)(b+ c -)(a+c-6-) SPECIFIC GRAVITY. THE comparative densit;y of various substances, expressed by the term SpecJfic Cravity, affords the means of readily determining the bulk from the know-n weight, or the weight fiom the lknown bulk; and this will be found more especially useful, in cases'where. the. substance is too large to admnit of being weighed, or too irregu!ar in shape.to allow'of correct measurement. The standard Wvit;h which. all solids and:liquids ate thus compared, is that of distilled water, one cubic foot of which i'eWigihs 1000 ounces avoirdupois; and the specific gravity of a'solid body is determined by the cdifference; between its weight in the air and in water. Thus, If the body be heavier than water, it will displace a quantity of luid equal to it in bulk, and will lose as much weight on immerbion as that of an equal bulk of the fluid. Let it be weighed first, therefore, in tle air, and then in water, and its weight in the air be divided by the difference between the two weights, and the quotient will be its specific gravity, that of water being unity. Exampl e. A piece of copper ore weighs 56+ ounces in tie air, and 43, ounceS in water: required its'specfif-e gravity. i 5625 —4875 - 1256 and 5625 + 12'5 45; the specific, gravity. If the body be lighter than water. it will float, and.displace a qhantity of fliid equal to it in weight, th'e bulk of which will be eqlal to that only of the part itmmeised,. A lhea vier' sulstance.... -,...... 62. SJpFacwx.IEIFIC (.A RIT-'. must therefore be attached to it, so that the two may sink in the fluid. Then, the weight of the lighter substance in. tle air must be added to that of the heavier substance in water, and the weight of both united, in water, be subtracted from the sumn; the weiglht of the lighter body in the air must then be divided by the difference, and the quotient will be the specific gravity of the lighter substance required. Example. A piece of fir -weighs 40 ounces in the air, and, being immersed in water attached to a piece of iron weighing 30 ounces, the two together are found to weigh 3-3 ounces in water, and the iron alone 25-8 ounces in the water: required the specific gravity of the wood. 40 + 25-8- 65.8 — 3-3 =62-5; and 40 -- 62-5 = 0'64, the specific gravity of the fir. The specific gravity of a fluid may be determined by taking a solid body, heavy enough to sink in the fluid, and of known specific gravity, and weighing it both in the air and in the fluid. The Idifference between the two weights must be multiplied by the specific gravity of the solid body, and the product divided by the weight of the solid in the air; the quotient will be the specific gravity of the fluid, that of water being unity. Example. Required the specific gravity of a given mixture of muriatic acid and water; a piece of glass, the specific gravity of which is 3, weighing 34 ounces when inimersed in it, and 6 ounces in the air. 6 - 3'75 = 2~25 x 3 = 6-75 -6 - 1'125,: the specific gravity. Since the weight of a cubic foot of distilled water, at the temperature of 60 degrees (Fahrenheit), has been ascertained to be 1000 avoirdupois ounces, it follows that the specific gravities of all bodies compared with it, may be made to express the weight, in ounces, of a cubic foot of each, by multiplying these specific gravities (compared with'that of water as unity) by 1000. Thus, that of water being 1, and that of silver, as compared with it, being 10'474, the multiplication of each by 1000 will give 1000 ounces for the cubic foot of water, and 10474 ounces for the cubic foot of silver. TABrE or SPECIrIC GRAVITIES —WATER- 1000. Metals. Mercury,. 138586 Crown glass,. 2'. 2488 Antimony,... 6712 Organic Bodies. Flnk t sstal. 8 Zinc,.....100 Oak wood.... 0925 Diamonds,. 3 Cast Iron,. 7207 Cork,.0 240 Tin,.... 7291 Ivory,. 1826 Liquids. Steel.. 7816 White wax,... 0960 Ether,..... 0715 Cast copper,... 8788 Alcohol,... 792 Bismuth,.. 9-882 Io~rgan1ic NBo-lJetaslic Oil of turpentine, 0'870 Silver,.. 10-474 Bodies. Sea water,. -C26 Lead,..11 852 Agate,..... 290 Milk,. 1030 Gold,...19258 Amber,.... 1018 Nitric acid,.. 1508 Platinum,..208837 Sulphur,.... 2-083 Sulphuric acid,. 1845 THE MECHANICAL' POWVER. 63 -Weights of given bulks of water and airfor calculating the absolute weights from the specific gravities of bodies. Cubic inch of distilled water (bar. 30, therm. 62) Logatithms. in grains......252-458 2-40219... foot............... in ounces avoir. 997-1369691 2'99875.......n................. iIn pounds do. 6;23210606 1-79463 Weight of 100 cubic in. of air in grains do. 30'49 1 148416 TI-E MECHANICAL POWERS, AND THEIR APPLICATION. TsIE simple Mechanical Powers are six in number, viz. the Lever, the Pulley, the W]heel and Axle, the Inclined Plane, the WeVdge, and the Screw. All machines are formed by combinations to a greater or less extent of these six elements. The mechanical effects, however, of the whole, are ultimately resolvable into tliat of the lever. By means of the Mechanical Powers a great weight may be sustained, or a great resistance slowly overcome, by the application of a small force. Or, a great velocity may be imparted to a small weight or resistance, by the use of a great force or power. The:Levero Levers are of three orders: In the first order, the fulcrum -is between the weight and the powv.er, In the second order, the weight is between the fulcrum and the power. In.:the third order,, the power is between, the weight and the fulcrum..The bent lever has no peculiarity except that of form, which is given to it for convenience in use. Its properties are those of the fi-rst order. In order to preserve an equilibrium between the power and the weight, they must bc to each other inversely as their distances from the fulcrum. Case 1. When the Lever is of thefirst order, or when the fulcrum is between the power and the weight. RUL,. Divide the weight to be raised by'the power to be applied; .l4'ITHIE MECHAANICAL POWERS. the quotient will give the difference of leverage necessary to support the weight in equilibrio. Hence, a small addition either.of le'ier-' age or weight will cause the power to preponderate. EXAMPLE 1. A ball weighing 3 tons is to be raised by 4 men, who can exert a force of 12 cwt.'Required the proportionate length of leve 60 60 3 tons 60-cwt.; and- 5. 12 In this example, the proportionate lengths of the lever to main- i tain the weight in equilibrio, are as 5 to 1. But, although the ball is sustained by a force of only one fifth of its weight, no power isj gained, for the weight passes through only one fifth of the space passed through by the power. EXAMPLE 2. A weight of 1 ton is to be raised with a lever 8 feet in length, by a man who can exert, for a short time, a force of rather more than 4 cwt. Required at what part of the lever the fulcruml:must.be placed. 20 cwt. 4 cwt. -5; i. e., the weight is to the power as 5 to 1; therefore, 4 cwt. 5x1 - 1 foot and a third fromn the weight. EXAIPLE 3. A Weight of 40 Ibs. is placed one foot from the filcrum of a lever. Required the power to raise the same when the length of the lever on the other side of the fulcrum is five feet. 40 x 1 8 x lbs., the power. Case 2. When the lever is of the second order, or when the fdlcrum is at one elnd of the lever and the power at the other, with the weight between them. RULE. As the distance -between the power and the fulcrum is to the distance between the weight and the fulcrum, so is the effeqt:to the power. EXAM.rPLE, 1. Required the power necessary to raise 120 lbs. when "the Weight is placed,six feet from the power, and two feet from the fulcrum. As 8: 2:: 120: 30 lbs., the power. EXAMPLE 2. A beamT 20 feet in length, and supported at both ends,: bears a weighlt of two tons at the distance of eight feet from'one end..Required the:weight:on eacl support. 40 cwt. x 8 feet 2.0w f 8 eet = 16 cwt. on the support that is furthest from the 40 x 12 weight; and ___ -24 cwt. on the support nearest to the weight 20 feet THEn MCHAMICAL POWERS 6. Case 3. When the'lever is of the thirdorder{ or the weight is at one end'of the lever, the fulcrum at the other, and the power is applied between them. RULE. As the distance between the power and the fulcrum is to the length of the lever, so is the weight to the power. EXAMPLE. The length of the lever being eight feet, and the weight at its extremity 60 lbs., required the power to be applied six feet from the fulcrum to raise it. As 6: 8:: 60: 80 lbs., Ans. The Pulley. Pulleys are of two kinds, fixed and movable. The fixed pulley affords no economy of powver, but merely changes its direction. The movable pulley changes its position with that of the weight, and effects a saving equal to half the power. An equilibrium is preserved between the power and weight, when' the weight is equal to thile product of the power and twice the number of movable pulleys. RULE. Divide the weight to be raised by twice the number of pulleys in the lower block; the quotient:will give the power necessary to raise the weight. EXAMPLE. Required the power to raise 600 lbs. when the lower block contains six pulleys. 600 6 2 50 lbs., the power. The Wheel and Axle. The wheel and axle act as a revolving lever; and in order to obtain an equilibrium between the power acting on the circumfrrence/of the wheel, and the weight or resistance acted on by the circumference of the axle, the power must be to the weight as tlhe i'adiuls of the axle'is to that of the wheel.. One or more radii of the wheel,' or winches, are often substituted for the. wheel in the simple! machine; and in compound machines the action is communicated by teeth or cogs, forming wheel-and-pinion work. RULE. As the radius of the wheel is to the radius of the axle, so is the effect to the power. EXAMPLE. A weight of 50 lbs. is exerted on the periphery of a wheel whose radius is 10 feet. Required the weight raised at, the extremity of a cord wound round the axle, the radius being 20 inches. 50 lbs. x 10 feet x 12 inches 20'inche' a — = 300 lbs., the weight. 20. inche's. gg66 THE MIECHANICAL POWNERS. I aaThe In1lined Plane.* The inclined plane acts as a mechanical power by sustaining a portion of the weight to be raised, while the direction of the -applied force is changed from the perpendicular to one more or less horizontal, and the weight moves upwards on it in a diagonal between them. Equilibrium is sustained wh.en the power is to the weight as the perpendicular height of. the inclined plane is to its inclined length or hypothenuse, when the power acts in a direction parallel to the inclination of the plane; but as the height ir to the base when in a direction parallel to the base. RULE. As the length of the plane is to its height, so is the weight to the power. EXAMPLE. Required the power necessary to raise 540 lbs. up an inclined plane 5 feet long and 2 feet high. As 5 2:: 540: 216 lbs., thepower. Tile leng~th, in the above rule, must represent that of the inclined surfa'ce, or of th ase, accordingly as the power acts parallel to either of these surfaces. The -Wedge. The wedge may be regarded as two inclined planes, united by a comnmon base, acting on two weights or resistances at once, or on a fulcrum and a weight, between which it moves, generally, in practice, by the impulse of successive blows. As in the inclined plane, equilibrium consists in the power being to the resistance as the back of the wedge is to its length, or to the: length of its side, accordingly as the resistance acts perpendicularly to the central line of length or to that of the side. Case 1. When:two bodies are forced from o/ne another by mcans of a twedge, in a direction parallel to its back. RULE. As the length of the wedge is to half its back or head, so is the resistance to the power. EXAMPLE. The breadtli of the'back or head of the wedge being 3 inches, and the length of either of its inclined sides 10 inches, required the power necessary to: separate two substances with a force of 150 lbs. As 10: 1: 150: 22~ lbs., the power. Case:2. CrT7en only one qf the bodies is movable. RULE. As the length of the wedge is to its back or head, so, is the resistance to the power. EXAMPLE. The breadth, length, and force, the same as in the last example. As 10: 3:: 150: 45 lbs., the power. The Screw. The screw is an inclined.plane, and may be supposed to be generated by wrapping a triangle, or an inclined plane, round a cylinder. T-he base of the triangle is the circumference of the cylinder; its height, the distance between two consecutive cords or threads; aanJ the hypothenuse forms the spiral cord or inclined plane. RULE. To the square of the circumference of,the screw, add the square of.the distance between two threads, and extract.the square root of the sum: this will give the length of the inclined plane. Its height is the distance between two consecutive cords or threads. When a winch or lever is applied to turn the screw, the power of the screw is as the circle described by the handle of the winch, or lever, to the internal or distance between the,spirals. Case 1. When the weight to be raised is given, to find the power. RULE. Multiply the weight by the distance between two threads:of: the screw, and divide -the prioduct by the circumference of'the circle described by the lever. The quotient is the power. EXAMPLE. Required the power to be applied to the end of a lever three feet long, to raise a weight of five tons with a screw of 1i inch between the threads. 11200 lbs. x 1'25 - 619 lbs., the power. 36 inches x 2 x 3'1416 Case 2. Whets the power is given, to ftsd the weight it will raise. RULE, Multiply the power by the circumference of the circle described by the lever, and divide the product by the distance between two threads of the screw: the quotient will be the weight The example is the converse of that in the former case. To HARDEN AND POLISH ALABASTER.-1. Take a strong solution of alum, strain it, and put it into a wooden trough sufficientlylarge to contain the figure, which must be suspended in it by means of a thread of silk; let it rest until a sufficient quantity of the salt is crystallized on the cast, then withdraw it, and polish it witha clean cloth and water. 2. Take white wax, melt it in a convenient vessel, and dip the cast or figure into it; withdraw, and repeat the operation of dipping until the -liquid wax rests upon the surface of the cast; -then let it cool and dry,'when it must be polished with a clean brush. 68 ToOTHED W HrLs. TOOTHED WHIEELS. The pitch (or the distance between the centres of two contiguous teeth) of cog-wheels is measured on the pitch-line, or extreme circumference of the wheel; and the distance between that line and the centre of the circle is reckoned as the radius of thle wheel. The following rules have been laid down for the diameters and number of teeth for wheels and pinions. RULE 1. As the number of teeth in the wheel + 2-25, Is to the diameter of the wheel, So is the number of teeth in the pinion + 16, To the diameter of the pinion. E-XAMPLE. Given the number of teeth -in the wheel =210, the diameter- of the wheel =- 25 inches, and the number of teeth in the pinioni - 30, to find the diameter of the pinion. As 210 + 2'325 25:: 30 + 1'5: 37102, = the diameter of tlhe pinion. RULE 2. As the number of teeth in the wheel + 2,25, Is to the diameter of the wheel, So is (No. of teeth in pinion + No. of teeth in eiheel) 2, To the distance of their centres. ExA-MPLE: Given the number of teeth in the wheel-210, the diameter of the wheel = 25 inches, and the number of teeth in the pinion = 30, to find the distance at which their centres: should be placed. 30 X 210 + As 210 + 2-2a5 25: -: 14'1342 inches, = the distancej of their centres. On the e Velocity of Wheels, Drums, Pulleys, &c. I When wheels are applied to communicate motion fiom one part of a machine to another, their teeth act alternately on each other; consequently, if one wheel contains 60 teeth alnd another 20, the one containing 20 teeth will make three revolutions, while the other makes but, one; and if drums or pulleys are taken in place of l wheels, the result will be the same, because their circumferences, describing equal spaces, renderotheir revolutions unequal; fronm this the rule is derived, namely, TOOTHED WHIEELS.;':Multiply the velocity of the driver by the number of teeth it: contains, and divide by the velocity of the driven: the quotient will be the number of teeth it ought to contain. Or, multiply the velocity of the driver by its diameter, and divide by the velocity of the driven: the quotient will be the diameter of the driven. If the velocities of driver and driven are given with the distance Qof their centres, Then the sum of the velocities: {'el.ocity ive:: distance velocity of driven of, centres radius of driven. radius of driver.'EXAMPLE 1. If a wheel that contains 75 teeth lmakes 16 revolutions per minute, required the number of teeth in another to work in it, and make 24 revolutions inthe same time. 75 x 16 Here 7 50 teeth. = Ans. 24 EXAMPLE 2. A wheel, 64 inches diameter, and making 42 revolltions per minute, is to give motion to a shaft at the rate of 77 revolutions in the same time; required the diameter of a wheel suitable for that purpose. 64 x 42 Here =4 - 349, inches. -.Ans. 77 EXAMPLE 3. Required the number 9of revolutions -per minute. made by a wheel or pulley 20 inches diameter, when driven by another of 4 feet diameter, and making 46 revolutions per-minute. 48 x 46 Here - - 110'4 revolutions. -= Ans. 20 EXAMPLE 4. A.shaft, at the rate of 22 revolutions per minute, is' to give motion, by a pair of wheels, to another shaft at the rate of 15~; the distance of the shafts from centre to centre is 451 inches; the diameters of the wheels at the pitch lines are required. 22 x 45'5 Here 22 + 15-5: 22:: 45'5 in.: 22 5 = 2669 in. 22 + 15'5 the radius of the driven wheel; which, doubled, gives 53'38 inches, the diameter. = —st Ains. Therefore 45,5 i5ches —.26 69 inches = 18 81 inches, the radius of the driver; which, doubled, gives 3762'inches, the diameter. =2d Ans. EXAMPLE 5. Suppose a drum to make 20 revolutions per minute, required the diameter of another to make 58 revolutions in the same time. Here 58 — 20 = 2'9, that is, their diameters must be as 2'9 to 1; thus, if the one making 20 revolutions be called 30 inches, the other:will be 3,0. 2'9 = 1.0845 inches diameter. 7 O TOOTHED WHEELS. EXAMPLE 6. Required the diameter of a pulley, to make:12revolutions in the same time as one of 32 inches making 26. 32 x 26 Here - 66-56 inches diameter. 12'5 EXAMPLE 7. A shaft, at the: rate of 16 revolutions per minute, is to give motion to a piece of machinery, at the rate of 81 revolutions in the same time; the motion is to be communicated by means of two gearing wheels and two pulleys, with an intermediate shaft; the driving wheel contains 54 teeth, and the driiving pulley on the axis of the driven wheel is 25 inches diameter; required the number of teeth in the other wheel, and the diameter of the other pulley. Let the drivenl wheel have a velocity of 36, a meau proportional between the extreme velocities 16 and 81,; 16x 54 then, 6 = -24, the number of teeth in the driven wheel.1st Avs. 36 x 25 And = 11'11 inches, diameter of the driven pulley.= 2d Ans. EXAMPLE S. Suppose in the- last example the revolutions of one of the wheels to be given, the number of teeth in both, and likewise the diameter of each pulley, to: find tile revolutions of the last pulley. 16 x 54 Here 24 36, velocity of the intermediate shaft.=Ans. l 36 x 25 Also, 11 12 81, the velocity of the machine. 11*11 GOLD LUSTRE Frod STONE-WARE. —Gold, 6 parts; aqua regia, 36 pir'ts. Dissolve: then add, tin, 1 part. Next add balsam of sulphuri 3 parts; oil of turpentine, 1 part. Mix gradually in a mortarr, and rub it in until the mixture becomes hard; then add oil of turpentine, 4 parts. It is then ready to be applied to a ground prepared for the purpose. To PETRIFY WOOD, &c. —Take equal quantities of gem-salt, rockalum, white vinegar, chalk, and pebbles powdered. Mix all these ingredients: there will happen an; ebullition. If, after it is over, you throw into- this liquor any porous matter, and leave'it there soaking four or five days, they will positively turn into -petrifactions. STEAM POWER AND THE, STEAM-ENGINE.'71 STEAM POWER AND THE STEAMI-ENGINE. STEAM is of great utility as a productive source of motive power; in this respect, its properties are-elastic force, expansive force, and l reductiol by condensation-. Elastic signifies the whole urgency'or power the steam is capable of exerting with undiminished effect. By expansive force is generally understood the amount of diminishijng effect of the steam on the piston of a steam-enginle, reckoning from that point of the stroke where the steam of uniform elastic force is cut off but it is more properly the force which steam is capable of exerting, when expanded to a known number of times its original bulk. And- contdensation, here understood, is the abstraction or reduction of heat by another body, and consequently not properly a contained property of the steam, but an effect:ploduced by combined agency, in which steam is the principal; because anyl colder body will extract the heat and produce condensation, but steam cannot be so beneficially replaced by any other fluid capable of maintaining equal results. The rules formed by experimenters, as corresponding with the results of their experiments on the elastic force of steam at giveni temperatures vary, but approximate so closely, that the following rule, because of being simple, may in practice be taken in prefer- 1 ence to any other: RULE. To the temperature of the steam, in degrees of Fahrenheit, add 100; divide the sum by 177.; and the 6th powler of the quotient will equal the force in inches of mercury.. EXAMPLE. Required the force of steam corresponding to a temperature of 312~. 812 ~ 100 2.32770= 159 inches of mercury. 177 To Estimate the Amount of'AdVantage Gained by Using Steam. Expansively in a Steanz-Enyine. When steam of a uniform elastic force is employed throu.ghout the whole ascent or descent of the piston, the amount of effect produtced is as the quantity of steam expended. But let thle steahm be shliut off at any portion of the stroke —say, for instance, at one half — it expands by degrees until the'termination of the strolke, and then exerts half its original force; hence tan accumulation of effect in proportion to the quantity of steam. -RULE. Divide the length of the stroke by the distance or space into which the dense steam is admitted, and find the hyperbolic logarithm of the quot ent, to which add; and the sum is'the ratio of the gain. EXA}rPLE. Suppose-an engine.-with a stroke of 6 feet, andthe 7 2 STEA~. PW A`N:D TAHE STAMIENGIN-:.; steam cut off when the piston has moved through 2; required the ratio of gain by uniform and expansive force 6. 2 =.3; hliyperbolic: logaritlim of' 3 10986 + 1 =' 2,0998, ratio of effect; that is, supposing the whole effect of the steam to be 3, the effect by the steadm beeinlg ctit off at 2 0986. Again, let.the greatest elastic force of steam in the cylinder of an engine equal 48 lbs. per square inch, and let it be cut off from entering the cylinder when the. piston has moved' 4 inches, the wholestroke being 18; required an equivalent force of the steam throughout the whole stroke. 18. 4'5 =4, and 48 4=1 2. Logarithmr of 4 + 1 - 388'629. c Then 2'38629 x 12 = 28.635 lbs. per square iiceh. In regard to the other case of expansion, when the temperature is constant, the bulk is inversely as the pressure; thus, suppose steam at 30 lbs. per square inch, irequired its bulk to that of -original wblk, when expanded soas to retain a pressure equal to that of the atmosphere, or 15 lbs. 15. + 30' 1 —— 5 = 3.times its original:bulk. It is. because of thle laten't hat in steam, or wvater in an an6rifboi state, that it becomes of such essential service in h1eating, boili:ig, diying, &c. In the heating of buildings its economy, efficiency, and si'mplicity of applica:ti'on are alike aclknowledged; the steai being simply conducted through all the deppartiebnts by pipes, by l extehnt of circulation condehises-thi-e latent heliat: bbeiiig thusil given to the pipes, and diffused by radiation. In boiling, its efici'enney is considerably increased, if advantage be taken:of- suffi'ciently inclosing the fluid, and reducing the pi essuie on- its surface, -by means of. an air-pump. Thus, water in a vacuum boils at:about a temperature of 98; and in sugar refining, where such means are employed, the syrup is boiled at 150~. The latent heat of Steam a't'-th-e common -ipi-essure of the atmosphere, according to very. accurate experiments,. is found. to- be 1000~; and we know that the sensible,. or thermometric heat. 2120. Now 2120- 32 = 180~, -and 1000o + 180- = 1180~; theref6re, steam at 2120 is simply highly. rrified water, and contains.11800 of heat; hence, to find the latent heat of steam at any. other temperature, subtract the sensible heat fi om 11800, and add 32 = the la6tent heat. EXAMPLE. Required the latent.heat of steam whose sensible heat 1180~ - a224~ 9560, And 9560 + 820 = 988'0 lateint heat. OAettbi ienchi of wnt'i,'prtodidees ab6i't''00:nubie in'ches of steam a C 1h es O St —--- ~ —.ii-i — STEAM POWER AND TIIE STEAM-ENGINE. 73 at 212~, or the common pressure of the atmosphere; but the boiling point varies considerably with the pressure on the surface of the fluid; thus, in a vacuum, water boils at about' 900; under common pressure, at 212~ and when pressed with a column of mercury 4 inches in height, at 216~; each inch of:mercury producing by its pressure a rise of about 1~ in the thermometer. The pressure or force of steam in the boiler (less than the weight upon the safety-valve) is generally indicated by a column of mercury in a bent Iron tube, which causes the range of the float to be only half the range of the mercury, 2 inches of mrercury being nearly equal to 1 lb. pressure of steam in the boiler, thus: Each, inch rise of the float indicates a pressure of nearly: 1 lb. - -Level of the mercury when there is no force of steam above the pressure of the: atmosphere. lb Calculate the Effect of a Lever and lWeight Upon the Safety- Valve qf a Steam-Boiler, &c. The lever, under all circumstances, is balanced by a known weight or weights, on the short end, making its point of rest on the valve' the centre of motion; so that the weiglht, added to that of the lever, is the effective weight upon the valve, independent of any other additional weight, thus: There are three different ways that it may be required to calculate the lever: 1.: Whe a certain pressure is reqsuired upon the valve, the distance qof te:weight uposn the lever, and the distance:of the valve from the cetre -of motiotL giveso, to find what weight will be reqsuired upon the lever at that distance. From the pressure on the valve in lbs. subtract the weight of the valve in lbs. and the effective weight of-tle level, multiply the remainder by the distance between thib fulcrum and the valve, and divide the product by the distance between the fulcrum and the '14 STEAM POWER AND THE STEAM-ENGINE. weight, and the quotient is the weight in lbs. required to be placed i upon the lever at that distance..2. hen a certai,, ppressure is required upon the valve, the weiglht upon the lever and distance of the valve frorm the centre of motion! given, to fild where that weight mutst be placed. From the required weight upon the valve in lbs. take the weight of the valve, add the effective weight of the lever, multiply the remainder by the distance between the fulcrum and the valve, and divide the product by the weight in lbs. upon the lever, and the quotient is the distance in inches from the fulcrum that the weight must be placed. 3. WUhen the distance of weight, distance of valve from the centre of motionl, and weight upon, the lever are given, to find whatpressure is upon that valve. Multiply the weight in lbs. upon the lever by the distance in inches to the fulcrum, divide the product by the distance between the fulcrum and the valve, and the quotient, plus the weight of the valve and effective weight of the lever, equal the weight upon the valve in lbs. EXxAMPLE 1. Suppose the lever A B (as above) to be 24 inches in length, and the valve C placed 5 inches front the centre of motion! A, what weight must be placed upon the lever 20 inches from A, to equal 80 lbs., on the valve C, the weight of the lever being 2 lbs., the weight D, which balances the lever, 4j lbs., and the weight ofi the valve 3 lbs.? 2 lbs. weight of the lever. 4'5 " to balance ditto. 3 " w wveighlt of the valve. 9'5 lbs. 80-9.5 x 5 Then -- 17'-625 lbs. 20 EXAMPLE 2. Suppose the weight upon the lever equal 17'625 lbs., it is required at what distance from A the weight must be placed to equal 80 lbs. at C. 80 — 95 x 5 = 20 inches. 17'625 EXAMPLE 3. Suppose, as before. that a weight of 17'625 lbs. is placed upon the lever 20 inches from A, required the pressure at C, the distance from the centre of motion being 5 inches, and the effective weight of the lever at that point equal 6} lbs., also the weight of the valve 3 lbs. 17'625 x'20 + 6'5 + 3 = 80 lbs. STEAM POWER AND THE STEAM-ENGINE. 7 To Find the Proper Diameter for a Safety- Valve. Multiply the bottom surface of the boiler, or surface immediately exposed to the action of the fire, in feet, by the multiplier opposite to the pressure in lbs. on each square inch of the safety-valve, and the square root of the product is the valve's diameter in inches at' thie narrowest part. If the boiler is to have two safety-valves, then( the square root of half the product equal the diameter of each. Pressure in lbs. Pressure in lbs. per- square inch. Multipliers. per square inch.; Multi9pliers..3.....................'356 15................ 315 4............... 353 201............................ 305 5...'348 25..2................'293 6........................... 344 30............................'289 7.. 33 9 35...............:282 8..............................336 40................. 275 10............... 329 45................'270 12.............................'321 50..........'64 In constructing steam-engines, the following simple'iule for obtaisiing the nossminal ihorse power is now generally adopted: The area of the cylinder in square inches multiplied by 7 lbs. pressure, multiplied into the speed of the piston in feet per minute, divided by 33000, equal the nominal horse power. Thtuv, area of cylinder x 7 lbs. X feet per minute -- nominal H. P. 33000 The lengthll of stroke and relative speed of piston, and number of revolutions per minute, will be found by the following table. In calculating the gross horse power developed in any cylinder, as shown by the indieator, it has been customary to allow one-tenth, and sometimes one-eighthl, for friction; this is now very properly abandoned, and the following rule;for calculating the indicator diagram should be always adopted: the mean pressure as shown on the card, multiplied into the area of the cylinder, multiplied into the speed of piston, in feet per minute, when the card was taken; this product, divided by 33000, will give the gross or indicated horse power: ft. in. ft. For 3 0 stroke 30 revolutions per minute = 180 per minute. 3 6 27 =" - 189 " 4 0 " 24~ = 196 " 4 6 " 22a " = 204 5 0 " 21 = 210 " 5 6 " 191 = 216 " 6 0 " 18" =222 " 6 6 " 17 " -226 " 7 0 6" 16 X — 231 "'7 6' 151 = 26 "8 0 " 15 " -240 ". 8 6 " 14-,6 = 244 " 9 0 " 13{ " = 247 " '7 6 ~ STEAM' PYO'wE AN-D TI-lE STEAMI-ENGINE. The Air-Punmp. The diameter of the air-pump should be a little more than half the diameter. of the cylinder, or the diameter of the cylinder in inches, multiplied by'6 will give the diameter of the air-pump in inches, the length of stroke to be one-half the length of stroke of the piston. The Condenser should never be less than half the capacity of the cylinder; and in engines where the pressure on the boiler ranges from twelve to twenty pounds on the square inch, a much larger condenser should be given. The' foot and delivery-valve passages should have' an area of onethird of the air-pump.'The Steam-Ports. The area of the steam-ports' on the cylinder should never be less than one-twentieth of the area of the cylinder. If the speed of the piston is above 250 feet per minute, the ports should never be less than one-fourteenth the area of the cylinders. The Cold- Water Pump. The capacity of the cold-water pump should be not less than one-thirty-sixth of the capacity of the cylinder. The Fly- JTheel. To find the weight of the fly-wheel rim the following practical rule is generally adopted: Horse power of the engine x 2000 (velocity of circumference of wheel in feet per second) 2 the weight of the fly-wheel in cwts. Th'e Fly- Wh eel, or Crank-Shaft. The nominal horse power of the engine and speed of the shaft being given, the diameter of this shaft, whether cast or wrought iron, will be found in the Tables of Strenqgth of Shafts. The Governor. To find the number of revolutions, divide 375 by the square root of the length of the pendulum; half of this quotient is the number of revolutions the balls ought to make per minute. To find the length of the pendulum, divide 375 by twice the number of revolutions; the quotient squared is the length of the pendulum. General Proportions of Locomotive -Engines. For. the area of the steam-ports when the stroke is 18 inches, the square of the diameter of the cylinder x'068 = the area in square inches. For the area of the eduection ports, the square of the diameter of the cylinder in inches x 1 28 = the area in square inches. The breadth of the bridges between the eduction ports and the induction = i inch and 1 inch. The diameter of the chimney = the diameter of the cylinder. For the area qf the fre-grate, the diameter of the cylinder in inches x'77 =-the area in superficial feet. For the elective heatitng-suoface of the boiler, the square of the diameter of the cylinder in inches x 5 -- 2 = area in square feet. STEAM POWER AND TiE STEAM-ENGINE. 7C For the diamneter of the feed-pump ram, the square of the diameter of the cylinder in inches x'011 = the diameter in inches. -For the cubical cbntent of he steam-roon, the square of the diameter of the cylinder in inches x 9-40=content In cubic feet..For the cubical content of inside fire-box abovefire-bars, the square of the diameter of the cylinder in inches —4=content in cubic feet. For the. inside diameter of the steam-pipe, the square of the diameter of the cylinder in inches x'03 = the diameter in inches. For the diaineter of the branch. steam-pip e, the square of:the diameter of the cylinder in inches x 021 = the diameter in inches. For the diameter of the top of the blast-pipe, the square of the diameter of the cylinder in inches x'01.7 = the diameter in inches. For the diameter of the feed-pipes, the diameter of the cylinder in inches x 141 -= the diameter in inches. For the diameter of the piston-rod, the diameter of the cylinder in inches 7 = the diameter in inches. For the:thickness of the piston, the diameter of the cylinder in inches x 2 -- -= the thickness in inches. For the diameter of the connectingq-rod at the middle, the diameter of the cylinder in inches x'21 = the diameter in inches. For the diameter of the plain part and inside bearing of the crasnkaxle, the cube root of the square of the diameter of the cylinder in inches x'96 = the diameter in inches. For, the diameter of the outside bearings of the crank for, axle, the cube root of the: product of the square of the diameter of the cylinders in inches x'396 = the diameter in inches. For the diameter of the crank-bearing, the diameter of the: cylinder in inches x'404 = the diameter in inches. ~ For the length-of the crain-k-bearing, the diameter of the cylinder in inches x'233 = the length in inches. Remarks on Steam-Engine Boilers: and their Proportions. For engines designed to give a gross indicator horse power of at least twice the nominal horse power, the grate surface should be *66 or'69 square feet per nominal horse power, but may be increased to'75 square feet, and should never be diminished to less than'60 square feet as a minimum. The area of opening over the bridges or through the tubes, should be -125- square feet, or 18 square inches per horse power, and may be increased to'143 square feet, or 20 square inches with advantage, particularly in tubular boilers, and should never be diminished to less' than 15 square inches, or'109 square feet per horse power. The area of chimney should be -076 square feet, or il square inclies, but may be increased to 13 square inches, and shouild never be diminished to less than 10 square inches per horse power. 7* is8 STEAM POWER AND THE STEAM-ENGINE. The heating surface in fire-places and flues should be 14 square feet per horse power, exclusive of all bottom surface, but may be increased to 15 square feet, and should never be diminished to less than 12 square feet per horse power. In calculating tubular boilers the whole surface of the tubes should be taken, and there should be a total of 17 square feet per horse power in the fire-places and tubes. In' engines designed to work to a gross power in the cylinder by the indicator greater than twice the nominal horse power, these proportions must be increased; or, if the reverse be intended, they may:be diminished in proportion. Of the Pressure of Steam, in Inches of Mercury, at Different Temperatures. I. IL III. IV. V. VI. Temperature, Fahrenheit. Dalton. Ure. Young. Macneill. Tredgold. 00 008 10 0'12 20 0'17'011 32 0'26 0-20 0-18 0-'I 40 0'34 0'25 0-20 0-'24 50 0 49 0'36 0-36 0'36 0'37 60 0'65 0'52 0'53 0'55 70- 0'87 073 0'75' 073 078 80 1'116 1'01 1-05 111 90 1'59 1-36 1-44 1 36 1'53 100 2'12.186 1-95 2-08 110 2'79 2-45 2-62 2'46 2-79 120 3'63 330'346 3-68 130 4'71 4'37 4'54 4'41 4-81 140 6-05 5-78 5-88 6-21 150'773 7-53 7-55 7'42 7 94 160 9'79 9-60 9-62 10'05 170 12-31 12'05 12-14 12-05 12-60 180 15-38 15'16 15-23 15-67 190 18-98 19-00 18-96 18-93 19'00 200 23-51 23'60 23-44 23-71 210 28'82 28-88 28-81 28-81 28'86 212 30'00 3000 30-00 30.00 83000 220 3518 83554 35-19 3492 230 44-60 43-10 42-27 42-63 42-00 240 53'45 51-70 51 66 50-24 STEAM POWErI AND THE' STEAMI-ENGINE. q9 Of the Temperature of Steam at different Pressures in Atmospheres. I. IL I. III. IV. V. I. Temperature Fahrenheit. French Acad. Dr. Ure. Young. Macneill. Tredgold. 1st At. 212'00 2120 2120 2120 2120 2d " 2505 250'0 240'3 249 250. + 3d ". 275 2 275 0 271 274. + 4th " 293'7 291 5 288 290 294.+ 5th " 3088 304'5 302 309. + 6th" 320'4 315'5 322.7th " 331'7 325'5 8th" 342'0 336'0 337 342. + 9th 350'0 345' 10th 358'9 11th 366'8 12th 3874.0 372.13th " 380'6 14th " 386'9 15th 892'8 16th " 98'5 17th 403'8 18th 408'9 19th 413'9 20th " 418'5 414 30th 457'2 40th 4 66'6 50th " 10'6 To PREVENT SPONTANEOUS COMBUSTION.-It is a fact better ascertained than accounted for, that fixed oils, when mixed with any light kind of charcoal, or substances containing carbon, such as cotton, flax, or even wool, which is not of itself inflammable, heat by the process of decomposition. and after remaining in contact some time, at length burst into flame. This spontaneous combustion takes place in waste cotton which has been employed to wipe nmachines, and then thrown away and allowed to accumulate into a heap. WVe have known an instance of the kind in a manufactory for spinning worsteds, where the waste wool, or "slubbings," as it is termed in Yorkshire, was thrown into a corner and neglected. It then heated, and was on the point of bursting into flame, when the attention of the workmen was directed to the heap by the smoke and smell. In cotton mills the danger exists in a still greater degree, and it is believed that the destruction of many cotton factories has been occasioned by this means. The cause of this peculiar property of fixed oils deserves more attention than has hitherto been paid to it. 80 STEAM POWER AND THE STEAM-ENGINE. TABLE T A B L E Of the Elastic Force of Steam, and Corresponding Temperature of the Water with wh.ich it is in Contact. c- -) P, sa o lbs. lbs. _ 14'7 30-00 212'0 1700 49 99'96 281'9 564 15 30-60 2128 1669 5 0 1()'200 2832 554 16 3264 216'3 1573 51 104'04 2844 544 17 34.68 2196 1488 52 106-08 2857 534 18 36-72 2227 1411 53 108-12 286-9 525 19 3876 2256 1343 54 110'16 288'1 516 20 4080 228'5 1281 55 112'20 2893 508 21 42'84 231'2 1225 56 114'24 290'5 500 22 44'88 233'8 1174 57 116'28 2917 492 23 46'92 236'3 1127 58 11832 292'9 484 24 4896 238'7 1084 69 120 36 294 2 477 25 51i00 2-1i0 1044 60 122040 295'6 470 26 53'04 243'3 1007 61 124 44 296 9 463 27 55 08 245'5 973 62 126 48 298'1 456 28 57-12 247-6 941 63 128'52 299'2 449 29 59-16 249-6 911 64 130.56 300 3 443 30 6121 251'6 883 65 132'60 301'3 437 31 63 24 253'6 857 66 13464 302 4 431 32 65.28 2555 833 67 13668 303'4 425 33 67 32 257'3 810 68 138'72 304'4 419 34 6936 2591 1 88 69 140'76 305 4 414 35 7140 260'9 1767 70 142 80 306'4 408 36 17344 262 6 748 71 14484 3'07'4 403 37 75.48 264.3 729 72 14688 308.4 398 38 77'52 265 9 712 73 148 92 309 3 393 39 79'56 267'5 695 74 6 15096 310'3 388 40 81-60 269 1 679 75 153'02 311'2 383 41 83'64 270'6 664 76 155 06 312 2 379 42 85 68 272 1 649 77 157'10 313 1 374 43 8 7'72.273'6 685 78 135914 3140 370 44 89 76 275'0 622 79 161'18 314'9 366 45 91'80 276'4 610 80 163 22 31.5'8 362 46 93 84 277 8 598 81 8 165'26 316 7 358 47 95 88 2792; 586 82 167[30 317'6 354 48 97 92 280'5 575' 83 169734 3184 350 This indes the pressu1re of the atmosphere.. * This includes the: pressure of the atmosphere, STEAM- POWE. AN)D THIE STE'AM.-ENGINE. 81 T A B L E-(Continued). lbs. - lbs. 86 175'46 321'0 339 100 204'01 332'0 295 87 177'50 321'8 335 110 224'40 3392 2 71 88 17954 3226 322 2 332 120 244-82 33458 251 89 181 58 323-5 328 130 265-23 352'1 233 90 183-62 324-3 325 140 285-61 857.9 218 91: 185-66 325'1 322 150 306'03 363'4 205 92 187-70 325'9 319 160 326;42 368,7 193 93 189'74 326'7 316 170 346-80 373-6 183 94 191-78 327'5 313 180 367'25 378-4 174 95 193-82 328-2 310 190 387'61 382'9 166 96 195-86 329-0 307' 200 408'04'387.3 158 97 197-90 329'8 304 T A B LE Of the Force and Temperature of Steam in Atmospheres. Atmos. Temp. Fah. Atmos. Temp. Fah. Atmos. Temp. Fah. Deg. Deg. Dog. 1 212-00 10 358'88 19 413'78 2 250 52 11 3'66'85 20 418-46 3 275-18 12 374'00 21 422-96 4 293'72 13 380'66 22 427-28 5 307'50 14 386'94 23 431'42 6 320-36 15 392'86 24 435'56 7 331-70 16 398'48 25 439'34 8 341-78 17 403 82 9 350'78 18 408'92 50 510-60 To WRITE ON SILVER WITH A BLACK WHIrCH WILL rNEVER GO OFF.Take burnt lead and pulverize it. Incorporate it next with sulphur and vinegar, to the consistency of a painting color, and write with it on any silver plate. Let it dry, then present it to the fire so as to heat the work a little, and it is finished. 82 STEAM POWE.It AND THE STEAM-ENGINE. TABLE, Of the Heating Powuer of various Comtbustible Substances, exhibiting the utnost Quanlity of' TVater evaporated by tihe Given lWeiqlets, and the snmallest Quantity of Air capable of produecinqg Total Coambustion. Pounds of Water wvhich a Pounds of Weight of Species of Combustible. Pound can Boiling Water Atmospheric air heat, from evaporated by 1 at 32~, to l)urtl 1 0o to 212. Pound. Pound. Wood, in its ordinary state,. 26 4 72 4'47 Wood charcoal,... 73 13-37 1l 46 Pit coal,. -.:... 60 10'90 9'26 Coke,..... 65 11 81 11'46 Turf..... 30 5'45 4'60 Turf charcoal;.... 64 11'63 9-86 Carburetted hydrogen,. 76 13:81 14-58 Oil,....... Wax,.... 78 14'18 15-00 Tallow.. Alcohol of commerce,... 52 9'56 11-60 To ESTIMATE DISTANCE.-Observe how many seconds elapse betweeni a flash of lightning and the thunder, and multiply them by 1142, the nurimber of feet sound travels in a second, the product will be thle distance in feet. The same process may be applied to the flash and report of a gun, or any other sound, provided we can ascertain the time at -which it is produced, and the interval that elapses before it reaches the ear. Illustration. Saw a flash of lightning five seconds before I heard the thunder: required the distance. 5 x 1142 _ = 3 mile distant. 3x1760 In the absence of a watch, the pulsations at the wrist may be counted as seconds, by deducting one from every seven or eiglit. PRMsMATIC DIAMOND CRYSTALS FOR WINDOWS. —A hot solution of sulphate of magnesia, and a clear solution of gum arabic, mixed together.: Lay it on hot. For a margin or for figures, wipe' off' the part you wish to remain clear with a wet towel. PERFECTLY BLACK HARD GLAss.-Plain paste, 600 parts; zaffre, 3 parts; manganese, 3 parts; iron, 3 parts. STEAM PowEn AN-D TIHE STETCAM-i-ENwGINE. 8 TABLE Of Nomiinal Horse PoweNr of Low PNressure Engianes. LENGTH OF S'rROKE IN FEIET. - 1 1 * 2 2 K 3 | 4 4'Y 5 5K 6 7 4'34 39'43'46'49'52'54'56'58'60'62'65 5'53'61'67'72'76'81'84 * 88'91'94'96 1'02 6'76'87'96 1'04 1'10 1'16 1'22 1'26 1'31 1'35 1'39 1'17 7 1'04 1195 1'31 1'41 1'500'58 1'651 1'72- 1'78 1'84 1'89 1'99 8 1'36 1'56 1'72 1'85 1'96 2'07 2'16 2'25 2'33 2'40 2'47 2'60 9 1'7L2 1'97 2'17 2'31 2'49 2'62 2'74 2'84 2'95 3 04 3'13 3'30 I0( 2'13 2'44 2'68 2 89 3'08 3'23 3'38 3'51 3'64 3'76 3'57 4'07 11 2'57 2'95 3'24 3'49 3'77 3'91 4'15 4 25 4'40 4'54 4'68 4'92 12 3'06 351 3'86 4'16 4'42 4'65 4'86 5'06 5'24 5'41 5'57 5'86 13 3'60 4'12 4'53 4'88 5'19 5'46 5'64 5'94 6'15 6'35 6'53 6 88 14 4'17 4'77 5'25 5'66 6'01 6'33 6'62 6'88 7'13 736 7'58 7'98 15 4'77 5'48 6'03 6'50 6'90 7'27 7'601 7'90 8'19 8'45 8'70 9'16 16 5'45 6'23 6'86 7'39 7'86 8'27 8'65 8'99 9'31 9'611'90 10'42 17 6'15 7'04 7'75 8'35 8'86 9'34 9'76 10'15 10'52 10:85 11'17 11'76 18 6'89 7'89 8'68 9'36 9'94 10'47 10'94 1'.'38 11'79 12'17 12'53 13'19 19 7'68 8'79 9'69 10'42, 11'17 11'66 12'199 12'6S 13'13 13'56 13'96 14'69 20 8'51 9*74 10 72 11 55 12'27 12'92 13'51/. 14'05 14'552 15'02 15'46 16'28 22 1030 11'79 12'97 13'98 14'85 15'63 16'62 17'30 17'65 18'18 18'71 19'70 24 12'26 145'03 15'44 16'63 17'67 18 61 19'45 20'23 209.5 21'63 22'27 23'44 26 14'39 16'46 18'12 19'52 20'75 21'84 22 56 23'75 24'68]' 25'39 26'14 27'51 28 16'68 19'09i 21'02 22'64 24'06 25'33 26'48 27'54 28'52 29'44 30'31 31'90 30 19'15 21'92 24'13 25'99 27'62 29'07 30'40 31'61 32'74 33'80 34'80 36'63 32 21'79 24'96 27'51 29'57 31'42 33'08 34'59 35'97 37'26 38'46 39'59 41'68 34 24'60 28'16 30'99 33'39 35'44 37'34 39'01 40'60 42'06 43'41 44'69 47'05 36 27'57 31'56 34'74 37'42 39'77 41'87 43'77 45521 47'15 48'67 50'11 52'75 38 30'72 35'17 38'71 41'69 44'66 46'64 48'771 50'72 52'54 54'23 55'83 58'78 40 4 04 3S'97 42'89 46'201 49'10 51'691 54'041 56'20 5s'21 60'09 61'86 65'12 42 37'53 42'96 47'29 50 94 54'13 56'98 59'581 61'96 64'18 66'25 63'21 71'78 44 41'19 47'15 51'90 55'91 59'35 62 54 66'461 68'00 70'44 72'71 74'85 78'79 46 45'021 51'51 56'72. 61'101 64'88 68'191 71'431 74'33 76'69 79'47 81811 86'12 48 49 02 56'11 61'76 66'54 70'70 74'421 77'82' 80'94 83'83 86535 89'08 93'78 50 53'191 60'89 67'02 72'191 76'71 80:761 84'44 87'82 90'96 9389 96'65 101'7 521 57'55 65'86 72'48 78'08 83'00 87'35 - 90'251 94'98 8'40 101'55 104'5 110'0 54 62'04 71'02 78'17 84'20 89'45]' 94'201 98'49 102'4 106'1 109'5 112'7 118'7 56 66'721 76'35 84'07 90'55 96'23 101'301 105'9 110'1 114'1 1178 121'21 127'6 58 71'58 81'93 90'18 97'14 103'2 108 6 1 13'6 1 18'2 122'4 16'3 129'2 136'7 60 76'60 87'68 96'50 103'9 110'4 116'3 121'6 126'4 131'0 135'2 139'2.146'5 62 81'79 93'62 103'04 111'0 117'96 124'18 129'81 135'03 139'86 14437 148'6 156'7 64 87'15 99'84 110'0 1 18'3 125'7 132'3 138'3 143'9 149'0 153'82 158'4 166'7 66 92'68 106'1 116'8 125'8 133'6 140'7- 147'3 i53 0 158'5 163'6 168'4 177'3 68 98'40 112'6 123'9 133'6 141'8 149'4 156'2 162'4 168'2 173'6 178'8 188'2 70 10-1'26 119'3 131'3 141'5 150'4 158 3 165'5 17'1 178'2 18460 189'4 199'4 72 110'30 126'2 139'0 1497 159'1 167'4 175'1 182'1 188'6 194'7 200'4 211'0 74 116'5 133'4 146.8 158'1 160'9 176'7 185'4 192'4 199'2 2085'7 211'6 223'4 76 122'9 140'7 154'8 166'8 178'6 186'6 195'0 20'29 210'1 216'9 223'3 235'1 78 129'4 148'2 163'1 175'6 186'7 196'5 205'4 212'1 221'4 228'5 135'2 247'6 80 136'2 155'8 171'6 184'8 1964 206'7 216'1 224'8 232'8 240'4 247'4 260'5 82 143:0 I 63'8!80'2 194'2 206'2 217'3 226'9 237'8 244'6 252'5 260'0 273'8 84!50'1 71'8 189'1 203'8 216'5 227'9.238'3 247'8 256'7 265'0 272'8 287'1 86 157'4 1801 198'2 213'6 227'0 237'8 247'4 258'2 269'1 277'8 286'0 301'0 88 164'8 188'6 207'6 223'6 237'5 251)'2 261'6 272'0 281'7 290'8 299'4 315'2 90 172'3 197'3 217'1 233'9 248'6 261'7 273'6 281'5 291'7 3041'2 313'2 329'7 8STEAMA PowER: AND TIHE STE:MaI-E'CNGINE. TABLE Of Nomional Horose Power of zHigh Pressure En7.gines. LENGTH OF STROKE IN FEET. 1, 1 22 2%~ 3 3%2 4 4 % 5 3K2 6 7 2. *5'29 32'35'37 38'40 *42 44 45 * 46 49 27 *39 45'50'54 57 60.63 *66 68'70 72'76 3'67 *65 *72'78 83 87'91'95 98 O.61 1'04 1.o1 32.8'89 98 1'06 1'13 1'19 1 24 1'-29 1'34 1'38 142 1'49j 4 1'02 1'17 129 1'38 1'47 156'62 68 1'74 1 80 186 1'9 47 1'29 1'48 63 1'75 1'86 196 2'05 2'13 2'21 2'28 2'35 2'47 5 1'59'83 2'01 2'16 2'28 2'43 2'52 2'64 2'73 2'82 2'88 3:06 572 1 93 221 243 262 278 2'93 3'12 3'18 3'30 3'42 3'51 3'69 6 228 2'61 2'88 3'12 3'30 3 48 3'66 3'78 3'93 4'05 4'17 4411 67 2'69 3'09 3'39 3'66 3'.90 4'08 4'23 4 44 4'62 4'77 4 89. 5`16 7 3'12 3'57 3'93 4'23 4'50 4'74 495 5'16 5'34 5'52 5 67 5'97 72 3'60 4'11 4'53 4'86 5'19 5'46 5'70 5'94 6 15 6 33 6'51 6'871 8 4'08 4'68 5'16 5'55 5 88 6'21 6'48 6'75 6;99 72 7241 7'80. 87~ 4'62 5'28 5'82 6'27 6'63 6'99 7-32 7 62 7'89 8'3 8'37 8'82 9 9 5'16 5'91 6'51 7'02 7 47 7 86 8'22 852{ 8'85 9'12 9'39 9'90 972 5'76 6'60 7"26 7 80 8'37 8'76 9'15 9'51 9'84 10'17 10'47 1001 o10 6'39 7'32 8'04 8-67 9'211 9'69i 1014 153 10'92 11'28 11'61 1221 10 7'05 8'04 8'88 9'54 10'14 10'68 11'16 11'61 12 03 12'421 12'78 13'47 11 7'71 8'85 9'72 10'47 11'31 11'73 12;45 12'75 13'20 13'662 14 04 14 76 11K/ 8'43- 9'66 10'62 11'46 12'15 12'78 13 80 13'921 4'61 14'91 15'33 16'14 12 9'18 10'53 11'58 12'41 13'26 13'95 14'58 15"18 15,721 16'231 16'71 17'58 12'S 9'9,6 11'40 12'57 13'53 14'37 15'15 15'84 16'47 17'04 17'58 18'12 19'08 13 1 10'80 12'36 13'59 14'64 15'57 16'38 166921 17'821 18'45 19'051 19'59 21-264 13,s 11'64 1332 14'64 15'78 16'77 17'67 18'48 19'20 19:89 20'52 21'15 22'26 14 125.1 14'S 1575 16'98 1803 18'99 19'86 20'64 21'391 22'08 22'74 23:94 14S 13'41 15'36 16'921 18'21 19'35 20'37 21'308 22'14 22'95 23 701 24'39 25'62 15 14'31 16'44 18'09 19'50 20'70 21 81 22'801 23701 24'57 25'35 26 10 27148 16 16'35 18'69 20'58 22 71 23'58 24'81 |5':951'6'97 27'.93 28'83 29'70 3126 17 18'45 21'1'2 1' 23'25 22'05 26'58 28'02 29'28 30'45 31'56 3255 33.57 35'28 18 20'67 23'67 26'04 28'08 29'82 31'41 32821 34'14 35'37 36'51 37'59 39'57 219 23'04 26'37 29'04 31'26 33'51 34'98 36'5, 38'04 39'39 40'68 41;88 44907; 20 25'53 29'22 32'16 34'65 36'81 38'76 40'531 42'15/ 43'65, 45:06 46 38 48'84 22 30'90 35'37 38'91 41'941 44'55 46'89 49 861 51'90 52'95 54 54 56'13 59'10 1 24 36,78 4209 46 32' 49-89 53'01 55'831 58 35 60'691 62'851 64'891 66'81 70'32 26 43'17 49'30 54'36 58'56 62'25 65'521 67'68 71'251 73'581 6'17 78'42 82:53 28 50'04 57"27 63'06 67'92 72'18 75'991 79'44 82'62 85'561 88'321 90'93 95'70 30'57 45 65 76 72339 77T97 82'86'87',211. 91'20 94'83 98'22 101:401 104'4 109:9 32 65'37 74'88 82;53 88'71 94 26 99'24 103:7 107'9 11 1'8' 115'4 1 18'7 125'0 34 73'80 84'48 92'9 100'22 106'3 112'0 117'1 121'8 126'2 1302 1134'0 141' 36 82'71 94'68 104 2 112'2 119'3 1256 1133:3 136'5 1414 146'01 150!3 158'2 38. 92'16 105'5 116'1 125'0 134'0 136'9 146'3 152'1 [ 157'6 162'7 167'5 176:3 40 i02'1 116'9 129'0 128'6 147'3 155'1 162'1 168'6 |174'6 |180'2 1856 195'3 42 126 i28'9 8 141'8 i52'8 1624: 170'9 178'7 185'9 192'5 198'7 204'6 215'3.44'123'5 14 1'4 155'7 167'7 178'1 187'6 199'4 204'0 211'3 218'1 1224'5'236:3 46' i35'0 154'6 170'1 183'3 194'6206 2143 223'0 230 0 238'4 245'4:258'3 i48' 147:0' 16'83 185'3 199'6 212'1 223'2 233'4 242'8 251'5 259'6 267'2 281'3.50 159'6 i82'6 201'0 216'51 230'1 242'3 253'3: 263'4 272'9 1281'6 289'9.305'1:52 172'6 197'6 217'4 224'-2 249'0: 262'01 270'71 284'9 295'2 304'6 13135.330'0 54 186 1 213'0 234'5 252'6 268'4 282'6 295'4 307'2 318'3 328'5 338'1,356'1 56: 20O0'1'299; 1 252:2 271'6 288'7' 303'9 317'7 330'3 342'3 353i4 363'6'382'8 58 214.7 245'8 270'5 291'4 309'6 325'81 34081 3554'6 367'2 378:9 389'7 40. 1'60 229(8. 263'0 289'5 311'7 731'2 348'9 364'8 379'2 393'0 405'6 417:'6 9439'5 Proportions of Coondensing;Engines?. _.. [ Diameter;' O l. of'Conch~ ~ ~~~~~~~~~: 1'-= denser. 0 ___/,_i__ lji _ -....... 412+2 6 4x1 x 3 8 1 3 7+x3 5-'x3 7 21 3+112 12 1 l+ M2 4 222 13 10 614+ 3 0 6 x1+ 9 1 6 8-x 3-2 6f x3i 2+ 4 2 2+1. 12 j 11 2 4-'2- 3 2 31 1 8.17 3 0 6 xlIl 1 1 6 9 x3i 7 x3 4i 6 2+2-+i 1i7 1 6 2 i 1 13 2 5 133+ 21 3+ 13t 12 19+ 4 0 10 x 1+ 12+ 2 0 10 x3+ 9 x3+f 5- 7 2 2+ 2 1 7 1+ I 1 1+ 2+ 5 3' 4- 2+ 49 15 15 24 3- 0 9Jx2+ 13+1 6 10 x4 9 x4 6 8 2ij2+1 62 1 1- 3 G 36 4 4 lx20 15 21 4 O 10 xl 13+ 2 0110 x4 9 x4 6/ 8 2j 1 2 2- i 1 1221 6 13S4j42t 4+ 160. 20127 3 0 11 x 2~15 1 6112 x5 12 x4 6S3 833 3 2 1 6 2i: 2 / 1, 2 7 146 3 5 12'x 20 20124 5 0 10 x2 115 92 6 12 x5 12 x4! 7 9 3 3 2[2 0} 2 2 1 I 2+ 7 416 3 5 18+ 25 27 5 0 10 x3 17+ 2 6 14 x5 12 x4+ 8 10 313+3 2 -6 2 2i 1+ 2 31 8 5 6 3+6 19+ 30 32 3 0 14 x 3 18 1 61~5 x 5[12 x5 7 9 33 33 1 62+ _2 1 2 3+ 8 5+6 6 6 24 x 25 m 30()29 6 0 11 x 3 118 3 0115 x 5a12 x 5 9 11 83i8 3 12 6 2+ 2j- 1+ 2 3+ 8 5)+6 13+16 "20 40 36 3 0 15 x 3{20 1 9 16 x6+ 15 x6 10 12 4 4 3 1 9 3 2 + 14 2+ 4 10 6 8 4 7 22 40 33 6 0 14 x3 21 3 0116 x61 15 x6 10 12 4 4 1 2 6 3 1. 2+ + 1+ 2+ 4 10 6 8 4 17 24 5040 4 0 16 x 3+22 2 0 17 x616 x6 |10 12 14+ 4 4 2 0 3+- 29j 1+ 2+ 4 11 6+19 147 24 1n 60 43 4 018 x4.24 2 0 21+i x 6 l8- x 6 ~11 14 4141 4 2 03-7 2 l'- 1 2 4+ 16 /8 17 24x381 70 40 7 0 16 x 3'25 3 6 21 x 6+ 18 1x 1216 4f14+4 4 3 03 2+ 1+ 2? 4+ 12 7 9 5 8 28 70 43 17 017 x3127 3 6124 x'6+119 16 [4} 4+3 0 -3 l 2+ + l1 2+ 12 17,9 15+18+ 29+ 80146 17 018 x4 29 3 6124 x7-120 x 13 16 5 5 14 3-01 2- ~ 1- 3 6 1+18 106 18+311 Dia. of piston-rod —~1L the dia. of cyl.1Air-pump Lod iron -L the diea. of air-p. Side-rods full + of piston-rods. cross-head ends=l " " " brass~. -". Connecting-rod at thinnest part } Depth in centre=*+ c " " copper i " " diameter of cylinder. Crank-pin } Thickness in centre=+ " " I diameter of cylinder. 86 STEAM POWER AND THE STEAM-ENGINE. TAB LE Of the Revolutions per Mile of DriVing Wheels, and Consunption. of Steanm and Water for each sized Wiheel; taking the steam admitted to each cylinder as exactly one cube foot, at a gross pressure of 98lbs. or 831bs. on the spring balance. WHEELS. Cylinder of Steam per iIile, and Water per Mile, Consumption, taking Steam at Revolutions taking Cylinder at 941bs above Diameter. Circumference, per Mile. one Cube Foot. atmosphere. feet. ft. in.'No. cube feet. ga!!orl. 10 31 5 168 672 14'0 9; 29 10 1'76'9 707'6 14'-74 9 28 3I 186'7 746 8 15 55 8~ 26 83 197 4 789'6 16'44 8 25 1! 210'1 840'4 17 5'7 23 6~ 224 897'6 18'69:' 21 11 240 960 20^0 6~ 20 5 258'6 1034 21'5 6 18 9'18 280'5 1122 23'37 65 17 3'33 305'6 1222'4 25'45 5 15 8'48:336 3 1344' 4 28' 0 4,, 13 11'1 379'0 1493'6 31 11 4 12 6'92 4203 1 680'4 35 0 38 11 9.'37 441'1 117922: 3733 38 10 11 ~94 480.1 1920-8 40 0 3 9 5'08 56f02 2243 46' 67 tNote.-As there are two cylinders at work in a locomotive, consequently there are four cylinders of steam for each revolution. M~IODELLING WAX. —-This is made of white wax, which is melted and mixed with lard to make it malleable. In working it, the tools and the board or stone are moistened with water, to prevent its adhering:; it-may be colored to any desirable tint with dry color. STEAM POW5ER AND TIIE STEAI-ENGINE. 8 7 T A BLE Of Pressure of Steam, exclusive of that of the Atmosphere. PRESSURE. PRESSURE. Tempera- Temperalbs. on In inches ture in de- lbs. on In in'ches tture in the of In atmo- grees of the of In atmo- degrees of' sq. tiich mercury, spheres. Fahrenheit sq. inch.I mercllry. spheres. Fahrenheit 1 2'04'068 2130 51 IC4'04 3'468 3010 2 4'08'136 216 52 106'08 3 536 302o 3 6'12'204 2190 53 108,12 3'604 3033 4 816'272 223 54. 110616 3'672 04 5 1020'340 225 55 11220 3 740 305 4 6 12'24'408 2)28 56 114'24 3 808 3061 7 14'28'476 221 57 116'28 3'876 307K2 8 16'32'544 234 58 118'32- 3'944 308K 9 18 36'60i2 236 59 120'36 4'012 309 10 2050 *680 229 60 122'40 4'080 310 I11 22 44 748 241 61 124'441 4'148 311 12 24'48'816 243 62 126'48 4'216 312 13 26 52'884 245K 63 128'52 4'284 313 14 2856e'952 2472 64 130'56 4'352 314 15 30'06 1'00 2492 65 132'60 4'40 315 16 32'64 1'0os8' 2oI 66 134'64. 4-488 316 17 34'68 1'156 253K,67 136'68 4'556 317 1 - 35-72 1'224' 55-s 68 13372- 4 624 3172 1-9 - 38-76 1'29r2 i z257 69 140'76 4692 3180 20 40'80 1'360 259 70 142'80 4'760 319 21' 4?'84 1'428, 261 71 144'84 4'828 320 22 410283 1496 2' 62s 2 72 146'88: 4'896 321 23 46'92 1 564 264 73 148'92 4'964 322 24 48 95 1-632. 266 74 15096 5032 32%2 25 1 51.'00' 1'700 267K 75 153'00 - 5'100 323K2 26 53'04 1'768 269 76 155504 5'165 324 27 52'03 1 836 270K'77 157'08 5'236 325 28 57'12' 1'904 272 78 159'12' 5'304 326 29 59'16 1'972 273 79 161'16 5'372 327 30 61205 2,040 275 080 163'20 5'440 327K a 1' I 63`24' 2'108, 276K 81 165'24 56508 328 32 65'23 2'176 278 82 167 28a 5 576 329 33 67'32 2 2244 279 3 169'32 5'644 330 34- 69 36 2'312 2803 84 171 36- 5'712 330o 3.5, 71'40 2'380 282 95 173'40 5'780 331 36 73'44 2'448; 283 86 175'44 5'848 332 37 75'48 2'516 284K 87 177'48 5'916 333 38' 77 5 2-584 286 88 179'52 5'984 333K 39 79'56 2'652' 287 89 181'56' 6'052 334 40 816Q0 8 2'720 288 90 183-60 61 20 335 41 83'64 2788 289 91 185i64 6'188 335 42 85'6S 2'856 290K 92 187'68' 6-256 336 43 87'72 2 921 292: 93 189'72 6'324 337 44 89'76 2'992 293 94 19176 6392 338 45 91i 800 3'060; 294 95 19380 6'460 338K 46- 93'84 3 128 295 96 195'84 6,528 339 47 95'88 3'196 290 97 169788 6'596 340 48 97'92 3'264 29 94 199'92' 6'664 340K 49. 9998 3332 299 99 201,96 6'732 311 5S 102D00 3'400 330 100 204-00 6'800 342 LINSEED OIL, CLARIFIED, FOR VaRNISIES.-Heat in a coppei boiler 50 gallons of linseed oil to 280~ Fah.; add 2; lbs. of calcined white vitiiol, and keep the oil at the above temperature for half an hour; then remove it firom the fire, and in twenty-four hours decant the clear oil, which should stand for a few weeks before it is used for varnish. 88 STEAM Pow ER AND) TIS STrEAM-ENGINE. TABLE Of the Pressure on a square and circular Inch, respectively, exerted by the elastieforce of Steam at various degrees of Temperature, with the HIeight of the column of liercury it will support. 1. PRESSURE ON A SQUARE INCH. I. PRESSURE ON A CIRCULAR INCH. aINCH'~~~~~~~~.)~ fi cc z( "I -!.2 1: F i 220 2- 1 963 5-15 222 2j 3'183 6'56 222 3 2356 6-18 224 3 3819 787 223 3~ 2'749 7'21 226 8i 4'456 9'18 225 4 3'141 8'24 228 4 5'093 10'5 227 4} 3'534 9'27 230 4 5729 11 8 228 5 3.927 103 232 5 6.366 13-1 231 6 4712 123 2 6 7 639 15.7 2320 612893 165 241 31018 21056 240 249 46 1 19 226 245 93 1209 24 9 241 10 7'854 20'6 247 10 12'73 26'2 242 10- 8'247 21'6 248 10} 18'36 27 5 243 1 1 8 2439 22 8 4 25 0 1 93 114 0 28 9 244 11 534 92 237 251 11 14-65 1301 245 12 9424 24107 252 12 15327 31'5 252 15 11'78 30'9 259 15 19'0 9) 39'3 26 20 15 432071 412 270 20 25746 520 5 269 25 19'63 51'5 278 25 31'83 65'6 276 30 23-56 612 8 287 30 38'19 78'7 283 35 27 49 12-1 294 35 4456 91 18 289 4 0 31461 82 4 300 40 5092 105 24 294 45 3534 92-7 305 45 5720 1 2618 243 1501 863 10 309 0 6366 11 1400 28 9 252a 15 1 11978 309 259 15 1 393 300 50 39~2'7 103 309 50 63~66 1 131 AMALGAMS —VTAiRNTSHES. 8'9 AMALGAMS. WHeN- mercury is alloyed with- any metal the6 compound: is calledE an amalgam of that metal; as, for example, an amalgam- of tin, bismuth — -&c. Amalgam for Electrical Machihe's. 1. Fuse 1 oz. of zinc with ~: oz. of tin, at' as low a temperature as possible; then add 13 oz. of quicksilvel, previously made hot; mnix, pour out, and, when cold reduce it to' powder, and triturate' it with- sufficient quicksilver to, bring it; to a proper consistence. 2. Zinc 1 part; tin 1; quicksilver 2. Melt together. 3. Zinc 2 parts-; tin 1; mercury 5. 4,. La Beaume's. Pour into a chalked wooden box 6 oz. of quicksilver; put into an iron ladle ~i oz. of beeswax, with 2 oz. of purified zinc, and 1 oz. of grain tin; set it: over' a brisk fire, and when the metals are melted pour them into the box, avoiding tlie dross. Wlien cold reduce it to powder, and mix it witli lard. Keep it in a box covered with tallow, and spread it on leather' for use. l Liquid Amalgam for Silveering Globes, &dc. Pure lead 1 oz; grain tin 1 oz.;: melt in a clean ladle, and immediately add 1 oz. of bismuth. Skim off the':dross, remove the ladle from the fire, and before the metal sets add 10 oz. of quicksilver. Stir together, avoiding' thlie fumes. Amalgcnm for Varnishing Plastic Figures. Melt 2 oz. of tin. with j oz. of bismuth, and add ~ oz. of quicksilver. When cold: giHiid it with whitef of egg, and apply to the figure. V A- R N I S II E S. Preparations: of Lac. Stick-lac consists of twigs of several kinds of trees encrusted with a resinous matter, produced by the puncture of an insect called the cocus- lacca. This, triturate'd' with water, and dried, forms seedlac. The- seed-lac, when heated and pressed in- cotton bags, forms shell-lac. Lac dye is the coloring matter extracited from stick-lac by water, and evaporated to dryness, with the' ad'dition' of eartlhy matters, and formed- into square;-cakes'-. Seedlac: and shell-lac are.chiefly used. in- varnishes, dissolvecl in rectified spirits,. or rectified wood na'plhtha.. The alcoholic solution is rendered-paler; so that it may be used for polishing light colored woods, by digesting: it in the sun, or near a fire, for two or three weeks, with good animal charcoal, and then filtering it througli paper in a funnel heated wit- hot water. Shelllac may be bleached by dissolving it' in a solu't-ion'iof' potash, or soda, and':passing' chlorine into the solution. 8* 90 VARNISHES. The precipitated lac is collected, and well washed. Kastner; directs 3 parts of carbonate of potash to. be dissolved in 24 of water, and 3 of lime added, and the whole digested in a close vessel for twenty-four hours. The clear liquor is poured off, and boiled with 4 parts of shell-lac. When cold, dilute with 4 times its bulk of water, and filter; then add chloride of lime, and afterwards diluted muriatic acid. With these preliminary remarks we come now to the lacquers, or varnishes. rThe Famous Brilliant Frenzclh Varnish jor Boots and Shoes. Take I of a pint of spirits of wine; 5 pints white wine; J pound of powdered gum senegal; 6 oz. loaf sugar; 2 oz. powdered galls; 4 oz. green copperas. Dissolve the, sugar and gum in the wine. When dissolved, strain; then put it on a slow fire, being careful not to let it boil. In this state put in the galls, copperas, and the alcohol, stirring it well for five minutes. Then set off, and when nearly cool strain through flannel, and bottle for use. It is applied with a pencil brush. If not sufficiently black a little sulphate of iron, and half a pint of a strong decoction of logwood, may be added, with -i oz. pearlash. Black'Varnish. Take any varnish, of the class you wish, 16 parts; lampblack 21 parts. Grind the black in a small quantity of the varnish, then! mix it with the remainder. Cabinet-mackers' Varnish. Pale shell-lac 700 parts; mastic 65 parts; strongest alcohol 1000 parts. Dissolve. Dilute with alcohol. Callott's Soft Etching Varnish. Linseed oil 8 parts; benzoin 1 part; white wax 1 part. Melt and keep it heated until reduced to two thirds. Pale Carriage Trarnish. Copal 32 parts; pale oil 80 parts. Fuse and boil until stringy; then add dried white copperas 1 part; litharge 1 part. Boil again, then cool a little, and mix in spirits of turpentine 150 parts. Strain. While making the foregoing, take of gum animet 32 parts; pale oil 80 parts; dried sugar of lead 1 part; litharge 1 part; spirits of turpentine 170 parts. Pursue the same treatment as before, and mix the two compositions while hot. Second Quality of Carriage Varnish. Take of gum anime 32 parts; oil 100 parts; spirits of turpentine 150 parts; litharge I part; dried sugar of lead 1 part; dried copperas I part. ~ Proceed as above. Copal Varnish. Copal 30 parts; drying oil 25 parts; spirits of turpentine 50 parts. Put the copal into a vessel capable of holding 200 parts, VARNISHES. 91 and fuse it as quickly as possible,:then add the oil, previously heated to nearly the boiling point. Mix well, then cool a little, and add the spirit of turpentine; again mix well, and cover up until the temperature has fallen to 140~ Fall.; then strain. To Dissolve Copal in Spirit. Take the copal and expose it: in a vessel formed like a colander to the front of a fire, and receive the drops of melted gum in a basin of cold water; then well dry them, in a temperature of about 95~ Fah. By treating copal in this way it acquires the property: of dissolving in alcohol. Black- Copal Varnish. Take lamp-black, or ivory-black, in fine powder, and mix it with the varnish. Blue -Copal Tarnish. Indigo, Prussian blue, blue verditer, or ultra-marine., These substances must be powdered fine. Proceed as before. PFine Pale Copal Varnish. Pale African copal 1 part. Fuse, then add hot pale oil 2 parts. Boil until the mixture is stringy, then cool a little, and add 3 parts of pale spirits of turpentine. Mix well. Flaxen Grey Copal Varnish. Ceruse, which forms the ground of the paste, mixed with a small quantity of Cologne earth, as much English red, or carminated lake, and a particle of Prussian blue, and color the varnish therewith. Green Copal Varnish. Verdigris, crystallized verdigris, compound green (a mixture of proper proportions, from a fourth to two-thirds, according to the tint intended to be given. The white used for this purpose is cefruse, or the white oxide of lead, or Spanish white. Proceed as before. SImproved Copal Varnish. Caoutchoucine (white anId scentless), strong alcohol, equal parts; copal in the proportion of two pounds to a gallon. Digest in a close vessel, without heat, fol one week. Pearl Grey Copal Varnish. White and black; white and blule; fol- example, ceruse nand lamp-black; ceruse and indigo. Mix them with the varnish, according to the tint required. Purple Copal Varnish. Prussian blue and vermilion, or any other blue and red; then proceed as before. 92 VARNISHES. Red Copal Varnish. 1. Vermilion, red oxide of lead (minium), red: ochire, or Prussian red, &c., and proceed as before. 2. Dragon's blood, brick red, or'Venetian red, &Tc., and proceed as before. Violet" Copal Var.nish. Vermilion, blue, white, in proportions required to color the varnish. White Copal Varnish. Copal 16 parts; melt, and add hot linseed oil 8 parts; spirits of turpentine 15 parts; finest white lead to color. Yellow Copal Varnish. Yellow oxide of lead, or Naples and Montpelier, both reduced to impalpable powder. Tliese yellows are hurt by contact with iron or steel. In mixing them, therefore, a hiorn spatula, with a glass mortar and pestle, must be employed. Or gum guttme, yellow ochre, or Dutch pink, according to the nature- and tone of the color to be imitated, and proceed as before.!Maslic Varnish. Gum mastic 5 pounds; spirits of turpentine 2 gallons. Mix with ma oderate heat (carefully applied), in a close vessel, then add pale turpentine varnish 3 pints. Mix well. Another. Mastic i pound; white wax I ounce; oil of turpentine l gallon. Reduce the wax and mastic small, then digest in a close vessel, with heat, until dissolved. Common Oil Varnish. Resini 4 pounds; genuine beeswax pound; boiled ile 1 gallon. -Mix withi heat, then add spirits of turpentine 2 quarts. Turpentine Varnish., Resin 1 part'; boiled oil I part. Melt, then add turpentine: 2 parts. Mix well. White Hard' Spirit Varnssh. Gunim sandarach 2: pounds; alcohol (65 op.) 1 gallon. Place them in a strong, well closed vessel, and apply the heat of warm water, with occasional agitation, until: dissolved; then add pale turpentine. varnish I pint. Mix well, and, let the whole res:t:for twenty-four hours, when it will be ready for use. White Spirit Varnish. Strongest alcohol 100 parts; sandarach 25 parts; tears mastic 6 parts; elemi 3: parts-; Venice turpentine 3, parts. Di1issoWe in a closely corked vessel. VARNI[SH ES. 93 VTarnish for Toys. Copal 7 parts; inastic 1 part; Venice turpentine - part; strongest alcohol 11 parts. Dissolve the copal first, with the aid of a little camphor, then add the mastic, &c., and thin with alcohol, as required. To Clean Varnish. Use a ley of potash, or soda, mixed with a little powdered chalk. Do not make tbe liquor too strong of the alkali. To Polish Varnish. Take 2 oz. powdered tripoli, put it in an earthen pot, with water to cover it; then take a piece of white flannel, lay it over 1a. piece of cork or rubber, and proceed to polish the varnis, always wetting it with thle tripoli and water. It will be known vwhen the process is finished by wiping a part of the work with a sponge, and observing whether there is a ilir evenr gloss. When this is t he case, take a bit of mutton suet and fine flour, and clean the worlk. Varnish for Harness. Take ~ pound of India-rubber; one gallon of spirit of turpentine; dissolve enough to make it into a jelly; then take equal quantities of good hot linseed oil, and the above mixture. Incorporate them well on a slow fire, and it is fit for use. A Tarirnishfor Fastening the Leather on Top -Rollers in Factories. Dissolve 21 oz. of gum arabic in water; and a like amount of isinglass dissolved in brandy, and it i3 fit for use. A Varnish to Preserve Glass from the Rays of the Sun. Reduce a quantity of gum tragacanth to fine powder, and let it dissolve for twenty-four hours in white of eggs well beat up; then rub it gently on the glass -with a brush. A fine Black Varnish for Coaches and Iron Work. Bitumen of Palestine 2 oz.; resin 2 oz.' umber 12 oz. Melt thenm separately, and then mix together over a moderate fire. Then pour upon them, while on the fire, 6 oz. clear boiled linseed oil, stirring the whole from time to time. Take it off the fire, and when moderately cool pour in 12 oz. of essence of turpentine. VIarnish.for Clock Faces. Spirits of wine 1 pint; divide it into four parts: mix one part withs an oz. of gum mastic in a bottle by itself; one part of spirit and J oz. gum sandarach in another bottle; and one part spirit and ~ oz. whitest part of gum benzoin. Mix and temper them to suit; if too thick add spirit; if too thin a little mastic; if too soft some sandairach or benzoin. When about to use it warm the silvered plate before the fire, and with a flat camel-hair pencil strolike it over till no white streaks appear; this will preserve it for many.years. 94 VARNISHES. Brown TVarnish. Rectified spirit 2 gallons; sandarach 3 pounds; shell-lac 2 pounds; pale turpentine varnish 1 quart. Put them into a tin bottle, cork securely, and agitate frequently, placing the tin occasionally in hot water till the gum is dissolved, then add a quart of pale turpentine varnish. Brilliant Amber Spirit Varnish. Fused amber, 4 oz.; sandarach 4 oz.'; mastic 4 oz.; highly rectified spirit 1 quart. Expose to the heat of a sand bath, with occasional agitation, till dissolved. The amber is fused in a close copper vessel, having a funnel-shaped projection, which passes through the bottom of the furnace by which the vessel is heated. Chinese TVarsish. Mastic 2'oz,; sandarach 2 oz.; rectified spirit 1 pint. Close the matrass with bladder, with a pin hole for the escape of vapor; heat to boiling in a sand or water bath, and when dissolved strain thlrough linen. Crystai TVarnish. Picked mastic 4 oz.; rectified spirit 1 pint; animal charcoal 1 oz. Digest, and filter. Picture Varnish. Chio turpentine 2 oz.; mastic 12 oz.; camphor ~ drachm; pounded glass 4 oz.; rectified oil- of turpentine 3 pints. This is for oil paintings. Canada Varnish. Clear balsam of Canada 4 oz.; camphelne 8 oz. Warm gently, and shake together till dissolved. This varnish is for maps, drawings, &c., which must be first sized over \with a solution of isinglass, taking care that every part is covered. When dry, the varnish is brushed over it. TiDngry's Essence Varnish. Powdered mastic 12 oz.; pure turpentine 1~ oz.; camphor ~ oz.; powdered glass 5 oz.; rectified oil of turpentine 1 quart. Conmnon Turpentine Varnish. This is merely clear pale resin, dissolved in oil of turpentine; usually 5 pounds of resin to 7 pounds of turpentine. Amber Varnish. Amber 16 oz.; melt in an iron pot, and add i pint of drying linseed oil, boiling hot, and add 3 oz. resin, and 3 oz. asphalte, each in fine powder. Stir till they are thoroughly incorporated; remove from the fire, and add a pint of warm oil of turpentine. Baflloon Varnish. Melt india-rubber: in small pieces with its -weight of boiled linseed oil, and thin it with oil of turpentine. ~AnISH.S. 95 Varnish for Xngraving on Copper. Yellow wax 1 oz.; mastic 1 oz. asphaltum i oz. Melt, pour into water, and form into balls for use. A softer varnish for engravers is made thus: Tallow I part, and 2 of yellow wax, or, with 2 oz, wax, 1 drachm common turpentine, and I drachm olive oil. Etching Varnishes. White wax 2 oz.; asphaltum 2 oz. Melt the wax in a clean.pipkin, add the asphaltum in powder, and boil to a propel' consistence. Pour it into warmv water, and form it into balls, which must be kneaded, and put into taffeta for use. Another. White wax 2 oz.; Burgundy pitch ~ oz; black pitch i oz.; melt together, and add by degrees 2 oz. powdered asphaltum, and boil,it till a drop cooled on a plate becomes brittle. Another. Equal quantities of linseed oil and mastic, melted together. Engraving Mixture for Writing on Steel. Sulphate of copper 1 oz.; sal ammoniac ~ oz. Pulverize separately, adding a little vermilion to color it, and mix with 14 oz. vinegar. Rub the steel with soft soap, and write with a hard, cleani pen, without a slit, dipped in the mixture. - Etching Fluids. For COPPrER-1. Aquafortis 2 oz.; water 5 oz. Blix. 2. C(allot's Eau Forte for:Fine'ouches.-Dissolve 4 parts each of verdigris, alum, sea salt, and sal ammoniac, in 8 parts vinegar; add 16. parts water, boil for a minute, and let it cool. For STEEL.-I1. Iodine 1 oz.,; iron filings ~ drachm; water 4 oz. Digest till the iron is dissolved. 2. Pyroligneous acid 4 parts by measure; alcohol 1 part. Mix, and add 1 part double aquafortis (sp. gr. 1-28). Apply it from lIto 15 minutes. Varnish for Engraving on Glass. Wax 1 oz.; mastic - oz.; asphaltum i oz.; turpentine - lldrachm. Another. Mastic 15 parts; turpentine 7; oil of spike 4. Le Blond's V'Varnish. Keep 4 pounds balsam of copaiva warmn in a sand or water bath, and add 16 oz. of copal, previously fused and coarsely powdered, by single ounces, daily, and stir it frequently. When dissolved add a little Chio turpentine. Sealinq TFVax Varnish. Black or colored sealing vax,'broken small, and sufficient rectified spirit to cover it; digest till dissolved. Black Japan. Boil together a gallon of boiled linseed oil, 8 oz. umber, and I oz. asphaltum, When sufficiently cool thin it with oil of turpentine. 96'VARN.ISirnEs. Brunswick Black. Melt 4 pounds asphaltum, add 2 pounds hot linseed oil, and when sufficiently cool add l gallon oil of turpentine. Varnish for Gun Barrels, after brownzing them. Shell-lac 1 oz.; dragon's blood l oz.; rectified spirit 1 quart. Dissolve and filter. Transfer Varnzisrh. Alcohol 5 oz.; pure Venice turpentine 4 oz.; mastic 1 oz. Hair arnzish. IDissolve 1 part of clippings of pigs' bristles, or horsehair, in 10 parts of drying linseed oil, by heat. Fibrous materials (cotton, flax, silk, &c.), imbued with tile varnish and dried, are used as a substitute for hair cloth. GClass Varnzish. This is:a solution of soluble glass, and is thus made: Fuse too-ether 15 parts powdered quartz (or fine sand), 10 parts potash, and I charcoal. Pulverize the mass, and expose it for some days to the air; treat the whole with cold water, which removes the foreign salts, &c.; boil the residue in 5 parts of water until it (lissol\ves. It is permanent in the air, and not dissolved by water. This varnish is used to protect wood, &c., from fire. Trarnish for Gilded Articles.. Gum-lac 4 parts; dragon's blood 4; annatto 4; gamboge 4; saffron 1. Dissolve each resin separately in 8 parts alcohol, and make a separate tincture with the dragon's blood and annatto, also in 8 parts alcohol each; then mix the former together, and add a sufficient quantity of the tinctures to give the required shade and color to the varnish. Gold VTarn-ishes. Turineric 1 drachmlltll ganboge 1 drachm; oil of turpentine 2 pints; shell-lac 5 ounces; sandarach 5 oz.; dragon's blood 7 driachms; thin mastic varnish 8 oz. Digest, with occasional agitation, for fourteen days, in a warm place; then set it aside to fine, uand pour off the clear. Another. Dutch leaf 1 part; ginboge 4; gum dragon 4; proof spirit 18. i acerate for twelve hours, then grind on a stone slab. Varnish for Water Color Drawings. Canada balsam 1 pint; oil of turpentine 2 parts, mixed. Size the draVwing before applying the varnish. Earthentoare Varnish. Flint glass 1 pa-rt; soda 1. Mix. Jifaailp. Mastic varnish 1 part; drying oil 2. Mix. PRPACTIOAL. TABLESo. 9 Another. | Mastic varnish 1 part; drying oil 1. Mix. -Another. Equal parts of mastic varnish, drying oil, and turpentine. Mix. IMetallic Varnish for CUoach Work, &c. Asphaltum 66 pounds. Melt, then add litharge 9 pounds; red lead 7 pounds; boil, then add boiled oil 12 gallons; yellow resin 12 pounds. Again boil, until in cooling the mixture may be rolled into pills'; then -add spirit~ of turpentine 30 gallons; lampblack 7 pounds. Mix:well. i mpermeable Varnish. Bo'ile.d oil 100 parts; finely powdered litharge 6 parts; genuine beeswax 5 parts. Boil until sufficiently thick and stringy, then i pour off the clear. Engravers'. Stopping-out Varnish..TAke lampblack and turpentine to malke a paste. PRACTICAL TABLES. Wz1mowET oF METALs —-WRouGHT IRON; SQUA.RE, ROUIND, AND FLAT,. Table 1. contains the weight of Square Iron in sizes, from i inch to six inches square, advancing by -I- inch; and from 6 to 12 inches square, advancing by ~ inch; and in lengths, from 1 foot to 18 feet.:The sizes are arranged inr the first column of each page, and the lengths along the top; the weight in lbs. immediately under the lengths, and in a line with the sizes. Table II. contains the weight of Round Iron in sizes from. inch to: inches diameter, advancing by ~ inch; and from 6 to 12 inches idiameter, advancing by i inch; and in lengths, from 1 foot to 18 feet. The sizes, lengths, and weights are arranged as in Table I.Table III. contains the. weight of Flat Iron in widths, from j inch to 6 inches diameter, advancing by l inch; in thicknesses, from l inch to l inch, advancing by i inch; and in lengths, fron 1 I to 18 feet. The widths, lengths, and weights are arranged as in the preceding tables, and the thicknesses alongside of the widths.. The tables are: all calculated to the nearest tenth of a pound. To the weights of bars of Wrought Iron add -,th prt for bars, of. soft steel; and from the same weights subtract,-t-th part. for bars of Cast Ir ono li_.us_~ ~... 98 PRACTICAL TABLES. TABLE I. S QUARE IRON. Size. 1 ft. 2 ft. 3 ft,.4 ft. 5 ft. 6 ft 7ft. 8 ft. 9 ft. Inch. lbs. lbs. lbs. lbs. lbs. lbs. lbs. l Ibs. lbs. 0-2 - 04 0 0 0 1-1 13 15 17 19' 05 1 0 14 1 9 2 4 2'9 3 8 388 483 0 8 1 7 2.5 3.4- 4 2 B51 5'9 6'8 7-6 1'3 26 4'0 5'3 6'6 79 9/2| 10,6 11.9 / 19 38 5'7 76 9.5 114 1383 152 171 2-6 52 7 8 104 129 15 5 18 1 20 7 23 3 1 34 6 8 101- 135 16.9 20.3 23'7 27.0/ 30'4 I- 4-3 8.6 12.8 17'] 21'4 25 7 29'9 34'2 38'5 1f 5.3 10o6 15 8 211] 26'4 31'7 37'0 4'22 47.5 J1 6.4 128 1921 25.6 32.0 38s3 447 51'1 5'75 l~ 7'6 152 22'8 304.38'0 45'61 53'2 60'8 68.4 1i 8'9 179/ 26'8 35'7 4461 536 62 5 71'4 80.3 1 10'4 20'7 31'1 41'4 51'8 621] 72'5 82'8 93 2 1 119 23 8 35 6 4175 59 4'713 832 951;106 9 2 13,5 27.0 40.6 54.1 6'76 811l 946 108. 2 1217 2j 15.3 305 458 611 76 3 916 1068 1221 1374 2, 17 1 34'2 51'3 68'4 85'6 102'7 119' 8 1369 154:0 21 1 91 38.1) 57-2'76-3 95'3 114-4 133 5'152'5 171.6 2~ 21.1 422 63'4 84.5 105'6 1267 147 8 169'0 190 1 2- 23.3 46.6 69.9 93.2 116.5 139.8 163'0 18631 209 6 2i 25.6! 51.1'~767 102 2 12~78 153.4 178'9 20451 230 0 24 27 9 559 838 111i8 139 7 167-6 195-7 223,5 251 5 3 30'4 608 91'2 12117 152'1 182'5 21291 243 3 273 7 31 33-0 66'0 99'0 132'0 165-1 198'1 231'1 26411 297 1 3. 35'7 71'4 1071 142'8 178'5 214'2 249'9 285'6 321'3 88 3885'770 115 4 1540 192'5 231'0 269'5 308'0 346'5 3~ 41i4. 82,8 124'2 165'6 20'70 248'4 289'8 331'3 372'7 34 44'4 88'8 133'3']77'7 222'1 266'5 310'9- 35563 399'8 34'47 5 95'1 142'6 190'1 237 7 285'2 332'7 380'3 427'8 3; 60-8 101 5 152'3 203 0 253 8 304 5 3553 406-0 456 8 PRACTICAL TABLES. 99 TABLE I. SQUARE IRON. Size. 10 ft. 11 ft. 12 ft. 13 ft. 14 ft. 15 ft. 16 ft. 17 ft. 18 ft. Inch. l bs. lbs. Ibs. Ibs. Ibs.- Ibs. Ibs. Ibs. lbs. 2-1 2'3 2'5 2-7 3'0 3'2. 3-4 3-6 3 8s 4-8 5-2 5-7 6-2 6.7 7.1 7.6 8-1 8-6. 8-5 9-3 11I0 11.8 12/0 13.5 14 4 15-2 132 14-5 15'8 17 2 1851 19'8 21'1 22-4 238 / 19'0 20'9 22-8 24'7 26'6 28'5 30'4 32'3 34-2 25*9 285 31;11 33*6 36-2 38-8 41-4 44 0'46*6 1 33'8 37 2 40'6 43 9 47'3 50'7 54'1 57*5 60-8 1. 42 8 47-1 51'3 55'6 59-9 64'2 68'4 72-7 77'0 1 52.8 581i 63 4 68 6 7-39 79.2 84-6 89'8 95.0 1 B369 70 3 76'7 83-1 89-5 95,9 102-2 1086.115'0 1t 76.0 83.6 )91.2 96.9 106.5 114,1 121:i 129"3 136-9 1t 89 3 98'2 1()071 116-0 1250 133!9 142 e 151'7 160-7 14 103'5 133'9 124-2 134-6 144-9 155:3 165'6 -176-0 186-3 1 118-8 130-7 142 6 154s5 166 4 178-2 1901 202 0 -213-9 2 135'2 148-7 162-2 175 8 189 3 202-8 216B3 229-8 243-4 2-. 1526 -167 9 183'21 198'4 213-7 228-9 244'2 259 5 274'7 42 1711 188-2 205 3 222 5 239-6 2567 2731t 2909 308 0 21 190'7 209'7 228-8; 247 9 266'9 286'0 305'1 324-1 343-2 21 211'2 232'3 253'4. 274'6 295'7 316-8 337'1 359'0 380'2 2A 232'9 256-2 279 5 302-8 326-1 349-4 372 396 0 419 3 2[ 255'6 281-2 306.71 3323 357r8 3883-4 409'0 434'51 460'1 2 7 1-279'4 3073 335'3 363-2 391-1 419'1 447'0 47560 502-9 3 304'2 334'6 365-0 395'4 425'8 456'2 486-7 517'1 547-5 3 3830'1 363-1 396 1 429'1 4621 495.2 528-2 561-2 594-2 38 357'0 39271 42841 464-2 499'9 535 6'5711 607 0 6427 1 3 1385-0 423'51| 462'01 500'5 539-0 577-51 616'0 6544'6 693'1 3.i 414'1 -45551 496-9 538 3 579'7 621-1 662-5[ 703'9 745-3 38~ 444-2 488-6 533'0 577 *4 62191 666-3 710-7 755-1 7995 3t 475-3 522-9 570-4 617'-9 6655| 713'0 6760 51 808'1 855'6 3; 507-6 558,31 609'1 659.8 710-6 761i31 812 1 862,91 913-6 10 RACTICAL TABLES.. TABLE I. SQU ARE IRON. Size. 1 ft. 2 ft. 3 ft. 4 ft. Sft. 6 ft. I7 ft. 8 ft. 9 ft. Inch. lbs.: Ibs. lbs. lbs. Ibs. Ibs. Ibs. lbs. Ibs. 4 54'1 108'2 1 62 3 21603 270' 41 324'5 378'6 432''7 486'8 4- (57'5 115:i',.172. t.230.1 287'66:!45.1 402.6 4601 517-7 4t 61'1 122-1 183 2 244.2 305.3 366.3 427.4 488-4/ 549.5 4. 64'7 129'4 194.1 258.81 3 23.5 388.2 452.9 517 61 582,3 4%. 68 4 136*9 205 3 273 8 342'2 410 7 479-1 547*6f -616-0 4 72 8-3 144-6 216 9j 28921 361-5 433.8 506.1 578.4 650-7 44{ 76!3 1:52'5/ 228-8 305.1 3811'-3 457,6 533'8 610'1 686'4 41 80'31 1'607 241'0- 321-3 401-7 482'0 562-3 6427 723'0 5 1 84'5 169-0 253-4 337-9 422-4 506-9 591-4 675-8/ 760'3 5%/ 88i8 177. 6 2'66'4- 35.5' 1;443'9 532.7 621-5 7103) 799'1 5% 93 2 186-3 279.5 372 7 41658 559%0 652-2 745.3 838'5 5 - 97' 7[ 195'a3 293 01 39061 488'3 585-9 683-6 781-3 878'9 5% 102'2 204'5 30()67 409'0 511'2 613*4 715-7 817 99 920-2 5% 1:07 0 21:3.S 3:20o'9 427'8 534'8 641 1 7-48 7 8.550 962 6 5%~ 111 -8 2ii235( 335.31 447'0 558-8 6705 782 -3 8940 10058 5 % 1' 16 7 233 3 3.50'0 466-7 583-4 7u00' 1 816'7 933-4 1050'0 -6 121.7i4 2.433 6.5 0 486-7 608 3 730'0 841-6 973. 10950 6 1:32-0 264'1 /'36 1 528-2 6602- 792-2 924-3 1056'3 1188-4 6' 1428 285'6 4284 571-3 7141 856'9 999-7 1142-5 1285'3 6, 1:54 0 3030(0 462'0] 616 0( 770-1 924'1 1078'1 12321 1'386'1 7 1656 331-2 49.6.9' 662-5 828 2 9993 8 11594 1325-1 1490-7 7i 177 71 355.3 533'0 710-7 888'4 10660 1243 1-421'4 l599'0 7;1 190l1 38083 57-0*4 7 60, 5 9507 1140o 8 133-1.0 15211 171112 7% 2030 406'0 609-1 8121 1015-1 1218.1 14212 1624 2 1827-2:8 216-3 432'7 649-. 865 3 1081'7 1298 0 1514'4 1730'7 1947'0 8% i 230 -l 460-.1 690 2] 920-3 11503 1380-411.610-5 1:8405- 2070o 6 8%:22442 488 4 7 32'74 976'911221i 1465'311709' 51195388 1;98'0 81 -258 8 517*6 776*,41035*2 129.40115:52.8 18116.2070 4 232921 9 273;8 547 0; 821.4 10956 1359 10-1642 3 8 19916 51 219003 25641.iO5 I:8 9 PRACTICAL TABLES. 101 TABLE I. SQUARE IRON. Size. 10 ft. 11 ft. 12 ft. 13 ft. 14 ft. 15 ft. 16 ft. 17 ft. 18 ft. Inch. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. 4 -540'8 594.9 649:0 703-1 757 -2 811-3 865*3 919.4 973-.5 4I 575-2 6327. 690'2'747'7 805-2 862-8 9203s 977.8 1035-3 41 [610'6- 671-6 7327 7 93'7 854'8 915'8 976-0 10379 10099'0 43 646'0 /711*'; 776'4 841.1 905.8s 970.5/1035:2 1099-9 1164-6 94 684 5 752'9 821'4 889'8 958 3 1026'7 1095'2 11 63'6 1232-1 44 723-1 795'4 8t;7 7 940'0 1012-3 1084-6 11569' 1229'2 1301o5 4 1 762-6 838-9 915-2 991-4 10677 1144'0 1220-2 1296 5 137 2'8 i 4; 803 3 883 7 9640 1044 3 1124-7- 1205 0 1285-3 1365-7 1446B0 5 844'8 92983 1013'8 1 098'2 11827 1267'2 13517 1436'2 1520'6 5j 887 8 976'6 1065'4 1154'2 1243'06 1331'8 1420'5 1509-3 1598! 1 5I 931*7 10'248 I1118-0 1211L2 13044 1397 5 14907 1583 9 16770j 5 976'611074'2 11711 19 269.511367'2 1464'9.1562-5 16602 17578 i 5 10224 1124. 6 1226-9 1329,1 1431 4 1533-6 1635'8 1738'1 1840'3 5t 10695 1176-5 1283 4 1390-411497'3 1604'3 1711' 2 1818'2 1925-2 58 1117'6 1229-3 1341 1 1452-8 156416 1676'3 1788 1 1899-9 2011-6 11600 1160 2834 14001 15167 16334 14750 1 1866-7 1983-4 2100'1 6 1229-6 1338' 3 1460:0 1581'6 1703' 3 18925'0 1946'6 2068-3 2190'0 61 1320-4 14524 1584 4 1716-5 18486 l980)6 2112-6 2244-7 2376-7 61 1428'2 1571'0 1713-8 1856-6 19994 12142'21 2285-1 2427-9 2570-7 61 1540-1 16941 18481 2002-2 2056 2 310-2 2464-2 2618-2 2772-2, 7 1656-3 18220 19876 2153 2 2318-8 2484-5 2650-1 2815-7 2981. -41 7 [ 17767 195484 2132 1 2309-7 2487-4 2665-1 2842-8 3020-4 3198s1 71 19014 2091-5 2'281-6 2471-8 2661.9 2852 0'3042-2 3232-3 342241 71 2030-2 2233-3 2436 3 2639'3 28423 3045'4 3248 4 3451-4 3654'4 8 2163'4 2379.7- 596'0 2812-4 3028.:7 3 245-0 3461 -413677'7{3894'0 81 2300'7 2530-7 27760, 8 2990.9 32209 3451 0 3681-1 3911' 4141'2 8% 2442-2 2686 4 2930 6 3174-9 3419-1 3663 3 3907'5 41517 4396-0 81 2588-0 2846'8 31056 33~64'4 3623-2 3882-0 4140-8 4399'6 4658'4: 9 2737-9 30117 3285-5 3559]3 3833'1 4106-9 4380-7 4654'5 49283 3 1i02o PRACTICAmL TABL.ES. TABLE IR. SQUARNE IR-ON.:. Size. 1 ft. 2 ft. 3 ft. 4 ft. 5 ft. 6ft. 7ft. sft. ft. Inch. Ibs. lbs. Ibs. lbs. Ibs. lbs. l bs. lbs. lbs. 9' 289-2 5784 -867-ill69 1446:1 1735 3'2024-5 2313-8 2630iI 9~ 305-1 610'1 91'52 1220'2 -1525'3 18830'321l354 24404' 2745 5 91 321-3 6427 964-0 12853 16067 19.28 0'22493 25707'2892 3 1 3379:6758 1501 3 1-8.351 7 1689 6 20275 236-5 -42703 4 -4041 i0 10[ 355'1i 710-2 1l065 4 142051 177.5'7 2130,88 2486-0 2841 1 31 9.6 2 10+ 372'7 745 11118'0114907l:18-634 223,6 0!'2608'i7 2981q4 3354'0 104 390-6 781'31 1171-911562'5 1953'1 2343'8,2734'4'3125'0 3515 7 11 409'0 817'9 11226'9 1635'8 204478 2453'8 2:862'*7 13271-7 3680'6 11+ 427 8 55'6 1283'4 117112'23139'1 2566'912994'7 3422 5 3850-3 11: 4'47-0[ 894'0 113411 11788'1 2235'1 2682'1 3129'2 3576'2 4023'2 11+:466'7 933-4 1400'1 18667 T333'4 2800'1 3266'8 3733-5 4200-2 12 486 7 973`3 1460-0 1946-6 2438 3 3291 993406-6 3893 2 4379.9 GLAZES.-Common earthenware is glazed with a composition containing lead, on which account it is unfit for many pharmaceutical;purposes. The following glaze has been proposed, among others, as a substitute: 100 parts of washed sand, 80 of purified potash, 10 of nitre, and 20 of slaked lime; all well mixed, and heated in a blacklead crucible, in a:reverberatory firnace, till the mass flows into a clear glass. It is then to be reduced to powder. The goods to be slightly burnt, placed under water, and sprinkled with the powder. GLAZE FOR PORCELAIN. —Feldspar 27. parts, borax 18, Lynn sand 4, nitre 3, soda 3, Cornwall china clay 3 parts. Melt together to form a frit, and reduce it to a powder, with 3 parts of calcined borax. SOL.VENT-FOR.OLD PUTTY AND PAINT. — Soft soap mixed with solution of potash or caustic soda; or pearlash and slaked lime mixed with sufficient water to form a paste.. Either of these laid on with an old brush or rag, and left for some hours, will render it easily,removable. PRACT$-AL TABLES. 103 TABLE I. SQUARE IRON. Size. 10 ft. 11 ft. 12 ft.- 13:ft. 1.4 ft.:15 ft. 16 ft. 17 ft. 18 ft. Inch. lbs. lbs. l sbs l. Ibs. Ibs. Ibs. bs. bIhs. 9.2892-2.3181 84 3417046 37 599 4049-14338 3 4627 5 4916 7 5206-0 9] 3050'6i33556'36607'3965 7 42708145758 48809 51860 54910 9 32'133'3534-7 38564 41773 449'8. 6 48 513 5462.6 5840,10 3379,2 37171 4055 0 4393.0 47V80.9'5068 8 54067 57446 60826 m,101 3551 4 3906'5 4261 6 46:16 s 49719'95327 0 5682 2 6037 3 63.924 10( 37,26-7 40994 4,472-1 4844-7 5217 455901 59628 6335-4 6708'1,10 "3906 3 4297 0 4687 5078'2 5468. 85859-4 6250-0 6644'7 7031.3 1-1,4089 6 4498 6 4907[4 5316 5 05725'4'6134-416543'416952' 3 7361'3 111 4278 1 47059 5133-7 5561 6 5989 4'6417 21845'0 7272' 87700'6 /1 44702 49173'5364' 5811s3 6258-36705-4 152-4 7599. 48046-4 11 46668 5133-5 5600i2 6066-9 6533'6 70003 7466-9 7933-6 8400'3 12 48666 B5353 2 5839 9 6326-5 6813-2,72998 778B65 8273 2 8759 8 ScoURING DROPS FOR REMOVTNG GRnEASE. —. Alcohol (pure) 6 oz., camphor 2 oz., rectified essence of lemon 8 oz. 2.- Camphene 3 oz., essence of lemon 1 oz. Mix. Some direct thlem to be distilled together. 3. French. Camphene 8 oz., pure alcohol 1 oz., sulphuric ether 1 oz., essence of lemon 1 dr. 4. Spirits of wine 1 pint, white soap 3 oz., oxgall 3 oz., essence of lemon ~ oz. BALLS, HEEL.-I. Melt together 4 oz. of mutton suet, 1 oz. of beeswsax, 1 oz. of sweet oil, I oz. oil of turpentine, and stir in 1 oz. of powdered gum arabic, and ~ oz. of fine lamlpblack. 2. Beeswax 8 oz., tallow 1 oz., powdered gum 1 oz., lampblack q. s. These are used not merely by;the shoemaker, but to copy inscriptions, raised patterns, &c., by rubbing the ball on paper laid over, the article to be copied. For copying ancient mnonumental brasses; a similar compound, colored with bronze powder ilntead:of ampblaek, is sometimes employed. ~104 PRACTICAL TABLIS: TABLE II. ROUN D IRON. Size. 1ft. ft. 3ft. 4t. 3 ft. 5 ft. 6 ft. 7 ft. 8 ft. 9 ft. Inch. lbs. lbs. lbs. lbs. lbs.: lbs. lbs. lbs. Ibs. 0 -2 03 0 5 0 7 0-8 1-01 1-2 1'3 1'5. 04 07 1-1 1-5 1-9 2.2 2. 6 30 3'4 0.7 1 3 20 2-7 3-3 40 40 6 53 60 1-0 2'1 3'1 4'2 5'2 6-3 7,3 8-3 9-4 1-'5 3'0 45 60) 7' 5 9'0 10 5 11'9 13'4 2 0 4 1 6-11 8-1 10-2 12-2 14-2 16'3 18-3.1 27. 53S 8:0 10'6 13'3 15'9 18'6 21'2 23'9 1~ 3-4 6'7 10-1 13-4 1608 20'2 23'5 26,9 30'2 1I 4-2 831 12.5 16 7 20'9 25'0 29 2 33-4 37-5 1- 5'0 10-0 15'1 20'1 25'1 30'1 35'1 40'2 45-2 1+ 6'0 11'9 17'9[ 23-9[ 29-9 35-8 418s 47'8 53'7 1i 7'0 14(,I 2101 28 0 35'1 42-1 49'1 56-1 63'1 1i 8.1 16-3 2441 32-5 40'6 48.8 56.9 65,0 73-2 1+ 9 3 18 7 28 0 37*3 46-7 56B0 65-3 74 7 84 0 2 10 6 212)- 31-8 42.51 53 1 63-7 74-3 84-9 95 5 2+ 12-0 240 360 480 5969 71-9 83 9 95-9 10'-9 2 13,5 26..9 40.3 53.8 67.2 80-6 94-1 107' 5 121-0 2+ 15'0 3001 44-9 60'0 74 9 89-9 104'8 119'8| 134'8 2j 16~7 33.4 50.1 66.8 83 4 100'1 116-8 133'5 150'2 2 18'8 36'6 54'9 /732 91'5 109'8 128'1 146/3 164-6 21 20'1 40'2 60'2 80'3 100'4 120 5 140'6 160 6 180'7 2j 21 *9 439 658 87 8 109 7 131 7 153 6 175Q6 197-5 3 23'9 47-8 71-7 95'6 119'4 14.3, 167-2 191'1 2150 3+ 25 9 5191 77-8 103 7 129-6 155.6 181-5 207'4 233-3 3+ 28'0 56'1 84-1 112-2 140-2 168'2 196'3 224!3 253-4 31 30'2 60'5 9071 121'0 151 2 181'4 211'7 241 9 272-2 3+ 32'5 65-0 97-5 130-0 162-6 195'1 227"-6 260 1 292'6 31 34-9 69-8 104'7 139'5 174'4 209'3 244'2 2790-1 314-0 3+ 83731 74-7 1120 - 149.31 186'7 224(0 261-3 2987 336'0 1 39'9 79'7 119'6 159'51 199'3 239'2 279 01 318'91 358'8 PRACTICAL TABLES. 245 TABLE II. RO U N D IRO N. Size. 10 ft. 11 ft. 12 ft. 13 -ft. 14 5 ft. 1ft. 15ft t. 17 ft. 18 ft. Inch. Ibs. Ibs. lbs. Ibs. lbs. lbs. lbs. bs. lbs. + i'l7 1i8sl 2'0 2-1 231 25 -26 2'8.3'0: 3'7 411 4-5 4'8,5'2 56 6 0 - 60 63 667 / 6'6 73 8'01 8'6 9'3 991 106 11'3 11'9 - i'10I4 1'5 12'5 13'6 14'6 15,6 16'7 17 13 18' 8 + 14,9 116:,4 17'9 19-4 20'9[. 224 23'9 25'4 26'9' 20'3 22'4 24'4 26'4 28'4 30'5i 3825' 34' 5'36'6 1''2;6 5 29 2/ -31'8 3,4 5 3872 39 8[1 42'5 45'1 47'8 1i 3386 387'0 40'3 W 437 47.'0 50'4 53'8 57'1/ 605 1j 41'7 45'9 50,1 54,2 584 626 6 668 70'9 75-1' 1: 50o2 55.2 60'2 65'2 70'3 753 80 3 85 3 90.3 1~ | 59'7 657 71-7 77'6:83,6 89'6 956 101.5 107-5 1f'l70'1 77 -1 84,1 91'1 98'1 105.2] 1112.2 119'2 126'2 I' 81-3 89.4 -97 5 1'05-7 113.8 1219 1230(0 138 2] 14638 I4 93-3 1027 112 0 1213 130l7 1400 149 3 1587 168-0 2 10'6-2 116'8 1'27'41 138 0 148'6 169 2 169'9 180'5 192-1 24-: 119'9 1391 9:143'9 15568 167 8 179'8 1818 193'8 205'8 2 134'4 147'84 161'3 1747 188'2 201'6 215'0 228'5 241'9 2 149'8 164'7 179'7 194'7 209'7 224'6 2396 254'6 269'6 2z 1669 1:183'6 200'3 216'91 233'6 250'3 267'0 283'7 300'4 ~- 182'9 201,2:219'5 237"-8 256'1 274' 4 2921' 311-0 3829'3 2t 200a 8 220-8 240 9 261-2 2811 301 1 321 2 341 3 361-4 2; 219'4 241:4 263'4 285'3 307'2 329,2 351'1 373 0 395 0 3 238.9 262-8 286'7 310'5 334'4 358'3 382'2 406'11 430'0 34 2:59.3 -285'2 311'1 337,0 363.0 388&9 414'8 440'7 46671:3!2804:308'4 33635 364'5 392,6 420'6.448'6 476'7 504'7:34: /802'4| 332'61 362'91 393 1 423:4 453'6 483 8 514'1i. 544-3 3:4: 325;1 357-6:390'1 422'7 455'2 4877 520('2 5562.' 585'2 31 348'9 383'7 418'6 455'5 488'4 523 3 558'2 5931 6279 34 373'3 410'7 448'0 486'3 522.6 560'0 597 -3 634'6 672'0:4 398;6 438-5 478-3 5182 558'1 598'0 637'8 6777 71'76 "'1O ~ PRACTICAL TABLES, TABLE 1]. ROUND IRON. Size. 1 ft. 2 ft. 3 ft.' 4 ft. 5 ft. ft. ft. 8 ft. 9 ft, Inch. lbs. lbs. Ibs. lbs. Ibs. Ibs. lbs. lbs. lbs. 4 42-5 84 9 127 4 169 9 212-3 2548 297 2 3397 382-2 4} 452 90I3 135 5 18o71 225-9 271 0 31062 361-4 4066 4 48 0 959 143'9 191'8 239-8 287 7 3357 3836 431;6 4| 508 1016 15241 20331 2541 3049 3557 406 4573'4t 53'81 107 5 161'31 215'0 268'8 322'6 3763 430-1 4838 41 56-8 113 6 170'4 227 2 283 9 3407 397 5 454'3 511'1 4} 60'0 119 8 179'7 23961 299'5 359'4 419-3 479'2 539'1 4j 631 1262 189-3 2a5241 3155 3786 4417 5048 567-8 5 668 1335 200[ 3 267 01 333 8 4005 467-3 5340 600-8 5 j 69-7 1395-5 209-2 278'9 348'-7 418'4 4881l 557'8 627 6 5r 732 1463 219 5 292.7 3'659 439.0 5122 585 4 6585. 5[ 7667 153-4 230 81 306 8 383-5 460-2 536-9 613'6 690-3 5- 80'3 160'6! 2409 321,21 401 5 481'8 562'1 642'4 7227 5] 8 0 16801 252-0 33601 420-0 504'0 588 0 672'0 75fi60 5 87-8 175'6 263] 3 351.1 438-9 52671 614-4 702'2 7 90]0 59 91,6 183[3 274-9 366'-5 458-2 549'81 641-4 733'1 8247 6 95 6 19111 286-7 382'2 477'8 573-3 668 9 764'4 860?0 6! 103-7 207 4] 311 1 414-8 518 5 622-2 725-9 829'6 933 3 6. 112 2 224-31:336 5 448-6 560'8 673'0/ 785 1[ 897'3[ 10094 6 1'210 241 9 362-9 483 8 604 8 725'8 846'7 967'6 10886 7 130.0 260o1 3901 520.2 650.2 780.3 910-3 1040'4 11704 7t 1395 279 1 418-6 558.2 697 71 837 31 976'8 1116'4 1255'9 7j 149-3'298 7 448 0 597 3 74161 896'011045 3 1194'6 13440 71 159'5 3189 4784 637 8 797 3 956'711116'2 1275'611435'1 8 |!69'9 33971 5096 6794/ 849'311019'1 1189tO 1358'81528'7 8+ 180'7 36141 542/ 1 722-8 903 5 1084'2 1264'9 1445'6 1626'3 18 191 8 383 -6 595 4 767.2 959'0l 1150.81342'6 1534 5 11726 3 8 203 3 406 5 609[ 8 813'0 1016 3 1219:6 1422'8 1626'1 182.9'3 9 215 0 430.1; 64B51 8602 1075B21 1290-2115053 1720'3 1935'4 PRACTICAL TABLES. 107o TABLE II. ROUND IRON. Size. 10 ft. I11 ft. 12 ft. 13 ft. 14 ft. 15 ft. 16 ft. 17 ft. 18 ft. Inch. lbs. lbs. lbs. Ibs. lbs. Ils. Ibs. Ls. Ibs. 4 424'6 467'1 5096 520 6945 6370 6764 7219 7644 4 451'7 496-9 542'1 587I3 632 4 677'6 722-8 761'0 813-1 41 4795 527 5 575'4 G6234 671I3 719-3 767 2 815-2 863'1 4 508-2 559'0 69 660'6 711 4 762-2 81310 863'9 914-7;4 537 6 591 4 645 1 698 9 752 806 8064 8602 913'9 967'7 41 567-9 624-7 681 738 /2 7950 851 8 908'6 965-4 102'2-2 4t 599'0 658-9 718 8 7787 8 8 86 8198 5 95841 10183 10782 481 6309' 694' 0 0 757 21 8202 8833 946 41009 5 1072-6 1135'7 5 667.5 734-3 801s 0 867 8 934 5 1001 3 1068'0 1134'8 1201.5 5j 697-3 767.0 836' 906'5 976'2 1046 0 1115 7/1185-4 12556 2 6 731'7 804-9 878-1 951-2 1024'4 1097 6 1170' 8 1243' 9 1317-1 5 767 0 813 7 920'4 99711 10;388 1150' 5 1227'2 1303'9 1380-6! 5 803 0 8833 963 6 1044-0 -1124 3 1204 6 1284 9 1365-2 1445.51 1 840-0 924 0 1008-0 1092-0 1176-0 1260-0 1344 0 1428 0 1512 O 1 877 8 9655 10533 L1411 112-28' 9 1,316 6 1404'4 1492'2 1580'0 5; 916 3 1008 0 1099' 6 119192 1282'9 1374-5 1466'1 1557 8 1649'4 6 95515 1051 J1 1146.6 124 22 1337*7 1432-3 1528-8 11624.4 1719'9 61 1037 0 1140 7 1244 4 1348'2 1451'9 1555-6 16.593 1763'0 1866'7 64 1121 6 1233 8 t1345 9 1458-1 1570'2 1682-4 1794' 6 1906'7 2018'9 6k 1209-6 1330 6 1451 5 1572 5 1693-4 1814 4 1935-4 2056 3 2177-3 7 13005 1430.5 1, 560'6 1690 6 1820-7 1950'7 2088s8 2210o8 2340.9 7 1I395415350 16745 18141 11953'6 2093 2 2232'7 23722 22511'8 q 1493%3 1642-6 1791 91941'3 20906 2239'9 9 23892 2538"6 2687' 9 7t 1594 6 1754 0 1913 5 2072-9 2232-4 2391 8 2551-3 2710 8 28702 8 11698618 s68 4 20383 I2208 1)23878 0 2547 8 27177 2887 6 3057'4 8-1 1809'0 1987'? 2168'412349 012529*7 2740 412891'1 3071-8 3252'84i |1918'1 2109:9 2301 7 ]249351 2685'3 2879 1 3068 9 3260(7 3452 5 8j 2032'6 2235'9 2439 11 26424 28456 3048 932522 34554 36587 9 215041 2365'4 2580'51 27955 3010'6 3225 613440'6 365567 38707 1WS PAdCTWAL.TAibL S TABLE IL R- O., JN D 1 RO N, Size l:ft I 2ft. / 7ft'ft. 4ft' 5.ft; 6ft. 7 ft. 9.ft....I i- _ Inrict. Ibt. Ibs., Ibs., Ibtb. Ibs, Ibs. Ibs.. Ibs, Wb9. 9 227-2 454 3 6815 9086.11135-8 1362 9 l190 1 1817 2 2044 4 9:j 239'6 479'21 7,18'8[' 958:4 11198'01 1437'6 116772 1916'8 2156'4 9$ 252 4 5605 8:s757 1 009 5 1261 9 1514'3 1766 20190.229114 10:( 266 3 5 32 6 7989 106512 1331-4 1597*7 186400 213013 283966 10 j 2789 5657'81 836' 8 1115 7 1394'6 1673 5 1952]5 2231'4125103 10:i: 1 292-7 6585 4 878'1 1170'8 1463'4 1756-1 2048'8 2341'5 2634-2 100 306'8 60036 920'4 1227'2 15340' 1840' 8 2147C6 2454-4 2'76-1'2 1:11 3212 642'4 9636 1284 9 1:6061 1927-3 22485 2569,7 2890-9 11$ 33610 672:0,1008 0 1344'0t 1i680:0 201601 2352 0 2688 0 302480 1:1i: 3511 1022 1053 3 1404. 4 117555:21066 2457 7 2808.8 3159.9 11 366'5 733'1 1099:6 1466-1 1832 7121992 2565-8 29323 3298 8 i8 I12 3822 76441146 5 15288 819110 22932 2675o 3057 7 3435939 BJRON-zr(: LiQurDs, Pon BRaosxzrNG COPPER MEDALS, FIGURES, INSTRUIMENTS,: &e.-1. Sal ammloniac 1 dri, oxalic acid 15 gr., vinegar 11 pint. After well cleaning the article, to be bronzed, wairm it gentl3; and,brush it over with the: liquid, using onlv a small quantitya at a time. When rubbed dry, repeat the application till the desired itint is obtained. [For copper medals, electrotype casts, &c,.] 2 Sal ammoniac 1 oz., cream of tartar 3 oz, salt 6 oz. Dissolve in a pint of hot water,, add 2 oz; of iiitre, and 2 oz. of nitlate'of copper:dissolved in. pint of water. 3.. Salt of sorrel I oz., sal ammoniac 2 oz., white vinegar 14 oz. [To give an antique appearance to bronze figures, &c.] 4. A diluted solution of tmuriate of platina. [For copper binding screws, and other small articles..] I 5.:A w:.eak solution. of- hydro-sulphuret of ammonia, or of silphurvet of potassiumll. [For e1lectrotype medals. Another method is the folloWingr: Immediately on, removing the electrotype east from the solution, bruslh it over dwith good bfack lead; then hieat it moderately, and brush it ovjer with a painting brush, the slightest Imoitgte b eing:usaed. nio'Wir~ ~ ~ ~ ~ PRATa'YIAL TA BEi.. -109 TABLE II. ROUND IRON. Size. 10 ft. 11 ft. 12 ft. 13 ft. 1 5 ft. 15 ft. 16 ft. 17 ft. 18 ft. Inch. lbs. lIbs. lbs. lbs. ]b. lbs. bs. lbs. hlbs. 9 22715- 2498.7 2725.8 2953o0 31801 3407 3 3634-4 3861. 6 4088s7 9~ 2396'0 2635'6 2875' 2 3114-8 13,54413594'0 3833'6 4073'2 4312-8 9 5 2,523'8 2776' 1:3028'5 3280 9 3533 3 3785'6 4038 0 4290'4 4542'8 10 26629|292921.3195 5 3461,7 3728'039943 4260,6 45269 4793-2 101 2789-2 3068'2:'347 1 3.626 0 39049'14183'8 4462'8 47417, 5020-6 104,92693219'63123 3804.9 409764390 84683'0 4975'7t5268'4 10 3068O0 o 74'8 3681 613988'4 4295'2 46020 14908'8 5215'6 5522'4 11:32122 35633413854'6 4175'8 4497'0,4818'2151395 5460-7'5781'9 11 3:13600 36960 4032'0 43681 14704'115040-1 5376'1 5712'1 604841 1 1 g; 3511*0 3862 1 42138 24564'449155 55266'6 56197 5968'8 6319'9 I1 ls665 4140319 439844765'0 5131 15498-0 5864-6 6231-1 6597'6 12:3822,1 4204-3 4586 5 49687 5350 9!5733 1 6115 3 6497,5!6879 SOLUTIONS USED IN ELECTROTYPE MIANIPULATIONS, (&C. 1. Acid Solution of Copper for the Decomposinzg Cell. Saturated solution of sulphate of copper 2 parts, sulphuric acid 2 parts, water 6 or 8: ptirts. id2. Gold Solution. Dissolve 2 oz. of cyanide of potassium (by;Liebig's. method) in a pint of warm distilled water, add j-oz. of oxide of gold, and agitate together. 3. Silver Solustion. Dissolve 2 oz. of Liebig's cyanide of potassium in a pint of distilled water; add I oz. of moist oxide of silver (precipitated by limle water from a solution of the crystallized nitre), and agitate together till the oxide is dissolved. 4. Solution in wOhich Steel Articles are dipped before Electroplating them. Nitrate of silver 1 part, nitrate of mercury 1 part, nitric acid (sp. gr. 1'384) 4 parts, Water 120 parts. 5. Solvstion, or Pickle, for immersing Copper Articles in before Electroplatinsq. Sulphuric acid 64 parts,' water' 64, nitric acid 32, muriatic acid 1. Mix. The. article, free from grease, is dipped in the pickle for an second or two. 10 ~110 PRACTICAL TABLES. TABLE IIL FLAT IRON., 1 ft. 2 ft. I ft 4 ft. 5 ft. 6 ft. 7 ft. 8 ft. 9 ft. in. in. lbs. lbs. lbs. Ibs. lbs. lbs. Ibs. lbs. Ibs. I 1- 0-8 1"7 2'5 3'4 4'2 5'1 5'9 6'8 7'6 1 1'1 2'1 32 4'2 5-3- 63 74 8'4 9'5 l 1'3 2'5 3'8 51 63:7-6 8'9 10-1 114 I 1i 1,5 30 4'4. 5 9 7'4 8'9 10'4 11'8 133 2 17 3~ 4 51 6'8 85 101 118 13-5 1592. 21 1 9 3 8 5-7 7 6 9'5 11-4 13 3 15-2 17 1.. 2i 2'1 4'2 6'3 8'4 10; 127 148 169 190 -. 2~ 2-3 4L6 7 0 -9 3 11 6 13'9 16'3 18-6 20'9 }; )3 2'55 511 76 10'1 127 15i2 17 203:22'8 /~/ 3~ 2'7 5'5 8'2 11'0 13'7 16 5 19'2 22'0 24'7 g 3s 3'0 5'9 8'9 11'8 14'8 177 207 237 26'6 3 32 6~3 9~5: 12.7 15,8 190 222 254 2'8-5 5 4 34 68 10-1 13-5 16-9 203 237 2770 304 4 3.6 72 10'8 14.4 180 21 5 25.1 28.7 32 3 I 44 38'76 114 15'2 19'0 228 26'6 30'4 34'2 4- 4t 470 8-0 12-0: 16 1 2011 241 28 1 321-1 36'1 i 5 4.2 8'4 12.7 16.9 21 1. 2 3 29*6 33-8.1 38!0 5 4!4 8'9 13'3 17'7 22'2 2'6 6 31'1 135'5 39'9 1 I5 4~6 9-3 1:3]9 18:6 23:2 279 32'5 372 41'8 5 49 971 14 6 19 4 24 3 29 2 34 0 38 9 43 7: 6 5'1 LOI 15 2 20 3 25-3 30 4 355 40-6 45 6. 1 1'3 25[ 3'8 5'1 6'3 7'6 8-9 10:1 11-4::,} 1- 1'6 3'2 48 6-3 79 95 111 127 143.1 1V 9 3-'8 5 7} 7'6 9'5 11 4 13 3 15 2 171 t 1 2-2 4,4 671 89 11 1 13'3 15'5 17:7 20'0 2 25 5'1'76 10'1 127 15.2 17 203 22'8 *.1 2~ 2'9 567 8-3 11'4 1453 17 20 220 8 2857 # 2- 32 63 95 127 158 190 22'2 -254 285S PRACTICAL TABLES. 111, TABLE.IIL FLAT IRON. 2 10 ft. 11 ft. 12 ft. 13 ft. 14ft. 15 ft. 16 ft. 17 ft. 18 ft. in. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbhs + 1 8 5 9 3 10'1 11'0 11-8 12 7 13 5 14-4 15-2 1 10-6 1116 127 137 14-8 158 169 17 9 190 + l 127 139 15-2 165 177 190 203 21 5 228 1 14-8 16-3 1717 192 207 222' 23/7 2 1 26'6. 2 16,9 186 203 22'0 23,7 25'4, 270 287 304 + 21 19-0 20 9 22-8 24' 7 26 6 28-5 304 32 3 34 2 i 2 21.1 23 2 2.53 27 5 29 6 81 7 33 8 35-9 38-0 + 2+ 23 2 25 6 27 9 30-2 32'5 34 9 37-2 39 5 41 8 3: 258 279 30,4 33.0 835'5 38.0 40.6 431 456 5 31 2 7.5 30o2 33o0 35-7 3885 41'3 43'9 46'7 49-4 i 31 29-6 32-5 3855 38 5 41-4 44*4 47 3 50 3 53-2 i 34 31 7 34-9 38-0 41-2 44 4 47-5 50 7 53 9 57 0 4 33.8 3r72 430 6 43-9 47o3 0so7 54'1 57.5 60-8 i 4+ 35 9 39 5 43 1 46-7 50 3 5399 57 5 61'0 64 6 ] 4- 38 0 41 8 45-6 49 4 53.2 57'0 60'8 64]6 6844+ 401 44'1 48 2 52'2 56'2 60-2 642 682 72'2 5 42 2 465 50 7 54 9 59-1 63-4 656 71-8 76-0 51 4 -4 48'8 53'2 57 7 62'1 66'5 7-1'0 75/ 4 799 + 4s 46"5 51 1 558 604 651 697 744 790 836 6 5+ 48'6 53'4 583 63-2 680 729 777 826 87*5 +41 6 50 7 55:8 60 8 655 9 70'9 76:0 81J1 86'2 9142 +-.1 12'7 13'9 15'2 16'5 17'7 19'0 20O3 21-5 22-8' 1+I 1'5'8 17'4 19:-0 20'6 22 2 23'8 25-3 28J 9 285 1 19 0 20 9 22,8 247 2626 28'5 304 32,3 342. 1 22 2 24'4 266 28'8 31'-1 333/ 355 377 3929 2 25 3 27i9 304 33 0 355 380 406 431 45 6 21 285 31-4 342 37-1 39'9 42 8 45'6 48J 5 51.3 22+~ 31-7i 3449 38'0 41-2 444 417 5 50'7 a539 57-0 .112 PRACTrICAL, TABLES. TABLE III. FLAT IRO N. 4o- lft. ft. ft. 4ft. 5ft. 6 ft. 7 ft. 8 ft. 9 ft. in. in. bs. lbs. Ibs. Ilbs, lbs. lbs. I bs. Ilbs. Ilbs. 21 3-5 ~ 70 10-5 13-9 17~ 4 20 9'24.4 27 9 31,4 3 38 7' 6 11'4 15'2 19'0 22'8 26-6 30'4 34-2/ 3 4-1 8-2 12'4 16'5 20-6 24`7 28'8 33 0 37'1.} 33 4-4 8-9 13/3 177 22.2 26:6 31.1 35.5 399 ~ 31 4'8 9'5 143 190 23, 28-5 333 380 428 4 51 101 152 203 253 30,4 355 406 456 3 42. 5'4 10'8 16.1 21.5 26'9 32-3 37 7 43'1 48'5 8 4 5-7 11-4 17-1 22.8 285a 3142 39 9 45.6 51,3 s 41 6-0 12)0 18'1 24'1 301 36 1 4281 9 482 542. 3, 65 63 127 19.0 25.3 31'7 38'0 444 507:57'0 3 5 6'7 13 3 20'0 26'6 33:3 39'9 46'6 53,2 59'9' 8 5j 7'0 13 9 20 9 27*9 341.9 418' 48-8 558 62 7 A 51 7 3 14-6 21 9 29 2 36 4 43-7 510 58 3 65a6 a 6 7 6 15 2 22 8 30 4 38 0 45-6 53 2 60 8 68A4 I 117.3.4 5.1 6:8 85a 10o1 11.8 13,5 15 2 + 1 2 1 4-2 6tS3 8-4 10 6 12'7 14'8 16'9 19'0 1 2 5 5 1 7-6 10.1 12- 15' 2 17'7 20-3 22-8 i 1.- 3 0 5 89 89 11 8 14-8 177 207 23-7 26,6 1 2 3'4 678 101 135 169 203 237 270 304 2j 3-8 76 11*4 15'2 19(0 22'8 2-66 30'4 34-21 4,I 2W:4 2 8'4 12'7 16-9 21,1 25-3 296 33-8 38 0 A I2-Bt 4 6 9-3 13891 181 232 27'9 32-5 37 2 41-8: 3 5 1 10-1 15 2 203 25'3 304 355 40-6 456 ji 3a 5.5 11-0 16 5 220 27-5 32 9 38-4 43-9 494: 3i 5 9 11.8 17 71 23'7 29 6 35-5 41-4 47'3 5329 3.1; 6'3 127 19'0 253 317 38-0 44.4 507 5701 4 68 1355 20o3 270 33 406 473 541 608 PRiACTICAL TABLES. 13 TABLE I}L -- FLAT I R O..3 5 l0 ft. 11 ft. 12 ft. 13 ft. 14 ft. 15 ft. 16 ft. 1I 7ft. 18 ft. in. lb$. lbs. lbs. lbs. lbs. lbs. lbs. lbs. bs. 3 21 34,9 36-3 41 8 45*3 48-8 52-3 55-8 b9*3 62 7 3'380'0 41'8: 45"'6 49.'4'53.2 57i0 608[:64'6 -68' 4 R 3f 41*2 45-1 49-4 53-6 57.7 61-8 65-9 700 74 2,[i 3i- 4444 48-8 53-2 6577 7 62,-1 6:6 5.71t 75-4 79.9 3 47-5 5 2 S 571 0 61 8 6 6 5 711 3 76 0 808 S 8551 4 14 50 7 55-8 60'8 65!-9 7 0 9 76 8:1 1 86 2 9112 14-1 53.9 59-3 647 70 0. 75-4 808 s6-2 91.6 970 - 4+ 57 (- 6 29 68'4 74'21 79 9 85 6 91-3 97,0 1027 4 4 60'2 662 72-2 783 843 90 3 96'3 102-3 108 4 65 38 69 76 0 82-4 88-7 95 0 101-4 1-7 114-. 5+ 66 5 73'26 79-8 86!5 t993311 99'8 1:06-5 113'1 119-8 5 69'7 76-7 83-7 90-6;97t6 104' 5 1111-5 118-5 125-5 + 5+ 72t 802 87 5 94-7 1020 109-3 116 6 123-9 131-2 3 76 0 83-6 91-2 98-9 105 5 114q1 121-7 129-3 136-9 i i -166- 18-6 20-3 22:0 23-'7 25'4 27(-0 28-7 30A4 + 14 21-1 23'2 25-3 27 5 29-6 31-7 33 S 35-9 38-0 25 3} 27-9 30-4[ 33'0 -35'- 38'0 4016 43'1 45'6 1 29 t; 32-5 35-5 38-5 41'4 4454 47]3 50-3 53-2E 2 33-8 37-2 40-6 43 9 47-'3 50'7 54q1 57'5 60'8 I 2+ 38 -C 41 -8 45-6 4 9 -4 53-21 5710 60:8 64-6 68'4 i 2 42'2 4-'6 -5 507 51 9 59' 1 63-4 65-6 71-8 760.1 i 2I 46-5 511 55"8 60-4'65-1 69i7 74-4 79-0 83 6 50-/ 55-8 608S 65-9 70oq9 76 0 81-1 86-2 91_2., >~t 3+ 54.0 604 65-9 71-4 769 82 4 87 9 93 3 98 8 +i 3~+ 59-2 65-1'71 76'-9 8-: 8 88-7 94-6 1o00:6 106 51: o+ 63-3 69-7 76-0 82-4 88 7 950 1014 10)77 114-0 + b| 4 ~ 67'6 74-4 84-1 87:91 94,'6:101:'4 1082 1 914'9 121'7 10* 11 4 PRAArICAL TABLES. TABLE' IIL FLAT IRON..-ft. a I ft. 2 ft. 3 ft. 4 ft. 5 ft. 6 ft. 7 ft. 8 ft. 9ft.' in. in. Ibs. Ibs. Ibs. lbs. lbs. lbs. Ilbs. lIbs. lbs. 4'7.2 14.4 21.5 28'7 35-9 431] 50.3 57'4 64,6 4 ~4 7'6 152 22-8 30'4 38'0 45'6/ 53'2 60'8 68'4 i 41 80 161 24'1 321 401 48-2 562 642 722' 5 8'4 16:'9 25,3 33 8 42'2 50'7 59'1 67'6 76'0 4 5;89 17 7 26 6 35-5 44.4 53.2 62.1 71.0'79.9 i 54 93 18'6 27-9 37-2 46'-5 55'8 65'1 74'4 83'7 4 54 9.7 19-4 292 38.9 486 583 680 77-7 875 6 101 20 3 3041 40-6 50 7 60-8'709 81 1 912 4I 1 2.1 4-2 63 8-4 106 12-7 148 16-9 1:9.0 i 1 26 53 7'9 10-6 13-2 15-8 18'5 21-1 23'8 4 14 3 2 6-3 9-5 12-7 15-8 19 0 222 256-4 28 5 14 3.7 7-4 111 148 18-5 22-2 25'9 296 33 3 42 2 4-2 8-4 127 16-9 211 25-3 29-9. 338 380,:'.24 4'8 9.5 14'3 19'0 23.8 28'5 33'3 38 0 42 8 I[ 2' 5:3 106:15'8 211 26-4 31-7 37.0 422 47.5 9 241 5,8 11'6 17'4 23-2 29'0 34 8 40'7 46-5 62 3 {, 3 6.3 12;7 190 253 31-7 380 44'4 50'7 576 E 34 69 13'7 20"6 27 5 34'3 41 2 48-1 54'9 618S 4 34 7-4 14-8 22-2 29'6 37 0 44-4 51 8 59-2 66'5 1 3. 7'9 15 8. 238 3107 39-6 47-5 55-5 6341 71.31 1 4/ 8:4 16.91 25-3 33-8 42.2 507 59.1 67'76.0 44~ 9' 0 18,0 26'9 35'9 44'9 53-9 62-9.71-8 80'8 4 44 9'5 19'0 28-5 3880 47'5 57 0 665 761 856 4 4. 100( 20-11 30-1 40-1 50-2 60 -2 7021' 80-3 95031 4 5; 106 2111 317 423 528 6341 7391 84 9 4 54' 11'1 2221 33'3 44-4 55'5 6655 r7'6 88'7 998 151 116 2321 34'9 465 58'1 697 81'3 9291 1046 4 ~s5. 12'1 24-31 36-4 48 6 60'7 729 850 9 72 109'3 PRACTICAL TABLES. 115 TABLE III. FLAT IRON. ft. 1 ft12. 13 ft. 14 ft. 15 ft. 16 ft. 17 ft. 18 ft. in. lIbs. Ibs. Ibs. lbs. lbs. Ibs. Ilbs. Ibs. Il bs. Ilbs. 4i 71 8 79-0 86'2 93'4 100'5 107'7 114 9 122 1 129-3:4i 41 76 0 83 6 91'2 989].106'5 114'1 121'7 129'3 136-9 I 41 80-3 88-3 96'3 104'3 112'4 120-4 128-4 136-4 144'5 5 84'5 92'9 101'4 109'8 118'3 126'7 135'2.143'6 152-1 ~ 51 88-7 97 6 106'5 115'4 124'2 13'13 142'0 150'8 159'7 ]~ 51 93-0 102 2o 1115 120'8 130-1 139'4 148.7 158.0 167'3 51 97-2 106-9 116-6 126-3 136 0 145 8 155-5 16652 174-9 6 101-4 1115 121-7 131*8 141-9 152-1 162-2 172 4 182 5 1 21r1 23-2 25.3 2775 29-6 317' 33'8 35-9 38'0 1~ 2614 29.0 31'7 34-3 37-0 3916 42-2 44-9 47-5 1i 31'7 34'8 38'0 412 444 47.5 50'7 53 57'0 fil lb 370I 407' 44'4 48'1 51'8;55-5 59/ 2 62'8 66'5 2. 42-2 46-5 50'71 54'9 60'1 63-4( 6"76 71'8 76-0.f 24 47'-5 52'3 57 0/ 6 1'8 66.51 713 76.0 80.8 85.5 41i 52-8 58-1 63-4 68'6 73-9 79'2 845; 89-' 95'0 21 58 1 63 9 69-7 75-5: 81 3 S7 1 92o9 981 104.5 j 3 63-3 69-7 76 0: 82-4 88-7 95-0 1014 107-, 114-0 t'4 68'7 75-5 82 4 893 96'1 103'0 109 9 116'7 123'6 3 73'9 8131 88 7 96 1 103-5 110 9 118-3 125 7 133-1 3 79-2 871 95-1 103-0 110 9 118-8 126-8 134-7 142-6 4 84-5.92-9 101-4 109-8 118-3 126-7 135.2 14306 152-1 |- 4~ 89'8 98-8 107'8 1167 125'-7 134,7 143.7 1]52'6 161,6 ~k 41 95 1 104-6 114'1 123-6 133'1 142'6 152'1 161'6 171'1: 141 1003 1104 1204 1304 140'5 1505 16051 170'61 1 806 1.i5 105'6 1 16-2 126-8 137 3 147 *9 158'4 169 0 179 6 190-1 1 5 110'9 122-0 133-1 1144 2 155-3 166'4 177-5 188-5 199-6 [-| *5:!1 116-2 127-8 139.41 1510 1626 174-3 185-9 197'5 209-1 / 5, 121 25 1336 1457 1-57 9.llo0 182s 2 194.3 206o5 2186 i'"c5I lC~ 1 1.4- 7 519 170,0, 1823 859192520. 11 6 PRACTICAL TABLES. TABLE IIIL FLAT IRON.:I~E _ I ~>;)1 v1 lft. 2 ft. 3 ft. 4ft.5 ft. 6 ft. 7 ft. 8 ft. 9 ft. in. i. lIbs. Ibs. Ibs. lbs, Ibs. Ibs. Ibs. Ibs. Ibs. - 6 12-7 253s 38-0 50 7 63-4 76 0 88-7 101 4 114-1 1 2-5 51 7 6 10 1 12-7 15*2 177 20-3 22-8 I 19 3-2 6 3 95 127 15-8 190 222 25-4 28-5 1 38 7 6 11 4 15*2 19 0 22.8 26.6 30-4 24-2 14 4-4 8-9 133 177 222 26-6 31-1 35"5 39"9 2'51 101 1 5) 2 20`3 25 3 30.4 35.5 40.6 45'6 57 114 17 1 228 285 34' 2 39'9 456 513 21 6 3 1 2 19rO0 253 31.7 38 0 44-4.507 57-0 r 2 7 0 131 209 27 9 34-9 41*8 48-8 55 8 62 7 3 7-6 15a2 22 8 30'4 38-0.45-6 53-2 60-9 68-4 I 31 8-2 16 5 24 7 33 0 41-2 49-4 57'7 65`9 74-2 3 s 89 17-7 26-6 35/5 44-4 53 2 62-1 -,71 0 79.9; 3p 9.5 19t0 285 38 0 47 5 57-0 66-5 76-1 85-6. 0 4 10.1 203 30'4 40-6 50'7 60-8 70'9 81 l 91-2 4a 108-~ 215'. 32.3 43o1. 53'9 64 6 75-4 862 970 41- 11*4 22 8 34-2 45-6 57 0 68-4 79 9 91 3 102-7 i 4i 120 24 1 36-1 48-2 60-2 7'22 84-3 96-3 108-4 [- 1 -85' 2 127 253 38,0 5047 63-4 1760 88-7 101-4 114-0 4 5 13 3 266 399 53-2 66 5 79 8 93 1 106- 119-8 13 9 279 418 558 69*7 83 7 976 111*5 125*5 * 5'8 14 6 2991 437 58-3 72 9 87-4 102 0 116,'6 131 2 6,Ic: 152 ~lj~8(: 30'4j~ 45~6. 608 76-0 91-2 106-5 1217 1367 9 1 14 5 1' 101Ol 15-2 20-3 25-3 304 35 5 40 6 456 1 2 6 8 13 5 20 3 27 0 33 8 40 6 47 8 54-1 60 8 1 -3 [10'1/ 203 30. 4 40-6 50'7 6081 7.0'9 81'1 91'2 1 4 13 5 27- 0 40- 6 54-1 676 81.1 94-6 108-1 121I7 1 5 169 338 507 67-6 84-5 101*4 118- 135'2 152 1 a-.1 6g G I20 3 40'76 60'8 881'1 1014 121 419 162-2 182'5 PRACTICAL TABLES. 117 TABLE III. FLAT IRON. _ 1 25.3 27,9 30'4 3_3'0 35.5 3s- 40o6 4381 45.6 1+ 31-7 34l 9 38f0 4121 44-4 4 147 50lf 539 571 0 _I 1 38 0 418 456 594 53,2 570 608 64.6 68_4 1 2 57'0 627 68'4 74'2 79'9: 85'5 913 9 7'0 102' 7' 2 633 697 76*0 82-4 8817 95.0 101-4 107'7 114 0 1 21 69-7 76 7 83 7 90-6 97-6 104-.5 111-6 118-5 125-5 3 76' 0 836 91'2 98'9 106-5 114-1 1217i 129-3 136 9 4 3+ 82-4 9306 98-9 107 1 1 15 3 123*6 131 8 140 0 148-3 -1 3+ 88'7] 97'6 106-5 115'4 124'2 133'] 142'0 150'8 159-7 / 1' 95-1 104-6 114-1 123'6 133-1 142'6 152-1 1t1-6 171-1 / 4 101-4 1115 121.7 1.31.8 141'9 152'1 162'2 172'4 182-5 + 4+ 107.7| 118*5 129.3 140. / 150-8 161 6( 172-4/ 183'2 193.9 ] 41 114-1 125'5 136'9 148-3 159-7 171'1 182-5 193'9 205'3 + 4+ 120 4 1324 144-5 156-5 168-6 180-6 192-6 204-7 216 7 + 5 126-7 1.39'4 152'1 164-8 177'4 190'1 202'8 21554 228'1 5+ 133-1 1464 159-7 173'0 186'3 199'6 212'9 226'2 239'5 / 5/ 139'4 153'3 167-3 181-2 195-2 209'2 223-1 237'0 250'9 + 5+ 145'7 160'3 174-9 189'5a 204'0 218'6 233'2 247-8 262'3 + 6 152-1 1673 182-.5 197-7 212-9 228-1 243-3 258-5 273-71 1 1I 50 7 558 60-8 65-9 70o9 76 0 81 1 86G2 912. 1 2 67 -6 74'4 81 81'19 94'6 10]'4 108' 1 114' 121 7 1 3 1014 1115 121 7 13117 141 9 152-1 162 2 172 4 182-5 1 4 1,352 148-71i62'2 175-7 i89.3 202.8 216.7 2298 243'3 I 5 169'( 185.9 202,8 219'7 236'6 253'51 270.4 287-'3 30.4'2 1 6 2028 2231 243 3 263'6 283 9 304-2 324*4 3447 365B0 118 TABLE OF GrPADIENTS. TABLE OF GRADIENTS And RPesistance per Ton for each. Vertical Rise. Gravity Vertical Rise. Gravity Vertical Rise. Gravity due to title to _ due to incline incline incline Ratio. Pr. Mile per ton. Rtatio. Pr. Mile. per ton. Ratio. Pr. Mile. per ton. one-in Feet. ls one in Feet. lbs. one in Feet. lbs. 100 52'80 22-40 74 -1138 32270 47 11234 47660 99 53-33. 22 626 73 72-32/ 306t85 46 115'04 48-684 98 538 22-858 72 7 33 3 31'111 45 117'33 49'777 97 54-43 23 0921 7 1 74-36 31.550 44 1 20'00 50'908 96 55-00! 233341 70 75-43 32 000 43 122078 52-092 95 55:60 23.579 69 76.49 32 464 42 125.71 53.333 94 5'5617 23'830 68 77 64 32 9401 41 128-78 54.634 93'! 56'771 2 4086 67 "78i81'33'432 40 13200 566'00 92 57 52 24.342 66 80o0 33 940 39 135'38 57'436 91;58'02/ 24-614:65 81'23- 34-460 1:38 1138 95 58'944 90 58'66 24-8881 64 82-50 35 0 37 142 70 60'540 89 59-33 25-168 63 8':'81 35'555 36 146'66 62'222 88 60-0 25-454 - 62 85-16 36'108| 3.5 150-84 64'000 87 60-69 25-746 61 86-55 36 7201 34 155 30 65-880 8:6 61-39 26-046 60 88 00 37 33 33 160 0 67 6880 85-16 62-00 26'303 59 89,49 37 966' 32 1650 7 0'0 85 62-1'2 26 353 58 91.03 38 620 31 170 32 72'216 84 62-86 26'6661 57 9263[ 39'2981 30 176'001 74'666 83 6361 26-988 56 9428 400 |29 82.06 77 240 82 64'39 27.317 55 96'00 40'726 28 188-56 80'00 81 6520)'27 718 54 97'77 411'480 27 195'55 82'960 80 66'0 28-00 53 99 62 42'2641 26 203'06 86'152 79 66'83 28-355 52 101-53 43'0761 25 21:120 89'60 78 67'69 28'718 8 51 103'52 43'920 24 220'0 93,336 77 68-57t 29'0901 50 105'60 44 800 23 229'56 97'368 76 69'47 29 4721 49 107'75 45'716' 22 240 101-816 75 70'40 29 867l 48 1 110 00 46 688' 21 251-43 106'666 To TAKE IMPRESSIONS FROM COINS, &c.-Make a thick solution of isinglass in water, and lay it hot on the metal; let it remain for twelve hours, then remove it, breathe on it, and apply gold or silver-leaf on the wrong side. Any color may be given to the isinglass instead of gold or silver, by simple mixture. VARIATrONS IN TIDES.-The difference in time between high water averages about 49 minutes each day. CAST-TRON PILLA RS. 119 TABLE of the Ultimate Breakinag IWeight, in toqns, of cast-iron pillars, calculated from Professor Hodgkinson's Formula. The length includes every half-foot fi'om I to 201 and the diameter every inch from 1 to 24. DIAMETER OF CAST-IRON PILLARS IN INCEHES. - -1 2 3 4 5 G6 tons. Itons. tons.: tons. tons. tons. 1 44-30 537 2312 6513 14544' 28038 14 22'23 269 1160 3269 7300 14073 2 13'63 165 711 2004 4476 8630 21 9-33 113: 487 137: 3064: 5905 8 3 6'84 83 357 1006 2247 [ 431 34 526 64 275 774 1729 3333 4; 4 19 5:1 219 617 1378 - 2656 44 343. 41-6 179 505 1127 2174 5 2-87 34.8 | 150 422 943 1 817 54 2144 29 6'12 359 02 1545 6. 2-11 25-D5 M110- 309 691 1333 64 -: 184 22.3 96 270 603 1163 7 162 19-6 84-6 238 532 1026 71 1*44 17.5 752 212 473: 912 8 11-29:15'6 |674 190 4-24 817 84 1-16 14-1 60-8 171 382 737 9 1'06 12'8 552 155 347 669 9'96 11-7 503 142 316 610 10 88 107 46-1 1 30 290 559 104& - 81 986 42'4 119- 267 515 11 75 9-11 39-2 110'246. 475 11j'69 8-45 36'3 102 228 441 12 -';'65 / 7'86 338 953 212 410 124 - 60 7'33 31-5 88-9'198 383 13'56 6'86 29-5 83-2 185 358 1834 53 6'43 27:'7 780 174 8 336'14' 50 6-05 26;0 73-3 1 63 315 144 -47 5'70 24'5 69'1 154 297 15 44 5-38 23. 15 65.23 145-6 280-8 154'-42 509:.21:90 61'69 137-7 265-5 16 40 4-82 20-75 58-45 130.5 251-6 164'377 4'57' 19-69 55'47 123'8 238-8 17 *358 -45 18-72 52'73 117-7 227'0 17 -341 4-14 17-82 50.19 112-1!216-1 18 *325 3'94 16-98 47'85 106-8 1 205'9 184'310 377 16-21 45-67 101-9 196-6 19 297 3-60 15-49 43- 64 97-45 187-8 194'284 3-44 14-82 41-76 93-241 179-7 20 -272 830 14-20 - 40'00 89-32 172-2 120 CAST-IiRON PILLARS. TABLE of the Ultimate Breaking Weight, in tons, of cast-iron pillars. (Continued.) DIMAIETER OF CAST IRON PILLARS IN INCHES. F 1:7 8 9 10 11 12 tons. tons. tons. tons, tons. tons. 1. 48838 78982 120691 176361' 248552 339982 1 24513 39643 60579 188520 124756 1'10648 2 1 5031 24310 37147'54282:76501 104643'2 1.0286 16635 25420 3.7145 52350 71607 3 7544 12202 18645 27246: 3 8398 52523'3 5:805 9388 14347 2.0965! 29546 40414 4 4.626 7482 11433 16707' 235461 32207 41 3787 6124 9358 13675. 19,273 26363 5,3166 5120 7824 11433 16113 22039 51'2692 4354 6653 - 9722.13703 18743 6 2322 3755 5738 8385 11818 16165 6 61,2026 3277 5008.7319:10315 14109 7 1787 2889 4415.:6452 9094 12439.71.1589 2570 3927 b5738 8087 11062 8'1 1424 2302 3519 5142.7247 9913 81,.1284 2077 3174 4638 6537 8942 9.1165 1885 2880 4209: 5932i 8114 94-.1063 1719 2627 3839 5411 7401 10 974 1575 2408 83519. 4959 6783 10 897 1450 2216 3238 4564 6243 11 828 1340 2048 2992 4217 5769 11'768 1242 1898 2774 3910 5349 12 ( 714 1156 1766 2581 3637 4975 12-4 666 1078 1647 2408: 3393 4642 13 623 1008 1541, 2252 8174 4343 13, 585 946 1445 2112 2977 4073:14. 550 889 1359. 1986 2799 3828 141 518 838 1280 [ 1871 2637 3607 15 489-1 7910 1208 1766 2489 3405 151- 462-6 1 48 1143 1671, 2354' -3220 1.6. 4383 7088 1083: 1583. 2230 3051 161 4159 6721 6 1028 1502',2117 2895 17 3953: 6394 977'0 1428 2012 2752 17. 3763 608.6 9301 1359 1915'2620.18: 3587 580 2 886'5 1295; 1826 2497 184 S 342 4 5538 846 2 1236 1743 -.2384 19' 3272 522 -808'7 1182 1665 2278 191 313'1 50674 773'8 1:131 1593 -2179 20 299-9, 4850 741'2 1083 1526 2088 Note.-Examp7e. Find the breaking weight of a east-iron pillar whose external:diameter is 17, and internal diameter 15 inches, and length 18 feet. CAST-IRON PILLARS. 121 T'ABLE of the Ultimate.Breaking Weight, in tons, of cast-iron pillars. - ___________ (Continued.) E= - ~ DIAMETER OF CAST-IRON PILLARS IN INCHES., 13 14 1.5 16 17 18 tons. tons.. tons. tons. tons, tons. 1. 453524 592195 759158 957714 1191290 1463470 1i 227638 297241 381039 480707 597950 734563 2 139588 182269 233660 294769 366664 450443 2.4 95522 124729 159895 201717 250912 308288 3 70064 91486 117281 147955 184040 226088 34 53912 70396 90243 113S46 141614 173966 4 42963 56100 7191i7 90726 112853 138638 4 / 35167 45920 58867 74263 92375 113481 5 29400 38390, 49213 62085 77228 94871 51- 25002 32647 41851 52798 65676 80680 6 21565 28158 36097 45538 56645 69586 [6 18821 24576 31505 39745 49439 60734 7 165933 21667 27776 35(040 43587 53545 74 14756 19269 24701:31163 38763 47619 8 13223 17267 22135 27924 34735 42671 81- 11928 15576 19967 25190 31333 38492 9 10824 14133 18118 22857 28432 34928 9i 9873 12892 16527 20850 25935 31861 10 9049 11815 15147 19109 23769 29200 104l 1 8329 10875 13941 17588 21877 26876 11 7695 10048 12882 16250 20214 24832 114 7135 9317 11944 15067 18743 23025 12 6637 8667 11110 14016 17434 21418 12J- 6192 8086 10365 13076 16265 19982 13 5793 7564 9697 12233 15216 18693 134-, 5433 7094 9094 11472 14271 17531 14 5107 6669 8549 10785 13415 16481 141 4811 6282 8054 10160 12638 15526 15 4542 5931 7603 9591 11930 14656 15.i 4296 5609 7191 9071 11283 13862 16 4070 5314 6813 8595 10691 13103 16 8 3863 5044 6466 8157 10146 12464 17 3671 4794^ 6146 7753 9424 11847 17-9 3495 4563 5850 7380 9180 11277 18 3331 43-50 5577 7035 8571 10750 181 3180 4152 5323 6715 8353 10261 1.9 3039 3968 5087 6417 7983 9806 191 2908 3797 4867 6140 7638 9383 20 2785 3 3637 4662 5881 7316 8987 Along the line marked 18 feet, and in the vertical lines numbered 17 and 15 inches, take the numbers 8751 and 5577;the difference of which, namely 3174, will be the breaking l. _ ~1 '122 CAST-IRON PILLARS.:TA:BLE of the Ultinzate Breaking Weigcjht; in lons, of cast-irion Pillars. (Continued.) DIAMETER OF CAST IRON PILLARS IN INCHES. I 19 20 21 22 23 24 tons. tons. tons. tons. tons. tons. 1 1777940 21385108 254-9140,:3013880'3536910 4122530 1a 892404 107838' 0 1279490 l512760 1775280 2069230 2. 547224 658904 784589 92763:0 1088610 128880 2'1 374471 450416 536902 6-3 4786 744947 86S292 3 274670 330374 393810 465605:.546409 636880 31 2:11350 254212 303024 358269 4120444 490059 4- 168428:202586 241485 28511 33a5059 390543 41 137865 165825 197666 233703 274260 319671 5 115257 1:138632 165251 195378 229286 267248 5f- s98017 117894 140532 166157. 194'8S8 227273 G'I6 184539 101684 1121210 143307 168177 196023 ~6j~ 73784 88748 105789 125047 146781 171085 7 65051 78243 93266 1102)70 129406 150832 71 57851 69584 82944 98067 115085 134140 8:: i 51840 62353 74326 878'76 10312G 120200 81 146763 56247 67047 79271 3028 -108430 9 42433 51038 60840 71930 84414 98390 09 38707 46557 55496 65614 77000 S950 10 35474' 42669 50862 60134 70571 82255 101 32651 39272 46814 55348 64954 75708 11: 30168 36286 43254 51140 60014 69951 111 27973 3 3645 40106 4741:7 55646 64860 12' 26020 31297 37306 44108 51763 60333 121I 24275 29199 34805 41150 48292 56288 13' 22710: 27315 32560 38497 45178 52658 131 21298. 25618 3&)537 36104 42370 49385 14 20021 24082 287 06 33940 39830 46424 141 18862. 22687 27043 31974 37523 43736 15; 17806 21417 25529 3018'4 35421 41286 154. 16.840 29255 24145 28547 33501 39.049 16 15955 19!1991 22823 27047 3140 36.997 161 15142: 1l821'3 21711 25669 30123 35:111 17" 14393 17312 20636 24398 28632 3,3374 171~ 13.701 16480 19644 23,225 27255 3:1768 -18 13060 15709 18725 22139 25981 30283 -18i 124'66 14994 17,873 21131 24799 28906 19 11913 14330 17081 20195 23700 27624 191 1139.8 13710 16343 19322 22676 26430 20- 10918 13133 15654 18508, 21721 25317 weight in tons. For practical pulposes the pillars, should.be calculated to bear one half nore-than the weightb to hich they are subjected. n, thl h TABLE OF STRENGTHS OF CAST-IRON SHAFTS. The cuibe of the diameter of a journal or shaft of sufficient strength is directly as the horse power, and inversely as the unmber of'voiutions of tlhe shaft per minute. MIr. Robertson Buchanlan' deduced from several experiments that a jourlial suitable to a' 50-hloise engine, nakling 5( revolutions per minute, should be 7i inches in diametel. It is front these data the following table has b1een computed. NUMBiER OF REVOLUTIONS OF SIIAFT PER MINUTE.I 10 15 20 25 30 35 40 45 50 55 60 65 70 80 1 90 895 100 110 120 130 in in. in. i.. in. in... in. in. in. in. in. i n. in| OI 0 16 5 5 5 4s- 4 4 4 4 4 31 3 3i 3+ 3-1 3- 3f 3+ 20- S- 7 8 6 7 6 6 6- 5i 5 51 5 5 4s 4 4 A 4 4- 4 * 4 3 0)11i 0 94 8 7 7 7 64 6 64 6 545+5 5+ s5 5 5 5j* 5 5 4 4 44 i| O40 k 1 0]~ 94 8~ 8 84 74: 7: 7 6 6 6 6. 61 6+ 6 54 5+ 54 5. 53 -5~:5:. 44| 50 12 1 1 110 9 81 8+8 7i 7 7 61 6i 6 61 6+ 6 6 57 54 5 541 60 132 11 104 9988487777+ 666+66 6 55 70 143.12 li1itlo 09 9 +1 9 8 848 8 7. - ~7 7 7 6 6. 6 6 6 6 6 6 1 o80 14i 12 + 11+1 101 9 910 7 84 848I8- 7 7 71 1 7. 7 6 6, 61 6 1 6 90 so1-+ 13- 12 0+ 1 0+ 91. S+ 98 8 7 74 7+' 7 7g1 64; 641 6 10 151 134 12+ Il 11 10 1 0 94 9 + 9 1 s 841 8 7 14 7- 6: 6+ 125 17+ 15 1 3+4 12 I 11+ 11 10- 104 10 9I 94 9+ 9 8 8 8+ 88 8 8 71 7+ 150 18+ 15i 141 13 -12s 12- 1 11+1 11 10+/ 10 10 9+ 9 9I 9 s 8i1 8 8 8 175 19 1 16+1 154 14*1 1 12+ 12 11 11I 10+4| 10+ l0 94 9 94 9+ 9 81 8 81 8i TABLE OF STRENGTHS OF CAST-IRON SHAFTS. (Continued.) NUMBER OF REVOLUTIONS OF SIHAFT PER MINUTE. 10 1 20 2 3 5 4045 50 5 60 65 70 5 0 5 90 95 100 110 130 in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. 200 20 174i 1541441 134 131 124 12-1 11- 114 11 104 104I 1014 10 9 94 94 9 84 8 O 2251 20-4J 18 1 16 151 1 144 131 134 120 124 11i 114 114 104 1010410)10 94 94 94 9 8- 250 21' 184l 1 154 15 14 l 13 12i 121 114 11 11i 11 1 104 104 104 10o - 10 9i 9 94 m 275 224 194 1/ 7 164 15 144 14 13 13 12 12 11 11 11 11 10 10 10 10 10 9 9 300 224 20 184 164 154 154 14i 134 134 13 12 1 1124 12 1 11 11 -1 1 10 10 104 10 9~ - 350 244 21 194 174 161, l15 154 144 144 1)34 1 2 124 12 12 11-1 1 4l 11 1' 0 104 1041. 400 2541 22 20 18 17 16 154 154- 144 144 13l 134 131 124 12 124 124 12 1 1141 11 104450 261 224 20 194 184 171 164 154 154 141 14 14 1 131 13 124 124 1241 124I 114 11 114 ~ 500 2741 2341 214 20 184 174 17 164~ 15 15 1 144 144J 134 134 13+ 13 12- 124 112 11I 114 5501 28 244 224 20i 19 184 17 17 164 154 154 15 14 141 14 131 13 1311 13 1 12- 124 114 6001 284i 2541 224 2145 20 1 9 184 1 164 164 15 154 15 141 144 13 134 134 13 124 124 6501 2941 254i 2314 214 204 194 184 18 174 161 161B 154 15+4 154 144 144 144 14 13t 13- 13 12i 70013041 264+ 2441 292 21 20 19i 18I 174 171 16i 164 154. 1.5- 154 14i 144 1441 14 134 138 1247501 31 274, 244 224 214 204 1 18 47 1745* 1i 15 154 15 14~ 144 14 134 1341 1800 316 27t 2651 2S} 22 21 20 19 If 18- 18 17 -; 16f 1-1 1 15+I 1 5 J 15 1. 1 141 13- 13411 TA:BLE OF STRENGTHS -OF WROUGHIT.-IRON SHAFTS. NUMBER OF REVOLUTIONS OF 31IAFT PER MINUTE. -20a.4 I 5 410 45I. I0 - - o 10 15 -20 2 30 35 40 45 50 55 60 65 70 750 80 5 9 8 5 1 00 110 120 130 in. in. in. in. in. m. in. Mn. in. in. in. in. in. n. in. n.. in. in. in. in. in. ~1064.|~ 6 ~ 5a 411t 3 4 F|4 3| i 3V| 34'4 34. 5 S% 34. 3A 33 3 21-:2 | 24. 1 c 2) 8 7 64 54. 5 5 a 4 4 44 44. 4 4 43 4 34 3 3 3 36 4 34 3 1 53 9i- - 6 - nj5 6 461s 5 4; 5 -9 i 5ww54B!4 t 4 ~1- 41 - 4i 5-4 4i 42 41 4 4 16 1. -5 -j -16 4 4 —a.1- s36 1 401i".'8 74 7 6 64 6 51 8 7 —r 6i 64.5L 5 4; r. 44%4. 4i'.. } ]; 9 4 4) II /- I }' 4 1 40104(). 98 8 8 74 q_.377 616646 6 54. W s 5 54.5 54L5615 |5i. -5 4: 4i 45 Z - 7~011-3 101 - { 94. 9:s4.7~- X7 6- 74]7 664.6. 6% 67~ 6 5-4 513 5}4 |54 5. 54 5 - 5Z 4 70 12i 10. 94. 91 84. 8 74. 74 1-7 64. 64. 64. 64 6 46 54.-6 41 4;5 _Ti% 590111 I3-1 1 10 9. 94. 8 8 14 8 74. 7-4 77l46. 6L s 6 64. 6 64 64.4 64. 54- 4 5T 4T 1001 34. 12 104. 104.W 1 9 96 9 84. 8 1-5 8 1 74. T 7 18 7 64 6 I-! 641-7 64. 6 4- 6 5 1!1251413 104. 104. 94. 96 | 84. 84. 84. 7 74. 7-74 7411 16.564-. |10 40 10 81.5 i 74. 7i 79 44 7& 7i-. 150 154 1312 112 112i l1:3 7 10 94. 9 8 7 -5 175 164-. - 14 1134. 126j114 104 101W10 | 16 || 8 74 8 / E 7, 200174. 1i54 134I 124 12 114.104. 104.t z104.i 84. 849. - 84. |8. 8 87-7 813 74 7 776 7 6 6 I 7-1 7-Y-6 7* 6-!5 1 dt 6. 4 F6. TABLE OF STRENGTHS OF WROUGHT-IRON SHAFTS. (Continued.) NUMBER OF REVOLUTIONS OF SHAFT PER MINUTE. 0 0 10 15 20 25 30 35 40 45 50 555 60 65 70 75 80 85 90 95 100 1.10 120 130 in. in. in. in. in. in. in. i i n. in. in. in. in. in. in. in. in. in. in. in. 22518 131.1 12+ 11i 11+ 11 1Q 1O- 91 — 9 - 1 9 9 8-8g 8/ 81 81 8 7+ 7+ 250 18 116+ 145 131 1216- l12j 11 11 -1 11 10 l9{ I9 95 91 9 81- 8+ - 8+ 1 27o51,9+ 16+ 1 5+ 14F13 1 2+ 1 112~ 11{112~+ r 1 01 5 lO A T1 10 0- 9j I8 9 8 8T++8 81-i7 n 45o['2o~-1 1 81. 16~ / 1lO 5 [ 5 1 41 21-3i1 3E1 24 1 2+ 1 +1 ]1 411&10 1[ 1'.[ 101 ] 0 i W 1 0 /9- 1 ~1 o lq- 9 l 5002 —:- 19P 17+ 1 13+ 181 1 0 1 4 /1'1 1-15-11o' 1o o1o1104 10+ 10' _ 10 35020+ 1+ 16+- 15+ 14+ 161 3 11+ 10 1+ 12 l2 i-1wlO lO 9+ 1 9+ 1A 8+ - 30021519 l 7+ 16 15+ 14 13 13+ 12+ 12+ 12 5.i1 1 10+ lOA1O+ 11 + 1 + 45002 223+ 1 209+~1 5 18 16+ 13 1 15 TH 1123 13 1 12+ 1+ 11 9- 11 1 1 1 10 1 0+ / 1 TV 1 123 12 1 1+ 1 50123 20+ 18 + 17. 5 1+ 113 1 21 1 1+ 1 1 113 13 31 12i2- 1 - 11 1 1 1+ 10 1 550124+,1: [21+ }19+ 17+ 16 16 15+ 1124+ — 12T 13+ 13+ 1312+ 1 1-r-111 11+ 11+-ii }wii+-[1 1 i0{ 1i0101 I 65025+ 122~lV+19- 17+ 17 [ 16+ [ 15-:t 1 4+ 14+ 13+ 1313 1012 l0j 121 1+15 1 400 21" w19 20+ 19618+ 14+ 1616 1+. 1 4 3 3113 - 112-,1 102i~J121-i t101 1 3 7506 22 3 2119 1 1 158 17 17 164 15 1+ 14+ 14-1313 -13- 1 12_ /12T91- 12. 12 1.11 91 1il 80027+ 2 20 211 20 197 18+_ 141 1l6, 13 A 115+ 14Ar4 1 1 134 13- 1134 12 1 3 1" 1 2- 1 2112 121 12 11M1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __14_ _14_ _ 12_ 1 2 I 121, 11151 13 119 (.1,5 loj 650 25~1 22t.8 20# 19 I- 17 16i I 5, 1,5 141 141 131 1 _t/ _17~ A.._____116~._116. 1_ 3 T-60a TEETHI OF CAST-IRON WHEELS. 127 TAB LE Showing the Strength of the Teeth of Cast-Iron Wheels at a givern Velocity. Thick- Breadth Strength of teeth in horse power at Pitch of ness of of teeth _ teeth in teeth in in inches, inches. inches. 3 feet per 4 feet per 6 feet per 8 feet per second. second. second. second. 3 99 1 9 -1 6 20 5'7 27 43 41.14 54.85 3-,.78 1 8 7'2 17 49 23 -'32 I 34 98 46 64 3-57 1 7 6-8 14 73 19-65 29'46 39'28.3 36 1'6 6'4 12 28 16.38 24-56 32'74 3.1.5 1-5 6 10 12 13 50 20-24 26'98 2- 94 1 4 5-6 8 22 10 97 16 44 21'92 2 73 1 3 5 - 2 6 58 8-78 13 - 1 6 17" -54 2'52 1'2 4-8 5-18 6;91 10-36 1 13'81 2-31 1'1 4-4 3 99 5-32 7-98 10'64 2-1 1-0 4 3-00 4-00 6'00 8'00 1-89'.9 3-6 218 2'91 4,36 5-81 1:-68 8 3-2 1 53 2'04 8'06 3'08 1-47 -'7 2-8 1-027 1 37 2-04 2-72 1-26 6 *24 64 86 1'38 1 -184 1 05 *5 2 -375 50 -75 1 00 i FURNITURE OIL.-1. Linseed oil I pint, alkanet ~ oz. Digest in at warm place till colored, and strain. 2. The same, with i pint of oil of turpentine. 3. Linseed oil 1 pint, alkanet root 1 oz., rose pink 1 oz. Let them stand in an earthen vessel all night. 4. A quart of linseed oil, 6 oz. of distilled vinegar, 3 oz. of spirit of turpentine, 1 oz. of muriatic acid, and 2 oz. of spirit of wine. 5. Linseed oil 8 oz., vinegar 4 oz., oil of turpentine, mucilage, rectified spirit, each ~ oz.; butter of antimony i oz.; muriatic acid 1 oz. Mix. 6. Linseed oil 16 oz., black rosin 4 oz., vinegar 4 oz, rectified spirit 3 oz., butter of antimony 1 oz., spirit of salts 2 oz.; melt the rosin, add the oil, take it off the fire, and stir in the vinegar; let it boil for a few minutes, stirring it; when cool put it into a. bottle, add the other ingredients, shaking all together. [The last two are I especially used for reviving French polish.] 7. Linseed oil 1 pint, oil of turpentine ~- pint, rectified spirit 4 oz., powdered rosin 1~ oz., rose pink -i oz. Mix. 8. Linseed oil 14 oz., vinegar 14 oz., muriatic acid 4'oa. Mix. 128 WEIGHT OF PPEis. TABLE Showing how to ascertain thAeweiphts of Pipes, of various Metal, and any diameter required. Thick- ness in Wrought Copper. Lead. parts of iron. an inch. 326 11 lbs. plate,'38 2 lbs. lead,'483 | 12 653 23." ".76 4 " ". 967, 976 9 35 " 1'14 5 - " " 1 " 45.- 13 46-. " ".152 8 " it 1933 1 627 58' " 1 9 9 " " 2.41 1i 95 70 " 2-28 11 " " 2 9 j37 2 277 80.- " " 2.-6 13 " " 3-383 2 6 93 " 3*04 15 " " 3.867 dRule. To the interior diameter of the pipe, in inches, add the thickness of the metal; multiply the sum by the decimal numbers opposite the required thickness, and'under the metal's name also, by the lengthl of the pipe in feet; and the product is the weight of the pipe in lbs. 1. Required the weight of a copper pipe whose interior diameter is 7~ inches, its length 61 feet, and the metal i of an inch in thickness. 7 5 +'125 = 7-625 x 1'52 x 6:25 = 72'4 lbs. 2. What is the weight of a leaden pipe 184 feet in length, 3 inches interior diameter, and the metal i of an inch in thicklness? 3 + 25 _ 3'25 x 3-867 x 18'5 = 232'5 lbs. Note: —Weight of a cubic inch of Lead equal *4103 lb. Copper, sheet " 3225 Brass, do. "'3,037 Iron, do. " 279 Iron, cast " 2''63 Tin, do. "'2636 Zinc, do. "'26 " GWater.". 03617 " To SOLDER TOnTOsSE-SHELL.-Bring the edges of the pieces of shell: i to fit each other, observing to give the same inclination of grain to each, then secure them in a piece of paper, and place them between hot irons or pincers;, apply pressure, and let them cool. The heat must not be so great as to buern the shell, therefore try it first on a: piece of white paper. WEIGHT OF CAST-IRON BALLS. 129 TABLE Of the UTeight of Cast-Iron Balls. Diameter Wei Diameter ht Diameter Weight in inches. in lS. in iiches. in ls. i iches. in lbs. 2 1.10 6 29-72 10 1.3771 2j 1*57 6j 3 162 10 148 28 2j 2'15 6~ 37.80 10 159.40 2l 2'86 61 42'35 1I- 17105;8 3372 7 47'21 11 183-29 3 4-71 7 52-47 11 19610( 38- I580 7A 58'06 11i 209'43 3~ 7-26 7t 64.09 11i 22340 4 8I 81 8 70 49 12 237'94 49 10-57 8- 77 32 121 253'1/3 4, 12'55 8+ 84-56 12~ 268971 48 14'76 8t 92'24 12 285'37 5 17'12 9 100-39 13.q02'41 r 19.93 9s 108.98 13 320.80 5-1 22'91 9~ 118'06 13+ 338,81 5: 26'18 91 127'63 13 357.93. 1. What will be the weight of a hollow ball or shell of east-ilon the external diameter being 9~, and internal diameter 81 inches i: Opposite 9~ are 118'06, and Opposite 81 are 92'24, subtract 25'82 lbs., weight required. 2. Requiring to remove a cast-iron ball 3'8 lbs. in weigh.t, and in diameter 6 inches, and replace it by one of lead of an equal weight, what must be the diameter of the leaden ball Weight of lead to that of cast-iron = 1'56, 6'53 Then — = V176 = 5-6 inches, the diameter. To TRANSFER ENGRAVINGS TO PLASTER CASTs.-Cover the plate with ink, and polish its surface in the usual way; then put a wall, of paper round it, and when completed pour in some finely powdered plaster of Paris mixed in water; jerk the plate repeatedly, to. allow the air bubbles to fly upwards, and let it stand one houn; i then take the cast off the plate, and a very perfect impression will i be the result,. TABLE OF THE WEIGHT OF FLAT AND ROLLED IRON, per foot in length. BiEAD'rH IN INCHES AND PARTS OF AN INCH. *14 31 3 3 3 21 2i 2 2 8 7- l 1 |i. ___: ___ _..._ I 1-68 1 57 1P47 1 -36 1 26 1-15 1-05 0 94 0-84 0 73 0 63 0'51 0052 0-42 0 -31 0 21 -3 252 2-36 2-20 2-04 1-89 1P73 1-57 1 41 1-26 1 10 0 94 0 86 0-78 0 63 0 -47 0- 31 l 3-36 3-15 2-94 2 73 2-52 2-31 210 189 168 147 1-26 1- 1 —-05 0,84 0-63 0-42 0 5.04 4-72 4-41 4-09 3.78 3-46 3-15 2 83 2-52 2-20 1 89 1 73 P157 1 26 0 94 0o63 o ~ 6-72 6-30 5-88 546 5-04 4 62 4-20 378 361 2-94 2 52 2 1 2 10 1-68 1 26 6 71.... ~ 1-. 3 5 2 28 5 13 -46 585 5 0156 5 4-72 l3 1^8'40 1787 7 -35 6 -82 6-30 5 l77 5095 4-72 4-20 67 3-15 2-88 262 2310 I 57 10-0 8 9 -45 882 8 113 756 693 6-30 5-66 5 04 4- 41 3-8 346 3- 15 2 52 11P*68 1891-02 10-29 955 8-8 82'OS 7 35 3616 1 5-' 88 5 4 t3 2 | 1' 1'154 205 20'9 4 0' 84'7 0 63 40 3 - 6 2 1 13-44 12-60 11-76 10,92 10-08 9-24 8-40 7-56 6-72 587 5-04 4-62 4-20 1' 1512 164-16 13-0 12 228 11-`34 10'39 955 8- 50 7-56 660 567 519 4-2 I 1 16580 15-75 14-70 13-65 12-60 11.55 10-50 9-45 8 40 7, 35 6-30 5-77 1 318'48:1732 16-16 15-01 13-S6 1 2-70 1 1-55 1039 9'24 807 0 1i 20-1818IS90 1b764 16-38 15:12 13-86 12-60 11-34.10-08 8-80 1-1 23-54 22-05 20-58 19-11 17 64 16-17 14,70 I13-22 2 26-88 2 520 23-52 21-84 20,16 18-48 16-80 15-12 3 40- 32 37-80) 35 28 32-'76 3 4- 47 -04 -I_ I_ _ II_3_!_I'3!4'I ~.oj. / CAST-IRON PIPES. 131 TABLE of the lWeight of Cast-Iront Pipes, in lengths. 4' n Weight. | o e Weight.. Weight. Inch. In.:l Ft. C. qr. lb. q Inch. In. Ft. C. qr. lb. Inch. In. Ft. C. qr, lb. 1 l+/ 3-, 12 6+ / 9 2-01611 I 9 1 50 7 -: 3: 21 9 9 2 3 20'} - I 6 t 12 41 d+ 21 - 9 3 2-21 9 72 8 41 1 4 9 4121 1 9 101 2 2: 6 1 8 12 9 6 5024' 6 2 0 7 + 9 3 7 i 6 2 8 2+ I 6 1 16 / 9 3 3 20 9 9 7 3 20 +Xj 6 2o10 +t I9 4.3 5 1 9 103 o0 + 6 3 10 1 9 6 2 4 12- 9! 5 1 16 3 I 6 | 6:6 9 6 9 3 +} 9 2 2 20 9 3 9 63-, 9 9 4 0 22 - 9 81 0 1 9 1 2 9l 50 10 1 19 110 21 i- 9 1 1 1 2 9 9 3. 6 9! 220 + 0 9 2 1 0 8 9 3 24 9 2 8 1 3 ++ 9 3 0 9 9 41 25 1 9 812 7 9 9 1 0 21 9 5 1 18 1I 9 11:2 12 ~-+" 9 1 2 1 4 1 9 57 i 1 6 13' 9 53 ~ -+ 9'2 0 8 +8:9 3 3 2 9 1 11 2 2 9 2 0 9 4 2 26 1 9 8: 3 16 1t 110. 9/ a 22 5 2 22 1 9 111~3 24. 9 I I 170 10 1 9 17 3 9 3 12 1 3 1 + 3 60 4 I9 12112 9' 9 4 00 o 9 7216 i 9 2 3 21 - 19 6 504 9 1 9 1 0 4+. 9 1 2 2 - 11 9 6 0 2 1 9 1 1 2114 19 204 019 8 0 26 14+4 ~ -1i 9) 6024: 9 2 2 14 9+ 1 9 4 0 1 8 9 73 14. 9 3 0 1222 1 9 5 1 0 9 19 23 9 11 9 86 1 2 6 1 9 1 1 2' 6 9-. 0. 9 6121 ti/o9 23 1 1710 9 94110 9 937. |: 9 3 124 11 1 9 13 0 26 5*-.' /'9 i1 3'10 9:" 6"2 14"' 1- ] 9 16: 3 5 2 9 2 0' 19 9 0:8'1 15~ 1 9 6 2 14 i 9 3 0 18 1 9 4 2 14 1 9 1O 0 10 9 3 3 7 + 9 53 7 1 9 3217 1 9 5 0O12 +1-39|742 0:0 1+09171 6 6,:[ a 9 2 0 199 16 I' 9 7'O 022 4- 9 222 21, 111 1 9'143 14 +9'101201 9 — 9.3-3-117 96.0 11 19:1140- 8 + 9. 4016 + | 9:171 7 / 1+[917314 1 9 5 2 20 1 9 9 3 20 11 9 21 3 4 13M MALLEABLE IRON. TABLE Of the weight of one foot length of Mallleable Iron. SQUARE IRON. ROUND IRON. Scantling. Weight. Diameter. Weight. Circumfer. Weight. Inches. Pounds. Inches. Pounds. Inches. Pounds. 021 0'16 1 0 26 - 0'47 0'37 1+ 0'41 084 066 14 0'59 134 1 03 0'82 1.89' 1 48 2 1'05 2'57 2'02 2* 1-34' 1 3'36 1 263 2j 1'65 4i 4-25 1* 3'33 21 201!i 5'25 1i 41' 2 3 2'37 6'35 1 - 4'98 31 2,79 1'' ~7'56 1i 5'93 34 3'24 8-87 I 1 6-96 3. 3-69 1'4 - 10'29 1i 8'08 4 4 23 41 11'81 9'27 41 5I35,2 13'44 2 10'55 5 6'61 2 17'01 221 13'35 75 799,24 21'00 2 16'48 6 9'51 25-41 2* 19'95 6 1118 I3 3024 3 2373 7 1296' 3- j 41'16 31 27'85 7 14-78 4 1 53'76 3* - 32 32 8 16-92 4I G68 01 3: 37209 8 19 21 5 84 00 - 4 4221 9 21P53 IG; 12096 4 6 53'41 10 26-43 I 1 164'64.5 65 93 12 31 99;'RESCO PAINTING. —Apply any colors that are not injured by lime (according to taste), on a fresh mortared or plastered wall. To TARE FAC-SIMILES OF SlGNATURES.-Write your name on a piece of paper, and while the,ink is wet sprinkle over it some finely powdered gunm arabic, then make a rim round it, and pour on it some fusible alloy, in a liquid state. Impressions may be taken from the plates formed in this way, by means of printing ink and the copperplate press. WVATCHMAKER)S OIL, WHICH NEVER CORRODES OR THIcKENs.-Take olive oil and put it into a bottle, then insert coils of thin sheet lead. Expose it to the sun for a few weeks; and pour off the clear oil. CoPPrEtas-CAsT-IRON PLATES. 13' T-ABLE Of the Dimenisions and Weight of Coppers, from 1 to 208 gallons. The Dimensions taken from lag to brim. 2 0.O ss | _ AE= o _,._1,U.1 9 1 -1+ 24; 15 22+ 291.29 43+ 12. 2 3 241 16 24 30 30 45 14 3 4+ 25 17 25- 32 3 6 54 1 5+ 4 6 25a 1 8 27 34 43 6416, 5 74i 26 19 28- 35 48 72 17+ 6 9 26+ 20 3 0 36 53 79+ 184 7 10- 26 21 3 31+ 37 58 87 9 8 12 27'22 33 38 63 94 1 20 9 13+ 271 23 3-4-+ 39 67 100+ 21 1 15 271- 24 36 40 71 106k 121+ 1 1 16- 2 27: 25 37- 45 104 156!22- 12 1 8 28 26 39 50 146 219 122- 1 3 19-L 28 27 40- 5a5 208 312 231 14 2 1 29 28 42 233 I 13+ 9 14 81 1 24 8 2 4238 10 ibs~. Ib oz. z~. 0o. ls. oz. 8!_.__,] 9 10tj148 19;242129 0 133'13~138 10 aTHE MANNER OF SOLDERING FERRULES FOR TOOL HANDLES, &C.Take your ferrule, lap round the jointing a small piece of brass wire, tnhen just wet the ferrule, scatter on the joining ground borax, put it on- the end of a wire, and hold it in the fire till the brass fuses. It will fill up the joining, and form a perfect solder. It may afterwards be turned in the lathe. CAST ENGRAVINGs. —Take the engraved plate you intend to copy, antd arrange a support of suitable materials round it, then pour on lit'the following alloy in a state of perfect fusion: tin 1 part; lead'64 parts; antimony 12 parts. These " cast plates" may be worked offt on a common printing-press, and offer a ready mode of procuring cheap copies of the works of our celebrated artists. 12 134 CocKS-LEAD —LEAD PIPE —COPPER TUBING. TABLE Of the Bore and Weight rf Cocks. Content of ock. Weight of Content of Bor Weight of Copper. Bore of Cock.: Bore of Cock. Copper. Cock. Copper Cock. Gallons. Inches. Pounds,. Gallons. Inches. Pounds. s0 o0~ 13 200o 20 30: 50 1 8 260. 3- 34: 80 2 12 340 i + 4:4 120 2+ 19 420- 3 56 150 23+ 26: 430! and; 38 70 upwards. Thriee-fourths of the diameter of the. bore, taken at the hinder part, will give the diameter of the cock at the mouth. T A B L E Of thie tVeight of Lead, per stuperfcialfoot. From one-sixteenth of an inch to one inch thick., Thickness. Weight, Thickns. Weight. Thickns. Weight. Thickness. Weight. inch. lbs. inch. Ibs. inch. lbs. inch. lbs. - 1-16th 3 1-8t h 7 - 1 3-4th 1 4-4 1-12th 5 1-6th 10.1-3rd 19+ 1 inch 59 1-lOth 6 1-5th 12 1-half: 2 9, Wleight of Lead Pipe of the usual thicknesses. Per foot in length. j-inch bore 1 lb. 1 oz. ".. 1lb. 8 oz. -1 lb. 12 oz. - 2 lbs. 1- - ".. 2 lbs.: - 2 lbs. -1 oz. — 2 lbs. 14.oz. 1+ -.. l. -lbs. I ozlbs. 4ls. 7 oz. 1+ ",.. 4:lbs. -4 lbs. 11 oz.-5 lbs.: 9 oz. 2 ".. 5lbs. 9 oz.; — lbs.: — 8 lbs, 5 oz. 2+ ".. 7 llb -8 lbs.i: 9 oz. -10 lbs. Weight of Copper Tuvbin.q. Of the usual thickness. When the inside. diameter is + of an inch, 3 ounces; + of an: inch, 5 ounces; 3 of an inch, 6 ounces; + of an inch, 8 ounces; and + of an inch, 10 ounces per foot. STRENGTH OF:MATERIALS. 135 STRENGTH OF MATERIALS. Materials of construction are liable to four different kinds of strain, viz., stretching, crushing, transverse action, and torsion or twisting:~ the first of which:depends Upon the b- ody's tenacity alone; the second, on its resistance to icomllpression; the third on its tenacity and compression combined; and the fourth, on that property by which it' oppose -any' acting force tending to change from; a straight -line, to that of a spiral direction, the fibres of which the body is Comnposed. In bodies, the. power of tenacity and resistance to compression, in the direction of their length is -as the cross-section: of their area multiplied:by the results of.experiments on similar bodies, as exllibited in the following table: TABLE IShowiing the lenacities, Resistances to Compression, and otker Properties of the colnl5mon lMaterials of' Constructioln. Absolute. Compared with Cast Iron. Names of Bodies. Tenacity Resistance Its Its in lbs. to compres- extens- stin per sq, [sio i st'egth I bility stifness inch. per sq. in. is Ash............ 14130 - 0 23 26 0.089 Beech......... 1222 8548 015 21 03 Brass............: 17968 - 10304 0'435 }0,9'049 Brick.......... 275 562 -- Cast iron......... 13434 86397 1 000 10 1'000 Copper (wrought).-. 33000 - - Elrm............. 9720: -1033 0 21 2'9 0'073 Fir, or Pine, white.. 12346 2028 0'23 2'4 0'1 it 4' red.... 11800 5375 0 /3 2'4 01 y' yellow. 11835' 5445 0'25 2:9 -0'08' Granite......... 10910 - - G8n-metal (copper } 35838 - _ 065 1.25 0.535'8,and:in 1.... MiWlleableiron. 56000 - 1.12 0 86: 1 3 Larch............ 12240 5568 0136 2'3 0058 Lead....:....... 1824 - 0096 25 0'0385 Mah ogan y,Honduras 11475 8000; 0'24 2'9.01487 Marble -551 6060 Oak.. 11880 9504: 0.25 2 -8 -:0,093 RIRope(lin. in circurn.) 20 _- ISteel........ 128000 - - - Tian (cast) 4.736 0182 0 7.75 o 025 Zihe!(sheet).... 9120 - 0365. 0.5 076 136 ROPES. AND CHANS —TIMBER. T A B L E Of the Conmparative Strength and Weight of -Ropes and Chains. ES 5 —.>- i;- a C 5 I ~r x ~.. ~3+ 2 A- 5 1 5- 10 23 + 43 10 0 44 + 4 8 116 10+1 28 49 11 11 q5 7+ Y- 10 2 10 11+ 30 I in. 56 13 8 5s 7 14 3.5 121 36 11 63 14 18 6+ 91 p 1 8 4 3+ 13 39 15 - 1 16 14 7 11+ 22 5 2 13- 45 /3- 79 18 11 8 15 1 27 6 43 141 48+- 1+ 87 20 8 8+ 19 t 32 7 7 151 56 1, 1 96 22 13 9+ 21 t 37 8 13+ 16 60 1+ 106 24 18 tNote.-It must be understood, and also borne in mind, that in estimating the amount of tensile strain to which a body is subjected, the weight of the body itself must also be takent into account; for according to its position so may it approximate to its whole weight, in tending to produce extension within itself; as in the.almost constant application of ropes and chains to great depths, considerable heights, &c. Resistance to Lateral Pressure, or Tranlsverse Action. TABLE Of D)ata, containing the Results of -Experiments o0n the Elasticity and Strengyth of variofus Species of Timber. Species of Value Value Species of Value Value Timber. of E. of S. Timber. ol E. of S. Teak,... 174 7 2462 Eln,... 50 64 101S3 Poona,.. 122'26 2221 Pitch pine,. 8868 1632 English Oak,. 105 1672 Red pine,. 133 1341 Canadian do., 155'5 1766 New Eng. fir, 158 5 1102 Dantzic do,. 86 2 1457 Riga do.,.. 90 1100 Adriatic do.,. 70 5.13838 Mar Forest do. 63 1200 Ash,... 119 ~ 2026 Larch,.. 76 900 Beech,..:98 1556 Norwayspruce, 105 - 41 1474 STRENGTH -'OF' MATEnRIA LS. 137 The strength of a square or rectangular beam to resist lateral pressure, acting in a perpendicular direction to its length, is as the breadth and square of the depth, and inversely as-the length. Thus, a beam twice the breadth of another, all other circumstances being alike, equals twice the strength of the other; or twice the depth, equal four timhes the strength, and twice the length,! equal only..half the strength, &c., according to the rule. To find the dimensions of a beams capable of o taintitninq a given wzeight, with a gives degree off:cflection, when supported at both ends., RULE. Multiply the weight to be supported in lbs. by the cube of the length in feet; divide the product by 32 times the tabular value of E, multiplied into the given deflection in inches; and the quotient is the breadth multiplied by the cube of -the. depth in inches..Note 1.-When the beam is intended to be square, then the fourth root of the quotient is the breadth andt depth required. Note'2,-If the beam is to tbe cylindrical, multiply the quotient by 17, and the fourth root.of the prodtuct is the diameter. EXAMPLE.:The distance between the supports of a beam of Riga fir is 16 feet, and the'weight it must be capable of sustaining in the middle of its length is 8000 lbs., with a deflection of not more than: of'arsn inch: what must'be the depth of the beam, supposing the breadth 8 inches? 16 x 8000 - 15175 -- 8 -= ViS97 = 12'35 in., the depth. 90 x 32 x x:75 To:determine the absolute.strenlgthlof'a tectasgnular beas of tiiiber a when supported at boths ends, alnd loaded imn the middle of its leeqth a~s beams in general oughlt to be calculated to, so that they amay be rendered! ca.tpaible of cilthstanl ding all acc'idental cases:of elneyency. R:ULE. Multiply the tubular value of S by four times the deptll of' the beam in inclhes, and by the area of the cross section in ilches: divide the product by the distance between the supports in inches, aI nid the qtuotient'will be the absolut-e strength" of the beam in lbs. Note I.-:lf the beam be not laid horizontally, the distance between the supports, for C calction1, must bIe the horizontal distance. Note 2.-One fourth of the weight obtained iby the -rule is the greatest weight that ought to be applied in practice as permanent' loai. Note 3.-If the load is to be applied at any other point than the middle, then the i st:ength will be as the product of the two distances is to the'square of half the length ot' i the beam between the supports; -or, twice the'distance'trom one end, multiplied by twice from the other, and divided by the whole length, equal the efflec'tive' length of the beam. EXAMPLE. In a building 18'feet in width, an engine -boiler of 5. tons is to be: fixed, the centre of which is to be 7 feet fromin the,wall; f'nd having two pieces of red, pine, 10 inches by 6, which I can:lay across the two walls for the purpose of slinging it at each end,: may I with sufficient confidence apply them, so as to effect this:'object? 12' 138 STRENGTH OF MATERIALS. 2240 x 5'5 -- 6160 lbs. to carry at each end. And 18 feet -7 -= 11, double each, or 14 and 22, then 14 x 22 14 x 22= 17 feet, or 204 inches, effective length of beam. 1341.x 4 x 10 x 60 Tabular value of S, red pine, 04 15776 lbs. the absolute strength of each piece of timber at that point. I' determine the dinmensions of a rectangular beam capable of supporting a required weight, with a given degree of deflection, when fixed at one end. RULE. Divide the weight to be supported. in lbs., by the tabular value of E, multiplied by the breadth and deflection, both in inches; and the cube root of the quotient, multiplied by the length in feet, equal the depth required in inches. EXAMPLE. A beam of ash is intended to bear a load of 700 lbs. at its extremity, its length being 5 feet, its breadth 4 inches, and the deflection not to exceed ~ of an inch. Tabular value of E = 119 x 4 x'5 = 238 the divisor; then 700.'238 - 2 94 x 5 -7-25 inches, depth of the beam.' o.find the absolute stren qth of a rectangular beam, when fixed at one cund and'loaded at the other. RULE. Multiply the value of S by the depth of the beam, and by the area of its section, both in inches: divide the product by the leverage in inches, and the quotient equal the absolute strength of the beam in lbs. EXAMPLE. A -beam of Riga fir, 12 inches by 4~, and projecting 6} feet from the wall; what is the greatest weight it will support at the extremity of its length? Tabular value of S = 1100 12 x 4'5 = 54 sectional area, 1100 x 12 x 54 Then, - 913884 lbs. When fracture of a beam is produced by vertical pressure, the fibres of the lower section of fracture are separated by extension, whilst at the same time those of the upper portion are destroyed by compression; hence exists a point in section where neither the One nor the other takes place, and which is distinguished as the point STRENGTH OF MATERIALS. 139 of neutral axis. Therefore, by the law of fracture thus established, and proper data of tenacity and compression given, as in the table (p. 135), we are enabled to formn metal beams of strongest section with the least possible material. Thus, in cast iron, the resistance to compression is nearly as 6- to 1 of tenacity; consequently a beam of cast iron, to be of strongest section, must be of the following form, and a parabola in the direction of its length, v the quantity of material in the bottom flange being; about 6~ times that of the upper. But such is not the case with beams of timber; for although the tenacity of timber be on an average twice that of its resistance to compression, its flexibility is so great that any considerable length of beam, where columns cannot be situated to its support, requires to be strengthened or trussed by iron rods, as in the following manner: and these applications of principle not only tend to dininish deflection, but the required purpose is also more -effectively attained, and that by lighter pieces of timber. To ascertain the absolute strength of a cast-iron beam of the preceding form, or that of strongest section. RULE. Multiply the sectional area of the bottom flange in inches by the depth of the beam in inches, and divide the product by the distance between the supports, also in inches; and 514 times the quotient equal the absolute strength of the beam in cwts. The strongest form ill which any given quantity of matter can be disposed is that of a hollow cylinder; and it has been demonstrated that the maximum of strength is obtained in cast iron when the thickness of the annulus or ring amounts to wth of the cylinder's external diameter; the relative strength of a solid to that of a hollow cylinder being as the diameters of their sections. ToRTOISE-SHELL GROUND FOR MIETAL.-Cover the plates intended to represent the transparent parts of the tortoise-shell with a thin coat of vermilion in seed-lac varnish. Then brush over the whole with a varnish composed of linseed oil boiled with umber until it is almost black. The varnish may be thinned with oil of turpentine before it is used. When the work is done it may be set in an oven, with the same precautions as the black varnish FORc.E IN PILE-DRIVING.-In a sandy soil the greatest force of a pile-driver will not drive a pile over fifteen feet. 140 STRENGTH o(Il?;.';.-it2TA B L E Showing the Weight or Pressure:a Beam:of Cast;Iron, ]: inc in breadth, will sustain,; without destroying: its ela'sticforce,,when it -is Supported at eac/h end, and loaded in the middle of:its length, ald also thie de/ct con iUn th;e n c'ddC d hich that weight will produce. Length. 6 eet. 7 feet. 8 feet. 9 feet,:10.feet. _' _.._.._ __ _. _._. _ 3 1278'24 1089'33 9.54'426 855'54 765'66 34: 1.739 *205 1143'2 28 1298 i365:1164 *46 1041'57 4 2272'18 1936 1'245 17.00'32 1520'405 1360'5 4~ 2875'16 2450 217 2146 *284 1924'36 1721'443 5 -3560:'144 3050'196 2650'256 2375 *32'21!~25'4 6 5112'123 4356. 163 3816'21 3420 27'3X060'33 7 6958'108 59291 14 /5194'183:4655'23 4165'29 8 9088'09 7744'123 6784 16.6080()1'203 5440 *25 /9 9801 1'-09 8586 142 7695:18 6885 122 io 12100 098 10600 i28 95060 *162 8500 -2 11 12826 1 17111495'15'10285 182 12 15264'107 1368.0 135 12240'17 13 16100 *1'125 14400'154 14 18600:'115116700'1143 12 feet. 14 feet. 16 feet. 18:feet. 20 feet. 6 2548 *48:2184 65 1912 -85 1699 1108 1530 1'34 7 1 3471 *41 2975 *58 2603('73.23141 93 2082 1-14 8 453" 36 3884!49 3396 64.3020 81 2720 1'00 9 57 3 -32 49 14'44 4302'57 3825'*72 3438 -89 10 7083t-) *28 6071'39 5312'51 4722 *64 4250 *8 11 8570'26 734.6'36 6428'47'5714 59 5142 7-3 i2 10192'*24 87 36'33 7648'43 6'i96 54 120 *67 13 11971 22 10260 31 8978 *39 7980 49 71 82 -61 14 138883 21 11900 *28 10412 136 9255 46 8330 5-7 1,0 159317 19 1136601.26 11952'34 106.24 43 9562'53 16 181.28 18 15536 124 13584 332 12080 -40 108s0 *5 i7'2050(0'17 17500 *23 1 53531 3'13647 *38 12282 -47 18 22,%93]'16 19656'21 17208j 28 1,5700'36 13752'44,,Note. —'Thi3 table shows the greatest weight that ever oughlt to be:laid upon a. beam for permanent load: and,; if there be any liability to jerks, &c., amplleallowance must be made; also, the weight of the beam:itself must be included.. STRENGTH OF MATERIALS. 141 bobfind the weight of a cast-iron beam of given dimensions. Rule. Multiply the sectional area in inches by the length in feet, and by 3 2, the product equal the weight in lbs.' Ex. Required the weight of a uniform rectangular beam of cast iron, 16 feet in length, 11 inches in breadth, and 1j inch in thickness. 11 x 1'5 x 16 x 3'2- 844'8 lbs. Resistance of Bodies to Flexure by Vertical Pressure. When a piece of timber is employed as a column or support, its tendency to yielding by compression is different according to the proportion between its length and area of its cross section; and supposing the form that of a cylinder whose length is less than seven or eight times its diameter, it is impossible to bend it by any force applied longitudinally, as it will be destroyed by splitting before that bending can take place; but when the length exceeds this, the column vill bend under a certain load, and be ultimately destroyed by a similar kind of action to that which has place in the transverse strain. Columns of cast ironl and of other bodies, are also similarly cir'cumstanced, this law having recently been fully developed by the experiments of Mr. Hodgkinson on columns of different diameters, and of different lengths. When the length of a cast-iron column with flat ends equals about thirty times its diameter, fracture will. be produced wholly by bending of the material. When of less length, fracture takes place partly by crushing and partly by bending. But, when the column is enlarged in the middle of its length from one and a half to twice its diameter at the ends, by being cast hollow, the strength is greater by 1th than in a solid column containing the same quantity of material. To determine the dimensions of a support or columhn? to bear, without sensible curvature, a giveln pressure in the direction of its axis. Rule. —Multiply the pressure to be supported in lbs. by the square of the.column's length in feet, and divide the product by twenty times the tabular value of E; and the quotient will be equal to the breadth multiplied by the cube of the least thickness, both being expressed in inches. Note 1.-When the pillar or support is a square, its side will be the fourth root of the quotient. a) If the pillar or column be a cylinder, multiply the tabular value of E by 12, and the hfurth root of the quotient equal the diameter..Ex. 1. What should be the least dimensions of an oak support, to, bear a weight of 2240 lbs., without sensible flexure, its breadth being 3 inches, and its length 5 feet. 4142 STRENGTH OF MATERIALS. Tabularl value of E - 105, and 2240. x 52 888 - 205 inches. Xe. 2. Required the side of a square piece of-Riga fir, 9 feet in length, to bear a permanent weight of 6000 lbs. Tabular value of E = 96, 6000 x 92 and 6-020 96 X 5 4 inclles-nearly. 20 x 96 T -AB L E,Of the Dimensions of CQlindrical Columns of GCast Iron to ssustain a c/iven load-or pressure:uith safety. LENGThI OR HEIGHT IN:FEET. ~,G[. 4 6 8 10 12 16 18 120 22.24 WEIGHT OP LOAD IN CWTS. 2 7 2 60.49 40.9325 26 22 -18 15 13 11 -2~ 119- 105' -91 77 65:55 47 40 341 29 25 3 - 178 163 - 145 128 111 97 84:73 64 56:49:3''247 232! 214 191 172 156 1-35. 119 106:94 -83 4' 326B 310 2881 266 242 220'198/ 1s78 160 144 1130 41 418 4001 379 354/ 327 301 275 251 2921 208 189 5 - 522 501 479 452 427 394 365 337 310( 285 262 6 607 *5.92 513 550 525 497 469 440 413 3886 — 360 7 1032 101'3 989 959 9'24 8 887 8- 808 765 725 686 8 1333 1315 1289 1259 1224 1185 1142 1097 1052 11005 959 9: 17l16 1697 1672 1610(1603(1561 1 515 1467 1416 i 36411311 10: 211-9 2100: 2077 2045 2007 1 964 1916: 1 865 1811 1755 1697 11 I:: 2570 2550t 2520 2490) 2450 2410 2358 2305. 2248 21.89 2127 12 130-5~0 3040 -:3020 2970 2 930 2900 283(1 278012730 2670 2600 Practica? Utility of the precedinqg Table. Ex. Wanting to support the front of a %building with cast-iron eolumnns 18 -feet in Iength, 8 inches in diameter,- and the metal 1 inch in ithickness;-;what weight may I confidently- expect each column capable of supporting: withotit -tecndency to defiection? STRENGTH OF MATERIALS. 148 Opposite 8 inches diameter and under 18 feet = 1097 Also opposite 6 in. diameter and under 18 feet = 440 657 cwi. Note.-The strength of cast iron as a column being 1'000o I" steel " 2 518 wrought irons " = 1'745 (oak),Dantzic "'1088 red deal " ='0785 Elasticity of Torsionl, or Resistance of Bodies to Twisting. The angle of fiexure by torsion is:as the length and extensibility of the body directly, and inversely as the diameter; hence the length of a bar or shaft beingf given, the power, and the leverage the pow-er acts with, being Iknown, and also the number of degrees of torsion that will not affect the action of the nlachine, to determine the diameter in cast iron, with a given angle of fextire. Rule. Multiply the power in lbs. by the lengtli of the shaft in feet,: and by the leverage in feet; divide the product by fifty-five times the number of degrees in. the angle of torsion; and the fourth root of the quotient equal the shaft's diameter in inches. Ex. Required the diameters for a series of shafts 35 feet in length, and to transmit a power equal to 1245 lbs., acting at the circumference of a wheel 2~ feet radius, so that the twist of the shafts on the application of the power:may not exceed one degree. 1245 x 35 x 2-5 --- 1981 - 6'67 inches in diameter. Relative Stresgth of Metals to resist Torsion. Cast iron.. =1 i Swedish bar iron.=105 Copper.... = 48 English do. =1-12 Yellow Brass.. = 511 Shear steel... =196 Gun-metal...'55 Cast do.. 2-1 Map Colors. YELLOW. 1. Dissolve gamboge in water. 2. Make a decoction of French berries, strain, and add a little gum arabic.. RED. I. Make; a..decoction of Brazil. dust in vinegar, and add a little gum and: alum. 2. Make an infusion of cochineal, and add a little gum. BLUE, A weak mixture of sulphate of indigo and water, to which add a little gum. GREEN. 1. Dissolve crystals of verdigris in water, and add a little gum. 2. Dissolve sap green in water, and add gum. 144- W or1GHT oF.Rno, -COPPER, ASD BtlAss. TABLE Of the Wteight of a S..perficiat Foot of Plate or Sheet lion, Copper, anqd Brass, in pounds. Iron. No. Iron. I Copper. Brass. No Iron. Copr. Brass. 3. - 1 2. 1 12.25 14.5. 13'7i. 16 2'9 2'9 2'75 T1 2'5 2 12 13 9- 13 2 17 2'18 2'52 2'4 - 5 3 11 1275 12'1 | 18 186 215 2 04'`7 5 4 10 11 6 11 - 19 1 7 1'97 1t87 3 10 5 8'74 10'1 9,61 20 1'54 1'78 169 O X 12' 0 6 812 9-4 893 2 2114, 1162 1'4 a i 1 D15 7 - 7'5 S'7 8'25 2 22 1'.261 *'45 1' 37 a. 17'- 1/ l ~t 8 6'86 t79.7 -54 t'.23; 1-I2" 1 3 1-23 t 20 9 6'24 7'2 6'86'241 1 161'1. i li ~ t22S 1I0 5'62 6'5 6' 18 25'9 1 04.99 -. 25 H 11 o5 58 S'5 E 26 8 92'88 -' 27'. 12 4.38 5o08 481 27: 7.2'.83.79. 30 1 3 3'5 4'34 4 12 28 64'74'7 -3 35 14'12 3'6 3'43 29'6-'64'61 1 40 15 2'82 3-27 3'1- 30.5'58'55 Note.-No. 1 wire gauge equal A5ths of an inch. i 4 4 77 ~ 1 "16 " 66 22'6 i 6 The great variety of thicknesses into which copper is manufactured, cause in trade the weight: to'be named whereby to determine the thickness required, the unit being that of a common sheet, so designated, viz. 4 feet by 2 feet, in lbs., thus: A.70 lb. plate is 3 jths of an inch in thickness.'i 46-,., 06 1 r6 6it 6' tii i 4 WErO}-IT O.t', It' M.A'TE11AsL.i I45 The thickness of lead is.also in common determined or understood by the weight;, the unit: being that of a- square or superficial foot; thus: 4 lbs. lead' is -'lth of an inch in thickness. 6 " t. 11 " 1i- " 15; ", Compiarative' tVeiyts of yDi#erent Bodies. Bar iron beingi.J, Cast iron being 1, Cast iron' ='95. Bar iron =.10 Steel = 1' 02 Steel _ 1 08 Copper- = 11i6 Brass - 1'16 Brass - 1'09 Copper = 121 Lead —:: 14-8 Lead = 1'56 1. Siippose I ha,:e an aiticle of plate. iron, thie weight of -:which is 728 1bs.; but wt ant the sainle of coppel;, and of simrilar dimenhsions,. what! -it11 be its wveigllt 728 x 1 16 = 844.48 lbs. 2. A:iodel of dry pine, weighing 32-1-lbs., allnd in which the iron for its:construction forms no material portion of the weight, what iay i anticipate-i4ts weiglit to be in cast iron? 32'5 x 16 520 lbs. Note.- It frequently occurs, in- teie formatioli or-ir cotistrudtiil of models, that neitherthe' quality nor condition of the timber can be properly estimated; and, in such cases, it! mal' be a neLr enoug.h approximatiot:' to,-reckon -15:lbs;; otfcast iroal to each lb."ofmodel. SILVERINGPowD)E., &c., foir siltvering copper, covering the worn pdrts of plate gdods;~i &. l. L; Niti'a~te:of siier 30 gr., common salt 30 gr., cream of tart'ar:3:-dr. —' Mx. Mdistenied with:: water,;:and rubbed on dial plates or other: copper articles, it- coats them with silver. 2. Silver precipitated firom its ittlric solution by copper 20 gr., ajhiii30;;igr., creamrA of - aortaia 2: d:., salt 2 dr. - 3. Precipitated silver ~ oz, common salt 2 oz., muriate of amm onia 2 oz., corrosive sublimate.-ll dr. MaIke- it -into:a paste with'water. Copper utensils are previously boiled with tartar and alum, and rubbed with this paste, thei made red hot, and afterI wards polished. 4. Dissolve nmiiiate of silver in: a -solution -of -hyposulphite of soda,:- and nix tixis with prepared," hartshorn,; or, other suitable,powder.` PLrATIN - o: SPRIfGas. —Platinum: jpart - goi- 1 2 parts. Add tlie platinuili to the gold in a state of fusion. 146 MENSURATION OF TIMBER. Tables by which to facilitate the Melensura-l tion of Timber. 1. Flat or Board Mieasure. Breadth Area of a Breadth Area of a Breadth Area of a in inches. liiteal tbot. in inches. lineal ftbt. in inches. lineal foot. ~ 0208 4 3334 8.6667 iz'0417 4+ -3542 81 -6875 8'0625 4+.375 8s'7084 1 ~0834 4f'3958 8'.7292 1+'1042 5'4167 9'75 1' 125 5+'4375 91'7708 1i * 1459 5D' 4583 9-'7917 2'1667 5+'4792 9t 8125 2+'1875 6 5 10 *8334 2+''2084 6{'5208 10+'8542 2+ * 2292 6A' 5416 10'87,5 3'25 6 *.5625 101. 8959 3+. 2708 I7 *5833 11. 9167 3+'2916 7I'6042 111 -9375 3+'3125 7+ 625 11+ 9583 I 7'6458 11+'9792 Application and [The of the Table. 1. Required the number of square feet in a board or plank 16+ feet in length, and 9+ inches in breadth. Opposite 9+ is'8125 x 16'5 =-13'4 square feet. 2. A board 1 foot 2+ inches in breadth, and 21 feet in length; what is its superficial content in square feet? Opposite 2+ is'2292, to which add the 1 foot. Then 1-2292 x 21 = 25-8 square feet. 3. In- a board 15+ inches at one end, 9 inches at the other, and 14- feet in length, how many square feet? 155 + 9 =2 -12+, or 1'0208; and 1'0208 x 14'5 = 14'8 square feet. To GIVE IRON A TEMPER TO CUT PORPHYRY.-Make your iron red hot, and plunge it into distilled water from nettles, acanthus, and pilosella, or in the very juice pounded out from these plants. PASTE FOR CLEANING METALS.-Take oxalic acid 1 part: rottenstone 6 parts. Mix with equal parts of train oil and spirits of turpentine to a paste. MENSURATION - OF - TIMBER. ~147 2. Cabic or Solid Meawsure. Mean y4 Cubic feet Mean 34 Cubic feet Mean Y4 Cubic feet girth in in each I girth in in each girth in in each inches. lineal foot. inches. lineal foot. Inches. lineal foot, _ lI f-oo 6. 25 14 1.361 22 3: 862 6i'272 141 1'41 221 3-438 6+ 294 141 146 224- 3 516 6+'317 14+ 1 511 2+ 3'598 7, *340 15 1 562 23 3'367 3 7+ *364 15+ 1i615 23;: 3'754 7~+ ~ 39 13.+ 1' 668 234 3 835 7 417 151 1'722 3as 3'. 917 8 444 16 1!777 24 4' 8'472 161 1833 24L 4084 8 2 *501 16+ 1P89 24+ 4 168 8+.531 161 1'948 241 4'254 9 *562 1 17 2 006 25 4: 4 | 9+ 594 17 2-066 25+4 4 428 9 ~ *626 174- 2-126 25 4 5 i6 9+ * 65i9 17 2 187 25+ 4 6 05-) 10 694 18 2 25 1 26 4 694 10+ *73 18 2' 313 261 478S5 10+ *766 184 2 376 264 4 676 10+t *803 18+ 2- 442 26 4 9t 11i 84 19 2' 506 27 5 062t 11 *-878 19 2- 574 27~ 5 158 11+' 918 19 2'64 27 * 5')52 li+'959 ] 19 2'709 27! 5 3 48 12 1 20 2'777 28 5' 4.44 12t 1.042 203 2 -89s 284 5.542 121 1o085 20 2;917.28 5.G4 134 1 129 20+ 2'99 28+ 5'74 13' 1>174 21 3 062 29 5'84 13 1*219 21+ 3t 136 29+ 5'941 131 1-265 21 3'2(,9 29 6'044 138 1'13 21i 3-285 29) 1'146 In lthe cubic estimation of timber, custom has established the rule of 1 the mean girt-being the side of the square considered as the cross sectional dimensions; hence, multiply the number of cubic feet per lineal foot, as in the Table of Cubic Measure, opposite the. + girth, and the product is tlhe solidit)y of the given dimensions in cubic feet. Suppose. the mean I girth of a tree 211 inches, and its length 16 feet, what are its contents in cubic feet? 3'136' x 16-= 50-176 cubic feet. .l118 4(C Y LINDERS —PIPES. CAST- METAL CYLINDERS. The Cylinders are solid, each 1.foot in length. Diameter. Iron. Copper. Brass. Lead. inches. bs. lbs. lbs. lbs. 1 2' 5 30 2'9 3'9 2 9'8 12-0 11-4 15 5 3 22'1 1 27 0 25'8 34' 8 4 9' 3 47 9 45' 8 61:9 5 161- 4 714- 9 716 96-7 6 8 8'4 107 8 103'0 139'3 7 1120 3 146 8 140'2 189'6 8 157 1 191 7 183'2 247 - 7 9 1,98 *8 2424 7 231 8 313-4 10 245'4 29-9'5 286'2 387"0 CAST-IRON PIPES. Ta/ble sh6oing the -lTeiclght of Pipes 1 foot long, of bores from' 1. isnch to: 1!2 inchleS in dliamneter, cldvcesiCineg by I of an inch i and of thticknesses from I: of an inch to 1:i inches, adlvaneing by i of an inch.. _ bore. 1 1 1 1 il. lbs. lbs. lb. lbs. I bs. lbs. lbs. lbs. 1bs. 1 3 1 511 74 10 0 12 9 161 19 6 23 5 27.6 1 3.7 6.0 8-6 11-5 14.7 183 221. 262 307 1 4-3 6-9 9'8 13'0 166 204 245 29'0 33 7 14 4 9 7.'8 111 14'6, 18'4 226 27 0 31'8 368 2 5'B 8 -8 123 161 20'3 24 7 29-5 34'5 39'9 24 6-1 9 7 13 5 17 6 22.1 26.8 31 9 37 3 49 0 24 67 10' 6 147 19'2 23'9 289 34.4 400 i 46;0 24 7 -4 11'5 160 20.7 257 831'1 368 14218 49,1 3 8 0 12 4 ]7 2 22.2 27.6 3 33 39 3 45:6 52.2 3+ 8-6 123 18 4 23'8 29'5 354 41-7 4831 55'2 31 92 9 142 196 25.'3 313 376 442 511 58'3 9'8 15-2 209 269 3311 397 46:6 5388 61:4 4'' 104 -'16,1 221 28'4 35 0 419 4-931 561.6 64,4 441 11'11 171 234 30'I0 36'9 441 516 594 676 4. 117 18 0 24!5 31"4 38'7 42 540 621 70'6 44 12-3 1889 2'5'8'330 40'5 483 565 64:9 736 5| 1291 19 98 27 01 34'5 4263 505 5893 676 76,7 51 13B5 20' 7 28 2 36'1 44'2 526 61-4 70:4 719 8 5 14 1 21 6 29-' 3716 4601 54 8 63'8 73"2 82'8 STRENGTH OF MATERIALS. 149 CAST-IRON PIPES. (Contiriued.) bore. 1 _ _ 1 1 f 1 in. Ibs. lbs. Ibs. Ibs. lbs. Ibs. Ihs. lbs. Ibs. 5+ 14-7 22'6 30'7 3391' 47-9 56'9 66-3 76'0 85-9 -6 -15-3 235 -'31-9 40'7' 49'7 59-1 68'7 78-7 88-8 61 16'0 24.4 33.1 42.2 51.5 61-2 71.2 81-2 92.0 6+ 16'6 253i 34'4 43-7 53'4 63-4 73'4' 84-2 95-1 6+1 17,2 26-2 35 6 45-3 55-2 65'3 76-1 - 87'0 98'2 7 17'8;27 -2 36.8 468 6- 568 67'7 78' 5 89-7 101i2 71 18.4.28.1 38-1:1 48.1 58:9 69-8 81'0 92'5 104:2 7+ / 19,01 29'0 3'91 49.9 60-7 72-0 83-5 95'3 107-4 7 19,6 29-7 40'5 51'-4 62'6 74-1 85a9 980 110-5 8 20-0 30-8 41-7 52'9 64'4 76'2 88'4 100'8 113-5 81 20-9 31'7 43'0. 54,5 66'3 78'4 90'8 103,5 116'6 8+ 21'7 32'9 44'4 56'2 68'3 80-8 9351 106'5 119'9'8 221 133'6.45'4 57 55 70O0 82'7 95'7 109'1 1227 9`'22'7 34'5 46'6 59'1 71'8 84'8- 98,2 1i118 125 8 91+ 23'3 35'4 47'9 60'6, 7'3'6 87'0 100'6 114'6 128'9 91,,23.9.36,4 49~1 62.1 75o5 891 103-1 117 4 1319 9 1 24-6 37-3 50"3 63'7 773 913 105-5 120-1 135-0 10 25-2 38-2 51.5 6562 79`2 934 108'0 122-8 138'1 101 25 8' 39'1| 52'8 66'71 81 0 95'6 110'4 125-6 141'1 10-'264 40'0 54'0 68'3 82'8 977 1129 128-4 144-2 10t 27-0 41 0'552'698 847 99-9 115-4 131 2 147'3 1:1 27'6 41'9 56.5 71 3 86'5 102'0 117 8 133'9 150'3 11i 28'2 4281 57 7 72'9 884 104'2 120'3 1367 153~4 11+ 28'8 43'7 58'9 1744 90'2 106'3 12'27 13974 156'4 11t 29.5' 446 601 175:9 92'0 1085 125 2 142 2 159'5 12 30'1 456 61 4 1775 93-6 110 6 127 6 145'0 162'6 Strenlgth of Jouriuals of Shafts. Mr. Buchanan's rule is: The cube root of the -weight in cwts. is |nearly equal to the diameter of the journal; it being prudent to make the journal a little: more than less, and to make. ta due allowance for wearing. IEx. What is tllhe diameter of a journal of a water-wheel shaft, 13 feet long, the weight of the wheel bei'ng 15 tons? By Mr. B.'s rule, V/l15 x 20- 6'7, or 7 inehes diameter. 13* 1:50 $STRENGTHII OF;MATERIALSBy Mr. Tredgold's rule, 336u Weight in the middle, x:13.- 873 /S73 91 inches diam. 600 V4, 6800 W eight equally distributed. 33600 x 13 = 436800 0 765 inches. -10 To re.sist:Torsion or Twisting. It -is obvious that the strength of revolving shafts* His directly as the cubes of their diameters and:revolutioins; and inversely as the resistance theiy have to overcome. Mr. Robertson;Buchanan, in; his Essay on the Strength of:Shafts, gives -the following data, deduced from several -experiments, viz.: That the fly-wheel shaft of Pa 50-horse-power engine, at 50 revolutions per minute, requires to be 7~ inches diameter; and therefore the cube of -this diameter, which is = -421S75, serves as a multiplier to all other shafts in the same proportion; and, staking this as a standard,:he gives the folioxving nultipliers, viz..: For the shaft of a steam-engine, water-wheel,.or any shaft connected with.a first power,.....400 For shafts in inside of mills, to drive Smalle.r machinery, or connected:witi the shafts above..200 For the small shafts ofa mill or machinery,.... From the foregoing, the following rule is derived, viz.: The number of horse, power a shaft is equal to.is direetly as the cube of the diameter and number of revolutions; and inversely as the above multipliers. Ex. 1. When the.fly-wheel shaft of a 45-horse-powver steam, engine makes 90 revolutions per minute, -hat is the diameter ofi the journal:? 45 x 400. = 200 V/200 = 580 inches diameter.:Ex. 2. The velocity of a shaft is 80 revolutions per minute, and' its diameter is 3 inches; what is its power? 33 x 80 -- =- 5'4 horse power. 400 EIx. 3. VWhat will be the diameter of the shaft in the:first example, when used as a shaft of the second.mover.5'8 - 4'64, or 45 200 46 inches diameter. 125'' 0 5Shafts here are understood as the journals of shafts, the bodies of shafts being generally made square. t'IThe diameters of the second movers wili be found by dividing the numbers in the table by 1"25, and the diameters of the third movers, by dividing the numbers by I'56. TABLE of the diameters of sh-afts, being the first movers, or ha.v2ing 400 for their multipliers. REVOLUTION:S. 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 INCHES DIAMETER.. 4 5 48 4 5 4 - - 37 — S 37 8 3 5 3 3 3 2 31 3, 2-9 2-9 28 27 27 2 6 2G6 26 2 5 5 5 -9- 5-1 4-7 4-4 4:1:399 373 36 3 3 3 33 32 3 1 3 3- 29 9 28 28 2'7 6 6- 5 5 5' 46 64-4 4 4 1 4 S3-5 38 - 7 36 5- 3 34 3- 2 32 3- 2 2-9 7 6r6 5 8 — 5;2 49. 4 6 4:4 42-: 4 3 9 37 36.3 6 3 5 34 34 33 34 3 3-2 3-1 341 S 6G 9 f6 5a-5 5 1 -1 148 S 4 —6 4-4 4 2 42 1 -4I 3- 9 3 8 37 36B 35 35 3-4 34.3:3 3.2 9 6 2: 6-3 57 5- 5 48 45 4-.4 42- 441 4- 348 37 37 36 3{6 35 3; 5 34 353 4- 3,9 3.7~ I:8:: 5,.6 I-4;~ ~3 3,6 3.5` -4: 10 74 6 45 59 64 54 2 4494" 4 4 4'2 41 4: * 8- 37 37 396 3- 6 3'5 3 4 ^: 12 9 6 9 6 3 5 8: 5-6 5-4 I 52 5- 48 -46 44 43 2 41 4 39 38 38 7 3-6 14 83 7 2 6- 7 -62 5 9 56 54 5 2 5 47 4 5 44 4 4 4 3 4-2 4 1 4- 4 39 3-8 16 8:7 7I6 7-1 66 61 5-8 5-6 5-4 5.'2 5. 4.8 4-7 46 45- 4.4 4-4 4-3 4-2 41 418 9i 7 9 7-*5 7 66 6-2 5-8 56' 564 5 2 5 49 4-8 4-7 4'3 4.5 4-4 4-3 -4.2 4t2;20 9 I3 8'1: 2 68 64 59 57 75 5-4 5.2 51 5. 4 4-6 4-6 4-5 45.4-4 4.4 25 10 8-5 8-1 4 71 6:8 6-3- 6- 5 9 56 6 5 5-4 5 3 5-2 5-1 4-948 447 46 4 6 30 107 93 84 9 74 71 69 67 6 63 5 9 5-8 5-7 5-6 5-5 5-3 5-2 51 -5 4-9 35o 11-4 9-8 8-:9 |84 7: —9 -4 71 69 6-66 65 6-3 6 1 59 5-7 5-6 5 5-4 53 a-52 6r2 40 111-7 -10-5 93 8 38:3 7 8 7 4 2 6.-. 67 6 - 6-4 62.- 5 9 5-8 567 566 56 55 45 192 10-6 9 -7 9s 8-7 8-1 176- 74 7-6 68 -7 5 64 62 6-1 6 5-9 58 5-'7 5-6 5012-6 1 1 10 9 -3 9 8-5 8- 7-8 7-4 7-3 -2 6-9 6 8 6-6 65 64 62 6G 5-9 58 55 134 114 104 9' 61 8 8 8' 4 8 75 7 3 72 6-7 6-6 6-5 6-3 62 6'1 - 601386] 12 108 10 9- 9' 86 82 77 776 7]4'7.3 -7i2 6-9 6-8 6-8 t 6-7 66 6' 4 6.42 152 STRENGTHI OF'WHEELS. It is a well known fact, tllat a cast-iron rod nrill sustain more torsional pressure than a mnalleableiron rod of the same dimensions; that is, a malleable iron rod will be twisted by a less weight than what is required to wrench a cast-iron rod of the same dimensions. When the strength of malleable is less than that of cast iron to resist torsion, it is stronger than cast iron to resist lateral pressure, and that is in proportion as 9 is to 14. From the foregoing, it is easy for the millwright to make his shafts of thle iron best suited to overcome tlie resistance to which they will be subject, and the proportioln of tlhe diameters of their journals, according to the iron of whllich they are made. Ex. What — will be the diameter of a malleable iron journal to sustain an equal weight with a cast-iron jouinal of 7 inches diameter? 7= — 343. As 14: 843:: 9: 220; now V/ 205 = 6'04 inches diameter. Strength. of lTVhcels. The arms of wheels are as levers fixed at one end, and loaded at' the other; and, conselquently, the greatest strain is upon tlhe end'of the: arm next the axle. For- that reason,' all arms of wheels should. be' strongest at that part. and tapering toward the rim. TThe rule for the bbreadth -and thickness of arms, according to. their length and number in the wheel, is as follows: Multiply tlhe opowe Or weight acting at the end of thie arm by the cube of -its length; the product of which,'dividesd by'265f6 times thle niumber' of arms multiplied by the deflection, will give the breadth and cube of the depth. Exr. Suppose the force acting at the circumference of a- spurwlieel to be 1600 lbs., the iadius of wheel 6 feet: ailll numbelr of' arms 8, and let the deflection not exceed htll of an inch. 1600 x 63 1265600-_ 1- = 163 = beadthll. and cube of the depth. 2656 x 8 x I 163 Let tile breadth be 2'5 inches; therefore 25 = 652; which is equal to the -cube of the depth. Now the cube root of 65'2 is nearly 4'03 inches: this, consequenltly,'is the depth or dimension of each armL in the direction- of the force. Note.-When the depth at the rim is intended to behalf that of the axes, use 1640 as a: divisor instead of'2656.'rhe teeth are as beams, or cantilevers, fixed at one end, and loatled at the other. The i rule applyinr directly to them where' the length of' the beam is the length of the teeth and the depth the thiclkness of the tecth.:. For the better explanation of the rule the following example is given. Ex. The greatest power acting at the pitchlline of the wheel is STRENX(TII OF WHEELS..153 6000.lbs., and the ticl;kness of the teeth 1~ inch. tlie length of tlle teeth- being 0'25: feet; it is required to determine the breadth of the, teeth. 6000 x 0;25 1500 1'21 x 1'52' —= 477 = 32;inches, the breadth reqiuired. I n-,order tlhat tihe teeth. may be e-apable of offering a sufficient resist:ance; after being w.orn by friction, ti!c breadth tlius: found should:be idoubled; theretore, in the:above exaimple, the breadth shoul: be 6 4,' or say 6 inchles. The.followi:ng idata re gleaned: fi'om experiments, Whichll are theeifore valuable, andlof much use: to the practical Ilechllanic: Rule. 3uiiltiply the blredth!of tlle teeth by the:square.of. ttlle thickness, and;divide; thle product by the lengtll;. tlhe.quotient will be thei propottional:strength in horse poweri, with a velocity of 2'27 feet per second. Ex.: What is -the power of a heel, the teeth of which are 6 inches broad, 1,5 inch Ithick, and 1: 8 inch long,, and revolving at the.velocity 6f3 3 feet: per second? 5.x 6 13'5 5.... x 65 = - 7'5,: strength at 2'27 feet per second, then 1'8 1.6 7'5 x.3 2'27: 75:: 3 = — = 991 horsepower 2'27 I Rule. The pitch is found by rriultip)lying the thlicIkness by 2'1, and the length is found by mntlti'i lying the thickness by 1'2. E'x. The thickness being 2 inches, what is the pitch and length, 2 x 2'1 =42, ptite h. 2 x 1 -2- 24, length, For, table of the proportions of wvheels, see next page. Ni ote.-The breadth of the teeth, as commonly executed by'the best mechanics,' seems to be from about twice to thrice the pitch. BEAN SHOT COPPER -Take copper, melt it, ancld pour it ili a small' stream into boiling water'. FEATHER SHnOT CorPER. —Take -copper, melt it, and pour it in a small stream into cold water. To PRESERVE WALLS vROMi DAMP:NESS.-WVhen the walls are about two feet high, use for one row of stonies or bricks a mixture of tar.! pitch, and fine sand, in thesame way as mortar. The comlposition must be previously melted to a'proper consistence. To PREVENENT IRO.N F ROoM RusTI.-;.WIarm your ir01on till you c-nnot b'eai y0u 1aland:-on iit- ithoutt burning:yourself.:Theni.rub it witlh new and clean white'vax. P:li-ti i;t again to the fire f till.t ihas soaked in the wax. When dln'e rub it over with a piece of serge. This prevents the iron 111o1i1 rusting: afterw.,ds. 1 54 ALLOYS. TABILLE of the Proport'ionqs of TWheels. 4'2 2 8' 2'40 13'8333 17'61 35'23 3 99 19 76 228 13'03 15'90 3180 3'78 18 72 216 1080 1427 28'54 8 57 17 68 204 9'63 12,72 2'5'54 336 16 6 4 192 8:53 11 27 22'54 1 315 1 6 180 75 0 991 19-82 294 14 5 6 16 653 8:63 17 26 2 73 13 5 2 156 563 744 14 88 252 12 48 144 480 684 1268 231 1 1 32 403 5 32 10'64 2'10 1' 4 120 333 4:40 8'81 189'9 108 270 3 57 7 14 1 68 8'32 96 213 281 5:62 14 7 2 8'8 1 63 2 215 4'30 1'26'6 24 72 12 10 19 3'18 105 5 2 1'6 0'83 1 10 220 3-57 1._'S 260 9 7r0 2 14' 2 ALLOYS, OR MISCELLANEOUS METALS. C1ha66 det's Medal M2etal. f 5 Copper 100 parts; tin 417.ast inmoulds fomed of 4 e2pel bone as1 h. Lead in Grains. Lead, melt it, and pour it in a small stream from a height of three or four feet into cold water. Bell.Metals. 1. Copper 25 parts3 tin 5. lix. 2. Copper 79 parts; tin 26. Mix. 3. Copper 78 parts tin 215. Mix. Co mmo Bell 7 Metal. Copper' 160 p)a8ts; tin 50. Mix Patrisiea BeXll Metal. Copper 72 parts; tin 26- iron 1C. Thi alloy is used for tfupel beadlls of sm all ornamentl clit i s stream from a eigt ofks. Batelh Metal. Brass 32 parts s; spelter 9. Mix. ALLOYS. 155 Another. Brass 35 parts; zinc 9. Mix. Brass, Copper 3 parts. Melt, then aidd zinc I part. Button Makers''iFine Brass. Brass 8 parts; zinc 5. Mix. Button Mak-ers' 06ommnonl Brass. Button brass 6 parts; tin 1; lead 1. Mix. Br iqAht Brass Color. Brass reduced to fine powder. Red Brass Color. Copper filings 3 parts; bole 2. Mix. Fine Brass. Copper 2 parts; zinc 1.. Mix. Brass for Wire. Copper 34'parts; calamine 56. Mix. To give Plates of Copper a Brass Color. Expose the plates, after being sufficiently heated, to the fumes of zinc. To Brass Copper Vessel.s I Argol 1 part; amalgam of zinc 1; muriatic acid 2; water to fill thle vessel. Mix. Brass or Hard Solder. Brass 2 parts; zinc 1. A little tin is occasionally added,,ewellers' Mfetal. Copper 30 parts; brass 10; tin 7. Mix. Fusible Alloys. 1. Bismuth 8 parts; lead 5;. tin. 3. This is fusible at boilini water heat. 2. Zinc, lead, and bismuth equal parts. This may be fused in a bit of writing paper, and will melt even in hot water. 3. Lead 3 parts; tin 2; bismuth 5. Mix. This alloy nmelts at 197~ Fah. In using this composition to make casts of seals, gems, I &c., it should be employed at- the lowest possible temperature at which it will keep fluid; for this purpose it is as well to let it! become pasty, and then forcibly impress the substances together. 4. Bismuth 2 parts; tin 3 parts; lead 5. Melt. This alloy fuses in boiling water. Gernzan Silver. 1. Nickel 1 part; zinc 1; copper 2. When intended for rolling into plates, use the following: 2. Nickel 25' parts; zinc 20; copper 60; to whichll may be added 3 of lead. 3. Pure copper 55 parts; nickle 23; zinc 17; iron 3; tin 2. Fine' Wlhite Geriman Silver. Iron 1 part; nickel'10; zinc 10; copper 20. Mix. 1:56 A LLOTS. GIernma Silver for CsCastigys, &c. Lead 3 parts nIlickel 20' zinc'20; Coppe- r. GO ix. Genignuine German Silver. Copper 401 parts: nickel'l:i: zinc 2.51;.iron 2,. Mix. G-ilding.MXetal. Copper 4 parts brass 1: tin 1. Fuse:together, Another. Copper 14 parts; zinc 6;, tin 4.:1 l7o Separate Gold:f-rol, Gilt Copper or Silver. Take a solution of borax in water,:apply to the::gilt surtace:and sprinkle over it some finely. powdered sulplhur; make the article red hot, and quench it iin w'ater; then scrape- off —thte gold, and: I recover it by means of lead. Gold it G.rains. Gold 3 parts; silver 1.;. Granulate-;by pouri'ng it in a small stream, from a moderate heiglht!, into:cold water;: then:dissolve the Silver with nitric acid, and washli ell in pure -water, next heat the grains, to give thelim- a proper histtrei. Comzmonon Gold. Spanish copper 16 parts:- silver 1: gold:2.' Melt together, Oniain'.s IAitlsible:lftl[tal. Tin 2 parts; lead 3 bismuth 5. Melt. This alloy-mnelts at 19f7~ Fallh. The addition of a little mercury renders it still more fusible. Alloy for Fl:ite Key Valves. Lead 4 parts; antimony-2.; Fuse.:' Pevter. 1. Tin 100 parts; antimony: 17.:Mix. 2.: Zinc 1 part; copper 3: lead8 8; tin 60. Melt the copper, then add the rest. 3a, Fsie0 Tin 50 parts; antimony4 4 bismuth 1: copper 1. Mix, as before. 4. French. Lead. 9 parts; tin 41.:.. Mix. Ki eller's l ei:al - i loyA; Tin:9 -parts; coplper 89; zinc: 2. GCliw M~etal. Brass 100lparts; splotter 13; tin 6, Mix. Aoth er. Copper 9 parts: tin 1. Pindchbecl'k. 1. Brass 2 parts:; copper 3. Melt under. cha-rceoa-dut. 2. Copper 5 parts: zinc 1 Melt the copper, tllen add the ziih; i.~~ -~~~Tin Filil"s. Take grain tin, melt it in an -iron ivessel,.: and stir it, while cooling, until it becomes a powder!: - theni sift!it,.~!.-.~ —---- ~ —-- ALLOYS. 15'7 | Tin, in; Graitns, Take CQrnish grain; tin, melt. it, and pou-r it -into a wooden. box, well iubbbed on the" inside with whiting_ or chailk close. theqcover,: and continue slaking it violently until- the tin is reduced to powder; then wash it in clean, water, and dry it immediately. Mosaic Gold, or lolbtl. Take copper and zinc, equal parts. Melt at the lowest temperature that will -'fuse the- formeri then mix bydstirring, and add more zinc, until the fused alloy becomes perfectly white; lastly, pour it into moulds. The proportion of zinc to the- copper is: from 50 to 55 per cent., exclusive of what:is lost by:the heat employed. Hard White Metal. Tin 1 part; spelter 3; brass 20.-: Mix. Turners'::Brass.. Brass 98 parts; lead 2. Mix, Titania, or Britannia JMetal. 1. Plate brass 2 parts; tin 2; -bismuth 2; antimony 2; copper 1; arsenic 1. Mix, and add this alloy, at discretion, to-melted:tin. I 2. Spanish. Of Spanish Titaniasmetal there are two kinds. The first is made thus: Antimony 4 parts;s tin. 2; arsenic' 1. The second is made in the following manner: Scrap iron 1 part;! antimony 2; nitre a little.:Melt- and-. harden -one pound of tin with::2 oz.. of this compositionl:. A little;arsenic improves:the color of this- alloy. TuZtenzag. Tin 2 parts; bismuth- l. Fuse. Type Metal. Lead 11 parts; antimony 2. Fuse. Ring- oGold Spanish copper 6 parts; silver 3; gold; Mix. Prince, Rupert'*s M~etal. Copper 2 parts; melt, and add zinc 1; part. White:Metal., Brass 1 part tin 2; antinony 4,. Another. Lead 20 parts; bismuth 12;:-antimony l. Fuse. Yellow D)ippinqg. iletal.' Copper 19 parts; spelter 6. Mix. A Metal that resemnbles Silver. Tinl oz.; copper 1 lb. This alloy will make a pale bell metal that will roll and ring very near -to sterling silver. Silver Dust.Take silver, dissolve it in nitric.acid, and precipitate it with slips of bright copper; wash the powder in spirits,-.;nl dry it. Ianitation.-latina. Pale brass 8 parts; spelter 5..- Mix. 14 1 58 ALLOYS. De.ssaussy' s Steel. Copper 100 parts; tin: 14. This alloy may be hardened and sharpened in a similar way to steel. Stereotype Metal. Lead 18 parts; antimony 4 parts; bismuth 2 parts. Melt. Another. Lead 16 parts; antimony 38parts; tin 5 parts; copper 2 parts. Another. Lead 20 parts; tin 8; antimony 1. Speculnan Metal. Copper 43 parts; tin 20. Mix. Another. Copper 7 parts; melt, and; add zinc 3 parts, tin 4. Prince's MKfetal. Copper 3 parts; zinc 1. Antotlher. Brass 8 parts; zinc 1,. Anothler. Zinc and copper, equal parts.' Mix. To make Irons resemble Gold. Take of linseed oil 3 oz.; tartar 2 oz.:; yolk of eggs, boiled hard and beaten, 2 oz.; aloes -i oz.; saffron 5 grains; turmeric 2 gralns. Boll together in an earthen vessel, and with it wash the iron, and it will look like gold. Should there not-be linseed oil enough more may be added. Queenz's, Mfetal. Lead I part; bismuth 1; antimrony 1; tin 9. Mix. Aiother. Tin 9 parts; bismutth 1; lead 2; antimony 1. Mix by melting. Aniother. Tin 1000 parts; regulus of antim6ny 80; bismuth 10; copper 40. Melt the copper, then expertly add the rest, and mix well together. Puriified Qiicksilver. Quicksilver 1 part; iron ilings 1. Distil in an iron retort, into a vessel containing water. Mock Gold. Platina 7 parts; copper 16; zinc 1. Fuse together. Br3onze. Metals. For medals, and small castings-copper 95 parts; tin 4. Another. Copper 89 parts; tin 8S; zinc 3. Antother. Ancient. Copper 100 parts;. tin 7; lead 7. ALLOYS. 159 Another. Kelly's. Copper 91 parts; zinc 6; tin 2; lead 1. Blanched Copper. Copper 8 parts; arsenic A part. fianheimt Gold. Copper 3 parts; zinc 1. Melt separately, then suddenly mix themr, and stir well. Red Tomebac. Copper 11 parts; zinc 2. Mix. FURNITURE PASTE. I. Melt 1 pound of beeswax with 1 pint of linseed oil, and add {oz. alkanet root; keep it at a moderiate heat till sufficiently coloied, then remove fiom the filre, add ~ pint of oil of turpentine, strain through muslin; and put it into small gallipots to cool. 2. Scrape 4 oz. of wax, and put it into a pipkin with as much oil of turpentine as will cover it, and ~ oz. of powdered resin; melt with a gentle heat, and stir in sufficient Indian red to color it. 3. Equal weights of beeswax, spirit of turpentine, and linseed oil. BRONZE POWDER. The best methods of preparing these powders are probably kept secret. The following are some of the published recipes:: 1. Gold leaf, or alloys of gold, reduced to powder by grinding them with sulphate of potash, or with']honey, and washing away the -extraneous matter with hot water, and drying the metallic:powder. 2. Dutch metal, and other similar alloys, treated in the same way. 3. Verdigris 4 oz.; tutty 2 oz.; sublimate 1 dr.; boliax 1 dr.; nitre 1 dr.:Mix them into a paste with oil, and -fuse the mixture in a crucible. This has failed in some hands, perhaps from the tutty being factitious. 4. Mix together 100 parts of sulphate of copper; and 50 of crystallized carbonate of soda; apply hleat till they unite. Powder the mass, when cold, and add 15 parts of copper filings; mix well, and keep it at a white heat for twenty minutes. Wash and dry the product. BALLS FOR SCOURING —BREECIIES BALLS, CLOTHlES BALLS. 1.: Bathbrick 4 parts; pipeclay 8; pumice 1; softsoap 1; ochre, umber, or other color, to bring it to the desired shade, q. s.; ox-gall Ito form a paste. Make it intd balls, and dry them. 2. Pipeclay 4 oz.; fuller's-earth - oz.; whiting 4 oz.; white pepper ~ oz; ox-gall sufficient to form it into a pasteo 3. Pipeclay 3 oz.; white pepper 1 dr.;. starch 1 dr, orris powder!l;dr. It may be kept in powder, or formed into balls, as above. 1 60 IMENxsUr;.riON oF CIRCLES. MENSURATION OF CIRCLES. TABLE of t~he Di'o;ters;, Cir'cunje ⪰~ d'uAd'eas' of circes. I 4 12'566- 12566 9 28/274 63'617 1963 000306 4 12 959 13 3641 4 28'667 65-396 ~ 3927 ]01227 4 13.351 1i 6 186' i -29-059 67:200 -, s~ 5890 *02761 4 13'744 15.0331! 29.452 69.029 ~ 73854 04909 1;14.137 15.904,i 29'845 7/ 882 1 *9S17 07670 I 14 5 29 168001 4 30-237 72; 759 - 11781 I 11044 1 14 9221 17-7'20 4 300630 74-662 - 1;3744.'15033 1 15 31.5 7 31'023 76.'588 4' 1651708'19635 5 15 5-1 I 13 466;16 1'8.40 -6 17 7 6771; 24850() 16'100: 20629 - 1:808:805 15 1'9635.30680 16493 21647 32 201 82-16 2 1598 *37121 16.886 2926901 4 33594 84 540: 2'356t2 44:172 4 17"2 178 23;'781 2 9586 86;590 6 2 55 25'51849 4 17 671 24850 4 33'379 88'664 4 27489 *60132. 18;064 25 9671l 33 772 90'762 2" 2 9452'69j{03-30 * 1'8s4507 27'108 I 34164 2885 1 38141'785 6 18849 28,274 11 34557 9033 4 3 534 994 19 242 29 464 -334 3950 97; 205 1: 3:927 1-227 + 1-9 635 0 679 35 343 994P02 4 4319 1.484 20027 91 31919 35735 101-623 4-712 1 767 1 20 420 -33. 183 36,1-28 1'03; 8' 9 5'105 2073'20'813 34'4711 36521 106*139 4 5 497 2 405 4 i21 2051 35-784! 4 36'913 108";434 5.890 2-761 -' 21 598 37 122 8 -37 30 i110753 2 628S3' 3 7141 7 21:991 38;484 12 -376991:13'097 6-675 3546 I 2293838 39'8711 4 38'091[ 115'46'6 7'068 3'976 - 227761'41-'282] 4 38:'4841i17'859 7,461 4430 4 23 1.691 -42-718 38'877 120 276 7854 4'908 2356'21 4421781 4 39.270 122:718 -8-246 54-11 i 123'954 145663 8'39:662 12 251'184 8639 5939 24 24 347 47'173 4 40"055 127'676 9'032 6-491 24-74048707 -40 448 130 192 3:9,,24 1 06T8 8 8'25"132, 50)2651 13 40.i840 132'732 4 9817 7 -669 4j o 1Qa 525'51i'8481 414t'233 135"297 I102l0o 8,2959 -' 25'918 534,561 4 41 626 137'886 10'602 89:946 4 26'310 55.o088[ 42 4018 140'500 10-995 9:621 12ti6-703 56:7451i -4-42-411 143'1:39 4 11;388 10;320 | 27 096| 58.4'26 } ~ 1 442804 145.802 11'781 11.0444 4 2i'489 60132i 443197 148489 I 12173 117 3 27 88 601211 4 3589 1 1 201 J 4 12 173 1 11793 | - 27'881t 6'862 } ~;143'5891 151'201 MENSURIATION OF CIRCI;ETS. 161 Diam. Circum. Arca. Diam. Circum. Area.: Diam. Circum. Area. 14 43-98 153-93 i19 59-69 283.52 24 75,39 452!39 44`37 156-69 6(08 287'27 17579 45711 44.'6 159 4S 4 6047I 29103 76'18 461'86 45'16 16'229 60h86 294-'83 16 757 466-68 4 4555 165-13 4 61-26 298-64 7696 471-43 4 45-94 167'98 165 302-'48 4 77-36 476-25 46-33. 170187 0 62'04 306'35 4 77-75 481'10 4 4-73 173'78 4 6243 310'24: } 78'14 485-97 15 47'12 176 71 20 62 83 314'16 25 78'53 490'87 4 47 51 179'67 63-22 318'09 i 7893:. 495'79 4790 18265 63 61 32s 06 1 7932 a50074 4 4830 185 66 4 6401 326'05 79'71 505-71 48'69 18869 4 6440 330'06 4 8010 510-70; 4908 191-74 4 64'79 334 10 80-50 515-72 4 49,48 194-82 6518 33816 80,89 520-76' 49'87 197'-93'65-58 34225 8128 525'83 16 50'26 20106 21 6597 34636 26 81 68 53093 50.'65 20421 4 66.36 35049 i 8207 53604 4 5105 207-39 1 66-75 35465 j 82-46 54118 51-44 210 59 4 67-15' 35884 82-85 546,35 4 51'83'213882 4 67 -54 363 051 5 83-25 551-54 4 5222 | 217'07 4 67-93 36728 | 4 83-64 556-76 52.62 220.35 4 6832 371-54 4 84-03 562-00 4 301 223.65' 68-72 375-821 84-43 567 26 -17 5340 226-98 22 69-11 380-13 27 8482'15'72 55 53 79 230'33 69-50 384-46 852-1 577-87 4 54-19 233-70 6990 388-82 85-60 583-20 i 54'58 237-10 4 70-29 393-20 4 86-00 588-57 54'97 240-52 70 68 39760 86.39 593 95 4 55 37. 243 97 4 71 07 402'03 4 8678 599'37 4 55"76 247-45 71-47 406t49 4 8717 604-80 56-15 250-94 4 71-86 410-97 4 87-57 610-26 18 56'54 254-46 23 72'25:~ 4:15-47 28 87 96 615 75 56i94 258'01 - 7264 420-00 4 88 35 621-26 57 33 261-58 73;04 424-55 4 8875 626-79 57-72 265-18 7.343 42913 89-14 632-35 4 15811 268 80 4 73-82 433-731 89- 1 637-94 4 5851 272-44 4 74'-21 -438-36 89-92 643-54 58'90 276-11 74'61 44 3-01 I 90:32 649-18 159-'29 279;8141 7i5-00 4047'69 4 9071 659' 83 14* 162 MENSURATION OF: CIRCUES. Diam. Circum.: Ara;i m. Circum. Area, Diam. dircum. Area. 29- 9110 660'52 34I 106-8 907-9:2 I 39 122-5 1194-59 t: 91 49 566622 1 1072 91461 1229 1202261 i:,91-89' 67195 107 5 921 i123-3 12(Q9 9 92'28'677-71 8 { 107-9 928'01' 123'7 1217'67 a92,67 6S;3'49 108'3 9;4'8'2 124-0 1229542 i9306' 689.29 { 108,7; 941 60 g 124'4 1233.-18 I 93"46;695,12 } 109-1 948.41 124-8 1240:981 -, Si93 85 lo00o98' - 109-5 955 25 125 2 124879 30,94t24 I706 680 35, 109 9 962-11 40 126`6 1256W64'94,64,,712 70I -i 110.3 968.99 ~ 126-0 126;4.50 - 9503 71186 { 1107 975-90 - 126 4 127-2 39:95'42 724'64 I X 111 1 992'84 i 126-8 1280'31 I 95,81 7'3061 i 111 5 989 80 12'7 2 1288-25:- 96'21 7I3661; 1119 99678 / 1276 1296.211:96.60 742 64 112 3 1003 71 1280 1304-20 | 6 9,9699: 748-69; 112-7 1010 81. 128-4 11312 21 31.97-38 75476 36 113-0 101-7:87 41 128 -8 1320'25'' 97!78 760-86 I 1134 1024 95 129 1 1328-32:9,98 17. 766 99 113 8 1032-06 129'5 1 336'401' r98'56 773'14 [ 114-2 103919'1 12:9-9 134451 98-96 779'31, } 114-6 1046'39: 130'3 1352'65! 99'35 785'51' 11-5:0 1053-52'i 130 7 1360 81'99 74 791 73 115 4: 1060783. 131-1 1369 00 10013 9797 18 106795 1315 13721 32 100 5 804 24 37 116I2 1075 212 42 131-9 138544 *' 100'9 810541 I 11'6-6 1082'48 8 132'3 1393:70' 4 1013 816-86 1 l 0O 1089 -79 i 132 -7 1401'981 t 10107 823'21! i 117 -4 1097'11i 133- 1 1410'291 ].102-1 82957 117'-8 1104-46 ] 1 i335 141862 9 102-4 835-97'1 118 -2 1111 84 i 133'9 1426'981' 102-8 842-39 I 1185 111-9 24 134:3 3 1435 361 p. 103 2 848 83 118-9 112666 1346 443 77 83'1036 855-30 138. 119-3 1134-11 43 135 0 1452i20 -} 104'0 86179 1197!1141-59 135-4 1460-651 104-4 868-30 B 120 1 1149 08, 135-8 1469 131:: 104'8 874:,84 1 1205 1156'61 - J 1362 1471'631. 1'05'2: 881 41 120-9/ 1164-1 -i 136-6 1486 171 1i05'6 888-00 1 121-3 1171-73 i 13-70 1494 -72 1060' 894-61; 121-7 117932' 137-4 1503301 106-4 901'25: 8 1,2-1 118694$ - 13 7-8: 51'1-90 Mt;SUR:AtItON -:OF CiRCE.,LES 16 Diarn. -Circui |. Area Dirin. 1Cilm. Area. Diam. Circum, Area. 44 138-2 1520 5.3 46 144,5 1661-90 48 150 7 180956 i i 138-6 1529 18 i 144 9 167095 g 1511 1818 99 i''189o0'1537'86 145.2 }1680-01 } 151.5- 1'828.46.139-4;''1546'55 145'6 1'689 i 0[ t 151.9 1837'93'139si8 -1555.28 146.0 41.69823 152.3 184745 140;1:1564'03.',146.4.17.7'307 371 152'7 1856'S 99 140'5 1572'81 1468 171654 1531 1866 55 - 1140'9 1581 61-72 17'25 73 15359 187613 45 141'S3'1590'43 47 147.6 117834'94 49 1538'9 1'885.74 jI''141;7 1599'28: 148'0 I'1I744 18 } 154:3' 1895'37. 142'1 1608'15 3 148.4 1753 45 154'7 1'90503'142;5 1617T04 148`8 1762 1 1'142.9 16125'.97 ~ 149.2 1771205 155:5 i 1924.41'.'143'3 1634192 149,'6:117 81'39. 155'9 1934'15;'143!7: 16 43'9 1508.0.1-79076 15- 6. 1 1943'91 14-4'1 1652'881 i 150'4 1800'14 I!15616 119530'9 Diam. Cilrum. Arein Aieain l Diam Circum..Areain Areainl in. inches. squar'e in. square feet. i. inihee. square il. square feet. 1____ _ —: -- - - 50 157'0 1963'-5.13'63' 5'172-7 2375"8 16'49' 157'8. i9$3l -:13 1'737 a-.173.5. 2397'4 1 164 ~ 158;6 2002'9.1390 i i 17I43 2419'2 168(0'+ 1 |5'l94. 2022. 8 1409, 4 2' 1T57:1 2441'0 1 695 51 1:160'2 2042'8 14'18 - 156''.9 -2463'0'17-10 I1610:206,2'9:14'32 17.617:I 24850 17-25 4 16171/'2083'0 -'14'4 6 17775.'21507'1. 17'41'1 64'16a'; 2 2103'3 146i0 17'8'82 -25294 17't56 52 16:16'3'3.' 212'3'7 |'1i474 | 7 |1790o 25517 17 77.2 4 16d4-1 i 2144.1 1.14 89 7. 1 7'98 2574'1 -17'87 + 164~9 1216471 15003 1 2 59'7I7 186 203'4: -1657 1'21854. 1517' 1814- 26193 -18'19 53 | 16,6W'5 22(0)61:l.15a3'2 8!1 a;8I.1822 2'2I 2'642,0 18'34 1 167'2 -22227,0 156:18'2.9'266'4 -9 1850 168;0 2248;0 1 1'6.1 182'317 26878 18683 - 4 168o'8 2269,0: 15' A.:: 184-5 271.08 18-82 4 1 169;6 2290'2 1 190 i 59 1 83 27339 18'98 4' ] 11704,2311'4 181 2 191 4 + 17'11',332, 82 1 162o0 1 1:'6.9:2780.5 193-0 -:'1 172'0 2354'2 -16,34'1877 21 803'9 19'47 164 MENSURATION 0F CIRCLFS. Diam. Circum.: Area in Area in. Diam. Circum. Area in - Area in in. inches. square in. - square feet. in. inches.. square in. square feet. 60 188.4 2827.4 19-63 H69 216.7- 37392 25.96 4 - 189'2 2851'0 1979 i 1217'5 3766'4: 26'15 ~- 190.0 2874.7 19-96,218-3 393,6 26.34 4 190 8 28985 2012 4 -2191I 38210 2653 61 191'6 29224 20'29 70 219'9 3848 4.26'72 4 192'4 2946'4 20'46 4 220'6 3875'9 26'91 - 193'2 2970'5 20'62 * 2214 3903 6 27 10 / 193'9 2994'7 20'79 2222 3931 3 27 30 62 194'7 3019'0 20'96 71 223'0 3959 2 27'49 ] 195'5 3043'4 21'13 I 223'8 3987'1 27 68 ~ 196'3 30679 9 21'20 ~ 224'6 4015,1 27'87 4 197 1 30925 21]47 i 225.4 4043 2 28 07 63 197 9 3117-2 2164 7 2 226'1 40715 28'27 4 198'7 3142'0 21'81 226-9 40998 2847. 199'4 31669 21'98 -j 227'7 4128 2 28'66 200 2 3191 9 22-16 4 228 5 4156-7 2886 64 201'0 3216'9 22'34 1 73 229'3 41853, 2906, i 201 8 3242'1 2251 4- 7 2301 421416 2926 4 202,6'. 3267'4' 22'68 230'9.4242.9 26'46 - 2034 32928 2286 231-6 427P 8 2966 65 204,2' 3318'3 23'04'i 74[ 232,4 - 4300'8 29'86 I 204.9 3343.8'23.22 233,2 4329.9'30-06 ~ 205'7 3369'5 23'39 ~ 234'0 4359'1 30'26 i 2065' 33953 23-57 234.8 4388,4 3047 66 207'3 34212 2375 75 2356 4417 8 30'6' i 208'1 3447'1 23'93 i 28364 444Q73 30'88 X208 9 3473,2 24'11 1 237'1:4476f9 31,09 4- S;209 7 3499.3 243.0 2379 45066.3130 67 210'4 3525'6 2448 76 2387 45364 3150 2 21'12 3552'0 24'66 4.' 2395 4566'3 31'71 i 2120.3578'4 24'84 1 4 2403 4596'3 31'91 2128: 3605-0 25-03 241 1 4626 4 32-12 68 213'6 3631-6.25 22 77, - 241'9 4656'6 32'33 f4 214'4 3658'4 25'40 I 4 242'6 4686,9 32'54 4 21.51 3685.2 25.59 - 2434. 47172 3275 4 215 9 3712'2:2577 244'2 4747 7:32'96 kMENSURAITION::OF:CIRCLES. ]165 II i liam. Circum.. A-Area'in -'Area in Diam. L-Circum.:Areain Aren.in in iichs. square i'squre'tet. il. Inhes. isquare in. sqare feet. - -_ hs- -_ _.... r _,st -._. _et i ____s i 78: 245 0 1 4778!3 3318 87 2733 5944 6 41-28 a 245 8 4809 0 3839 2741.1 5978 4152 E 246 6 4839-8:3 33;' 8 2'74 8 6013'2 I 41 75 2 247:4 487(07 53381 8 2756- 6047'6 41'99 79 248 1 4901 6 3403 2764 60821 -2 2489 4932 7 34;24: 277- 2 61167 429 47 249-7 49663 9 3446 I 278 0 61514! 41271 2505 4995-1 346S 278.8 6186.2 42 95 80 2513 54026 5 3490 89 2796' 662211 4320 24521 5058'0 35 1124 280`3 62561 1 43'44 2528 5089-5 354834 281, 6291'2 43968 a 2536 5121 2 35 56 j 2819 63264 143892 81 254'4 51$530 3578 990 2827 63617 1 44 1'7 255''2 5184'8 3600 J 28315 63971 44'42 2.56`0 5216;8 36 2-2 A 284'3 643256 44466 2568 524898 36544 1 2851 6468 2 44'81 82 2 57 6 5281-0 36 67 91 2858 i 6503 8 45-16 2583 5313',2 3'6,90 28166 65396 45'41 259.1 5345 6 37 12 1 2874 65755 46 6 I 2599 5378'0 37 34 282' 6611 45'91 83 2607 54106 37'57 92 28909 6647'6 46416 261 5 55443812 37679 28958 6683'8 46'41 262 3 5476i.0 38 02 29085 672090 46166 26311 5508 8 38'25 291 3 6|756'4 46"91 84 263 8 5:5417 38'48 93 292"1 6792'9 47'17 41 2646 55748 38'71 I 2929 68294 47'43 2654 560(79' 3'94 3 293'7 68661 47'68 1 2(662 5641;1 39 07 I 2945 6902 9 4793 85 267 0 5674'5 39.40 94 295'3'6939'7 4819 267:8 5'7079 3963 4 296'0 69767 4845 268:6 574141 3987 i 268 10138 48:70 2693 57750 4010 | 297'6 7050'9 48'96 86 2701 | 5'808'8 4033 95 2984 70882 49'22 } 270`9 58426 4057 + 299:2:7t1255 49'48 27 1"7 5876''5 4(180 j 300!0 71630 49'64 272-5:591005 4I104 i 3i008 72d0o 5 50soo 9; 1:66 MENSURATION OF CIRCLES. Diam. Circum. Arain Area in I Area in in. incihes. j square in. square feet. in. inche.s. square in. square feet. 96 301.5 7238.2 50.26 1121 380.1 11499.0 79.85 1 30 3 7275'9 50'52 [ 122[ 383'2 11689,9 81-18 ~ 303'1 7313'8 50'i 78 123 386*4 11 882'3 82 51 303'9 7351-7 51'05 124 389-5 12076'3 8386 125 392'7 12271'8 85'22 97 301)'7 - 7389'8 5135 * 3055 (7427'9 51'57 2 3063 7466.2 584 126 395'8 124690 8659 A 3070 75045 5211 127 398'9 12667'7 87'97 ) t 30'0:3 7 504'6 2 |:a2t 128 4021 12867'9 89-36 129 405'2 13069 8 90"76 3086 7581.5 5203 130 408.4 13273'2 92'17 3 0 8 - 6 75sv.5 52.06, 0:3094 7620-1' 5291 A 3102 5387658,8 53IS'1 l31 411-5 13478.2 92.59 132 414'6 [ 13684-8 95 03 99 f 311 0 7697 7 5345a 133 417*8 13892-' 96-47 3I 3 11'8 7736:6 53 72 134 420'9 14102' 6 97 93 312-5 7 775 6 5 39 9 135 4A-1 14313-9 99'40 9 13133 7814'7 54'26! 00 314-1 7854-0 54'51 100 141 780 136 427-2 14'5267 1 00'88 4101 / 831 0117 563 [137 I430'3 114741'1 [102'36 102 320,4 8091:2 56.744 a 436 15 1038 139 436'6 ]15174'7 105'37 103 323~5 8332~3 57~86 l~-140' 439 8 15393-8 106'90 104 326'7 8494'9 58'99 105 329'8 8659'0 60'13 141 442,9 15614-5 1108-43 106 30330, ~88247' 61'28 1421 446-1 15836'8 109'97 101 336'1 / 8992'0 1 62'44 1431 449-2 160606 111-53 108 3,392 91609 63-61 144 452-3 16286"0 113'09 109 13424 9331-1 64-80 145 455-5 165130 114'67 1 110. 3 Y455 9503,3 65'99 146 458'6 16741'5 11626 111! 3148'7 | 9676 9. 67-20 147 ] 461'8 16971'7 11786 112 351'8 9852'0 68'41 148 4649 17203 1 113 3550 1-0028.7 /696.4 149 4680 174366 12108;1 114 3581 3a 10207~0, 701'88 1150 47162 176715 12271 115' 3612 10386-9 7 2'13 116 364-4 10568-3 7339 151 474.3 179079 12436 I 117:367'5 10751 3 74'66 11521 477'5 [ 18145'9 126"01 118. 370'7 1,10935'9 75'94 153 14806 1 8385'4 127'67 1 19: 373-8 11122-0 1: 77'23 1541 -483:8 18626-5 129-35 i 120 376'6 111309'7 7854 155 4869 188692 13103 CIRCUMFERENCES AND AREAS OF CIRCLES. 1 67 TAB LE Of the Circumferences and Areas of Circles, from 1 to 50 feet, advancing by an inch. Diamo Circumference Diam. Circumference ft. & in. in feet and mi. Area in feet. ft. & in. in feet and in. Area in feet. lft. 3 1i'7854 3 13 4* 14'1862 1 3 4' 9217 4. 13 7 14'7479 2 3 8 10690 5 13 10: 163206 3 3 11! 12271 6 14 1 1569043 4 4 21 1'3962 7 14 4* 16'4986 5 4 5* 1.5761 8 14 7- 17'1041 6 4 8 1'7671 9 14 11 17'7205 7 4 11* 1 9689 10 15 2* 18'3476 8 5 2 2.1816 11 15 56 18.9858 9 5 5i 2 4052 ft. 15 8* 19.6350 10 5 9 2'6398 1 15 11 - 202947 11 6 2* 28852 2 16 2* 20'9656 2 ft. 6 3 3.1416 3 16 51 21 6475 1 6 6-,s 34087 4 16 9 22'3400 2 6 9* 3'6869 5 17 01 23'0437 3 7 0* 3'9760 6 17 3~ 23'7583 4 7 34 4'2760 7 17 6 24'4835 5 7 7 4 5869 8 17 9- 25'2199 6 7 10* 49087 9 18 0 25 9672 7 8 18 5`2413 10' 18 3* 26'7251 8 8 4~ 5'5850 11 1S 27'4943 9 8 7* 5'9395 6ft. 18 10- 28'2744 10 8 10* 6 3049 11 19 1i 29-0649 | 11 9 1* 6'6813 2 19 4.-29'8668 3ft. 9 5 7'0686 3 19 7~ 30'6796 1 9 8s 74666 4 19 10 — 31 5029 2 9 11 7'8757 5 20 1 32'3376 3 10 2*. 8 2957 6 20 47 33-1831 4 10 5* 8'7265 7 20 81 34'0391 5 10 8* 9'1683 8 20 11* 34'9065 6 10 114 9'6211 9 21 2* 35'7847 7 11 3 10'0846 10 21 5* 36'6735 8 11 6* 10'5591 11 21 8* 375736 9 11 9* 11-0446 7ft. 21 11 38'4846 10 12 5 11'5409 1 2,2 3 39'4060 11 12 3* 12'0481 2 22 6- 40'3388 4 t. 12 6* 125664 3 22 9 41'282.5 1 12 9* 13'0952 4 23 0* 42'2367 2 13 1 13'6353 5 23 2- 43-2022 1 68. CIRCUMFERENCES,AND AREAS OF..CIRCLES.. Diam. Circumference Diam. Circumference ft. in. in in leet and in. Area in feet. f.ie: in teet and in Area in feet, 6' 23'6-t -2 44'1787 3 45 41 9934021' 7 23 11 45"1656 4 35 7 100'8797 8 24 13 46'1638 5 35 1(- 10236S.9 9 24 4J.: 4717,30 6 36 103'8691 10' 24' 8 7 427 36 48926 105 7i794 11' 21 10 49'22'6' I 8 36 74 106'9013 S ft. 25 1 0 50.265 9 36 107 108'434'2 1 2.5 4 5613178 10 37 2- 109.9772 2, 225 i 52.3816 11 37 51 111531'9 3- 25 11 53'4562 12ft. 37 8t 113-0976 4I 26 24 54-5412 1 37 114 114'6732 5 26 54 5b 6 3177 2 38 24 116-2607 6 26 84 56'745.1 3 38 54 117 8590 7 26 11 5,7'8628 4 38 84 119'4674 81 27 2 58 9920 5 39 0 121'08-76 9. 27 54 60,Q1821.L 6 39 31 1227187 10; 27 9 612826 7 39 64 124.3598 11.. 28 04 62'4445 8 39 94 126'0127 9ft., 28 34 636174 9 40 o0 127 6765 1 1 28 64 64-8006 10,.- 40 4 12938594 I 2 28 91 65'995,,1 11 40 64 1319Q360 3,8 29 0' 67 2007. 13ft./1- 40 19 1o3273-26 4 29 34f 68'4166 1 41 14 134'4391 5 29 7 69-6440 2; 41 44 136'1574 6. 29 104 7 88. 3 41 74 137'8867 7 30 1 72'130,9 4. 41 104- 139'6260 8 30 44 73-g3910 5 42 14- 141'3771 9 30 7 74'6620 6 42 4; 143'1391!10- 30 119 75'9433 7 42 8 144'9111 11 31 1- 72362 8 42 114- 146-6949 10 ft., 31 5 8'540 9 43 24 148'4896 1 31 84 79 8540 10-, 43 54 150'29432 31 114 811795 11.' 43 84 152'1109 3 32 24 825160 14 ft.1:, 43 11 153-9384 4 32 54 83'8627 1' 44 24 155'7758 5 32 84 85'2211 2 44 6 157 6250 6 32 114 86-5903 3. 44 94 159'4852 77 33 21 87'9697 4 45 0O 161-3553 8 33 64 89'3608 5 45 3 163'23,73! 9 33 94 90'7627 6 45 6-4 165i1303 10. 34 04 92-1749 7 45 94 167 033:1 11. 34 34 93 5986 8 46 04 168'94,79 I l7t.- 34 64 95'0334 9 46 4 170'873.5 1 34 94 96'4783 10 46 74 172'8091 2 -35 04~ 97 93471 1'1 46 114 174'7565 CIRCUMt1ERENCES ANPD AUREAS OF CIRCLES, 169 Diamlm Cirltumnference.e iam. Circumference ft, & in. in:feet and in. Areti in:foet. ft. & in. in feet and in. Area in feet. 15 ft. 47 1 -1767:150 9 58 10 276'1171 1 147 4j. 178'6832 1-0 - 9 2 278'5761 2 1 47 7 180'6684 8 1 11 59 5* 281'0472 3 47 -107 182-6545 19 ft 59 8j 28835294 4 48 2k 1846555 1 59 11 2860210 1 5 48 59 1866684 2 60 2* 288'5249 6 48 8f 188'6923 3 60 5i 291'0397 7 48 11 1907T260 4 60 8: 2935641 8 49 2 192'7716 - 5 60 1 1 296'1.107 9 49 5k 1948282 6 61 3: 298'6483 10'49 8j 196'8946 61 6: 301'2054 1-1 li50 0 198-9730 8 61 91 3037747 16- ft. 50 83 201;0624 9 61 0 3806'3550 1 50 6* 203'1615 10 62 3 8 308'9448 2 50 9f 2052726 11 62 6 31 1-54-69 3 51 0* 207 3946 20rft. 62 9j 314'1:600 4 51 s3 209'5.264 1 63 1* 316'7824 5 51 6* 211 6703 2 63 4j 319i4173 51 10 213'8251 3 63 7 322.0630' 7 52 1* 215'9896 4 03 114, 324'7182 8 52 4- 218'1662 ) 5 6 1 327:'3858 9 52 7* 220-5307 6 64 4t 330'0643 10 52 10* 222'5510 7 64 7 3832-7522 I 11 53 1 224'7603 8 64 11 335'44525 17 ft 53 4t 226'9806 9 65 2* 338'1637 i 53 8 2. 229'2105 10 65 5* 340'8844. 53 11* 231'4525 11 65 8: 343'6174 3 -549 2i* 2337055 21ft. 65 11* 346'3614 4 54 51 235'9682 1 66 2* 349'1.147 5 54 8 238'2430 2 66 5 351'8804 6 54 1 1: 240'5287 3 66 9 354'6571 7 55 2* 2428241 4 66 0*; 3574432;8 55 6 245'1316 5 67 3*. 360.2417 *9 55 9* 247-45()0 6 67 6* 363'0511 10 56 01: 249'7781 7 67 9- 365-8698 11 56 3* 252'1184 8 68 0O 368'7011 18 ft 56 6* 2544696 9 68 31 371.5432 1 56.9 25688303 10. 68 7 3743947 2 57 0* 2592033 11 68 101 3772587 3 57 4 261'5872 22 t. 69 1* 380'1336 4 57. 7I 263'9807 1 69 4~ 383'0177 5 57 101 266'3864 2 69 7* 3859144 6 I58 11 26B'8031 3 69 10 388'8220 7 5'8 41 271'2293 4 70 1* 3917389 I 8 58 7* 27b 6678 5 70 5 3946683 15 170 CRCmUMV1J1UMIis s AND AREA9 O1 CaWcLUS. Diam.n Cireumferenppe Diam. Circumferencee ft. & il. In ftet andt in. Area in fet, ft. & in in feet andi i. Area in feet. 70 8 3 8976087 3 82 - 5~ 541'1896. 7 0 I1 1 400'558! 4'82. 85 544'6299 1 2 - 4 03-204 1 5; 82 11 548 08301 9 71 5% 406'4935 6 83 3 5-515471 10 1:8 8 409'4759 I 83 6% 555'0201 11 71 11% I 412'4707 8 83 94 558'5059 23/It. 72 3 1 415'4-766 9 84 0O 562'002'7 1 72 6% 418'4915 10 84 3% 565 -5084 2 72 9 1 421'5192 11 84 6- 569'0270, 73 0i 424'5577 7 ftl. 84 9% 572 5566 4 73 3% 3 427'6055 1 85 1 576'0949 5 73 6% 430'6658 2. 85 4} 579'6463 6 73 9 1 433'7371 3 85 8s 583'2085 7 -74 1 436'8175 4 85 113 586'7796 8 74 4% 439'91(6 5 86 1 590'3637 9 74 7%. 443-0146 6 86 41 59395e87 10 74 10Q 446-1278 7 86 7% 597 5625 11 75 1: 4492536 8 86 1 861 601'1793 24ft. 75 41 452 3904 9 87 2% 6048070:1 75 7 4555362 10 87 5. 608 4436 2 75 11 I458'6948 11 87 8t 612'0931 3 76 2% 461'8642 28 ft. 87 11 615 7536 4 76 5% 465'0428 1 88 2% 619'4228 5 76 8% 468-2341 2 88 5- 623-1050 6 76 11 471-4363 1 3 88 9 626'7982 7 77 2% 474'6476 4 89 0% 630 5002,8 77 5% 477'816 I 5 89 3% 6342152 9 77 9 481-1065 6 89 6% 637-9411 10 "78 0% 4843'506 89 9% 641'6758 11 78 3% 487-607 3 8 90 0t 645'4235 25ft. 78 63 4-908750 9 90 % 649 1821 1 78 9%-. 494-1T51.6 10:90 6%-: 65249495 2 1 79 0% 497'441 1 11 90 11% 656'7300 3 79 3S 500'7415 29ft. 91 11 660 5214 4 79 7%. 504'0510 1 91 4-% 664'3214 5 79 11 507 3732 2 91 7% 668 1346 6 W80) 11 510(7063 1 3 91 10I % 671-957T 87 -8 4% 514'0484 4- 92 1%- 675 -7915 80 7 517-4034 5 92 418 679'6375 9 80i 10, 520-76922'! 6 92 8%- 68-34943 10 81 1% 524-1441 - 92 11%- 68:7'-3598 11: 81 5 527'5318 i 8 93' 2%f 691-2385 26Pft. 8 8% 530-9304 9 9 51 695-1280.It 81 11% 534'3379 10 9s3 8- 699'0263 2 82. 2;| 537-7583, 11 93 11%: 702'9377T _.~:] ~ ----'81' —--- CIRCUMFERENCEs AND AREAS OF CIRCLES. 171 Diam. Circumference Diam. Circumference ft. & in. in feet and in. Area in feet. ft. & in. in feet and in. Aiea in feet. 30ft. 94 2 706'8600 90 106 0: 894'6196 1 94 6 710-7909 10 106 3+ 8S'90413 2 91 81 714-'7350 11 106 6+- 903'4763 3 95 99 718'6900 34 ft. 106 91 907 9224 4 95 3+ 7122'6537 1 107 0+ 912'3767 5 95 6 7266305 2 107 4- 916'8445 6 95 99 730'6183 3 107 7j 921'3232 7 96 0+ 73-16147 4 107 10 925'8103 8 96 4 738'6242 5 108 1] 930'3108 9 96 7i- 742'6447 6 108 4 934-8223 10 9t1 1(0 746'6738 7 108 7 9393421 11 97 1,- 750'7161 8 108 10- 943'8753 31f 97 4+ 751'7694 9 109 2 9484195 1 97 7+ S 7'588311 10 109 5 952'9720 2 97 10 762 9062 11 109 8 957'5380 3 98 2 766'9921 35ft. 109 11+ 962'1150 4 98 57 771'0866 1 110 2+ 966'70u1 5 98 8+ 775'1914 2 110 5+ 971'2989 6 98 1 1 779-3131 3 110 8 K 975'9085. 7 99 2+ 783 4403 111 0 980'5264 8 99 s5 7875808 5 111 3_ 985'1579 9 99 8+ 791-7322 6 111 6-s 989'8003 10 100 0 795 8 22 1 7 111 4 + 994 4509 11 100 3- 800'0654 8 112 0+ 999'1151 321. 100 6+ 804 2496 9 112 3+ 1003'7902 1 100 91 808 442 1 112 64 1008'4736 2 101 07 812'6481 11 112 10 1013 1705 3 101 37- 8168650 36ft. 113 1] 1017'8784 4 101 6s' 821 0904 1 1113 41 1022'5944 5 101 10 825'3291 2 113 7+ 1027'3240 6 102 1j 829.5787 3 113 10+ 1032'0646 7 10 2 4- 833'8368 4 114 1 I 10368134 8 102 7~ 838'1082 5 114 4+ 1041'5758 9 102 10- 842'3905 6 114 8 10463'491 10 103 1t 846'6813 7 114 11- 1051 1306 11 103 47 850'9855 8 115 2} 1055'9257 33ft. 103 8 855'3006 9 115 5: 10607317 1 103 11Il 859'6240 10 115 9+ 1065-5459 2 104 2I 863'9609 11 115 11+ 1070'3738 3 104 5+ 868-3087 37 ft.. 116 2 10i75-2126 4 104 8+ 872-6649 1. 116 6 1080'0594 5 104 1l+ 877'0346 2 116 97- 1084'9201 6 105 2 881'4!51 3. 117 0- 1089'7915.:7 105 6 885'8040 4 117 3} 1 0946711'8 105 9- 890'2064 5 117 6i 11099'5644 12 CIRCUMrFERENCES AND AREAS'. OFi CiRmOLES. Diam. CSircumferrence Diam. Circumference ft & in.. infet and in. Area in feet. ft. & in. in feet and in. Area in feet. 6 117 99 1104-4687 3 1'29 7 1336 4071 7 118 01 1109'3810 4 129 10 1341-8101 8 118 4 1114'3071 5 130 1t 1347.2271 9 118 7! 1119'2440' 6 130 41 1:352-6551 10 118 101 1124 1891 7 130 7t 1:358'0908 11 119 11 1129'1478 8' 130 101 13635406 38ft, 119 43 1134'1176 9 131 1 1369 0012 1 119 719 1139'0953 10 131 5 1374'4697 2 1,19 10:' 1144-0868 11 131 81 1379-9521 3 120 2 1149'0892 42- t 1'31 11 - 1385-4456 4 120 5 1154'0997 1 132 21 1390'2467 5 120 81 1159'-1239 2 132 51 1-396-4619 6 1,20 114 1164'1591 1 3 132 81 1401'9880 7 121 2~ 1169'2023 4 132 11- 1'4075219 8 121 51 1174'2592 5 133 3 1413'0698 9 121 8 1179-3271 6 133 61 1418-6287 10 121 11-i 11844030 7 133 91 14241952 11 122 3- 1189-4927 8 134 01 1429'7759 39.ft. 122 6} 1194-5934 1 9 134 3 4 1435:3675 1 122 91 1199'7195 10 134 61 1440'9668 2 123 01 1204 8244 11 134 91 1446-5802 3 123 3j 12099577 43 t. 135 1 145'22046 4 123 6: 1215-0990 I'1 135 4j 1'457'8365 5 1 23 91 12202542 2 135 71 1463.4827.6 1'24 11 u 1225-4203 3 1 35 101 1469'1-39.7 7 1'24 4} 1230'5943 4 136 1- 1474'8044 8 124'71 1235-7822 5 136 4 1480-4833 9 124 101 1240'9810 6 1:36 7 1- 1486. 1731 10 125 1t 1246-1878 7 136 11 1491'8705 11 125 4} 1251-4084 8 137 2~ 1497T'5821 40 ft. 125 74 1256'6400 9 1'37 51: 1503:-3'046 1' 125 11 1261'8794 10 137 81 1'509'0348 2' 126 2+ 1267-1327 11 137 111 1:514-7791 3 1'26 51 1272-3970 44 ft. 138 21 1520:5344 4 126 81 1277-6692 1 138.5 1526-2971 5 126 111 1282'9,553 2 138 9 1532'0742 6 127 2t 1288'2523.3 139 O01 1537-86'22 7 127 51 1293-5572 4 139 31' 1543'6578 8 127 9 1298 87'60 5 139 61 1549'4776 9 128 0+ 1304.20,A7 6'139 91 11555,2883 10, 1,28 31 13.09-5433 I 7 140 0' 1561:1165 11 128 61 1-314'8949 8 140 31 1:566'9591 41 ft. 128 94 1320'2574 9 140 7, 1'5728'125 129 0o 1.3.25'6276 0.:140 101- 15786735 2129 31:1331'0119 1 141 1} 1584' 5488.: CIRCUMFERENCES ANDt ARnEas o' CIRCLES. 173 Diam. Circumference I Diam, Circumference ft. & in. ill feet and in. Area in feet. ft. & in. in feet and inl Area in feet. 45ft. 141 4- 1590'4350 7- 149 5-, 1778'2795 1 141 7+ 1596'3286 8. 149 8+ 1 784'5148 2 141 10+ 1I602 2366 9 15O O0 1790'7610 3 142 1- 1608 155 10 150 3j 1 7 97 01:45 4 142 5 1614'0819 11 150 6- 1803'2826 5 142 8t 1620(0226 48ft. 150 9+ 1809:5616 6 142 11i 1625 9743 1 151 0- 1815'8477 7 143 2+ 1 631 9334 2 151 3S 1822 1485 8 143 5 1637'9068 3. 151 664 1828'4602 9- 143 8 lt 438912 4 151 10-( 1834 7791 10 143 11s 1649 8831 5 152 1 1841'1127 II 144 3 1655'8892 6 152 4 1847 4571 46ft. 144 6 -1661906 1 7 1152 7+ 1853'8087 1 1.44 t9} 1667'9308 8 152 10 1860'1750 2 145 0 1673'9698 9 153 1I 1866'5521 3 14s 3+ 1680'0196 10 153 4j 1872'9365 4 145 6j 1686-0769 11 153 8+ 1879-3355 5 145 9+ 1692-'1485 49 ft. 153 11i 1885'7454 6 146 1+ 1698 2311 1 154 21 1892'1724 7 146 4} 1704'3210 2 154 5+ 1898'5041 8 146 74 1710'4254 3 154 8+ 1905 0367 9, 146 101 1716'5407 4 154 11- 191,1'4965 10 147 1. 1722'6634 5 155 26 1917 9609 11 147 41 1728-8005 6 155 6 1924'4263 47ft. 147 _7 173849486 7 155 9+ 1930:9188 1 147 11 1741-1039 8 156 0+ 1937'3159.2 148 2+ 1747-2738 9 156 3+ 1943'9140 3. 148 5~ 1753'4545- 10 156 6+ 19504392 4 148 8+- 1759'6426 11 156 9+ 1956'9691 5 148 11~ 1765'8452 50ft. 157 0+ 1963'5000:6 149-.2 1772'0587 To PRESERVE STEEL GooDS. -Caoutehoue 1 part; turpentine 16 parts. Dissolve with'a gentle heat, then add boiled oil 8 parts. Mix by bringing'them to the: heat of boiling water; apply it to the steel with a brush, in the way of varnish. It mavybe removed with turpentine. The oil may be'whiolly-omitted. SIZE. —-Oil size is made by,grinding yellow ochre or''burnt red ochre with boiled linseed oil, and thinninlg it with oil of turpentine. Water. size (for burnishledgilding) is parchment size ground with yellow ochre..SILICA AND CAIRBON.-Silica is- the base of the mineral world. Carbon the base of the organized. i~~~n~~u~u~~l15 174s IvoavIVORY. HTow to Soften it.-Take 3 oz. spirits of nitre, and 15 of spring wateir; mix together; drop'in the Ivory, and let it soak. In'thriee or four days it will be so soft as to obey youi fingers. How to Dye Ivory whern Sof'tened-If you desire to dye Ivory when thus softened, dissolve, in spirits of wine, such colors as you wish to use. When the spirit of wine is sufficiently tinged with the color you have put nge in e ill our Ivory, and leave it there till it is dyed to suit you. Then take out the Ivory- and give it what form you please. How to Hardet Iuvory.-To harden the Ivory afterwards, wrap it up in a'sheet of white paper, cover it with dry, decrepitated salt, and lay it by for twenty-four hours, when it will be restored to its original hardness. To're- Whiten Ivory which has TaLn'ed a Brown Yellow. —There are two ways of doing this, namely: 1. Slack some lime in water, into whichl drop the ivory; decant it gently, and boil till it looks quite white. 2. To polish it afterwards, set it in th.e turner's wheel, and after having worked it, take some rushes and pumice stone, mix a subtile powder with water, and rub till it becomes perfectly sliooth: then heat it by turning it, over a piece' of linen or sheepskin, and when hot rub it with a little whitening diluted with olive oil; then rub it with a little dry whitening alone, and finally with a piece of soft white rag, and the Ivory will look remarkably white. How to Dye Ivory Black. Immerse the Ivory in a boiling solution of logwood, then take it out, and wash it in a solution of'copperas. B/ute.-There are two ways of reaching this color. The first is to soak the Ivory in a solution of verdigris in nitric acid, which will make it green; then dip it into a solution of boiling hot pearlash, and it will turn blue. The second way is as follows: Immerse'the IvOry in' a solution of sulphate of indigo and water, partly neutralized with'potash. Green.-Stecp blued Ivory in a solution of nitro-muriate of tin, and then in a decoction of fustic. Another and a more instantaneous plan is to immerse it in a solution of acetate of copper. Yellow. —Steep the Ivory in, a bath of neutral chromate of potash, and afterwards in a boiling solution of acetate of lead. Red.-Soak the Ivory for. a short time in a solution of tin, and then in a decoction of cochineal. Violet.-Moisten the Ivory with a solution of tin, as before; then immerse it in a decoction of logwood. Purple.-Soak the Ivory in basolution of sal ammoniac into four times.lts weight of. nitroius acid. Fluid for lVarking Ivory.-Take nitrate of silver, 2 parts; nitric acid, 1 part; water, 7 parts. Mix. Etching:Fluid for Ivory. —Take' of diluted sulphuric acid, and diluted muriatic acid, equal parts. Mix. CENTRRE. 175; Etching Varnish for Ivory.-White wax, 2 parts; tears of mastic, 2 parts. Mix. To Gild Ivory. —Immerse it in a solution of nitro-muriate of gold, and then, while yet damp, expose it to hydrogen gas. Wash itl afterwards in clean water. Another plan of gilding Ivory is by! immersing. it in a fresh solution of proto-sulphate of iron, and( afterwards in a solution of chloride of gold. To Polish Ivory.-Use a rubber and putty and water. The hardest, toughest, whitest, and most translucent ivory has the preference in the market; and the tusks of the sea horse are considered to afford the best. Ivory has the same constituents as the teeth of animals: three-fourths being phosphate, with a little carbonate of lime; one-fourth cartilage. With regard to dyeing Ivory, it may in general be observed, that the colors penetrate better before the surface is polished than afterwards. Should any dark spots appear, they may be ~leared up by rubbing theni with chalk; after which the Ivory should be dyed once nmore to produce a. perfect uniformity. of shade. On taking it out of the boiling hot. dye bath, it.should be plunged immediately into cold water, to prevent the chance of fissures being caused by the heat. CENTRE, In a general sense, denotes a point equally remote from the extremes of a line, surface, or solid. Centre of Attraction Of a body, is that point into which if all its matter is collected, its action upon any remote particle would still be the same. Centre of Equilibrium. Is the same, in respect to bodies immersed in a fluid, as the centre of gravity is to bodies in free space. C]entre of Friction Is that point in the base of a body on which it revolves, into which if the whole surface of the base and the mass of the body were collected, and made to revolve about the centre of the base of the given body, the angular velocity destroyed. by its friction would be equal to the angular velocity destroyed in the given body by its friction in the same time. Centre of Gravity Of any body, or system of bodies, is that point upon which the body, or system of bodies, acted upon only by the force of gravity, will balance itself in all positions; hence it follows, that, if a line or plane, passing through the centre of gravity, be supported, the body or system will be also supported. 176 HESION. Cettre: of GOyration Is that point into which, if the whole mass were collected, a given force, applied at a given distance,:would produce the same angular' I elocity in the same time as if tile bodies were disposed at their Irespective distances. This point differs from the Centre of Oscillation only in this, that, in the latter case,-the motion is produced by the gravity of the body; but, in the' former, the body is put in motion by some other force, acting at one place only. COHESION Is that species of nttraction Nwhich, uniting.particle to particle,, retains together the comp(nenlt parts of the same mass; being thus distinguished from Adhesion. or that species of attraction -which takes place between the surfaces of sinmilar or dissimilar bodies. The absolute cohesion of solids is measured by the force necessary to pull them asunder. Thus, if a rod of iron be suspended in a vertical position, having weight attached to its lower extremityi till the rod breaks, the whole weight attached to the rod, at the time of fracture, will be the measure of its cohesive force, or absolute cohesion. The particles of solid bodies, in- their: natural state, are arranged in such a manner, that they are in equilibrium in respect to tlhe' forces whicli operate on thei; therefore, when aily new force is;: applied, it is evident thalt the equilibrium will be destroyed, and that the particles will Iove aamong themselves till it be restored.i When the new force.is Ippli d to pull the body. asunder, the body. becomes longer itl the directioni of the foice, which is called the extension; and its area, at right angles to tile direction of the force, contracts. When the fo rce is applied to compress the body, it becomes shorter in the direction of the force, which is called the compressionw; and the area of its section, at right angles to the. force, expands. In eitheri case, a part of the heat, or any fluid that; occupies the pores or interstices of the body,' before tle new force was made to act upon it, will be expelled'. PLATIN-A-MooI:. -Zinc two parts: platinum onre part. -Melt. nd: reduce tile alloy to powder, whlich' mlast b-e treated with diilute sulphuric acid until all tile zinc is vwashed out; then wash it with water, digest it in a lev of'po;tash, anildagain wash it with water. Thiis powderl possesses fthle property of converting alneollol into i inegar. THE VELOCIToY OF SoUND. —It has been ascertainl(d, by careful investigation, that sound passes in water at-a speed-of 4,708 feet per seeonid. LAws OF ELASTIC FLUIDS. 17 Mf:ECHANITCAL LAWS OF ELASTIC FLUIDS. Boyle's or aariotte's Law. The elastic force of a gas or air at a given temperature is inversely proportional to the space which it occupies. Let p - elastic force of a- gas when it occupies the space s. 1-)= do. do. S..' p:==_ps P.S The elastic force of any gas at a given temperature is proportional to its density. The density of any body is the weight of a cubic unit of it, usually one cubic foot. Let p = the elastic force when the density is d. Andk = do. do. unit...p = k& Dalton'.s and Gay-Lussac's Law. All gases, under the same pressure, undergo equal expansions *oi equal increments of temperature. It was ascertained by these eminent philosophers, that 1.00 measures of air expand to 137-5 measures on being heated from 320 to 212~ of Fahrenheit's thermometer, hence 37=5 - increments of 100 measures for 180 degrees of heat. 37-5 do. 1 180 do.!00.. 375 do. 1 1 do. 1_1 - 480 a Let V = volume of any gas at the temperature t. It cldo. do. t' Then, (-t'- 32) ccurately. 1 + a (t - 32)' a y. 1 + a (t'- t) I very neatly..Amonton's Law. This law is the relation between the elastic force, the density, and the temperature, of any gas. If, then, the volume of a gas be constant,t its elastie force will increase; and, if the elastic force is aonstant, its volume, will increase for every increase of temperature. It is important to connect these quantities by an equation. ~c178 ILAws OF ELASTIC FiLUIDS. Put p = elastic force of a gas at the temperature 0~ and density d. Then, p kd (I - aH) where k is a constant quantity depending on the nature of the gaws, and a= - When a light and heavy gas are once mixed, they do not exhibit any tendency to separate; in this respect they differ from mixed liquids. I)alton's.Ex'cperiment. The vessel a conttains a light gas, as hydrogen, and. - the vessel b contains a heavy gas, as carbonic acid; the; - two gases are. allowed to communicate by a narrow tube c, an interchange speedily takes place of a part; c of their contents, vwhlich their relative position might' be supposed would prevent. Contrary to gravity, thie heavy gas ascends, and the light one descends, till, in a few hlours, tile gases become perfectly mixed, and the proportion of the two gases is thie same in both vessels. Gases diffuse into the atmospliere -and into each other with different degrees of rapidity. The velocity with which air will rush-into a vacuum is 1348 feet per second. To determine the velocity with which the air of the atmosphere'will rush into a space containing rarer air Let v - velocity of air, of density (d), rushing into a void. V = velocity of air rushing into air of density D.... v=. iel _DThere will always be a current so Iong as (D) and (d) are unequal. Iloniunating Gases. Pure hydrogen burns with too feeble a flame to be employed for the purpose of illumination. Carburetted hydrogen has the property of precipitating its carbon; in the act of burning, its solid particles become incandescent, and diffuse a strong light. The more carbon the gas contains the more brightly does it burn. Two measures of hyvdrogren gas, with one measure of the vapor of carbon, form the carburetted hydrogen found in coal mines, and is also evolved in ditches, from decomposing vegetable matter. Another kind of carburetted bhydroen, called olefiant gas. is formed by two measurIes of hydroren and two rneasures of gaseous carbon. This gas burns with a brighter flame than the common carburetted hydrorgen. The best substances for flurnishing a gas rich' in luminiferous materials are pit coal, resin, oil, fats of all kinds, tar, wax, &c. The volume of gas discharged from the end of a pipe is directly -proportional to thle square of its diameter, and inversely as the square root of its length. LAWS OF ELASTIC FLUIDS. 1 9 Let n = number of cubic feet of gas discharged per hour through a length of pipes I feet and diameter D. 3162 Da This formula is applicable only when tho gas is transmitted through the pipes, without being let off in its wav by burners. If the main send off branches for burners, then, for the same length, the diameter may be reduced; or, for a like diameter, the length may be increased. STAINS, TO RxEmovE.-Stains of iodine are removed by rectified spirit. Ink stains by oxalic acid or superoxalate of potash. Iron moulds lby the same; but if obstinate, it has been recommended to moisten them with ivk, then remove them in the usual way. Red spots on black clotlh, from acids, are removed by spirits of hartshorn, or other solutions of ammonia. Stains of Markinlg ink, 0or Nitrate of Silver, to remove. L WVet tile stain with fresh solution of chloride of lime, and after tenl or fifteen minutes, if the marks hlave become white, dip tIle part in solution of ammonia or of hyposulphite of soda. In a few minutes wasvh with clean water. 2. Stretch the stained linen over a basin of hot water, and, wet the mark with tincture of iodine. BROWNING, OR BRoNZING LTQUIDS, FOR GUN BARnnLS.-r I, Aqua.fortis 4 oz., sweet spirit of nitre ~ oz., spirit of wine, 1 oz., sullphate of copper 2 oz., water 30 oz, tincture of muriate of iron, oz.. xis. 2. Sulphate.of copper 1 oz., sweet apirit of nitre 1 ox., water 1 pint Mix. In a few days it will be, fit for use. 3. Sweet spirit of nlitre 3 oz., gmln benzoi I,~. oz., tincture of muriate of iron' oz, sulphate of copper 2 dr;' spilit of wine -} oz. Mix, and add 2 1bs. of soft watee. 4. Tincture of mturiate of i on.. oz., spirit of ni, tric etherl,- oz., sulphate of copper 2 scruples, rain water, pint. The above are applied with a sponge, fter. cleaning the barre-l withl lime alnd water. When dry,. they -are polished with. a stiff brush, or ironl scratch brush. BRONZING LIQu1S) For TIN CASTINGS..-W'ash them ovel,, after being well cleaned- and wiped, with solution of 1 part sulphate of iron, and 1 of sulplhate of copper. in 20 parts of water; afterwards with a solution of 4 parts verdigris. in 11 of distilled -vinegar: leave for an, hour to dry, and then polisli with a soft brush. and colcothar. SoTLViNTS FroR GUTTA PERCH. -Benzolbe readily dissolves. it:: so da chloroform and bisulphure$ t of'catbon. 180 SQUAItES, Cvus, SCQVARI RooTs, ANsPe CUBE RooTS, T A13L E Of Squares, Cubes, Square and Cube Roots of Nrumbergs. Number. Square Cue,. Square Rot. tubee fRoot. Number, 1 1 1 1 1'0 1 2 4: 1414213 1'25:992 2 3 9 27 1'732050 1.4422'5 3 4 16 64. 2'0 1-58740 4 5 25 125 2'236068 170997 5 6 6 36 216 2'449489 1-81712 6 I - 49 / 343 2645751 1'91293 7 8 64 512 2 828427 2'0. 8 9 81 729 I 3'0 2'08008& [ 9 10 100 1000 3'16227.7 2]15443 10 11 121 1331 3-316624 3'22398 11 12 144 1728 3 464101 2'28942 12 13 169 2197 3.'605551 2 35133 1S 14 196 2744 3'741657 2'41014 14 15 225 3375. 3'872983 2'46621 15 16 256 4096 4'0 251984 16 17 289 4913 4-123105- 2'57128 17 18 324 5832 4'242640 2.62074 18 19 361 6859 4'358898 2'66840 19 20' 400 8000 4 472136 2',71441 20 21 441 9261 4'582575 2'75892 2.1 22 484 10648 4'690415 2'8020,3 22'23 529 12167{ 4 795831 2,84386 i3n 24 576 - 138.24] 4.898979 2'88449 2'4 25 625 15695 5.,0.2'92401 25 26 676. 17576 5099019 2'96249 -26 27 729 19683 5'196152 0 27 28 784 21952 5'291502 303658 28 29- 841 24389 5 385164 3'07231 29 30 900 27000 54 77225 310723 30 31 961 29791 5567764 3'14138 31 82 1024 32768 5'6568054 3'17480 32 33 1089 35.9;37 5'744562 3'20753 33 34 1156 39304 [,5830951 3'23961 34 3: 1225s 5 42875. 916079 3 27106 35 36 1296 I 46656 6-0 3'30192 36 397 1369 50653 6'08276:2 3'33222 37 38 1444 54872 6l,64414 3'36197 38 3.9 1 521 59319 6'244998 3'39121 39 40 1600 64000.6'324555 3 419.95 40 41 1681 68921 6'4:0.3124 3'448'21 41 SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. 181 Number. Square. Cube. Square Root. Cube Root. Number. 42 1764 74988 6'480740 3'47602 42 43 1849 79507 6'557438 3'50339 43 44 1936 85184 6'633249 3'53034 44 45 2025 91125 6'708203 3'55689 45 46 2116 97336 6'782330 3 58304 46 47 2209 103823 6'855654 3 60882 47 48 2304 110592 1 6'928203 3 63424 48 49 2401 117649 7 0 3'65930 49 50 2500 125000 7'071067 3'68403 50 51 2601 132651 7'141428 37'0842 51 52 2704 140!;08 7'211102 3'73251 52 53 2809 148877 7'280109 3-75628 53 54 2916 157464 7 348469 3'77976 54 55 3025 166375 7'416198 3 80295 55 56 3136 175616 7'483314 3'82586 56 57 3249 185193 7.549834 3-84850 57 58 3364 195112 7'615773 3'87087 58 59 3481 205379 7'681145 3'89299 59 60 3600 216000 7 745966 3'91486 60 61 3721 226981 7'810249 3'93649 61 62 3844 238328 7 874007 3'95789 62 63 3969 250047 7'937253 3'97905 63 64 4096 262144 8 0 4'0 64 65 4225 274625 1 8062257 4-02072 65 66 4356 287496 1 8124038 4-04124 66 67 4489 300763 8-1853.52 4 06154 67 68 4624 314432 8'246211 4'08165 68 69 4761 328509 8'306623 4'10156 69 7o 4900 343000 8'366600 4'12128 70 71 5041 357911 8'426149 4-14081 71 172 5184 373248 8'48528'1 4'16016 72 73 5329 889017 8'544003 4-17933 73 74 5476 405224 3'602325 4'19833 74 75 5625 421875 8'660254 4'21716 75 76 5776 438976 8 717797 i 423582 76 77 5929 456533 8'-74964 4'25432 77 78 6084 474552 8-831760 4'27265 1 78 79 6241 493039 8 888194 4'29084 79 80 6400 512000 8'944271 4'30886 80 81 6561 531441 90 4'32674 81 82 6724 551368 9'055385 4'34448 82 83 6889 571787 9'110433 4'36207 83 84 7056 592704 9-165151 4'37951 84 86 7396 636056 9-273618 4'41400 86 871 1569 658503 9'327 37 9 4'43104 87 16 182 SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. Number. Square. Cube. Square Root. Cube Root. Number. 88 7744 681472 9'380831 4 44796 88 89 7921 704969 9'483981 4'4.6474 89 90 8100 729000 9'486833 4,48140 90 91 8281 7 53571 9'539392 4'49794 91 92 8464 778688 9'591663 4'51435 92,93 8649 804357 9'6443650 4;53065 93 94 8836 830584 9'695359 4'54683 94 95 9025 857375 9'746794 4'56290 95 96 9216 884736 9 797959 4'57885 96 97 9409 912673 9'848857 4'59470 97 98 9604 941192 9'899494. 4'61043 98 99 9801 970299 9'949874 4'62606 99 100 10000:1000000 10O 4'64158 100 101 10201 1030301 10'049875 4'65700 101 102 10404:1061208 10 099504 4'67232 102 103 10609 1092727 10'148891 4'68754 103 104 10816 1124864 10 198039 4,70266 104 105 11025 1157625 10'246950 4 71769 105 106 11236 1 191016 10,295630 4'73262 106 107 11449 11225043 10'344080 4'74745 107 108 11664 1259712 10,392304 4 76(220 108 109 11881 1295029 10 440306 4-77685 109 110 12100 1331000 10.488088 4:79141 110 111 12321 1367631 10'535653 4'80589 111 112 12544 1404928 10'.583005 4;82028 112 113 1.2769 1442897 10'630145 4.83458 113 114 12996 1481544 10 677078 4'84880 114 1i5 13225 1520875 10'723805 4'86294 115 i116 13456 1560896 10'770329 4.:87699 116 117 13689 1601613 10'8166,53 4 89097 11.7 118 13924 1643032 10'862780 4'90486 118 4119 14161 1685159 10'908712 4:91868 119 120 14400 1728000 10'954451 4'93242 120 121 14641 1771561 11'0 4 94608 121 122 14884 1815848 11'045361 4,95967 122 123 15129 1860867 11,090536 4'97318 123 124 15376 1.906624 11'135528 4-98663 124 125 15625 1953125 11180339 5 0 125:126 15876 2000376 11'224972 5'01329 126 127 16129 2048383 11,269427 5 02652 127 128 16384 2097152 11'313708 5'03968 128 129 16641 2146689 11 357816 5'05277 129: 130 [ 16900 2197000 11.401754 5'06579 130 131 17161 2248091 11'445523 5 07875 131 132 17424 2299968 11'489125 5 09164, 132 i133 17i6809 2352637 1 1 532562 5'10446 133 134 17.96- 2406104 11'575836 5'11722 134 SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. 183 Number. Square. Cube. Square Root. Cube Root. Number. 1.'35 18225 2460375 11'618950 5'12992 135 136 1:8496 25154.56 11 661903 5 14256 136 137 18769 2571353 511704699 5 15513 137 138 19044 2628072. 11 747340 5'16764 138 139 19321 2685619: 11'789826' 5'18010 139 140 1'9600'2744000 1 1'83215'9 5:19249 140 141 19881 2803221 1' 1 8S4342 5'20482 141 142 20164 2863288 11'9:16375 5 21710 142 143 20449 2924207 11 9'58260 5'22932 143' 144 207:36 2985984 120' 5'24148 144 145 21025 3048625 12'041594 5'25358 145' 146 21316 3112136 12'083046f 5'26563 14c6 147 21'609 3176523 12'124355 527763 147: 148 21901 3241792 12-165525 5-28957 148 149 22201 3307949 12'206555 5 30145 149 150 22500 3375000 12'247448 5'31329 150 151 22804 314295 12-288205' 5'32507 151 152 23104 3511808 12-S3288 28 5-33680 152 153: 23409 3581577 12 369'3:16 5 -34848 153 154 23716 3652264 12'409673 5'36010 154 155 24025 3723875 12'449899 5 37168 155 156 24336 3796416 12 4899'96 5'35 8321 J 156 157 24649 3869893 12'529964 5'39469 157 158 24964 3944312 12'569805 5'40'612 158 159 25281 401.96793 126 09520 5'41750 159 160 25600 4096000 12'649110 5 42883 160 161 25921 4173281 12 6'88577 5 440'12 161 1B62 2 6244 4251528 1272 79229 5'45136 162 163 26569 4330747 1'2'767145: 5 46255 163 164 26896 4410944 12 806248 5'47370 164 165 27225 4492125 12'845232' 5 48480 165 166 27 556 4574296 128'84098 5'49586 166 167 27889 4657463 12'922848 5'50687 167 168 28224 4'41632 1;2'96148"1 5'51784 168 169 28561 4826809' 13 0 552877 169 170 28900 4913000 13 038404 55'3965 170 171 29241 5000211 1;3 076696 5'55049 171 172 29581 5088448 13'1'1:487,7 556129 172 173' 29929 5177717 1: 1:52946 5 57205 173 174 30276 5268024 13'1900906 55827'7 174 175 30625 5359375 13 228756 5'59344 175 176 30976 5451776 13-2664'99 6560407 176 177 31329 5545233 13 304134 5'61467 177 178 31684 5639752 13'341664 5-62522 178 179 32041 5735339' 13 379088 5'63574 179 180 32400 6832000 13.416407 5.64621 180 1.81 32761 5929741 13 453624 5`65665 181 184 SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. Number. Square. Cube. Square Root. Cube Root. Number. 182 33124 6028568 13490737 5'66705 182 183 33489 6128487 13'527749 5'67741 183 184 33856 62.29504 13'564660 5'68773'184 185 34225 6331625 13'601470 5'69801 185 186 34596 6434856 13'638181 5'70826 186 187 34969 6539203 13 674794 5'71847 187 188 35344 6644672 13'711309 5'72865 188 189 35721. 6751269 13'747727 5'73879 189 190 36100 6859000 13'784048 5-74889 190 191 36481 6967871 13 820275 5'75896 191 192. 36864 7077888 13'856406 5'76899 192 193 37249 7189057 13'892444 5'77899 193 194 37636 7301384 13'928388 5'78896 194 195 38025 7414875 13'964240 5.79889 195 196 38416 7529536 14'0 5'80878 196 197 38809 7645373 14'035668 5'81864 197 198 39204 7762392 14'071247 5 82847 198 199 39601 7880599 14'106736 5'83827 199 200 40000 8000000. 14'142135 5'84803 200 201 40401 8120601 14'177446 5'85776 201 202 40804 8242408 14-212670 5 86746 202. 203 41209 8365427 14'247806 5'87713 203 204 41616 8489664 14 282856 5'88676 204 205 42025 8615125 14'317821 5'89636 205 206 42436 8741816 14'352700 5'90594 2.06 207 42849 8869743 14'387494 5'91548 207 208 43261 8998912 14'422205 5'92499 208 209 43681 9129329 14'456832 5'93447 209 210 44100 9261000 14'491376 5'94392 210 211 44521 9393931 14'525839 5'95334 211 212 44944 9528128 14'560219 5'96273 212 213 45369 9663597 14'594519 5'97209 213 214 45796 9800344 14'628738 5'98142 214 215 46225 9938375 14 662878 5'990172 215 216 46656 10077696 14'696938 6-0 216 217 47089 10218313 14'730919 6'0()0924 217 218 47524 10360232 14'764823 6'01846 218 219 47961 1()503459 14'798648 6 02765 219 220 48400 10648000 14'832397 6'03681 220 221 48841 10793861 14'866068 6'04594 221 222 49284 10941048 14'899664 6'05504 222 223 49729 11089567 14-933184 6'06412 223 224 50176 11239424 14-966629 6'07317 224 253 50625 11390625 15'0 6-08220 225 226 51076 11543176 15'033296 6 09119 226 127.51529 11697083 15'066519 6'10017 227 228 51984 11852352 15'099668 6-10911 228 SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. 185 Square. Cube. Square Root. Cube Root. Number. Number. /. 229 52441 12008989 15-132746 6'11803 229 230 52900 121'67000' 15'165750 6'12692 230 231 53361 123263'91 15 198684 6'13579 231 232 53824 12487168 15'231546 6'14463 232 233 54289 12649337 15-264337 6'1 5344 233 234 54756 12812904 1i5297058 6'16224 234 235 55225 12977875 15-329709 6'17100 235 236 5'5696 13144256 15'36229-1 6'17974 236 237 56169 13312053 15'394804 6'18846 237 238 56644 13481272 15 427248 6'19715 2'38 239 57121 13651-9-19 15.459624 6-20582 239 240 5760'0 13824000 15'491933 6-21446 240 241 58081 13997521 15 524174 6'22308 241 242 58564 14172488 15'55-6349 6 23167 242 243 59049 14348907 15'588457 6'24025 243 244 5953'6 14526784 15'620499 6-24879 244 245 60025 1]4706125 15 65247:5 6 25732 245 246 60516 14886936 15'684387 6'26582 246 247 61009 1506922:3 15'716233 6'27430 247 248 6150'4 15252992 15'7480175 6'28276 248 249 62001 15438249 15 77973-3 6'29119 249 250 62500 15625000 15'811388 6 29960 250 251 63001 15813251 15-842979 6'30799 251 252 63504 16003008 15-874507 6'31635 252 253 64009 16194277 15-905973 6 32470 253 254 64516 16387'064 15'937377 6'33302 254 255 65025 16581375 15 968719 6'34132 255 256' 65536 16777216 16' 6'34960 256. 257 66049 16974593 16e-031219 6-35786 257 258 66564 17173512 16'062378 6 36609 258 259 67081 17373979 16'09347'6 637431 259 260 67600 17576000 6 -1 24515 6'38250 260 261 68121 17779581 16'155494 6 39067 261 262' 68644 1798472:8 16.186414 6-39882 262 263 69169 18191447 16'217274 6.40695 263 264 69696 1839:9744 16'24807'6 6.41506 264 265 70225 18fiO9625 16'278820 6 42315 265 266 70756 18821096 16'309'506 6 43122 266 267 71289 190341'63 1 6 340134 6-43927 267 268 71824 19248832 16-370705 6 44730 268 269 72361 19465109 16401219 645531 269 2'70 7290'0 19683000 16 4'31676 6 46330 270 271 73441 19902511 16'462077 6-47127 271 2712 739'84 20123648 16-492422 6-47922 272 273 74529 20346417 16'52:271'1 6 48715 273 274 750'6 20570824 16'5562945 6 49'506 274 275 75s625 20796875 16-583124 6 50295 275 16N* 186 SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. Number. Square. Cube. Square Root. Cube Root. Number. 276 76176 21024576 16'613247 6'51083 276 277 76729 21253933 16'643317 6-51868 277 278 77284 21484952 16'673332 6-52651 278 279 77841 21717639 16'703293 6-53433 279 280 78400 21952000 16'733200 6'54213 280 281 78961 22188041 16'763054 6'54991 281 282' 79524 22425768 16 792855 655767 282 283 80089 22665187 16'822603 6'56541 283 284 80656 22906304 16'852299 6 57313 284 285' 81225 23149125 16-881943 6'58084 285 286 81796 23393656 16'911534 6'58853 286, 287 82369 23639903 16'941074 6'59620 287' 288 82944 23887872 16'970562 6'60385 288 289' 8'3521 24137569 17'0 6'61148 289 290 84100 24389000 17'029386 6 61.910 290 291 84681 24642171 17-058722 6'62670 291 292 85264 24897088 17'088007 6'63428 292 293[ 85849 25153757 17'117242 6'64'185 293 294 86436 25412184 17'146428 6'64939 294 295 87025 25672375 17-175564 6 65693 295 296 87616 25934336 17 204650 6'66444 296 297 88209 26198073 17 233687 66(7194 297 298 88804- 26463592 17'262676 6 67942 298 299 8940.1 26730899 17'291616 6'68688 299 300 90000' 27000000' 17320508 6'69432 300 301' 90601 27270901 17 349351 6'70175 301 302 91204 27543608 17'378147 6'70917 302 303 91809 27818127 17 406895 6'71657 303 304 92416 28094464 17'435595 6-72395 304 305 93025 28372625 17'464249 6'73131 305 306 93636 28652616 17'492855 6-73866 306 307 94249 28934443 17'521415 6'74599 307 308 94864 29218112 17'549928 6'75331 308 309 95481 i 29503629 17'5783'95 6'76061 309 31.0 96100 29791000 17 606816. 6'76789 310 311 96721 30080231 17'635192 6'77516 311 312 97344 30371328 17 663521 6'78242 312 313 97969 30664297 17'691806 6'78966 313 314 98596 30959144 17 720045 667968$ 814 315 99225 31255875 17-748239 6'80409 315 316 99856 31554496 17 776388 681128 316 317 100489 31855013 17 804493 6'81846 317 318 101124 32157432 17 832554 6'82562 318 319 101761 32461759 17'860571 6'83277 319 320 102400 32768000. 17 888543 6-83990 320 321 103041 33076161 17-916472 684702 321 322 103684 33386248 17'944358 6'85412 322 SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. 187 Number. Square. Cube. Square Root. Cube Root Number. 323 104329 33698267 17'972200 6'86121 323 324' 104976 34012224 18'0 6'86828 324 325 105625 34328125 18'027756 6:87534 325 326 106276 34645976 18.055470 6.88238 326 3.27 106929 34965783 18'083141. 6'88941 327 328 107584 35287552 18'110770 6'89643 328 329 108241 35611289 18'138357 6'90343 329 330 108900 35937000 18-165902 6-91042 330 331 109561 36264691 18'193405 6'91739 331 332 110224 36594368 18 220867 6'92435 332 333 110889' 36926037 18'248287 6'93130 333 334 111556 37259704 18'275666 6'93823 334 335 112225 37595375 18'303005 6'94514 335 336 112896 37933056 18S330302 6 95205 336 337 113569 38272753 18 357559 6'95894 337 338 114244: 38614472 18'3847476 6'96581 338 339 114921' 38958219 18'411952 6 97268 339 340 115600 39304000 18 439088 6 97953 340 341 116281 39651821 18'466185 6.98636 341 342 116964 40001688 18'493242 6'99319 342 343 117649 40353607 18-520259 7'0 343 344 118336 40707584 18'547237 7 00679 344,345 119025 41063625 18'574175 7'01357 345 346 119716 41421736 18 601075 7 02034 346 347 120409 41781923 18'627936 7 02710 347 348 121104 42144192 18'e54758 7 03384 348:349 121801 42508549 18'681541 7 04058 349,350 122500 42875000 18'708286 7 04729 350 351 123201 43243551 18'734994 7 05400 351 352 12'3904 43614208 18'761663 7-06069 352 353 124609 43986977 18'788294 7 06737 353 354 125316 44361864 18'814887 7 07404 354 355 126025 44738875 18 841443 7 08069 355 856 126736 45118016 18'867962 7 08734 356 357 127449 45499293 18'894443 7 09397 357 358 128164 45882712 18'920887 7'10058 358 359 128881 46268279 18-947295 7-10719 359 360 129600 46656000 18-973666 7-11378 360 361 130321 47045881 19'0 7 12036 361 362 131044 47437928 19'026297 q'12693 362 363 131769 47832147 19'052558 7 13349 363 364 132496 48228544 19'078784 7'14003 364 365 133225 48627125 19'104973 7 14656 365 366 133956 49027896 19'131126 7'15309 366 367 134689 49430863 19 19157244 7'15959 867 368 135424 49836032 19'183326 7 16609 368 369 136161 50243409 19'209372 - 7 17258 369 188 SQUARES, CUBES, SQUARE RooTs, AND CUBE ROOTS. Number. Square. (Cube. Square Root. Cube Root. Number. 370 136900 50653000 19'235384 7'17905 370 371 137641 51064811 19'261360 7'18551 371 3'72 138384 51478848 19-287301 719196 372 373 139129 51895117 19-313207 7'19'840 373 374 130876 52313624 19'339079 7'20483 374 375 140625 52734375 19-364916 7 21'124 375 376 141376 53157'376 19 390719 7'21765 376 377 142129 53582633 19'416487 7-22404 377 37'8 142884 54010152 19'442222 7 23042 378 379 143641 54439939 19-467922 7 *23679 379 380 144400 54872000 19-493588 7324315 380 381 145161 55306341 19519221 724950 381 382 145924 55742968 19'544820 7'25584 382 383 146689 56181887 19-57038.5 7 26216 383 $84 147456 56623104 19 595917 7-26848 384 385 148225 57066625 19'621416 7 27478 385 386 148996 57512456 19'646882 7'28107 386 387 149769 57960603 196723 1 5 i'28736 387 388 150544 5'-'411072 19'697715 7-29363 388 389 151321 58863859 19'723082 7'29989 389 390 152100 59319000 19'748417 7'30614 390 391 152881 59776471 19'773719 7-21238 391 392 153664 60236288.19'798989 7'31861 392 393 154449 60698457 19'824227 7'.32482;393 394 155236 61162984 19:849433 7' 33103 394 395 1i56025 61629875 19-8.74606' 7 33723 395 396 156816 62099136 19-899748 7 34342 396 397 157609 62570773 19:924858 7' 4959 397 398 158404 63044792 19 949937 7 35576 398'399 159201 63521199 19w974984 7'36191 399 400 160000 64000000 20'0 7'36806 400 401 160801 64481201 20;024984 7:37419 401 402, 161604 64964808 20'049937 7;38032 402 403 162409 65450827' 20-074859 7-38643 403 404 163216 65939264 20'09975,1 7'39254 404 405 164025 66430125 20'124611 7'39863 405 406 164836 66923416 20-149441'I 40472 406 407 165649 67419143 20-174241 7'41 07 408 166464 67917312 29'199009 7-41685 408 409 16728i 68417929 20'223748! 742291 409 410 168100 68921000 20'248456 7%42895 410 411 16892.-1 69426531 20'273134 7'43499 411 412 169744 69934528 20'297783 7'44101 412 413 170569 70444997 20322401 7'44703 413 414 171396 70957944 20'346989 1 45303 414,415 1722'25 71473375 20'371548 71 45903'415:416 173056 71991296 20;396078 71 46502 416 SQUARES, CUBES, SQI:A,.CE Roo'r, AN)D CU3E RoOTS. 189 Number. Square. Cube. Square Root. Cube Root. Number. 417 173889 72511713 2094205757 1 47099 417 418 174724 73034632 20'445048 5'47696 418 419 175561 73560059 20'469489 71 48292 419 420 176400 74088000 20'493901 7'48887 420 421 177241 74618461 20'518284!7 49481 421 422 178084 5151448 20-542638 1750074 422 423 178929 i5686967 20'566963;7 50666 423 424 179776 16225024 20'591260 7 51257 421 425 180625 176165625 20'615528 7 51847 425 426 181476 177308776 20639767 7 52436 426 427 182329 77854483 20'663978 7-53024 427 428 1831.84 8402752 20'688160 7'53612 428 429 184041 1 8953589 20'712315 7'54198 429 430 184900 9507000 20'736441 7 54784 430 431 185761 80062991 205760539 7'55368 431 432 186624 80621568 20'784609 7555952 432 433 187489 81182737 20'808652 7'56535 433 434 188356 81746504 20'832666 7 57117 434 435 189225 82312875 20'856653 7 57698 435 436 190096 82881856 20880613 7'58278 436 437 190969 83453453 20 904545 7 58857 437 438 191844 84027672 20 928449 7'59436 438 439 192721 84604519 20'952326 7 60013 439 440 193600 85184000 20'976177 7560590 440 441 194481 85766121 210 7'61166 441 442 195364 86350888 21 02.3796 761741 442 443 196249 86938307 21-047565 7'62315 443 444 197136 87528384 21'071307 7'62888 444 445 198025 88121125 21'095023 7'63460 445 446 198916 88716536 21'118712 7'64032 446 447 1'99809 89314623 21'142374 7'64602 447 448 200704 89915392 21'166010 7-65172 448 449 201601 90518849 21'189620 7.65741 449 450 202500 91125000 21'213203 7'66309 450 451 203401 91733851 2t1236760 7'66876 451 452 204304 92345408 21-260291 7'67443 452 453 205209 92959677 2'1283796 7-68008 453 454 206116 93576664 21'307275 7 685'73 454 455 207025 94196375 21'330729 7'69137 455 1 456 207936 94818816 21'354156 7'69700 456 457 208849 95443993 21'377558 7'70262 457 458 209764 96071912 21-400934 7 i 0823 458 459 210681 96702579 21-424285'7-71384 459 460 211600 97336000 21'447610 7'71944 460 461 212521 97972181 21-470910 71 72503 461 462 213444 98611128 I 21'494185 7 73061 462 463 214369 99252847 21-517434'73618 463 190 SQUARES, CUBES, SQUARE ROOTS, AND CUBE RPOOTS. Number. Square. Cube. Square Root. Cube Root. Number. 464 215296 99897344 21-540659 7'74175 464 465 216225 100544625 21'5603858 7-4731 465 466 217156 101194696 21'587033 1775286 466 467 218089 101841563 21'610182 7 75840 467 468 219024 102503232 21'63330 717 63993 468 469 219961 103161109 21'656407 7'6946 469 470 220900 103823000 21'679483 1 77498 470 471 221841 1044871111 211702534 1778049 471 472 222784 105154048 21'725561 7 78599 472 47,3 223729 105823817 21'748563 7 79148' 473 474 224676 10649642-4 21'771:541 7179697 474 47 5 225625 107171875 21'794494 1780245 475 476 226576 107850176 21'817424 7 80792 476 477 227529 108531333 21'840329 1781338 417 478. 228484 109215352 21'863211 1781884' 418 479 229441 109902239 21'886068 7 82429 479 480 230400 110592000 21'908902' 7182973 480 481 231361. 111284641 21'931712 1783516 481 482 232324 111980168 21'954498 1'84059 482 483 233289 112678587 21'977261 1 84601 483 484 234256 1113379904 22'0 7 85142 484 485 235225 114084125 22'022715 7 85682 485 486 236196 114791256 22'045407 7 86222 486 487 237169 115501303 22 068076 17'86761 487 488 238144 1'16214272. 22'090722 7 87299 488 489 239121 116930169 22'113344 7 87836 489 490 240100 117649000 22'135943: 7 88378 490 491 241081 118370771 22'158519 7'88909 491 492. 242064 119095488 22'18107,3 1 89444 492 493 243049 119823157 22'203603 7189979 493 494 244036 120553784 22 226110 7 90512 494 495 245025 121287375 22,'248595 179104.5 495 496 246016 122023936 22'271057 7 91578 496 4.97 247009 122763473 22 29,34!96 1792109 497 498 248004 123505992 22'315913 7 92640 498 499. 249001 124251499 22'338307 7 93171 499 500 {250000 125000000 22 360679 7 93700 500 501 251001 125751501 22'383029 7 94229 501 502 252004 126506008 22'405356 1794157' 502 503 253009 127263527 2'2 427661 7'95284 503' 504 254016 128024064 22'449944 1795811 504 505 255025 128787625 224722472205 96337 505 506 256036 129554216 22 494443 7 96862 506 507 257049 130323843 22'516660 1,'97387 5071 508 258064 131096512 22'538855 7 97911 508 509 259081 131872229 22'561028 7 98434 509 510 260100. 132651000 22'583179 7.98956 510 SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. 191 Number. Square. Cube. Square Root. Cube Root. Number. 511 261121 133432831 22-605309 7 99478.511 512 262144 134217728 22-627417 8'0 512 513: 2.63169 13500569.7 22:649503 8'00520 513 514 264196 135796744 22'671568 8'01040 51'4 51.5 265225 3659087'5 22"693611 8:01559 515 516 266256 137388096 227'15633 8'02077 51'6.517 2.67289 13818841'3 22'737634 8'02595 517 518 268324 13899.1832 22'759613 8'03112 518 5.19 26,9361 1399798359 22'781571 8'03629 519 520 270400 140608000 22'803508 8'04145,520 521 271441 141420761, 22;825424 8'04660 1:521 522 272484 142236648,22'847319 8 05174 522 523' 273529 1430556'67,, 22:869193 8'05688'523 524 274.576 1438.77824 22'891046 8'06201 624 525 275625 144703125 22'912878 8'06714 525 526 276676 145531'576 22'934689 8'07226 526 527 277729 146363183 22'956480 8'07737 6527 528 278784 147197952, 22'978250 8'08248 528 529 27:9841 148035889 23'0 8 08757 529 530 280900 148877000 23'021728 8'09267 530 531 281961 149721291' 23'043437 8'0977'5 i:"531 532 283024 150568768' 23'065125 8'10283'532 533 284089 151419437; 23'086792.810791.533 5 34 285156 152271 3304 23'108440 8'11298 ~ 534 535 286225 153130375 23'130067 8'11804'535 536' 287296 153990656 23'151673 8'12309 536 537 288369 154854153 23'173260 8-12814 537 538 289444 155720872 23'194827 8'13318 538 539 290521 156590819 23'216373 8" 13822 539 540' 291600 157464000 23'237900 8"14325 540 54.1 292681 158340421 23'259406.814827 541 542 293764 159220088 23'280893 8'15329 542 543 294849 160103007 23'3023'60 315830 543 544 295936 160989184 233238'07 8.16331 544 545'297025 161878625 23 34'5.235 8'16830 545 546 298116 162771336 23 366642 817'330 546 547 299209 163667323 23 388031 8 178.28 547 548 800304 164566692 23'409399 8S18326 548 549 301401 165969149 23 430749 8'18824 549 550 302500 166375000 23'452078 8 19321 550 551 303601 167284151 23 4733:89 8'19817 551 552 304704 168196608 23 494680 8'20313 552 553 305809 169112377 23'515952 8S20808.553 554 306916 170031464 23'537204 8'21302 554 555 308025 170953875 23 558438 8 21796 555 556 309136 171879616 23 579652 8'22289 556 557 310249 172808693 23-600847 8'22782'557 1c2. S.QUARES, CUrES, SQUANRE ROOTS, AN.D CuBI, RPOOTS, NIumber Square I Cube Square Root. Cube Root. Number. 558 311364 173741119 1 23622023 8-232l4 5 58 559 312481 174676879 23-643180 8-23766 559 560 313600 175616000 23'664319 8 24257 560 561 9314721 176558481 23 685438 8 24747 561 562 315844 177504328 23 706539 8 25237 562 563 316969 178453547 1 23-727621 8 25726 563 564 318096 179406144 23748684 8226214 564 565 319225 180362125 23'769728 8'26702 565 666 320356 181321496 23'790754 8'27190 566 567 321489 182284263 23 811761 8-27677 567 568 322624 183250432 23-832750 8 28163 568 569 323761 184220009 23 853720 1 828649 569 570 324900 185193000 23'874672 829134 570 571 326041 186169411 23 895606 8 29619 51l 572 327184 187149248 23-916521 8-30103 572 573 328329 188132517 23.937418 8 30586 573 574 329476 S189119224 23 958297 8 31069 574 575 330625 190109375 23 979157 8'31551 575 576 331776 191102976 24'0 8 32033 576 577 332929 192100033 24'020824 8-32514 577 578 334084 193100552 i 24'041630 8'32995 578 579 335241 194104539 24'062418 8'33475 579 580 336400 195112000 24'083189 8'33955 580 581 337561 196122941 24'103941 8 34434 581 582 338724 197137368 24'124676 834912 582 583 339889 198155287 24'145392 8'35390 583 584 341056 199176704 24'166091 8'35867 584 585 342225 200201625 I 24'186773 8'36344 585 |586.343396 201230056 24'207436 8'36820 586 587 344569 202262003 24'228082 8 3729'6 587 588 345744 203297472 i 24'248711 8'37771 588 589 346921 204336469 1 24'269322 8'38246 589 590 348100 205379000 24'289915 8'38720 590 591 349281 206425071 24'810491 8'39194 591 592 ) 350464 1 207474688 24'331050 8' 39667 5 92 593 351649 208527857 24'351591 8-40139 593 594 352836 209584584 1 24-372115 8-40611 594 595 1 354025 210644875 24'392621 8-41083 595 596 355216 211708736 1 24'413111 8'41554 596 597 j 356409 212776173 24'433583 8-42024 597 598 8357604 213847192 24'454038 8'42494 598 599 358801 214921799 24'474476 8'42963 599 600 360000 216000000 24-494897 8'43432 600 601 361201 21.7081801 24'5'15301 8'43900 601 602 0 62404 218167208 24'535688 8'44368 602 603 363609 219256227 24'556058 8'44836 603 604 364816 220348864 24-576411 8'45302 604 SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. 193 Number. -Sqluare. Cube. Sqluare Root. Cube- Root. Number.. 605 366.025 221445125 245,96747 8-45769.605 606 367236 222545016 24;617067!8.46234 -606 607 368449 22364-8543 24.'637370 8:.46700'607 608s 369664 22475571.2 24.657656 8-47164.608 609 - 370881 225866529 24'677925 8:47628:609 610 372100 226981000 24,698178 8'48092.610 611 373321 228099131 24'718414 8'48555 -61:1 61:2/ 374544 229220928 24'738633 8 49018 61-2:6133 -375769 230346397 24'758836 8 49480 [61-3:614 376996 23147:5544 24-779:023 8-49942 614 615: 378225 232608375:24-799193 8'50403 6-15 616 379456 233744896 24 819347 850864 6.16 617 380689 234885113 24'8.39484 8'51:324 617 61.8 381924 236029032:24'85:9605 8,51784 6.18 6-19 383161 237176659 24-87.9710 8:'52243 619.620.:384400 238328000 24-899799 8,:52701 620 621'385641 239483061 24'919871 8,531.60 621 622 3 386884 240641'848 24'939927 8'53617 622 623:388129 241804367 24'959967 8"54075 623:624!389376 242970624. 24'979992 8'54531 624 625 390625 244140625 25'0 8. 54987 625 626 [ 391876 245314376 25.019992 - 8'55443 626 6.27 393129 246491883 25,039968 8-.55898 627 628 1 394384 247673152 256059928 8'56353 628 629 395641 [248858189 2507-987,2 8'56808 629:;630 3969001 250047000 25'099800 8,57261 630 6.31 [398161.251239591 25'119713 857715 631 6032 399424 252435968 25'139610 8'58168 632:633' 400689.253636137 25159491 8,58620 633;634 401956:254840104 25:'179356 8.'59072 634 635 403225.25604787-5 25-199206 8'59523 635 636 404496:257.259456 25219040 8?59974 636 637.405769 25847.4853 25'238858 8'60425 637 638.407044.259694072 25'258661 8'60875 638 639 408321 26091711-9 25'278449 86.1324 639 640.409600 12621.44000 25'298221 8'61773 640 641 410881 263374-721 25-317977 8"62222 641 642 412164.26460.9288 25'-337718 8.62670 642 643 413449:265847707 25'357444 8.62118 643 644 414736:267089984 25'377155 8"63565 64.4 645:416025 268336125 25'396850 8'64012 64.5 646.417316;269586136 25'416530 8.'64458 646.647 418609.270840023.25-436194 8'64904 647 648 419904 - 2720977.92 25455844 8'65.349 648 6.49 421201.273359449 25'.475478 8:65794.:649.650 422500.274625000 25.49.5.097 8.6.6239 6.50 651 423801 275894451.25'514101 8.'66683 651 17 194 SQUA;RES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. Number. Square. Cub6. Square Root. Cube Root. Number. 652 425104 277167808 25.534290 8'67126 652 653 426409 278445077 255'53864 S867569 653 654 427716 279726264 25-573423 8'68012 654 655 429025 281011375 25'592967 8'68454 655 656 430336 282300416 25'612496 8'68896 656 657 431649 283593393 25 632011 8 69337 657 658 432964 284890312 25'65151Q 8'69778 658 659 434281 286191179 25'670995 8'70218 659 660 435600 287496000 25'690465 8'70658 660 661 436921 288804781 25 709920 8 71098 661 662 438244 290117528 25729360 87 1537 662 663 439569 291434247 25'748786 8 71975 663 664 440896 292754944 25'768197 8'72414 664 665 442225 294079625 25 -787593 8 72851 665 666 443556 295408296 25'806975 873289 666 667 444889 296740963 25'826343 8'73726 667 668 446224 298077632 25'845696 8'74162 668 669 447561 299418309 25'865034 874598 669 670 448900 300763000 25'884358 8 75034 670 671 450241 302111711 25 903667 875469 671 672 451584 303464448 25 922962 8'75903 672 673 452929 304821217 25'942243 8 76338 673 674 454276 306182024 25'961510 8'76771 674 675 455625 30754687.5 25'980762 8'77205 675 676 456976 308915776 26'0 8'77638 676 677 458329 310288733 26 019223 8'78070 677 678 459684 311665752 26'038433 8 78502 678 679 461041 313046839 26'057628 8'78934 679 680 462400 314432000 26'076809 8'79365 680 681 463761 315821241 26'095976 8'79796 681 682 465124 317214568 26'115129 8'80227 682 683 466489 318611987 26'134268 8'80657 683 684 467856 32001350-4 26'153393 8'81086 684 685 469225 321419125 26'172504 881515 685 686 470596 322828856 26'191601 8-81944 686 687 411969 324242703 26'210684 8'82373 687 688 473344 325660672 26'229754 8'82800 688 689 474721 327082769 26'248809 8 8322'8 689 690 476100 328509000 26'267851 883655 690 69 1 477481 329939371 26'286878 8'84082 691 692 478864 331373888 26'305892 8'84508 692 693 480249 332812557 26-324893 8'84934 693 694 481636 334255384 26'343879 8'85359 694 695 483025 335702375 26'362852 8 85784 695 696 484416 337153536 26'381811 886209 696 697 485809 338608873 26'400757 8'86633 697 698 487204 340068392 26'419689 8'87057 698 SQUARES, CUBES, SQUARE R, AOOTS, AND CUBE ROOTS. 195 N umber. Square. Cube. Square Rout. i Cube Root. Number. 699 488601 341532099 26'438608 8'87480 699 700 490000 343000000 26'457513 8'87904 700 701 491401 1 344472101 26'47 6404 8'88326 701 702 492804 345948408 26495282 8-88748 702.703 494209 347428927 26'514147 8'89170 703 704 495616 I 348913664 26'532998 8'89592 704 705 497025 350402625 26 551836, 8.90013 705 706 498436 351895816 26 570660 - 90438 706 707 499849 353393243 26 589471 1 8 90853 707 708 501261 354894912 26 608269 891273 708 709 502681 356400829 26'627053 8'91693 709 710. 504100 357911000 26'645825 8,92112 710 711 505521 359425431 26'664583 8'92530 711 712 507944 360944128 26'683328 8'92949 712 713 508369 362467097 26'702059 893366 713 714 509796 36399434-1 26'720778 8'93784 714 715 5511225 365525875 26'739483 8'94201 715 716 512656 367061696 26B758176 8 94618 716 717. 514089 368601813 26 776855a 8 95031 717 718 515524 3701362332 26 796522 8 95450 718 719 516961 371694959 26o814175 8o95865 719 720 518400 373248000 26832815 896280 720 721 1 9841 374805361 26 851443 8 96695 721 722 521284 376367048 26'870057 8'97110 722 723 522729 377933067 26'888659 8'97524 723 724 524176 379503424 26-907248 8'97937 724 725 525625 381078125 26'925824 8 98350 725 726 527076 382657176 26'944387 8'98763 726 727 | 528529 384240583 26'962937 8'99176 727 728 529984 385828352 26'981475 8'99588 728 729 531441 387420489 27'0 90 729 730 532900 389017000 27.'018512 900-11 730 731 534361 390617891 27'037011 9'00822 731 732 535824 1 392223168 27'055498 9'01232 732 733 537289 393832837 27-073972 9'01643 733 734 538756 3954469041 27 0924341 9'02052 734 735 540225 397065375 27'110883 9-02462 735 736 541696 398688256 27'129319 9 02871 736 737 543169 1 400315553 27'147743 9-03280 737 738 544644 401947272 27'166155 9'0368S 738 739 1 546121 403583419 27'184554 9'04096 739 740 547600 405224000 27'202941 904504 740 741 54'-081 406869021 27 221315 9'04911 741 742 550564 408518488 27 239676 9-05318 742 743 552049 410172407 27-258026 9'05724 743 744 553536 411830784 27'276363 9-06130 744 745 555025 413493625 27-294688 9'06536 745 196 SQUARES, CUBES), SQUARE ROOTS, AND CUBE RooTs. 1Number.! SqlUare, Cube. Square Root. Cube Root. Number. 746 556516 415160936 27'313000: 9 06:942 746 747 558009 416832723 27'33130) 9 07347 747 718 559504' 418:508992 27 349588 9-07751 748 749 5 61001 420189749 27 3678'64 908156. 749 750 562500 421875000 27'386127, 9'08560 750 751 564001 423564751: 27'404379 9'08963 751 752; 565504 425259008 27 422618: 9 09367 752: 753 567009 426957777 27'440845' 9'09770 753 7'54; 568516 428661064 27'459060 9'10172 754 755 570025 430368875' 27'477263 9'10574 755 756 571536 432081216( 27'495454 9'10976 756 757 57 3049 4337980938 27'5'13633 9'11378 757 758 574564 435519512 27'531799: 9'11779' 758 159 576081 437245479 27'549954' 9'12180 759 760 577600 438976000 27'568097 9'12580 760 761 5791-21 440711081 27'586228 9'12980 761 762 580644 442450728 27'604347 9'13380 762 763 582169 444194947 27 622454 9'13779 763 764 583696 445943744 27 640549 9'14178 764 765 55:852'25 447697125 27'658633 9'14577 765 766 586756' 449455096 27 676705 9'14975 766 7'67 588289 451217663 27'691764 9'15373 767 768 589824 452984832 27-712812: 9'15771 768 769 591361 454756609' 27'730849: 9'16168 7'69 770 592900 456533000 27'748873' 9'16,565 770 771 594441 458314011' 27-766886: 9'16962 771 772 595984 460099648 27'7-84888 9'17358 772 773 597529 461889917 27'802877; 9'17754 773 774 599076 463684824 27'820855 9'18150 774 775 600625 465484375 27 838821: 9'18545 775 776 602176 467288576 27'856776 9'18940 776 777 603729 469097433' 27'874719 9'19334 777: 778 605284 470910952 27'892651 9'19728 778 779 606841 4727291389 27'910571 9'20122 779 780 608400 474552000 27 928480 9'20516 780: 781 609961 476379541; 27 946377' 9'20909 781: 782 611524 478211768 27'964262 9 21302 782: 783 613089 480048687 27-982137: 9'21695 783: 784 614656 481890304: 28'0 9'22087 784; 785 616225 483736625 28'017851 9'22479 785 786 617796 485'587656 28'035691 9'22870 786 787 619369 487443403 28-053520 - 923261 787 788 620944 489303872 28'071337 9'23652 788 789 622521 491169069 28'089143 9'24043 789 790' 624100 493039000 28-106938 9-24433 790 791 625681 494913671; 28'124722 9 24823 791 792- 627264 496793088 28'142494 952213 792 SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. 197 Number. Square. Cube. Square Root. Cube Root. Number. 793 628849 498677257 28-160255 9'25602 793 794 630436 500566184 28'178005- 9 25991 794'795 632025 502459875 28'195744 9'26379 795 796 633616 504358336 28 213472 9 26767 796 797 635209 506261573 28'231188 9'27155 797 798 636804 508169592 28.248893 9.27543 798 799 638401 510082399 28 266588 9 27930 799 800 640000 512000000 28'284271 9'28317 800 801 641601 513922401 28'301'943 9'28704 801 802 643204 515849608 28'319604 9'29090 802 803 644809 517781627 28'337254 9'29476 803 804 646416 519718464 28'354893 9'29862 804 805 648025 5.21660125 28'372521 9'30247 805 806 649636 52'3606616 28'3901:39 9'30632'806 807 651249 525557943 28'407745 9'31017 807 808 652864 527514112 28'425340 9 81401 808 809 654481 529475129 28 442925 9 31785 809 810 656100 531441000 28'460498 9 32169 810 811 657721 533411731 2847806.1' 9'32553 811 812 659344 535387328 28'495613 9'32936 812 813 660969 537367797 28'513154 9'33319 813 814 662596 539353144 28 530685 9 33701 814 815 /664225 541343375 28'548204 9 34083 815 816 66585'6 543338496 28'5657-13 9 34465 816 8171 667489 545338513 28'583211 9 34847 817 818 f669124 - 547343432 28 600699 935228 818 819 670761 549353259 28'618176 9'35609 819 820 672400 551368000 286'35642 9 35990 820 821 674041 553387661 28'653097 936370 821 822 675684 555412258' 28'670542 9386750 822 823 677329 557441767 28'687976 9-37130 823 824 678976 559476224 28'705400 9 37509- 824 825 680625 561515625 28-722813 9'37888 825 826 682276 563559976' 28'740215 9'38267 826 827 683929 565609283 28 757607. 9'38646 827 828 685584 567663552 28-'74989 9 39024' 828 829 687241 569722789 28-792360 9'39402 829 830 688900 571787000' 28'809720 9'39779 830 831 690561 57385619-1 28'827070 940156 831 832 692224 575930368 28 844410 9 40533 882 833 693889 578009537 28 8'61739 9'40910 833 834 695556 580093704 28'879058 941286 834 835 6907225 58218287'5 28'8'96366 9-41662 835 836 698896 584277056 28-913664 9 4'2038 836 837 700569 586376253 28'930952 9'42414 837 838 1702244 588480472 28'948229 9'42789 838 839 703921 59058'9719 28 965496 9 43164 8S9 17* 198 SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. Number, Square. Cube. Square Root. Cube Root. Number. 840 705'600 592704000 28'982753 9'43538 840 841. 707281 594823321 29.0 9:43913 841 842: 708964 596947688 29'017236 9'44287 842 843. 710649 5990771 07 2.9'0344.62 9'44660 843 844 712336 601211584 29'051678 9'45034 844 845 714025 603351:125 29'068883 9 45407 845 846 715716 6054:95736 29'086079 945779 846 847 717409 60764-5423 29'103264 9'46152 847 848 719104; 609.800192 29'120439 9'46524 848 849 720801 611960049 29'137604 9'46896 849 850 722500 614125000 29'154759 9-47268 850851 724201 616295051 29'171904 9 47639 851 852: 725904 618470208 29'1.89039 9'48010 852 853 7t27609 620650477 29'206163 9'48381 853. 854. 729316 622835864 29 223278 9 48751 854 855 731025 625026375 29'240380 9'49122 855 856 732736 627222016 29'257477 9'49491 8.56. 857 734449 629422793 29'274562 9'49861 857 858 736164 631628712 29'291637 9-50230 858 859' 737881 633839779 29'308701 9'50599 859 860 739600 636056000 2.9'325756 9'50968 860 861 41321 638277.381 29 342801 9'51336 861 8.62. 43044 640503.928 29'.359836 9'51705 862 863 744769 642735647 29.9376861 9.52073- 863 864 746496 644972544 29.393876 9-52440 864 8.65 748225 647.214625 29 410882 9'52807 865 866 749956 649461896 29 427877 9'53174 866 867 751689 6517143.63 29 444863 9'53541 8.67. 868 7-53424 65397.2032 29'46.1839 9'53908 868 869 755161 656234909 29'478805 9'54274.869 870 756900 658503000 29 495762- 9'54640 870 871 758641 660776311- 29'512709 9'55005 871 872: 76038.4 663.054.848 29'529646 9'55371 8772. 873 762129 665338617 29-546573 9'55736 873 874 763876 667627624 29'56349.1 9'56101 874 875 765625 669921875 29-580398 9'56465. 875 876 7 6376 6722213.76 29 597297 9,56829 876 877'769129 614526133 29'6.14185 9 57193 877 878 770884 676836152 29'631064 9 57557 878 879 77:2641 679151439: 29647934 9-57920 879 880 774400 681472000 29.664793 9'58283 88088.1 776161 683797841 29'681644 9'58646 881 882 777924 686128968 29-698484 9059009 882; 883 779689 68846'5387 2:9q715315 9-59371 883 884 781456 690807104 297 3:21.37 9'59733 884 8.85 783225 693154125 29'748949 9'60095 885 886[ 784996 695506456 29'7657-52 960456 8:86 SQUARES, CUBES, SQUARE. ROOTS, AND CUBE ROOTS. 199 Number.: Sqquare. Cube. Square Root. Cube Root. Number. 88a7 786769: 697864103- 29'782545: 9'60818 887. 8,8.8 788544 700227,07-2. 29;7.99328. 96117-9 888 88:9 7903.2-1. 702595369- 29.816103.- 9'61539. 889. 890 792100'~ 704969000- 29i832S867' 9;61900 890 8.91 793881: 707347971: 29,849623 9162260: 891 892. 7:95664: 709732288 29'866369 9'62620,892 893- 797,449., 712121957- 29.883105; 962979' 893 8941 799236[ 714516984, 29'899832 9'63339 894 895 80.1025 [ 716917375: 29'916550 9'63698 895 896 8028.16 719323136- 29'933-259 9'64056 896 897] 804609] 721734273, 29'949958 9'64415 897 898. 806404' 724 15 0792: 29'9666:48 9'64773 898 899. 808.201. 72657.2699- 29'9838328- 9'651i311 899 900. 810000- 729000000 30'0 9 65489 900 901 81:1801] 731-432'701 30'016662, 9'65846 901 902; 8:1'3604: 7338.70808. 30'033.314. 9.66204 902 903: 8.15409.: 736314327. 30'049958 9'66560 903 904- 81.7216: 7387:63264. 30'0665.92 9'66917. 904 905, 81 90265: 741;217625a 30.083217 9.67274 905 906! 820836} 74367741-.6 30'099833. 9 67630. 90 907: 822649: 746142643 30'116440 9'67986 907 908! 824464 748613.312 30'133038, 9.68341 908 909: 82.B6281 751089429. 30 149626 9.68697 909 9.1:0 828100 753557:1000 3016.6206 969052, 910 9-11 829921 75605803:1 30'182776: 9-69406 911 912:. 831744: 758550528 30'199337 9'69761 912 9138 833569 761.048497 30'215,89 9'70115 913 9:14 835396: 763551944' 30'232,2432- 9'70469; 914 915D. 837225' 76606087-5 30'248966' 9'70823 91 916. 8-3:9056 7685.7-5296 30-26,5491 9'71177 916 917.' 840889-| 771:0,952131 30.282007 9715.30 917 91,8' 842t,2.24:, 7736,20632 30-2985:14 9'71883 918919 $844501.1 776151-559 30.3415,012. 97223,6: 919 920' 846400: 778688000 3033.1501 92588 920 9s21 8,48.2414 781 2299-6-.L 30'.347-981 9'72941 921 922. 850084. 7837.777448 30'3.6,44.5.52: 927.3. 92 92.3 851929.: 7863:304,67 30'38091,5 9'7.3644 923 924 853776. 788889024 3.039r73.68 9-738996 924 925' 855625 791453'125: 30'41:38.12 974347 925 9261 8.5.747 6: 79402277'6 30'430248. 9'74698S 926 927,85:9329:.'796597983 3.044667:4' 97-5049 927 928:8 861184- 7991787.52 30'463092. 975399 928 92.9' 863041 801.7650.89 30-479501 9'75750 929 9.30: 864900' 804357000 30,495901 9-76100 9.30 93.1. 8.667'61 80695,4491 30.512292 976449 931 938.2. 868624: 8095.57568 30'5286. 75 9'676799 932 93:3. 870489:., 812:166237 30545048 977148 933 200 SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS. Number. Square. Cube. Square Root. Cube Root. Number. 934 872356 814780504 30'561413 9'77497 934 935 874225 817400375 30'577769 9'77846 935 936 876096 820025856 30'5941-17 9'78194 936 937 877969 822656953 30-610455 9'78542 937 938 879844 825293672 30'626785 9'78890 938' 939 881721 827936019 30'643106 979238 939 940 883600 830584000 30-659419 9'79586 940 941 885481 833237621 30-675723 9-79933 941' 942 887364 835896888 30-692018 9-80280 942 943 889249 838561807 30 708305' 980627 943 944 891136 841232384 30'724583 9 80973 944 945 893025 843908625 30-740852 9-81319 945 946 894916 846590536 30-757113 9-81'665 946 947 896809 849278123 30'773365 9'82011 947 948 898704 851971392 30'789608 9'82357 948' 949 900601 854670349 30'805843 9'82702 949 950 902500 857375000 30'822070 983047 950 951 904401 860085351 30'838287 9'83392 951 952 906304 862801408 30'854497 9'83736 952 953 908209 865523177 30'87'0698 9-84081 953 954 910116 868250664 30-886890 9'84425 954 955 912025 870983875 30 903074 9`84769 955 956 913936 873722816 3()'919249 9'85112 956 957 915849 876467493 30'935416 9-85456' 957 958 917764 879217912 30'951575 985799 958 959 916681 881974079 30 967725 9'86142 959 960 921600 884736000 30-983866 986484 960 961 923521 887503681 31,0 9`86827 961 962 925444 890277128 31 016124 9-87169 962 963 927369 893056347 31-032241 9'87511 963 964 929296 895841344 31-048349 9'87853 964 965 931225 898632125 31'064449 9'88194 965' 966 933156 901428696 31'080540 9'88535 966 967 935089 904231063 31'996623 9'88876 967 968 937024 907039232 31-112698 9'89217 968 969 938961 909853209 31'128764' 9'89558 969 970 940900 912673000 31-144823 9'89898 970 971 942841 915498611 31'160872 9'90238 971 972 944784 918330048 31-176914 9'90578 972 973 946729 921167317 31 192947 9-90917 973 974 948676 924010424 31'208973 9'91257 974 975 950625 92685937'5 31'224990 9-91596 975 976 952576 92971417'6 31'240998 9'91935 976 977 954529 932574833 31 256999 9-92273 977 978 956484 935441352 31"272991 9'92612 978 979 958441 938313739 31-288975 9;92950 979 980 960700 951192000 31'304951 993288 980 SQUARES, CUBES, SQUARE ROOTS,, AND CUBE ROOTS. 201 N umber. Square. Cube. Square Roout. Cube Root. Number. 981 96 2361 944076141 31'320919 9-93626 981 982 964324 946966168 31'336879 9'93963 982 98'3 966-289 949862087 31'352830 9 94300 983 984: 968256 952-763904 31'368.774 9'94637 984 985I 970225 955671625 31'384709 9'94974 985 986/ 972196 95-8585256 31'4006836 995311 98:6 987 974169 961504803 31'416556 9-95647 987 98-8 976144 96443.0272 31 432467; 9'95983 988 989/ 978121 967'61669 31t'448370: 9'96319 989 990 980100 97029900) 31'464265 9'96655 990 991 982081 973242271 31'480152 9'96990 991 992 984064 976191488 31-496031' 9'97'326 992 993 986049 979146657 31'511902: 997661 993 994 988036 982107784 - 31-52776.: 997995 994 995 990025 985074875 31'5436-20 9'98:330 995 996: 992016 988047936 31: 559467 9'98664 996 997 994009 991026973. 31 57 5306: 9'98999 997 998/ 996'004 994011992i 31'591138 9'99332 1 998 999 998001 997002999 31'606961 9'99666 999 1000 1000000 1000000000 31623776 10- 10(00 SILVER, TO PURIFY AND REDUCE.-Silver, as used-in- the arts and coinage, is alloyed with a portion of copper. To purify it, dissolve the. metal in nitric acid slightly diluted, and add common salt, which throws. down the whole of, the silver in the- form- of chloride. To reduce it into a metallic state several methods are used: 1. The chloride must be repeatedly washed with distilled water, and placed in a zinc cup; a. little diluted sulphuric acid being added, the chloride is soon reduced. The silver when thoroughly washed is quite pure. In the absence of a zinc cup, a porcelain cup containing a zinc plate may be used. The process is expedited-by warming: the cup. 2. Digest the washed chloride with pure copper and ammonia. The quantity of ammonia. need' not be sufficient. -to dissolve the chloride. Leave the mixture. for a day,. then wash the silver thoroughly. 3. Boil the- washed and moist chloride in solution of. pure potash, adding a little sugar: when fwashed it is —quite pure. WELDING COMPOSITION.-Mix borax with 1!).th of sal ammoniac, fuse the mixture, and pour it on an iron; plate. When, cold, pulverise it,. and mix it- with: an equal weight of: quick lime, sprinkle it oln ilon. whlich is heated to redness, and replace it in. the fire. It may be welded below tlhe usual heat. 202 BLACKING RECirES. BLACKING RECIPES. Liquid Blacking, for Boots and Shoes.-1. Ivory black, 3 oz.; molasses, 2 oz.; sweet oil, l oz. Mix to form a paste. Add gradually 4 oz. of oil of vitriol, and then half a pint of vinegar, and 1l pint of water, or sour beer. Some prefer mixing the oil of vitriol with the sweet oil. 2. Ivory black, 2 lbs.; molasses, 2 lbs.; sweet oil, ~ lb. Mix, and add - lb. oil of vitriol, and enough beer or vinegar to make up a gallon. 3. Ivory black, 3 lbs.; molasses, 4 lbs.; vinegar, 1 pint; oil of vitriol, 8 oz.; water, 1 gallon. 4. Ivory black, 2 lbs.; neat's-foot oil, 4 oz. Mix, and add 3 quarts of sour beer or vinegar, and a spoonful of any kind of spirits; stir till smooth, and add 2 oz. of oil of vitriol, and sprinkle on it i drachm of powdered resin. Then boil together 3 pints of sour ale with a little logwood, and I oz. of Prussian blue, 3 oz. of honey, and 8 oz. of molasses. Mix, but do not bottle it for two or three days. 5. Ivory black, 8 oz.; brown sugar, or molasses, 8 oz.; sweet oil, 1 oz.; oil of vitriol, ~ oz.; vinegar, two quarts. Mix the oil with the molasses, then add the oil of vitriol and vinegar, and lastly the ivory black. Blackinlyfor D)ress Boots -1. Gum, 8 oz.; molasses, 2 oz.' ink, 1 pint; vinegar, 2 oz.; spirit of wine, 2 oz. Dissolve the gum and molasses in the ink and vinegar, strain, and add the spirit. 2. To the above add 1 oz. of sweet oil, and I oz. of lampblack. [These are applied with a sponge, and allowed to dry out of the dust. They will not bear the wet.] 3. Beat together the whites of 2 eggs, a table-spoonful of spirit of wine, a lump of sugar, and a little finely powdered ivory black to thicken. Blackinc, wsithout Polishinq.-Molasses, 4 oz.; lampblack, ~ oz.; yeast, a table-spoonful; 2 eggs; a tea-spoonful of olive oil; a teaspoonful of turlpentine. Mix well. To be applied with a sponge, without brushing. India Babber Blacking.-Ivory black, 60 lbs.; molasses, 45 lbs.; vinegar (No. 24), 20 gallons; powdered gum, 1 lb.; India rubber oil, 9 lbs. (The latter is made by dissolving, by heat, 18 oz. of India rubber in 9 lbs. of rape oil,) Grind the whole smooth in a paint mill. Then add, by small quantities at a time, 12 lbs. of oil of vitriol, stirring it strongly for half an hour a day for a fortnight. Paste Blacking. — 1. Oil of vitriol, 2 parts; sweet oil, I part; molasses, 3 parts; ivory black, 4 parts. Mix. 2. This may be made with the ingredients of liquid blacking, using sufficient vinegar, in which a little gum has been dissolved, to form a paste. Make it into cakes, and dry it. BLACKING RECIPES. 203 3. (Bailey's Blacking. Balls.) Bruised gum tragacanth, 1 oz.; water, 4 oz. Mix, and add 2 oz. of neat's-foot oil, 2 oz. of fine ivory black, 2 oz. of Prussian blue. Mix, and evaporate to a proper consistence. Blacking for Harness.-1. Isinglass or gelatine, 1 oz.; powdered indigo, 1 oz; soft soap, 4 oz.; logwood, 4 oz.; glue, 5 oz. Boil together in 2 pints of vinegar, till the glue is dissolved; then strain through a cloth, and bottle for use. 2. Melt 8 oz. of beeswax in an earthen pipkin, and stir into it 2 oz. of ivory black, 1 oz. of Prussian blue ground in oil, 1 oz. of oil of turpentine, and I oz. of copal varnish. Make it into balls. To be applied with a brush, and polished with an old handkerclhief. 3. Molasses ~ lb.; lampblack, 1 oz.; yeast, I spoonful; of sugar candy, olive oil, gum tragacanth, and isinglass, 1 oz. each; a cow's gall. Mix all together witll 2 pints of stale beer, and let it stand before the fire for an hour. lieel Balls.-I. Melt together 4 oz. of mutton suet, 1 oz. of beeswax, 1 oz. of slweet oil, ~ oz. oil of turpentine, and stir in 1 oz. of powdered gum arabic, and ~ oz. of fine lampblack. 2. Beeswax, 8 oz.; tallow, 1 oz.; powdered gum, 1 oz.; lampblack, q. s. Heel balls are used not merely by the shoemaker, but to copy inscriptions, raised patterns, &c., by rubbing the ball on paper laid over the article to be copied. BLACKLEAD PENCILS.-The easiest way of producing, not only blacklead, but all sorts of pencils, is by the following process, which at once combines simplicity, cheapness, and the finest quality. Take white or pipe-clay: put it into a tub of clean water, to soak for twelve hours, then agitate the whole, until it resembles milk, let it rest two or three minutes, and pour off the supernatant milky liquor into a second vessel, allow it to settle, pour off the clear, and dry the residue on a filter. Then add blacklend, any quantity. Powder it, and calcine it at a white heat in a loosely covered crucible, coo1, and carefully pulverize, then add prepared clay, prepared plumbago, equal parts. Water to mix. Make them into a paste, and put it into oiled moulds of the size required, dry very gradually, and apply sufficient heat to give the required degree of hardness; lastly, the pieces should be taken carefully from the moulds, and placed in the grooves of the cedar. The more clay and heat employed the harder the crayon; less clay andl heat of course produces a contrary effect. The shade of black may also be varied in the same way-. Each mould must be made of four pieces of wood, nicely fitted together. BLACK FOR MINIATURE PAINTERS.-Take camphor, and set it on the fire, and collect the soot by means of a saucer or paper funnel Inverted over it. 204 STRAIN AND ~STRESS -OF MATERIALS. STRAIN AND STRESS OF MATERIALS. Let A B be; a: beam. of timber, firmly fix- Fig.-. ed in a wall at A, and a weight, W, mea- 17////// sured in pounds avoirdupois, acting at the' A-:D 3 extremity B, at right angles to A B. If A:Bbeone foot, and the weight W be / one pound, then the strain produced at A is called a unit of strain. /// If the beam A B be (l) feet long, and the weight be (TW) pounds, then the units of strcwin produced at A, by the weight acting':at B, will be I TW. And the units of strain which the weight W produces at, any other part of the beam D, are measured by TV. B D. Let A B = 10 feet, and the weight V be:equal to 11-2 lbs., and BD) - 7 feet.' The un'its of strain at A = 112 x 10.- 11.20. The units of strainw-at:D =.112 x - = 7.84. The greatest strain on the beam is at A, at which place' the beam would break if it was equally strong throughout. If the weight'Tbe uniform ly distiibuted Fig 2 over the whole length of the beam A B, as'/j in fig. 2, the units of strain at A will be only one-half as great as that produced by A-' -B the weight W 7acting as in fig. 1. The units qof: strain at'A, which are pro-; duced'by-the-beam itself, are equal to the' /// weight of the beam multiplied by half. its length. The beam.A B, fig. 3, is equally Fig. 8. strong between the points. A and., l when the underside of it is a common:B parabola.. Hence, firom a square beam, one-third' part of it may be cut off without di-t./ minishing its strength.:: Fig. 4. If the weight iV be uniformly distri~'7;.. ".. buted over'the whole length of the /;,,,,... beam A B, as in fig. 4; then the beam is equally strong when'the underside of it' is a: straight'line. In this- case;. one half the' beam: maybe;cut.away without diminishing its strength. -TRAIN AND.TRESS OF MATERIALS, 205 a:| XFig. 5 Let the weight W (fig. 5) be sustained by a beam A B, which rests on two props at C and D. The pressure on the prop at C is equal to IV. B E: A B. The pressure on the prop at D is equal to W. A E: A B. The units of strain at -E are equal to W. A E. B E: A B. The units of strain at G are equal to W. A E'. B G: A B. The units of strain at F are equal to W. B E. A F: A B. The greatest strain, which is produced by the weight W, is at E. The units of strain at the middle of the beam, produced by the W. AAE weight Waacting at E, are equal to 2 Let A B = 18 feet, and a weight of 112 lbs. be placed: at E, which is 8 feet from A. Apply these numbers to the above formuhl and their results. 10 x 112 The pressure on the prop at C is equal to 1 -- 62.5 lbs. 18 8 x 112 The pressure on the prop at D is equal to -- - -= 49.8 lbs. 18 The units of strain at E are equal to 1 8 x 497.77. 18 8 x 112 The'Units of strain on the middle are equal to = 448. When the weight TV is laid on the middleof the beam A B, the WV. A B units of strain on the middle are equal to -- * 4 If the weight TV be uniformly distributed along the beam A B the units of strain on the middle of it will be equal to.'; 8 which is only one half the strain that is produced by the weight having been laid on the middle. Fig. 6. - - - -------------— I —-l~i.18_ 206 STRAIN AND STRESS OF MATERIALS. When the beam A B (fig. 6), supports a weight W, at E, it is equally strong between the points A and B, if the upper sides, A E, B E, be two parabolas whose vertex is A and B respectively. Fig. 7. @n- --- - -.1~_ - L1- M D Let the weight T have a bearing EF (fig. 7), equal on both sides of the centre G, and also let the weight be equally distributed on the bearing E F. W. AB W.EF The units of strain at G are equal to 4 8 Now, if the weight W were a sphere, and were laid on the middle of the beam at G, the units of strain at G would be equal to' 4 If the same weight be formed into a cube, whose side is EF, the units of strain at the centre G will be less than in the case of the i W. BEF sphere by 8' Fig. 8. Let A B be any beam suspended vertically from the point A (fig. 8): and let the sectional area be constant'/j/,l from A to B, where a weight WT'lbs. is acting to A extend the beam. Put a = area of the section of the beam in square inches. I = length of the beam in feet before the B weight is applied to elongate it. e = the elongation produced by the weight W;TV E= weight which would be necessary to make /iT e equal to 1. The quantity Eis called the modulus of elasticity of the material of which the beam is composed. In the case of the beam being compressed by the weight TV acting in the opposite direction, Put c =compression produced by the weight W. C force which is necessalry to make c equal to half of (1). The quantity C is called the modulus of elasticity of the material, when it is.subject to compression. E= and C=at e aW Units of work done to elongate the beam e feet ---- 9 2~~~ STRAIN AN-D STRESS. OF MATERIALS. 207 Units of work done to compress the beam c feet- WiMcan results of experiments on four diferent kicds of Cast-iron bars, i0 feet long and i squarLe ince in section. Weight laid on bar Etension of bar in 12 W per square inch inches i n e. Set of bar in inches. The value of 1 W lbs. I inches. 1054.009.... 117085 1581.0137.00022 115131 2108.0186.00055 113308 3161.0287.00107 110150 4215.0391.00175 107802 5269.0500.00265 105377 6323.0613.oo00372 103142 73076 73..00517 100496 8430. 0859.00664 98;139 9484.0995.00844 95316 10538.1136.01062 92762 11591.1283.01306 90347 12645.1448.01609 87329 13700.1668.02097 82133 14793.1859.02410 79576 Hence, the breaking weight per square inch of section is 14793 lbs. — 6.6 tons nearly; and the ultimate extension is.1859 inches, or d{b of tlle whole length, 10 feet. If we deduct the set'0209 from'1859, we shall have'165 inches for the elongation produced by the weight 14793 lbs. 14793 x 10 x 12.. E= modulus of elasticity = 10758545..165. Breaking weight = 6.6 tons x area of section in square inches. If the weight 5269 be taken, the modulus of elasticity will be considerably increased. Deduct.00175 the set fiorn.05, leaving.04825 inches for the elongation due to the weight 5269 lbs. 5269 x 10 x 12.E -- modulus of elasticity = 2C -- 13104249..04825 This difference in thle modulus of elasticity arises from the circumstance of the law of elasticity not being proportional to thle weight. 208 STRAI-N AND STRESS OF MATERIALS. TABLE Of the Tensile Strength of Wrought Iron. The Bar was 10 feet long and I square inch section. The value of Weight laid on the Extension of the Bar 12 W Bar W. or value of 12 6. Set of Bar. e lbs. inches. inches. 1262'00520 242665 3785'01690 0005 223998 63.09 *02772'Q005 227608 8833 ~ 03790'0005 233061 11356 - 04854'0005 233966 1388.0 *05950 0007 233285 16404'06980 0007 235016 18928'081.70' 00130 231675 21452' 09310'00270 230415 23975'10570 -00410 226824 26499 ~12040 00680 220092 29023 - 14500 0120 200157 30284 - 1991 0 ~0120 302841990 12 ~after bearing the 23660 eight 17 hours. 31546 24200 1083 130357 ditto ~ 24490'1111 aLfter five minutes. 35332 2'04 1'874 17320 The bar broke with a weight of 24 tons per square inch of section. Hence the tensile force of wrought iron is nearly four times as great as the tensile force of cast iron. TAB LE Of thle Compressive: Strenqgth of WIroughst I'on. The Bar was 10 feet long anld t square inch section. Weight laid on the Decrement of length, Weight laid on the Decrement of length, or Bar, or ( W). or the value of 1i c. bar, or (WI). the valUe of12c,. Ibs. inches. lbs. inches. 5098' 028 23018'119 9578'052 25258 ~130 14058'073 27498 ~142 16298'085 29738 154 18538'096 31978' 174 20778'107 34218'214 STRAIN AND STRESS OF MATERIALS. 209 The crushing force of wrought iron is 12 tons per square inch. It is a curious fact, that cast iron is decreased in length nearly double what wrought iron is, by the same -:weight; but the wrought iron bar will sink to any degree with little more than 12 tons per square inch, whilst cast iron will bear 43O56 tons to produce the same effect. A wrought bar will bear a compression of {-3 of its length, without its utility being destroyed. Cosrnpression of Cast Iron. Mean results of experiments on tour different kinds of Cast Ilon, 10 feet long, and I square inch in section. Weight laid on the Decrement of length, 11 wv bar (1W). or the value of 12 c. Set of bar in inches. c lbs. inches. inches. 2065'01875'00047 110119 4129 03878' 00226 106485 6194'05978'00400 103617 8259'07879' 00645 104822 10324'09944'00847 103819 12388'12030'010875 102980 14453'14163 ~01405 102049 16518'16338 ~01712 101101 18583'18505 02051 100420 20464'20624 02484 100114 24777'24961 ~ 03220 99263 28906'29699 ~04300 97331 33031'35341 06096 93463 The crushing or compressive force of cast iron per square inch is 43'56 tons, which has been obtained from eleven kinds of cast iron. But the tensile force of cast iron is 6-6 tons; therefore the compressive force is equal to the square of the tensile force, or (6'6)a. Transiverse Strength of Beams. A Ki B To find the neutral line, forces of' 1'//~////,//Z1 extension, forces of compression, moC ments of extension, and moments of compression of a beam subject to transverse flexure. Let the form of the section of the beam be that of the figure A B D E, where BC, HE, represent sections of r:s (-tG: the top and bottom ribs, FG that of the vertical one connecting them, and D O pass through the neutral line. Put a, a' - NI; NK, respectively. 18* 210 STRAIN AND STRESS OF MATERIALS. c, c' D H, A C, respectively. b, b' = E, AB, do., = the thickness of the vertical rib. ff'= tensile and compressive forces of the material, in a square inch of section, as exerted at a distance (a) on opposite sides of the neutral line. For the determination of the neutral line f 6ba2- (b-il) (a-c)2 -f 6' a't2- (b'-i) (a'-c)} And a + a' = D, where 1) is the whole depth of the beam. For moderate strains per square inch f =f'., ba2- (b6-) (a-c)2= b' (D-a)2 —(b'-,-) (D —a-c')2 Moments of extension= f = ba- (b-l) (a-c)3 Moments of compression = b' a'3-(b'-il) (a'-c')3 If W be the weight laid on the middle, and I equal length between supports, =W =.f ba+ba'. -(b_ - (ai)(a-c)- (6b'-) (a'-c.)} 1~ If the form of the Isection be this, Then b' =- Therefore, for the neutral line b a2- (b-l) (a - )2 = (D -a). Moment of extension.=f {ba3_(b-__(a e)} Moment of compression = 3 And W - 3.f ba + a'3 -(-b )(a — C) 4 3a If the form of the section be this, Then b= 8 Therefore, for the neutral line 0 J~ fa2 = b' (D -- )2 - (b' - ) (D - a c)2 f la2 Moment of extension =.K STRAIN AND STRESS OF MATERIALS. 211 Moment of compression b'a' -(6'-A)(a' -')3 4 3a And 4 - 3 - f{ B aS + b' a-3-(b. — B)(d ~,)s} If the form of the section be this, Then, b = and b' = K Therefore, for the neutral line 2a=D or the neutral line is in the middle of the section. Moment of extension 12 N 2 Moment of compression f12 2 T W - 2lB D2 Transverse Strength of Cast-Iron Bars. Breaking Ultimate Mean of Length of Bar between supports, with its weight laid deflexion in experidimensions. on middle. inches. ments. lbs. 4~ feet, with 1 inch square 440 1'79 3 9 feet, with 2 inches square 1338 3'0035 6 131 feet, with 3 inches square 2861 4' 667 5 61 feet, with 3 inches square 6117 1'2916 3 From the three last experiments we find 2_f= 1490. 3 2 D2.'. W- 1490 x For a cast-iron beam, where W is the breaking weight in lbs., B is the breadth of the beam measured in inches, D the depth of the beam measured in inches, and l the length of beam between supports measured in feet. 212 STRAIN AND STRESS OF MATERIALS. The best dimensions of a beam, whose section is given in the figure, are when the bottonl flange contains six- times as much area as the top flange. And the breaking weight of such beams may be found by the following admirable rule: ~/z2X//.//y//X//// z;//~: Multiply the sectional area of the bottom flange in square:inches, by the depth of the beam in inches, and divide the product by the distance between ithe supports, measured in feet, then 2'14 times the quotient will giv-e the breaking weight in tons. A cast-iron bar is not weakened by passing half the breaking weight over it 96,000 times, with a velocity of 81 feet per minute. Deftection of Beams. Let the beam be supported at A and B, and weight TW applied at the middle C. 3 13 D DCC E / = the modulus of elasticity of the material. -= breadth of beam in in.'- TAW D = depth of beam in inches. = I length of beam in inches. c Let the beam be supported at A, and a weight TIV applied at. the other B extremity. BC' 4 "1} Rule for finding the ultimate deflexion of a cast-iron beaus: Ultimate deflexion -D C in inches - for first figure. 40.D 612 Ultimate deflexion B C in inches = D for second figure, where I is measured in feet and D in incihes. These values f'or the ultimate deflexion are Independent of the breadth'of the beam. Find the ultimate deflexion of a east-iron bar, the distance between the supports being 24 feet, and depth 4~ inches. STRAIN AND STRESS OF MIATERIALS. 213 3 12 3 x 242 Ultimate deflexion = = 9 — 6 inches. 40D02 40 x 4 If the weight Wbe n-iforrnly distributed along the beam, the deflexion will be in all c es l of the deflexion which is produced by the weight acting or he middle, or in the case of having only one support, acting at the extremity. Traisverse -Flexure of a W]rought-Iron Bar by Pressure acting Ho6rizontally. Length of bar 14 feet 7. inches, depth of bar in direction of pressure 1'515 inches, breadtll 5'523 inches, distance between supports 13 feet 6 inches. The experiment was continued to the limit of perfect elasticity, or to that point at which the elasticity was sensibly injured. Weight applied, acting Deflexions after Sets after five Ratio of weights to hiorizontally.' five minutes. minutes. deflexions. lbs. inches. inches. 28 -051 *0 549 56 112 0 500 112 ~ 232 O 483 168 - 344 -001 488 224 -458 002 489 280 - 571 - 003 490 336 ~684 ~003 491 392 -800 ~004 490 448 - 916 ~ 006 489 c504 1.005 007 501 " 560 1- 124.008 498 C 6-16 1 222' 010 504 0 672 1-332' 011 504 728 1-434 -017 508 784 1'547'019 507 840 1-693' 019 496 896 1-823 -019 492 952 ] -933 -020 493 1.008 2 044 -021 493 1064 2-165 -022 491 214 STRAIN AND STRESS OF MATERIALS. To find the weight which a wrought-iron beam is capable of bearing without injuring its elasticity. 1073 3 D2 fD2 W_,$ — lbs. = ~ tons, nearly. B and D are measured in inlches,' and l in feet, being the distance between the supports. What is the weight that can be laid on a wrought-iron bar, 20 feet long,.3 inches broad, and 6 inches deep, without injuring its elasticity? 3 x 36 108 40 40 The deflexion of a wrought-iron beam, supported at each end, and loaded in the middle, when the elastic limit is obtained. 12 Deflexion in inches -'0167 x The length, 1, is measured in feet, and D, the depth, in inches. Taking the bar given in the last example, 400 Deflexion ='0167 x - 111 inches. 6 A X~ Hollow Rectangular Beams.,/////,, Let A B CUD be the section of a hollow reetangular beam. Let A D = D, anda cd d A B- =B, andab —= b 3 D e' where W is tile weight applied at the middle /. between the supports, andf is a constant dependD~ mfi f, 2 iing on the nature of the material. FLUID FOR ETCHING ON CoPPER.-Verdigris 4 parts; salt 4; sal aminoniac 4; alum 1; water 16; strong vinegar 12. Dissolve with heat. AcrD FOR ETCHING ON STEEL.-Pyroligneous acid 5 parts; alcohol 1; nitric acid 1. Mix the first two, then add the nitric acid. STRAIN AND STRESS OF MATERIALS. 216 TAB LE Of Experiments on the Transverse Streingth of Rectanzglar Tubes of Wroeught-Iron, -spported at each end, and the weight laid on the middle. Breaking Weight of tubes weights, exciin- External External Thickness of Distance between between the sup- sive of the rdepth of breadth the plates of the suppolrts. ports. weights ofthe the ofthes the tubles. tubes. tnbes. tubes. Feet. Tons. Inches. Inches. Inches. 30{0 42' 62 cwt. 57.' 24 16 * 525 7*s 72 2 36 lbs. 4.54 6 4'1325 30-0 23-09 cwt. 22- 84 24 16 *272 7*5 35'53 lbs. 1 409 6 4 I065 3'75 9'65 " 1. 1 3 2 061 3-75 4-34 "'3 3 2 103 4510) 130'36 cwt. 114'76 36 24 75 3 75 9 65 lbs. 1I1 3 2 1061 30s0 39 cwt. 54.3 24 16 150 In several of-these experiments the tubes gave way by the metal at the top becoming wrinkled. In similar tubes the strength, and consequently the breaking weight, is proportional to (1'9) power of the lineal dimensions. From these experiments the breaking weight may be obtained as follows: W=4lD BaS —bd:in tons. The breadths and depths are measured in inches, and the length in feet. If the thickness of the metal be equal to t inches completely round the section, Then, W-= B 3-(B 2 t) (D- 2 )I the breaking wreight in tons for a wrought-iron tube, whose form of section is..... What is the breaking weight of a rectangular tube 40 l I feet long, depth 2 feet 6 inches, thickness of plate z inch, and breadth 18 inches? / W= 4800- 1s 801 -- 5 x x29.53 4800 216 STRAIN AND STRESS OF MATERIALS, -1 {01 486000- 449267 = 22'96 tons.'From a great numbler of well arranged experiments, on the sti ength of iron beams and tubes, it follows that they may be safely:readuced in -strength from the middle towards the extremities in the ratio indicated by theory. I)'C Let A B be a beam supported at its extremities E and F, and put F equal to the necessary strength at the middle of the beam. i (1C2 CID2 Then, the necessary strength atD -x A C2 The tensile force of wrought iron is to its compressive force as 2 to 1. Hence, the plate on the upper side of hollow wrought-iron tubes should contain an area twice as great as the plate on the under side. Strength of Cast-Iron. Pillars. The breaking weight of solid cylindrical cast-iron pillars. In solid pillars, with their ends rounded, and rnoveable, d436 Breaking weight in tons- 149 x... (1) In solid pillars, with their ends flat, and incapable of motion, Breaking weight in-:tons = 4416. x (2)* where I is the length of pillar'in:feet, anwd d the diameter in inches. In hollow pillars of cast-iron, where D, d are the external, internal dia-meters, and l the length: both ends of the: pillar were moveable. _D3.6 _ d3;G Breaking weight in tons = 13 x di~ In hollow cast-iron beams, whose ends were flat'and firmly fixed, Br'eaking weight in tons = 44'3 Of three cylindrical'pillars of steel, wrought and cast iron, and w:ood, all of the same length and diameter, the first- having its ends * Formula (1) was obtained from the mean result of'eighteen pillars, varying in length from 121 times the diametier daw to 1-5 times.'The formula (a) \Vas derived from eleven pillars, with flat ends, varying in length from 78 to 25 times'the diameter. STRAIN AND STRESS OF MATERIALS. 2 1 rounded, the second with one end round and the other end flat, and the third with both ends flat, the strengths are as 1, 2, and 3. These formula and results were obtained from experiments on pillars, varying in length from 121 times the diameter down to 15 times. Effects of Temperature upon the Strength of Cast-Iron. The strength of cast-iron is not reduced when its temperature is raised to 600~, which is nearly that of melting lead; and it does not differ very widely whatever the temperature may be, provided the bar be not heated so as to be red hot. EXA.MPLE. Find the strength of a hollow cylindrical cast-iron pillar, 14 feet long, 6-2 inches external diameter, and 4'1 inches internal; the pillar being flat and well supported at the ends. 141'7 = 88-801 6'23'6 — 712'22 and'4-13" - 1607.'. Breaking weight in tons =44-3 x -' d3 /1.7 443 x 12'22 - 160'7 88'801 = 275 Comparative Strelgqth of Long Pillars. If the strength of cast-iron pillars be 1000, then wrought-iron will be 1745, cast-steel 2518, Dantzic oak 108-8, and red deal 78'5. The strength of similar pillars is as the square of their linear dimensions. Resistance to Torsion. Let I = length of prism from the fixed end to the point of application of the lever used to twist it. r = radius of prism, if round. b, d = breadth and thickness, if rectangular.;V= the Weight acting by means of the lever to twist the prism. L = length of the lever to which the weight W is applied. 0 = angle of torsion. R = resistance to torsion at the time of fracture. C = constant for each species of body. 7r-= 3'14159, &c. For a cylinder, 2 LI W-= C rOr4 and2 WFL- IRr. For a square, 6 L I W= CO d4 and 6 WL= /21R d3. For a rectangle, 3 I W(b2 + d2) C 0 b3 dSan&3: Wi:L V4b2 + d2= R b'd - -- "-'-~ - --— 19 218 STRAIN AND STRESS OF MATERIALS. The Ultimate Resistance of a Cast-iron Beam to Torsion. In a cylinder, W:L - 51055 r3. In- a square prism, WL = 7660 d3. b5 d2 In a rectangular prism, W L = 10834 Vb,+d2. All the dimensions are taken in inches. Strength of Ropes. The cohesion of hempen fibres is 6400 lbs.- for every square; inch of section. circumference squared Breaking weightin tons = the circumference being measured in inches. Ex. —Find the breaking weight of a rope 6 inches in circumerence. 36 Breaking weight = - = 9 tons.!Fr a common cable, circumference squared Breaking weight in tons circumference squared.5 These axe practical rules and easy of application. PROCESSES FOR STAINING WOODS. Mahogany Color (1Dark).-BJoil 4 lb. ot maclder ana z oz. oI log-.woocl in a gallon of water; then brush the wood well over with Athe hot liquid. When dry, go over the whole with a solution of 2 drachms of pearlash in a quart of water. Mahogany Color (Ligtt).-Brush over the surface with diluted initrous acid, and when dry apply the following, with a soft brush::Dragon's blood, 4 oz.; common soda, 1 oz.; spirit of wine, 3 pints.'Let it stand in a warm place, shake it frequently, and then strain. -Repeat the application until the proper color is obtained. To Stain Maple a Mahogany Color.-Dragon's blood, ~ oz.; alkanet, i- oz.; aloes, 1 dr.; spirit of wine, 16 oz. Apply it with a sponge or brush. Rose wood.-Boil 8 oz. of logwood in 3 pints of water until reduced to half; apply it; boiling hot, two or three times, letting it dry between each. Afterwards put in the streaks, with a camel's hair pencil, dipped in a solution of copperas and verdigris in a decoction of logwood. LOGARITHMS. 219.Ebony. —Wash the wood repeatedly with. a solution of sulphate of ironi; let it dry, then alpply a hot'decoction of logwood and nutgalls for two or three times. When dry, wipe it With a wet sponge!; and when dry again, polish- withl linseed oil. Red. —. Take a pound of Brazil wood and mix it with a gallon of stale urine. Pour over the wood while boiling hot. Before it dries it should be laid over with alum water. 2. A fine red may also be obtained by a solution of dragon's blood in spirits of wine. Yellow.-Nitric acid, lightly diluted, will produce: a fine yellow on wood. Sometimes, if the wood is not in proper condition, it will create a brown.' Care must -be taken that the acid used be not too strong, or it will render the wood nearly black. Blue.-Take of alum 4 parts; water 85 parts. Boil. Purple.-To produce this color, take of logwood 11 parts; alum 3 parts; water 29 parts. Boil. Mahogany. —l. Linseed oil 2 pounds; alkanet 3 ounces. Heat them together and macerate for six hours, then add resin 2 ounces; beeswax 2 ounces. Boiled oil may be advantageously used instead of the linseed oil. 2. Brazil-wood (ground); water sufficient; add a little alum and potash. Boil. i 3. Logwood 1 part; water 8 parts. Make a decoction, and apply it to the wood; wvheh dry, give it two or three coats of the followinfg varnish: dragon's blood 1 part; spirits of wine 20 parts: Mix. To take Stains out of MIahogany. —Spirits of salts l; parts; salt of lemons 1 part. Mix, then drop a little on the stains, and rub them until they disappear. To Stain MHusical fInstrduments:.-Crimson: Boil one pound of ground Brazil wood in three quarts of water for an hour; strain it, and add half an ounce of cochineal:; boil it again for half an hour gently, and it will be fit for use. Purple: Boil a pound of chip logwood in three quarts of water for an hour; then add four ounces of alum. L O G A R ITH X S. Logarithms literally signify ratios of numbers; hence Logarithmic Tables may be various, but those in common use for the facilitating of arithmetical operations generally are of the following corresponding progressions, viz.:Arithmetical, 0, 1, 2. 3. &c., or series of logarithms. Geomretrical,, 10, o, oo, oo, &c., or ratio of numbers. And thus it may be perceived, that if the log. of 10 be 1, the log. of any number less than 10 must consist wholly of decimals, because increasing by a decimal ratio. Again; if the log. of 100 220 LoGARLTIMIS. be 2, the log. of any intermediate number between 10 and 100 must be 1, with so many decimals annexed; and in like manner, the log. of any intermediate number between 100 and 1000, must be 2, with decimals annexed proportionally, as before. Application and Utility of Common Logarithmic Tables. The whole numbers of the series of logarithms, as 1, 2, 3, &c., are called the indices, or characteristics of the logarithm, and which must be added to the logarithm obtained by the table, in proportion to the number of figures contained in the given sum. Thus. suppose the logarithm be required for a sum of only two figures, the index is 1I if of three figures, the, index is 2; and if of four figures, the index is 3, &c.; being always a number less by unity than the number of figures the given sum contains. EXAMPLES. The index of 8 is 0, because it is less than 10. The ihdex of 80 is 1, because it is less than 100. The index of 800 is 2. because it is less than 1000. The index of 8000 is 3, because it is less than 10,000, &c. The index of a decimal is always the number which denotes the significant figure from the decimal point, and is, marked with the sign, thus, -, to distinguish it from a whole number. EXAMPLES. The index of'32549 is - 1, because the first significant figure is the first decimal. The index of'032549 is - 2, because. the first significant figure is the second decimal. The index of'0032549 is- 3, because the first significant figure is the third decimal, &c., of any other sum. If the given sum-for which the logarithm is required contains or consists of both integers and decimals, the index is determined by the integer part, without having any regard to the other. 1. To find the logarithm of any whole number under 100. Look for the number under N in the first page of any Logarithmic Table; then immediately on the right of it is the logarithm required, with its proper index. Thus the log. of 64 is 1'806180, and the log. of 72 is 1'8573332. 2. To find the logarithm of any number between 100 and 1000, of any Semn not exceeding 4 fi/ures. Find the first three figures in the left-hand column of the page under N, in which the number is situated, and the' fourth figure, at the top or bottom of the page; then the logarithm directly under the fourth figure, and in a line with the three figures in the column on tile left, with, its proper index, is the logarithm required. Thus, the log,. of 450 is 2:653213, and the log. of 7464 is 3 872972. Or, the log: of 378'5 is 2'578066, and that of'7854 is- 1'895091. LOGARITHMS. 221 3. To find the number indicated by a given logarithmn. l Look for.the decimal part of the given logarithm in the different. columns, and if it cannot be found exactly, take the next less. Then und:er N in' the: left-hand column, and in a line with the logarithm found, are three figures of the number required, and on the top of the column in which the found logarithm stands is one figure more; place the decimal point as indicated:by "tlie logarithmic index, whiqh determines the sum, properly valued, as required. If the logarithm cannot be found exactly in the tables, subtract from it the next less that can be found, and divide the remainder by the tabular difference; the quotient will be the rest of the figures of the given number, which, being annexed to the tabular number already found, is the proper number required. Ex. Required, the number answering to the logarithm 3'233568. Given logarithm... 3233568 Next- less is the log. of 1712=-3-233504 Remainder..... 64 64 Tab. Diff. = 253, and - 2'5 253 HIence the number required = 1712'25. For practical purposes in mechanics, logarithms are seldom resorted to, unless for the raising of the powers of numbers or! extraction of their roots. These operations, when tables are at hand, they very much facilitate; involution or the raising of powers, being:performed simply by multiplication, and evolution, or the extraction of roots, by division, as in simple arithmetic. Ex. 1. Required, the square or second power of 25-791. Log. of 25'791 - 1411468 Multiplied by 2 the power required. Logarithm 2,822936 indicated number or square required - 665'175. Ex. 2. What is the cube of 30'7 146? Logarithm = 1'48'7345 Multiplied by 3 the power required. Logarithm 4-462035 indicated number or cube required - 28975'7..Ex. 3. Required, the square root of 365. 2-562293 Log. = _ — 1'281146 indicated number or root = 19'105. 19* 222 LOGARITHMS. EX,. 4. Find the cube root of 12345. Log. =401491 = 13,63830 indicated number or root = 23,116. For TABLE OF LOGARITHMS, see p. 483. ENGRAVING IN ALTrO-fELZEVO.-In the common operation of engraving, the desired effect is produced by making incisions upon the copper-plate with a steel instrument of an angular shape, which incisions are filled with printing-ink, and transferred to the paper by means of a roller, which is passed over its surface. There is another mode of producing these lines or incisions, by means of diluted nitrous acid, in which the impression is taken in the same iway. Another method of engraving is done upon a principle exactly the reverse, for instead of the subject being cut into the copper, it is the interstice between the lines which is removed by diluted aquafortis, and the lines are left as the surface, from which the impression is taken by means of a common type-printing press, instead of a copper-plate press. This is effected -by drawing with common turpentine varnish, covered with, lampblack, whatever is required upon the plate; and when the varnish is thoroughly dry, the acid is poured upon it, and the interstice of course: removed by its action upon the uncovered I part of the copper. If the subject is very full of dark shadows, this operation will be performed with little risk of accident, and I with the removal of very little of the interstice between. the lines; but if the distance between, the lines is great, the risk and difficulty is very much increased, and it will be requisite to cut away the parts which surround the lines with a graver, in order to prevent the dabber with the printing-ink from reaching the bottom, and I thus producing a blurred impression. It is obvious, therefore, that the more the plate is covered with work, the less risk there will be in the preparation of it with the acid, after the subject:is drawn, and the less trouble will there be in removing the interstice, if any, from those places where there is little shading. GLASS, SOLUBLE.-Mix ten parts of carbonate of potash, fifteen of quartz (or of sand free from iron of alumina), and one part of charcoal. Fuse together. The mass is soluble in four or five parts of water; and the filtered solution evaporated to dryness: yields a transparent glass, permanent in the air. 6 feet. 5 feet. 5 feet. 4| feet. 4 feet. Radius of beams. hu.:a cc Co c C O W ca cW w, ". cD cs. Parallel bars in L-D CID W co tll.:clopll LIZ) t~~ tl) LIZ) pr *'feet.. feet.,.Or~~~~~~~~~~' c. A' Cr <> <> ~> is CS ~ I~~:r~ 3~ OO~ - ~C Length of radius'.o c z:rl o-..r c:P o rods-in ieet and. 0 C) to _os _-.t.. inches.. 8 feet. 7 feet. 7 feet. 6- feet. Radius of beams e 6~ feet. in f ~et.n. c - P F FP A CO CO P tP COj c c P CCO: C: CC C A s as c C sc Parallel bars in c k& Ip~33L~P~r~h-L 1P130 N~r IP~L IPESH tp4: 14- O q4o — L 0+1< ppo Ip,4c 0 -t S t4- 44S tH= P4i1 feet...o L i~ L, C... C'. - o WW W feet. 4- C Length of radius -.. r'ods in feet and cx t) 00, O C1 CO 16 t O C-, 5 co CS-C O Cs C;R C 1l0 M, r l tZ) 00 t-C to C.) co co -4 C<> ~n M co MC., 12 feet. I li feet., I 11 feet.1-0 feet. Radiis of beams. in feet. -;rr. -Z M C Cs7 -r -Zr Cs s CS; C) C s C.) Cs CS VI C CS (:1 OCs 0C c C. Csr C x c c Parallel bars in;A~.0s~i~s~ _. foss~ c~r~OF. 00feet. Length ofradius, - Ce z CO O X) 1 3) 0 FF O Ca O s O | X w r - c- - c o |~ C O o lCS2bo'3 |rods in feet and < A m GO- Cs ~- -,Zt co Cs 0 CM A C) C> C s co GO - to cs VI- WS C c - C CS W inches. s4.sco~~nsf-s J~ onns- ~ sp~ n s. nk ss on-~fson~t s n43 s Os P nSfnn5 nt.SsC 224: CAPILLARY ATTRAC'(ION. CAPILLARY ATTRACTION. If a number of glass tubes, open at both ends, be immersed, the water will'-rise to the same height in each tube, so long as -thediameter of the tube exceeds'the fifteenth of an inch; in all tubes less thali this, the water will rise higher in the tube whose diameter is the least. Such tubes, whose diameters are less than one fifteenth of an inch, are called Capillary tubes, fronm the Latin word capillzs, signifying a hair. Phenomwena of Capillary Attraction. Let P Q R S be a vessel containing water to the line P S. The A- C' )D E 7 G I water will rise in the capillary trubes A B C to. V..,:. l 7.' Is theb heights m n n, o, whicli are'inversely' proportional 0 K A to their diameter. If B be P —,,- S br oken at a, the water will not rise to the top of, -e w it, but will stand at b,'-a Q little below the top, Whatever be'the length oi diameter of the tube. And,, if the tube be taken out of the water and' laid horizontally, the water will recede from the end that was inimmersed.'If a tube D be composed of two different bores, the water will rise to the height p; and if another tube, E, of the same form and size, —be- immersed with'its -smaller end downwards, tile water wvill rise in it to the same heigllt p. If the vessel Fv u be- plunged into water, and by exhaustion the water is raised: to the capillary tube F't i, it will afterwards asceld to tehe height, which is just the same as in' a capillary tube G of the' saiie bore as F t u, and length Fx. In tubes of the same matter, immersed in the same fluid, the product'of the elevations by the diameter is a constant quantity. In a glass t,ube, immersed in water, this constant'has been found' by Muschenbrock,'039; by' Weitbrecht'0428; by Monge,'042; by.Atwood'053. VFrom these numbers, the: diameter of a tube may be found, in which thev water will rise, by capillary attraction, the height 7 inches.'039 Diameter= ~'0056 inches, nearly. The constant quantity, here referred to, is called the modulus of Capillary attraction. The following moduli are from' Brewster; they were obtained CAPILLARY ATTRACTION. 225 Iwith a glass.tube of'0561 of an inch diameter, by means of an improved apparatus: Name of Fluid. Modulus.| Name of Fluid. Modulus. Water,. 0327 Oil of hyssop,.0195 Very hot water, 0301 Oil of rosemnary,. 0193 Muriatic acid,.. 0248 Oil of bergamot,.. 0192 Oil of boxwood,.. 0240 Oil of amber,... 0192 Oil of cassia,., 0236 Oil of anise seeds,. 0192 Nitrous acid, -. 0232 Oil of Barbadoes tar, 0191 Oil of rapeseed,. 0227 Laudanum,. 0191 Castor oil,. 0226 Oil of cloves,. 0187 Nitric acid,.... 0222 Oil of turpentine, 0187 Oil of spermaceti,.. 0220 Oil of lemon,.. 0187 Oil of almonds,... 0217, Oil of lavender,. 0184 Oil of olives,. 0215 Oil of camomile,.. 0184 Balsam of Peru,.. 0212, Oil of peppermint,. 0184 Muriate of antimony, 0209 Oil of sassafras,.. 0184 Oil of rhodium,. 0205 HIighland whisky,.. 0184 Oil of pimento,.. 0203 Brandy,..... 0183 Cajeput oil,.... 200 Oil of wormwood,. 0183 Balsam of capivi,.0200- Oil of dill seed,.. 0182 Oil of thyme,.0199 Oil of ambergris,. 0181 Oil of bricks, distilled 01 Oil of juniper,. 0180 from spermaceti oil, O il of nutmeg,.. 0180 Oil of caraway seeds, *0198 Alcohol.. 178 Oil of rue,... 0198 Oil of savine,.. 0174 Oil of spearm'nt,. 0197 Ether,...... 0160 Balsam of sulphur,.0196 Oil of wine,.0153 Oil of sweet fennel t 0195 Sulphuric acid,.. 0112 seeds,. These experiments were made with a tube, carefully cleaned and dried after each experiment. A dry tube will raise the water to a less height than a wet one. When capillary tubes are plunged into mercury, it falls instead of rising, as is the case with other fluids; and its fall is such, that when it is multiplied by the diameter of the tube, the product is a constant quantity'015 (Cavendish). When water is made to' pass through a capillary tube of such a bore that the fluid is discharged only by successive drops, the tube, when electrified, will-furnish a constant and, accelerated stream; and the acceleration is proportional to the smallness of the bore. A jet of warm water will rise to a much greater height- than a jet of cold water, though the water in both cases moved through the 226 WooDS. same aperture, and was influenced by the same pressure. A syphon which- discharges cold':water o;My "by drops, will- furnish warm water in an uninterrupted stream. Let CEEB,- A D EB, be two plates of glass, having their sides EB joined together with was, and s}D: Ftheir surfaces smoothiand clean; and also: their sides, A D), CF, separated slightly so as to form the angle ABC. If this apparatus be plunged in - a vessel, so'that IN CG represent - tlhe water's surface,- then' the wdater will risee between the'plates- of'glass, by capillary attraction, -to the height FE G, so that the boundary- of the water on the planes FEBC, DEBA, will be the hlyperbolas G E and IE, having for their asymptotes the surface of.the fluid and the line ENH. The height, n mn, to wliich the water will rise,: is regulated entirely by the R saine laws which prevail' in the case of the tubes; calling the' distance, ii o, betweeil the plates thie diameter ofl the tube. c ence the height, nvm, is equal to the height in a tube-whose diamneter is equal to n o6; and so on for any, other' point. All phenonena of capillary attraction are exhibited' equally both in air and in vacuo, and they are entirely independent of tie thlickness of the mateiial composing the tubes and plates.:The elevation and delpression is not proportional to the density of the liqiid; wwater stands much higher'in -a glass tube:than alcohol WOODS. Howz to Polish Wood. Take, a piece of pu-mice-stone and water, and pass repeatedly over the work until the rising of the grain is cut dovwn. Th'en take pow'dered tripoli and'b'oiled linseed oil, and polish the work to a bright surface. To Gather and Preserve Woods. Woods should be gathered and exposed in a dry situation, to a heat of from 900~ to 150' Fah., until sufficiently dry. The larger kinds are more easily,chipped before drying. STEAM-ENGINE. 227 To Preserve Woodwork. Take boiled oil and finely powdered charcoal; mix to the consistence of- paint,,:and give the woodwork two or three coats with it. This composition is well adapted for casks, water-spouts, &c. To produce Figures on Wood. Slack some-.lime. in: stale wine. Dip= a brush in it, and form on the wood figures to suit your fancy. iWhen dry, rub it well with a rind of pork. STEA M -EGINE. To Estimate, by means of an Indicator, the Amount of Effective.Power produced by a Stcam-Engine. Rule. Multiply the, area.of the piston in square inches by the average.force of the steam, in lbs., and by the velocity of the piston in feet per minute; divide.the product by 33,000, and Cloths of the quotient equal the effective power. Ex. Suppose an engine with a cylinder of 37- inches diameter, a stroke of 7 feet, and making.17 revolutions per minute, or 238 feet velocity,:and the average indicated pressure of the steam 16'73.bs.: per square inch; required the effective power. Area _ 1104'4687 inches x 16'73 lbs., x 238 feet. 33000 133'26 x 7' _- 13326,793'282 horse power. 10 To' determine the proper Velocity for the Piston of a Steam-Engine. Rule. Multiply the logairithm of' the zth part of the stroke at which the steam is cut off by 2'3, and to the product of this' add ~7. Multiply the sum by the distance in feet the piston has travelled When the steam is cut off, and 120 times the square root of the product will equal the proper velocity for the piston in feet:per minute. Ex. Let the steam be cut off in, an. 8-feetstroke when the piston has travelled th4 of the length; required its proper velocity. Logarithm of 4 060206 Multiplied by - 23'8 1'384738 To which add.7 2'084382,/~415694"76; - 2'04 x 120 - 245 feet;, velocity per minute. 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 228 PROPERTIES AND EFFECTS OF HEAT. TABLE Of Approximate Velocitiesfor the Pistons of Steam-Engines. CONDENSING ENGINES. NON-CONDENSING ENGINES. Length Number of Length Number of of stroke Velocity in feet revolutionsper of etroke in Velocity in revolutions in feet. per minute. minute. feet. feet per minue. per minute. 2 160 40 1 186 62 2j 1771 351 2 200 50 3 192 32 2j9 212 421 3A 203 29 t 2 2 171 39 4 214 261 3 - 222 37 49 2201 24A 3+ 231 33 5 230 23 4 236 291 5 [ 2836 21 1 4 243 27 6 240 20 5 2471 241'7 245 175 253 23 8 256 16 6 264 22 Of thle Parallel Motion in a Steam-Engine. When the power from the piston is communicated.by means of a beam or lever moving upon an axis, the parallel motion becomes a very important portion of the machine; for then it forms the link of connexion, and by its properties renders the action of alternate circular motion, and reciprocating vertical motion, mutually agreeable, thereby properly insuring to the piston-rod a truly direct line to that of the cylinder; but to effect this, the greatest degree of exactitude, of the various parts is required, otherwise extra friction is created, and the effective power of the engine proportionately diminished. THE PROPERTIES AND MISCELLANEOUS EFFECTS OF HEAT. Linear Expansion of iMetals from 32~ to 212~.-FARADAY. Zinc, 1 part in.. 322 Gold, 1 part in. 682 6 Lead, ".. 351 Bismuth, "'. 719 Tin, pure, ".. 403 Iron, ".. 812 Tin, impure, ".. 500 Antimony, ". 923 Silver, ".. 524 Palladium, ".. 1000 Copper, ". 581 Platinum,'. 1100 Brass,. 584 Flint Glass,..1248 PROPERTIES AND EFFECTS- OF HEAT. 229 TABLE Of the Expansion of Water by Heat. —By DALTON. Temperatute. Expansion. Temperature. Expansion. 12~ Fahrenheit. 100236 1220 Fahrenheit. 101116 22 100090 132 101367 32 100022 142 101638 42 100000 152 101934 52 100021 162 102245 62 100083 172 102575 72 100180 182 102916 82 100312 192 103265 92 100477 202 103634 102 100672 212 104012 112 100880 TABLE Of the Heating Power of various Combustible Substances, exhibiting the utmost quantity of Water evaporated by the given Weights, and the smallest quantity of Air capable of producing total Combustion. DR. URE. Pounds of water Weight ofatSpecies of Combustible. which a pound-, Pounds ofboil- mospheric air at can heat from 0~ ing water evapo- 32 to burn 1 to 212~. rated by i pound.; pound. Smallest qan tity, PerfectlY dry wood, 35 00 636 Smlest qaniy Wood in its ordinary state, 26 00 4 72 4' 47 Wood charcoal,. 7300 13'27 11'46 Pit coal,...... 60'00 10*90 g9'26 Coke,...... 65 00 1 11 81 11' 46 Turf,........ 3000 5.45 4'60 Turf charcoal.. 6. 00 11' 63 9 86 I Carburetted hydrogen gas, 76'00 13'81 14'58 Oil,... Wax........ 78'00 14'18 15'00 Tallow. Alcohol of the'shops.. 5260 9*56 11'60 20 2380 PROPERTIES ANDD EFFECTS OF HEAT. T A.B L'E Of boiling points of water holding variozus proportions of salt in soluttion. Parts of Degrees of Degrees of Degrees of Salt. [Fahrenheit. Reaumner. Centigrade. Saturated solution... 3687 226 6 86-2 107 8 333a4 224 9 85 7 1072.... 6:3030 223' 7 852 1065 "c "'s.2728 22~25 84,7 1058 24'25 22-1 4 84'1 105'2 6. 21'22 220.2i 83'6 104 6 St" t. Is' 18-18 -2 1'9 83 103I 9 6C...* 15'15 217.9 82,6 103-3 46.. L. 12 12 216'7 82,1 102.6'{... e 9'09 215'5 81 6 102 49 Cd... 6 06 214-4 81.1 10113 Sea-water... 30. - 03 2132 80.5 1007 Common water.... 0 212 80 100 Expansion of iqiguids in Jfolzmse from:329 to 2120 Fahrenheit. 1000 parts of water become 1046 " " oil " 1080 mercury " 1018 spirits of wine" 1110 " It" air " 1373 Of the Linear lDilatation of Solids by Heat. Dimensions which a bar takes at 212~, whose lezngth at 32~ is 1'000000. Cast iron,... 1'00111111 Cast brass,... 1 001:87'50 Steel (rod),... 100114470 Silver, 1'0018900 Steel, not tem- 01 7 Tin,.... 10028400 pered,'.. Lead,.... 100284836 Ditto, temper- 100136900 Zinc. 1-00294200 ed yellow,. Glass from 320 } Ditto, at a high- to 212',00086130 er rate,01235 Glass from 2120 1 91827 Iron,.. 100118203 to 920,1827 Soft iron, forged, 1'00122045 Glass from 3920 Golpd,,. 1 001 0000 to 572 ~ 1 Copper,.. 100191000 EXPANSION OF ATMOSPHraIC AIR. 231 Of Capacities of Bodiesfor Heat refeirred to WTater as the Standard. Water,.... 1 0000 Iron,..1300 Olive oil,....7100 Hardened steel,.. 1230 Linseed oil,..... 5280 Steel softened by fire,. 1200 Oil of turpentine,... 4720 Soft bar iron.... 1190 Quicksilver..... 0330 Brass...... 1160 Ice....... 9000 Copper,..... 1140 Pit coal,.....'2777 Zinc,....... 1000 Chalk,....... 2700 Ashes of charcoal,. 0909 Sea salt,... 2300 Silver....... 0S20 Sulphur,.. 1900 Tin...... 0704 Ashes of cinders,. 1855 White lead.... 0670 Black leadj..... 1830 Gold....... 0500 Ashes of elm wood,.. 1402 Lead,..... 0420 T A B L E Of the Exypansioa of Atmospheric Air by Heat. Degrees of Degrees of Deglees of' Fahrenheit. Bulk. Fahrenheit. Bulk. Fahrenheit. Bulk. 832 1000 650 1077 100 11 52 35 1007 70 1089 120 1194 40 1021: 75 1099 140 1235 45 1032 80 1110 160 1275 50 1043 85 1121 180 1315 55 1055 90 1132 200 1364 60 1066 95 1142 I 212 1376 The pressure or gravity of the atmosphere, being equal to a column of water 34 feet ill height, is the means or principle on which rests the utility of the common pump, also of the syphon, and all other such hydraulic applications. In the pump, the ianternal pressure on the surface of the liquid is removed by the action of the bucket; and as by degrees the density becomes lessened, so the.water rises by the external pressure to the above-named height; and at such height it will remain, unless by some derangement of construction taking'place, the atmospheric fluid is allowed to enter and displace the liquid column. But observe, if the temperature of the water or other liquid be so elevated that steam or vapor arise through it, then, according to the vapor's accumulation of density, may the action of the pump be partially or wholly destroyed; and the only means of evasion in such cases is to place the working bucket beneath the surface of the liquid which is required to be raised. 282 DEGREES OF THE THREE TIIERMOMETRICAL SCALES. TABLE Of the Degrees of the three Thernmometrical Scales, Above Boiling Point of Water. P c D C < - 9:0 s, | E I E 392 201601 356 180 144 320 160 1280 284 140 112 248 120 96 3195 6 14 1 155 1 2 8.3 119 391 3 4 319 282 247 198 17 8a 158 a 138 1182 a 5 388 352 316 280 244 39 15 1 142 12 0 60 3971 I 351 3 15 253 247 199 179 3 1579 2 19 84 1197 386 3509 314 278 242 3 158 349 141 231 5277 1 21 9 385 196 115 9 3156 136 941 16 384 348 312 276 240 383 195115611 347 1751140 311 1551124l 275 13510811 239 115 92 382 346 310 274 238 381 194 345 174 309 154 273 134 237 114 380 155 344 139 8 123 272 107 236 91 19 173 307 153 271 133 25113 378 154 3423 13306 122 270 106 2345 90 378 342 13 306 270 234 377 192 341 172 305 152 269 132 233 112 376 191 153 340 171 137 304 151 121 268 131 105 232 l 89 I'! 11 1 1 1i 1 1 375 3391 303 267 231 374 190 152, 338 170 136 3021150 120 266 130 104 230 110 88 373 337 3011 265 229 372 189 1 336 169 b300!149 264 129 228 109 371 335 l3S: 299 119 263 103 227 370 188 334 168 2981148 262 128 226 108 369 150 134 297 118 261 102 225 86 368 187 332 167 296'147 260 127 224 10 7 3671 149 331 133 295 117 259 101 223 85 366 330 294 1 258 222 365 185]14811 329 165 132 293 11451116 1257 1251100 221 105 84 364 328 292 256 220 363 184 327 164 291 144 255 124 219 104 362 147 326 131 290 1151 254 99 218 83 361 183 325 163 289 143 253 123 217 103 360 146 324 130 288 1141 252 98 216 82 359 182 323 162 287 142 1 251 122 215 102 358 145 322 129 286 1131250 97r 214 81 357 181 {321 161 11285 141 249 121 213 101 To convert the Degrees in the, three Scates into each other. To convert Centigrade or Reauntmur's into Fahrenheit's Degrees.-Multiply the number of degrees by 9, divide the product by 5 fbr Centigrade. or by 4 for Reaumur's; add 32 to the quotient, and the sum will be degrees of Fahrenheit. To convert Fahrenheit's into Centigrade or Reaumurr's Degrees. —Subtract 32 from the number of degrees, and divide the remainder by 9; multiply the quotient by 5 for Centigrade, or 4 for Reaumur's; the products will be the required degrees respectively. DEGREES OF THE THREE THERMOTMETRICAL SCALES. 233 Comn1arative Table of the Degrees of the-three Thlermometrical Scales, Fahrt Cent. Rea.' Fahr' t Cent. Rea. Fahr't Cent. Rea. aFnalrt Cent. Rea. IFalhr't Cent. Ren. 212 100 80 1 167 75 60 122 5040 1 7-7 25: 20 32 0 0 211 166 121 16 121 1 210 99 79 165 74 9- 120 49 39 7 24 19 30 - 1 1 209 164 119'74 29 - 208 98 163 73 118 48 3 23 1828 2 207'78 162 72 58 117 8 72 22 27 -.206 161 72 116 71 1 26 205 96 7 160 71 57 115 46 7 70 21 17 25- 4 - 3 204 159 114 69 24 203 95 76 158 70 56 113 45 36 68 20 16 23 - 5 - 4 202 150 112 67 22 201, 9475 156 69 5 111 44 35 1 66 19 21 6 198 7 5 854 0 3 62 15 2 1 - i200.!155 1 10 65 20. I 199 1548 109 64 18 19 1' 19 1153 67 108 4 l 1 1.631 - 197 91 152 107 2 1 17 196 91 51 66 53 1106 4 1 33 61 1613 16 9 7 195 150 105.: 160;' 15.. 194 90 72 1149 65 52. 104 40 32; 59 15 12 1 4 -10- 8 1938 148 103 I 58 I 13 1 192 8971 1:47 64 1 102 39 314 7 14 11 -141 191 1346 101 3 56 1 1 190 88 111 5 63 100 38 5513 10-1 1 70 30 1 10 -14 189 70 14 62 50 99 374 1 3 12 1 3 13 184 139 94. 49 4 18384967 1138594: 9:.' 27 489 3 -16 182 1 3 3;7 i. 92 47 2 -13 181.83 658 91.33 46 8' I -17' 63 6 46'G26 6 0 - 14 180.8266 135 46 9032 5 0 179 183 8 9 44 178l 8 65 133 56 88 3 24 6 5 _2.1915 177 I 132 87 42 171~6 80 64 131 55 4: 86 30 24 41 5 4 -4 -20-16 175 130 85 40 5 174'9 63 129 54 84 29 39 4 6 -21[ 173 128 83 8 38'7.1 17,'78 127 53 82 28 37 3 8 -22 62 12 42 81 22 2 2 - -23 169:7-6,61 19 124 51 41'9:26 21 34 1] 11 -241 16:8.1~,123'78 33 -12 d ~~~~~~~-13 -25-20 20* 234 WEIGHT OF SUBSTANCES OF'CONSTRUCTION. TABLE of the Weight of Substances of C6onstruction, showing- the weight of a cubic inch, and a cubic foot, in ounces andpounds avoirdupois, and also the number of cubic inches in one pound, of the! substances most used in construction. Weight of a cubic foot. Weight of a cubic inch. Names of Bodies. - Number of cubic inches in oz. in lbs. in oz. in Ibs. in a pound. Copper, cast,. 8788 549'25 5'086'3178 3'146 Copper, sheet,. 89'1'5 557 18 -; 159 *3225 3.103 Brass, cast,.. 8396 524'75 4 852 3037 - 3'293 Iron, cast,. 7271 445 43 4'203 263 3802 Iron, bar,.. 7631 47693 4410 276 3623 Lead,. 11344 709'00 6456'4103 2'437] Steel, soft,. 7833 489c56 4527'2833 3'530 Steel, hard,. 816 488 50 4517 2827 3537 Zinc, cast,. 7 1-90 449'37 4156'26 3'845 Tin, cast,.. 7 292 - 455'75 4-215 *2636 3790 Bismuth,... 9880 61950 5 710 *3585 2 789 Gun-metal,'. 8784 54900 0075 3177 3147 Sand,... 1520 95'00 *8787 -055 18190 Coal,.. 1250 78-12 7225 0452. 22'120 Brick,.... 2000 12500 1] 156 - 0723 13.824 Stone, paving,. 2416 151 00 1'396 -0873 11 443 Slate,... 2672 16700 1544 -0967 10-347 Marble,... 2742 171-37 1 585 -0991 10-083 White lead,. 3160 197 50 1-826 -1143 8-750 Glass,. 2880 180-00 1-664'1042 9'600 Tallow,... 945 59-06 -5462'0087 29-258 Cork,. 240 15'00 138 0197 115'200 Larch,.. 544 34'00 315 0201 50-823 Elm,. -.-. 556 34'75 321 0201 49-726 Pine, pitch,. 660 41-25 382 024 41-890 Beech,.. 696 4350:403 0252 39-724 Teak,... 745 46'56 431 -027 37-113 Ash,. -. 760 47'50 440 -0275 3.6'370 Mahogany,. 852 53-25 493'0308 32-449'Oak,,.. 970 60'62 561 -0351 28-505 Oil of turpentine, 870 5437'503 0315 31-771 Olive oil,.. 915 57-18.529 -0331 30-220 Linseed oil,. 932 58'25 539 0337 29'655 Spirits, proof, 927 57:93 536 03352 29-288 Water, distilled, 1000 60-50 -578 -03617 27-648 sea,.. - 1028 64 25 594 -0372 26-894 Tar,....' 1015 63-43' -87 -0367 27'242 Vinegar,. 1026 64-12 593 -037 26-949 Mercury,.. 13568 848 00 7 851 -4908 2,037 I _: — ____848'001'4908 | 2'037 [ SOLDERS. 235 Conducting Power of Materials used ini the Construction of Houses. As observed by MS. Hutchinson. Slate,.... 100 Oak wood,..... 3366 Keene's cement, 19-01 Asphalt,...... 4519 Plaster and sand, 18-70 Chalk (soft),.. 5638 Plaster of Paris,. 20'26 Stock brick,.6014 Roman cement,. 20-80 Bathstone,.... 61'08 Beech wood,. 22'44 Fire brick,..1.... 70 Lath and plaster,. 25'55 Lead....... 521-34 Fir w'ood,... 2760 Air and gases are very imperfect conductors. Heat appears to be propagated through them almost entirely by conveyance, the heated portions'of air becoming lighter, and diffusing the heat1 through the mass in their ascent as in liquids. Hence, in heating a room with hot air, the hot air should be introduced at the lowest! part. The advantage of double windows for warmth depends, in a great measure, on the sheet of air confined between theim through I which heat is very slowly transmitted. Capacity of Bodiesfor Transmitting Heat. The capacity which bodies possess of transmitting heat, does not depend upon their transparency; or bodies are not all transparent to heat in the same proportion that they are transparent to light. The following plates of an equal thickness of'1031 inches allowedvery different proportions of heat to pass through them. Of 100 rays transmitted from an Argand oil lamp there were: Rock salt,.... 92 Emerald,....... 29 Mirror glass,.... Gypsum,.. 20 Rock crystal,.. 62 Fluor spar,.... 15 Iceland spar,... 62 Citric acid,.... 15 Rock crystal, smoky & brown 57 Rochelle salt,. 12 Carbonate of lead,.... 52 Alum,... 12 Sulphate of barytes, -... 33 Sulphate of copper, 0 SO L D E R S. -For Lead.-Melt one 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,- 1; bismuth, 1. Melt thenl together and run them into slips. Resin is also used with this solder. For Gold.-Pure gold, 12 parts; silver, 2; copper, 4. For Brass.-Brass, 2 parts; zinc, 1. For Iron.-Good tough brass, with a small quantity of borax. For Pewter.-Bismuth, 2 parts; lead, 1; tin, 2. For Copper.-Copper, 2 parts; zinc,' 1. For Silver. —Silver, 5 parts; brass, 6; zinc, 2. Hard Solder.-Copper, 2 parts;'zinc, 1. Soft Solder.-Tin, 2 parts; lead, 1 part. 236 GRADATIONS OF TEMPERATURE. TABLE T A B L E Ofproportions for making Shafting with Half-lap Couplings, shou*ng length of Neck and sizes of Coipling-box. (Mlanichester Rules.) Diameter of Length of Diameter of Len Length of Diameter of Neck. Neck. Coupling. Lap. Box. Box. Inches. Inches. Inches. Inches. - Inches. Inches. 2 4 38 2i 5 5+ 2+ 4+ 3i 21 6 6 2+ 5 4 3 6+ 6 21 5I 4+ 3+'7'7 3 6 4+ 3' 7+ 7+ 3+ 6+ 5 3 8 8s 3+ 6- 4'7 6 4 8+ 9+ 4b t ~6+ 4+ 9 10+ | 5 8 f7 6 9+ 1II a j 81 8+ 5+ 11 12+ 6 9 9 6 12 13+ 6+ 9+ 9+ 6+ 13 14i'7 10+ 10+'7+' 1,4 16 7 11 - 8 12 12 8 16+ 18 8+: 12+ 12+ 8+ 17 19 9 13+ 1-3 9 18 20 9+ 14 I10 14+ 14:10 18 22 11 15 16 11 20 24 12 16 17 - 12 21 26 Gradatiaons of Temperature. The following are interesting facts in the range of temperature: 1660 Greatest artificial cold. (Faraday.) 150 Liquid nitrous oxide freezes. 122 Liquid sulphuretted hydrogen fieezes. 105 Liquid sulphurous acid freezes. 91: Greatest artificial cold measured by Walker. 56 Greatest natural cold observed by a "V erified" thermometer. (Sabine.)'70 Greatest natural cold observed at Fort Reliance by Back. ] (Doubtful. ) B 58 Estimated temperature of planetary space. (Fourier.) 447. Sulphuric ether freezes. 15 39 Mercury freezes. 30 Liquid cyanogen freezes. (Faraday.) 13 Mean temperature at the Pole. (Arago;). 11 A mixture of two-parts alcohol and one part water freezes. 7 A mixture of equal parts alcohol and water freezes. GRAI)ATIONS OF TEMPERATURE, 237 / 20~ Strong wine frieezes. 28 Vinegar freezes. 30 Milk freezes. 32 Ice melts. 4[ Mean temperature at Edinburgh. 50'7 Mean temperature of London. 60 Mean temperature at Rome. 81'5 Mean temperature at the equator. 98 Heat of the human blood. 98 Ether boils. 100 Phosphorus melts. 173 Alcohol boils. 117 Highest natural temperature observed of a hot wind in Upper Egypt. (Burckhardt.) 133 Wood-spirit boils. 142 Spermaceti melts. 151'31 Beeswax melts. 212 Water boils. 226 Sulphur melts. 2 242 Nitric acid boils. 283 A compound of equal parts of tin and bismuth melts. I 442 Tin melts. 460 The surface of polished steel acquires a pale straw color. Q 476 Bismuth melts. 0 554 Phosphorus boils. 560 Oil of turpentine boils. 580 The surface of polished steel acquires a uniform deep blue. 590 Sulphuric acid boils. (Dalton.) 594 Lead melts. 600 Linseed oil boils. 635 -Lowest ignition of iron in the dark. 662 Mercury boils. 700 Zinc melts. 752 Iron bright red in the dark. 810 Antimony melts. 884 Iron red hot in the twilight. 1077 Red heat fully visible in the daylight. 1141 Heat of a common fire. (Daniell.) 1869 Brass melts. 1873 Silver melts. 1996 Copper melts. 2016 Gold melts. 2500 Steel melts. 2786 Cast-iron melts. 3080 Platinum melts. The line of perpetual congelation has a variable altitude in different climates. 238 PROPERTIES OF NUMBER1S. At the equator it is 14760 feet. At the Alps " 8120 " In Iceland " 3084 " At the polar regions ice is perpetually observed at the surface of the earth. PROPERTIES OF NUSMBERS. 1. A PrTime NiYmber is that which can only be measured by 1 or unity. 20 A Colmposite- Nunmber is: that which can be measured (or divided without a remainder) by some number greater than unity. 3. A Perfect Number is that which is equal to the sum of all its divisors, or aliquot parts: thus 6 _. +.6 +. 4. If an odd number divides all even number, it will also divide the half of it. 5. If the last digit of any number be divisible by 2, the whole number is divisible by 2. 6. If the two last digits be divisible by 4, the whole number is divisible by 4. 7. If the three last digits be divisible by 8, the whole number is divisible by 8. 8. If a number terminate with 5, it is divisible by 5; and if it terminate with 0, it is divisible by either 10 or 5. 9. If the sum of the digits constituting any number be divisible by 3 or 9, the wahole is divisible by 3 or 9; and if also the last digit is even, the whole number is divisible by 18. 10. If the sum of the digits of a-ny number be divisible by 6, and the right hand digit by 2, the whole is divisible by 6. 11. If the sum of the 1st, 3d, 5th, &c., digits of any number be equal to that of the 2d, 4th, 6th, &c., that number is divisible by 11. Thus 327943 contains 11 = 29813 times exactly. 12. If a square number be either multiplied or divided by a square, the product or quotient is a square; and conversely, if a square number be either multiplied or divided by a: number that is not a square, the product or quotient is not a square, 13. The product arising from two different prime numbers cannot be a square number. 14. The product of no two different numbers prime to each other (that is, 1 being the common measure) can make a square, unless each of those numbers be a square. 15. The square root of an integral number, that is, not a comnplete square, can neither be expressed by an integer nor by any rational fraction; so with the cube root of an integer. 16. Every prime number greater than two, is made: up of 4 times some number, + 1 or - 1.; that is, of one of the forms 4w + 1, or 4n- 1. PROPERTIES OF 1 NUMBERS. 239 17. Any prime number greater than 3, divided by 6, will leave a remainder of 1 or 5: that is, every number greater than 3, is one of the forms 6n + 1, or 6n- 1. 18. The number of prime numbers is infinite. 19. A square number cannot terminate with an odd number of cyphers. 20. If a square number terminate with 4, the last figure but one will be an even number. 21. If a square number teirminate with 5, it will terminate with 25. 22. No square number can terminate with two equal digits,;except two cyphers, or two fours. 23. No number whose last digit is 2, 3, 7, or 8, is a square number. 24. If a cube number be divisible by 7, it is also divisible by the cube of 7. 25. The difference between any integral cube and its root is always divisible by 6. 26. Neither the sum nor the difference of two cubes can be a cube. 27. A cube number may end with any of the natural numbers. 28. All the powers of:any number that end with 6, will terminate with 6: so with the numeral 5,. TABLE Of the first Nrine Powers of the first Nine Numbems. 1st 2d S3d:{ 4th th 6th 7th 8th 9th 2 14 8 16 32 64 1288 256 51 12 3 9 27 81 243 729 2187 6561 19683 4 16 64 256 1024 4096 16384 05536 262144 6 25 125 625 3125 15-625 78126 390625 1953125 6 36 216 1296 7776 46656 279936 1 9616 10077696 7 49 343 2401 16807 117649 823543 5764801 40353607 8 64K 12 4096 32768 262144 2097 152 16777216 134217728 9 811 729 6561 59049 531441 4782969 43046721 387420489 24)0 USEFUL NUMBERS. T A B L E Of Useful Ntumbers.' c..... 8'1415927 V/'2 -.. - 1'4142136 Log....... 4971499. 7071068 Log. i-r.. 11447299 2. * 44428829 1 7r.... 0'3183099 2'2214415......9.8696044. 0 4501582 1 if 1. 013212, 1'2533141./.'P -...17724538..2.. 2533141 1.. / 0'5641896 0 7978846........... -- 2'7182818 Log...... 04342945 Modulus of common logarithms......' 434294482 Log. of ditto......... 96377843 g............... 3219084 v~f-............... 5.67363 Logr....... 1'5077222 Inches in a French metre.....9... 39 3.7079 Log. of ditto........... 15951741 Feet in ditto....,..,. 3'2808992 Log. of ditto........... 05159929 Square feet in the square mrtre...... 10764297 Acres in the Are......... 0024711 Lbs. in a kilogramme....... 2 20548 Log. of ditto.. 03435031 Imperial gallons in a litre...... 02200967 Lbs. per square inch in 1 kilogramme per square millimetre.1422 Cwts. ditto, ditto........... 12'7 Volume of a sphere whose diameter is 1... 05235988 Arc of 1~ to rad. 1. 0017453293 Arc of 1' to rad. 1..... 0000290888 Are of 1" to rad......... 0000004848 Degrees in an are whose length is 1..... 57 2957800 Grains in 1 oz. avoirdupois........ 437 SURFACE OF BOILERS' TUBES. 241 Grains in 1 lb. ditto......... 7000 Grains in a cubic inch of distilled water, Bar. 30 in., Th. 62~....... 252-458 Cubic inches in an ounce of water..... 1'73298 Cubic inches in the imperial gallon... 277'276 Feet in a geographical mile...... 6075-6 Log. of ditto............. 37835892 Feet in a statute mile........ 5280 Log. of ditto.........3..'37226339 Length of seconds' penldulum in inches... 39'19084 Cubic inches in 1 cwt. of cast iron...... 430-25 -. " c " 1Bar iron..... 397-60 <( " " Cast brass.. 368'88 cc " " Cast copper..... 352'41 cc " " Cast lead..... 27280 Cubic feet in 1 ton of paving stone... t, 14'835 " " Granite.... 13-505 " " Marble.13070 | " " Chalk.. 12'874 " " Limestone....... 11'273 " "4 Elm:.... 64'460 Honduras mahogany.. 64000 to c': " Mar Forest fir..... 51650 " " Beech...... 51'494 Riga fir....... 47762 "c " Ash and Dantzic oak. 47- 158 *6 "4 Spanish mahogany... 42066 | " " English oak.... 36205 To find the weight in lbs. of 1 foot of common rope, multiply the square of its circumference in inches by........ 044 to'046 Ditto for a cable......... 027 TABLE Surface of Boilers' Tubes oflifferent Lengths and Diameters. Diameter. Length. Surface. Diameter. Length. Surface. In. Ft. in. Sq. ft. In. Ft. in. Sq. ft. 21 | 5 0 3 27 3 66 51 "?, 5 3 342 " 6 8 52 c " l 5 6 3'6 " 7 0 55 "c l 5 9 3'.5 7 6 5'89 6 0 3'9 " 8 0 6928 3c 6 0 47 8 6 6'67 99 6 3 4'9 21 242 RECIPES FoR MAIxKINO G LASS. RECIPES FOR M{AKING DIFFERENTI KINDS'OF:GLASS. 1.: IBottle: Glass.-I. Dry glauber salts, 11 pounds; soaper salts, 12 pounds; ha'llf a bushel of waste soap ashes; sand, 56 pounds; glass skimmings, 22 pounds; green broken,glass, 1 cwt.; basalt, "25 poun:ds. Tllis mixture affords a dark green glass. 2. Yellow or white sand, 100 parts; kelp, 30 to 40; lixiviated wood ashes, from 160 to 170 parts; fresh wood ashes, 30 to 40 parts; potter's clay, 80 to 100 parts; cullet, or broken glass, 100. If basalt be used, tile proportion of kelp may be diminished. 2. Green n-incldow, or Broad Glass.-Dry glauber salts, 11 pounds; soaper salts, 10 pounds;, half a bushel. of lixiviated soap waste; 50 pounds of sand; 22 pounds of -glass pot skimmings; 1 cwt. of' broken green glass. 3. Crown Glass.-300 parts of fine sand; 200 of good'soda ash; 33 of lime; from 250 to 300 of broken glass; 60 of white sand; 30 of purified potash; 15 of saltpetre; (1 of borax;)' of arsenious acid. of. Neairly White Table Glass. —I. 20 pounds of potashes; 11 pounds of dry glauber salts; 16 of soaper salt;:65:of sand; 140 of cullet of the same kind. 2. 100 parts of sand; 235 of kelp; 60 of wood ashes; 1~ oft manganese; 100 of broken glass. 5. White Table Glass. —1. 40 pounds of potashes; 11 of chalk;'76 of sand; Ei of manganese; 95.of white:ullet. 2. 50 — of.purified potashes; 100 of sand; 20 of chalk, and 2 of saltpetre. 6. Crystai Glass.-l. 60 parts of purified potfshes; 120 6f sand; 24 of chalk; 2 of saltpetre; 2 of arsenious acid; L- of -manganese. 2. Purified pearlashes, 70 parts; white sand, 120; saltpetre, 10; ~ of arsenious acid;. - of manganese. 3. 67 of sand: 23 of purified pearlashes; 10 of sifted slalked lime; i of manganese; 5 to: 8 of red lead. 4.. 120 of white sand; 50 of red lead:; 40 of purified pearlash; 20 of saltpetre; i of manganese. 5. 120 of white sand; 40 of'pearlash purified; 35 of red lead; 13 of'saltpetre; -~ of angaese.' 6. 30 of the finest sand; 20 of red -lead; 8 of pearlash purified; 2 of saltpetre;. a little arsenious acid and manganese.'7. 100 of sand; 45 of red lead;'85 of purified pearlashes; - of manganese; s of arsenious acid. 7. Plate Glass -1. Very jwhite sand, 300 parts; dry purified soda, 100 parts; carbonate of lime, 43 parts; manganese, 1; cullet, 300. 2. Finest sand, 720 parts: purified soda, 450;'quicklime, 80; saltpetre,'25; -cullet, 425. A little borax has also been prescribed; much of it communicates an exfoliating property to iglass. - PRIME NUMBERS. 243 TAB LE Of Prime lumnbers to 5000. 2 197 461 751 1051 138.1 1697 3 199 463 757 1061 1399 1699 5 211 467 7161 1063 1409 1709 7. 223 479 76'9 1069. 1423. 1721 11 227 48.7 773 1087 1427 1723 13 229 491 787' 1091 1429 1733 17 233 499 797. 1093 1433 1741 19 239 503 809. 10'7 1439. 1747 23 241 509 811 1103 1447 1753 29 251 521 821 1109 1451 175.9. 31 257 23 823 1117 1 1453 1777 37 263 541 8 1123 1459 1783 41 269 547 829 12 11 1787 43 271 557 839 1151 1481 1789 47 277 563. 853 1153 1483. 1801 53 281 5,69 857 1163 1487 1811 69 283 571 859 1171 1489 1823. 61 293 577 863 1181 1493 183i' 67 307 587 877 1187 1499 1847 71 11 593 88.1 1193 1511 1861 73 313 599 883 1201 1523 867' 79 3.17' 601 887 1213 1531 1871 83 331 607 9(7 1217 1543 1873 89 337 613 911 1223 1549 1877 97 347 617 1 919 1229 155 187 101 349 619 929 1231 1559 1889 103 353 631 937 1237 1567 1901 107 359 641 9.1 1249 1571 1907 109 367 643 947 1259 1579.. 1913 113 373 647 95 1277 1583 1931 127 379 653 967 1279. 1597 1933 1'31 383 659 971 1283 1601 1949 137 389 661 977 1289 1607 1951 139 397 673 3S'.3 1291 1609, 1973 149 401 677 991 1297 161,3 197.9 151 409 683 997 1301 1619. 1987 157 419 691 1009 1303 162.1 1993 163 421' 701 1013 1307 1627 1997..167 431 7.09 1019 1319 1637 1999 173 433 719 1021 1321 1657 2003 179 439 727 1031 1327 1663 2011il 181 44 733 1033 1361 1667 2017 191 449 739 103.9 1367 1669 2027 193'457 743 1049. 1373 1693 2029..a 244 PRIME fNUMBERS. 2039 2399 2789 3203 3581 3967 4371 2053 2411 2791 3209 3583 3989 4391 2063 2417 2797 3217 3593 4001 4397 2069 2423 2801 3221 3607 4003 4409 2081 2437 2803 3229 3613 4007 4421 2083 2441 2819 3251 3617 4013 4423 2087 2447 2833 3253 3623 4019 4441 2089 2459 2837 3257 3631 4021 4447 2099 2467 2843 3259 3637 4027 4451 2111 2473 2851' 327 1 3643 4049'4457 2113 2477 2857 3299 3659 4051 4463 2129 2503 2861 3301. 3671 4057 4481 2131' 2521 2879 3307 3673 4073 4483 2137 2531 2887 3313 3677 4079 4493 2141 2539 2897 3319 3691 4091 4507 2143 2543' 2903 3323 3697' 4093 4513 2153 2549 2909 3329 3701 4099 4517 2161 2551 2917 3331 3709 4111 4519 2179 2557 2927 3343 3719 4127 4523 2203 2579 2939 3347 37'27 4129 4547 2207 2591 2953 3359 3733 4133 4'549 2213: 2593 2957. 3361 3739 4139 4561 2221 2609 2963' 3371' 3761 4153 4567 2237 2617 2969 3373'3767' 4157 4583 2239 2621 2971 3389 3769 4159 4591 2243 2633 2999 3391 3779 4177 4597 2251' 2647 3001' 8407 3793 4201 4603 2267 2657 3011 3413 3797 4211 4621 2269 2659 3019 3433 3803 4217 4637 2273 2663 3023 3449 3821 4219 4639 2281 2671 83037 3457 3823 4229 4643 2287 2677 3041 3461 3833 4231 4649 2293 2683 3049 3463 3847. 4241 4651 2297 2687 3061 3467' 8851 4243 4657 2309 2689 3067 3469 3853 4253 4663 2311 2693 3079 3491 3863 4259 4673 2333 2699 3083 8499' 3877' 4261 4679 2339 2707 3089 3511 3881 4271 4691 2341 2711 3109 3517 3889 4273 4703 2347: 2713 3119 3527 3907 4283 4721 2351 2719 3121 3529 3911 4289 4723 2357 2729 3137. 3533' 3917 4297 4729 2371 2731 3163 3539 3919 4327 4733 2377 2741.3167 3541' 3923 4337 4751 2381 2749 3169 3547 3929. 4339 4759 2383 27'53 3181 3557 3931 4349 4783 2389 2767' 3187. 3559 3943 4357 4787 2393'2777 3191 3571' 8947 4363 4789 SOLID INCHES AND SOLID, FEET. 245 4793 4817 4877 4919 4943 4969 4999 4799 48.31 4889 49.31 4951 493; 5003 4'801 4861 4903 4933 49.57 4987 5009 4813 4871 4909 4937 4967 4993 TABLE Of Solid Inches and Solid Feet. Feet. Inches. Feet. Inches. Feet. Inches. Feet. Inches. 1= 1728 26=44928 51= 88128 76=-131328 2 3456, 0 27 46656 52 88956 77 133056 3, 5,184: 28 48384 53 91584 78. 1347844 6912 29.50.112 54 9-3312 79 136512: 5 8640 30 51840 55 95040 80 138240 6 10368 31 53568 56 96768 81 139968 7 12096 32 55296:'57: 98496 82 141696 8 13824-: 33 57024 58$ 100224' 83 143424 9. 15.552 34 58752 59 101952 84 145152 10 17280 35 60480 60 103680 85 146880 I'1 19008 306 62208, 601 105408 86 14860,8.: 12 20736 3.7 639.306. 6,2 107136 87. 1503306 13g 2'2464 38 65664 63 108864 88 152064 14 24192 39 67392 64 110592 89 15379,2 15 25920: 40 69120 65 1:12320 90 155520 16 27648 41 70848 66 114048 91 157248 17 29376. 42..72576 67 1157706 92 1'58976 18 311.04 43. 74304 68 117504 93 16070.4 19.32832 44 76032 6.9' 119232 94. 162432 20' 34560:. 45 77760 7.0: 120960 95. 164160 21. 36288 46 79488 71 122688 9.6 165888. 22' 38010 47 81,216 7,2: 124441,6 97 167616 23 39744 48 82944. 73, 126144 98 169344 24 41472. 49. 84672. 74. 12787.2 99. 171072 25 43200 50, 86400 7 -5 129600 100Q 172800. TA B LE ShowtQing the Weight of Cast-Iron Plates, 12 inches square, and front 4. of an inch to 1 inch. thick. Width in 4 4 4 One Inch-es, 125 *25- 375 5. 62, 75 *85. Inch. lbs. oz. lbs. oz. lbs oz. lbs. oz. lbs. -oz: lbs. oz lbs. oz. lbs. oz. 12 4 134 9 104 14 8 19 54 24 24 29 0 33 134 3S8 104 21* 246 DIMENSIONS OF WHEELS. To find the Horse Power that a Cast-Iron Wheel is capable of transmitting. Multiply the breadth of the teeth or face of the wheel in inches by the square of the thickness of one tooth, and' divide by the length of the teeth, for the strength at a velocity of 136 feet per minute. Thus a wheel with the breadth of teeth ='7 inches, thickness= 14, and length = 2, ought to transmit 7'35 horse power. For'75 x 1'96 14 =- 196, and = 7'35. The strength at any other velocity is found by multiplying the power so obtained by any other required velocity, and by'0044. Thus, the wheel as above, at the velocity of 320 feet per minute, would be capable of transmitting 10'3488 horse power. TAB LE Of the Dinmensions of Wheels in Actual Use. Pitch in Number Bre'dth No. of Horse Power. inches. Character of Wheel. teeothf inches. revolu - teeth. inches. tions per minute. Actual. Calculated. 1~ Spur Wheel,.. 72 4 120 8 75 21 Spur Wheel,. 95 6 25 1j 1'676 3* Bevil Wheel,. 40 7 30~ 20 24'34 2 Cog Wheel,.., 60 6 40 12 15'82 5~ Bevil Wheel,.. 70 12 10 60 67'396 2i Spur Wheel,. 90 8 12 6!972 3* Internal,.80 9 20 41 48'8 3 Cog Spur Wheel, 60 8 30 121'117 6 Spur Wheel,. 30 14 7 21'26 4 Spur Wheel,. 100 10 8 25 296 2* Spur Wheel,. 33 7 55 23 *25 21 Spur Wheel,. 108 7 20 25 26 2* Internal,.. 100 7 10 87 90'4 5 I nternal,.... 60 12 12 55 535 5 Spur,.... 41 10 20 61 *50 476 Spur,..... 50 12 23 65 71'3 31 Bevil Wheel,. 35 10 24 26 25'6 4 Cog Bevil Whee], 50 10 28 33 32-6 4 CogSpur Wheel, 35 9 20 18 16'3 6 On WaterWheel,. 112 14 12 110 168 41 Spur Wheel,. 55 10 16 56 5456 55 10 16____ ___ TRANSVERSE STRENGTHI OF BoDIES. 247 TABLE Showing the Circumference of a Rope equal to a Chain made of Iron of a given Diameter, and the Weight in Tons that each is proved to carry; also the weight of a Foot of Chain made froms Iron of that dbnension. Rope's circum- Chain Diameter in Proved to carry Weight of a linear ference in inche~ inches. in tons. foot in Ibs. avoir. 3 +&i 1 1'08 4 2 41 1 3 16 2 5} i 4 2-7 6 ~& o 5 3.3 7 s 4-6 8~ i~~ ~ ~ &-1~~91 15.5 9 1 3 7-.2 919 & 1l 6 15 8 4 10~ 1 inch. 18 9 4 The Transverse Strength of a body is that power which it exerts in opposing any force acting in a perpendicular direction to its length, as in the case of beams, levers, &c., it is inversely as their lengths, and directly as their breadths, and the square of their depths. But, if cylindrical, as the cubes of their diameters. That is, if a beam 5 feet long, 2 inches broad, and 3 inches deep, can carry 1798 lbs., another beam of the same material, 10 feet long, 2 inches broad, and 3 inches deep, will only carry 899 lbs., being inversely as their lengths. Again, if a beam 5 feet long, 2 inches broad, and 3 inches deep, can support 1798 lbs., another beam of the same material, 4 inches. broad, and 3 inches deep, will support double that weight, being directly as their breadths. A beam of the same material, 5 feet long, 2 inches broad, and6 inches deep, will sustain 7192 lbs., being as the square of their depths. 248 EQUIVALENTS. AND SPECIFIO GRAVITIES. T. A -BL E Showingz the Eq.uivalents and Specific Gravities of sixty-two Simple Substwances. ~'o. l~ame of I I?~": 1 Nameof Nameof Substeace. Substance. _ 33 D. _. Hydroge~n, H. 1 0689 METALS Oxgen,.. Oor. 8 1- 026 Continued. Nitrogen,. N. 142 1- 529 Chlorine, C. 35'5 2444 Chromium, Cr. 28*191 5*9 Carbon,.. 612 44 1 Mercury,. Hg. 203 113'5 Iodine,.-. I. 1261 4-948 Silver,.. Ag. l108 3 10'5 Sulphur,.. S. 16*1. 1; 99 Gold,..Au. 200 193 Phosphorus,. P. 15'7 1*7 Platinum,. Pt. 98 8421'5 Fluorine,. F. F 187 Tin,... Sn. 58 9 7 29 Bromine, Br.; 8'4 3' Cobaltt. Co. 29 5 7'83 Boron,.. B. 11 Manganese, Mn. 27 *7 8'0 Selenium,. Se. 40 4'5 Nickel,. -. Ni. 295 88 Antimony,'Sb. 646 6 67 Arsenic,. As. 37 *7' 1 5 7,E TALS.- Palladium, Pd. 53 35 11 5 Rhodinna,. RR. 52-2 11 Potassium,.. 392'865 Asmiuni,. Os. 99.7- ]10 Sodium,. Na. 25 *972 Iriidium,. Ir. 99-8 18'68 Lithium,..L. 10 Cadmium,. Cd. 55'8 86 Calcium,.. Ca. 20-5 M~olybdeMagnesium,. Mg. 127 num,.. Mo. 47'9 89 6 Silicon,.. Si. 22. Tuingsen, orl Alunminum,. Al. 13.' 7 Wolfriam, W. 94&8. 17 Iron,... Fe. 28 7:-7 Vanadium, V. 68.5 Lead,... Pb 103 7. 1135: Uranium,:. U. 217,2 Copper, ~./ Cu 31 7 8,8 Titanium,. Ti., 24.5 Columbium,. Cm. 1848 Cerium,. |Ce. 46 Glucinum,.'G. 26 Niobi,: Yltrium,.. Y. 32 Pelophlim,. Pe. Zirconiurn,. Zr. 34 Norinum,. No. Thiorinuin,. Thl 60 Didymium, D. Strontium,. Sr. 4358 Lantalnum, Ln. 48. Barium,.. Ba. * 86 Jerbium. I Tb. Bismuth,. Bi. 71 5 Erbium,. Telluriu'm,' Te. 64 2 Rutnhieium, Ru. 52 Zinc,... Z. 32 3 From 6'8 to 7.1 PENDULUMS. 249 The Feeding Properties of different Veyetables. In comparison with 10 lbs. of hay. Hay,... 10 Carrots,... 35 Clover hay,. 8 Cabbage,. 30 to 40 Vetch hay,. 4 Pease and beans, 2 to 3 Wheat straw,. 52 Wheat,... 5 Barley straw, o 52 Barley,.. 6 Oat straw,. 55 Oats,... 5 Pea straw,.. 6 Rye,... 5 Potatoes,. 28 Indian corn,. 6 Old potatoes, 40 Bran,... 5Turnips,. -. 60 Oil-cake,.. 2 Thus 2 lbs. of oil-cake is worth as much as 55 lbs. of oat straw. PENDULU IS. A pendulum that vibrates seconds, or 60 in the latitude of Lohndon, is 39'1393 inches long; and 4/39-1393 x 60=375'36, which serves as a constant number for other pendulums; thus, 375'36 divided by the square root of the pendulum's length, gives the.number of vibrations per minute; and divided by the vibrations per minute, gives the square root of the length of pendulums. EXAMPLE I.-Required the number of vibrations a pendulum of 25 inches long will make per minute. 375'36 = 75'072 vibrations per minute. 4/25 EXAMPLE 2.-Required the length of a pendulum to make 80 vibrations per minute. 375'36 = 4'6922= 22'014864 inches long. 80 TABLE containing the Length of Pendrblms to vibrate SeconLds in various parts of the Wtorld. Inches. Inches. At Sierra Leone,.. 3901954 At New York,.. 39'10153 " Trinidad,.. 39'01879 " Bordeaux,... 39'11282 " Madras, 39'02630 " Paris,'... 39 12843 " Jamaica,... 39'03508 " Edinburgh,.. 3915540 " Rio Janeiro,.. 39-01206 " Greenland,.. 39'20328 A pendulum vibrating half seconds in the latitude of London is 9'8 inches in length; and for quarter seconds, 2'5 inches. TA BL E.Showing the Symbols and Equivalents of Binary Compounds. EquivaName of Compound. Symbol. lent. Remarks. Water,. O..... 9 Binoxide of.h.ydrogen,. 02 H 17 Easily decomposed by the metals and metallic.oxides. Protoxide of nitrogen,.. 0 N 222 Supports combustion; its taste is sweet and pleasant. z Binoxide of nitrogen,.. N 302 Transparent and colorless, produces orange red vapors in the W atmosp eric air and oxygen.:03 N 38-2 1, mos I;er.:a., ~ gree:2 CZ IH4yponitrous acid,.::09 - o N 38 2 It s colorless at0 degrees, but green at common temperatures. Bicarburet of nitrogen, C2 N 26.44'Called cyanogen, cannot support combustion. Nitr'ous acid s..... 04 N:462.Its vapor.is a deep red color,:and is rapidly absorbed: by water. c Nitric acid... N 54 2 Extrermely acid and caustic, emits suffocating fumes. Ammonia,..:H N 17 Sometimes called spirits of hartshorn. or volatile alkali. w Sulphurous acid,.. 02S 321 -Does not support respiration or combustion. Used'in bleaching and diseases of the skin. Sulphuric acid,...S 40 1 Sometimes called oil of vitriol. Very acid and corrosive. Protoxide of carbon,. * iC 0 1412 Inflammable, transparent, colorless. Burns with.a blue flame. Carbonic acid,... 2212 Non-supporter of combustion or respiration. Transparent:and colorless. Hydruret of carbon, H C S 712 Sometimes called olefiant gas. Burns with a rich yelIow flame. Bihydruret of carbon, H2 C 812:Fire-damp, which causes the explosions in coal mines. Bisuiphuret of carbon, S2 C 3832 Very volatile. Evaporating rapidly at natural temperature. Bicarburetted hydrogen, C2 1H Boracic acid,. B 03 5 Easily ffised. Much used with soda as a flux. Chlorous acid,..,4 C' 67 5a Explodes at., a low temperature, dang brous to obtain. The fumes with phosphorus to be carefully aivoided. Chloric acid,.. C1 75'5 Dissolves zinc and iron. Hydrochloric acid,.. H C1 36'5 Muriatic acid. Great affinity foriwater. Possesses:a aacrid, pungent, suffocating odor. Quadrochlorine of nitrogen,.... C140 1562: Detonates with violence when exposed' to heat Its odor is penetrating and insupportable. Nitro-muriatic acid,. Composed of chlorine 1, water 1, and nitrous acid 1. Known by the name of Aqua Regia, from its power of dissolving Z: gold... Iodic acid, I 05 166 Obtained from iodine and nitric acid. Teriodide of nitrogen, I3N 393'7 Detonates by a slight pressure. Hydriodic acid,... 12I7:5 Acts powerfully upon mercury. Hydrofluoric acid,.. HF 19X7 Its vapors highly irritating. Produces ulceration on the skin. d Phosphoric acid, P2 0a'71-4: Phosphorous acid,. p2 03 55'4 Powerful taste, and:a disagreeable fcetid smell. It is a powerful deoxidating agent. Precipitates: gold, silver, mercury, and platinum in the metallic form. Phosphuretted hydrogen, I-3p2: 34'4 Transparent and colorless. It detonates with oxygen when heated to 300~, or when the electric spark passes through it. Selenious acid,... OSe 56 Selenic acid,..0.. 3 Se 64' Bears a great resemblance to sulphuric acid. Selenureted hydrogen, 1. Se 41 Protoxide of iron, 0 Fe 36 Peroxide of iron,.. 0 F e 80 Or sesquioxidc. The brown rust of iron consists of this oxide. The color is red. TABLE Showing the Symbols and Equivalents of Biniary Compounds. (Convtinued.) EquivaName of Compound. Symbol. lent. Remarks. Black oxide of iron,.. 04 Fe3 116 This compound is formed when iron is oxidated in the air, or in contact with water at a high temperature. Protoxide of lead,.. O Pb 111-7 Commonly called litharge. Used in flint glass. Dinoxide of lead,. O Pb2 2154 Quadrotisoxide of lead,. 04 Pb3 3431 Called red lead. Much employed as a pigment. Binoxide of lead,... 02 Pb 1197 Dinoxide of copper,.. O0 Cu 7114 Called red oxide of copper. Native production. Found in X copper mines in crystals of a red color. Protoxide of copper,. O Cu 39'7 Called black oxide of copper, or copper black. Binoxide of copper,.. 0 Cu 47'7 Protoxide of zinc,.. 0 Z 40'3 The only combination of oxygen and zinc we know. Sesquioxide of antimony, 03 Sb2 153-2 Occurs native; commonly called oxide of Antimony. Antimonious acid,... 4Sb2 161'2 It combines with alkalies by fusing themn together. Antimonic acid,.... 05Sb2 169-2 Protoxide of tin, or stannum,....... 0 Sn 66'9 Sometimes called black oxide of tin; great attraction for oxygen. Binoxide of tin..... 02 Sn 74'9 Occurs native, generally associated with oxide of iron. Bisulphuret of tin,.. Sa Sn 91-1 Formerly called mosaic gold. Used in the arts- to ofurnish a -bronze, termed bronze powder. Chloride of tin,... C1Sn 944 Powerful deoXidating agent. Used in calico printing, and as BIch lo id' tin,'.. ~ 129.9 a mordant fixing colors. Bichloride of tin,... C12 Sn 12 199 Called permuriatc. Used in dyeing and calico printing. Pri otoxid e of bismuth, O Bi 79'5 Cilhlorride of bismuth,.'CBi 107 It was formerly called butter of bismuth. gPi'toxide of manganese,. O Mn 357 Pesqtuixide of manganese, 0'Mn2 7 9 4 It occurs native; pure and as a hydrate Ried0xide of manganese, 04 Mn 115-1 Bioxi'de. of mangLanese, 02 OMn 437I Used in'the preparation of oxygen and chlorine. It is used to give a dark coating to earthenware. Protoxide of cobalt,.. O Co 37-5 Commonly called smalt when combined with a little silica t and potassa. In this state it is much employed in coloring l glass and glazing of earthenware. -Arsenious aeid..,.. 09 Asas 99'4 Extremely poisonous, either internally' orexternally. Arsenic acid.. As2 115'4 Considered as noxious as arsenious acid, or mlore so. Sesqioloxi deofarsenic,. i3As' 78'4 Deleterious. Killed Gehlen in 1815. It has an offensive 1 odor. Burns with a blue flame. h',otoslph-uret of uarsenic, S As 53'8 It occurs native. Called realgar; used as a pigment. Z Sesquisulphuret of arsenic, S As2 123'7 It occurs native; a brilliant. yellow. Used as a pigment,. nown as "King's yellow." Used in calico printing to deoxidate indigo. Sulphuret of mercury, S Itg 219'1 Artificial Cinnabar. When powdered it is vermillion. Bisulphuret of mercury, S2 Hg 235-2 Called calomel, Chloride of mercury, C1 Hg 238'5 Protoxide of silver, 0 Ag 116'3 Prochloride of silver, Cl Ag 143'8 Exposed. to the sun becomes purple. It occurs native, and Protoxide of gold,. O Au 208 mnuch formed in chemical operations. Binoxide of gold,... O0 Au 216 It has a dark green color. Teroxide of gold,... O2 Au 224 Sometimes called suric acid. i. _ ____~~___ _~~_____ ~ ~ 02 u 224 So d. c~ TABLE Wiowlng the Synbols and Equivalents of Binary Compounds. (Continued.) VEGETABLE ACIDS AND SALTS. EquivaName of Comptound. Synbol. lit. Remarks. Acetic acid,.... C H?3 61'48 Pungent and ageeable odor; ystallizes at a low temperature; blisters the skin. ~te/ttribe acid,.. 05 C H12 66-48 Solution in water very sour; crystallizes in prisms. Citric acid,... O4 C4 H2 58-48 Nearly like tartaric acid. Oxalic Acid,. C2 36-24 Powerful poison; two or three drachms produce death. It is like Epsom salts in appearance. o BenzoieeC. 09C.4 H5 114-68 Is very white; its odor is fragrant and peculiar. Burns 0 wii a yellowr flame. ~allic acid,.O...1 C7 H3'85.84 Emnployed as a, re-agent. It takes fire when exposed to heat'and produces salts of iron, the basis of black ink. Hydrocyanic acid, HC2 N 27 44 i Priissic acid; dangerous poison. Cyanic acid,.. 0 C2 N 34-44 It is liquid, volatile, and poisonous. Ferrocyanic acid, C8 H2 N Fe 109'32 1 Made into Prussian blue, Not poisonous in small doses thers Produe y the action of different acids on alcohol. Sulphuric ether,.. O C4 H5 37- Used to produce artificial cold; is very inflanmnable. Nitrate of iron,.... N + O Fe'90 2 Deliquesoent, and attracts oxygen from the air. Sulphate of iron, 03S + O Fe'761 Made fron iron pyrites, Copperas and green vitriol are Carbonate o iron,... - ~ C + 0 Feprauired from'this salt. Carbonate of iron,..) C + O Fe 58 a12 Attraots oxygen from the air, and assumes the appearance.o the rust of iron.. ~_._......-.-. Acetatc of iron,.: Muchl employecl in dyeing and calico printing. Nitrate of lead,.. N + 0 Pb 16U59 Used in calico printing. Sulphate of lead, 03 N + 0 Pb 151'8 Phosphate of lead,. Insoluble. Carbonate of lead,.. C + 0 Pb 133'82 Usually called white lead. Acetate of lead,.. A + O Pb 16;3'18 Sugar of lead, used in dyeing and calico printing. Subacetate of lead,. A + 02 Pb2 274-88 Called Goulard's extract.. Chloride of lead,.. ~ Patent yellow; is a mixture of chloride and oxide of lead. Nitrate of copper,. N + O Cu 93'9 Deliquescent, and kept in close vessels. Sulphate of copper, 03 N +:0 Cu 1 98 Blue vitriol, employed as an escllalrotic. Acetate of copper, A + 0 Cu 91-18 Crystals of a bluish green color. Nitrate of zinc,... 05N + O Z 945 Crystallizes in four-sided prisms. Deliquescent. Sulphate of zinc,.. 03 N + O Z 804 Wlite vitriol, rhombic prisms colorless. Carbonate of zinc,. G2 N + 0 Z 62-42 Acetate of zinc,..A + 0 Z 91'78 Crystallizes ini rhomboidal prisms with shining lustre. Nitrate of mercury, 05 N + 0 Hg 265 2 Sulphate of mercury, 03N + O Hg 251-1 Bichloride of mercury, Cl2 Hg 274 Corrosive sublimate-dangerous. Nitrate of silver,. 05 N + 0 Ag 170'5 Darkens when exposed to light. Common marking ink is composed of this and a little mucilage. Sulphate of silver,.. 0N + 0 Ag 152-54 Phosphate of silver,.. P' d- O Ag Alcohol,... C;2. H 23.24 Common sugarL,... 0" C12 IIl 172'44 Starch sugar,.. 4 C2 Hl2 199'44 Uric acid,.... 0 C H-I N'2 91-12 Urea,.... 2 (H20 C N) 6064 I z~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I 256 RECIPE lOnr, DYIEING I-IATS. RECIPE FOR DYEING I-IATS. The bath for dyeing hats, employed by the London m!anuftcturers, consists, for 12 dozen, of 144 Pounds of logwood; 12 " green sullilhate of iron or copperas,! "i veredigris. The copper is made of a semi-cylindrical shape, alnd should ibe surrounded with an iron: jacket, or case, into.which steam may be i admitted, so as to raise the temperature of the interior bath to 190~ Fall., but no higher; otherwvise the heat is apt to affect the stiffening varnish, called the gum, with!which. the body of the hat has beent imbued. The logwood having been introduced and digested for some time, the copperas and verdigris. are added:in successive quantities, and in the above proportions, along with every successive two or three dozen of hats suspended upon the dipping maclline. Each set of hats, after being explosed to the bath, with.occasional airings, during 40 minutes, is taken off the pegs, and laid out upon the ground to be more completely blackened by the peroxydizement of the iron with the atmospheric oxygen. In 3 or 4 hours the dyeing is completed. When fully dyed, the hats are well washed in running water. A skilful operator furnishes the following valuable information Irelative to the stiffening of hats. He says: All the solutions of gums which I have hitherto seen prepared by hatters, have not been perfect, but in a certain degree a mixture, more or less, of the gums, which are merely suspended, owing to the consistency of the composition. When this is thinned by the addition of spirit, and allowed to stand, it lets fall a curdylooking sediment, and to this circumstance may be ascribed tlhe frequent breaking of hats. My method of proceeding is, first, to dissolve the gums, by agitation, in twice the due quantity of spirits, whether of wood or wine, and then, after complete solution, draw off one half the spirit in. a still, so as to bring the stiffening to a proper consistency. No sediment subsequently appears on dilutingr this solution, however much it may be done. Both the spirit and alkali stiffenings for hats made by the following recipes,.have been tried by some of the first houses in the trade, and have been much approved of: Spirit StifeZniiq.-:-7 pounds of orange shellac; 2 pounds of gum sandarac; 4 oz.; of gum mastic; ~ pound of amber resin; 1 pint of solution of.copal; 1 gallon of spirit of wine, or wood naphtha. The shellac, sandarac, mastic, and resin, are dissolved in the spirit, and the solution of copal is added last. Alkali stigening. —7 Pounds of common block shellac; 1 polln of,amber resin; 4 oz. gum thus; 4 oz gum mastic; 6 oz. borax; ~ pint of solution of copal. LIQUEFACrTON. OF GASES. 257 The borax is first dissolved in a little warm water (say 1 gallon); this alkaline liquor is now put into a copper pan (heated by steam), together with the shellac, resin, thus, and mastic, and allowed to boil for some time, more warm water being added occasionally until it is of a proper consistence; this may be known by pouring a little on a cold slab, somewhat inclined, and if the liquor runs off at tile lower end, it is'sufficiently fluid. lf, on the contrary, it sets before it reaches the bottom, it requires more water. Whhen the whole of the gums seem dissolved, half a pint of wood napltha must be introduced, with the solution of copal; then the liquor niust be passed through a fine sieve, and it will be perfectly clear and ready for use. This stiffening is used hot. The hat bodies, before they are stiffened, should be steeped in a wveak solution of soda in water to destroy any acid that may have been left in them (as sulphuric acid is used in the making of the bodies). If this is not attended to, should the hat body contain any acid when it is dipped into the stiffening, the alkali is neutralised, and the gums consequently precipitated. After the body has been steeped in the alkaline solution, it must be perfectly dried in the stove before the stiffening is applied; when stiffened and stoved, it must be steeped all night in water to which a small quantity of tile sulphuric acid has been added; this sets the stiffening in the hat body, and finishes the process. A good workman will stiffen 15 or 16 hats a day. If the proof is required cheaper, more shellac and resin must be introducedl. TABLE Of Ps.essures at which certaisi Gases are Liquifled. Gias s the name given to those elastic fluids which aro permanent under a considerable liressure, and at the tempserature zero. *Ilccomes liquid. Calculated boiling!Name of Gas. point, barometer 30 inches. At Under a pressure of. Snllphurous Acid,. 59 F. 3 atmospheres 4~ Fall:r Chlorine,.. GO 4 2 Ammonia,.. G " G4 Sulphuretted I-Iydrog. 50 17 " 142 Carbonic Acid,.. 32 " 229 Ilydrochloric Acid,. 50 50 " 249 Deutoxide of Azote,. 4 50 " 254 22* TABLE. I Showing the Proportionate Streng th of Wheels in Hose. Power, with a Velocity of 2'27 Feet per Second. w C S CO C ~ | C -C'aC a,' 672 325 43 39 11760 2599 518 7762 103127 12954 1558 656 3125 426 375 110'95 1244'38 488'76 732'7 977'00 1221'9 1466-7 32.25 x 435 1' 6'30 3'00 40'9 3'60 102'25 |225'34 459'38 675'66 900'00 1126o00 135,22 Formula — 1 — - _ 1-_60 6'00 2'875 39'00 3'45 96'30 1212'00 424'2' 636'34 848'4 1016'5 1272'68 strength, at 2'27 feet per s econd 5'77 2'750'37"34 3,30 -85'87 1t89'11 378'32 572'26 756'5 945'79 1134-84, 2 f 5'51 2625 358 3'15 77'14 16555 33964 5110'0 679.64 848.20 975-42 Ftpe; Strength t.perS 5-25 2'500 34 2813 00 71'41 157'41 324'58 476-5 629'26 786'9a 934 74 Then as 2'27: 117'60::5:259'9 h. p 4-72 2-250 30'68 270 57'50 126'70 253'39 371'14 506'60 633,19 The thickness of cog multiplied by O 4-46 2'125 305 8 2:55 54'00 11894 242,24 357'26 474'77 594'71 2'1 equals the pitch, and the thick42 2'00 272 0 2'40 45'33 99'84 199'25 299'50 399'40 ness of cog multiplied by 1'2 equals 3-83 1'875 25'19 2,25 39.35 86.60 173.34 259'53 346-70 3~68 1'750 23'92 2'10.34'60 76'16 152'42 288'00 304'84 g 35 4 1-625 229 6 1t95 30,56 67.32 134.62 2019 4 - 3'1 5 1 500 420'7 1 80 1 2558 56'30 I1l 2 30 11 69 00 5688 13712518426 315 1 21'33 46'70 93850 13a590 2-625 1'250 17l00 1-50 17.70 38.10 77.00 246 1'125i 15'99 135 -14'98 33'00 65'99 2-10 1-00 401365 10 1137 253400 501 00 2-10 1-00 13365 1-20 11l~337 25,00 50,00 KNOT TABLE. jThe minuttes and seconds of tine in? which a vessel passes over the measur ed kno2t beintg knzown, look for the corresponding nsnnber inl this table, uewhich will be the rate of the vessel in knots per ihoz1ur. Sec. 3m.| 411. 5rn. 6r. 7m. Sm. 9m. 10ml. lm. 12m. 13m. 14rn. 0 200 00 15-000 12'000 10'000 85:71 -7500 6'66 6'000 5 454 5000 4-615 4 285 1 19890 14'938 11-960 9 972 8, 551 7'484 6654 5'990 5 446 4'993 4-609 4'280 2 19-780 14-876 11'920 9,944 8*530 7-468.6,642 5'980 5-438 4'986 4'603 4'275 3 19-672 14-815 11'880 9-917 8-510 7'453 t 6-629 5970 5429 4'979 4.597 4.70 4 19-564 14'754 11'841 9'890 8-490 7-438 6'617 5-960 5-421 4'972 4'59'1 4-265 5 19'460 14'694 11-803 9'863 8 470 7,.422'6605 5 950 5,413 4-965 4 45:85 4-260 6 19-355 14-634 11-764 9'830 8-450 7 407 6-593'5940 5 405 4-958 4-580,4-255 7 19,251 i4'575 11-726 9.'809 84-'430 7-392 6-581 5'930 5-39 7 4-951 4-574 4-250 8 19-150 1'4-516 11 688' 9 783 8-413 7-377 6-569 5 921 5-389 4'945 4'568 4-245. 9 19-047 14-457 11,650 9'756 8-391 7,362 6'557 5'911 5-381 4-938 4'562 4-240 10 18-947 14400 11-613 9 -729 8-372'7346 6'545 5 901 5-373 4-931 4'5561 4-235 11; 18 —848 14-342 11 575 9 703'88352 7-331 6-533 5-891 5-365 4'924 4-551 4-230 12 18-750 14-285 11i538 9-677 8333 7-317 6-521 / 5882 5'357 4-918 4-545 4-225 13 18.652 14-220 11501 9.651 8314 7-'302 6-509 5-872 5"349 4-911 4.539 4-220 14 18-556 14-173 11 465 9-625 8-29.5 7-287 f 6-498 5863 5-341 4-904 4'534 4-215 15 lS-461 14-118 11-428/ 9-600 8-275 7-272 6-486 5-853 5-333 4-897 41-528 4-210 16 18'367 14-063.11'392 9I574 8-256 7-258 6-474 5-844 5-325 4-891 4-22 4-206 17 18274 14-008 11',356 9'549.8-238 7-243 6-463 5-834 5-317 4'884 4-516 4-201 18 18 —181 13-953 11- 323 9-524 8-219 7 229 6-451 5825 5-309 4-878 4'1 1 4-196 19 18-090 13-900 11-285 9'490 8'200 7-214 6-440 5'815 5 301 4'871 4-505 4-191 20 18-000 13 846 118250 9 473 8-181 7 200 6428 5-806 5-294 4'864 4-500 4-186 Cxt KNOT TABLE —(Continued). Sec. 3n. 4m. 5mn. 6m. 7l. SM. 9n. lOm. 11. 12m. 13m. 14m. 21 17 910 13,793 11-214 9'448 8163 7-18,5 6-417 5 97 5285 4858 4-494 4-181 22 17-823 13-740 11-180 9-424 8-144 7.171 6-405 5.787 5-278 4.851 4.488 4.176 23 17 734 13'688 11'145 9'399 8127 7157 639i 5-778 5'270 4'845 4'483 4-171 24 17 647 13-636 11'111 9'375 8108 7 142 63s 5i'7759 5'263 4'838 4417 477 166 25 17-560 13 584 11'077 9'350 8'090 7-128 61 5760 5255 2 4'832 4-472 4'161 26 17'475 13.5383 11-043 9.326 8 071 7-114 6360 5-750 5'247 4'825 4'466 4-157 27 I17~ 391 13.483 11,009 9.302 8.053 7-100 6 349 5741 240 4.819 4460 4-152 28 17 307 13'432 10,975 9.278 8 035 7086 6338 57 732 5,232 4.812 4.455 4.147 29 17-225 1 338 3 109 142 9-254 8-017 7.072 6 327 5'723 5-224 4'806 4'449 4-142 30 ~17 143 13333 10 909 9 230 81000 7 059 6'315 5714 5,217 4'800 4-444 4-137 7 31 1706-1 13 284 1.0876 9'207'7982 7045 6 304 5705 5210 4793 443'8 4'13 3 32 16-981 13'235- 10'843 9-183 7 964 7 031 6'293 5-696 5-202 4-787 44:33 4'128 33 16'901 13:186 10-810 9-160 7 947 7017 282 5687 5195 4780 4-428 4'123 34 16 822 13'138 10'778 9'137 7 929 7 004 6-271 5678 5187 4774 4422 4'118 35 16'744 13'091 10'764 91-13 7912 6-990 6-260 5669 5179 4768 4417 4114 36- 1-6B67 13,043| 10-714 9 090. 7895 6-977 6'250 5 666 5172 4-761 4-411 4-110 37 16-590 12-996 10:682 9 068 7-877 6-963 6-239 | 5-651 5164 4.755 4-406 4.105, 38 165o14 12-950 10 651 9 045 7-860 6-950 6'228 5'642 5-157 4'749 4:400 4100 39 16-438 129'303 10-619 9-022 7-843 6 936 6'217 5-633 5-150 4743 4'395 4'095 40 16,363 12S857 10.588 9 000 7-826 6923 6 207 5 625 5-142 4.738 4.390 4.090 41 16-289 12'811 10'557 8'977 7-809 6909 6196 5616 5135 4730 4'384 4'086 42 16 216 12-766 10'526 8'955 7`792 6896 6185 5-607 5'128 4-724 4.379 4 081 KNOT TABLE-(Concluded). Sec. 3m. 4111. 5m. i 6mn. i 7m. Sll. 9m. 10m. 11n. 12m, 13m. i 14m. ___ _ V! I _._ _,_! 43 16143 12'711 1 10495 8933 775 688 6174 5598 5121 4-718 4374 4077 44 16'071 ] 12,676 10'465 8'91'1 7-758 6870 6'164 5-590 5'114 4-712 4S368 4072 4-5 16000 12631'10434 1 8-889 7741 6'857 6"153 5'581 5106 4706 4363 4067 46 15,929 12'587 10'404 | 8-867'7725 6-844 6'14.3 5'57'2 5'099 4;700 4 358 4 063 47 15-859 12'543 10'375 8-845 7 708 6'831 6-132 5'564 5'091 4-693 4 353 4-058 48 15789 12.500 10-345 8-823 7692 6818 6122 5-555 5084 4-687 4347 4054 49 15-721 12-456 10-315 8.801 1 675 6,805 6'112 5547 5077 681 4342 4049 50 15-652 12-413 1 10.286 8-780 76-59 -6792 6.101 5.538 5070 4-675 4337 4044 5 1 15-584 12-37 1 10,256| 8.759.7643 6.779 6"091 5-530 5.063 4.669 i 4'332 4040 52- 15-.517 129329'9' 10227 8'73277-627 6766 6081 5521 5-056' 4663 4326 4025 1 53 15'450 12'287 10-198 8- 716 7-'611 654 6-071 5-513 5'049 4 657 4 321 4 031 54 15-384 12'245 10'169 8-695 7'595 6'741 6-060 1 504 5 042 4"651 4'316 4-026 55 15'319 12'203 0 l-.140 8'675 7579 6'7 29 1 6050 5496 5 035 4-645 4311 402 51 15254 121'62 10'112 8'654 7-563 6-716 i 6040 5487 5028 4639 406 4-017 57- 15-190 12'121 10'084 8'633 7-547 6"704 6'030 5 479 5-020 40633 4301 4 013 58 15-125 12'080 10'055 8 612 7-531 6-691 i 6-020 1 5471 5013 4 627 4-295 4008 59 15-062 12 040 1.0'027 8-591 7515 (6'67: 6-010 5 463 5'006 1 4621 4290 4004 262 CEMENTS. CEMENTS Shell-lac Cement, or Liquid Glue. -Fine orange shell-lac, bruised, 4 oz.; highly rectified spirit, 3 oz. Digest in a warm place, frequently shaking, till thle shell-lac is dissolved. Rectified wood naphtha imay be substituted for spirit of wine, where- the smell is not objectionable. This is a most useful cement for joiining almost any material. Shell-lac Cement, without Spirit..-Boil 1 oz. of borax in 16 oz. water; add 2 oz. powdered shell-lac, and boil in a covered vessel till the lac is dissolved This is cheaper than the above, and for' many purposes, answers very well. Both are useful in fixing paper~ labels to tin, and to glass when exposed to damp. KIeller's Armenian Cement, for Glass, China, &c.-Soak 2 dr. of' cut isinglass in 2 oz. of water for 24 hours; boil to 1 oz.;add 1 oz. of spilit of; wine, and strain through linen. Mix this, lwhile hot, with a solution of 1 dr. of mastic in 1 oz. of rectified spirit, and triturate with j dr. powdered gurn animoniac, till perfectly homogeneous. 1Dr. Ure's Diamond Cemenent.-Isinrglass, 1 oz.; distilled water; 6 oz.; boil to 3 oz., and add 1~ oz. of rectified spirit. Boil for a minute or two, strain, and add, while hot, first, -J oz. of a nlilky emulsion of arunmoniac, and then 5 dr. of tincture of mastic. Hoenle's Cement, for Glass or Earlhensware.-Shell-lae, 2 parts; Venice turpentine, 1 part. Fuse together, and form into sticks. UCheese C'ement, for Earthe,2sare, &c -'Mix together white of egg, beaten to a froth, qllick-lime,,rnd grated cheese. Beat them to a paste, which forms an excellent cemlent. Curd Cement.-Add ~ pint of vinegar to ~ pint of skimmed milk. Mix the curd with the whites of 5 eggs well beaten, and sufficient powdered quick-lime to form a paste. It resists water, and a moderate degree of heat. Uenment for joiningl Spar and Marble Ornaments, &d.-Melt together 8 parts of resin, 1 of wax, and stir in 4 parts, or as much as may be required, of Paris plaster. The pieces to be made hot. Hensler's Cement.-Grind 3 palrts of litharge, 2 of recently burnt lime, and 1 of white bole, with linseed oil varnish. This is a very I tenacious cement, but it takes considerable time to dry. Singer's Cemenf, for Electricarl Machli.es and Galvanie Trough.s.-| Melt, together,5 lbs. of resin, and 1 lb. of beeswax, and stir in I lb of red ochre (higrhlv drlied, and still warm), and 4 oz. of Paris plaster, continauing the iheat a little above 212~, and stirring constantly till all frothinge ceases. Or (for troughs), resin, 6 lbs; dried red ochre, 1 lb.; calcined plaster of Paris, - lb.; linseed oil, - lb. C0IENTS. 263 ornmposition for welding Cast Stel. —Take of borax, 10 parts sal ammoniac, 1 part; grind or pound them roughly together then fuse them ls a metal pot over a clear fire, taking care to continue the heat until all spume has disappeared from the. surface. When the liquid appears clear, the composition is ready to be poured out to cool and concrete afterwards, being ground to a fine powder, it is ready for use. * * To use this composition' The steel to be welded is first raised to a "bright yellow" heat, it is then dipped among the welding powder, and again placed in the fire, until it attains the same degree of heat as before; it is then ready to be placed under the hammer. Caest-Irmo,, ement. —Take of clean iron borings, or turnings, 1 cwt.; of sal-ammoniac 8 oz.; and 1 oz. of flour of sulphur. Mix them thoroughly, and add sufficient water. If the cement is not to be immediately used, care should be taken to keep the mixture soaked in water; if left dry, the cement will heat, and be spoiled. Cement for Steam Pipe Joints, &c., with Facced Fanges —To 2 parts of white lead mixed, add 1 part of red lead dry; grind, or otherwise mix them, to a consistence of thin putty; apply interposed layers, with one or two thicknesses of canvas or gauze wire, as the necessity of the case may require. Gahes.-i-. A very strong glue is formed by throwing a small quantity of powdered chalk into mnielted common glue. 2. To make a glue which will resist the action of water-boil one pound of common glue in two quarts of skimmed milk. Botany Bay Ceme&t.-Take 1 part of Botany Bay gum, and melt and mix it with 1 part of brickdust. Cap Cement.-As Singer's; but 1 pound of dried Venetian red may be substituted for the red ochre and Paris plaster. Bottle Cement.-Resin 15 parts; tallow 4 (or wax 3) parts; highly dried red ochre 5 parts. The common kinds of sealing-wax are also used. Tuon~es ceseent.-Beeswax 1 oz.; resin ~ oz.; pitch A oz..Melt,,,ad stir in fine brickdust, 4cwppersmitlh's. (Sement.-Powdered quick-lime, mixed with bul-!ookl's blood, and applied immediately. Enginee s' Ceement. —Equol weights of red and white lead, with dryving.oil,. pread on tow or canvas. This is an admirable composition for-uniting large stones in cisterns, &c. qoute CUemenet far,Closing the Jobn.ts of Isron Pipes. —Take of iron b0oirings, coarsely,po0wdered, 5 lbs.' of powdered sal-ammoniac 2.oz.,; of sullphur i oz.; and water sufficient to moisten it. This composition:hardens rapidly,;;but if time, can be allowed it sets more firmly winthout -the.svllphur. 5 must be used as:soon as mixed, and ram.med tightly -into the joints... 264 CEMENTS. Cement for Steam Pipes.-Good linseed oil varnish ground, with equal weights of white lead, oxide of manganese, and pipeclay. Gad's Hydraulic Cement.-Powdered clay 3 lbs.; oxide of iron 1 lb; and boiled oil to form a stiff paste. Cements for Masonry of' Chaimbers of Chlorine, &c.-Equal ptarts of pitch, rosin, and plaster of Paris. Roman CemenCt. —1 bushel of slacked lime;'-, lbs. of green cop-1 peras; and - buLshel of fine gravel sand. The copperas should be dissolved in hot water. It must be stirred with a stick, and kept stirred continually while in use. Care should be taken to mix a.: once as much as may be requisite for one entire friont, as it is very difficult to obtain the same shade or color a second time. It oiiglit; to be mixed the same day it is us:ed. TThis is the English Roman cement. The genuine Roman cement consists: of the pulvis puteolauus, ocr puzzolene, a feriuginous clay from Puteoli, calcined by th-e fires oi Vesuvius, lime, and sand, mixed with soft water. The only prepa-I ration which the puzzolene undergoes- is that of pounding and I sifting; but the ingredients are occasionally mixed with bullock's blood and suet, to give the composition greater tenacity. Seal Engravers' Cement. —Resin 1 part; brickdust 1 part. Mix with heat. Mcarine Cemtent, commonly called Marine Glue.i-Cut caoutehouc into small pieces, and dissQlve it, by heat and agitation, in coal ilaphtha. Add to this solution powdered shell-lac, and heat the. whole, with constant stirring, until combination takes place; then pour it, while hot, on metal plates, to form sheets. When used, it must be heated to 280 Fallh., and applied with a brush. Liquid Glue. -Dissolve bruised oralnge sheil-l1ac in ~ of its vweilght of rectified spirit, or of rectified wood naphtha, by a gentle heat. It; is Very useful as a general cement and substitute for gilue. Anothel kind may be made by dissolving 1 oz. of borax in 12 oz. of so-ft water, adding 2 oz. of bruised shell-lac, and boiling till dissolved, stirring-it constantly. Bank Note Glue.-Dissolve 1 lb. of fine glue, or gelatine- in Water; evaporate it till most of the water is expelled; add ~ lb. of brown sugar, and pourl it into moulds. Some add a little lemon juice. It is also made with 2 parts of -dextrine, 2 of water, and 1 of spirit. M[aissiat's Cement, as en Auii'-Tight Coveringq for Bottles, &c.Melt india-rubber (to which 15 per cent. of wax or tallow may be added), and gradually add finely powdered quicl-lime, till a chiange of odor shows that combination has taken place, and a pilr'oer coiisistence is obtained'. Cement for Attaching Mfetal Letters on Plate Glass.s-Copal vaiirnish 15 parts; drying oil 5 parts; turpentine 3 parts; oil of turpentine CEMENTS. 265 2 parts; liquified glue' 5 parts. Melt in.a water bath, and add 10.parts of slacked lime. Japan, se Cement.-Mix rice flour intimately with cold water, and boil gently. French Cement.-Mix thick mucilage.of gum arabie.with powdered starch.: Storie Cement.-River'sand 20 parts; litharge 2 parts; quick-lime 1 part. Mix, with linseed oil. Plumbers' Cement.-Resinl part;briCk-dust:2 parts. Mix, with.heat. Pacriian Ueezeint.-Gum arabic 1 oz.; water 2 oz.; sufficient starch to thicken. C6ement for Floors.-The following style of floor is well adapted for plain country dwellings: Take two thirds of lime, and one of coal' ashes, well sifted, with a small: quantity of loam clay; mix the whole together,:temper it well with water, and make it up into a heiap; let it li sixor sieven days, and:then temper it again. After this, heap it up for three or four days, and repeat the tempering very high, till it'becomes smooth, yielding, tough, and gluey. The ground being then levelled, lay the floor therewith about 2. or3 inches thick, making it smooth with a trowel. The hotter the seasoni is the better; when thoroughly dried it —makes a capital floor. Should a better looki7nq floor be desired, take lime of rag stones, well tempered with white of eggs; and cover the floor half- an inll thick with it, before the under flooring is too dry. If this be well done, and the floor thoroughly dried, it will look, when rubbed withi a little oil,.as transparent as metal, or glass. Common Paste.-To a table-spoonful of flour add gradually half a pilit of cold water, and mix till quite smooth; add a pinch of powdered alum (some add a small pinch'of powdered rosin), and boil for a few moments, stirring constantly. The addition of a little brown sugar, and a few grains of corrosive sublimate, will, it is said by: practical chemists, preserve it for years. Soft Cement. —Melt yellow wax with halfits weight of common I turpentine, and stir in a little Venetian w-ed, praeviously well dried and finely powdered. This cement does very well as temporary stopping for joints and openings in glass and-other apparatus, where the heat and pressure are not great. Lutes, or Cements, for Closina the - Joints of Apparatus.-Mix Paris plaster with water to a soft paste, and apply it immediately. It bears nearly a red heat. It may be rendered impervious by rubbing it over with wax and oil. Avother.-Slacsked lime, made into a paste with white of egg, or a solution of gelatine. Another. Fat.Lute.-Finely,powdered clay, moistened with water, and beaten up with boiled linseed oil. Roll it into cylinders, 23 .266 CEMSNT S. and press it on the joints of the vessels, which must be perfectly dry. It is rendered more secure by binding it with strips of linen moistened with white of egg. Another.-Linseed meal beaten to a paste with water. Another.-Slipsof -i moistened bladder, smeared with white of egg. Fire and Waterproof -Ce mentt;-To half A pint of mill put an equal quantity of vinegar, in order to' curdle it; then separate the curd from the whey, and mix the latter with four or five eggs, beating the whole well together. When it is well mixed add a little lime through a sieve, until it has acquired the consistence of a thick paste. With this cement broken vessels may be united. It resists water, and, to a certain extent, fire.:Fiie Lutes.-The following composition will enable glass vessels to sustain an incredible degree of heat: Take fragnents of porcelain, pulverize, and sift them well, and add an equal quantity of fine clay, previously softened with as much of a saturated solution of muriate of soda as is requisite to give the whole a propelr consistence. Apply a thin and:uniform coat of this composition to the glass vessels, and allow it to dry slowly before they are put into the fire. Another.-Equal parts of coarse and refractory clay, mixed with a little hair, form a good lute.:A Cement for Stopping the Fissures of Iron Vessels.-Take two ounces of muriate of ammonia, 1 ounce. of flour of sulphur-, and 16 ounces of cast-iron filings, or turnings. Mix them well in a mortar, and keep the powder dry. When, the cement is wanted take one part of this and twenty parts of clean iron filings, or borings; grind them together in a mortar, mix them wnith water to a proper consistence, and apply them between the joints. This cement answers for flanges of pipes, &c,, about steam-engines. Genuzine Armnenian Cemenet.-" The jewellers of Turkev, who are mostly Armenians," says Mr. Eton, a very intelligent traveller, and at one time a resident and consul in that country, " have a singular method of ornamenting watch cases, &c., with diamonds and other precious stones, by simply glueing'or cementing them on. The stone is set in silver or gold, and the lower part of tlie metal'made flat, or to correspond with the part to which it is to be fixed. It is then warmed gently, and the glue applied, which is so very 1strong that the parts th'us cemented never separate. This glue, which will firmly unite bi;c of glass, and even polished steel, and may of course be applied to a vast variety of useful purposes, is thus made:-Dissolve five or six bits of gum mastic, each the size of a large pea, in as much spirits of wine as will suffice to render it liquid; in another vessel dissolve as much isinglass, previously a little softened in water (though none' of the water must be used), in French brandy, or good:rum, as will make; a two: ounce phial of C E;MENS. 267 very stlrong glue, adding two small bits of gurm galbanum, or ammoniacurn, which must be rubbed or ground till they are dissolved. Then mrix the whole with a sufficient heat, kleep the glue in a phial closely stopped, and when it is to be used set the phial in boiling water." Anotlher.-Thick isinglass glue 1 part; thick mastic varnisll 1 palt. Melt the glue, mix, and lkeep it in a closely corked phial. For use, put the phial in hot water. Elastic Cement for Bells.-DissolVe in good brandy a sufficient quantity of isinglass, so as to be as tlhick as molasses. A very strong Carpenters' Gl'.ic.Dissolve an. ounce of the best isinglass, wittl a mnoderate heat, in a pint of water. Take th is solution, and strain it through a piece of cloth, jand add to it a proportionate qulantity of the best glue, which has been previously soaked for about twenlty-fourhlours, and a gill of vinegar. After the whole of the materials have been brought into a solution, let it once boil up, and strain off the impurities. This glue is well adapted for any worlk which requires particular strength, and where the joints themselves do not contribute towards the combination of tile work; or in small filIets and mouldings, and carved patera, that are held on the surface by the glue. 4. Glue for I-ilaying Brass or Silver Strings, &e. —.Melt your glue as ulsmal, and to every pint add of finely powdered rosin and finely powdecred brickdust two spoonfuls each; incorporate the whole, together, and it will hold the metal much faster than any common glue. A. strong Glue that will resist MIoisture. — Dissolve gum sandarac and mastic, of each 1 of an ounce, in ~ of a pint of spirit of wine, to which add I of aln ounce of clear turpenti.ne. Nlow take strong glile, or that in which isinglass has been dissolved; tllen, putting the grumns into a double glue-pot, add by degrees the glule, conlstamitly stilring' it over the fire till the wvhole is well mixed. tlien |striailn it through a cloth, and it is ready for use. Toul may now return it into the glue-pot, and add -, an ounce of very finely powdered glass; use it quite hot. If you join two pieces of wood tog'etherv witlh it youl may, whlen perfectly hard end 1dry, immerse it inl water and tile joint wNill not separate. A Paste for laying Cloth or Leather on Table Tops.-To a pint ofi the best wlleaten flour add two table spoonfuls of finely powdered rosin, alnd: one spoonful of powdered aluill. 3Mix thlem well together, put them into a pan, and add by degrees rain water, carefully! stirring it till-it is of the consistence of tllinnish cream; put it into a saucepan over a clear fire, keeping it constantly stirred, that it may not get -lumpy. - When it is of-a stiff consistence, so that the spoon will stand uipright in i iti it is done enough. Be careful to stir it well from the bottom, for it will.burn:if not well attended to. Empty it out into a pan, and cover it over till cold, to prevent a 268 CE IENTS. skin forming on the top, which -would make it lumpy. This paste is very-superior for the purpose, and adhesive. To use it for cloth or baize spread the paste evenly and smoothly on the top of the table, and lay your cloth on it,-plessing and smoothing it with a flat piece of wood; let it remain till dry; then trim the edges close to the cross-bandingr. If you cut it close atfirst it will, in drying, shrink and look bad where it meets the banding all round. If used for leather, the leather must be first previously dampened, and the paste then spread over it; then lay it on the table, and rub it smooth and level with a linen cloth, and cut the edges close to the banding with a short knife. Some lay their table-cover with glue instead of paste, and for cloth perhaps it is the best method; but for leather it is niot:proper, as glue is apt to run througlh. In using it for cloth, great care must be taken that your glue is not too thin, and! that you rub the cloth well down with a thick piece of wood made hot at the fire, for the glue. soon chills. You may by this method cut off the edges close to the border at once. Cement Stopping.-Mix equal quantities of sawdust, of the same wood required to be stopped, and clear glue; and with this stop up the holes or defects of the wood. Where the surface is to be japanned or painted, whiting may be used instead of saivdust. Be sure to let the stopping dry before you attempt to finish the surface. Malhoygany-colbred Cement. —Melt two ounces of beeswax, and half an ounce of rosin, together; then add half an ounce of Indian red, and a small quantity of yellow ochre to bring the cement to the desired color. Keep it in a pipkin for use. A Cemuent to stop falws or Cracks in,Wood of any Color. —Put any quantity of fine sawdust, of the same wood your work is made with, into an earthen pan, and pour boiling water on it, stir it well, and let it remain for a week or ten days, occasionally stirring it; then boil it for some time, and it will be of the consistence of pulp or paste; put it into a coarse cloth, and squeeze all the mois-' ture from it. Keep for use, and when wanted mix a sufficient quantity of thin glue to make it into a paste; rub it well into the cracks, or fill up tlie holes in your work with it. When quite hard and dry, clean your work off, and, if carefully done, you will scarcely discover the imperfection. Fireproof Stucco for Wood, &c.-Take moist gravelly earth (previously:wasled), and make it into stucco with the following composition: Pearlashes two parts; water five parts; common clay one part. It has been tried on a large scale and found to answer. Terra Cotta -Potter's clay, Ryegate sand, and water, each a sufficient quanitity. Model and bake. Pew's Conmposition for covering Blildings.-Take the hardest and purest limestone (white marble is to be preferred), free from sand, clay, or other matter; calcine it in a reverberatory furnace, pulverize and pass it through a sieve. One part, by weight, is to be mixed ANALYSIS OF CERTAIN ORGANIC SUBSTANCES. 269 with two parts of clay well baked and similarly pulverized, conducting the whole operation with great care. This forms the first lpowder. Thle second is to be made of one part of calcinaed and pulverized gypsum, to wlich is added two partes of clay, baked and pulverized. These two powders are to be combined, and intimately incorporlated, so as to form a perfect mixture. When it is to be used, mix it withl about a fourth part of its weight of water, added gradually, stirrilg the mass well the whole time, until it forms a thick paste, in which: state it is to be spread like mortar upon the idesired surface. It becomes in- time; as:hard as stone, allows no moisture to penetrate, and is not cracked:, by heat. When well prepared it will last any length of time. When in its plastic or soft state, it may be colored of any desired tint. TABLE Of Analysts of certain Organic Subsance;i, firom the best aut horities. Carbom. Hydrogen. Oxygen. Nitrogen. Total. Sugar..... 42'220 6 600 51175 - 100 Star.ch,. 14 2.5( 6 674: 49'076 100 Gum,.I 42,682 6'374'50'944 100 Lig11in, 52'53 5609 417 8 100 Tannin,. 52.590 3'825 43'585 100 Indigo, 73. 260: 2'500 10'43 13'81 100 Camphor,. 7. 3338 10'67 14'61' 34 100 Caoutchouc,. 87-2 12 8 - 100 Albumen, 52883'7540 23872 15705 100 Fibrin, 53836 7'021 19'685 19 934 100 Casein, 59781 7429 1I1409 21 381 100 Urea,.. 18:9 - 9 26' 52 i5'. 100 Gelatine,... 881 7 914 27 207 [6'998 100 Picronmel, 54 5453 1 82 43 65 100 Hlo1dein, 4.... 42 64 476 18 100 Emelin,... 64'57 7'7 22'95 4'3 100 Vera tril,.. 66'75 8 54 19'60 5'04 100 Cinchonin,.. 7781 737' 5 93 889 100 Quinin, 75 76 75 2 - 8'61 8'11 100 Brucin,... 70 88 6 66 17 39 5 07 100 StryChnin,.. 7643 6 70 11 06: 5'81 100 Narcotin,.. 65'00 550 02699 2'51 100 I Morphin,... 72340 6366 16'299 4'995 100 23* 270 TOOTHED WHEELS. TABLE To Calculate the Pitch of a Toothed Wheel, when the radius and numzber.of teeth, are given; and the RADIUS,'when the pitch and: number of teeth are given, fromt 10 to 159 teeth. o 3 Radius. d l Radius. Radius. I Radius. Radius. 10 1 618- 40 6-373:70 11:144 100 15'918I 130 20-692 11 1-774 41 6.532 71 11.303 101 16 077[ 131 20 851 12:1 1-932] 42 6691 7:21 11 463- 102 16'236 132 21'010 13 2'089 43 6-850:73 11'622 103 1.6'3.95 133 21169 14 2-247 44 7'009 74 11'781 104 16'554 134 21-328 15 2405 45 7'168 75 11 940 105 16'713 135 21 488 16 2-563 46 7'327 76 12'099 106 16-873 136 21 647 17 2'721 47 7'486 77 12-258 107 17-032 137 21 806 18 2-879 48 7'645 78' 12'417'/ 108 17-191 138 21-965 19 3-038 49 7'804[ 79 12'576 109 17-3a50 139 22-124 20 3-196 50 I 7 963 80.,12'735 110 17'509 140 22-283 21 3-355 51 8.122 81 12'895 111 17'668 141 22-442 22 3-513 52 8'281 82 13'054 112 17'827 142 22'602 23 I'672 i53: 8'440 83'1 13-2131 113 17'987 143 22'761 24 3-830 54 8'599 84 - 13'370 114 18-146 144 22.920 25 - 3-989 55 8758 85 13-531[ 115 18'305 145 23'079 26 4'148'56 8'917 86 13-690 116 18-464 146 23'238 27' 4'307 57 9'076 87t 13-849 117 18'623 147 23'397 28' 4-465 58 9'235 88 14'008' 118 18'782 148 23'556 29 4624 59 9'394 89- 14'168' 119 18'941 149 23-716 30- - 4788] 60 9553 90 14'3271 120 19'101 150 23-'875 31' 4'942, 61 9'712 91 14'486- 121 19'260) 151 24'034 32' 5101 62 9;872 92 14'645 122 19'419 152 124-193 33 5.260 63 10-031 93 / 14804 123 19.578 1538 24352 34' 5'419 64 10'190 94 14'963 124 19'737 154 24,511] 35 I 5'578 65 10'349 95 15'122 125 19896 155 24'620 3-6 5737 66 10 508 96 15'281 1'26 20'055 156 -248'30 37 5'896 67 10.667 97 15'440 127 20'214 157 24-989 38i 6055 68 10,826 98 15 600 1'28 20.374 158 25.148 39[ 6'214'69 10'985 99 15'759- 129 20'533 159: 25'307 RULE 1.-Divide the required radius by the radius opposite the given nu-mber of teeth in the table; the quotient will be the required pitch of the wheel. Example. To find the pitch of a- wheel whose radius is 43 inches, that shall contain 90 teeth: Required radius 43 -: 14'327 = 3-inch pitch. RULE 2.-Multiply the radius opposite the given number of CABLES. 271 teeth in the table, by the pitch required; the product will be the required radius of the wheel. Example. To find the radius of a wheel that shall contain 48 teeth of 21-inch pitch: In the Table, radius 7 645 x -25 - l- 19- inches nearly. CABLES. TABLE For finding the Strain 1that may safely be applied to a good H eempen Cable. Circum. Pounds. Circumfer. Pounds. Circumfer. Pounds. 6' 4320' 10 25 12607 5 14 50 25230' 6'25 4687'5 10-50 13230'' 1475 26107'5 6'-50 5070 1075 13867'5 15 27000 6f75 5467 5 11 14520' 15 25 27907'5 7' 5'880' 11-25 15187'5 1550 288307'.25 63.07'5 1 150 15870 1l 75 297675 7'50 6750' 1175 16567 5 16 30720' 7" 75 7:207'5 12* 17280' 16 25 31687'5 8' 7680' 1225 18007'5 16 50 32670' 8'25 8167'5 12 50 18750- 16*75 33667 5 8,50 8670' 1275 19507'5 17* 34680 8 75 9187 5 13. 20280' 17 25 35707 5 9. 9720' 1 325 21067 5 17:.50 367509V25 10267'5 13,50 21870 11775 371807':8 9'-50 10830' 13775 22687'5 18 38880' 9'75 11407 - 5 14 23520 - 18.25 39967' 5 10.' 1O00 i 1425 24367 5 To ascertain the: Strength of Cables.-Multiply the square of the circumference in inches by 120, and the- product is the weight the cable will bear in pounds, with safety. To ascertain the Strenqth of Ropes.-Multiply the square of the' circumference in inches by 200, and it gives the weight the rope will bear in pounds, with safety. To ascertain the Weight of Ma7aila. Ropes and' Hawsers.-Multiply the square of the circumference in inches by 03, and the product is the weight in p ounds of a foot in length. This-is but: an approximation, sufficiently correct for many pur-, poses. 27 2 CABLES AND ANCHORS. T ABLE Showzing the Size of Cables and Anchors proportional to the JTonnaqe of Vessels. Cables. Chain Ca- Proof Weight of Weight of Weight of Tonnage of Circumfer. bles.- Diam. in Anchor in a fathom of a fathom ot vessels. in inches. in inches. tons. pounds. chain. Cable. 5 3- T' 66 5' 2 1 8 4' 1 i 84 8- 4 10 4-'. 2f' 112 11' 4'6 15 -' 1 4 168 14' 6-5 25 6' 9 5' 224 17' 8-4 40 6- * 6' 336 24' 9-8 60 7' 1I'7' 392 27' 11-4 7. 7 i -. 9' 532 30' 13. 100 8'./1 10' 616 36' 15. 130 / 9. 12' 700 42' 18 9 150 9'.1' 14'- 840 50' 21180 10 1' 16' 952 56' 25 7 200 11' 1'-i- 18 1176 - 60' 28'2 240 12' 1 20' 1400 10 33'6 270 12' 1 T 21' 1456 178' 36'4 320 13' 1* 22' 1680 86' 42-5' 360 14 1. 25 1904 96' 45'7 400 14- 14 27' 2072 104' 49* 440 15 1 17 30' 2240 115' 56' -480 16 1-. 33' 2408 125' 59 5 520 16's 1' 36 2800 136' 63'4 570 17' 1. 39' 3360 144' 67' 2 620 17*a 1 14 42' 3920 152' 171-1 680 18 1'1 45' 4200 161' 75' 6 1740 19 1'43 49' 4480 172' 84'2 820 20' 14 52 5600 184- -93'3 900 22' 1.14 56' 6720 196' 112'9 1000 24' 1' 60' 1168 208' 134'6 HEMPEN ROPE-COPPER RODS. 2.73 TA-B L-E For finding the Strain that may be applied to a Hempen Rope with safety. Cireum. Pounds. Circumfer. Pounds. Circumfer. Pounds. V1 200' 3 50 2450' 61 7200' 1'25 312'5 3'75 2812'5 6-25 7812'5 1'50 450' 4' 3200 6-50 8450' 1P*75 612'-5 4 25 3612 5 6G*75 9112-5 2. 800- 4'50 4050' 7 9800' 2'25 1012'5 4 75 4512'5 7 25 10512-5 2'50 1250 5' 5000' 7*50 11250' 2'75 1512-5 5'25 5512-5 775 12012'5 s3 1800 5 50 6050' 8 12800' 3-25 2112.5 5 75 6612.5 T A B:L E Of WVeight of Copper Rods or Bolts, from ~ to 4 inches in diameter, and 1 foot in lenglth. Diam. Pounds. Diameter. Pounds. Diameter. Pounds.. -,I,,I _._.1 1892 11 3 8312 2 170750.5-6 2956 1.3 4.2688 2 18s9161' 4256 1' 4I7298 24' 20-8562.1.{ 5794 1l5w 5'2140 24 22'8913. 7s567 1 5 7228 24 250188 A9 *9578 1' 6'2547 3' 27'2435 1'1824 1' 6'8109 38. 29o5594.11B 1 4307 19 7'3898 3'i 31'9722 1'7027 1 7'7 993.1 3'4 34'4815.13 19982 1 9'2702 3. 37'0808 4 23176 1.4 10'6420 3 39.7774. 2'6605 2- 12'1082 34. 42'5680 1 3'0270 2- 1 13'6677 34 -45'4550 1- 3 34170 24' 15}3251 4:. 484330 274 COPPER PIPES-CSTERINS. Weiglit of a copper rod 12 inchles long and 1 in. diameter 3:039 lbs. Weight of a brass rod 12 inches long and 1 incli diameter ='2'86 lbs. TAB L E Of the Weighlt of i:veted Copper Pipe.,- from 5 to 30 inches in diams)eter, from 3 to 5. thick, and 1 foot in length. In 5 3 12'4 9' 4 30' S 19- 4 60:142 5- 4 16 880 | 10 4'32208 19 v 7o i233 5'~ 3 13'628 11' 4 3o20 () 20 a 78208 5'4 4 18 395 12' 4 38a 4.56 21 82 984 6- 3 14'765 13- 4 41 456 22 2 5 86 771 6' 4 19 908 14' 4 44'640 23 5 90 571 6' 3 15'897 15 4 47 646 24 5 94 308 6 I 4 21'415 15 5 59'588 25 5 98 122 3' 17 034 16,' 4 50)752 26 5 101 897 T -4 22.932. 16 5 63.470 27 5 105 700 7 4 24'447 1 17 4 53'856 28 5 109'446 8' 41 25-961 171 5 67.'344 299 5 113'221 8' 4 27'471 18 4 57037 30 5 116997 9 4 28985 18 5 71258 The above weilgits includc tlle laps onl the sheets for riveting and caulking. The weights of the rivets are not added-; the qntsbher per linear foot of pipe depends upon the distance they are placed apart, and I their size upon the diameter of the pipe. Tr A B L E Shozsing the C(apacity of Cisterns in Gallons. For each 10 Inches in Depth. Feet Feet Feet Feet Dlam. Diam. Diam. Diam. f2 1 19 5 5 122'40 S 313'33 12 1 705' 24:30.6 65 148'110 Si4 353-72 13 827.4 3 44-:'06 6 176 25 9:39)656 14 1 959'6 84 5997'6 i 20685 9 46140 1 5 110156 4 78-33 7 2X988 1 0 489-20 20 19584 4_ 1 99-14 7 275-40 11 592'40 25 30599 WEIGHT OF COPiER,' LEAD, AND IRON, 275 T A B L E Containing the weight of a Square Foot of Copper and Lead in-lbs. avoirdupois, from I- to ~ an inch. in thickness, advanicing by 3. Thickress. Copper. Lead, Thickness. I Copper. Lead. 13 1'45 1'85 1 and 3 13 07 16'62 A- 2.90 370 7 "! 14152 18.47 4s35 5594 15.97 2031 ~ 5' 80 7-39 - 17 41 22 16 and 7q'26 9 24 3 8 "18 ]' 87 24'00 ~ " 8.871'1108 1 " -' 0 20 32 25.85 - 10'16 12'93 2177 27 70 11'61 1477; 23 22 29'55 TAB LE Of the Wleight of a Square Foot of Sheet Iron in lbs. avoirldtpois, the thickness being the nunmber on, the wire gauge. —No. 1 is A- of anL inch; N/o. 4, ~; No. 11, ~, &c. No. on wire gauge. 1 2 3 4 6 S avoirdu., 12 5. 1 12 11 I 0 9 8 7 7 1To_ 01;1 I guge, 9 10 1 11 i 12 13 1 14 15 16 rounds: voirdu., 6 5 68 5 4B62 - 4-31 4 39) 3 gauge,. I 17 s18 19 20 21| 22 avoirdu. 26 5 218 1 4 93 162 145' 1 37 | T A 3 I:E Of the TIeight of a Square Foot of Boiler Plate Iront, ftom~ -j tol I inch thick, in lzbs. avoirdupois. 7Q-'3 31 5 7 1_' -) 5 7-510 125 15 17 5 20 2'25 251 275 30 32a5 35 3751 40 276 PUMPs, TAB L E Shiowing the. Q:caltity of TVaterper Lin:earr Foot in Pumps, or Vertical'PIipes of different Diamneters. Diameterof Number of Numberof Diameter Number of N Numeer of pump in alonsper biet per otlrrrrmt) gallons peer cutlsc teet inches. linear foot. linear f!ot. in inches. linear teot, per linear foot. 2'136'0218 8 2 176 34S90 21'172'0276 8 2-314 3712 2~'212 0340 8~ 2456 3940 24'257 0412 84 2603 4175'306'0490 9 2754 4417 3}1 *359'0576 94 2909 -.4666 3~ 416'0668 94 3068 -4923 34'478'0766 94 3'232'5184 4'544'0872 10 3 400 5454 441' 614'0985 104 3 572'5730 44'688 1 1104 1 0 3-748 6013 44 767'1230 104 3929 6302 5 850.1363 l 4.114 *6599 51'937'1503 1 1 4'303'6902 54 1.0298'1649 11 4496'7212 54 1 I24'1803 11 4694 *7529 6 1'224'1963 712 4896 853 64 1'328 2130 12 5312'8521 64 1436:2304 13 5746'9217 64 1'549'2489 134 6 -196'9939 7 1 1666'2672 14 6'664 1 0689 74 1 787 2S66 15 7650 12271 1912'3067 1 16 8I704 1 3962 7i 2;042'3275 18 11'016 17670 Exacmples illustrative of the Utility of the Table. 1. Required the quantity of water lifted by each stroke-of the bucket of a 94-inch pump, the length of tlae stroke being 21 feet. 30'68 x 2'25 = 6'903 gallons, eaci:hstroke. 2 What length of stroke with a 6-inch pump will be necessary to discharge 44 gallons of water per minute, the number of strokes being 18 in the givent time? 44 - 2 feet, the: length of stroke. 1-224 x 18 3. eWhat must be the diameterctapable of raising 25 cubic feet of water per minute, the length of the stroke being 24 feet, and making 16 effective strokes per minute? SCREWS. 277 25 2._ 25....-625, or 10I inches, nearly. 2'5 x 16 Properties of Atmnospheric Air. —It is by the oxygen of the atmosphiere that comnbustion is supported. The common combustibles of nature are chiefly compounds of carbon and lyd ogen, which, during combustion, combine with the oxygen of the atmosphere, and are converted into carbonic acid and watery vapor, different speoies, of fuel i'equii'ing different quantities c f oxygen. The quantity required for the combustion of a pound of pcoal varies from two to three lbs. Sixty cubic feet of atmospheric air will produce 1 lb. of oxygen. The pressure or fluid properties of the atmosphere oppose bodies in passing through it, the opposing resistance increasing as the square of the velocity of the body, and thle resistance per squa re foot in lbs. as its velocity in feet per second, multiplied into'002288. Thus, suppose a locomotive engine in a still atmosphere;, at a velocity of 25 miles per hour, presents a resisting frontage of 20 feet; required tlie amount of opposing resistance at that velocity. 25 miles per hour equal 36 67 feet per second. Then 36-672 x 002288 x 20 = 61-5 lbs., constant opposing force. TABLE Showog: thle NTheimbee of Threads to a Inzch in V-thkread Screws. Diam. in inches, + |i 116 + i 1| ii i-} Diam. in inches, 1 1 1 l 1 2 2L 2+ 2+ 3 31 3+ No. of threads, 6 5 5 4 4| 4 4 3'1 3~ 31 3+ Diam. in inches, 1 3 4 4+ 4 1: 5 5 51 5 6 No. ofthreads, 3 |3 2j 24 2t4l2+ 212f 2 2+ 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~. 24 27 8 CWOODs-GASES. TAB LE Of the coanponent pacrts of one English pound aeoirdapois of 7000 grains of thefollowaing vacrieties of Wood. [MiJSIIET.] Water, Color and degree of'saturation |Description of Wolod. iyd. gas, Carbon. Ashes. of the alkalloe principle. escrrptioofloo.'Carb. acid I Oa'-,... 5382 6 1587 8 29 6 grey, sharply alkaline. Asl,.. 5688'2 1258'0 53'8 whitish blue, shrp. alk, Birch,. 5650'2 1 2244 125'4 brownish red, shrp. alk, Norway Pine, 5630'9 1344-3 24'8 brown, not at all alk. Mahogany,. 5147'0 1784'4 68 6 grey, sharply alkaline. Sycanmore,. 55440 13814 74'6 pure white, weakly alk. Holly,... 5524'4 1394-3 81 3 pure white, sharply alk. Scotch pine, 5816'7 1151'9 31'4 brown, perceptibly alk. Beech,... 57373 13959 66'8 greyish wlite, shrp. alk. Elm,. 5576'6 1370-2 53'2 grey, partially alkaline, Walnut, 5496 5 1446 4 b671; pure white, light as i down, weakly alk. American Maple 5553'2 1393-1 53-7 dark grey, shatply alkl Do. Blaeech, k 259 1301'8 72'3 brown, sharply alkaline. Beech, Laburnum,.. 51964 1721:0 82'6 white & grey, partly alk, Lignumn Vitro, 5083:0 1 8800 35'0 grey, sharply alkaliuea Sallow,.. 56260 12948 792 light gr'ey, sharply alk. Chestnut,.. 53413 16296 291 grey, shar ply alkalie.. T A B L F, Of PThopertses of Oases. Atmospheric air being the statinrlld of comparlson, or 100oo; Names. peNrill Names. Specific gravity, gravity; Hydriodie acid gas,, 4340 Caltbonic oxide gas,. 972 Chlorine ".. 2500 Carburetted byCarbonic.".. 1527 di'ogenl.. 97 2 Nitrous oxide " 5. 2 7 Prussic acid ". 937i Cyanogen ". 1805 Ammvniacal ". 5,90; OxSgen ". I111 Steam of water ". 623 Hydroger. 69, SCREWS-ATMOSPHERE 279 T A B L E Of Change Wheels for Screw-cuttinlg; the leading Screw being of ~ inch pitch, or containing 2 threads ivn an inch. Numberof c I " teeth.'n [ Number of teeth in Number of teeth in a3U 3 ii z,,. g. ~~ ~ u.;:' >tee1 h a Ii / e0 in 4 40 i2 6 1 0 0 ~ ) 80 70 9~ 9o" 90' 1 0 j 951 2( 40 90 20o 90 S- i 60/7 00 8 22 60 201 0 2- 8 o ~ 1 5o170 1 20 75!- 2'2 so[12o 20'ao t80 10 11 /60 }. 20 /,o/~ S2o2 80 1so'3o0 2o 140 01 -I 20 son1 50 95 20 100 1 80 40 8[ 40 10 20 011 0 951 20 100 1 80 50 81 90 85 20 90 24 19:120 20 130 1 80 60 81 60 970 20 9 2 0 a 60 100 20 120 1t 80 70 9g 90 90 20 39 201 40 90 20 90 4 840 980 139 80 60 20 11021 8 70 130 20 140 24 40 90 104 60 7 90 20 8( 2 7 640 10 200 120 421 80 100 15 950 90 20 150 28 780 120 20 150 2t 80 110 11 60 55- 20 1201 22" 80 130 20 140 53 0 1200 12 90 980 20 120] 2381 40 95 20 100 t; 40 1301 1211 60 815 20 901 24 65 120'20 130 36i 8 140 13 90 90 20 130: 23 60 100 20 150 31 180 150 131 60 90 20 90 251 30 85 20 90 4 140 80, 1731 80 100 20 110 2 70 130 2850 140 41-i40i 85 14 9090 20 140 27 40 90 20 120 41I 40 90' 14'- 60 90 20 951271 40 100 20 110 418~ 40 95 15 90 28 75 140 20 150 5'401 i001 16 60 80 20' 1910 281- 30 90 20 95 751 40 110+ 1 8 80 10 20. 130 30 70 140 20 150 6 40 120 1 168 80 110 20 1 150 39 30 80 20 120 61-2 40 130 1 7 45 85 2 0 90 IS 3 40 110 20 120 140 1 140 171 80 100 20 140 34 31) 85 20 120 71-140 0160120 120i 35 60 140 20 150 Temperature and Height of the Atmosphere at various heights. Water heavier than the Height. Temperature. air. Level of the sea,... 60~ 860 times. One mile above,. 43 1,083 Two miles above,.. 26 1,363 " Three miles abcve,.. 9 1,716 " Four miles above,... -8 2,160 Five miles above,.. -25 2,719..~~~~~~~~~ 280- WEIGHT OF WATER IN PIPES. TABLE Showzing hozw to discover the Qtantity and Weight of Water in Pipes of a gy given size. Diameter.in I Quantity in cubic Quantity in imperial Weight in lbs. -ilches, inches. gallons. avoirdupois. ~ I: 14'14 0'051 0'51 1 56 55 0'205 2'05 1~. 127'23 0-460 460 2 226-19 0"818 8'18 21 353-43 1'278 12'78 3 508 94 1'841 18'41 3~ 692 72 2'506 25 06 4 904-78 3-272 32-72.4j 1145'11 4'142 41'42 5 1413-72 5.1113 51 13 5~ 1710-60 6'187 6187 6 203575 7-363 73'63 J6 2389'18 8'641 86 41 7 2770'88 10.'022 100.22 7 3180-86 11'505 115'05 8 3619-11 13'090 130'90 s8 4085-64 14-777 147'77 9 4580'44 16'567. 165 67 9~ 5103'52 18'459 184'59 10 5654'87 20-453 204-53 10i 6234'49 22.550 225 50 11 6842'39 24'748 247'48 1 1+ 7478'56 27 049 270'49 12 8143'01 29'452 294-52 This table shows the quantity and weight of water contained in one fathom of length of pipes of different bores from 1 inch to 12 inches in diameter, advancing by half inch. The weight of a cubic foot of water is taken at 1000 ounces avoirdupois, and the imperial gallon at 10 lbs. Multipliers used for ascertaining the quantity of Tallow, Oakuzm, and Oil that can be. contained in'anks for use of Steam-vessels. Tallow,. 59 lbs. in a cubic foot. Oakum,.. 11 lbs. in a cubic foot. O.il....-. 6'23 falls. in a cubic foot. Coal,... 45 cubic feet to a ton...,..,. _ PROPERTIES OF BODIES. o TABLES, combining the.Specific Gravities and other Properties of Bodies. Water the standard of Comparison, or 1000. PROPERTI:ES OF METALS. PRnoEITIES oF STONES, EARTIIS; &CNames. 5 53. ea _ __: Names. Platinum,.. 19500 3280- 3 5 _ - 3 8 Marble,average 2720170 -0013 9'25 C PurIe gold, * * *19258 2016 I1 1-8 - I3 10-0. Granite, do. 2651 165'68 12~:6'2 _ Mercury,...13500 I - ~ I_ Purbeck stone, 2601162 -56 13t9.0 C Lead,.. 11352 612' 319' 81 8 7 1'0 6 1:8 Portland do.,. 2570160 62114 4-5 Pure silver,.. 10474 1873 - 2 2 2-4 2 9-7 Bristol do, 25411 59 6214 - Bismuth.. 9823 476'156 1 -45 - - 2'0 - - Millstone,. 2484 155 25141; Copper; cast 8788 1996'193 S51- - - - Paving stone,. 2415 150 93 14 5 7 wrough;,. I8910 - 15 08 5 3 28 1 8 9 Craigleith do., 2362 147 62j15 5.0 W Brass, cast,.. 7824 1900 210 8 01 - - to any/ _ Grindssto ne,. 21,43133 93.16axo 6 | d egree Chalk, British, 27 81 1381112 0'5 sheet,.. 8396 - - 1223 6 6 - - -86 Briel,... 0001250017 08 I7on cast..7264s 2786 125'787 ~ to any Coal, Scotch, 1300 81'15227 - Iro, Cast, degree Newcastle, 1270~ 79 37 27 - bar, * *. 7700 -'137 25'00 4 8 4-7 4 3-7,, Staffordsh. 1240 77 50 29 Steel, soft,... e7833 - 133 58'91 - _- Cannel,. 1288 77 37129 - hard,... 7816 _ _ to any _ a' 7816 - degree Tin, cast,.. 7291 442 *278 211 8 4 1 2 5 30 Zinc, cast,... 7190 773'329 a506 7 8 1 6 7 3-6 _~~~~~~~~~~~~~~~~~~~. [ 282 fSPECIFIC GRAVITIES AND OTHER PROPERTIES OF BODIES. Properties of Woods. ms. -.,Comparative Names. D ae' D bC.~ h English oak.. 934 58 38~/ 11880 100 100 100 Riga do,... 872 54 414 12888 93 108 125 Dantzic do... 756 47 48 12780 117 107 99 American do.,.. (;72 42 53 10253 114/ 86 64 Beech,......5 82 48 45 12225 77 103 138 Alder,.8.... 800 46 482 9540 63 80 101 Plane,.....60 40 55 10935 78 92 108 Sycamore..... 604 38 59 9630 59 81 11I Chestnut...... 610 38 59 10656 67 89 118 Ash,.... 845 52' 43 14130 89 119 160 Elm,..... 673 42 53 9720 78 82 86 Mahogany, Spanish, 800 50 45 7560 73 67 61 " Hon duras, 6 37. 40 55 11475 93 96 99 Walnut,.... 671 4/2 53 8800 49 74 111 Teak,...... 7 50 46 482 12915 126 109 94 Poona,...... 640 40 55 12350 99 104' 82 African oak,. 944 59 38 17200 101 144 138 Poplar,..s 83 34 66 5998 44.50 57 Cedar,..1 33 i 68 7420 28 62 106 Riga fi..... 753 47 48 9540 98 80 64 Memel do.,..... 546 34 66 9540 114 80 56 Scotch do.,.5..2 58 33 68 7110 55 60 65 Christ. white deal, 590 37 60 12346 104 104 104 Americal white spruce,. 551 34 66 10296 72 86 102 Yellow pine,.... 461 28 80 118.3 95 99 103 Pitch pine,.. 660 41 541 9796 7 3 82 92 Larch,..,.... 530 31 72 12240 79 103 134 Cork...... 240 15 149 Fusing Point of various Metals. The fusing points of the more refractory substances are only to be ascertained approximately, on account of the doubtful accuracy of the indications' given by the pyrometers at very high temperatures. The pyrometer constructed of platinum is the mlost. delicate, although the rate of its expansion must be uncertain as it approaches its own fusing point. The following are considered to be:the fusing points of metals: SPEcIFIc GRAVITIES AND OTHER PROPERTIES OF LIQUIDS. 283 Fahr. Fahr. Platinum,.... 30800- Silver,..... 18300 Wrought iron,. 2910 Zinc,...... 700 Steel,.. 2500 Lead........ 590 Gold,.. 2190 Bismuth..... 500 Cast iron, 2i00 Tin,..... 450 Copper, 1920 A dull red heat is estimated as 1480~; a bright red heat as 18300; and a white heat as 23700 to 2910, Fahll. TABLE of' Properties of Liquids. Names. b. Names. -, I -I ~I. Acid, sulphuric,. 1850 18 5 Oils, expressed: " nitric,.. 1271 12'7 7 linseed,... 940 9' 4 " muriatic,. 1200 12-0 sweet almond,. 932 9' 3 " fluoric,. 1060 10' 6 whale,... 923 9'2 " citric,.. 1034 10 3 1hempseed, 926 9.3 " acetic,.. 1062 106 olive,.... 915 9'2. Water from the Oils, essential: Baltic,... 1015 10' 2 cinnamon, 1. 043 10 4 Water from the I lavender, 894 8.9 Dead Sea,.. 1240 12'4 turpentine,.. 870 8-7 Water from the amber,. 868 8-7 Mediterranean,. 1029 10*3 Alcohol,.. 825 8 2 Water, distilled,. 1000 10:0 Ether, nitric,.. 908 9'1 Proof spirit,.. 922 9 9'2 Vinegar,... 1009 101 Axle Grease. 1. The popular axle grease of the celebrated Mr. Booth is made as follows: Dissolve -~ lb. common soda in 1 gallon of water, add 3 lbs. of tallow and 6 lbs. of palm oil (or 10 lbs. of palm oil only). Heat I:them together to 200~ or 210~ Fah.; mix, and keep the mixture constantly stirred till the composition is cooled down to 600 or 70~. 2. Another and thinner composition is made with ~ lb. of soda, 1 gallon of water, 1 gallon of rape oil, and g lb. of tallow, or palm oil. 3. The French compound, called Liard, is thus made:-Into 50 parts of finest rape oil put 1 part of caoutchouc, cut small. Apply heal, until it is nearly all dissolved. 4. Mankettrick's lubricating compound consists of 4 lbs. of caoutchouc (dissolved in spirits of turpentine), 10 lbs. of common 284 FUSIBILITY OF METALS. soda, 1 1b. of glue, 10 gallons of oil, and 10 gallons of water. Dissolve the soda and glue in the water by heat, then add the oil, and lastly the caoutchouc, stirring them until perfectly incorporated. 5. The following is tile ordinary kind of axle-grease in common use:-1 part of fine black lead, ground perfectly smooth, with 4 parts of lard. Some recipes add a little camphor. TABLE Of Fusibility of Mietals. As given bY M. Thenard. 1. -Fusible below a red heat. CENTIGRADE. Mercury.. - Potassium ~58 j Gay Lussac and Potassium,.... + 58 { llThenard. Sodium,. 90 Do. do. Tin,...... 210 Newtonl. Bismuth,. 256 Do. Lead,.. 260 Biot. Telluiu,.. a little less fus. than led Klaproth. Arsenic,.. undetermined Zinc.... 370 Brongniart. Antimony,... a little below a red heat 2.-Infusible below a red heat. PYROMETER OF WEDGWOOD. Silver,.... 20~ Kennedy. Copper,.... 27 Wedgwood. Gold,..... 32 Do. Cobalt,...... to melt than iron [ron,... 130 Wedgwood.,,..... 158 Sir G. McKenzie. Manganese,... 160 Guyon. Nickel,.... 160 Richter. Palladium,'Palladium,. Nearly infusible, and to Cerium, Infusible at the forge Osmium,.fulnace. Fusible at Iridium,.m th e oxyb - hy droge Rhodium,. pipe. PlRtinum,. blowpipe. Columbium,.....~~~~~~~~~~~~~~~-. DISCHARGE O' WATER. 285 TABLE onutaining. the vQuantities of Water, int cubic feet, that woill be dischargedouver a we:r per icn?ute, for every inc/k in its breadth,. so/tei the depths of the Water from2 the suiface to the top edge of the wasteboard do szot exceed eighteen, ilches. c2 1'140 1:' 211 11 14'707 15632; 6 5'925 6'295 15 23'419 24-88a3 | 8 9' 122 9' 692 ]7 28-'258 0 30024 2 11'884 10 564 1 48 370786 32 710 < 4 Of the C 42omposito of d89ere5 t Gunpowders. KINDS. Nitre. Charcoal. Sulphur. Royal Mills at Waltham Abbey, England,..,... 75 15 10 France, national establishmi't. 75 12'5 1 2'5 Frenc, for sportsmen,... 8 12 10 6 Frenc, for minig,. 65 2315 20 United States of Am 9erica,. 75 12-5 1 1235 prussia,........ 75 13'5 11' 5 Russia,..... 73'78 13'59 12`63 Aust8ia (musket),2 9..... 2 17 16 Spain...'64 1 8 764 30'78 12-710 SVedell,........ 76 1 5 97 Switzerland (a rolnd positier 76 14i 10 Chinese, glnd 75 14-4 9910 ThFraneore tical prp (as abovei 75 13-23 1177 Thieoretical propor. I(as azbove) 7 5 13~23 117 286' MAN AND HtORSE POWitre Alloys. Alloys having a Density greater than the' Alloys having a Density Iess than the Diean ot'their Constituents. Mean of their Cooistituents. Gold and zinc. Gold and silver. Gold and tin. Gold, and iton. Gold and bismuth. Gold and lead. Gold and antinmony.~ Gold and copper. Gold and cobalt. Gold and iridium. Silver and zinc. Gold and nickel. Silver and lea(L Silver and copper. Silver and tin. Silver and iron. Silver and bisniuth. Iron and bismuth. Silver and antimony. Iron and antimony. Copper and zinc. Iron and lead. Copper and tin. Tin and lead. Copper and palladium. Tin and palladium. Copper and bismluth. Tin and antimony. Lead and antimony. Nickel and arsenic. Platinum and molybdenum. Zinc and antimony. Palladium and bismuth. TAB13 LE Shovwilg the estimated Power of iXMa.n or Horse as applied to Mtacahinery. Lbs. avr. at Lbs. avr. at the rate of the rate of Application of the Power. 220 feet per one foot per minute. minute. A man is supposed to be capable of lifting or carrying..... 27 273 6000 A man' is sutpposed to be capable of turning the winch of a crane with a force equal to........ 300 When the united efforts of two men are applied to' the winch of a crane, the han- or dles being at right angles, each'man exerts a force equal to........ 33-499'7350 A man is supposed to exert a power in pumping equal to 1 7335 3814 In ringing, a man exerts a force equal to 38955 8570 And in rowing...... 40'955 9010 The power of a horse is equal to... 150 33000 ,EX.Axb wiot, n OF WIND. 287 T AB LE Of tee Spe', i Ar4 Force of Windl, at dlifferent velocities. Velocity of tle WVintl in Force in lbs. avoir- Common Appellations given to dupois per square the Wind. fbot. VMiles peuro. Feetper second. 1 1'47'005 Hardly perceptible. 2 -2'93' 020 3 4'40 044 5 Just perceptible. 4 1 5 87 107 23 Gentle, pleasant wind. 5 733 8123 10 14'67' 492 15 22'00 1'107 10 22 (30 492i F Pleasant, brisk gale. 20 29'34 1'968 25 36'67 3'075 30 44'01 4'429 Higih windS, 35 51.34 6'027g 40 58'68 7'873 45 66101 9963 Very high 50 73'35 12-300 A storm or tempest. 60 88'02 17'71.5 A great storm. 80 117'36 31'490.- A hurricane. 100 146'70 49'200, A violent hurricane, which wrenches and tears up trees, forces dwellings and minor buildings fironm their foundations,anddrives them before it. Note. —The following rtle is used to.find the force of wind acting perpenditcularly upon a surface:-Multiply the- surface in feet by the square of the velocity in feet, and the product by'002288. The reslkit is the force in pounds avoirdupois. TABLE showing the Heig/ht of the Boiling Point, Fach., at diferente Heights of the Barometer. Barometer. Boii'ng Point. Barometer. Boiling Point. Inches. Degrees. Inches. Degrees. 31 213'57 28 ~ 209-55. 301- 21279: 28 2086.9 30 212-00 271 207-84 29', 211'20 2 - 20696 20 210-38 j In a vacuum water boils at 98~ to 1000, according as the vacuumr is more or less perfect. 288 POWER9 OF AtRit[OuS SciEsaii Or F T-A B L E Of the sizes of Nuts, equal in sirenygth to their Bolt.. Diam. of Short diameter i Diam. of Short diameter of Diam. of Short diameter bolt in in. of nut in in. bolt in in. nut in inches. bolt in in. of nt:lin in. * 7:X \1 1S 21 22: j ~ lbs. s.2 4 1-1 11 3j 2-* 5 1 1 10 2' 3 12 1 3'6 1Ono 2 26 4 3 6 111 2 in 21 4. I 4 Note.-The depth of the head should eqlual the diameter of the bolt; the depth of the aut shouldxeceed it, in the proportion of 9 or 1o to 8. TA BL E Slowiing thee Power of various Species of Ftuel. Species of Fuel.' ut a 0 5!. t. -,a.). d Q..._ lbs W3a 42 o | htS. lbs. Ilbs. lbs. Cakling coal,. 9800 84 -45 8 22 Coke,.. 9000 7-7 8-1 900 Splint cOal,. 7900 6I5.925 1028 Oak wood, dry,. 6000 5 13 12M2 136 Ordinary oak, 3600 307 20 31 22 6 Peat compact, ofoY- 250 a325,0 2:8 225 250 RATIO OF::T-HEI't:ARDNEfSSES. OF BODIES. 289 Of the Ratios of the Successive Hardllesses of Bodies. _ - Substances. ) Substances. > Diamond from Ormus, 20 1 37 Sardonyx, - 12 2'6 Pink Diamond;... 19 3-'4 Occidental amethyst, 11 2-7 Bluish Diamond,. 19 3'3 Crystal,..... 1 2-6 Yellowish Diaond,. 19] 3'- 3 Cornelian.... 11 27 Cubic Diamnond,.. 18 32 Green Jasper,...11 27 Ruby...... 17 42 Reddish yellow do.. 9 26 Pale rubty fron Brazil, 1 6 35 Schoerl], 10 36 Deep blue sapphire, 16 8 Tourmaline,. 10 30 Do., palel... 17 3-8 Quartz... 10 2-7 Topaz,:..... 15 4-2 Opal,... 10 2'6 Whiitish topaz,.. 14 3-5 Chrsolite,. 10 37 Rttby spinell,.. 13 3'4 Zeolite,..... 8 2'1 Bohemian topaz,. 11 2'8 Fluor..7 3' 5 Emerald,.... 12 28 Calcareous spar, 6 27 Garnet5..... 12 4'4 Gypsum,. 5 23'~ Agate...... 12 2 Chal k, 3 2-7 Onyx,....12 26 Ductility antl Me~lleability of Jfetals. Ductility is flie property of being drtawn out in length without breaking. This property is possessed in a pl)e-eminent degree by gold and silvel; as also by many ottler mliet als by glass in the liquid sta&te, and by- many semi-fluid resinous ~and gummy substances. Tlie: spider; and the silkworml exhibit the finest natural exercise of ductility, upon the peculiar TViscid secretions from which they spin their threads. When a body can be readily extended in all directions under the hammer it is said to be malleable; and when into fillets, under the rollingpress, it is said to be lamlinable. There appears, therefore, to be a real difference between due. tility and malleability; for the netals which draw into the finest wire are not those whicll afford the thinnest leaves nnder the hammer, or in the rolling press. Of this fact iron affords a good illustration. Among the metals permanent in the air-seventeen are ducetile and sixteen are brittle. But the mnost d~uctile cannot be wire-dlawn or laminated to any considerable extent without being annealed from time to- time during tlle progress of the extension, or rather the slidingof t|le particles alongside of each other, so as to loosen their lateral coliesion. 25 290 DUCTTLITY OIF METALS. TABLE Of the Ratio of the )Ductility and [l alleability of l~etals. Metals ductile and Brittle metals in Metals in the order MIetals in the order I malleble, illalpha- oetli heir wire-drawing oftheir lmninable betical order. dulphabetictility. ductility Cadmium. Antimony. Gold. Gold. Copper. Arsenic. Silver. Silver. Gold. Bismuth. Platinum. Copper. Iron. Cerium? Iron. Tin. Iridium. Chromium. Copper. Platinumni. Lead. Cobalt. Zinc. Lead. Magneisium. Columbium. Tin. Zinc. Mercury. Iridium. Lead. Iron. Nickel. Matnganese. Nickel. Nickel. Osmium. Molybdenum. Palladiumn 2 Palladium l Palladium. Osmlum, Cadmiumn Cadmnium l Platinum. Rhodiunm. Potassium. Tellurium. Silver. Titanium. Sodium. Tungsten. Tin. Uranium. Zinc. Gontcltclingy Pozoers of Various Sabstances: The conducting power of wood is very low; the softer woods:bring lower in this respect than those whlich are harder. Of metals,. and!some other substances, the following is the order, acevrding to Despretz:,Gold,......1000 Tin,. 304:Silver...... 97 3 Lead,..... 180 Cope,...... 898 Marble........ 24 Platinumr,... 381 Porcelain,.... 12 Iron,. 874 Tile,...... 11 Zinc,...... 36 Radiating Power of Varciouzs Substances. Bodies that brave polished surfaces radiate heat less than those that are roughened, and metallic surfaces less than those of more imperfect conductors. The following are the proportions of some of eaclh, according to Leslie: Lamp-black,... 100 Rough lead,.... 45 Water,...... 100 Mercury,... 20 Writing-pal;. 98 Polished lead..... 19 Glass,...,. 90 Polished iron,. 15 Tissue-paper... 88 Tiln, silver, copper, and gold, 12 Ice,.. SCALE OF PtOOFs FOR CHAIN RIGGING. 291 Reflecling Potwers of Various $Sbstances. Heat is reflected from the surface on which its rays fall, in the same manner as light; the angle of reflection being opposite and equal to that of incidence. The metals are the strongest reflectors of heat, in the following order, according to Leslie: Brass,....... 100 Lead,....... 60 Silver.9...... )90 Tinfoil rubbed wit mer., 10 Tinfoil,. 85 Glass,.. 10 Block-tin,... 85 Glass, waxed or oiled,. Steel,. 70 Power of Various Seubstances to 7ransmit Heat. All bodies capable~ of transmitting heat are, more or less, transparent, though their powers of transmitting heat and light are not in the same relative proportions; as the following list of the relative powers of equal masses, determined by Melloni, will show: Air........ 100 Rape-seed Oil.. 2 Rock salt, transparent,. 92 Tourmarline, green, 7 Flint-glass,. 67 Su-lphuric Ether,. 21 Bisuiphuret of Carbon,. 63 Gypsum,. 20 Calcareous spar, transparent,62 Sulphluric Acid,... 17 Rock-crystal,. 62 NNitric Acid,.... 15 Topaz, brown,... 57 Alcohol...... 15 CrofnVglass... 49 Ahlim, in crysstals,.. 12 Oil of tulpentine,... 31 Wate r.. 11 TAB LE Shlowing tAe Scale of Proofs for Chain -Rigging close-linked, &c.; the extrenle Length of Linhks 9not to exceed five diameters of their size ise I'on. Diam. of Testing Max. Minimum Diam. of Test. Maximum Minimum Links. Weight. Strain. Strain. Links. Wght. Strain. Strain. Inches.'ros.. Tons. Tons. Cwt' Inches. Tons Tons. Tons. Cwt. 1t 31j 75 68 0. I 55 14 13 10 1j.27 64 58 0 i 4 12 10 15 1Q 22i 54 49 0 3: 10 8t nearly. 18 45 41 0 3 7 6 18 1 15I 37 34 0 2)I 6 5 2 TT 1 12 30 280 l 10 4 3 0 0 10q 26 25 0 1I 3 2 14 9j 23 22 0 P A none broken. none broken. n-r| - 17 20 20 0 1} 1 14 8 i 17 16 0 y:v 13 ll- 0 19 292 MASONRY. MASOINRY. Of the ditferent kinds of i~asonry. Masonzry,'in the general acceptation of the- term, is the art of cutting or squaring stones, to be applied lo the purposes of build ing; or, in a more limited sense, it is the'art of joining stones toge tlier with mortar, or otherwise. The ancients enumerate seven different mnethods in which they arranged the stones of their buildings. Vitruvius thus classes them: three of hewn or squared stones, threw of unhewn, and one a mixture of both imethods. 1. Nlet masonry. This is represented in fig. 38, within the area D E F G, where the stones are squared and placed upon one of the angles, their joints.A 1-; —. thus forming a net-like appearance. This method, though very neat, is wanting in firmness and strength; for the oblique position of the stones, in regard to each other, gives them a tendency to separate.B.' rather than to form a compact assemblage Fig. 1. of parts thatr unite in supporting each other. Whenever this form of masonry is employed, it is consequently necessary to klieep th'e'work together by a border of stones, having some other arrangement, one that is not only capable of supporting itself, but of overcoming the resistance of the net-like form. This is shown in the same figure at A B'C; and where tlie network is merely a casing of stone to the, brickwork of a wall, it will be found to answer tolerably well, and loolks very neat. 2. Bound masonry is that represented in fig. 2, and is remarkably strong. The Iperpendicular joints in each course fall directly in the middle of the stones composing the course below and above it;. and while it has every requisite of solidity, the joints have, at the same time, a regular and pleasing appearance. Fig. 2. Fig. 3. 3. Greek masonry is that represented in fig. 3, where everalternate stone, as shown at A ID, EF, and BC, is made of thel whol:e thickness of the wall, and serves to bind together the stones MASONRY. 293 which compose the external and internal faces of the building; and this may be called double binding, asfrom-the perpendiculal joints being somewhat similarly situated to that in bound masonry, it has also an additional binding, by extending to the courses above and below it thus forming a compact and durable wall, which resists every effort to separate in any direction. 4. ilasonlr by equal cor?'es. This method of uniting stones is shown in fig. 4, and only differs fiom the bound masonry in its being composed of'unhewn stones, or rather in being formed of stones that are not so aeeui'ately cut, nor the edges so perfectly squared; it being only necessary that the external face should be level, and the horizontal joints at equal distances fiom each other, care being taken at tlie same time that the perpendiculars are so situated as to bind'the courses above and below them. Fig. 4. 5. Masonry by unequal courses. This is represented in fig. 5, andl is, like the last, formed of unbewn stones, without any regularity as to their size, it being sufficient that each course is made to bind with the A IPreceding, and the only liegclarit y observed is in the joining which sepatIates cach course, the courses themselves being of unequal thickness, as shown at A BC D., 6. Mllasonry filled up in the middle,, as shown in fig. 6, is formed of unhewn stones of unequal courses, and the middle, as at D, is filled up with stones thrown in at random among the mortar. Fig. 6 7. i. Compound masonry is, as its name imports, a mixture of the other kinds. It is repslesented infig. 7, where the external course A. B is formed of bound masonry, and the corresponding internal course is at some distance from it, but held to the former by means of iron cramps, as shown at a, b, c, d, c, f, the space between being filled in with small stones or flints thrown into the mortar. 25 _ - -- 26~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 294 MASONRY. The JMethods of Joining Stone. As the strength and durability of masonry depend as much on the method employed, and the care taken in making all the joints to correspond accurately with each other, as in the quality of the material employed, some remarks will be required in explanation of the methods of joining stone. We shall, tlierefore, enumerate the several means adopted by workmen, and, where nlecessary, notice the purposes to which each method is best adapted, giving some cautions to secure success in practice, and to save the workman unnecessary labor and trouble. The joints in masonry are either secured by the means of mortar, cement, or plaster of Paris, or the courses are held together by cramps, joggles, mortice, and tenoning, or dovetailing. 1. Joining by mortar, or cements. It is absolutely necessary that the joints should be perfectly smooth, and touch in every part; and the' stones must be so square as to bed well on eachl other, that is to say, they must not have such irregular faces ias to roll, or, in technical terms, be winding to each other. The greatest care must be taken by the workman to have his mortar of a proper consistence-not too thin, as in drying it would shrink from the work, inor too thick, for that would prevent the stones from bedding properly. The best way in irregular masonry, or in that composed of small stones thrown, as it were, between the regular work, as in compound masonry, is to saturate fresh lime with water, and, while hot, to pour it on the work, which hardens and consolidates the whole into one solid mass. This method is much used in. joining soft stones and brickwork, and is calculated to promote the strength and solidity of the work. 2. Joining by cramps. Cramping is performed by inserting into the two pieces of stone, which are to be bound together, a piece of iron or some other metal, the ends of which, bent at right angles, are inserted in a cavity cut in each stone, the cavities being so large as: to admit the iron easily; melted lead is then poured in to fill the vacant space, and, when cold, a chisel is driven into it, so that it may press close to the work; for all metals expand by fusion, and obstacles may prevent them from contracting in cooling. Cramps composed of copper are, in many cases, very preferable to those made of iron, for they are less likely to oxidize, or rust, or to be affected by the lime or mortar. It would be of advantage to coat the cramps, if made of iron, with some substance that would defend them from the effects of damp. We may here remark, that the chlanlnel made to receive the cramp should be dovetailed, to prevent the lead:from coming out, which it is otherwise apt to:do, in the course of time. The only objection to the use of copper cramps, in preference to iron, is their expense, which in large public works is not of any importance, and, for common purposes, iron answers very well; but the more malleable or tough the iron IMAsoNra. 295 i the better it is, as it is more calculated to resist the different:temperatures to which the work may be exposed. 3.:Joninig by joggles. The method of securing the joints -of m1asonry by means of joggles is chiefly adopted for securing the joints of columns or pillars; and consists in sinking a cavity in the twopieceslin such a manner as to make them correspond with each other, and inserting in that cavity a piece of metal, stone, or even wood, so that any lateral thrust may not be able to separate them. This method may, with very great advantage, be applied in the construction of domes, and w-orks of the same nature, where it is necessary to avoid the lateral thrust as much as possible. We may here take the opportunity of mentioning a plan proposed by Dr. Hutton, in his edition of Oznamare's Mathematical Recreations, for talking away the lateral thrust of domes and cupolas. The following is the problem proposed, and the solution given: "How to construct a hemispherical arc/i, or what the architects call an are' en cul-de-fo'ur, which'sh/ll have no thrust on its piers.' Let A B, fig. 8, be two contiguous voussoirs, which we will suppose to be three feet in lengtli, and eighteen inches in breadth. Cut out on the contiguous sides two cavities, in thie form of a dove-: tail four inches in A a depth, -with an aperture of the same exi~ r~~~~~~ig. s. tent, a, 6, five or sixF inches in length, and as Imuch in breadth. This cavityTwaill serve to i'eceive a double key of cast-iron, as shown in fig. 9, or of common fbrged-iron, which is still more secure, as it is not so brittle. These two voussoirs will thus be connected together in such a'manner that tley'can -[ I not; be separated without breaking the dovetail at V the re-entering angle; but, as each of its dimensions Fig.. in this place will be fourinches, it w-ill be easily seen that an immense force would be required to produce that effect; for we'are taught. by vwell-known experiments on thestrength of iron, that it requires a force of four thousand five hundred'pounds to break a bar.f'foi-ged iron an inch square, by the arm of a lever of six.inches; consequently, two hundred'and eighty-eight thousand pounds would be necessary to break a bar of sixteen square inches, like that in question. Hence there is reason to conclude, that these voussoirs will be connected together by a force of two hundred and eightyv-ight thousand pounds; and. as they will'never:eixperience,,an effort to disjoin them nearly so great, as might easily be proved by calculation, it follows that they may be considered as one pi'ece." They might be still further strengthened in a'very considerable degree, forhe height othehese dovetails might be made double, and 296 MASONRY. a cavity might be cut in the middle of the bed of the upper voussoir, fit to receive it entirely: the dovetail could not then be broken without breaking the upper voussoir also; but it may be easily seen that, to produce this effect, an immense force would be required. The second method proposed by Dr. Hutton is more properly by the aid of joggles. Let A and B, fig. 10, be two,..Ad \ contiguous voussoirs, 0, T\ and C, fig. 11, the inA_0 Y verted voussoir of the next course, which Fir. 10. ought to cover tile joint between A and B. Each of the voussoirs A and B being divided into two parts; as a b and cd; then if atc ab and c d we sink an hemispherical l Q BC hi O t cavity, in which to introduce a globe of! /!..-'! very hard marble, and in the upper voussoir,. fig. 10, wve sink similar cavities, b c; this, Fig. 1. whmhen laid: on be, fig. 11, will form a secure joint without any lateral thrust; and the two courses cannot be separated without a force adequate to either break the solid stone, or disunite the marble globe; a force almost inconceivatle, or at least one far superior to that produced by the arch; the wthole dome, or cupola, is, in fact, one solid mass, and can exert no lateral thrust upon the,walls on which it is raised. Marble globes are recommended, because iron is. liable to rust; but, if the joggles were made of iron, and covered with pitch before they were placed in the cavities, there would be little to fear from rust; and particularly as the iron is inclosed in the substance of the stone, anrd'quite excluded from the action of atmospheric causes. Little need be said in'this place as to morticing and tenoning, or dovetailing, except tlhat they differ slightly from the same operations in joiners' work; for, as cement is used in the joiiing, they need not be so accurately cut, and are made shorter and thicker than those formed by the joiner, it being sufficient that the parts of each piece to be joined enter into each other at most five or six inches, even in large masses of stone. In small pieces, an inch or an inch and a half is sufficient; for, if the tenon or dovetail be too long, it will decrease the solidity of the joint. For greater security, a small channel is frequently cut in the shoulder of the joint, and melted lead is poured into it, which, filling up the space round the tenon or dovetail, makes the joint more secure, and the work firm and solid. In laying some sorts of stones, it is desirable, as far as possible, to place them in thle same' direction as they had when in the quarry, or, as it is termed by workmen, bedways of the stone; for, if laid in other directions, they are liable to peel and split by the action of the atmosphere. [BRICKL.AY'NG. 29. B RI CKLA YIN G. Foundations. The best soils for building upon are gravel, chalk, and stone rock. Those most to be guarded against are sands, bog earth, clays, -and made earth (no matter how hard). Where these occur, avoid -piling (except in water works); plank the foundations through the centre of the walls, place long tassels in the piers, lay ilr chain bond, let the plates be stout, and in one piece, the whole length of each wall; all that is required is to' so bind the building that it may settle altogether, and not partially. In doubtful foundations, it is advisable to have,: trench duer out to the depth of 2 feet to 3 feet below the footings of the brickwork, and about twice the width of the footings, which is to be filled up with cotncrete, composed of stone lime ground and ballast, or coarse gravel, to be mixed with water, in the proportion of one of limtle to five or six of gravel; immediately that it is made up it must be shot into the trench fromn a stage, 6 feet above, which will cause it to fall in a solid mass; and inl a few hlours afterwards it will be as firm as a rock. It is strongly recommended to have good plates; whatever may be' slighted in other parts these should not be neglected-they are thie soul and support of a building, and cannot, if put in too snall, be taken out'and replaced, as other timbers m-ay; tlie difference in large houses will rarelv amount to twenty —five dollars. Bond the work-English bond-using:all whole bricks, a course of stretchers and header's alternately. Particular care must be taken that all the internaljoints of brickwork are well flushed up with mortar; too frequently the wo]ikmen are apt to neglect doing it; the consequence is, that all the interior joints are hollow, and allow the damp to penetrate to the inside, no matter how thick the wall may be. Another serious defect in brickwork is in not properly bonding the facing to the backing, particularly if the facing be malms or bricks, which cost an extra price; generally the headers, are only bats or half bricks, inst:ead of being a stretcher' or a whole'brick to bond in with the brickwork at the back; there ought to be at least one stretcher in every 3 feet to each course, if there be not the wall will split or divide into two thicknesses. In building arches of a large span, it is advisable to build themr in: half brick rims, with vertical or6 radiating bond every 3 or 4 feet in the gitt; if this latter precaution be not' adopted, the conseqlience will be, that when the centre is' struck, the rims will divide and weaken the arch, and perhaps cause a total failure. In selecting bricks, clap them together-if they ring well, and, C,298 IBRICKLAYING, when broken, sliow that they are btrnt tllrough, they will answer the purpose. A hard clamp burnt gray stock is all that is wanted for strength; for wvater-worlks and foundations use clinker burnt marl stocks. Avoid samnel or place bricks, and cluffy stocks, and generally prefer hand tempering to pugging the clay. In mixing of mortar, it is advisable to see that the laborer properly turns uptlle mnortar, and that the lime is thoroughly incorporated with the sand tllroughout;; avoid usizg too much wate;ras it drowns the linle and weakens it; in large works it is best to mix the lime and sand in a mill-cement must be mixed in small quantities. TAB LE S/iowving the Qucanttit?/ of Eartth to be removed, the Lm7onber (f Bricks aand Gallons in one foot in depth 01or length. Da Brick rim. 1 Brick irh. Diamir. in Ihe __ _- ontents clear. Feet cube Number of bricks. Feet cube Number of bricks. in of dof e lai egallons. ft.. in. digging. laid dry. in morta digin. laid dry. in mortar. 0 9 1 8 23 19 40 GO 50 2 10 2 4 28 23 49 570 58 5 1 3 ( 31 33 2 7 5 9 80 66 00; 1'6 4 0 38 31 7 1 90'74 1 1 9 4 9 43 35 83 102 82 i 5 "2 0 5 9 48 41 9 ]112 92 19k 2 3.71 53 44 11 0 122 100 25 1 2 6 83 58 4S 12 6 132 108 30,1 3 11 0 68 57 1 5 9 154 126 i 44 3 6 14 2'79 5 6 19 6 174 142 60 4 0 17 7 89 3 23 8 194 159 78 4 6) 21 6 100 82 1 28 3 214 176 100 5 0 26 o 110 90 33 2 234 12 129 5 6 30 7 120 98 38 5 254 209 149 6 0 35 8 130 107 44 2 276 226 176 6 6 41 3 140 1 15 50 3 296 242 206 7 0 47 2 150 123 56 7 316 260 239 7 6 53 5 160 131 63 6 336 2'76 275 8 0 60 1 170 140 1 70 9 358 9292 1 31! 8 6 67 2 180 148 785 37 8 308 / 35-L 9 0 I 4 7 191 156 1 6 398 1 326 a396 1 0 OI 90 8 212 174 1 103 9 438 360 489 I_ I_,_ _ _ _ _ _. _ _l I. _ _ _ _ _ BitCRLAVtNO.'29' Llr the measurement of brickwork no allowance is to'be made in quantity for small or difficult works. Flues to be measured solid. Timbers inserted in walls not to be deducted. Two inches to be allowed for bedding plates, where no brickwork is over them. All cuttings to be measured superficially, excepting to bird's mouths and squint quoins, which are to be run. The net quantity of brickwork being found, it is to be reduced to the standard thickness of a brick and a half, and brought into statute rods of 51 yards square, or 272 superficial. Ovens, coppers, and solid walls, of irregular thickness, to be cubed and brought into the standard thickness, by multiplyirig by 8 (the number of 1I inches in a foot), and dividing by 9 (the number of 1~ inches in a brick and a half, or 13~ inches, the standard thickness). Facings of all descriptions to be measured and charged extra, per foot superficial. 272 feet superficial is a rod of brickwork, 1~ brick, or 13~ inches thick, the standard thickness, to which all brickwork, of whatever thickness, is reduced.:806 cubic feet, or 11~ cubic yards, equal to 1 rod of reduced brickwork. 4352 stock bricks to 1 rod reduced, 4 courses 1 foot high. 4533 ditto, if the 4 courses measure 11~ inches high. These calculations are without allowing any waste, which is more than amply compensated in dwelling-houses, by not deducting flues aind bond timber; in such work, 4300 stocks, or 4500 place, are sufficient,. 5371 bricks laid dry to 1 rod. 4900 ditto in wells and circular cesspools. A rod of brickwork contains 235 feet cube of bricks, and 71 feet of mortar (4 courses to a foot); which will weigh, upon an average calculation, 15 tons. A rod of brickwork requires 1 cubic yard of chalk lime, and 3 single loads or yards of drift; or 1 cubic yard of stone lime, and 38 single loads or yards of sand; or 36 bushels of cement, and 36 of sharp sand. 16 bricks to a foot of reduced brickwork. 7 ditto to a foot super of facing. 1-0 ditto to a foot super of gauged arches. 30 bricks on edge, and 45 bricks flat, to 1 yard of brickY:ogging. 36 stocks laid fiat, and 52 ditto on edge, to 1 yard of paving. f36 paving bricks laid flat, and 82 ditto on edge, ditto. A load of mortar, 27 feet cube, requires 9 bushels of lime and: I yard eof:and. A hod contains 20 bricks. Lime and sand loses one third of its bulls when made into mortar -likewise.cement and sand. 8300 PLASTERING. The proportion of mortar, or cement, when made up, to the lime, or cement andisand before made up, is as 2 to 3. Lime, or cement and sand, to make mortar, require as much water as is equal to one third of their bulk, or about 5~~ barrels for a rod of brickwork built with mortar. P LAST E P IN G. Thickness of Compo. Inch yards. a~ inhe yards. Y2 inch yds. 1 bushel of cement will cover. 1-... 1... 24 1 do. and l of sand do.. 2... 3.. 1 do. and 2 do. do.. 3.4.. 4:. 1 do. and3 do. do.. 41... 6... 9 (4 inch is the usual thickness.) 1 cwt. of mastic and 1 gallon of oil.. 1.. 2. 2 1 cubic yard of chalk lime, 2 y:ards of road drift or sand, and 3 bushels of hair, will cover 15 yardls of render ond set on brick, and 70 yards on lath, or 65 yards, plaste;r and:renlder 2:coats:andset on brick, and 60 yards on latll; floated wo'rk will require about the same as 2 coats an'd set. 1 bundle of:laths and 500 nails will cover aboutt 44 yards.;[ortar. 1 hundred of lime contai:ls 25 striked bushels, or 100 peceks. It is a measure 3 feet square, and 3 feet 1 inch deep. 1 chaldrion of lime is equivalent to 57'765 cubic feet, or rather llore than:2 hundred. 18 heaped bushels, 22 striked bushels, or I yard cube, a single load of sand, Ilortar, &c. 1 double load is equal to 36 heaped bushels. 1 hod of mortar is equal to 1134 cubic inches, or 8 duodecimal inches, or 9 x 9, and 14 inches long. 2 hods of mortar snake a bushel nearly. Cemn'ent. 1 barrel of cement is 5 bushels, and weiglls 3 cwt.: 1 rod of1brickwork, in cement, requires 36 bushels of cement and 36 bushels of sand. 1 yard, or 9 feet stuerficial of 14 inches, or 1-7 brick-work, in cement, requires about 24 bushels. 1 yard superficial of pointing to brickworkl in cement, requires about one eighth of a bushel. 1 yard square of plastering, in cement, requires three foutlthls of a bushel. PLASTERING. 301 UCarpcntry and Plastering are measured by the square foot orI yard; or, in moulded and ornamental work, by the linear foot. In extensive work the square of: 100 feet is. also used. Paving is measured by the square yard. Diggiirtg, dc. 23'~ cubic feet of' sand, 17~ ditto clay, 18 ditto' earth, 13 ditto chalk, equal to a ton. A cubic yard of earth, before digging, will occupy about 1: cubic yard when dug. 27 cubic feet, or 1 cubic yard, contains 21 striked bushels, which is considered a single load, alnd double these quantities a double load. 11.8: cubic feet of night soil, 1 ton. 21,tons of ditto, is the quantity: a cart, contains.; 6 feet longl, 3' feet 3: inches wide,, by 2 feet 4 inches deep, or 45 feet cube. Coarse. Stz~f. Coarse stuff, or lime and: hair, as it is sometimes called, is p'repared in the same way as common mortal,'with the.addition, of hair procured from the tanner, -vwhich must be well mixed with the mortar by means of a three-pronged. rake, until the hair is equally distributed throughout the composition, The' mortar should be first formed, and when: the lime and sand: ha.ve been: thoroughly imixed, the hair should be added by degrees,' and the whlole so tlioroughly united that the hair shall app.ear to be equally distributed.throughout. Fine Stuzd. This is made by. slaking lime'with a sm-all portion of water, after which so much water is added as to give it the consistence of cream. It is then allowed to settle for some time, and the superfluous water is poured off, and the sediment is suffered to remain till eraporation reduces it to a proper thickness for use. For some kinds of work it is lecessary to add':a. small portion of -air. Stucco for'Inside of JValls. This stucco consists of fine stuff already described, and a portion of fine washed sand, in. the proportion of one of. sand -to three of fine stuff. Those parts'of interior walls. are finished. witl this stucco which are intended to be painted.. In using this material, great care must be taken that the surface be perfectly level, and.to I secure this it must be well worked.with a floating tool or wooden trowel. This is done. by sprinkling a little water occasionally. on the stucco, and rubbing. it in. a circular direction, with thle float, till the surface.lias attained a high gloss. The durability. of tlhe work. very much depends.upon the care with which this process is done, for if it be not thorouglly worked it is apt.to crack.. 26 Gcaoge Stlr This is chiefly used for mouldings andcl cornices which are run or formed with a wooden moulcd. It consists of about one fifth of plaster of Paris, mixed gradually with four fifths of fine stuff. When the work is required to set Very expeditiously, the propolrtion of plaster of Paris is increased, It is often necessary that the plaster to be used should have the prop)erty of setting immediately it is laid on, and- in all such ceases gauge stuff is used, and consequently it is extensively employed for cementing ornaments to walls or ceilings, as well as for casting the ornaments themselves.'inigs' iSteceo. To fifteen pounds of tIle best stone lime add fourteen pounds of bone ashes, finely powdered3 and about ninety-five pounds of clean, washed sand, quite dry, either coarse or fine, according to the nature of the work in hand. These ingredients must be intimately mixed, and kept from the air till wanted. When required for use, it musit be mixed up into a proper consistence for working with lime water, and used as speedily as possible. Parker's Cement. This cement, which is perhaps the best of all others for stucco, as it is not subject to crack or flake off, is now very commonly used, and is formed by burning argillaceous clay in the same manner that lime is made; it is thein reduced to powder, by the process described in a previous part of this work. The cement, as used by the plasterer, is sometimes emplloyed alone, and sometimes it is mixed with sharp sand; and it has then the appearance, and almost the strength, of stone. As it is impervious to water, it is very proper for lining tanks and cisterns. Heamelein's Cement. This cement consists of earthy and other substances insoluble in water, or nearly so; and these may be either those which are in their natural state, or have been manufactured, such as earthenware and china; those being always preferred which are least soluble in water, and have the least color. When these are pulverized, some oxide of lead- is added, such as litharge, grav oxide, orI iinium, reduced to a fine powder; and to the compound is added a quantity of pulverized glass or flint stones, the whole being thoroughly mixed and made into a proper consistence with some vegetable oil, as that of linseed. This makes a durable stueco or plaster, that is impervious to wet, and has the appearance of stone. The proportion of the several ingredients is as follows:-to every five hundred and sixty pounds of earth, or earths, such as pit: sand, river sand, rock sand, pulverized earthenware or porcelain, add forty pounds of litharge, two pounds of pulverized glass or flint, one pound of minium, and two pounds of gray oxide of lead. MIix ------— ~~ —------------ _ PLASTERING. 303 the whole together, and sift it through sieves of different degrees of fineness, according to the purposes to which the cement is to be applied. The following is lthe method of usillfr it:-To every thilty pounds' weiglit of the cement inp lpo der add about one qulrt of oil, either linseed, alnut, or some othieir vegetable oil, and mix it in the same inanner as any other molrtar, pressing it gently together, either by! treading on it, oi1 with the trowel; it ihas then the appearance of mioistened sand.'- Care mnust also be taken th-tt ilo mnore is mixed at1 one time tlhan is ieqired fort use, as it soon hardens into a solid mass. Before the cement:is':aplied, the face of the w\all to be plastered shounld be brushed Oiver with oil, particularly if it be applied to brick, or any other substance that quickly imrbibes tlle oil; if to wood, lead, or any substance of a' similar nature, less oil may be used. iMalthia, or G' reeA: 2ifctstZ This is made by mixirgf lime and sand in the manner of mortar, and making it into a pioper consistenlcy witli milk or size, instead of water. Plaster in ii7nitations of ls arble. This species- of work is exquisitely beautiful whllen done with taste and judgment, anid is so like marble to the touch, as well as appearance, that it is scarcely possible to distinguish the one from thle other. We shall endeavor to explain its composition, and the nmanner in which it is applied; but so much depends upon the workman's execution, that it is impossible for any one to succeed in an attempt to work with it without some, practical experience. Procure some of the purest gypsum, and calcine it until the large masses have lost the brilliant sparkling appearance by which they are charaecterized, and the. whole mass appears uniformly opaquec. This calcined gypsum is reduced to powder, and passed through a very fine sieve, and mixed up, as it is wanted for use, with Flanders glue, isinglass, or some other material of the salme kind. This solution is colored with the tint required for the scagliola, but. when a lmarble of various colors is to be imitated, the se-eral colored comipositions required by the artist must be placed in sdearate vessels, andl they aiLe then mingled together in nearly the same manner tliat tlie painter mixes his color on the pallet. Hatring the'wall or column prepared with rough plaster, it is covered with the composition, and the colors intended to imitate the marble, of whatever kind it nmay be, are applied when the floatilng is going on. it now only remains to polish the worki, which, as soon as the eomposition is hard enough, is d one by rubbing it with.pumicestone, the work beinli kept wet with water applied by a sponlge. It is then polished with Tripoli and charcoal, with a piece of fine linin, and finished with a piece of felt, dipped in a mixture of oil and Tripoli, and afterwards with pure oil. 304 PLASTERING. Copnpositiown. This is frequently used, instead of plaster of Paris, for the ornameDntal p)arts of buildings, as it is rnore durable, and becomes in time as larld as stone itself. It is of great use in the execution of the decorative parts of architecture, and also in the finislings of picture fialmes, being a cheaper method than carving, by nearly eighty per cent. It is mnade as follows: Two pounds of the be-'. whitening, one pound of glue, and half a pound of linseed oil are heated together, tile conmpositionl being continually stirred until the differe;nt sub. stances are thoroughly incorporated. Let the compound cool, and then lay it on a stone covered with powdered whitening, and heat it well until it becomes of a touph and firm consistence. It may thell be put by for use, covered with wet cloths to keep it fresh. When wanted for use it must be cut into pieces, adapted to the size of the mould, into vwhich it is forced by a screw press. The ornaient, or cornice, is fixed to the frame or wall with glue, or with white lead. To make Glass Paper. Take any quantity of broken glass (that with a greenish liue is thle best), and-pound it in an iron mortar. Then take several sheets of paper, and cover them evenly with a thin coat of glue, and, holding them to the fire, or placing them uponl. a hot piece of wood or plate of iron, sift the pounded glass over them. Let the several sheets remain till t;he glue is set, and sliake off the superfluous powder, which will do again. Then hang up the papers to dry anid harden. Paper made in this nmanner is miluch superior to that generally purchased at the shops, whicl chiefly consists of fine sand. To obtain different degrees of fineness, sieves of different degrees of fineness:must be used. To make Stone Paper. As, in cleaning wood-work, particularly deal and other soft woods, one process:is sometimes found to answer better than another, we may describe the ilalnner of'manufacturing a stone paper, wvhich, in some cases, will be preferred to sand paper, as it produces a good face, and is less liable to scratch'the work. Having prepared tthe paper as already described take any quantity of piowdered punmice-stone, and sift it over tile paper through a sieve of moderate fineness. When the surface has hardened, repeat tlhe process till a tolerably thick coat Ihas been formed upon the paper, which, when dry, will be fit for use. Y,~~~~~~~~~~~~~~~~~~~~ WOODWORK, CAnPENTRY, ETC. 305 WOODWORK, CARPENTRY, &c. Decay of Woodl. Some woods decay much more rapidly than others; but they will all, in some situations, lose their fibrous texture, and with it. their properties. To ascertain the causes which act upon woods, and effect their destruction, is an important. object both to the builder and to the public. Cause of the Decay of Tizmbcr. All vegetable as well as animal substances, when deprived of life, are subject to decay. If the trunk or branch of a tree be cut horizontally it will be seen that it consists of a series of concentric layers, differing from each other in color and tenacity. In distinct species of trees these layers present very different appearances, but in all cases the outer rings are more porous and softer thail the interior. Wood is essentially made up of vessels and cells, and the only solid parts are those coats which form them. These vessels carry the sap which circulates through the tree, gives life and energy to its existence, and is the cause of the formation of leaves, flowers, and fruit. But when the tree is dead, and the sap is still in the wood, it becomes the cause of vegetable decomposition by the process of FERMENTATION. There are five distinct species of vegetable fermentation-the saccharine, the coloring, the vinous, the acetous, and the putrefactive. We are indebted to Mr. Kyan for the discovery that albumen is the cause of putrefactive fermentation, and the subsequent decomposition of vegetable matter. Circumstances favorable to Vegetable Decomposition. Wood is not equally liable to decay under all circumstances. When thoroughly dried it is not so quickly decomposed as when in its green state, for in the latter condition it has in itself all the elements of destruction, and it is scarcely possible, to prevent the effect if it be then used in building. But supposing the timber to be perfectly seasoned, it is more liable to decay under some circumstances than in others. Timber is most durable when used in very dry places. When timber is constantly exposed to the action of water; the decomposition effected will depend upon the nature and chemical composition of the substance. A portion of wood may be soluble in water, but other parts are not; so that after a definite period, the continued action of water upon a piece of timber ceases, and if it can sustain the influence of this cause until that period there is no termination to its endurance, except from those casualties which it might have been able to bear in its original state, but cannot after the removal of that portion of its substance soluble in water. 26 806 WOODWORK, CARPENTRY, ETC. Should a piece of timber that has been for a long time exposed to water be brought into the air and dried, IT WILL BECOME BRITTLE AND USELESS. When wood is alternately exposed to the'influence of dryness and moisture it decays rapidly. It appears, from experiments, that after all the matter usually solutble in water has been removed, a fresh maceration and contact of the air produces a state of matter in that which is left which renders it capable of solution. A piece of timber may then in this manner be more and more decomposed, until at last the whole mass is destroyed. The builder is sometimes compelled to use wood in places' where it will be.exposed to alternate dryness and moisture; fencing, weather boarding, and other works, are thus exposed. In all these cases he may anticipate the destructive process, and provide against it. The wood used in such situations should be thoroughly seasoned, and then painted or tarred; but, if it be painted when not thoroughly seasoned, TIHE DESTRUCTION wILL BE HASTENED, for the evaporation of the contained vegetable juices is prevented. There is one other circumstance to be considered-the influence of moisture associated with heat. Within certain limits the decomposition resulting from'moisture increases with the temperature. The access of the air is:not absolutely necessary to the carrying on of this process, but water is; and as it goes on, carbonic acid gas and hydrogen gas are given off. The woody fibre itself is not friee from this decomposition, for, as tlhe carbonaceous matter is abstracted by fermentation, it becomes more susceptible of this ichange. This statement is proved by the circumstance, that when'quicklime is added to the moisture the decomposition is accelerated, for it abstracts'carbon; but the carbonaite of lime produces no such effect: a practical lesson may b'e learnt from this fact; if timbers be bedded in mortar, decomposition must follow, for it is a long time before it can absorb sufficient carbonic acid to neutralize the effect, and the dampness which is collected by contact with the wet mortar increases the effect. When the wood and the lime are both in a dry state'no injury results, and it.is well known that lime protects wood from worsms. When the destructive; process first becomes visible it is bvy the swelling of the timber, and the formation of a mould, or fungus upon its surface. This fungus'or cryptogamic plant rapidly increases, and:soon covers over the wholef surface of a piece of timber, having a white, girayish-white, or brownish hue. When the seeds of destructioli are thus onsce sown they cannot be readily eiradicated. Heat and moisture may be considered the prominent'causes of the rapid decomposition of vegetable substances. When wQod is completely and constantly covered with water this effect is not produced; and wev have an example in the fact, that, although those parts of a vessel which are subject to an occasional moisture. are liable to dry rot, yet those: parts which are constantly beneath the wvater are not ever thus affected; and although the -head Qf a pile, which may be nowh and then wetted' by the casual rise of the tide, WOODwORK, CARPENTRY,.ETC. 307 and is then dried again by:the sun, may be decomposed, yet those parts which are always covered with water have been found in a solid state after CENTURIES of immersion. MBeans of Preventing Deceay. Something may be done towards the prevention of decay by felling the timber at a proper season. A, tree may be felled too soon or too late, in relation to its age and to the period of the year. A tree may be so young that no part of it shall have the proper degree of hardness, and:even its lieart-wood may be no better than sap-wood; or a tree may be felled when it is so old that the wood, if not decayed, may have become brittle, losing all the elasticity of maturity. The time required to bring the:scveral kinds of trees to maturity varies according to the nature of the tree and the situation in which it may be growing. Aluthors differ a century as to the age at which oak should be felled, some say one hundred, and others two hundred years; it must, then, be regulated according to circumstances. But it is also necessary that the timber trees should be felled at a proper season of the year; that is to say, when their vessels are -least loaded with those juices which are ready for the production of sap-wood and foliage. The timber of a tree felled iz sp rivng or ii1 autumnn would be especially liable to decay; for it would contain the element of decomposition. Midsummer and midwinter rce the proper times for cutting, as the vegetative powers are then expended. There are some trees, the bark of which is valuable, as well as the timbei'; and as the best time for felling is not the best for stripping the bark, it is customary to perform 1these labors at different periods. The oak-bark, for instance, is generally taken off in earl' spring and the timber is felled AS SOON 0 AS THE: FOLIAGE iS DEAD; and this method is found to be highly advantageous to the durability of the timber. The' sap-wood is hlftrdened, ahd'all the available vegetable juices are expended in the production of foliage. Could this plan be. adopted with other trees, it W-ould be desirable; but the barks are not sufficiently valiable to pay' the expense of stripping. Seasonsing Tinber. Supposing all tthese precautions to be taken'in flIling timber, it is still necessary to season it; that is, to adopt som'ie means by which it may be dried, so as to tlhrow off all the juices which are still associated with the fibres of the wood. As soon as the timber is felled, it should be removed to some dry place; and, being piled in such a manner as to admit a circulation of air, remain in log for some time, as it has a tendency to prevent warping. The next process is to cut the timber into scantlillgs, and to place these upright in some dry situation, where there is a good current of air, avoiding the direct rays of the sun. The more gradually the --— ~ ~~~era s.Th m ual h 308 WOODWORK, CARPENTR~Y, ETC. process of seasoning is carried on, the better will be the wood for all the purposes of building. Mr. Tredgold says, "It is well known to chemnists, that slow drying will render many bodies less easy to dissolve; while rapid drying, on the contrary, renders the same bodies more soluble. Besides, all wood, in drying, loses a portion of its carbon, and the more in proportion as the temperature is higher. There is in wood that has been properly seasoned a toughness and elasticity which is not to be found in rapidly dried wood. This is an evident proof that firm cohesion does not take place when.the moisture is dissipated in a high heat. Also, seasoning by heat alone, produces a hard crust on the surface, which will scarcely permit the moisture to evaporate from the internal part, and is very injurious to the wood. "For the general purposes of carpentry, timber sh7ould not be uised in less lhan two years after it is felled; and this is the least time that ought to be allowed for seasoning. For joiners' work it requires four years, unless other methods be used; but, for carpentryv, natural seasoning should have the preference, unless the piessuie of the air be removed." Many artificial methods of seasoning timber have been proposed; and a brief notice of some of those which have been found most useful will be required. Seasoning by a Vacuum. All the vegetable and animal juices are kept in their particular vessels by the pressure of the atmosphere: remove that pressure, and the animal fluids could no longer be retained by the veins and arteries; and the vegetable fluids would exude and appear on the surface of the plant. Place a small piece of wood beneath the S'eceiver of an air-pump, and exhaust the air, and in a short time the wood will be covered with drops of the liquid which can no longer be retained, as the atmospheric pressure is removed. Mr. Langton thought that this might be applied to the extraction of those vegetable juices in timber, known to be the cause of its decay. All arrangement was therefore adopted, by which large masses of timber might be inclosed in a vessel having such maclinery as would be necessary to exhaust the air, heat being at the same time employed so as to vaporize the exuded juices. The vapor is conveyed away by pipes surrounded by cold water, and is condensed into liquid having a sweet taste. This process is deserving of more attention than has hitherto been given to it. Water Seasoning. It has been stated, by various writers, that wood immersed in water for about a fortnight, and then dried, is better suited for all the purposes of the joiner. There canll be no doubt that immersion in water tends to neutralize the effect of the saccharine matter, by dilution or an almost absolute removal. This process has also the effect of rendering the wood less liable to crack and warp; but, if W OODWORIK, [UARPENTRY, ETC. 309 we judge by Duhamel's experiments, it injures the strength of the material, and should not, therefore, be adopted in any instance where the timber is to be employed by the carpenter. Evelyn recommends boards that are to be used for flooring to be seasoned in this way: "Lay your boards," he says, "a fortnight in water, (if running the better, as at a mill-pond; head;) and then setting them upright in the sun and wind, so as it may pass fieely thrlllough them, turn them daily; and thus treated, even newly-sawn boards Will floor far better than those of a many years' dry seasoning, as they call it." Timber intended for ship building may be immersed in sea water; but that which is to be used for houses ought to be placed in'fresh water; for if timber, or any other building material, be impregnated with salt, it will ever be wet, for salt attracts moisture' so readily that it may be used approximately as a hygrometer. Plaster or mortar made with salt water will always sweat with a moist atmosphere; and timber intended for the house carpenter, if impregnated with salt, will always be damnp, or covered with a crystallized efflorescence. Much injury, however, is sometimes done by not thorougzhly inmersing the timber; the carpenter should therefore be careful when he enmploys this method of seasoning, that the timber is entireldIy covered with water, and that it be not exposed to its action for too long a time. Seasoning by Smnoking and Charrinzg. Authors who have wviitten upon the seasoning of timber liave,spoken of the effects of smoke, and the carbonization of the surfiace. We have adopted the same arrangement,'but it will'be necessary to caution the reader against a misconception of a very inaccurate expression. Timber cannot- be seasoned:by either smoking or charring, but seasoned timbers may be made more capable of resisting the effects of certain situations by these processes. Should a piece of timber, containing the vegetable juices, be smoked or charred, it would be a means of accelerationg decomposition; for preventing all means of evaporation, the common sources of protection would become sources of destruction. But when timber is to be used in situations where it is liable to be attacked by worms, or' to produce fungi, it. may be desirable'to smoke- or to' char it. Seasoni-ng by Boilintg or Steaminng. Timber is sometimes seasoned by steaming or boiling, both of which means are frequently adopted by ship-builders. The strengthll of timber appears to be somewhat inmpaired by these processes, but it is generally less liable to slhrink or crack. Duhamel states that he boiled a piece of wood, and then dried it upon a stove, but in drying it, it lost part of its substance, as well as the water contained, and, upon a repetition, he found that it had lost still more of its weight. Four hours' exposure to steam or boiling water is sufficient for timbers of ordinary dimensions, and the drying afterwards goes I _______________ _____ _ _ - 310 VWOODnwoaR, CA:,PENTRY, ETC. onl,ery rapidly, but it should be done as gradually as possible. The joiner frequently. finds it necessary to steamn or boil ood, to bend it into a particular curve, and also the ship-builderl. It has been sta.ted bv writers on ship-building, that boiling increases the durability of timber;: and, in piroof of this, they inform us that the planks ii- the bow of a ship, which are bent in this way, are never affected by the dry rot. It may now be inquired whether, after the most perfect seasoning,' timber is secured against the process.of decay?.To this questioii a negative answer must be given. However well the timber may be seasoned, it will certainly rot if placed in a damp situation, the rapidity of the decomposition depending upon the nature and state of' the wood, and the activity of the destroying agent. As thej builder seldoom attempts any other seasoning thllan that which depends upon drying?. his timbers, it is absolutely necessary tlhat he should carefully avoid the rise of damp, and adopt every means inl his power to prevent this evil. Timbers are usually placed in con-f tact with.' walls, but it must not be supposed that this is suflcientl to keep therm froml the access of damlp, for they are frequently thle conductingl media. Brickwork very readily absorbs moisture, and also throws it upwards, so that the ends of timbers are in contact with the very source of inischief. To prevent the rise of damp upwards, it is common touse, for a few feet above the foundations, cement, a substance impervious to water, instead of mortar, or to place between tile courses zinc or slate. But that these pllans may be effectirve, the basement walls should be surrounded with an open area, for, if in contact with the earth on their sides, tliey can be of! no value. To prevent dampness fiolWm entering in friont, the briclkwork should be covered with compo, or somle substance impermeable to twater. Another thing to be considered, for the security of tilmbers, is to arrange, in every plan, of a building, for a perfect circulation of air. Ventilation is a most important requisite in the construction of a' building, although it is generally a matter of very little importance in the consideration of those who have to plan or construct buildings. The ventilation of roofs is by no mneans difficult, but there are often so many obstacles to the ventilation of flooring that the designer will not give sufficient attention to his subject to provide against them. These things, however, are not matters of speculation, to be. attended.to by those lwho have no higher employment, but are absolutely necessary for the construction of a work that is intended to survive the builder. The attention of scientific men has been recently directed to the experiments mlade by Mr. Kvan. Having made a great number of experliments with a view to ascertain the primary cause of vegetable deconmposition, lie was at last convinced that albumen was that' cause, and that to neutralize its effects would be to prevent decomposition. Some plan was required similar to that adopted in.tanning. The gelatin in animal bodies is quite as liable to decom WOOD\tORt, CARPnTNrY, RTC. l31'position as the albumen of vegetables; but when tannin, the infilSioni of oak bark, is combined with it, the destructive properties are lost, and the a'nimal matter becomes durable, and almost incapable of-decay. Reasoning upon this effect, Mr. Kyan imagined that it might be possible to prevent vegetable decomposition by causing the albumen to form a conmbination with somne other substance; anWd, knowing the affinity of corrosive sublimate for the albumen, he entered uplon a series of experiments,:which led him to propose the use of that substance as a protection for timber. Mr. Kyan inferred that, as wood consists of various successive layers, in which the albumen, or juices containing albumen, circulated freely, it is quite certain, as these juices within the wood, with the watery parts, fly off by the leaves, that the albuimen remains behind, and it is probable that this albumen, which from its-nature is peculiarly prone to enter into new combinations, is the thing in wood which begins the tendency to decomposition, and produces ultimate decay, whether that decomposition is attended with the formation of cryptogamic substances, or whether in the less organized form, the change occurs with the simple production of what hals been called the dry rot. Mr. K. conceived, therefore, if albumen made a part of wood, the latter would be protected by converting that albumen into a compound of protochloride of mercury and alburnen; and he proceeded to immerse pieces of wood in this solution, and obtained the same result as that which he had ascertained with regard to the vegetable decoctions. SHaving done so, it became necessary to employ various modes of experiment, as well as comparative experiments. Now it is not clear: in what part of the wood the vegetable albunen may be found, though it exists more especially in that part of the tree which is denominated the alburnum or sap, and is found between,the heart-wood and the innermost layer of barlk. The experience of all practical men has confirmed the opinion, that this portion of wood is the first to decay. It is.probable that, as the alburnum becomes successive layers of wood, it loses a quantity of albumen; or that, in consequence of the pressure which takes place by the addition of each successive layer, it becomes so situated as to lose-a part of its exposure to the vessels where a change imay occur, and therefore becomes, in some measure, protected; for, that which is one year alburnuni or sap, may be, and indeed, generally is, proper wood the next, The mode in -whicli the application of the solution takes place is in tanks, which, may be constructed of different dimensions; from twenty to eighty feet in length, six to ten in breadth, and three to eight in depth. The timber to. be prepared is placed in the tank, and secured by a cross-beam to prevent its rising to the surface. The wood being thus secured, the solution is then. admitted from the cistern above, and for a tinme allnrenains perfectly still. In the course of ten or twelve hours, the water is thrown into. great agitation by the effervescence occasioned by the expulsion of the air 112 12 WooDW.oil'K,. CARPENTRY't,ETC. fixed in the. wood,.by the force -with whichl the fluid is drawn in by chemical affinity, and by the escape of that portion of the chlorine, or muriatic acid gas, whicih is disengaged during the process. In the course of twelve hours this cormmotion ceases, and ill the space of seven to fourteen days, varying. according to the diameter of the wood, the change is complete, so that as the corrosive sublimate is not an expensive article, the album.en. may be converted into an indecomposable substance at a very moderate rate, and the.seasoning will take place in the course oftwvo or three weekls." Mr. Kyan's method of seasoning has been already tested, under circumstances so severe, that they may be said to have proved its efficiency. A piece of oak was five years in the fungus pit in Woolwich yard, London, a place notorious for the rapid and almost instantaneous destruction of vegetable matter, and it was a.s sound when taken out as when put in. This was the most severe test to which the method could be subjected, and its having sustained tlhe trial is a proof of the value of the discovery. It has, however, been: objected to the process, that the impregnation of timiber.withl corrosive sublimate must unfit it for use in ship-building; but Mr. Kyan has furnished'evidence to the contrary, ancd proves tliat salubrity is one advantage. We strongly recommend' the builder to make experiments himself upon wood prepared by Mr. Kyan, by using it in places where decay is rapid. 1Framzing of Timbers. When timbers are framed together, it is with the intention. of supporting some weight, or resisting the strains to lwhich the materials may be exposed in-the situations where they are to be placed. Horizontal or vertical timbers are not alwaVns of themselves sufficiently strong to sustain the pressure to which they may be subject, but they need assistance, and it then becomes a question, horw can the materials intended to -assist be best applied, and what are the smallest scantlings that can be adapted? Two things must be studied-stability and economy. It has been often stated that these two results cannot be accomplished bv the same arrangement, but as the forces which are to be opposed have usually, a direct application, so the system by which they are to be resisted lay, usually, be of a simple construction. Compo.sition and Resolution of Forces. Two great mechanical principles lie at the base of all proper.attempts to estimate the nature of the forces which may be exerted upon substances in particular situations; these principles are called the composition and the resolution of forces. Thle resolution of forces is the means of finding any two. or.more forces which may resist or contr-ol thle pressure of any one force. The comlposition of forces consists in finding the direction and amount of one force that is capable of producing. the samle effect as WoODWOrvu, CArPEN'TRir'FYTC.;3'13:twno ol more forces acting in different directions. This is, in fact, only the reverse of the resolution of forces, and the two are, strictly speaking, but one principle; and if the one process be understood, the other must be almost so'of necessity. Nor may the student pass over this part of the woi, urider a fear that it is too mathematical for Iim to understand, for he can never be certain that the roofs or other framing wllich lie may design will support the weights they are intended to carry, if he does not know how to calculate thie'action of tile weights or forces by which they may be pressed. Let B'D, fig. 1, be the kling-post of a: roof, and let B A, B C,'beI Fig. 1. the rafters: they are framed together for the purpose of carrying *some weight; and the question is tlis-are tlley su fficiently strong to carry the weight which is to be placed upon them l To determine this we must refer to the resolution of forces. Let us supose some somedetermined -w eight' to rest upon the point B. Tlen, by some scale of equal parts, draw a line B d, equal to the number of pounds, hundred weights, or tons, resting upon thle point 13. and' draw d a parallel to B C, and de parallel to B A. Now measure'the line a B by the same scale, and it will give the number of pounds, hundred weights, or tons, by Which A B is strained, and c B will.give tihe strain upon BC. B3ut,l in t.he drawing affixed, the rafter B C is longer than the rafter'13 A; but this does not at all affect the weight, for it remains the same, whatever may be the length of the beam which carries it; but' it is necessary to remember that, by increasing the'length of the beam, it is rendered less capable of i supporting the weight, and a proportionate increase of dimensions must be allowed. But should the direction of the beau' be changed, a very different result will be obtained, for in every ease! | the pressure will be increased or decreased. The strain upon the, beam B A, fig. 2, will now be measured by the line a b, and that!. upon B c by b c. In fact, a very slight alteration of position may, under certain circumstances, enormously increase or'decrease ai strain. It will be scarcely necessary to explailn how two or more 27 314 WooDNORai, CArPEN'TRY,: ETC. Bl ~ forces may be comnposed, and the single force, acting in a certain dirlection, be calculated.,,i \\Gd Leaving the subject of tlhe composition and resolud-A // \?'Z \tion of forces, after LA - \\a statement of the principle, we may proceed to explain D \\ the construction (\, and arrangement of those palrts of a ig. 2. 2.. building which belong to the carpenter, And, first of all, we may speak of roofs. T.he Construction of Roofs. The simplest method of constructing a roof is to place horizontal timbers firom wall to wall, but this method is only suited to very short bearings, and does not readily throw off the water which may fall upon its covering. Tile Egyptians constructed flat roofs. To prevent this inconvenience, a roof may be made as an inclined plane; and such a construction has advantages, though its want of uniformity and beauty, antd also its want of strength, propoltioned to the amount of timber. employed, are objections to its use; but still'it is stronger than the' flat roof, and readily calrries oft the water that many fall upon it. The best form foi a roof'is tlat in which there are two sides, equally inclined to the horizon, and resting in a line called the ridge of the roof. The angle -rwhich thej inclined side forms with the horizon is called the pitch. In coun-i tries where there is a cold climate, and snow is apt to fall in large quantities, tle' roof is high; in warnm countries the roof is low. In Gothic architecture the roof is generally high pitched, and it is so consonant with the style that it often forms a prominent feature in these buildings. Thete are not so many advantages in high pitched roofs as most persons suppose, and there aredmany disadvanltages. The additional force of the wind upon a high roof is a serious. objection, and when parapets are employed it is so far from preventing the effects of a heavy fall of rain or snow that the gutters are so filled' that the pipes cannot carry off lthe mwater fast ernough, or, being stopped by thle dirt carried -down by the velocity of the water, an overflow is occasioned.- The height of roofs is now generally between one third and one sixth of the span. It is the carpenter's business to frarle the timbers of roofs, and sometimes he is required to design fthem, and he should therefore know how to obtain the stenfthl and othler qulalities required, with the smallest possible iamounlt of timber. WOOl)wsornK, CAIPENTRY, ETC. 315 A piece of timber, in whatever -way it may'be placed, except when vertical, will bend or sag, thait is: to say, its upper side will form itself into a concave surface. The more horizontal the timber is placed thle more it will alw'ays sag, anid as the distance between the points on which it rests is inlcreased, so it has greater liabilities of bending. To prevent this effect as much as possible, arrangeinents must be made for the support of the beam in some intermediate points. Now, it may be supported fiom either above or below. If there should be anlly vwalls between those on whlich the ends of the timber rest, these will be sufficient for all the pllrposes required; if not, the same result must be.produced by a system of fiaming. The timbers which compose a roof are known by different names, according to the uses for which they are employed, and the situations in \which thely are ilaeced. The principal timbers of a roof are the followinlg, but they are not all used in every roof: the tiebeams, wall-plates, collar-be'amls, king-posts, queen-posts, struts, principal rafters, common rafters, ridge-piece, collar-beams, purlins, and pole-plates. The TIE-BEAlt (A), fig. 3, is a horizontal piece of timber, which Fig. 3. extends from wall to wall, and rests upon the WALL-PLATES (B) at each end. It is employed for the purpose of connecting the feet of the principal rafters (C), which would otherwise have a tendency to push out the walls by their own weight, and the wveight of the materials placed upon them. In roofs of large span, it is necessary that the tie-beam should be well supported in some point or points, between the ends on which it is supported, for if this be not done it will sag and draw either one:or both of tle principal rafters towards its centre, and thus destroy the staibility' of the framing. The KING-POST (D) is sometimes used for this purpose. It 31 6 WoODwORnK, CARPEN-TRY, ETC.. is a piece of timber- placed in a -vertical position, connecting the point whelre the two principal rafters meet, and:the centre of the tie-beam. When the king-post is not thought to be sufficient to support the pressure;which may be on the framing. QUEEN-POSTS (B), fig. 4, may be used, which are.pieces of timber placed in an upright position, if Fig. 4. supporting severally the two rafters, and equidistant from the centre of the truss. Tile horizontal piece.of timber (C) which connects the heads of the queen-posts, is called a straining-beam; and that which connects their base, so as to prevent the struts from pushing them nearer to each other, is called a straining cill. Those pieces wvhich are placed in pairs, to assist in supporting. the principal raftl;ers, are called struts; they are frequently used to unite the rafters and: the base of the king-post. Any horizontal timber above the tie-beam is called a collar-beam. The ridge-piece (H) is that piece of timber which forms the apex of the roof, and is supported by the heads of the principal rafters or the king-posts, and in its turn supports one end of the common rafters. A poleplate is a beam over the walls, supported by the principal rafters or the tie-beam, and is intended to carry the lower ends of the common rafters. Purlins' (E) are horizont.al timbers, between the pole-plates. and' ridge-piece. The small spars (cc), which are parallel. to the principal. iafters, and are supported by the ridgeplate, purlins,: and pole-plates, are called common rafters. The Dimensions of Timbers used in a Roof. However accurately a roof may be designed, it is unfit for its pum-pose if the dimensions of the parts be. not accurately proportioned.. Too accomplish this, some experience-is required, and a WOODWORK, CARPENTRY, ETC. 317 knowledge of tlie strength of timbers, under particular circumStances. I There are two things to be secured-a sufficient strength to supPolt the weights to be carried without sagging, and to do that Wvithout burdening the walls or other parts of the building over ivhichl the roof is thrown. This is not always an easy task, for ioofs are sometimes to be made in suchi forms as prevent the adoption of those means which would otherwise immediately accomplish the object. Sometimes a very large roof must be made flat, at other times a lantern-light must be provided in its centre; and, in a third case, it may be necessary to erect a dome. In designing for these and other roofs, attention should be paid to the character and success of similar worlks already executed, and the artist should study the points of similarity and difference between these and his own work, so as to provide against dangers, which may peculiarly affect his building. _Examples of Roofs. Fig. 5 is a roof, the rafters of which are only supported by a collar-beam (C), which acts in part as a tie; but this arrangement is so feeble, that it should never be used over a space where the span is more than fifteen feet. Fig. 5. In fig. 6 there is the addition of a tie-beam. (A), and the strain is here thrown from the collar to the tie-beam; the former being compressed,'the latter in a state of tension. As there is no arrangelI~~~ 1:Fig. 6.:meat in this truss to support the tie-beam, and to prevent- it from sagging, it is unfit for a span of more than twenty-five feet. 27-* 31;8 WOODWORK, CARPENTRY, ETC. To prevents the -inconveniences resulting from the sagging, of the tie-beam, a king-post (P) and struts (SS) may be introduced, as Fig. 7. shown in fig. 7. This form of roof is very well adapted for a span of twenty-five feet. For a span of thirty to five-and-forty feet, the truss represented Fig.: 8, in fig. 8 is very well suited, and is now very commonly adopted by architects and builders. Floors. The timbers which. support the flooring boards, and the ceiling of a room beneath, are called, in carpentry, the'naked flooring. There are htllree kinds of naked flooring-single,: double, and fraimed. Single flooring is that in vlhich there is but one series of joists, as shown in fig. 9, where A AA A are joists, and B the flooring-boards. To make a single floor as strong as possible, the joists should be thin. but deep, sufficient thickness being always allowed for the nailing of the flooring boards. Two incihes by six is' tlie smallest WOODWORK, CARPENTRY, ETC. 319 Fig. 9.. dimension. for joists; for a length of tw'enty feet they should be about three inches thick, and twelve inches deep. Sometimes the joists cannot have in a particular place a bearing upon the walls, and then a piece of timber is framed between the nearest joists. This is done where flues, fire-places, and stairs interfere. The timber thus used is called a trimmer, and the two joists on which it is supported are called trimming-joists, and should be made a little stronger than the common joists. Thus, in fig. 10, Fig. 10. AA are common joists, BB trimming joists, and C a trimmer. When the bearing is more:than seven or eight feet, the joists should be strutted; that is to say, short pieces of board should be fitted between the joists, so as to form a continued line from wall to wall. These struts greatly strengthen the floor, and prevent the joists from sinking; but it is not desirable to mortice them into the joists, as that process has the effect of weakening the joists themselves. Fig. -11. 320 VOODWORcK, CARPENTRY, ETC. Double flooring is that in which there are two tiers of joists, the bindingjoists, as A A, inl fig. 11, which in fact support the floor, and the bridging joists B B. In this kind of flooring, the binders extend from wall to wall, and the bridging joists are notched down upon them. Beneath the binders we have a third tier of timbers (D), which are pulley-morticed into tlhe binders,- and are called ceiling joists. When the binding joists are framed into a large piece of timber, called a girder, the floor is said to be a double framed floor. Thus in fig. 12 A is the girder, B a binding joist, C a1 bridging joist, D D Fig. 12. ceiling joists, and E flooring boards. This kind of floor is decidedly the best when it is necessary to provide for a good and even ceiling, for although single floors miay be made very strong for a great bearing, yet thl ceilings are always liable to crack. It is not easy to obtain timber for girders of much more than twenty feet scantling, and they are therefore trussed. Trusses are used in both floors and roofs, but we have not thought it desirable to interrupt the course of explanation we have given, by a reference to any plarticulars concerning this bral-nch of carpenter's work; yet it is necessarly that we should now make a few remarks upon it. Trusses. When timbers are so framed together as to support weights, they are called trusses. It fiequently happens that a piece ofi tim;ber, in itself incapable of supporting a weight, mnay, when cut into scantlings of different dimensions, and fiamed together, not! only carry that weight, but also support a much greater. The bow I and string roof, invented- by Mr. Smart, is an example in point. Fig. 13. WOODWORK, CARPENTRY, ETC, 321 Let A:A, in fig. 13,- be a piece of timber, which we will suppose to be insufficient of itself to carry a particular weight; from this cut the pieces o, s, e, b, and o s, d, c. Then let these pieces be raised as in fig. 14, and a key be placed between them at the apex; and it will form a very, strong truss, which may be made still more capable of resisting a strain, by the application of struts. Fig. 14. The principal rafters of a roof are so called because they are trussed. It is not necessary to truss all the rafters in a roof, and it would be very expensive to do so; and therefore trusses are placed at particular distances from each other, according to the \weight to be carried; and they are formed in different ways, according to the span over which they are thrown. It: has been already stated, that girders are sometimes trussed, and should always be. when their bearing is much more than twenty feet. We have often seen trusses which, so far from strengthening the girders, have decidedly weakened them. Large girders are sometimes sawn down the middle, and when reversed, are bolted together with slips of wood between them. It has been supposed that this strengthens, and is adopted for this purpose; but the su pposition is erroneous, though the plan is certainly a good one, for it allows a free circulation of air between the pieces, and facilitates the emission of any dampness that may be in the timber. A strong girder may be made as strong, in fact, as any truss of the same depth, by bolting two pieces of timber together, or by conifiing them with iron hoops, the ends of the girder being smaller than the centre, so as to allow the hoops to be driven tightei, and confine the beams. In fig. 15 we have given a representation of a strong trussi Fig. 15. girder, the truss post and the abutment pieces being made ofi wrought iron. Of Connecting Timbers. It is sometimes impossible to obtain timbers of the length required for the several parts of a building, and it is then necessary to join two or r.xore pieces together, so as to form them into one piece, and to injure the stability as little as possible. This process 322 WOODWORK, CARl'ENTRY, YrTC. is ealled scarfing, and the parts of the joints which come in contact are called scarfs, and are usually connected by iron bolts. There are many ways of scarfing, every builder adopting that one whichll appears to hinm the best under the circumnstanees in wlich the timber is to be employed. Two or three different methods may be mentioned, leaving the workman to examine those vlhich lie nay happen to meet with in practice, and the various designs which have been given by writers on the art of building. Fig. 16 shows the mlleans of scarfing without diminishing the.... I. Fig. 16. length of the pieces. This is done by the introduction of a third piece, lhaviig thle form of steps,. and all the pieces being United together by bolts and plates. Fig. 17 is a representation of a scarfing, which is very simple, and frequently used, though there is a considerable loss of timber. Fig. 17. Thle pieces to be iunited are connected by iron bolts, an iron piate being placeed on botl sides. Fig. 18 represents a form of scarfing, adapted to a beam, which; Fig. 18. in consequence of the indentation. T25tber Partitionzs. Rooms and passa.es are often separated by timber partitions, i which are so formed as to be covered with lath and plaster. In fg. 19 we have given a design for thle framin of a artition with a door:through it: A A are th..e dBoor-posts, B the head, C the sill, DD ale braces which support the -quartering, and are assisted by .WOODWORK, CAnrsNmrRY, R'rc. 323 t H. Fig. 19o;the struts, EE. It will be quite evident from a glance at the drawing, that the door-posts help to sustain the braces and struts; while they in return prevent the fall'of the door-posts. Braces may be introduced in various ways, but strength is the object for which they ought to be introduced, a circumstance whichl is very frequently entirely forgotten by, carpenters. In some instances, it may be found desilable to introduce a simple truss into a design for partitions. The carpenter usually connects his timbers either by notching, or by molrt ice and tenon. Dovetail joints are sometimes used in carpentry, but they ought never to be adopted, for they will always draw when the timber shrinks, and the oblique surface of tle dovetail tends to force the timbers apart, acting as though it were a wedge. Gluing Joints. In general, nothing more is necessary to glue a joint, after the joint is made perfectly straight, or, in technical termls, out of winding, than to glue both edges while the glue is quite hot, and rub them lengthwise until it has nearly set. When the wood is spongy, or sucks up the glue, another method must be adopted, one which strengthens the joints, while it does away witeh the necessity ofusing the glue too thick, which should always be avoided fir thie less glue there is in contact with the. joints, provided they t;ouch, the better; and when the glue is thick, it chillls quickly, and calnnot be well rubbed out from between the joints.. the method to which we refer is, to rub the joints- on the edge with a piece of soft chalk, and, w-iDping it so as to. take offt any lumps, glue it in the usual imanner; and it wvill be found, whenr the wood is poieous, to hold much faster than if'used without chalking Of the dle7renret A~ethods.. of joie-ihg, Toodoork. Many workmen, are not aware. of thle proportion which a piece male t.o'fi into another slloulcl: have towat rds that into which it is 324 WOODWORK, CARPENTRY, ETC. fitted, so as to produce the greatest strength with the least possible waste of material' or how to proportion a joint, so that it shall not fail or give way before another. In too many instances, the method of joiningo woodwork is regulated by no other rule than the fancy of tile workman. It is not difficult to explaini why joiners' work so frequently fails; why the parts separate With a trifling strain; or,: from being bound too tiglitly together, fly and split in al directions. It is not so frequently fiolm the: bad execution of the work, as:firom the want of an adequate estimate of the strength required to resist the stress on the joint. We shall, then, describe the several kinds of joints, or the methods of fralming and joining, timber; and, under each head, give such directions, founded on the. principles of mechanics, as will enable the workman to proceed with some degree of certainty; and not, as is too frequently the case with artisans, observe no other rules than those which custom has authorized, and practice made famliliar. Dovetailing. We have given, in the cuts, several examples of dovetailing. The parts which fit into each other are kniown by different names; the projecting piece, represented in fig. 20, is called the pin of the dovetail; and the aperture into which it is fitted, as shown in fig. D 0 L n Fig. 20. Fig. 21. 21, is called the socket. Now thle strength of a dovetail depends upon so proportioning the pin and the socket as to enatble them to support, rather than destroy, each other. Let A B C D, fig.: 20, be a scantling, which is required to be joined to another, by means of a single dovetail. The strength of the joint depends on1 the form of-the dovetail, as well as on the proportion it bears to the parts cut away. We shall endeavor to lay down the principle on whllich the greatest strength may be secured. Having squared the end ofi the scantling, and gauged it to the required thickness, A I K L M, divide IM into three equal parts, at K and L. Let KIL be the small end of the dovetail, and make the angles I K G and M L H equal to about 75 and 80 degrees respectively; and make GE and EI F paralllel to A N and B 0. Then cut away the parts A I K GE N, and BI M L F O, anld hlaving formed the socket to corrdispond, by WObDWORK, CARPENTRY, ETC. 325 marking the form of the dovetail on the top of the piece A B C D, fig. 21, and cutting away accordingly, the pieces imay be fitted together, as shown in fig. 22. It may be here observed, that the bevel of the dovetail, that is, the angle I IK G, fig. 20, may be either more or less than has been mentioned, according to the texture of the wood. Hard, close-grained woods, not apt to rive or split, will admit of a greater bevel than those which are soft, or subject to split; thus the bevel of a dovetail in deal must be less thani ini hard oak, or in mahogany. It is a great fault to make a Fig. 22. dovetail too beveling, for instead of adding to the strength of the joint, as some persons suppose, it weakens it; for. provided the bevel is sufficient to prevent the possibility oft pulling the pieces apart, tlie less the bevel that is given the b'etter. It must have been observed, that there is a great difference between the dovetail made by the cabinet-maker and by the joiner; the former has very little bevel, the latter very much; the former looks neat, and is at the same time strong while the latter, appearing} to ai'm at strength, looks clumsy, and is at the same time nmuch the weaker of the two. Fig. 23 represents the dovetail in common use for drawer-fronts. When it is required to hide the appearance of the joint in front, the board A BCD is cut with the pin, and AEF PB with the socket. The pins in this sort of dovetail are in general from about three quarters of an inch to an inch apart, according to the size of the pieces to be joined. A.... B i Fig. 23. Fig. 24. Fig. 24 represents the pin part of a lap dovetail, which, when put together, shows only a joint, as if. the pieces were rebated together, as shown in fig. 25. A B C D represents the pin, E F G H thie socket, and when put together the line H G is only seen as a joint; and if the corner AB is rounded to the joint G H, it will appear as if only mitred together. This kind of dovetail is very useful for many purposes where neatness is required, such as in making boxes. 326 WOODWORK, CARPENTRY, ETC.:E` ( Fig. 25. Fig. 26. Fig. 26 represents a still neater. dovetail; and, as the edges are mitred together it is termed a mitred dovetail; and is the same as that showra in figf. 6, except that instead of the square shouldelr, or rebat, ill A B13. it is cut into a mitre, and the other piece is made to correspond. Aliother rery neat as well as expeditious method of joining pieces! of wood, and it is somewhat analogous to dovetailing, is shown in fig. 27. The 6oint is first formed into a mitre, and the pieces are then keyed together, either by making a saw kerf in a slanting direction, as at A B, or by cutting out a piece, as at C D. in the form of a dovetail. The first method, A B, is called, amongst workmen, keying together; the second, C D, key-dovetailing. CA ID Fig. 27. Fig, 28. The last method to be mentioned is that shown in fig. 28, and may be termed mitre dovetail grooving; the part A B being formed with shoulders cut to the required bevel, aRnd a piece left for the pin dovetail, which is inserted into the socket dovetail, made to correspond to it in the piece CD, which has been previously formed into a mitre. This method, though not much employed, may be used with great advantage in manv instances, particularly when it is required to join pieces together the'lengthway of the I grain. Mortice and Tenon. Under this head, we shall endeavor to give some rules necessary to be observed in attempting to proportion the parts of the, mortice,.and tenon, so that they may be equally strong, or that the tenon may not be more likely to give way than the checks of the mortice; for this is the principal thing to be avoided. The workman WOODWORK, CARPENTRY, ETC. 827 frequently allows too little substance for the tenon, lest he should weaken the mortice; and sometimes he falls into the opposite error;:facts which clearly prove that he is not acquainted with a means of obtaining a maximum of strength with a given quantity of material. Figs. 29 and 30 represent a simple mortice and tenon. The dotted Fig. 29. Firg. 30. lines show the parts to be cut away. To show-.the thickness of the tenon, and consequently the width of the mortice, we have here one tenon and two shoulders, that is, three parts; one of. which is to be allowed for the tenon, and two for the shoulders; and thlis will in general be found the best proportion, for if the tenon be more. than that, it will weaken the shoulders of the mortice. Now if we have, as is frequently the case, two tenons in one piece, as represented in fig. 31, there will be five parts, two tenons, and three shoulders; so that each tenon will be one fifth of the thickness of the stuff, for the shoulders are all equal to the tenons. This rule may - be generally observed, unless the tenon is at a considerable distance from the end of the stuff, and then something. more may be allowed for its thickness, as tile mortice is then not so liable to..___..' split; but it should in no case, how- Fig. 31. ever sound the timber, or tough the material, be more than two out of four parts; that is to say, it can never be safe to.make the tenon more than half tile thiclkness of the stuff, and that only under, particular circumstances, when the mortice is near the middle of the scantling, for the piece in which the mortice is cut would, in other cases, be considerably weakened. There is frequently injoiners' work a shoulder at the bottom of the tenon that fits into the piece. —=~ —C —--- C —in which the mortice is cut, as re-..-7 —presented in fig. 32; and the tenon I-.:-l-_ —-. is divided into two parts, as there I: showin, whieli, -when the stuff is wide, is a good method, as it strengthens the piece in which the mortice is cut, without weakening, in the same proportion, the mortice itself; and we would suggest, in Fig, 32,. outhis cfromse, that te piece C, ut 4 out -from between the tenons AB Fg 3828 WooDwoRK, CARPENTRY, ETC. and D C, be ne'arly, if not quite, -one third of the distance AD; for if mucli less, the piece left between the mortice will add but very little to the strength of the piece in which the mortice is made; and the tenon would be stronger in proportion to the mortice-piece than necessary. It may be here observed, that if the width of the tenon be much more than four times its thickness, additional strength will be gained by dividing the tenons into two or more parts, as shown in the figure, particularly if we allow a. small piece at the bottom of the tenon, as represented in the draw~ing. Groovievn and Lapping. This method of joining wood-work is analogous to that of morticing and tenoning. When it is required to join two boards together by means of a tongue and groove, the groove should never exceed one third of the thickness; and often, if the piece for the tongue be formed of- hard wood and liable to split, one quarter of the thickness will be sufficient. When a panel is let into a groove in the style, the joiner is often guided by the thickness of the panel itself, which- should never be) less than one third the thickness of the style. In making a groove nacross the grain, as for partitions, it will be best, i n most cases, to make it about a fifth or sixth of the substance of the stuff. But, if the groove be formed into a dovetail, one quarter the thickness will be better, and the dovetail should be made a little tapering, but not too much. It should, in fact, be so made as to go almost homle without requiring a blow from a hammer or mallet to drive it into its place until it has nearly attained it; and all joints should be easily separated with a gentle blow before they are glued. In a lap-joint, that is, in lapping two pieces together, supposing them of equal thickness, half the substance of each should be cut away; and, if of unequal thickness, the lap should be made in the thinner piece, of about two thirds or three quarters of its thickness, according to the substance of the thicker piece; thus endeavoring in thls, as in all other cases, to avoid weakening one piece more than another. Bending and Glubing-up. In bending and gluing-up stuff for sweep-work, much judgment is necessary, and, as the methods are various, we shall mention a few which the workmalln may apply, as occasion may require, one method being preferable to another, according to the nature of the work in hand. The first and most simple method is that of sawing kerfs or notches on one side of the board, thereby giving it liberty to bend in that direction; but this method, though very ready and useful for many purposes, weakens the work, and may cause it to break when strains are thrown on the piece. But a tolerably strong, sweep may be made in this manner, if, after sawing the kerfs WOODWORK, CARPENTRY, ETC. 329 (pa-rticular care being taken'to make them regular and even, and to saw them at regular depths), some strong glue be rubbed into each kerf. When bent into - the required sweep, a piece of strong canrvas should be glued over the kerfs themselves, and the glue be left to harden in, the position, to which the stuff is bent. Another method is to glue up the stuff in thin thicknesses, in a cawl or mould, made with two pieces of thick wood cut into the required sweep. This method, if done with care, that is, making| the several pieces of equal thickness throughout, of wood free from knots, is perhaps the best that canll be devised for strength and accuracy. It is also a practice sometimes to glue up a sweep in three'thicknesses, making the middle piece the contrar y aof the grain to the outside and inside pieces, which run lengthwise. This method, thoagh frequently used for expedition, is much inferior to the above, as the different pieces cannot shrink together, and con- I sequently the joint between them is:a-pt to give way. A solid piece, if not too thick, may be sometimes bent into. the form required. If'a piece of timber be well soaked upon the intended outside of the eurve, it" may be bent:into position, and if kept in that position till cold will; retain:the curvature that is given to it. The only other method of forming a curve, necessary for us to, mention, is that of cutting out solid pieces to the required sweep, I and' gluing them upon one another till they have the thickness required, taking care that the joints are alternatelyin the centre of each piece below it, soaimething in: -the manner of courses of bricks one above the other. In this case, it will be necessary, if the work be not painted. to veneer the whole with a thin piece, after it has been thorouglily dried and planed level, and then made somewhat rough wvith-either a rasp or toothling-plane. But the joiner must adopt one plan or another, according to circumstances. Scribing. Scribing is the operation by which a. piece of wood-work is made,to fit again-st; an irregular surface.; Thus, for instance, the plinth of a room- is made- to meet or correspond with the unevenness of the floor. To determine the portion which is to be cut off from a partition, or any wood-work where a floor or ceiling is irregular, it is only necessary to open the compasses to a width equal to the greatest distance between the plinth and the floor; and, passing one leg over the uneven surface, the other leg will leave a mark on the plinth. If the wood be cut anwy on that line, a surface will be obtained which will make a good joint with the floor or ceiling. But the chief use of: the art of scribing is to enable the j6iner so to connect the- moulding -of panels or cornices, that when placed together, they shall seem to form a regular mitre-joint. This method has certainly one advantage oyer the common method of mitring, for, if the. stuff should: shrink, little or no alteration will be made in the appearance, but, under the same circumstances, a mitre 28* 330 WOODWORK, CARPENTRY, ETC. would open,,and the joint would be shown.' The method adopted is this: To -cut one piece of the moulding to the required mitre, and then, instead of cutting the other to correspond, with it, cut away the parts of the first piece to the edge of the first moulding, which will then fit to the other moulding, and appear as.a regular mitre. Finishing of Jooiners' Work. Joiners' work is generally intended to increase the beauty of a'buiilding. When a joiner works in wainscot, oak, or mahogany, his Chief object must'be to obtain a surface perfectly smooth and evenh. When the framing is glued together, the glue which oozes out, and may be spilt upon the work, must be allowed to remain a few minutes and chill, and may then be carefully scraped off with a chisel; and the parts which cannot be thus cleaned may be washed with a sponge dipped in hot water and squeezed nearly dry. This not only saves trouble in operations which follow, but prevents staining, always produced when glue is suffered to remain till quite hard, particularly on wainscot, which turns black in every joint or place where the glue. is suffered to remain. After this operation, which, though it may appear tedious to some workmen,'will be found a saving of time, the work should remain till p lerfectly dry; and, when the joints and other parts have been levelled with a smoothing plane, the whole surface may be passed under a smooth scraper, and finished with fine glass-paper. It will'be sometimes necessary, when the grain is particularly cross, to d'amp' the entire surface with a sponge "to raise the grain," and then again to apply the glass-paper.. The work will then be ready for polishing with wax, or varnishing, and the good appearance of the work will be in proportion to the time and trouble expended in the process. In cleaning pine, the same precautions must be taken for the removal of glue left upon the joints, or spilt upon the work, as already described. This being done, the work may be cleaned off.with a piece of glass-paper that has been rubbed with chalk, or, in -some cases, with a piece of hearthstone. The work is then ready "for the painter; but' as there are knots. and other places wvhere the turpentine contained in the wood is apt to.ooze out, either with or Iwithout'the increase of heat, and thus spoil the appearance of the finisliing, those parts are done over with a: composition, and the'process is called priming. This is properly the painter's business; but it must sometimes be done by the joiner, for the sake of saving his work.' The composition used for this purpose is made with red lead, "size, and turpentine, to which is sometimes added a small quantity of linseed oil. Priming has also'the advantage of prev enting the knots from being seen through the paint. Somne workPmen' omit in this composition the oil and the turpentine, but the size of itself is apt to peel off, and does not thoroughly unite itself with the wood. TERMs USED IN BUILDING. 331 Another method of cleaning-off pine is sometimes adopted. When the surface has been made quite smooth with the plane, it is rubbed with a piece of chalk, and the whole is cleaned with a piece of fine pumice-stone, as in the former process it was done with glasspaper; but.if the grain should be still rough, the work may be damped. with a sponge. and the operation repeated when dry. As, in finishing interliorl work, it is now customary to imitate the graining of different kinds of wood, it is necessary that the joiners' work should be well finished; for if a- good even surface be not provided, it will be impossible for the painter to produce the effect he desires. Every defect in the ground will, in fact, be more visible under a delicate graining than when the surface is covered with successive coats of color; but, even in the latter case, work, well prepared will not only look better, but the color will not be so apt to chip and peel off as when the surface is not properly levelled. TERIMS USED IN B UILDING. Abaczus.-The upper member of the capital of a column, that on which the architrave rests. It has different forms in the several orders::In the Tuscan or Doric, it is a square tablet; in the Ionic, its edges are moulded; in the Corinthian, its sides are- concave, and frequently enrliched with carrving. Abutmenlt.-That part of a pier from which the arch springs.: Acanthus.-A plant whose leaves are carved on the Corinthian and Composite capital. They are differently disposed, according to circumstances; and the leaves of the:laurel and parsley are sometimes employed in. their place. Acroterium.-A pedestal on the angle or apex of a pediment, intended as a base for sculpture. Altitude.-The perpendicular height of anything in the direction of the plumb line. The length of a body is measured on the body itself, and remains constant, its altitude varies according to its inclination to or from the perpendicular. Alto Relievo.-A sculpture,. the figures of which project from thel surface on which they are carved. Amphiprostylos -An order of Grecian temples, having columns in the back as well as the front. Amphitheatre.-A double theatre, employed by the ancients for public amusements. The colosseum at Rome, built by Vespasian, is one of theseO. Annulet.-A:small square moulding, used to separate others; the fillet which separates the flutings of a column is sometimes known by this term. 382 7TER.fS USED IN BUILDING. Antce. -Pilasters attached to a wall, receiving- an entablature, and: having bases and capitals differing according to the order employed, but. always unlike those of the columns. Ailntepagmenta.-A term in ancient architecture, the architraves round doors Apoplqyge.-That part of a column which connects the upper fillet of the base and the under one: of the capital with the cylindrical part of the shaft. Arceostyl-os — That style of building in which the columns are distant from one another from four to five diameters. Strictly speaking, the term should be limited to an intercolumlniation of four diameters, which is only suited to the Tuscan order. A'rch.-Such an arrangement, in a concave form, of building niaterials, as enables them, supported by piers or abutments, to carry weights and resist strains. Arch-buttres.e. —Sometimes called a flying buttress; an arch springing from a buttress or pier against a wall. Architrave.-That part of the entablature which rests upon the capital of a column, and is bene-ath the frieze. It is supposed to represent the principal beam of a timber building. Area.-This term is applied to superficies, whether of timber, stone, or other mnaterial, and is the superficial measurement; that is, tie length multiplied into the breadth. The word area sometaimes signifies an open space. Arris -The line in which two surfaces meet each other. Ashier.-C-ommlon freestone, as it comes from the quarry, generally about nine inches thick, but of different superficial dimensions. Ashleriag. —Quartering, to which laths are nailed. Astracgal.-A small moulding with a semicircular profile, sometimes: plain and sometimes ornamented. Attic Order. -A term used to denote the low pilasters which are placed over orders of columns or pilasters, and frequently employed in the decorations of an attic. B. Banuster.-A. small pillar or pilaster, supporting a rail.:BBalastrade.-A series of balusters, connected by a rail. Band.-A square member. To distinguish the situation in which it -is placed, or tile order in which it is used, an adjective is frequently prefixed; thus, a dentil or a modillion band. Base.-The lower division of a column. The Grecian Doric has no base,'and tle Tuscan has- only a single torus on a plinth. Bead.-A circular moulding, which lies level with the surface of the material in which it is formed. When the moulding projects, or seversal are joined, it is called reeding. Beak. —A small fillet in the under edge of a projecting cornice, TERMrs USED IN BU'LDING. 338 intended to prevent the rain from passing between the cornice and fascia. Beam.-A piece of timber in a building laid horizontally, and intended to support a weight, or to resist a strain. Bcam-filling.-The masonry, or brickwork, between beams or joists. Bearer.-A vertical support. Bearing.-The length between bearers, ol' walls; thus, if a beam rests on walls twenty feet apart, the bearing is said to be twenty feet. Bed 31louldings.-Those mouldings between the corona and the frieze. Bevil.-An instrument used by workmen for taking angles. In formn it resembles a square, but the blade is- moveable about a centre. When the two sides of any solid body have such an inclination- to each, other as to form an angle greater or less than a right angle, the body is said to be beviled. Bond.-A term used to signify the connection between the parts of a piece of workmanship. In bricklaying and masonry, it is that connection between bricks, or pieces of stone, which prevents one part of the building from separating itself from another. Bond Timber.-Timber laid in walls to tie or bind them together..Brace.-A piece of timber placed in an inclined position, and used in partitions or roofs, to strengthen the framing. When a brace is employed to support a rafter, it is called a strut. Bressummer.-A beam, or iron tie, intended to carry an external wall, and itself supported by piers or posts. Bricknoggin. —Brickwork between quartering. Buttress.-A mass of stone or brick-work intended to support a wall, or to assist it in sustaining the strain that may be-upon it. Buttresses in Gothic architecture are used for ornament as well as strength. Cabling.-Cylindrical pieces filling up the lower part of the flutes of a column, Camber.-To give a convexity to the upper surface of a beam. Cantalivers.-Pieces of wood or stone beneatli the eaves to support them, or-mouldings above them. Capital.-That part of a column or pilaster benieath the entablature; or, in other words, the uppertmost member of a column or pilaster. The capital is variously formed, according to the order: Thus, we have the Tuscan, Doric, Ionic, Corinthian, and Composite capitals, and many others, that have been invented since the times of the Greeks and Romans. Caryatides. —Figures of women, introduced to support an entablature, instead of columns. 334 TERMS USED IN BUILDING. Catsement.-Applied to a window whiclh is hung upon hinges in place of lines and weights. Casting.-The warping or shrinking of timber or wood-work, occasioned by an insufficient strength,- or by an unequal exposure to the weather, and want of proper seasoning. Cavetto.-A concave moulding, the quadrant of a circle. Centering.-The framing upon which an arch is turned. Clampinqg.-Wlhen one piece of wood-is so fixed into the end of anotl!er as to plevent it fiom splitting, or casting, it is said to be clamped. The pieces may be united with a mortice and tenon, or, with a groove and tongue. Collar Beam.-A beam framed between two principal rafters. Console.-An~ ornament cut on the key-stone of'an arch, sometimes in the form of a scroll, at other times to represent a human face. GContent.-The amount of any substance in rods, yards, feet, or inches whether solid or superficial. Coping.-The stone which covers the top of a wall or parapet. Corbel.-A bracket, or piece of timber projecting firom a wall: in Gothic architecture, usually carved with some grotesque figure. Cornice.-The combination of mouldings which finishes or crowns an entablature.-The term is also applied to the mouldings which finish the walls and ceiling of a room, hall, or passage, filling up the angle which they make. Crown.-A term applied to the uppermost or highest part of an, arch, that in which the key-stone is fixed. Ciy7na. —A'moulding with a v-ayed or crooked profile, partly convex, partly concave. It is called by workmen an ogee. When the hollow part of the moulding is uppermost, it is called a cymarecta;; when the convex part is above,.a cyma-reversa. D. ]Dado.-That flat part of the base of a column between the plinth and the cornice. It is of a cubical form, and from thence takes its name..Dentils.-Square blocks' ihtroduced as ornaments into cornices of the Doric, Ionic, and Corinthian orders. A small circular piece is sometimes cut out, and at other times they are fluted. )iee.-A square cube. Door Frame.-The case in which a door opens and shuts, consisting of two uprights and one horizontal piece, connected together by mortices and tenons.'Dormer -A window made in the sloping part of a roof, or above the entablature.:.Devetailed —When two. pieces of wood are fastened together, by letting the pieces of one into apertures formed in the other, of a TERMS USED IN BUILDING. 335 shape somewhat resembling a fan or dovetail, they are said to be dovetailed. 2Drops. —Ornaments in the Doric entablature resembling bells placed immediately under the triglyphs. Dwarf TWall.-A wall that has a less height than that of the story in which it is used. Eaves.-The edge of a roof or slating which overhangs a wall, and is designed to carry off the water, without flowing down the vall. Echinus.-A moulding, the profile of which is the quadrant of a circle turned outwards, or in some instances a conic section. It is said to resemble the shell of the chestnut. Ellipse.-That curve called by workmen an oval. Entablature.-That assemblage of mouldings, &c., which are supported by the column. It consists of three parts-the architrave, frieze, and cornice. Entasis.-The swelling of a column. EzEstqlos —That intercolumniation in which the columns are placed two diameters and a quarter from each other. Eye.-A term sometimes used in architecture to denote a small window in a pediment. The middle of the Ionic volute, that is, the circle within which the different centres for drawing it are found, is known by the same name. F. Farade.-The face or front of a building; strictly speaking, the principal front. Fascia.-A flat broad member, in architecture, but of small projection. It is used to denote the flat members into which the architrave:is divided, and these are called fascima. Feather-edged.-Boards or planks thicker at one edge than the other. Fillet.-A small square moulding, of slight projection. In carpentry, it means a piece of wood to which boards are nailed. Flashirngs.-Pieces of lead so let into the wall as to lap over a gutter. Flattiny.-Painting, which has no gloss on its surface, being worked with turpentine. It is used for finishing, Flutes. —Vertical channels cut in the shafts of columns anid pilasters, sometimes meeting one another at a sharp edge, and at other times having a fillet between them. Flyers.-Stairs which rise without windinding Flue. —The aperture of a chimney. Footings.-The courses of brick or stone at the foundation of a wall.: ci 336 TERnMs, UVSED IN, B.UILDING.; Frieze.-The flat member in entablature, separating the architrave from the cornice. F- rring.-A means of restoring an irregular framing by the addition of small pieces of wood nailed to the framing itself. -Fust.-The shaft of a column. G. Gable.-The upright triangular end of a building at the ends of a roof, Girder. —The largest piece of timber in a floor, that into which the joists are framed. Groin.-The intersection of two arches. Groove.-A rectangular channel cut in stone or timber; such as that which is cut in the stiles to receive the panel of a door. Grounds.-Those pieces of wood imrbedded in the plastering ofI walls to which skirting and other joiners' finishings are attached. Guttce.-See "Drops." Gutter.-A valley between the parts of a roof, or between the roof and parapet, intended to carry off the rain. II. Half Round.-A moulding in a semicircular form, projecting from the surface. Headers.-Bricks laid with their short face in front. Hips. —Those pieces of timber placed in an inclined position at the corners or angles of a roof. ImTpost.-Tlle combination of mouldings which form the capital of:a pier. Insulated.-A term applied to a columin which is unconnected with a: wall, or to:a building, that stands detached from others. Intercolumaniation.-The space between two columns. fintertie.-Small pieces of timber placed horizontally between, and framed into, vertical. pieces to tie them together. J. Jambs-The side pieces of an opening in a wall, such as doorposts, and the uprights at the side of window frames. Joggle-piece.-A post to receive struts. Joists.-Those pieces of timber which are framed into a girder, bressumner. or otherwise, to support a ceiling or a floor. TERMS USED IN BUILDING. 337 K. lKey-stone.-That stone in the top or crown of an arch which is in a perpendicular line with the centre. Kfin-post. —The centre post of a trussed framing, intended to support the tie-beam and struts. K lnee.-A- piece of timber bent to receive some weight, or to relieve a strain. Lantern.-A frame in the dome or cupola of a building to give light. The term is applied to some kinds of fanlights, that is, the frame over a door to light a passage or corridor. Lining. —That joiners' work which covers an interior surface. Lintels.-The pieces of timber which lie horizontally over the jambs of windows and doors. M. Mantel. —The cross-piece which rests on the jamb of a chimney. Metopa.-The interval between the triglyphs in the Doric order. Minute.-The sixtieth part of the diameter of a column. Modillion.-An ornament in the Ionic, Corinthian, and Composite orders. It is a sort of bracket, and shoullld be placed under the corona. Module.-The semi-diameter of a column, and is divided into thirty minutes. It is the measure by which the architect determines the proportions between the parts of an order. 3Iortise. —A method of joining two pieces of wood; a hole being made in one of such a size as to receive the tenon or projecting piece formed on the other. Mosaic.-A term applied to pavements, and other work, when formed of various materials of different shapes and colors, laid in a kind of stucco, so as to present some pattern or device. The ancients were very successful in the execution of Mosaic, and many fine specimens remain to this day. Mullion.-Upright posts or bars which divide the lights in a Gothic window. N. Naked.-This term is applied, in architecture, to a plain surface, or that which is unfinished; as the naked walls, the naked flooring -that is, uncovered. The word is sometimes applied to flat surfaces before the mouldings and other ornaments have been fixed. Newel.-The centre round which the stairs wind in a circular staircase. 29 R "T 8 TERMS USEoD IN BUILDING.' Nosings. —The rounded and projecting edges of the treads of stairs. 0. Obelisk.-A slender pyramid. Ogee.-A -mouldi:ng, consisting of a portion of:two circles turned in contrary directions, so that it is partly concave and partly convex, and somewhat resembles the letter S. Order.-An assemblage of parts having certain proportions to one another. There are five orders of architecture-Tuscan, Doric,: Ionic, Corinthian, and Composite-all of which *were invented by the ancients,:and: are now employed by the moderns. Oval.-A curve line, the two diameters of which are of unequal length, and -is allie:d. i.n form to the ellipse. An ellipse is that figure which is produced: by cutting a cone or cylinder in a direction oblique to its axis, and passing through its sides. An oval may be formed by joining different segments of circles, so that their meeting shall not be perceived, but form a continuous curve line. All ellipses are ovals, but all ovals' are not ellipses; for the term oval may be applied to all egg-shaped figures, those which are broader at one end than the other, as well as to those whose ends are equally curved. Ovolo.-A convex projecting moulding whose profile is the quadrant of a circle. Panel.-A compartment inclosed in a frame, inito which it is framed or grooved. Parapet.-A low wall generally about breast-high, on the top of bridges or buildings. Pargetting.-Rough plastering, commonly adopted for the interior surface of chimneys. Pedestal.-That arrangement on which columns are sometimes placed: it is divided into three parts-the cornice, the die, and the base. Pedirnent. —A low triangular crowning ornament in-the front of a building, andi over doors and windows. Pediments are sometimes made in the form of a segment of a circle. Pier.-A square) or other formed mass, used to strengthen or support a building; it sometimes signifies that mass of stone or brickwork between the arches of a bridge, and from which they spring, or against which they abut. -'But the termn is usually employed to designate: the: solid part between the: doors or windows of a building. Pilaster. -A'square pillar insulated, or engaged to the wall, and is: usually; enriched with a e capital; and.base. Piles.-Large timbers, usually shod with pointed iron caps, TERM-3S USED IN BUILDING. 339 driven into the ground for the purpose of making a secure foundation. Pillar. — An irregular, insulated column. & It differs from a column in having no architectural proportion, being, either too massive or too slender. Pinnacle.-A small spire used to ornament Gothic buildings. Pitch of a Roof.-The proportion obtained by dividing the span by the height; thus we speak of its: being one half, one third, one fourth. Plinth -The solid support of a column or.pedestal. Plumb-line. —An, instrumlent to determine perpendiculars; it consists of a piece of lead attached to a string. Porchk. —The vestibule or entrance to a building. Portico.-A kind of gallery or piazza, frequently erected in front of large buildings. Posts. —Square timbers set on end; the term is especially applied:to those which support the corners of a building, and are then framed into the bressummer or cross-beam, under the: walls. Prick-ing-up.-The first coat of plaster worked upon laths. Prqfille.-The outline; the contour of a part, or the parts com. passing an order. Pugging.-The stuff: laid upon sound boarding to prevent the passage of sound from one story to another. Puncheons. —Sholrt pieces of timber employed to support a \weight when the bearing is too distant. Purlines.-Those pieces of timber which lie across the rafters to prevent them from sinking. Putlogs.-Pieces of timber used in building a scaffold; they are those which lie at right angles to the line: of wall, and rest on the scaffold poles or ledgers. Pgramid.-A solid massive edifice which rises from a square or rectangular base, and terminates in a point called the vertex. Quarter Round.;-See -" Ovolo." Quarters. -Pieces of timber used in an upright. position for partitions. Quarters may be either single or double; the single are generally two inches. thick, and four inches broad; the double are fdurl inches square. The quarters are never placed at a greater distance than fourteen inches from each other. Quirk -A piece, of ground taken out of a plot. The term is also applied to a particular form of moulding, one which has a sudden convexity.y, Quoins.-The corners of a building; they are called rustic quoins when. they project from: the wall, and have their edges cha mfered off. 340 TERMS USED IN BUILDING. Rabbet or Rebate.-A groove or channel in the edge of a board. Rafters.-Those timbers which form the inclined sides of a roof. Raking.-Means literally inclining, and is applied to those mouldings which, instead of maintaining the horizontal line, are suddenly bent ouit of their course. Rails.-Those pieces in framing which lie in a horizontal position are called rails; those which are perpendicular are called stiles; hence two rails and two stiles inclose a panel. The term is also applied to those pieces in fences or paling which go from post to post. Relief-The projection which a figure has from the ground on which it is carved. ]ieturn.-That part of any work which falls away from the line in front.:Ridge.-The highest part of a roof, or the timber against which the rafters pitch. Riser -That board in stairs set on edge under the tread or step of the stair. Rustic.-This term is applied to those courses of stone-work, the face of which is jagged or pecked so as to present a rough surface. That work also is called rustic in which horizontal and vertical channels are cut in the joinings of the stones, so that when placed together an angular channel is formed at each joint. Sash. —The framework which holds the squares of glass in a window. Sash-frame.-The fiame which receives the sash. Scantling.-The measure to which a material is to be or has been cut. Scotia.-A semicircular concave moulding, chiefly used between the tori in the base of a column. Scribing.-Fitting wood-work to an irregular surface. Scroll.-A carved curvilinear ornament, somewhat resembling in profile the turnings of a ram's horn. Sill.-The horizontal piece of timber at the bottom of framing; the term is chiefly applied to those pieces of timber or stone at the bottom of doors or windows. Shaft.-The body of a column; that part between the base and capital. Shore.-A piece of timber placed in an oblique direction to support a building or wall. Skirting.-The narrow-boards placed round an apartment against the walls, and standing vertically on the floor. Sleepers.-Pieces of timber placed on the ground to support the ground-joists, or other woodwork. TERMs: USED IN.- BUILDING. 341 Sofit.-A term applied to a frame or paneling overhead, or to a lining, such as that which is fixed in the underside of the tops of windows. Stiles.-The upright pieces:in framing or panmeling..Struts.-Pieces of timber which- support tile rafters.. Senmser. -A large piece of timber: supp:orted by piers- or; posts:; when it supports a, wall,,it is called a breast-summer, or bressummer. T.. Tenon.-A piece of wood so formed as to be received into a hole in another piece called a mortice. 1/'oat.-Thlat: hollow..vllifilchl terminates thellupper, end of the shaft:of:a columtniToLngue.-That projecting piece at the end of. a:.:board:vhichk-is formed to be inserted into a groove. Torme.-A. moulding that has a convex semicircular or semielliptical profile. Transomn -A: piece that is fi;rmed across a double window-light. Ti-ellis. —An open fi'amiing, pieces crossing eacli other so as to form diamond or lozenge-shaped, openings. Tryglyphs.-Ornaments in: the'Doric frieze consisting of a square projection: withl two angular chalnnels, tlle."edges:of each forming half a channel. Theyv are:: placed immedialtely over: the centre of:'a column; their width is. generally- one. module. T-rimmzers.-Pieces of timber framed at right angles to the joist for chimneys, and the well-holes of stairs. Tympanum. -The space;inclosed:. by the:inclined and horizontal sides of a pediment. V. Valley -The space between two. inclined sides of a roof. Vaults. — lTihdrgro und buildings with arched ceilings; whlether circular or elliptical. Fertex. —Thie top. or summit of a pointed'body; as of a cone. Volute.-Tlhe scroll' in the capital of the Ionic order.' Voussoi:s. —The stones which compose the face'e of' an, arch; having a somewhat wedge-shaped form. W., Wall-plates.-The timbers built up with a wall, to carry the joists.; Weather:-boarding.-Weather-edge boards; fixe&d v:erti'cally,:so; as to,i.ap: over'one aanothier:. Well-hole —The aperture.eft in flors to bring:up-th e-stairs.:: 29* 342 GLUES —PAPERS. GLUES. Parchment Glue. Parchment shavings 1 pound; water 6 quarts. Boil until dissolved, then strain and evaporate slowly to the proper consistence.'Use a water bath if you want it very light colored. Japanese Cement, or Rice Glue. Rice flour; water, sufficient quantity. Mix together cold, then bring the mixture to a boil, stirring it all the time. Observe to boil it in a vessel that will not color it. Japanners' Gold Size. Gum ammoniac 1 pound; boiled oil 8 ounces; spirits of turpentine 12 ounces. Melt the gum, then add the oil, and lastly the spirits of turpentine. Gold Size. Yellow ochre 1 part; copal varnish 2 parts; linseed oil 3 parts; turpentine 4 parts; boiled oil 5 parts. Mix. The ochre must be in the state of the finest powder, and ground with a little of the oil before mixing. Glue Liquid. Glue, water, vinegar, each 2 parts. DissolVe in a water-bath, then add alcohol 1 part. An excellent cement. Tr'ansparent Liquid Japan for Metal. Copal varnish 35 parts; camphor 1 part; boiled oil 2 parts. Mix. Portable Glue for Draughtsmen, &c. Glue 5 parts; sugar 2 parts; water 8 parts. Melt in a waterbath, and cast it in moulds. For use, dissolve in warm water. Waterproof Glue. 1. Glue 1 part; skimmed milk 8 parts. Melt and evaporate in a water-bath to the consistence of strong glue. 2. Glue 12 parts; water sufficient to dissolve. Then add yellow resin 3 parts, and when melted add turpentine 4 parts. Mix thoroughly together. This should be done in a water-bath. PAPERS. Fire-proof Paper. Take a solution- of alum and dip the paper into it, then throw it over a line to dry. This is suitable to all sorts of paper, whether plain or colored, as well as textile fabrics. You must try a slip of the paper in the flame of a candle, and if not sufficiently prepared dip and try it a second time. PAPEns. 343 Black Edge Paper. Blacklead 11 parts; common ink 22 parts; dissolved gum-arabic 1 part. Mix. Then with a sponge lay the color on the edge of the paper, previously placed in the cutting-press, rub it in with a piece of cloth, and burnish. The edge of the paper must be rendered perfectly smooth before applying the black. To Stain Paper or Parchment. Red.-Brazil 12 parts; water 70 parts; alum 5 parts. Boil. 1. Blue.-Sulphate of indigo. Water to dilute. 2. Prussian blue 2 parts; inmuriatic acid 1 part. Water to dilute. 3. Logwood 4 parts; water 30 parts; sulphate of copper 1 part. Mix. Green. —Crystals of verdigris 2 parts; vinegar 1 part. Water to dilute. Yellow. —French berries, water, and a little alum. Boil. Purple.-Logwood 2 parts; alum 1 part; water 20 parts. Boil.' The addition of a little gum to the above renders them suitable for coloring maps, &c. Paper for Draughtsmen, &c. Powdered tragacanth 1 part; water 10 parts. Dissolve and strain through clean gauze, then lay it smoothly with a painter's brush on the paper, previously stretched on a board. This paper will take either oil or water colors. Copying Paper. Lay open your quire of paper (clean white, of large size), take the brush and cover it with the following varnish, then hang it up on the line; take another sheet and repeat the operation, until you have finished your quantity. If not clear enough, give each sheet another coat when dry:-Canada balsam, turpentine, equal parts. DMix. Liq'uid Gold, for Vellum, c&c. Take gold-leaf and grind it with gum-water; then add a small quantity of bichloride of mercury, and bottle, for use. Liquid Silver, for Vellums, &c. Take silver-leaf and grind it, with gum-water or glair of egg. Palper that Resists Moisture. Take unsized. paper, lay it flat on a clean surface, and brush it over with a solution of mastic in oil of turpentine; or plunge it into the solution and hang it up to dry. This paper possesses all the usual qualities of writing paper, with the advantage of resisting moisture. To'Detect the presence of Plaster in Paper. Calcine the paper in a close vessel, and dilute the residue with 344 PAPERS. vinegar, in a silver spoon; if sulphuretted hydrogen is disengaged, which blackens the spoon, the presence of a sulphate (plaster) will be shown. This adulteration has lately become very common among the paper-makers, with the view of increasing the weight. Waxed Paper. Take cartridge or other paper, place it on a- hot iron and rub it with beeswax, or make a solution of the wax in turpentine, and apply it with a brush. Useful for making water or air-proof pipes, for chemical experiments, &c. To:extract Grease Spots from. Paper. Apply a little powdered pipe-clay, on which; place a- sheet of, paper, then use a hot iron. Remove the adhering powder with a' piece of India-rubber. Papier MdchE. Take paper,.any quantity. Boil it well, then pound: it to apaste, and mould. Used in making toys, snuff-boxes, &c. To Gild the Edges of Paper. Armenian bole 4 parts; sugar candy 1 part. White of egg to mix. Apply this composition to the edge of the leaves, previously firmly screwed in. the cutting-press; when nearly dry smooth the surface with the; burnisher; then take; a. daimp, sponge and: pass over it, and with a piece of cotton-wool. take the leaf from the cushion and apply it to the work; when quite dry burnish, observing to place a piece of silver or India paper between the gold and the agate. Tracing Paper. Nut oil 4 parts; turpentine 5 parts. Mix, and: apply it to the paper, then rub it dry with wheat flour, and brush it over: with oxgall. This will bear writing on. Lithographic Paper. Give the paper 3 coats of thin size, 1 of starch, and'1: of solution of gamboge. Each to be applied with a sponge, and allbwed to dry before the next is applied. Hydrographie;Paper. This name has been given to paper which may be written on with water. It may be made by rubbing paper over with a mixture of: finely powdered galls and sulphate of iron heated till it becomes white. The powder may be pressed into the paper by passing it between rollers, or passing, a heavy iron -over it. A mixture of dried sulphate of iron and ferro-prussiate of potash may be used for blue writing. Or the paper may be imbued with a strong solution of one:ingredient thoroughly dried, and the-other applied in powder. Paper which has been wet with a solution- of ferro PAPERS. 345 prussiate of potash also serves for writing on with a colorless solution of persulphate of iron. Iridescent Paper. Nut-galls 8 parts; sulphate of iron 5; sal-ammoniac 1; sulphate of indigo 1; gum-arabic i. To be boiled in water, and the paper washed with it exposed to ammonia. To give Paper the Appearance and Toughness of Parchment. Dip white unsized paper for half a minute in strong sulphuric acid, and afterwards in water containing a little ammonia. When dried it will look like, and be as strong as parchment. Photographic Papers. The following papers should be the finest satin post, of uniform texture, free from the maker's mark, specks, and all imperfections. The papers nmust be prepared by candle-light, and kept in the dark till used. 1. Signple Nitrated Paper.-This is merely paper brushed over with a strong solution of nitrate of silver. In brushing over the paper it must be crossed. Its sensitiveness is increased by using spirit of wine instead of water. This paper only requires washing in water to fix the drawing. 2. Muriated Paper. —The paper is first soaked in solution of copper salt, pressed with a linen cloth or blotting paper, and dried. It is then brushed over on one side (which should be marked near the edge) with the solution of nitrate of silver, and dried at the fire. The stronger the solution the more sensitive the paper. If dipped in a solution consisting of 35 grains of chloride of barium and 2 oz. of distilled water, richer shades of color are obtained. 3. Iodized Paper.-Brush over the paper on one side (which should be marked) with strong solution of nitrate of silver (100 gr. to 1 oz ); then dip it in a solution consisting of 100'gr. of iodide of potassium dissolved in 4 oz. of distilled water. Wash it in dis. tilled water, drain, and dry it. 4.'Bromide Paper.-Soak the paper in a solution composed of 40 gr. bromide of potassium dissolved in 1 oz. of distilled water; then brush it over with a strong -solution of nitrate of silver, and dry in the dark. 5. Calotype Paper.-Dissolve 100 gr. of crystallized nitrate of silver in 2 oz. of distilled water, and add 2 fluid dr. and 40 minims of acetic acid. Mix these at the time of using with an equal measure of cold saturated recently prepared solution of gallic acid. Brush iodized paper with this solution, and mark the side; in half a minute dip it into water, and press it between blotting paper. It is then ready for the camera, where it remains from half a minute to 5 minutes. When removed from the camera dip it into water, 346 PAPERS. press it between blotting paper, and wash it with a -soluttion.lo 10t9) gr. of bromide of potassiurm' in 8 or 10 oz. of, water. 6. C(rornotype Paper.-Soak the paper in a solution of bichromate of potash (in which solution a little sulphate of indigo is sometimes added to vary the color), and dry it at a brisk. Tfix the drawing careful immersion in wvarm water is all that is requiredl It is not sufficiently sensitive for the camera. 7.' Conipound 6hromotype Paper.-Dissolve 10 gr. of biclihromate of potash1, and 20 gr. of sulphate of copper, in: an ounce: of weater. Wasll the paper in thlis solution, and dry it. After ti;he paper has been exposed to thle sun, with the article. to be copied superposed upon it, it is washed over in the dark witll a solution of nitrate of silver of moderate strengtl. A vivid pictulre inakes its appearance, which is sufiiciently fixed by washing in pure rwater, This is for copying englgravingcs,: &c. Another method is to blrtsh writing paper over with a solution of 1 dr. of sulplhate of copper in~ 1 oz. of water; and when dry with a strong, but not satturated, solution of bichromate of potash. 8. Cyanotype Paper.-B1rush the paper over with a solution of ammonio-citratc of iron. Expose the paper in the usual wa,- then wash it over with a solution of ferro-cyanide of potassium. 9. Crysotype Paper.-Wash the paper with solution of ammoniocitrate of iron, dry it, and afterwards brush it over with a solutionl of ferro-cyanide of potassium. Dry it in a dark room. The imange is brouglht out by brushing it over with a neutral solution of gold l or silver. 10. Catalisotfype.-Steep paper inh water, with a: drop or two of hydrochloric acid, absorb the superfluous moisture withl blotting paper; brush over with:a mixture: of ~ dr. syrup of iodide of: iron, 2~i dr. of water, and a drop or. two of tincture of iodine. Dry wi:v hi blotting paper, and brush over withl a solution of 12 gr. of nitrate of silver to 1 oz. of distilled water. It is then. ready for the cnmera. The picture is fixed by washing in water, and afterwards in a sol- I tion of 20 gr. of bromide to 1 oz. of-potassium. 11. _cPaperfor Positive Photographs. —Most of the preceding give negative pictures, the lights and shadows' being reversed; in the following they are correct: Dissolve 40 gr. of muriate of ammonia in 4 oz. of water. Wash highly glazed paper in thissolution, dry it, and brush it over with the following solution: Dissolve 120 cgr. of crystallized- nitrate of silver in 1I oz. of distilled water; and add 1 oz. of alcohol; after it has stood a few hours filter it. Expose I the paper thus, washed to the sunshine, till it is darkened; if Inottiled, wash it a second time, and expose it again. Before using thec paper make up the following: solution' Hydriodate of barytes 401 gr.; water 1 oz.; pure sulphate of iron 5 gri Mis, filter, add a drop or two of diluted sulphuric acid,. and when settled decant the clear liquor for use. VWash the- paper over -in this, solution, expose BRONztIN. ti~G it in the. damp state,: vith the erigraving or other object on it to the light, and fix the drawing by washing with water onlly Photograishs. To copy objects, lay them on a plate of clear glass, fixed' in a frame; place the prepared paper over them; and fix a back, with a cushion attached to it, so as to press the paper closely on the glass. The-glass is then exposed to the light, and the drawing afterwards fixed, as described above. For feathers, lace-work, and other objects which freely admit light through them, the nitratedl paper and less sensitive muriated papers may be used. For copying'engravings, leaves, and other. botanical objects, or entomological specimens, the more sensitive mouriated papers, or the bromide paper, or other sensitive kinds, may be used. Engravings should;:be vetted, and placed with their face to the prepared. side of the paper, anld kept in close contact with it. Leaves should lhave their under surface next the glass. For the camera, the most sensitive samples of the muriated papers,' made with not less than 100 gr. of nitrate of silver to the ounce, are selected. The calotype is still more certain. The papers intended for the camera lrequire to be very:carefully prepared. Glass is:used instead of:paper, after being coated with white of egg, or collodion, with which.the compounds of silver are mixed, or over which they are brushed. B R-O N' Z I:NG. Bronzi.tZg: Sculpture, Wood, &c.'Bronze.of a good q.uality'acquires, by oxidation, a fine green tint, called patina agtiqza., Corinthian brass receives, in this way, a beautiful clear green color. This, appearance is. imitated by an artificial process, called bronzing. A solution: of sal-ammoniac and salt of sorrel in vinegar is used-for bronzing metals. Any number of layers'nay be applied, and the shade becomes deeper: in proportion to the number applied. For bronzing sculptures of wood, plaster figures, &e., a composition of yellow ochre, Prussian blue, and, lamp-black, dissolved in glue-wate;'is employed. Bronze. 1. Copper 83 parts; zinc 11 parts;.tin 4 parts. lead 2 parts,'Mix. 2, Copper 14 parts; melt, and add zinc 6 parts; tin 4 parts. Ancient Bronze. Copper 100 parts; lead and tin each 7 parts. Mix. 348 ENAMELS. To give an Antique Appearance to Bronze Figures. Salt of sorrell part; sal ammoniac 4 parts; white vinegar 224 parts. Dissolve, and apply with a camel-hair pencil, just sufficient to damp the bronze, previously warmed. Repeat the operation if required. Keller's Bronze. Copper 91 parts; tin 2 parts; zinc 6 parlts; lead 1 part. Mix. Bronze Powder. Bichlloride of mercury 1 part; borax and nitre each 8 parts; tutty 16 parts; verdigris 32 parts; oil to make into a paste. Melt. Beautiful Red Bronze PoYwcder. Sulphate of copper 100 parts; carbonate of soda 60 parts. Apply heat until they unite into a mass, then cool, powder, and add copper filings 15 parts. Well mix, and keep them at a white heat for twenty minutes, then coo], powder, and wash and dry. Bronzing Fluid for Gunbs, &c. Nitric acid sp. gr. 1 2, nitric ether, alcohol, muriate of iron, each 1 part. Mix, then add sulphate of copper 2 parts; dissolved in water 10 parts. ENAMELS. Whilte Enamel. Tin 2 parts; lead 1 part. Calcine, then take of the above oxides 1 part; crystal 2 parts; manganese a small portion. Grind well together, fuse, and pour the mass into cold water; dry, grind again to powder, and fuise; repeat the process four or five times, observing great care to prevent any contamination from smoke, or iron, or copper. Another. Arsenic 14 parts; potash 25 parts; nitre 12 parts; glass 13 parts; flint 5 parts; litharge 3 parts. Blue Enamel. Fine paste (not metallic) 10 parts; nitre 3 parts. Oxide of cobalt to color. Green Enamel. Frit 1 pound; oxide of copper. ounce; red oxide of iron 12 grains. MAaRBLE STAINING. 349 Fluxes of Enaazel Colors. 1. Flint powder 1 part; calcined borax 1 part; flint glass 3 parts; red lead 4 parts. Keep them in a state of fusion, in a'Hessian crucible, for three hours; then pour into cold Water, dry, land powder. 2. Glass powder 11 parts; white arsenic 1 part; nitre I part. Mix. YellowJ EYznamel. White oxide of antimony 1 part; white lead 2 parts; alum aild sal-ammoniac each 1 part. Mix in fine powder, and apply just suf-.ficient heat to decompose the ammoniac. Black' Enamel. Clay 2 parts; protoxide of iron 1 part. Mix. M A R BLE ST.A ININ G. To Stain Marlble. It is necessary to heat the marble hot,- but not sufficiently so as to injure it, the proper heat being that at whichl the colors nearly boil. Bl/ue.-Alkaline indigo dye, or tuirnsole with alkali. Red.-Dragon's blood in spirits of wine. Yellow.-Gamboge in spirits of wine. Gold Color.-Sal-ammoniac, sulphate of zinc, and verdigris, equal parts. Green.-Sap green, in spirits, with potash. Brown.-Tincture of logwood. C:'rimson.-Alkanet root in turpentine. The marble may be veined according to taste. To stain marble well is a tedious and difficult operation. To Staint White Maarble. Apply with a brush a strong alcohol tincture, made from the root alkanet. To Clean Marble. Chalk (in fine powder) 1 part; pumice 1 part; common soda 2 parts. Mix Wash the spots with this powder, mixed with a little water; then clean the whole of the stone, and wash off with soap and water. To Extract Oil fomn Stone or Marble. Soft soap I part; Fuller's earth 2 parts; potash 1 part; boiling water to mix. Lay it on the spots of grease, and let it remain for a few hours..~~~~~~8 350 COMPOUND COLORS IN DYEING. COMPOUND COLORS IN DYEING, Are produced by mixing together two simple ones; or, whill is'the same thing, by dyeing cloth first of the simple color, and then by another. These colors vary to infinity, arcording to the proportions of the ingredients employed. From blue, red, and yellow, red-olives, and greenish-greys are made. From blue, red, and brown, olives are made from the lightest to the darkest shades;'and by giving a greater shade of red,' the slated and lavender-greys arecmade. - From blue, red, and b!ack, greys of all shades are made', such as sage, pigeon, slate, and lead-greys. The king's or prince's color is duller than usual; this mixture produces a variety of hues or colors almost to infinity. From yellow, blue, and brown, are made the goose-dung and olives of all kinds. From brown, blue, and black, are produced brown-olives, and their shades. From the red, yellow, and brown, are derived~ the orange, gold color, feuille mort or faded leaf, dead carnations, cinnamon, fawn, and tobacco, by using two or three of the colors as required. From yellow, red, and black, browns of every shade are made.:From blue and yellow, greens of all shades. From red and blue, purples of all kinds. are formed. ~:Dyer's Spirit. Aquafortis 10 parts;: sal-ammoniac 5 parts; tin 2 parts. Dissolve. Japan:Grounds. Red.-Vermlillion makes a fine scarlet, but its;appearance in japanned work is much improved by glazing it with a thin coat of lake, or even rose pink. Yellow.-King's yellow, turpeth mineral, and Dutch pink, all: form very bright yellows, and the latter is very cheap. Seed-lao varnish assimilates with yellow very well; and when they are required very bright, an improvement may be effected by infusing tinrmeric in the varnish which covers the ground. Grcee.-Distilled verdigris laid on a ground of leaf gold produces the brightest of all greens; other greens may be formied by mixing Kilng's yellow and bright Prussian blue, or turpeth mineral and Prussian blue, or Dutch pink anrid verdigris. Blte.-IPrussian blue, or verditer glazed with Prussian blue or smalt. 1Whitec.-White grounds are obtained with greater difficulty than any other. One of the best is prepaled by grinding up floclk-white, or zinc-white, with one sixth of its weight of starch, and drying it; it is then tempered, like the other: colors, using the mastic' varnish for common uses; and that of the best copal for the finest. Par1 ~ ~ ~ - -- ~-~~~; —~. I- - - t a-LISfhlESp. f 351 ticular care should be take tliat the'copal for this use be made of the clearest and whitest pieces.'Seed-lac may be used as the uppermost coat, where a- very delicate white is not required, taking care to use such as is least colored. Black.-Ivolry-black, or lamp-black'; but if the lamp-black be usedit should be previously calcined inl a closed crucible. Black grounds may be formed on metal, by dr'ying linseed oil only, when mixed with a little lamp-black. The work is then exposed in a stove, to a heat which- will render the oil black. The heat should be low at first, and increased vely gradually, or it will blister. This kind of japan requires no polishing. It is extensively used for defending iron articles from rust. P O L I St I E S. To Polish Brass -Inlaid WIork. File the brass very clean with a' smooth file; then take some tripoli powdered very fine, and mix it with the linseed oil. Dip in this a rubber of hat, with which polish the work until the desired effect is obtained. If the work is ebony, or black rosewood, take some elder-coal powdered very fine, and apply it dry after you have done with the tripoli, and it'will produce a superior polish. The French mode of ornamentilg with brass differs widely from ours, theirs being chiefly water-gilt (or molzu); excepting the flutes of columnQ, &c., which are polished very high with rotten stone, and finished with elder-coal. To Brass Plates of Copper. The plates previously sufficiently heated, expose themn to the fumes of zinc, To Clearn Brass. 1. Finely powdered sal-ammoniac; water to moisten. 2. Roche alum 1 part; water 16 parts. Mix. The articles to be cleaned mustbe imade-warlm, then rubbed with either of the above mixtures, and finislihed with fine tripoli. This process will give them the brilliancy of gold. To Brass: Vessels of Copper. Argol' 1 part; amalgam of zincl part; muriatic acid 2 parts; water to fill the' vessel, Boil. Method of Cleaning Brass Ornaments. Brass ornaments that have not been gilt or lacquered may be cleaned, and a very brilliant color given to them, by. washinig them 352 BOOKBINDErs' RECIPES. with alum boiled in strong ley, in the proportion of an ounce to a pint, and afterwards rubbing them -with strong tripoli. French Polish.: Alcohol 260 parts; copalVarnIish 13 parts; sandaraclh (powdered) 1 part; mastic (powdered) 1 part; shell-lac (powdered) 24 parts. Mixs, and digest in a moderate lheat,, in a strong- close vessel. To French Polish. The varnish being prepared (shell-lac),:-tle article to be polished being finished off as smooth as possible 1with glass paper, and your rubber being prepared as directed below, proceed to the operations as follows: The varnislh, in a narrow necked bottle, is to be applied to the middle of tile flat face of the rubber, by laying the rubber on the mouth of the bottle and shaking up the varnish once, as by this means tile rubber will imbibe tile proper quantity to varnish a considerable extent of surf:ace. Thle rubber is then to be inclosed. in a soft linen cloth, doubled, thIle rest of the clothl being gathered up at the back of the rubber, to formu a handle. Moisten the face of tile linen with a little raw linseed oil, applied with tile finger to the mniddle of it. Placing your work opposite the light, pass your rubber quickly and lightly over its surface until the varn ish becomes dry, or nearly so; charge your rubber as before with varnish (omitting the oil),'and repeat tile rubbing, until three coats -are laid on, when a little oil may be applied to the rubber, and two coats more given to it. Proceeding inll this way, until the varnish has acquired some thickness, wet the inside of the linen clotli, before applying the varnish, with alcohol, and rub quicly, lightly, and uniformly the lWhole surface. Lastly, wet the linen cloth with a little oil and alcohol without varnish, and rub as before till drvy. To makhe the Rubber.-Roll up a strip of tliick woollen clotil wlrhich has been torn off, so as to form a soft elastic edge. It should form a coil fiom one to three inches in diameter, according to the size of the work. BOOKBINDERS''RECIPES. Japan Coloring, for Leather' Book- Covers, &c. After tile book is covered and dry, color the cover with potashwater mixed with a little paste, give it two good coats of Brazil wash, and glair it. Put the book between wands, allowing the boards to slope a little. Dash on copperas water, then with a sponge full of red liquid, press out on the back and on different parts large drops, which will run down each board, and make a fine shaded red. When the cover is dry wash it over two or three times with Brazil wash, to give it a brighter color. BOOKBIN'DERS' RECIPES. 353 Blzie Sprinkile for BdokbBinders. Strong sulphuric acid 8 ounces; Spanish indigo, powdered, 2 oz. Mix in a bottle that will hold a quart, and place it in a water-bath'to promote solution. For use, dilute a little to the required color in a teacup.; Blue icarble for Books, dc. Color the edges with King's -yellow, and wlien dry tie the book between boards. Throw on blue spots in tlie gurn trough, wave theini withl the iron pin, ald apply the edges thereon. Bi.own CGolor for _Mfarblinjg or Sprinkling Books.1. Logwood cliips' 1 part; anilatto 1 part; boil iiln water 6: parts. If too light, add a piece of copperas about the size of a pea.2. Umber, any quantity. Grind it on a slab with ox gall and a little lampblacL. Dilute witll ale. Gold Sprinkle for Books. Put into a marble mortar half an ounce of pure honey and one book of gold leaf, rub them well togethier until tlley are very fine; add half a pint of clear water, and mix them well together: whlen the water clears, pour it off, and put in more, till the honey is allextractedi and nothing remains but the gold.:Mix one grain of corrosive sublimate in a teaspoonful of spirits of wine, and when disssolved, put the same, together wvith a little gum-water, to the gld, antd ibottle it close for use. The edges of the book may be sprinkled or colored very dark, with green, blte, or purple, and lastly with the gold liquid, in small or large spots, very regular,'shakinng tile bottle before using. Burnish the edges when dry, and dover tllem with- papler to prevent the dust falling thereon. This:'lprinkle will lhave a most beautiful appearance on extra work; ladies may use- it for' ornamenting tlleir fancy work, by putting it on with a pen or camel's-hair brulsh, and wvhen dry burnish it with:a dog's tooth...Marble for Leather Book-coveris. Wasli tie covei' and glai'i it; take a spollge charged with wateri, having the book between wands, and drop, the: water' from the s:ponge on the different parts of the cover, sprinkle very fine withl vinegar blaclk, then wTith birownn, and lastly wvith vitriol water. Obsei've to sprinkle on tihe colors immediately after each other, and *to wash the cover over wit a clean spon'ge and water. Clhinese Edye for Books. 1. Color the edge with ligiht' liquid bliei and dry; tlien take a sponge charged withl vermillion, ltand dab on spots according to fi'acy; niext thirow on rice, and finiskh the edge: with dark. liquid 2. Color light blue on different paits of the: edge with a-sponge;! do the same wliere'theire are vacancies; with yellow and Brazil red; 354 BoOKBINDERs' RECcIcEs. dry and dab on a little vermillion in spots; then throw on rice, and finish with a bold sprinkle of dark blue. Burnish. fWax lAlarblefor Leatlher Book-covers, &c. This marbling must be done on the fore edge, before the back of the book is rounded, or becomes round, when in boards, and finished on the head and foot. Take beeswax and dissolve it over the fire in an earthen vessel; take quills stripped of' their featheris, and tie them together; dip the quill-tops iil'the wax, and spot the edge, with large and small spots; take a sponge charged with blue, green, or red, and smear over the edge; whlen done, dash off the'wax, and it will be marbled. This will be useful for stationery work, or for folios and quartos. Egyptian 3racrble for Leather Book"-corers. 1. Yello.-Boil quercitron bark. with water and a little powdered alum, over a slow fire, until it is a good strong yellow. Pour the liquid into a broad vessel, sufficiently large to contain the cover when extended. Before the liquid is cool, take the dry cover, and lay the grain side flat on the color; press it lightly that the -whole may receive the liquid; let it soak some time, and then take it from the vessel. The book must be covered in the usual manner, and permitted to dry from the fire. Glair the book; when dry, place'it between the wands; take a sponge and water, and press large spots thereon; dip a quill-top into the vinegar black, with it touch the water on' the cover indifferent parts, which will have a fine effect- when managed with care.'Let it stand a few minutes, then take off the water with a clean sponge. 2.-Green:. —Color the cover in a large vessel, as mentioned before, with Scott's liquid blue; when done, put it, into a vessel of clear water for an hour. Take it out and press out the water, then cover tie'book. Glair'the cover; when dry, place it between twands, and drop weak potash water from a sponge thereon;: dip the quill-top into the strong black, and touch the water. with it. This must be repeated till you have a good black. When dry, clear it with a sponge and water. 3. Red.-Boil Brazil, dust in rain-water on a slow fire, with a little powdered alum and a few drops of solution of tin, till a good color is produced. Dip a piece of calf leather into the liquid, and you may ascertain the color wanted. If too ligllt, let it boil till it is reduced to one half of the quantity'; take it fiomn the fire, add:a few more drops of the solution of tin, and pour it into a large vesselI Put the dry.cover on the liquid, and let it remain for a quarter of an hour, then press out the water. Color it over with a sponge and the quercitron bark-vater, and cover the book. Glair the cover, place it between wands, dash on water with a brush, also potash water; and,'lastly, finish it with the strong; vinegar black,.with.the quill-tolp. Observe that, too much black is not put BooxBINDERs' RECIPES. 355 on; the intention of the marble -is to show the red as transparently as possible. French Afarble for Books. Provide a wooden trough, two inches deep, six inches wide, and the length of a super-royal sheet. Boil in a brass or copper pan any quantity of linseed and water, until a thick mucilage is formied strain it into the trough' and let it cool; then grind on a marble slab any of the following colors in small-beer: Prussian blue. king's y'ellow, rose pink, vermillion, flakee white, lamp-black,'brown umber, green, blue, and yellow, orange, red, and yellow, purple, red, and blue,. brown, black, and yellow, or red. The. lamp-black and umber must be burnt over the fire to deprive them of their greasy nature. For each color you must have two cups, one for the color after grinding, the other to mix it with ox-gall, which must be used to thin the colors at discretion. If too much gall is used the colors.will spread; when they keep their place on the surface of the trough, when moved with a quill, they are fit for use. To prevent the water entering between the leaves of the book, tie it tight' between cutting-boards of the same size, and place the troughl in a steady situation, to prevent the colors from moving. Having all things in perfect readiness for marbling, supposing you begin with the blue, throw on with the brush bold spots of blue, sprinkle very fine with the white on the blue spots, fill up the spaces with red and yellow, by dipping first the quill-top into the yellow, and touching the gum therewith; then with the red. The red and yellow may be waved or drawn round the blue spots wvith an iron pin, or as the marbler may think' proper, according to fallcy. HIold the book with its edge downwards,. and press it lightly on the colors so disposed on the gum, and the edge will be immediately marbled. The colors that may remain on the gum must be taken off,;by applying paper thereon, before you prepare fol marbling again. In this manner you may marble the edges to resemble the end-papers, which will have a pleasing effect. Chinese Jfarble for Leather Book-covers, dic. Color the cover of the book dark brown, and when dry put it into the cutting-press, with the boards perfectly flat; mix v-whiting and water of a thick consistence and throw it on, in spots or streaks, some large and some small, which must remain till dry. Spot or sprinkle the cover -with liquid blue, and lastly throw on large spots of liquid red. The colors must be dry before washing off the whiting. Orange Sprinkle for Book.s. Color the edge with King's yellow,, mixed in weak gum-water, then sprinkle with vermillion mixed in the same manner. 356' BOOKBINDERnS''RECIPES. Green-Spri2zkle for Books. 1. Yellow the edge, then sprinkle with dark blue. 2. French berries 1 part; soft water 8 parts. Boil, and add a little powdered alum; then bring it to the required shade of green, by adding liquid blue. Green lMarlcble for Leather Book-covers, &c. The edoc must be marbled;with a good bright greein only. When tlle color is prepared with the os-gall, and ready for use, a fewV drops of sweet oil must be mixed: therein, the color throw;n on witli a brush, in large spots, till the gum is perfectly covered. The oil will make a light ede roilnd each spot, and have a good effect.: Blue, green, and brown- may be also used separately iri like nanner. Sheets of paper may be done, liaving a trough lage enougli, and the sheets damped as for printing, before maibling. Spirits of turlentine inay be sprinkled oa1 the colors, which will umake white spots.'Binder's Thread'lrar:blie.'Yellow, the edge; when dry, cut:'pieces of thick thread-over the edge, which will:fall on different parts irregularly; give it a fine dairk sprinkle, and' shake off the thread. Rice JMarble, for Leather B'ook-covers, &c. Color the cover with spirits of wine and turmleric, then place on rice in a regular manner; throw on a very fine sprinkle of copperas-water till tlie cover is nearly black, and let it remain tfill dry The cover: may be spotted with the red liquid or potash-water, very freely, before the rice is thrown off the boards. Orangre Cblor for Jlfa.bling or Sprinzkling Bo'oks, &c. G(rounid Brazil-wood 16. parts; ann'btto 4 partsi; alum-, -sugar, and gum-arabic, each 1 part; water 10 parts. Boil, st'ain, and bottle. Tree Jlarble; for Leather Book-covers.:. A marble in the form of trees- may.be done by. bending the boards a little on the centre, using the same method as the common marble, having the cover pleviously preparied;.. The end of a candle may be'rubbed on different parts o6f the boards, which will f6rm knots. Viregar.Blck for Bookbinders, cc. Steep iron filings or rusty iron in good vinegar for two or three days, then strain off the liquor. To Spriiskle Books. Take a stiff brush made of. hogs' bristles, perfectly clean, dip it in the color; squeeze out the superfluous liquid; then rub a BOOKBINDERS' RECIPES. 35'7 folding-stick across the,brush, and a fine sprinkle -will fall on the edge of the book, which should be previously screwed tight in the cutting-press. Repeat the operatioin until the color is thlrown equally on every part of the leavres. The brush should be held in the left hand, and the stick in the right. Purple Sprinkle for Bookbinders. Logwood chips 4 parts; powdered alum 1 part; soft water 241 parts. Boil until reduced to sixteen parts, and bottle for use. 2. Brazil dust (fine), and mix it with potash-water for use. Soap Marble for Books. This is applicable for marbling stationery, book edges, or sheets of paper for ladies' fancy work. Grind, on a marble slab, Prussian blue, with water, and a little brown soap, to a fine pliable consistence, that it may be thrown on with a small brush. Grind King's yellow, in the same manner, with water and white soap. When green is intended for the ground color, grind it with brown soap, and King's yellow with white soap. Lake may be used for a ground color, and PIussian blue ground with white soap; brown umber for a ground color, and flake-white ground with white soap. Any color of a light substance may be grousnd for marbling. Spotted iMVarble for Books, &c. After the fore-edge of the book is cut, let it remain in the press,:and throw on linseeds in a regular manner; sprinkle the edge with any dark color, till the white paper is covered, then shake off the seeds. Various colors may be used. The edge may be colored with yellow or red before throwing on the seeds and sprinkling with blue. The seeds will make a fine fancy edge when placed very thick on different parts, with a few slightly thrown on the spaces between. Brown Sprinkle for Leather Book-covers, &c. Pearlash or potash I part; soft water 4 parts. Dissolve and strain. BRed Sprinkle for Binders. Brazil-wood (ground) 4 parts; alum 1 part; vinegar 4 parts; water 4 parts. Boil until reduced to seven parts, then add a small quantity of loaf-sugar and gum. Bottle for use. Black Sprinkle for Leatler Book-covers, &c. Green copperas 1 part; soft water, hot, 6 parts. Dissolve. Stone l}farble for Leather Book-covers, dc. Glair the cover, and when dry put the book into the cuttingpress, with the boards sloping, to cause the colors to run gently. I 358 CRAYONS. down. Tlirow on weak copperas-water with a Ubrush:; dip:a sponge into' the strong potash-waiter, and press ouft: the color fromni the sPong(e onI differelt parts of the baclk,' so that the colors may run Idown each side fom thle back. Wherie the bown has left. a v lacancy apply vitriol-water in the same manner. The book must remain till perfectly dry before wasling it. CRAYONS. LitheOqracphic Crayon& 1. Take white wax 4 parts; gum-lnac 2 parts. Melt over a gentle fire, then add dry- tallow soap in shavings 2 parts. Stir: until dissolved. Next add lwhite tallow. 2 parts; copal varnishl 1 part; lampblack 1 part. Mix well, and continue the heat and stirring until, on trial by cooling a little, it appears of a propelr quality, which should be that it will bear cutting to a fine point, and trace delicate lines without breaking. 2. Take dry white tallow soap 6 parts; white wax 6 parts; lampblack 1 part. Fuse in a covered vessel. 3. Take lampblack l part; tallow soap 2 parts; shell-lac 2: parts; -wax 4 parts. Mix, with heat, and mould. 4. Take dried tallow soap 5 parts; wax 4 parts; lampblack 1 part. Mix as before. Crayon1s. 1. Slhell-lac 6 parts; spirit 4 parts; turpentine 2 parts; color 12 parts; pale clav 12 parts. Mix. 2. Pipe-clay, color as required, water to mix. Form into a stiff paste, and roll it into crayons. To Fi'x Crayon Colors. Paste your paper on canvass, in a frame, in the uslual way, then brush over the back two or three times with' the' following mixture, and when the last coat.is dry give the face- of the.picture one or two coats in the same way. This vill -male it resemble an oil painting. Spirits of turpentine 10 parts; boiled oil 6 parts. Mix. To render permanent Chialk or Pencil Drawings. Laly the drawing on its face, and give the back two or three thin coats of the followincg (No. 1) inixture; let it dry, and turn it with. the chalk upwards, and give tliat side one or two coats also; lastly, if you choose, give it one or two coats of No. 2. 1. Isinglass: or gum-arabic 5 parts; water 12: -parts. i Mix. 2. Canada balsam 4 pmarts; turpentine 5 parts. Mix. Wash to fix Blackdead Pe2zcil Drdawings. 1. Isinglass 1 part; Water 50 parts. Dissolve with heat, and filter., G LDIN. 359.2.:Take skimmed milk, and strain. For use, pour the liquid on a surfac'e sufficiently large, and take the drawing by the corners,lay it flat on the wash, then carefully remove it, and place it on a slanting surfaeeito drain and dry. This will also answer for chalk drawsings. G I L DI N G. To Gild or Silver Leather. Finely powder resin, and dust it over the surface of the leather, then lay on the leaf, and apply (hot) the letters' or impression you wishi to transfer; lastly, dust off the' loose metal with a cloth. Thle cloths used for this purpose become; in time, very valuable, anid are often sold to the refiners for, $5 to; $7.: To gild ont Calf and Sheep Skin. RWet the leather withy the white of eggs; when dry rub it with your hand and a little olive oil, then put the gold leaf, and apply the hot iron to it. Whatever the hot iron shall not have touched will go off by brushing. To gild Copper, Brass, &c. (Patent.) Fine gold 5 parts; nitric acid (sp. g. 1'45) 21 parts; hydrochloric acid (sp. g. 1:15) 17 parts; pure water 14 parts. Digest with heat in a glass vessel until all the gold is dissolved, and till red; or yellow, fumes cease to rise. Decant the clear liquid into some convenient vessel, and add water, 500 to 600 parts. Boil for two hours, let it stand to settle, and pour off the clear into a suitable vessel. For use, heat the liquid and suspend the articles (previously well cleaned) by means of a hair or fine wire, until sufficiently coated with gold, then well wash them in pure water. To gild Glass and Porcelain. 1. Apply to the part a surface of gold size; when nearly dry lay.on the leaf. 2. Gold powder 2 parts; borax 1 part; turpentine to mix. 3Mix and apply to the surface to be gilded with a camel-hair pencil; when quite'dry,. heat it in a stove until the borax vitrifies. Burnish. Platinla, silver, tin, bronze, &c., may be applied in a similar manner. To give Iron the color of Copper. Take 1 oz. of copper-plates, cleansed in the fire; 3 oz. of aquafortis; dissolve the copper, and when it is cold use it by washing your iron with it by the help of a feather; it is presently cleansed 360 GILDING. and smooth, and will be of a copper color; by much using or rubbing it will wear off, but may be renewed by the same process. A way of Gilding with Gold upon Silver. Beat a ducat thin, and dissolve in it two ounces of aqua-regia; dip clean rags in it, and let them dry; burn the rags, and, with the tinder thereof, rub the silver with a little spittle; be sure first that the silver be cleansed fiom grease. Gilder' s tVax. 1. Yellow wax 3 pounds; verdigris 1 pound; sulphate of zinc 1 pound; red oxide of iron 2~ pounds. Powder the last three articles very fine. 2. Yellow wax 7 pounds; colcothar 7 pounds; verdigris 3 pounds; borax + pound; alum 1 pound. To dye in Gold Silver IMedals, or Laminas, through and through. Take glauber salt, dissolve it in warm water, so as to form a saturated solution. In this solution put a small proportionate quantity of calx, or magister of gold. Then put and digest in it silver laminas cut small and thin, and let them lie twenty-four hours over a gentle fire. At the end of this term you will find them thoroughly dyed gold color inside and out. To gild Silks, Satins, &ec. Nitromuriate, of gold, in solution, 1 part; distilled water 3 parts. Mix. Lay out any design with this fluid, and expose it, while wet, to a stream of hydrogen gas; then wash it with clear water. To make Transparent Silver. Refined silver one ounce; dissolve it in two ounces of aquafortis; precipitate it with a pugil (a quantity that may be taken up between the thumb and finger) of salt, then strain it through a paper, and the remainder melt in a crucible for about half an hour, and pour it out, and it will be transparent. To make Copper into a Metal like Gold. Distilled verdigris 4 oz.; Tutite Alexandrinve prseparatse two oz.; saltpetre I oz.; borax -~ oz. Mix all together with oil, till they be as thick as pap; then melt it in a crucible, and pour it into a fireshovel, first well warmed. Mercurial Plating. Quicksilver 4 parts; nitric acid 4 parts; finely powdered cream of tartar 2 parts; finely powdered salt of sorrel 1 part. Dissolve the silver in the acid, then add the rest, and stir until dissolved. This imparts a pleasing silvery appearance to articles formed of copper, by merely applying it with the finger. Grecian Gilding. Take sal-ammoniac and bichloride of mercury, equal parts, GLAss STrAINS. 861dissolve in nitric acid, and niike a- soluti'on of gold with this fluid, lay it on the silver, and expose it to a red heat; it Will then be gilade-d. To gill.7 or siluveer Writinyg. Let there be a little gunl and lump-sugar in thle inlk:yoti: write with; when dry, breathe on it andapply. the leaf. To whitenl Coopper throughout. Take thin plates of copper, as thin as a knife, heat them six or seven times, and quenchl themi iin water; then melt them, and to eaellh poutid aadd 4 olnces of saltpctre aind 4 ounces of arsenic, well powdeired and mixed. and first melted apart in anotlhet crucible, by'gentle degrees; then talke them out, and powder them; then' take Venetian borax and' wlite tartari of eacli an ounce and a half; then melt these, wit thle former powder, in. a crucible, and pour them out into sonme iron receiver; it will appear as Clear as crystal, and is called crystaalli'un flz ixm arsenicum. Of this. clear matter, broken into little pieces, thlrow into t (he melted copper (by small pieces at a time, staying fiave or six-, minutes bctweC1ci ech linjection) 4 ounces; l:hen all is tllrownvi in, increase the fire, till: all be well imelteil togetlier fr a quiarter'of an hour; then' poui it out into an ingot. lbo gild Steel. Apply an ethlerial solution of gold. This is equally adapted to lettering, as wholly covering the object. It may be applied with a pen, or otherwise. G L A SS S T A I N"S. Red Stain for' Glass. 1. Rust of iron 100 parts; glass of antimlony 99 parts; yellow glass of lead 98 parts; sUilpliiret of silver 3 parts. Mix.:;.i- WhViitei bhiai'd: ei'amiiel' 1O00 Pats:; rierd chllkl' 50 pasrts,; peroxide of'eopperi S p'atm'ts: R- dttee to fine powdei, anRd:-mix. Blue Glass. Plain paste 300 parts; zaffire 3 parts; manganese' 1 part. If tlie cglass should- be of t-oo dee-p a; blue, ause less zaifre an'd manganese; if too purple, omit the manganese altogether. -Black Staii fori Glas-s. 1. Black scales of iron. 29 parts;. white crystal glass 4 parts; antimony 2 parts; manganese I pAir, lti-negar to mix. 2 -.-; Gla:' of afitin.on. -1 part;, o. xielt of coppefie 2: parts'; crystal j glass8: piatss.x'- MIix;1 362 FACTrrious STONES. Orange Stain for Glass. Precipitated silver powder, yellow ochre, red ochre, equal parts. Turpentine to mix. Bfrown Stain for Glass. White glass 2 parts; manganese 1 part. Mix, Fleslh Colorfor staiininy Glass. Red lead 1 part; red enamel 2 paits. Mix with alcohol, YellouJ Stain for Glass. Chloride of silver 1 part; burnt pipeclay 8 parts. Reduce to fine powder, and mix. This stain must be applied to the. back of the glass. To Marble a Glass Globe. Grind well on a stone, miniumr for red, tnrmelic or rather' cerussa citrina, for yellow, smalt for blue, erdigris for green,. eeruse, or chalk, for white. Work each in oil separate, and with a hlog's hair pencil, single or mixed, as you think fit, scatter the same into the: glass, and roll it, or dispose the colors, a ts yoike. Tien, last of all, fling a little mead amongst them, whicIh cdveirs all. For the Magic Lantern, paint the glasses with transparent cors', tempered with oil of spike. FACTITIOUS STONES. Factitious Amethyst. 1. Take strass 5000 parts; oxide of manganese 37 parts; oxide of cobalt 25 parts; purple of Cassius 1 part. Fuse for twenty-six hours, and cool slowly. 2. Take paste or strass 10,000 parts; oxide of manganese 25 parts; oxide of cobalt 1 part. Factitious iEnmerald. 1.L Oxide of chrome I part; green oxide of copper 20 parts;;strass 2300 parts Fuse with care for twenty-six hours, then cool,slowly. 2. Strass 10,000 parts; acetate of copper 150 parts; protoxide of i-on 3 parts.- As before..3. Strass 6600 parts; carbonate of copper 60 parts;.glass of anti- -mony 6 parts. Fuse with care. 4. Strass 500 parts; glass of antimony 20 parts; oxide of cobalt;3 parts. As before. Artificial Coral. Yellow resin 4 parts; vermnillion 1 part. Melt.. This' gives a very pretty effect to'glass, twigs, cinders, stones, &c., dipped into: FACTIT IOUS SiTONES. 363 I it. It is also useful for a cement for ladies' fancy wo;l, such as grottoes, &c., Paste resembling the Red Corndelian. Plain paste 1000 parts; glass of antimony 500 parts; calcined vitriol 63 parts or less;:manganese 4 parts. Melt together. Padte resembling tlhe White Cornelian. Plain paste 1000 parts; yellow ochre 8 parts; calcined bones 31 parts. As before.Factitious Opal. 1. Strass 500 parts; holirn- silver 10 parts; calcined magnetic ore' 2 l)arts; chalk marl 25 parts.:Mix:in fine powder, and fuse withll great care. 2.:Plain~ paste 100 parts; calcined bones 6 parts. Factitious Oriental Ruby. Strass 7000 parts; prvecipitate of Ca-ssius and nitric peroxide of, iron. each 16,5 parts,; golden sulphuret of antimony 160 parts' manganese calcined with nitre 150 parts; rock crystal: 1000 parts. Mix in fine powder; and carefully melt. ]Fcictitious Sapphire. 1. Oxide of cobalt 1 part; strass SO parts. 2. Paste or strass 2300 parts; oxide of cobalt 34 parts. Fuse carefully for thirty hours. 3. Plain paste 100 parts; smalts 12 parts; manganese 1 part. As I before. 4. Plain paste 10 pounds; zaffie 3 drachms; precipitate of gold and tin 1 drachm. As before. Factitiotis Topdz. 1. Strass 1000 parts; glass of antimony 42 parts; purple of Cassiusl part. Fuse for twenty-four hours, and cool slowl]. 2.: Strass 4000 parts; saffron of Mars 40 parts. As before. iTo solder together Rubies. Apply them to a strong flame by means of the blow-pipe, and wvhen sufficiently soft unite thiem with care; they will neither lose their color nor. weight.,Factitious Ruby. Strass 40 parts; oxide of manganese 1 part. Mix, and treat as for topaz. tVheite Crystal, or Factitious D)iamond. Manganese 1 part'; rock crystal 2800 parts;'bora 1900 parts; white lead 5700 parts. Mix in fine powderi then fuse in a clean cruiible, pour it into water, dry, powder, and repeat the process two or three times. 36.4 FACTr. ITIo.Us $TOES. q'o'p'ositioq for Fixed Brilliants. Meal gunpowder 16 parts; zinc, or steel, or cast-iron boringas 6 parts. Mix. Paste re~seme bling Tlzegar -Garniet. Plain paste 1000 lparts,; glass of antimony 500 parts;. calcined iron 16 parts. Add the antimony last. Gold or Yellow Paste. Take plain paste (made without the saltpetre) 10.0 parts; oxide of iron 1 part. Fuse. Fact.itious Lap:iz Lazuli. Plain paste -1000 parts; calcined bones 73 parts; -zaffre ~ parts; magnesia 5 parts. If it is desired to vein it with gold-.: gold 1).pow der and borax, equal parts vnei the' cakes to'taste, and.tlien hleat them sufficiently hot for cementationl. Foils for C1rystal.?,.2Paste., Put two or three laye'rs of tin-foil in to the socket made for Ath' stone, heat it gently, and fill it with quicksilver, let it: rest two or three minutes, then pour it out, anldl place in thle stone. Factitious Yellow Diainond. Strass 500 parts; glass of antimony 10 parts. Fuse. Another. Strass 500:parts; chloride of silver 25 parts. Mix, and fuse. Strass, or ~Alyence Base. 1. Pure rock crystal, or flint, 8 parts; salt of tartar 25 palts. Powder, mix well, bake, and cool, ithen put it in'to a basin'of water, and add dilute nitric acid until effervescence ceases; collect, wash, and -dry the powrder.;.next.a. dd fine white-leaad 12 pafrts. Levigate and well'wash it with -pure wvq-ter', thebn of thle above, mixture dried 12 parts; calcined boprax 1 part. Triturate them together, melt in a clean crucible, and pour the mixture into cold water; dry, powder, and melt it in the same manner, a third time, always'in-'a'fresh crucible, observing to,:seliarate any lead -that ma be revived.''To the third frit,'ground to powder, apdd:purified, litre g part. Remelt, and a mass of crystal will be -found in the cru.cibl]e of a beautiful and diamond-like lustre. 2. Arsenic 1.part;- borax 23 parts; pure pearlash 180 parts; minium 525 parts; rock crystal 338 parts. - Mriix, a-s before. 3. Arsenic 1 part; borax 30 parts; potash 105 parts; carbonate of lead 709 parts; fine white sand 315 parts. Mix with care. 4. Arseic: -1 part; borax 835 parts,; potass 3.25 parts; minium W9.00. par.ts; rock crystal 5,80-parts. Treat as.bef6re. 5.:Rock, crystal 400 parts; pure white. lead.945 parts; pure potash; 140 parts; borax 41 parts. ~~~~~ __i~~~~~~~~~~~-I INKS. 365 6. Pure potash 2 parts; fine white sand 15 parts; litharge 20:parts. See also Paste. INKS. Indestructible Itk.c 1. Powdered copal 25 parts; oil of lavender 200 parts; lampblack 2 parts; indigo 1 part. Dissolve. 2. Asphaltum 1 part; lamp-black ~ part. Melt, then add oil prepared for printers' ink, by boiling and burning until sufficiently stringy, 1~ part. Mix together, and add spirits of turpentine 3 or 4 parts. We would propose this inkl, made with less turpentine, so as to be sufficiently thick for stamping,. as the most perfect preventive of fraud, as when applied to the surface of an engraving, or letter-press, nothing Will remove it tliat-will not also discharge the ink of the stamp. It will stand the action of the alkalies, chlorine, acids, &c., even in a heated state, when: they-will at once destroy the texture of the paper. Lithographic In&k. 1. Take Venice turpentine 1 part; lamp-black 2 parts; tallow 6 parts; hard tallow soap 6 parts; miastic in tears 8 parts; shelllac 1'2 parts; wax 16 parts. Melt, and pour it oult on a slab. 2. Take dry tallow soap 5 parts;.mastic in tears 5 parts; Scotch soda 5 parts; shell-lac 25 parts; la mp-black 2 parts. Fuse the soapaind -lae, then iadd the remainder. For use, this ink must be rubbed down with water in a saucer (warmed), until an emulsion is formed of a proper consistence to flow easily fiom a pen or pen'cil. Bl2ue Writing Fluid. 1. Ferrocyanide of iron, powdered, and strong hydrochloric acid,- each:2 parts.: -Dissolve, and dilute with soft water..-2. Indestructible.-Shell-lac 4 parts; borax -2 parts; soft water 36 parts; boil in a close vessel till dissolved; tlien filter, and take of gum-arabic 2 parts; Soft water 4 parts.. DiSsolve, and mix the two solutions together, and boil for five minutes as before, occasionally stirring to promote- their union; when cold, add a sufficient quantity of finely powdered indigo and lamp-black to color.; lastly, let it -standifor two or three liours, until, the coarser powder has sub- I sided, and bottle for use, Use this fluid with a clean pen, and keep it- in;glass or earthen:inkstands, as many substances will decomppose:it while in the liquid state. When dry, it -will resist; the action of water, oil, turpentine, alcohol, diluted sulphuric acid, diluted hydrochloric acid, oxalic acid, chlorine, and the caustic alkalies and alkaline earths. 31* Red Ink for Writing. Boil over a slow fire 4 ounces of Brazil wood, in small raspings or chips, in a quart of water, till a third part of the water is evaporated. Add during the boiling 2 drachms of alum in powder. When the ink is cold steam it through a fine cloth. Vinegar or stale urine is often used instead of water. In case of using water adding a very small quantity of sal-ammoniac would improve this ink. FineBlack Wr iting: ink. Take 2- gallons. of a. strong decoction of Slogwood, well strained, and then add 1~ pounds blue:galls in coarse powder; 6 ounces sulphate:of iron; 1 ounce acetate of copper; 6 ounces of well ground sugar; and 12 ounces of gum-arabic. Set the above on the fire until it begins to boil, then set it away until it hias acquired the desired black. Black Ibnk i mproved To 1 pint of common black ink add 1 drachm of impure carboQnate of potassa, and in a few minutes it will be a jet black B.e careful that the ink does not run o.ver, during the efferv.eseence caused by the potassa. Greenz nk.: 1. Cream of tartar 1 part; verdigris 2 parts; water 8 parts. Boil until reduced to a. prqper co1or." 2. Crys.tallized acetate of copper 1 ounce; soft water 1 pint. Mix. X1.2arking Ink. Lunar caustic 2 parts; sap green: and gum-arabic each 1 part; distilled water. Dissolve. Th.e Preparfration7.-Soda 1. ounce; water 1 pint; sap, green.,+ drachm. fDissolve, and wet the linen (where you intend to' write) with this mordant, then well dry it. Printing Ink. 1. (Very fine.)-Balsam of capaivi 9. parts; fine lamp-black. 4. parts; indigo 1- part; dry yellow soap 3 parts. Grind perfectly smooith. 2. (Extemporaneous.)-Balsam of capaivi, lamp-black to color. Grind well together with a little: soap. 3. Take linseed oil; heat it in a proper vessel. until it begins: to boil, then remove it from the fire, and kindle the vapor; allow it to burn till; it- becomes stringy when tried between the fingers, then add-gradually to every quart black resin 1 pound. Dissolve, andia add very cautiously drv brown soap in shavings, 4~ ounces to every quart. Set it' upon. the fire, and'stir the mixturesuntil the combination is complete;: next, put into a suitable pot, finely ground indigo 1 ounce; fine Prussian blue 1 ounce;: fine' lamp-black 188. ounces. For every pound of-resin employed pour the liquid on the color, well mix, and lastly,: subject it to the action of a mill. Indoelible. Inkfor!Marking Linen.1. The juice of sloes 1 pint; gum A ounce. This requires no mordant, and is very durable. 2. Nitrate of silver 1 part; water 6 parts; gum 1 part. Dissolve. If too thick dilute with warm soft water. Autograephic Ink for Lithographers. White soap 25 parts; white wax 25 parts; mutton suet 6. parts; lamp -blaek 6 parts; shell-la 10 parts; Iastic. 10 parts. Mix with lheat, and proceed as for lithographic ink. To restore TVriting# effaced with Chlorine. 1.: Expose it to the vapor of sulphuret of ammonia, or dip it into a solution of the sulphuret. 2. Ferrocyanide of potass 5 parts; water 85 parts. Dissolve, and immerse the paper Jn tfie:fluid, then slightly acidulate the solution with sulphuric acid. To give an:appearance of Age to Writing. Infuse a drachm of saffron in half a pint of ink, then write with it. Perpetual Ak for Tombstones, lItarble, ~ic. Pitch 11 parts; lamp-black 1 part; turpentine sufficient. Mix, with heat. Blue ]idk.:Take sulphate of indigo, dilute it with water- till it produces the color required. It is with sulphate very largely diluted, that the faint blue lines of ledgers and other account books are ruled. If.the ink were: used strong, it would be necessary to. add clialk to it to neutralize the acid. The sulphate of indigo may be had of the woollen dyers. Copying Ink. Add 1 ounce of moist sugar to every pint of common ink. Red Permanent Ink. Vermillion 4 parts; sulphate of iron 1 part; drying oil to mix. An.y other color will answer besides.red. This ink will resist most of the usual reagents.. Black Permanent Ink. Nitrate of silver 2 parts; distilled, water 28 parts; sap green 1,,part. Dissolve. -. For the: Mordalt.-Common soda. 2. parts;: gum-arabic I1 part:; Soft water 8 parts. Mix, and moisten the linen wvith this: fluid, and well dry before using the ink. 368 INKS. Yellow Intk. 1. French berries 1 pound; alum 2 ounces; water 1 gallon. Boil and strain, then add gum-arabic 4 ounces..2. Water 30 parts; Avignon berries 7 parts; gum and alum each 5 parts. Boil for one hour, and strain. B-lue Ik. for Ruling. Take 4 ounces of vitriol, best quality to 1 ounce of indigo; pulverize the indigo very file; put the indigo on the vitriol, let them stand exposed to the air for six days, or until dissolved; then fill the pot with chalk, add half a gill of fresl gall, boiling it before use. iBlack Ink for Rulinzg. Take good black ink, and add gall as for blue; do not cork- it, as it will prevent it from turning black. Red Ink for Ruling. One pound of Brazil wood to one gallon of the best vinegar; let the vinegar silnmrer before.you add'the wood, then let them simmer together for half an hour, then. add three quarters of a pound of alum to set the color; strain it through a woollen or cotton cloth, cork it tight in a stone or glass bottle. For ruling, add half a gill of fresh gall to I quart of' red ink, then cork it up in a bottle for use. Ind'ian Ink. 1. Takefinest lamp-black,.and make it into a thick paste with th!n iin nglass; size, then mould it; attach the gold leaf, and scent with a little essence of musk. 2. Take lamp-black, make it into a thick paste with gum-water, and mould it. Carbon Ink. Dissolve real India ink in common black ink; or add a small quantity of lamp-black, previously heated to redness, and ground perfectly smooth with a small portion of the ink. 6old and Silver Ink. Fine bronze powder, or gold or silver leaf, ground with a little sulphate of potash, and washed from the salt, is mixed with water and a sufficient quantity of gum. Gluten ink. Dissolve wheat gluten, free from starch, in weak acetic acid of the strength of'common vinegar; mix 10 gr. of lamp-black and 2 gr. of indigo with 4 oz. of the solution, and a drop or two of oil of cloves. WAXES, 369 Ink for writing on Zinc Labels-H:orticultural I1k. 1. Dissolve 100 gr. of chloride of platina in a pint of water. A| little mucilage and lamnp-black may be added. 2. Sal-ammoniac 1 dr., verdigris 1 dr., lamp-black ~ dr., water.0:dr. Mix. ChIromle Ink. Extract of logwood ~ oz.-:; gum~ oz.; water a pint. Dissolve also in 12 oz. of water;- oz. of yellow chromate of potaslh (or i oz. each of bichromate and bicarbonate of: potash). Mlix the two solutions. The ink is,ready for immediate use. Ink for writing on Steel, Tirn Plate, or Skheet Zinc. Mix 1 ounce of powdered sulplhate of copper and J ounce -of powdered sal-ammoniac, with 2 ounces of diluted acetic acid.; adding lamp-black or vermillion.'W A X E S. Black Sealing-wax. 1. Shell-lac 2 parts; yellow resin 3 parts; ivory black 2 parts. Powder fine, and mix by melting carefully. 2. Yellow resin 15 pounds; lard 1 pound; beeswax 1 pound; lamp-black 3 pounds. Mix with heat. Soft Sealing-wax. ~Yellow resin 1 part'; beeswax 4 parts; lard 1 part; Venice turpentine 1 part; color to fancy. Mix with ma gentle heat. Gold Colored Sealing-wax. 1. Bleached shell-lac 1 pound; Venice turpentine 4 ounces. Melt, and add gold colored talc as required. 2. Bleached shell-lac 3 pounds; turpentine 1 pound; Dutch leaf, ground fine, 1 pound or less. Mix with a gentle heat. The leaf' should be ground or powdered sufficiently fine without being reduced to dust. Green Sealilg-wax. Shell-lac 2 parts; yellow resin 1 part; verdigris 1 part. Powder and mix by heating slowly. Scezted Sealing-wax. 1. Balsam of Peru 2 parts; sealing-wax composition 130 parts. Mix, with a gentle heat 2. Sealing-wax composition 99 parts; essence of musk 3 parts. Add the latter when the wax is cooling, and stir well. 370 WAXES. 3. Wax composition 96 parts; oil of lavender 4 parts; oil of lemon 3 parts. As before. Blue Seall7g-wax. Shell-lac 2 parts; smalts 1 part; yellow resin 2 parts. Powder, and mix carefully with heat. fed Sealig-wax. -1. Shell-lac 2 parts; resin 1 part; vermillion 1 part. Powder fine, and melt over a slow fir.e. 2. Yellow resin 14 parts; Venetian turpentine 4 parts; beeswvax 1 part; red or orange lead 5 parts. Mix, with heat. 3. Oil of turpentine 1 part; lard 1 part; vermillion 2 parts; gurn-lac 12 parts. Mix, with -a gentle heat. 4-. (Very. fine.)-Shell-lac - 4,parts; Venice turpentine 1 part;. vermillion 3 parts. Mix. Engravers' Border Wzax. Beeswax 1 part; pitch 2 parts; tallow 1. Mix. Black Bottle Wax. Common resin 20 pounds;. tallow 5 pounds; lamp-black 41 pounds. Mix, with heat. Red Bottle WlTax.. Common resin 15 pounds; tallow 4 pounds; red lead 5 pounds. Mix, withll heat, Any color may be employed..Marbled Sealin~g-wax. Take wax of different colors and melt them in separate vessels, ] and when they begin to cool a little stir therm all together, andi form the mass into sticks. ENGINEER'S FIELD BOOK: CONTAINING FORMULE FOR THE VARIOUS METHIODS OF RUNNING AND CHANGING LINES, LOCATING SIDE TRACKS AND SWITCHES, ETc, AND T A B I3 E S OF 1RADII ANtD THEIIR LOGARITHMS, NATURAtL AND LOGARITHMAIC VERSED SINES, AND EXTERNAL SECANTS, &c, tOGETHEII WITH A TABLE OF NATURAL SINES AND..TANGENTS,' ETC, TO EVERY DEGRtE AND MINUTE OF THE QUADRANT. AND LOGARITHMS OF NU]MBESR FROM 1'TO 10,000. CHARLES HA SL FTT,' RECOMM 0ENDATIONS. Office of the 0. e& M. R. R. Co., Cincinnati, May, 1854. Having examined Mr. Haslett's Field Book for Railroad Engineers, and made use of the rules he has laid down in many instances in field work, on the division of which I have had charge, I am satisfied of its superiority to any similar work yet published, in comprehensiveness and the ready application of the rules. The introduction of versed sines and extriliali secants into the calculations very much reduces the time and labor required. by the usual method of -calculatiosti for- loca;ting. lines. I recomrheiid it'to engineers, as bein.g a book combining accuracy and a ready application to field practice. J. B. CUMMINGS, Enineewr Eclsterbn 9Div. Ohio and MIiissssi ppi R. t. I most fully concur in recommending Mr. HaIInslett's work to the attention of Engineers, believing it' better than anything of the kind yet published. N. A. GURNEY, Chief- Eiyinees, indiana.South W- estergi R..; B. C. A. HASErT,' ESQ.-Dat Sir: —I leave exaiiined witll considerable care the work you propose to publish for.the use of engineers in the field, and I have no hesitancy in sayiiig that it will be the most useful- of any vwork: of its:clhlrac'ter yet offered to the' public. Yours very truly, A. S.: O;aooi; JDivision Enginee;, Ohio andl lississiopi B?.?. I concur with Mr. Cummings in ithe opinion that Mr. Haslett's mode of locating: lines very much reduces the time and labor required by the'snual mIethod. S. S. POST. Chief Ening.i;eer, OhMo and M]ississippi B. B. From statements received from engineers of the Ohio and Mississippi Railroad who have used Mr. I-Iaslett's method, I have every lieason to believe it to be an improvement in simplicity and accuracy over the old methods commonly in use. O. M. MITCHELL, COs.. iEngineer, Olio and lississippi B. B. From the foregoing recommendation, with a hasty examination of the tables, I concur in the opinion of Messrs. Post & Mitchell. E. GEST, Engineer. PREFACE. 3 P R E F A.C E. iN presenting this work to the public, the Author, claims for it the' adaptation of a' new principle in t-rig.onometrical analysis -of the formulas gen!ei ally usved in field calculations. Experience has shown, that versed sines and external' secants -as frequently enter into calculations on curves as sines and tangents; and by their use, as illustrated in the examples given in this work, it is believed that many of the rules in general use are much simplified, and mainy calculations concerning curves and running lines made less intricate, and results obtained with more accuiray:fid: fari:less trouble, than by any methods laid down in works of this kind. The examples given have all been suggested by actual practice, and will explain themselves. It has not been thought necessary to enter into all the details of demonstration, as this book is int(n led expressly for use in the field; and engineers seldom have time to enter into tedious geometrical demonstrations, when direct application of rules is required. As a book for practical use in field work, it is confidently believed that this is more direct in the application of rules and facility of calculation than any work now in use. In addition to the tables generally found in books of this kind, the author has prepared, with great labor, a Table of Natural and Logarithmic Versed Sines and External Secants, calculated to degrees, for every minute; also, a Table of Radii and their Logarithms, from 1~ to 60~. Rules and examples are also given for running curves without the use of an instrument; also for locating turnouts, side tracks, switches, &c. Having been for several years engaged in surveys and locations of railroads, and practically convinced of the great saving of time V 7 4 PanrAcr. and trouble gained by using the rules and principles given in this book, the Author submits it, without further preface, to the pro; fession, fully confident that its use will be practical proof of its metl.ts, The tables and examples have been prepared with great care, and their accuracy may be relied upon. While the Author claims a fair share of originality in the following work, bhe wlotild acknolwledge malny valuable suggestions derived from Mifflin's Diagramsj as also from HIenck on Compound and Reversed Curves, authors to whom he would refer those Wtishing to follow the subject at greater length. On the manner of working an instrument Mifflin is veryclear and concise. This work ds designed especially for practical field:: engineers, already fani,~ia~ nuith mri6unor details. C. H Cincinnati, 1855. TI-IE ENGINEE R'S FIELD BOOK. FORMULXE FOR RUNN-INTG LINES, LOCATING SIDE-TRACKS, &c. PROPOSITION I. FIG. 1.* To change the origin of a curve so that it shall terminate in a tangent parallel to a given tangent. Suppose the curve A C to have been described containing 60~ of curvatwlre, and that the distance G D equal 50 feet. We llave by logarithms: SiIne 60~ (total amount of curvature),. 9937581 Is to R... 10 000000 So is G D, 50 feet,.. 1698970 To AB = 57'73 feet,.. - 1'761439 GD 50 Or by nat. sines = 860573. sin. It 60' 86603 Produce the tangent fiom A to B.= 5773 feet; then make the * The diagrams in this workl are not drawn to any exact scale, but are designed to represent inerely the acbstract gceom6trical rela tion of lines. 31 6 FORIULzE FOR RUNNING LINES, curve B D equal A C; that is A M'C = B N D; then the tangents will..be -parallel..This rule will apply to the.origin of a compound curve, using the total amount of curvature run. PROPOSITION II. FIG. 2. Havinq a curve A B terminating in a tangent D F, it is required to find the radius of a curve that will give a tangent C G parallel to D F at any given dcistance therefrom, as at D C say 30 feet. _FqFr. XD Let A M be the given radius- 114.6 feet, the are A B 800 feet, containing 40~, and D C perpendiculai distance 30 feet. By logaithms: As' versed sine 40~.... 9369133 Is to R..10 000000 So is D C - 30.feet.... 1477121 To M N=difference of radii given and required- 12822.... 2'107988 Then we have 1146 + 128 - 1272 = radius of a 4b 30' curve. Then say: 1146: 1272::800:8.88 _:arc AC. This case is equally applicable to changing the last radius used in a compound curve terminating in a parallel tangent. -LOC-ATIN'G SIDE TRAncKs,;-ETc. 3Ew - PROPOSITIO'N III. FIG. 2. Ivn case the preceding method should- consnume too much of thie tangent C -, it s requtred to change the origis'of th:e curve, also the):leth of raditus, so th'at he required tangent may commence opposite to B, running parallel to B H. J In this case the radiating point will be changed from M towards A and B, the radius shortened, and the point A moved towards K. Let the required distance between tangents, the given radius, and curvature be, as in Proposition II., then we have by logarithms: As the external secant of 40~.:. 9484879 Is-to radius... 10000000 So is 30 feet =.... 1477121 To difference of radii - 9823.. 1'992242 30 By natural external secants = —-0- 40 - 981~306407 And 1146 — 98 = 1048 = radius of a 50 28' curve. Then, as 1146: 1048:: 800:'732 = length of 50 28' curve. 98 (natural tangent of 40 ='83910) 82 feet. Produce tangent 82 feet from A to K, and curve from thence 732 feet: of-a 50 28' curve. 6:2.* 3-78 FORMULAE FOR RUNNING LINES, PROPOSITION IV. Fi,. 3. Havinsg qocated a cutrve.wit l a giu.en: radiuis, terminating in a gi~ven poi-nt, it is reqred to chni fhe ol.teoigin of.the csrve, acso the,rqdius,;so as to pass,through the same' termintatingl pon.t, with a different direction of tangent.:',~or.,, a —_'f Let the -given radius M B equal 2292 feet; the given are B D equal 1000 feet, containing 25~ of curvature; the given tangents D F and D E make an angle of (say) 4~, D F being 400 feet, and E F = 28 feet. 28 We have 4~ = angle E D F, consequently the angle ~4 x 1'75 L D = 25~ + 40 = 290. By logarithms: As versed sine 29~. 9'098229 Is to versed sine 25.. 8'9711703 So is radius- given B M = 2292 3'360217:To radius required C L = 1714 feet. 3'233991 By tables 1714 feet = radius of 3~ 20j' curve. PROPOSITION V. FIG. 4. Having pr.odjced the two tangents to the r inte.rsection,, it is i retqired to connect' them by a' cure passing a given distancefro'm the v'ertical pon int. Given the angle L CB = 31~ 44', and CE =50 feet. -to find the LoaCATING. SIDE TRACKS, ETC. $79 radius M A. By geometry, the angle AM =~- L C B I 5~ 52'. By logarithms we have: As external secant 15~ 52' = - LC B 8'597789 Is to 50.... 1'698970 So is R. 10'000000 To MAA=1262=R. of a-:32' curve 3'101181 50 50; Byq ayrt:~u.al exGerjnz see~ians:s -X —:.....o 1262 ft. CASE 2D. To ifindthe tangent A C, or C B; or point of curve. By -logarithms': As IR.. 10. -000000 Is to AM = 1262... 3'101181 So is tangent "159 52'. 9 453 668 To. AC = 388'8.... 2'554849 By natural tangents: 1262 x (natural tangent 15~ 52' = 26546) = 388 feet = C A = C B. 380 FORMULAE FOR RUNNIING LINES, PROPOSITION VI. FiG. 5. Having located a curve connecting two tangents, it is required to move the middle of the curve any given distance, either towards or from the vertex. Let the angle TLG = 36~ = whole amount of curvature; the are A B C 1 200 feet; the radius A N- C N 1910 feet, and I B = B F = 10 feet. It is required to find the radii H M and E O. We have by logarithms: External secant 18~ = half of 36~ = AN L 8'737153 Is to 10. -...... 1000000 So is R. 10 000000 To difference of radii - 183 feet.. 2262847 10 By natural external secants: = 183 ft. 054595 1910 + 183'= 2093 = M R = radius of a 2~ 44' curve; and 1910-183 = 1727 = O E= radius of a 3~ 19' curve. By natural tangents: 183 x (natural tangent 18~ = 32429) = 59'4 = HA = A E. LOCATING SIDE TrrAcK$, ETC. 381;PROPOSITIOIN VIL. FIG. 6. It is required to locate a tangent from an inaccessible point on a curve. Lgt A B. C be the given curve with a R. of 1637 feet curving 3~ 30' per 100 feet; C the inaccessible point.': Assume a point B, if eonvenient, at a given distance, say 300 feet,:from C.' Throw off a tangepnt, and measure, at right angles therefrom,'B E -= external secant of arc B C; then to find by logarithms the distance B E, we have: As radius..... 10000000 Is to O C167... 3214122 So is external secant 10~ 30' angle C 0 B. 8231221 To B.E-= 2788. 1-445343 By natural external secants: 1637 x (nat. ex. sec. 10~ 30' -'017030) = 27'88. Measure the line B E = 27 88 feet at right angles to B IH. Set the instrument over E, and' turn off the angle BEC = 79~ 30' = complement of 10~ 30'. E CF will be,the direction of the tangent required. CASE 2D. Suppose there be no convenient accessible point between A and C, produce the curve to D, measure the external secant D F as before, place the instrument at F, and turn off the angle DFC. This will give the direction of the tangent F C as before. oASE eD.. Should the lines AI and IG be more practicable -for operating 382 FORMULXA FOR RUNNING LINES, than the curve A B' C, calculate and produtce the tangent from A to I, the vertex: of the curve AB C,: and turn off the angle K I F A O C, and make I C _ A I, as calculated. CASE 4TH.'Again, should the last method be found impracticable, and the chord AD: clear from obstructions, measure the chord AD, and turn off tangent from D. Suppose angle K AD -25~, then we have 1637 x (nat. sine' 25~ 42,262) 2 A D 1384 feet. Note.-The arc A B C D contains 500 curvature. PROPOSITION VIII. FIr; 7. It is required to find a curve which will connect two lines with/out produci ng the tangents to an intersection. The principle involved in this diagram affords an easy mode of solving a very interesting geographical problem. Suppose A E is a mountain near the sea or a very extensive level. Measure with' an'instrument for taking vertical angles the depression or'"dip' of the horizon K E B-' 130 H; then external secant'K E B x radius:-of earth —.A -E height of mountain. Loct1rS Stit Tt'Acts, ETC. 383 Let thie line be either a cur.re L A,14 A, or a tangent D A, as the case may be. Suppose it impracticable, by reason of buildings or other obstructions, to produce the tangent to: a vertex x. At A lay off with the instrument a right angle to tangent, andl produce it till it meets F B produced in E. Measure this distance, and the angle A E B; then its complement A 0 B will be the amount of curvature required to curve on to the tangent B F. Suppose the angle A E B = 656, then A O B 25~. Let A E be 120 feet) then we have by logarithms: As external secant 25~. 9'014427 Is to 120 feet..... 2'079181 So is R... 10000000 To 0B- 11608 a4~ 56~' curve. 83064754 And 1160'8 x (tangent 25~ = 46631) = EB = 541'28 feet. Then will be 25~ of curvature — 4~ 56~' = the rate of curvature, give the length of curve between the two given points A and B - 506'2 feet. PROPOSITION IX. nFG. 8. To draw a tangent to two curves already located, Let the curve CR A G H, of 2000 feet radius, be located from! tangent C K; and let E S BI) D be a curve of 2605 feet radius, located from tangent E F. We are required to find the points A and B having a tangent common to both. Suppose R to be the point in the first curve, and S the point in the second. There being obstructions in the way, we will run the zigzag line RL PS, making RL tangent to R, and PS tangel.t' to S, Suppose PR L Q-= 20' and TPS 15~; let R 1; - 1100 feet, L P = 1300, and P S = 1400. Assume radius N R as a meridian; that is, suppose NR to be due north. Then will R L be due west, L P south 700 west, P S south 85~ west, and radius S M north 5~ west. These artificial courses, then, will show the relatiue bearings, with Twhich we obtain the following traverse: Ceurse. Distance.! Northing. Southing. Easting. Westing. North.. 2000 2000 0000 0000 0000 West. 100 0000 { 0000 0000 1100 S. 70~ W. 100 0000 444'63 0000 1221'60 S. 85~ W. 1400 0000 I 122'02 0000 1394'66 N, 5 -W. 2605 2595 07 0000 0000 227 05 45095 07 1 56665 0000 394-331. I~~~~~~~ 384 FORgWULE- oa~ RIUNN-sINXG LiNES, Difference northing and southing (459507 -- 56665)- 402842;i then4028. = 971882- natural tanfent R N G 44~ 23, - course 4028'42...?/ of XN N. 440 23',vcst, all ngle S aig D 9= 2 892, or 44~, 23 -- b. i-IL —r~-,Pri~~Tz;Lsi~~-.~.u~.i~ir~ui; — ~-;~-~r;L-:j.....-L ~~-~:. —- -~C1.~-LI~Lbdld~-P-S~-~~~XCI --- LOCATING SIDE TRACKS, ETC. 385 To calculate M N make the difference:of latitude 4028-42 = cosine 44~ 23', and the required distance N M = iadiUi. Then we have 4028'42 402842 by natural cosines o sie 4 -4 23' - = 5636'7 = M N. Or by logarithms: As cosine 1R N M - 440 23'. 9854109 Is to R... 10000000 So is difference of latitude 4028'42. 3.605134 To NM - 5636'7.. 3'751025 The triangles A N I and B M I being similar, we have by logarithms (Davies' Legendre, book II., prop. X)-that is, by "composition and division:" As N M = 56367... 3751025 Is to R... 10'000000 So is sum of radii'4605 = (2000 + 2605). 3'663230 To cosine AN I = B M I = 35~;13'. 9 912205 Having now determined the angle R N I = 44~ 23', and the angle A N I -350 13', the angle R N A becomes to their difference = 9~ 10'. Therefore continue the curve from R toywards A, 90 10! of curvature, and we have the tangent point A required. Again, we have S M I-_ 390 23', and the angle B M I= 35~ 13, consequently curve from S to B 4~ 10' of curvature, and we have the tangent point B required. Now to find the length of tangent A B, multiply the sum of the radii 4605 by the natural tangent of 35~ 13', and we have the length required. CASE 2D. FiG 9. i3i I` ~~~-.. To —<;H 386 Tfo>auuL~i vonR- RUNNIN'I-NGEL, Suppose the two curves to be connected by a common tangent, instead of running in opposite directions as ain: Case 1st, curve-the same way,' as GIHS and iC DEL. -:It is required to find the position of the tangent S D. Assiume the points H and E; from IH lay off tangent H I; from: E lay off tangent E F; join F and I by a straight line, if convenient, or by a traverse, if there be obstructions. Let A H be an artificial meridiari and, as in Case 1st, calculate the distance~ A B, ak-o its course = angle H A G; this will give also the angle E B A. Suppose radius A H- 1432'5, tangent HI- 500 feet, angle 3I I F = 6~, I F. 1000 feet, N F T - 8~, E F 600 feet, and radius EB =2865 feet. We will then lhave the -following traverse, by which to find the course and distance of A B: Course. Distance. Northing. Southing. Easting. Wetting. North.. 1432'5 1432'50 East... 500- 5 00:S. 84~ E... 1000 104'50:l 984-60 S. 760 E.. 600 145'20 582'20 Total 1432 50 3029'77 2066 80 692 72 Difference of latitude 1597'27; departure = 1374'08. Departure 13740 ='86026 = natural tangent 40~ 42 = diff. lat. 1597'27 8 course A B = angle H A G. Diff. lat. 1597 27 1597 27. -. —-- = 2106'86 =distance cosine course cosine 40 -42''75813 A B. diff. radii 1432'50 Then a = - 4 ='67992 = natural cosine 47~ 10' = AB 2106'86 D B A = S A G. Now 47~ 10'-400 42' 6~ 28' = — HAS. Then curve from H 6~ 28' = 162 feet nearly to S.. Now A B makes with B E an angle of 400 42' + 80 + 60 - 54~ 42'. Hence we must curve frcm E to D 54~ 42' -47~ 10' = 70 32' curvature = 377 feet distance. The points S and D will be the termini of the required tangent. Then difference of radii x natural tangent (D B E - 47~ 10')= 1432'5 x 1'07864 = 1545 15 A K SD = D length of tangent. Now when the two curves'are so situated as to be seen the one fiom the other, assume two points as near as you can judge to the true termini of common tangent. Cause about a dozen small LOCATING SIDE TRACKS, ETC. 387 istraight stakles or pins to be set up'endway about twenty feet apart.from one of the assumed points or curves. Then set the instrument at the other, and see how tangent from instrument strikes the rosw 6f stakes. Note the difference, and move the instriument until tangent therefrom strikes as tangent to the row of stakes. Make a point where it-does. Set-the instrument over said point, and in like manner see how tangent from instrument strikes the other curve. Thus Vve dispense -with all the previous calculation. PROPOSITION X. FIG. 10. Having located two curves connected by a tangent, as in Case 2d, Prop. IX., it, is required to throw out the tangent, and introdice instead a curve with given radius. Let the radius AS = 1432'5 feet, B D = 1637 feet, and their common tangent S D = 220 feet. It is required to find on the two curves two' tangent points, X- and Y, from-which, if the required radius: (say 2865 feet) be drawn, it will pass through the points A.ind B, intersecting in the centre P, equi-distant from X and Y. Now in the triangle B AK we have given, difference of radii 388 FORMiULA: F'OU RIJNNINO tIINEs, BRIK - 1637 1432'5:.= 204'5; lso, A I.:-s: S;D A 220, to finid.tihe angle. K A B, its complement K B A.= -.SA G,*- and the distance -A:B. B-K 2-04.,5tu - A ThS en: B -_- - 92954 =h natural tangent of -420 54~' K ARB,. AK 22 Therefore its complement K B SA -SAG =47,:5-'. -Now B K X secant K B A 204'5:X 1-46880-1 - 30037- A B13; call it 800 feet. Aganini in the triangle: B A P A e haxVe AB —300, A P2865 - 1432'5 = 1432'5, B P = 2865 - 1637 _ 1228. To find the angles A B P, B P A, and B A P, make A P 1432'5 feet the base, and let Q be the foot of the perpendicular from B. Then by trigonometry we have: A:P: B:P + BA:: B P —BA: PQ.-QA-, or 14325: 1228 300: 1!228 + 300: 122 0 989 -- P Q -— QA. Then 1432,5 + 989.8 14325- 5 98918.. -_=PQ = 1211.15, and -= QA 2213Q5. ~2.~~ 2 A Q. 221-.35 Then = =.73783 = nat. cos. of B A P 40 97'. AB 300 PQ 1211,15 Again, - - = 1228 = 98628 = nat. cos. Of BP A = 9O 30'. Now YB P being a straight line, the angle YB A - 42~ 27' + 9~ 30' = 510 57' and X AP being a straight line, the angle X;A G BAP= 420 27'. Now the angle SAG being 47~ 5~' the angle S-A X will equal 47~ 5j'- 42~ 27' = 4~ 388', and Y B D = 51~ 57' _475' = 4~ 51'.: We therefore move back from S 4~ 38~' of curvature, or 116 feet to X; also from D 4e 51i' of curvature, or 139 feet to Y; we then have the points X and Y, which are to be connected with a 2~ curve of 2865 feet radius. PROPOSITION XI. - FIG. 11. Having located a compound curve terminhatin~g in a given tangent, it is reqeuired to change the p. c. c., also the length of the last radius, so as to pass through the samne terminating point with a given difference in the direction of the tangent. Let the given curve H A be a 2~ of 2865 feet radius compounded to A B, a 20 30' curve 2292 feet radius, 800 feet in length, and containing 20~ of curvature; it is required to move the p. c. c. forward from A towards B13, curving therefrom with a shorter radius than 2292 feet, passing through the fixed point B on to tangent with 2~ 30' additional curvature. The following method, though not perfectly accurate, will be. Beeause';the three,angles in. thie triangle tK A, B =1801., Also.the sm.of the angies on one side theline BG = 1800. Subtracting g from. 1800.the angle.A and;th. riitagleY it:. K,, we hae' left the angliet at - whib treatihig fri;''m 1"80" the, igib A (as-before) arnd the igik angle- S Ak: K, we havie the:;angle: S8: A G;'hence the aiigle K B:A' the angle''S'A G. LOCATING SIDE TRACKS, ETC. 389 found sufficiently so for most practical purposes. Had' the 20 curve H A been continued 800 feet farther, to a point C, the varia-. tion B C would be equal' 28 feet.* Now by compounding to a 2~0'30 curve I turn off'Wvit tlle instrument: for'l te' cord AB 2~ 200 — 160 more than I would for the chord A C; for = — 2 but if the instrument'set at the required point P, with a backsight on A, 200 + 20 30' and a foresight on B, I turn off + 2 11 15', that is 3~ 15' 2 instrumental deflection over and above that:required for a contiiiuous 20~ curve to C; the curve P B will therefore be shorter than AB in the ratia of 3~ 15' to2~; hence the proportion: 3': 2:: 800.: 492 = length of curve I'e B. A P then will equal 800 -492 = 308 feet of 20 curve; but 308 feet of a 20 curve gives 6~ 10' of curvature; hence P B contains 22' 30'- 6~ 10' = 160 20"'of. curvature. in,492: feeet distance; then we have 16333. 3-31980 = 3S0 19','or 1728 feet radius for the 4'92 curve P B. It will be sufficiently accurate, however, to continue the-;2~. curve 310 feet to P, and then run 490-feet of a 3~ 20' curve. Were H A a tangent' by making. A P the same length and rate of curvature' as: above, the curve P B would be: the same also. * 2- x 1.75 x 8 =28. 33* 390 FORMUAIE zR RUNNING- LINES, PROPOSITION'XIL.. taving l cadted a comipond curve terminating inn a tangent, it i.ts required to change the point of compound curvatuire io tlhat the curve will term2inate in a tangentt parallel to a given tangent at diny required distance perpendicular thereto. RUL I)ivide the required distance between parallel tangents by the difference of radii of the two last branches of curve. From the cosine of total amount of curvature in last branch subtract this quotient; tlie- remainder will be the natural cosine of amount of curvature: required for last radius. Given a curve 600 feet long, 2865 feet radius, compounded with a curve of 1910 feet radius 400 feet long, then tangent; required to fix point of compounding, to give parallel tangent 30 feet outside or inside of tangent given: S- ='03141 955 400 feet curvature = 12~ cosine 12~ _ 97815 less' 03141 cosine of curvature required 18~ 47' - 94674 18~ 47' - 1'~ =_ 6~ 47' curvature to be used in moving p. c. c. back to throw a tangent 30 feet inside. If we move tangent inside 6~ 47' - 2~ = 339 feet. Length of 2" curve = 600 - 339 = 261. Length of 3~ curve = 400 + 226 = 626 ft. The entire length of curve by alteration becomes 261 + 626 = 887 instead: of 1000 feet as before, admitting of more tangent at the end. This- last rule is applicable whien- tlie imoement of: the p: c. c. iI retrograde or from tihe terminating tangent, thereby increasing thej armounit: of curvature in last curve, and diminishinig that' of':thel pre'- t ceding curve. |Wlien i is' re-quir ed to move th`e p iint c. c.'forard, diminiishing the amount. of curvature in last curve, add the quotient of the required distance divided} by difference of radii, t tthe cosine of given amount of cuirvaturi;e;: and: the sum will be the cosine' of the amount of cur'vature required in thef last curve. Fin&dthe distance'as before; a-d' move the point forward: the difference of curvature,: always reckoning, said difference! according to the rate of curvature- back of p. c. c. PROPOSITION XIII. FIG. 12. Havingr located a curve between. two tangent. points, it is; proposed to lengilhen tlie radii at thie two teromini, and shorten the radius in the' middle. Let thed proposed curve, be one of 1:146. feet radius-=' 5~, 800 feetin length, and' containing 400 of curvature. It is proposed'to: in trioduce at each end: 100 feet of a 2' 30' curve =- 2292: feet radius.Required the other radius. From the t. p. to the centre is 400 feet, or 20~ of curvature. LOCATING SIDE TRACKS, ETC. 391 Introducing 1006 feet: of'a-: 2 30' 2292 feet R., thiere will be 20 30' of 2292: feet ra'dius +q 1,~ 30' of:ashoi tei:radius. By logarithms: As sin e i~o: 30'. 9'4781l42 Is-: to 2~: 30' -,. 8 639680 So is diff: radii- 11:46 feet- OM. 3'59:185 To difference between given radius and required = 167 = 0 L.. 2220723 Given radius being 1146, radiuis re quiredwlil:lBe 16 —1:67 = 979 = L = a~ 57 51i' cur ve. 392 FORMULA: FOrt RUNNING LINESY PROPOSITION. XI-V. - LEMMA.; FIG; 1 3. To divide a given angle ~into two parts,:so:that; the tangents of the angles will be in a given ratio. Let'therequired ratio be as threeto fiveand the. given angle ADC = 300;:let the straight line A B C be=:8. Make AC a chord of 60,- or:twice.30.. Describe' the:circle A C D passing 4q13. through A and C. At B, a distance of three from A, erect the perpendicular B D, produce it to L, tlien A D B and B D C will be the angles required. For B. D is common to two right angled triangles, and hence the tangents of the vertical angles are as AB to B C. To calculate the required vertical angles let A D B = x, B DC = y, then A E L = 2 x, and L E C = 2 y - central angle. Then A E C = 2 (x + y) = 00~. Erect a perpendicular friom E upon H, then will A E H = H E C = x + y. Then L EM (H being produced to M) equals x + y 2x=-y —x; then EL.= E C =R., and LM =BH. Then H C: (L M = 13 H):: sine (x + y): sine (y- x). LOCATIsaNG SIDE -TRACKs, E~rn 393 But H C is half of A C, and BH- is half B;C-;-A:B, therefore (B C + AB) * (BO- 2A Bj:: sin e (' +-y) i sine (y-x t), hat is, as thlie sm f lie num bers expressiilg the ratio is to difference, so is sine of the gien angle to sine of the difference required. By logarithms: As 3 + 5 = 8..... 0'903090 Is to 5 -3. -2. 0301030 So is sine 30... 969890 To-.sine y —x= 7~ 10' 38".. 9'096910.' y + x =-r _ (30, y-x - o 10' 38", thierefore 2y "= 3~ 10' 38", y=- 18~35' 19", x = 11- 24' 41. PROPOSITION XV. FiG. I4. Froim two fixed points, having produced tangents uniting in a vertex at unequal distancesfrom them, it is requizred to6 lcate a'ompoiunid curve. Suppose the tangents produced to A, and let A E'= 50'7 feet, 0 /e I FORMULA FOR RUNxNING INEs E; B- 6206 feet, the angle F E A - 40'.. Required the radii of a c. c. to join A and l, and also the point of compound curvature. We observe the'external: secant E C is common to bothl curves. Now by construction of the tables we:have: external secant a = tangent a x tangent ~ a; radius being unity. The angles EBCC and EA C are measured by half their arcs C B and C A. 40' Call these angles x and y respectively. Then x -+ = 2 - 20'; then 620'6 X tangent x = 505'7 x tangent y, or 620'6: 5057 = tangent y: tangent x. Then, by previous proposition 620-6 + 505'7: 620-6 5057.:: sine (x +::y:= 20~): sine (x —y) or, 1126 3: 114'9:: sine 20~: y —x. Neither of the radii being given, we will assume the condition, that the, p.c. C shall be in line with the vertex E and the centres 0 and D. We have by logarithms: As 11263... 3051654 Is to 114'9 2... 2'060320 So is sine 20.... 9534052 To sine (x- y)-2~.... 8'542718 Nowx+ y = 20. Thenx 90 +:$2'- 11~; consequently C O B 18~, and A D C = 22'. Now we have the length of tangent and curvature given, to find the radius. By logarithms: As tangent 18~.. 51'17T6 Is to 620..... 2'792812 So is R......10000000 To O B = 1910 feet.. 3281036 To find AD: As tangent 22~.... 9606410 Is to 5057.. 2'703895 So is R....10 000000 To AD = 12516 = 4' 34': t... 38097485 1910 (external secant 18'~= 051462) —1251'6 x (external secant 22 = -078535) -- C E = 982 feet. PROPOSITION XVI. FIG. 15. Let B be a point in a curve whose radius B F is given, and let D be another fixed tangent point. It is required to find point of c. c., the curve A B being produced, from which to start a curve to terminate in tangent at D, also the radius of last curve. Given the angles M D B, M B D, distance B D, and radius B F. Imagine the simple curve B C L to be run with a given radius B F till L N becomes parallel to D M. Now by the nature of a curve, LOCATING SIDE TRACKS, ETr..9,upon whatever point on the curve the transit be placed, the difference between backlsight on B' and foresight on I, is alw-ays the 13 13+D same, namely,- 2 - ow at the true point of c. curvature C, the difference between backsight on B and foresight on D is also equal to +, therefore the transit reading the same on D as on 2 L, C L D must be in the same straight line. Hence whenever the nature of the ground will admit of it, e;'ect a flagstaff at D, curve round fiom B towards L until taking a back-1 sight the foresight necessary to fall upon L shonld strike the flagstaff' at D. The transit will then be at the point of c. curvature sought. Then measure C D, and make this proportion: sine H C L*: i C ID R: x=OD. Suppose H C L = 8~, and the distance C D = 600 feet. Then by substituting in the above proportionj we have by logarithms: As sine 8~ = H C L. 9143555 Is to - CD -300... 2'477121 So is R...10-000000 To.x = O ID 28556. 3'333566 Because HCL = CaOD. 396 FORMnULXE onR RUNrNING LINES, W.7~hen the ground' will'not admit; of this methqd, ascertain by:measurement or calculation the distance. from-:B to D.. B + D. B -D* 2 (B F) xB+ = B L. Now angle L B D = -- The tri2 2 angle will then have the two sides B D and B L, and the included angle B, to find the angle L D B = CD B.t Now in the triangle B C D we have the angle B C D ='to the supnB + D plement of,2 also the angle at D, consequently the angle at B. These angles, together with base B D, determine the chord C D; from which, with the angle -ICL, calculate R as before. H CL becomes known from the fact, that C BD gives CBM = GC B, B+D C IB being B-C B D. This taken from ---- will give the angle H C L required. PROPOSITION XVII. FIG. 16. Between two tangents to locate a curve passing through a given point. Suppose A B and C D to be the tangents permanently fixed with B~D B-D * Beeaus. N B.L (isoscelcs) -- exterior angles at N and -M = - 2-D, B being = B +. D B —D NBD, andB —-:1 — =LBD. t L D = 13 (C L BB= -D)-L B D which will make all the angles known. 4~ -~ —-' — ~~- ~ LOCATING SIDE TRACKS, ETG. 397 reference to some agreement between individuals; and let F be the given point at which it is necessary to keep a given distance from some building or other object. Suppose A B and C D produced to:meet in E.'he angle 0 E D, and consequently its half E B D, are lknown. The distance I E is also known. Let the angle O E D = 60~, let I F -= 17'5 feet. It is required to find the point B, so that the angle F B I shall _ 30~. By natural sines: 17. = 35 -= FB= —HD. sine 30~ Now. 4/(35 + 17 5) x (35 — 175) - 5) /5 5 x 17'5 = 30'3 = I B. Suppose I E measures 462 feet. Then B E will equal 462 + 30'3 = 492'3. By similar triangles F B: B E:: B I: B K, or 35: 492'3:: 30'3: 426 2 = B K -- D)K. Then B D = 852'4 and B H -- 8524 — 35 = 817'4. Now we have by geometry.,$/B Ht x B F = B A, or 4/817'3 x 35 1691 — B A. Hence A B+ BE -A E, or 169"1 + 492-3 - 661 4. To find radius: AE 661'4 taigent 803~- 0.775 -= 11455 a-R. Now suppose it is inexpedient to produce the tangents to a vertex, the angle O E D being; known, find the point B as before, and turn off E B D = O E D, measure B D, and calculate by trigonometry the side E D,B E, and also B A as before. Again, suppose the angle at E is not known, neither is it practicable to measure a direct line between the two tangents, calculate by traverse the true course and distance between any two con- i venient points on the tangents by Proposition YIII., from which calculate the position of E. Without ascertaining the distance to E, the radius A G can be calculated thus: A F2 2002 4000 2 F —A G, or. let A:F = 200, then - 3 1146 - AG. 2 I F 17,5 x 2 3*5 Therefore commence at A, and run 800 feet of a 5~ curve to C. PROPOSITION XVIII. FIG. 17. Given the length of a common tang/ent -D G = a, and the angles of intersection it and m, to determine the common radius CE - CF = radius of a reversed curve to unite the tangents HD and B L. Now D C = R x tangent -n, and C G = R x tangent A al; we have therefore D G tangent = 800 ft., n = 16~ and mi- 120. * The sum of two quantities multiplied by their difference is equal to the difference of their squares. 398; FORMULA FOR RUNNING LINES, a 800 800 I tan-1. -n z+tran. q- + =- tan. 8~+tan. 6 - 14054+ 10510 - 3256'7 x'14054 - D C = I) A 45769,825 7 x.10510= C G G B 342-27 799-96 Suppose it to *be required to introduce 200 feet of tangent I between the curves, that portion of the tangent D G taken by the two curves will be 600 feet. Then we have: 800: 600:: 3256'7: 2442'5 radius: 800: 600:: 457'69: 343a27 ) 800 600:: 342'27 256 70 gen 599.97 =. " ON REVERSED CURVES, TURNOUTS, ETC. FIG. 18. AF = 98 feet, A D = 102 feet, and D E = 102 feet. Let G - gauge of track, and R - radius of turnout, x: -distance u w 1 (Y ~ LOCATING SIDE TRACKS, ETC. 399 on chord from A, the origin of curve, to F, the point of frog; then -will x 1'2. G. Now suppose R = S- 00 feet, and G -6 feet, then will x - 4/ x 80() x 6 = 4/9600 — 98 feet nearly. Or let x distance on main tracik to a point opposite of the frog. Then will x _/G(2I - G) or V6(2 x 800) 6)- 4/6 x 1594 4/9564,- 9779 feet. HIence the following rule is sufficiently:correct for all practical purposes: AfIaltiply twice the radius by the gauzge of track, extract the sqitare root of the product, and tee have the distance from origin of curve to pooint of frog. Formula for angle of frog: G R- versed sine of curvature to frong angle of frog. Ex. 8 -'0075 7 2'. 800 IMake the movable' end of the -switch rail sucli'a distance from the origin of the curve, that the departure of a curve of that radius for that distance will be equal to the opening of that rail at the movable end, say 5} inchles. With an 800' feet radius, the distance from origin of curve to 2'7x2'7 2 11 opening of switch rail will be 27 feet,:for 160 = 11 1600 24 inches nearly. It will appear therefore that theiopening of a 20 feet rail, with an 800 feet radius curve commencing at the other end, will be only 20 x 20 3 inches, for. 1600 - 3 inches. If we consider the movable rail as a movable tangent, and the origin of the curve as the opening of the rail, the angle of frog and length of curve will be obtained by Proposition XII. EXAMPLE. A 20 feet rail, with 5~ inches opening, makes an angle -with the main track -- 1~ 8', then on 6 feet gauge the distance from opening to.other side= — 5 feet 6~ inches 554 feet. Then by Proposition XII. we have: cosine 1 8' — 99974 554'00692 800 *99282 -— cosine 6~ 52' angle of frog. 400 FoRtULmA FOR RuNNING LINEs, And 60 52' -1~ 18' =5~ 34' - amount of curvature between opening of rail and point of frog. By the first method, when the distance between tracks -— 13 feet we have 4 /13 x 800 -- 102 feet tnearly for distance f0rom origin of curve to point of reversion. But if the point of reversion be made at the point of frog, the distance between nearest rails of tracks being 7 feet, wehave 6have: 7 800: 933'3 =radius of curve with which to leave frog, and 6: 7:: 98: 114' 8 distance from frog to end-of turnout. Or making the movable rail tangent, and its opening 5j inches, angle of opening being 1~18', the point of reversion beingmade at frog, to find the angle of frog, we have: cosine 10 18' = 99974 654 0.4 _.00780 *99274 = cosine 6~ 55' nearly the same as before. TURNOUTS ON CURVES. FIG. 19. Suppose the turnout is on a curve running in the same direction, say a 20, with a radius of 2865 feBt. - Now an 800 feet radius gives a 7~ 10' curve, and 7~ 10' - 2~ = 5~ 10' relative departure from main track. But the radius of a 50~ 10'- 1109 feet; then 4/2 x 1109 x 6=x=1153 —distance from origin of curve to point of frog. Therefore to make a turnout from a 2~ curve and running the same way would require 115 feet. -7; ~7 B LOCATING SIDE.T:RACKS. ETC. 401 If it were required to keep the distance the same as on a straight line, it would be necessary to make the 77 10' curve a 9~ 10' curve of 625 feet radius. If the 2~ curve run in the opposite direction of the turnout, and the radius was 800 feet, then the convergence will be 7~ 10' + 2~ = 9~ 10' curve, and the radius of a 9~ 10' curve'being 625 feet, we have: x = 4/2':x 625 x 6 = 500 = 86 -6 - distance from origin of curve to point, of frog. Wthen the m'?nain track is a curve, and it is required to get on to a side track' running parallel thereto. Note.-In treating of tulrnouts,When the main and side track ate curves, the movable rail is considered a part of the curve used for turnout, according to method Ist. Let E M be the main track on a curve of 2865 feet radius. It is proposed with a turnout fiom E, with a curve of 800 feet radius, to fall upon the side track B N, distant 13 feet from the main track, and running parallel thereto. Now 2865 feet radius denotes a 2~ curve, and 800 feet radius is a -7~ 10' curve. Therefore the diver-:gence of the curve E F from the curve E M is equal to (7~ 10'- 2~) = 5~ 10' curve; and the radius of "a 5~ 10' curve being 1109 feet, the divergence of the curve E F from the curve E Mi is equal to that of a curve of 1109 feet radius. By simil';rreasonng,::the -conv!ergence,/Of'.th'e: cuirve: F B to wards being parallel withl —:E I4:is 9~; 10'. per: hundred feet, which may be expressed by a radius of 625;feet from:tangent. Then: we have 1109+ 625 — 1734: 1109:: 13:~ 8:31_ -distance of'point of reversion from: main tralck. Now since x- 2 RH. G, we have by substituting i/ 2 x 1109 x. 8 31-= 135:'7 = distance from origin of curve to point of reversion, radius used being 800 feet. The radius of relative curvature being expressed in.:tilhe fimula, we have the proportion 1109:625:: 13577: 76'56 distance from reversion to 2d track. Suppose it be required to put the side track on the opposite side, then we have 1734: 625:: 13: 4'68 - distance of point of reversion from side track. Then we harve the formula 1/2 x 625 x 4'68 = 76'48 distance from origin of curve to point of reversion. Then 625: 1109:: 7648: 1357 = distance from point of reversion to side track. ON RUNNING'CURVES BY OFFSETS, OR WITHOUT THE USE OF AXN INSTRUMENT FORR MEASURING ANGLES. FIG.. 20. From a tangent E A let. it be required to run a curve A B C D, Shaving for its radius 0 C. To do this we have only to find H C and its half M C = G B. 34* 402 FoRMULL.& FOR RUNNING LINES, Suppose the chords A B, B C, C D are equal- in: length; being 100 feet each. The chords, and consequently the.arcs, being equal; the angle H B C is twice the angle G A B. But G A B is measured by half the arc A B =- B C, consequently the angle H B C is measured by the whole arc B C. But the angle B O C is also measured by the arc B C, consequently the angles I B C and B O C are equal. Now triangle B (O C is isosceles, and BRI being equal to BC triangle I H B C is:isosceles also; consequently the two: triangles are similair and we have the proportion: B C2 H C:BC:: B C:B O, consequently H C -, or HC= 10000 ATherefore M C = G hence the following rule:B Therefore M C = G B= 2; hence the following rule: The square of the uniform length of chord divided by radius will give the linear deflection from chord produced to curve, or half of this will give the deflection from tangent produced to curve. EXAMPLES.'0000 Suppose A 0 = 2500 feet, then 250 — = H C = 4 feet, and G B = 2 feet. LoCA'rI-G SIrD: TrLAcKs, ETC. 403 Suppose A O 2865 feet, the radius of a 2~ curve, then we have 10000 XI C- - 349 or 3'5 feet nearly; and G B=- of 3-'5=1-75. 2865 Since the angle G A B - l~ the deflection for 1~ per hundred feet is 1'75, or 0~ 1' 60'029,. and one minute for one foot = 60 *00029, as by tables of natural sines. Case 2d. Suppose we run the curve around to a point which we will call station 10, or 1000 feet from beginning. The point Q, which is less than 100' feet distant from station 10, say 50 feet, being at station 10 + 50. Suppose this a 2~ cirve compounded at station 10 + 50 to a 3~ curve of 1910 feet radius. Now the instrument setting on station 10 with a backsight on station 9, the instrumental deflection to 10 + 50, 150 feet, will be 1~ 30'. Now since 1~ per 100 feet is 1-75, that of 1~ 30' will be 2'62 feet. But the last chord being but 50 feet, or half of a hundred, the deflection will be half of 2'62 = 131; hence we have the following rule: Multiply together half the curvature in degrees = instrumental deflection between the baclksight and point required, the lenigth of the last chord and 1-75, and the product is the distance from clioi'd produced to point required. Ca.se 3d. Suppose the curve from 10 + 50 to station 11 is a 3~ curve of 1910 feet'radius. Nowv the deflection from chord to tangent, from station 10 to station 10 + 50, is 0~ 30', and the deflection from tangent to chor'd between 10 + 50 and 11 is 0~ 45', therefore the entire deflection = 30' + 45' 1~ 15'. Now 1~ 15' in a hundred = 1'75'x 1i = 218, rand for 50 feet willbe =- 109 feet. Find station 12 by Case 2d, thus 2~? (= instrumental deflection for 150 feet) x 1'75 = 3'93 = deflection from chord produced to station 12 on cu'ive. Continue the curve around as at first, observing to measure from curve to tangent the same deflection as from tangent to curve, or half the usual chord deflection; the tangent point being supposed a full station. If not a full station, ascertain the tangent point by Case 2d, and the next full station on tangent by Case 3d. Hlaving produced two tangents to an intersection at U, it is required to connect tlhem nwit/h a curve of given length. FIG. 21. When the angle..made by tangents is not greater than 15~ the distance from vertex to the' two ends of the curve will not differ materially from half the length of the curve. 404 FORMNULA: FOR RUNNING LINES, 2Ld'.1 Suppose tile tangent D C produced 100 feet to E, measure C X =I 100: feet, measure E X. Now suppose it is 21 feet. 21 Now the deflection of 1~ for 100 feet is 1-75, and 75 12 curvature. Suppose it is required to divide the curve into 6 stations. Then 21 21- 3,5, the deflection for 2.; in 100 feet. Hence it is a 2~ curve. I Or 12~ divided by 6 stations gives a 2~ curve also. The deflection being — 1 75 from tangent to curve. Between twofixed points to supply the intermediate points by ordinates front the chord. FIG. 22. By what hasbee reiousy demonstrated,6 3 2 middle ordinate By what has'been:previously demonstrated, the:iddle ordinate 4 to. 4 will be expressed by 2R. At 3 the deflection from tan2'R LOCATIN[XG S-riE TRACKS,- Erc. 405 X l2 x 2 gienit run each way from 4 to curve; is 21 at 2it is 4x4 Hence the ordinate 4 to 4- 2. Or 2 R being a common 214 denominator, its relative value may be expressed by 4 x 4. At points 3 and 5 on chord the distance will be (4 x 4) — (1 x 1) = 3 x 5 l 15. At 6 and 2 (4 x 4) —(2 x 2):-=2 x: 6 - 12. At 7 and 1-(4 x 4)-(3 x 3) -1 x 7 7. The ordinates are as follows: x 7: 2 x 6 -12 3 x 5 i:15 4 x 4 - 16. Then we observe that the sum of the two factors is equal, namely the length of chord. Hence the following rule: Multiply together the two segments of the chord or distance, divide by twice the radius, and the result is the distance from chord to curve. Suppose for example the radius 5000 feet, then at pointsl and7 tie: have: 10000 - 10000 ='7'feet — offset at station 1 from 10000 10000 end. 200 x 600 For 2 and 6 I=- 12- = 2d offset. 10000 300 x 500 For 3 and 5 15 =l 3d offset, 10000 400 x 400 and the entire length being 8 stations 10000 16 greatest or middle ordinate. Had it been a 1~ curve of 5730 feet radius, the ordinates would have been: 1 x 7 x st- 6-12 2 x 6 x -- = 1050 x 5 x s -'13'12 4 xl 4 x -14-00-middle ordinate;, and so in proportion to any other rate of curvature in degrees. Hence when the rate of curvature is in degrees and no minutes, we have the following rule: Multiply together the distances in stations each side of the point, and the rate of curvature, deduct fr om this product i of itself, the remainder will be the ordinate required. * The departure in 100 ft. of a 1~ curve fromn tangent being -= 5 = 8 of a foot. 406 FoR,NrULE rFOR IUNNINrNG, LIEES, CASE 2D. Suppose that between the points 0 and 8 there occurs a point of c. c., for instance at 3 or 5, the curves compound from a 5000 feet radius to a 4000 feet radius. 300X 300 By 1st method - 100 1 25=distance from end of chord 8000 to tangent run from p. c. c., and -0o — 0 = 25 — distance from 10000. other end to said tangent. Measure from ends of chords respectively 11'25 and 25 feet; on this line, at a distance 300 feet froml 11 25 offset, and 500 feet from 25 feet offset, -would be the point of compound curvature sought. Or imagine either curve produced to a point opposite the end of the other; calculate by Proposition XI., and measure the distance between the two curves, then on the new chord find the p. c. c. as by simple curves.- Thus: 300 x 300 300 x 300.... — = 2'2a 8000 10000 Measure 2'25 from the old chord, and you have the direction of the new. Having found the p., c. c. calculate the offsets fiom each chord separately. The above rule for ordinates, although not perfectly accurate, I considering the divisor always - 2 R, while it is variable, is sufficiently near for centres to grade by, when the chord subtends not more than 20~ curvature. This rule will alho apply to placing centre points between stations. Thus: On a chord of 100 feet, radius 1000 feet, let it be required to locate a point 30 feet from one end and 70 feet fiom the other. 30 x 70 Then we have -= 105. 2000 FOR SPRINGING RAILS. Let L - lenatth of rail and I = length of radius. Then: ( L —— ) — L = spring in feet. 2R L2 X 14 - LR - = spring in inches. tt LOCATING SiDE T1AiC KSi, ETC. 407 12 spring in eighlths of an inch. ( 2L ) = spring in sixteentlbs of an inch. EXsAMPLE. Let the rail be 20 feet long, and the radius 1200 feet. Then 24 x 202 9600 1200 1200 HIence the rule: 24 times the square of the length of rail in feet divided by length of radius in feet, will give the spring in middle in sixteenths of an inch. To find the' length of chordfor any rate of curvature (less thanl 8~) not specified in the Table of Chords (p 414.) EXAMPLE. Let it be required to find the length of chord corresponding to 800 feet of curve for a o 0' curve. 7~ curve gives. 769 01 70 15' curve gives 766-79 Difference.. 222 Then 15: 10;: 2'22: 1'48, and 769'01- 1 48 = 767 53; or 15: 5:: 222:'74, and 766'79 + 0'74 767 53. The result, as obtained by the table of sines, is 767'54, only y1. of a foot difference. That is, sine 28~ 40' x radius 800 x 2 ='67'54. Suppose now it be required to find the length of chord corresponding to 950 feet of a 6~ curve. 900 feet gives length of chord. 867 45 1000 it ". ~ 955 37 Sum.... 1822-82 Mean 950........ 911 41 Now sine 280 30' x radius 955'37 x 2 = length of chord =- 91 171, being only 3- of a foot difference, so that this table wvill be suffi cient for ordinary purposes. For common rates of curvature for a less distance, say 650 feet, the variations from the true length would be scarcely perceptible. 408 FORnrL;e FOR RU.NN-IN: LINES, &c. P1)ROrBLEMI.-L —t A and C be two fixed tangent points, the positions of whose tangents aere determined by the- angles.D A C =m 18~, B CE = -2 = 6~, an2d the perpendicular distance _D C = p 463ft.* JRequired the amount of curvature in the arc A B, its reversio7n B C, and the le;sth of the common radius OB: MB by which the arcs A B: anid B C are described. D \ I =sn / Let -- nat. vers. sine D A C, and- n ats. vers. sine B CE. Let.r = nat. vers. sine (A O B - m) = (B M C- n). Or curvature A B - z + x, and curvature B C = n + x.. + 7, v. s. 180 +v. s.6~ To find x we lave, x- = 2 - 2 2 - 0.048944 + 0!005478 - 2Q4 8 —--------— =~ 027211 nat. vers. sine 130 23' 48". Thlerefore are A B 1:8~ +- 13 23' 4S" =3 10 23!- 48". and B C =- 6~ + 13~ 23' 48" = 19~ 23 48". Then by principles from which-Proposition XII1. is derived, to find 0 B _- R, we have perpd. dist. D C 7) hvve llase - o1 twice nat. vers. sine AB - nat. vers. sine (mic-n) - O p = 463'5 463 5 _ nat. v. s. 31 2.3' 48" X 2-' nat. v. s. 12' - 0 146420 X 2-0'021852 - - 17104- OB = radius of-a 3' 21' curve. 0'270988'310 24' 190 24' Then — 3 2 gives 937 ft. arc A B, and B-,gives579 ft =arc-BC. 30 21? 3a21' 0 If D C cannot be measured, measure A C and calculate D C. Thus if A C = 1500 ft. we have 1500 x sine 18' = 1500 x 0080902 = 468'58. t D G E being equal to A O B, A O B -m m = A L G = CL G. Therefore = nat. vers. sine A L G = 10' 2' 48'. TABLE OF RADII AND THEIR LOGARITHIMS. 409 (TIIE CURVATUTrE IS SUBTENDED BY A CIIORD OF 100 FEET.) 0 DEGREE. ( 1 DEGREE. 2 DEGREES. LOgrithii M. - Rdldiu.L i atogarithms. 111fill ~ ~MI I.Ltar I; 0 5730 0-758128 0 2865 8-457115 1 8j43775 5566274 1 5636 8750949 1 2841 8453511 2 171887 5'235244 2 5545 8'743888 2 2818 30449987 3 114592 5'059153 3 5457 8786940 3- 2795 8-446891 4 85944 4'934214 4 5872 38730106 4 2772' 3442876 5 68755 4-887804 5 5289 -7283867 5 2750 8-439387 6 57296 4-758123 6 5209 38716707 6' 229 8-8435928 7 49111 4'691176 7 5181 8-710206 7 2707 8 482498 8 42922 468833184 8 5056 30703772 8 2686 08429089 9 88197 4'582801 9 4982 03697432 9 2665 8-425708 10 34877 4'586274 10 4911 0691183 10 2645 8 422856 11 81252 4-494881 1 11 4842 30685020 1 1 2624 83419028 12 28648 474578C9 12' 4775 8678947 12 2605 3415727 18 - 26444 4-422331 13 4709 8 672958 18 2585 8-412448 14 24555 4-390146 14 4640 3-667057 14 2'566 084G9197 15 22918 40360183 1 5 4584 8 661220 15 2547 30405967 16 21486 40332154 16l 4523 83655469 16 2528 83402763 17 20222 4-005825 17 4465 38649792 17 2509 3099581 18 19099 4'281002 18 44-0T 8 644189!18 2491 8 396424 19 18098 4-257520 19 4852 8 638656 19 24708 8893288 20 17189,4285245 20 4297 6830194 20: 2456 83890176 21 16870 4'214055 21 4244 8 627800 21 2488 8'-87085 22 15626 4'198852 22 4192 3-622470 22 2421 80384016 23 14947 48174546 2 4142 3-617196 20 2404 8'380968 24 14824 4'156064 24 4093 1 612005 24 2 2087 08377948 25 13751 4'188884 25 4045 80606866 25 2 1 374937 26 18222 4'121802 26 89971 G601787 26 2855 30371954 27 12782 4'104910 21 8952 3-596766 27 2809 3'3689S9 28 12278 4089117 28 3907 8,591808 28 2023 83-66046 29 11854 4078876 29 8863 3-58896 29 2307 3363121 00 11459 4-059154 30 3820 30582044 01 2292 0860217 81 11090 4044912 31 3778 8577246 01 2277 8357831 82 107430 4'031125 2 8787 3-572499 2 2261 80354466 03 10417 4-017760 03 8697 8567804 0 30 2247 3 351618 184 10111 4-00479 11 4 8657 8'563160 84 2232 3348789 85 9822 89922061 85 8619 08558564 05 2218 803457 361 9549 39799731 86 0581 1 554011 36 2204 0 843187 37 9291 38968072 087 8544 8549516 37 2191 30840411 38 9047 8-956493 08 8508 545063 8S 2176 80887655 89 8815 3945209' 89 8473 8540654 89 2162 884915 40 8594 03984216 408 348 8'536289 40 2149 8-382198 41 8385 8923490 41 8404'531968 41 j 2185 3329487 42 8185 3913029 42 8370 13,527690 42-1 2122 8-826799 43 7995 8-902806 43 888 0 38523452 43 2109 - 324126 44 7818 8'892824 44 8306 3'519257 44 2096 l8321471 45 7689 -888063 45 3274 |.0515100 45 2088 83818831 46 7473 3'873519 46 8243 8510985 46 2071 l'3162C8 47 7814 3'864179 47 213 1'506907 47 2059 I 318600 48 7162 3855036 48 3183 0'502868 48 2046 8'311008 49 7016'846081 149 3154 8'498866 49 2034 81 808480 50 6876 08807308 50 8125 8494900 50 2022 83-05869 51 6741 8'828708 51 097 8'490970 51 2010 -308823 52 6611 3'820275 52 3669 0'487075 52 1999 30800797 53 6486 30812002 5301 3042 8'488205 58 1987 3298274 54 6866 8'808885 854 3016 30479389 54 1976 8029577i 55 6250 I!795915 55 2989 08475596 55 1965 08293280 56 6189 38788091 56 2964 03471836 56 1953 80290809 57 6021 3780403 57 2988 8'468108 1 57 1942 2-288349 58 5927 3-7T2851 58 2913 3'46441 58 1931 3285902 59 - 5827 087654261 59:2889 8460748 59 1921 83-288470 60 I 5730 0 8758128 60 - 2865 3457115 601 1910 18281051 35 410- TAtBLE Or RADII AND TnEIR LOGARITHMaS. 3 DEGREES. 4 DEGREES. 5 DEGREES. M. RRadius. ogaritm. Rdius. Logarim,. M. Radius. Logaithm. 0 1910 83281051 0 1433 3156151 0 1146 83059290 1'1900 8.278646 1 1427 3-154544 1 1142 8'057845 2' 1889 8-276258 2 1421 8'152548 -2 1139 38056407 8.1879 8-278875 8 1415 8-1,50758 3 1185 1'3-055010 4 1869 38271508 4 1409 3,148975 4 1131 8'058542 5 1858 8-2(9155 5 1403 8-147100 5.1127 38'052115 6 1848 8:262814 6 1898 8-1'45481 6 1124 38050696 7 1889 8 264486 7 1892' 8143670 7 1120 83049279 8 1829 83-62170 8 1386 381'41916 8 1116 3'047808 9| 1819 38259867 9 1381 8-140170 9 1118 8'046461 10 1810 38257576 10 1875 38188480 10 11C9 3'045359 ii 1800 83255297 11 1370 38 186697 11 1106 3'043662 12: 1791 83253029 12 1864 8 184977 12 N1132 83042268 13 1781 3'250771 13 1359 38183251 18 1( 89 3'040879 14 1772 8 2485380 14 1354 8-181539 14 1(95 3'089495 15 1763 8'246297 15 1848 38129838 15 1092 38'088114 16 1754 8'244077 16 1343 8-128184 16 1088 3'086740 17 1745 83241867 17 1388 8126441 17 1085 8'085368' 18 1786 38289669 18 18338 3124756 18 1081 8-034002 19 1728 3-237481- 19 1828 8:123075 19 1078 3'082636 20 1719 8'285305 20, 1322 38121404 20 1075 8'031281 21 1710 38283140 21 1817 8,119787 21 1071 83'029927 22 1702 38230985 22 1312 38-118078 22 1068 3'028577 23 1694 83228841 28 1307'83116428 23'1065 3'027280 24 1686 3'226707 24 1302 8-114773 24 1061 38'25890 25 1677 83224584 25 1298 8-318184 25 1C58 3'024552 26 1669 8'222479 26 1293 38111401 26 1055 8'028219 27 1661 3 220869 27 1288 31(9871 27 1052 3'021889 28 1658 38218277 28 1283 38108249 28 1048 8'620565 29 1645 3'216128 29 1278 8-106632 29: 1045 3'019248 830: 1687 83214122 80 1273 8C105022 80 1042 8'017927 31 1630 38212060 81 1269 83103418 81 1089 38'016614 32 1622 83210007 82 1264:8.101818 82 1086 83015805 88 1614 8'2079638 883 1260 83100225 83 1036 8'018999 84- 1607 8'205980 84 1255 38098638 84 1030 8'012698 835 1599 83208906 35 1250 3'C97056 35 - 1027 38011400 86 1592' 3'201892 86 1246'8-(95481 86 10134 8'010107 387 1584 8-199891 37 1241 8 C,98910 37 /1021 3'008817 88 1577 83'197890 88 12387 30C92374' 38 1017 83007582 89 1570 381959083 89 1232 3-090788 39 1014 38006249 40 1568 83193925 40 1228 -089286 40 1011 3'004072 41 1556 83191957 41 1224 8 3*87689' 41 1008 30038698 42 1549 8189996 42 1219 38086147 42 9 10C6 8'002427 43 1542 3'188045 48 1215 1 084610 43 1003 3'001159 44 1585 8'186108 44 1211 38083079 44 1000 2'99897 45[ 1528 8'184168, 45 1207 1'081558 45' 99639 2'998686 46 [.1521 38182244 46 1202 3-080088 46 994'0 2'997381 47 5115 3180827 47 1198 38078518 47 99111 2-996128 48] 1508 38178419 48 1194 38077002 48 19883 2'994880 49 1501 3-176519 49 1190 38075503 49: 9854 2'998634 50 1495 83174627 150 1186 8.074005 50 982-61 2'992898 51 1489 38172742 51 1182 8-072511 51 979'8 2'991156 52 1482 81.70868 52 1178 8 071022 52: 97711 2'989921 58 1476 83169001 53 1174 1 8069587 58 97483 2'988690 54 1469 83167142 54 1170 8'368059 1 54 971-5 2'9874683 55 14638 38'16590 55 -1166 183066584 55 968-7 2-986199 56 1457 3'163447T 56,1162 80651!16 563 96631 2'986018 57 1451 -'161612 7 1158 — 8063648 57 96384 2'983801 58 ]1445 38'159784 58 1154 38062194 58 960'7 2'982587 59 1489 3-157963 59 1150 8-060738 59t 958-0 2'981377 60'.;' 1433 8 156151 60: -1146 3 8059290 60: 955'4 2'980170:. _ _. _ _ _ _~~~~~~~~~~~~~~~~~~~~~~______________________________________ TABLE OF rADII- AND TIJEpt LOGAnITHIMIS. 411 6 DEGREES. T{ 7 DEGPEES. 8 DEGPREES. RM.- adins. Lgh M. Radiii. Logaithm. M. I Radiu....Logarithm. 0 955-4 2 9381180 | O 819 0 2 91829o | 0 711683 2.8,55385 1 95217 2-918967 I 1 817-1 2-912266 1 715'3 2-854488 2 950-1 2-977T66 1 2 815-1 2-911234 2 713-8 2-853583 3 947'5 2-972569 3 8132 2-9102!8 7 12'3 2-852684 4 949 29 2-975' 4 8113 29 9183 1 4 110'9 2-851787 5 942-3 2-974186 i 5 809.4 2-93161 }1 5 709.4 2-850891 6 939-7 2-972997 6B 831 o 2-917142 6 77019 2-849999 7 983712 2-971814 7 895'6 2'93'6124 0 — -5 2-849107 S 934-6 2-97;,638 8 813 7 2-935111 8 705-0 2-848219 9 932-1 2'939456 819 8 9 2934097 9 7038- 2-847829 10 929(6 2-96S282 10 890' 2-933;93 10'702-2 2-846445 11t 93)11 293T1111 11 1 793 1 2992 82 11 II 70087 2-845562 12 924.6 2-93594' 1 963 29 i1076 12 699-3 2,84469 13 942'1 2-964i778 13 794'5 2'9 0073 13 69719 2-84.9799 14 919.6 2-986186 1 14 926 2 ~899073 14 696-5 2-342921 15 91712 2962458 15 1 79108 2 893075 1 15 695-1 2-842344 16 914-8 2'-9'18063, i6 789-0 2'-97T78 16 93-T7 2-841169 171 912-3 2-960150 V 171 7872 2S9308a5 -17 692-8 2-840296 13i8 9'9 2-99301l 18 884 2S9G9-1 18 690-9 2-839424 19 9J7'5 2-957854 19 83-6 2'94103 19 689-5 2-838554 201 905'1 2-956711 2) 7818 2-893118 1 20. 688-2 2837687 21 932'8 2-955572 21 7801 2-892184 1 21 686'8 2-S86S21 22 93004 2-954434 22 78-3 2891151 22 685-4 2-835956 231 89SO 2-95833800 23 17766 2-890171 | 23 684-1 2-835093 24 89517 2-952168 I 24 7T74'8 2'889193 24 682-7 2-834232 25 893-4 2-951040 25 778- 2-888218 25 6814 2833873 26 891'1 2-949915 126 77113 2-887244 126 680-0 2382515 271 888-8 2-948192 i 27 7696 6 2-836272 27 678-7 2-831659 28, 886-5 2 947673' 23 7679 28853038 28 677-4 2-330805 29 884'2 2-946555 29 7;66;2 2'-84336 2 676'0:2'829953 30 882'0 294552 830 76'44 2-883371 80 6747 2-829102 81 879'7 2'944380 81 1762-8 2-882409 81 67834 23828258 38 8177.5 2-943223 8!i2 61-1 2s881445 81 6721 2-3217405 38| 875-2 2-942116 33 1759- 4 2'3180490 883 6T071 2826560 84 8T730 2 941015 4l 757S 8 2'879534 14 669'4 2-825715 I85' 8703 1 2'939914 35 756 1 2'8] 8580 35 668-1 2'824873 36, ]8686 2'938819 36 7544 2'S77627 86 668-9 2'82'32 37 866-4 2937722 i 87 17528 287618 87 665- 2-123193 38 864-2 2'936633 i 38 1 751-2 2'875730 38 664-13 2-822355 89.8621 298355413` 9 749-5 |2874783 8 39 66380 2982159 40 859'9 2934459 i 40 174719 2'-88840 1 40' 6617 2820685 41 8571 2'988.387 1:411 14683 2'872900 1411 660'5'2319S52 42 855('6 2'932295 I 421 T44-7 2-871959 11 42.659-2 2-819321 483 1 8583'5 2-'931218 i 43 743'1 2'871022 i43 6579 2818191 441 851'4 i 29:30142 44/, 741' 5 1 2-370086 44 656-1 2-817363 45 8498 i 2929070 45 789'9 2'869153 45 655-4 2!816537 46 847'2 2'928300 1 46 73883 2'868221 46 54'42 2-815712 471 845-1 2926983 i 47 7368'7 2-867291 41 65:3'0 1 281488 48 843-1 2'925867 48 785' 1 -8661363 48 65117 2-814063 49 841'0 2'924806 149 733-6 2868438 1 49 65()5 2-81324 50 8390 2'928747 50 7820 2'864514 50 6493 2'812428 51 836'9 2922691 51 7380-5 2'863593 51 648'1 2'811611 52 834-9 2'921637 5 7128-9 2-862678 2-1 646-8 2-810796 58 8329 2'923585 58-3 727'4 28617;56 53 645-8 2-809982 51 830'9 2'919536 54 125'8 286S0840 1 54 644'4 2809169 551 8239 2'918489 55 724-'83 2'859932 8 64f'2 2;808358 561 98269 2'91T446 56 22'8 2'859914 156 642-0 2807594 l57 1 824'9 2'916408 1! 57 1 721-3 2'858104 |57 640-8 2-806741:58 822'9 2'915365 I581 719-8 2-857196 58 639-6 9'805935 59 8210.o 2.914327 i 59 71s.- 2.856289 59 | 638-5 29815180 60 819'0 2918295 60 7168 2-855885 60 63873 2804327 412 TABLE OF RIADII AND T:Inm LocA.RuTnI1s. 9 DEGREES. 13 DEGREES. 11 DEGrEES. M. 1adis. L...garithn. I M.I Rad.li Logarit.... M. Rda.' L..g.irhm. 0 683T' 1 2-8048327 0:578' 21' 58674 0 5211- 2113897 1 6386-1 2-803526 1 5727 2-751953 1 529 2'716742 2 634.'9 1 2-802124 9 511 2 -151232, 2 520-1 2 2 634-9 2 28'82724 2:2:!571' 5 ] 2"'5232 520'1 1 S2'716087 3 633-8s 2-801926!3 538 f'256514 3 519-3 2115484 4 632'6 2-301128 4 i 699 21551 4 515 2114781 5 631'4 2-800332!5 1 569 0 21755: 79 5 517'8 2'71-1130 6 63'[.3 2-'799538!6.: 568'0 2754364 6 517.0 27184479:7: 629-1 21793745: 71 5' 1 21.56 50 3 7 516-2 2'712830 8: 62-0 2'9791053 8 566'2 2'7529037 8 515-4 2'712181 9, 8233 2-7911683 9 / 565.22/' 25295 9 51417 2111533 11 625 17 2-968314 10 564'3 2751514 10 513'9 2'710887 11 624-8:21795587 11 568' 4 2150804 11 513'1 2'710241 121 6205::2.194801 112 562' 2.750096 12 5124 2'719596 18 3 622.3`2-79491T 13 5616 2-149389 13 511.6 2708953 14 621.2 21'793234 14 560.6 21748683 14 510.9 2.708310 15 620'1 2192452 15 5597 2417918 i 15 510-1 2'707668 16 I 6190 2'791673.16 55s.8 2147214 i 16, 509.3 21370170 IT 6119 2-79)1894 l17 B5519 2'746572 17 508-6 21706387 18 616'-8 I 21793117 IS 557'0 21745870 18 50739 2'705748 19 615'7 2'-89340 1'9 i 5561 21745170 19: 507-1 2-705110 20 614'6 [2'188566 o20 55532 21744471 20 506'4 21704413 21 613-5 2 1781794:'21 ]554-3 2-143773 21 505'6 2-103831 22 612'4: 2'181821'22 553-4 2143076 I 22 504'9 2703202 20 611-3 2 1.786252 238 552-6 2.742380 23 504.1 2132568 24] 610-2'I 2'7854.82 24 551'7 2-741686 I24 50334 2'701934 25 6&'9'1 2'784715 25 5'50-8 2'14991 i 25 5027 2-'101302 26 [618'1; 2'7839-38: I[26[ 549'9 "2-'740800'/26 [401'9 2-43'700671 271 63010 2-183183 i 271 519'0 2'789609 27 501'2 2'700040 28 605'9 2'782420 I 28 548-2 23788918 28: 500'5 2-699410 29 6-19 1'781657 29 5473 21738229 29 499'8 2-698782 39 60338 2780897 30 546'4 2137541 31 499'0 2'698154 81 602'8 2'780138 31 545'6 2-736854 81[ 498'8 2'6971527 82.1 6017 2'7979 337 2 544'1 2'736169 /32 497'6 2'696901 33! 6007 6 i0 21'8622 38/3 5438' 2135'78484 33:'1 496'9 26962'76 841 599'6 21'77863 34 543'0 2'734890 I 81 4912 2'695652 35 593-6 2177711138 35 542'1 2-7134118 835[ 495-5 2'695029 36 59r'5 21'776360 36 5413 2-733486 36 494'8 2'6944907 87 5965 2'7715618 37 543-4 2'732756 3187 494'1 2'693785 38 595-5 2-4T584 88 539-6 2-732077 38 493'4 2'693165 39 5941 2'774108 39 5388 2131398 39 492'71 2'692545 40 593'4 2'773361 40' 5319 2-130121 140 4920 2'691926 41 59)'4 2-172616 41 5311 2-7130045 41 491'3 2'6918308 42 591'4 2'771870 42 536'3 2129370 42 493-6 2'690692 431 593'4 2'771124 43 535-4 2-128693 43 489'9 2'690076 44: 589-4 2'703383 1 [44 584'6 2'7128023 44 489-2 2'689460 45 588'4 2'769642 45 53'8 21'727351 45 4885 2688846 46[ 58714 21768912 46 532-9 2'126684 46 487'8 2'688233 47 586-4 2'768163 47 532-1 2'726010 4 4871 268620 48 585-4: 2'-167426 48 531'3 2-725342 48 464 268008 45 486'4 2'687008 497 5b84-4 2766689:49' 531)'5 24724674 ]49 485'7 2'686398 50 /583.4 12765955 /50 52917 2-124008 150 485'0 2685788 51 582-4 2'76152' 5:11 528-9 2'723342 51 4844 2685119 52] 581-4 2'764489 i52 528-'0 2-722677 52 483'7 2-684570 53 580-4 i2-763158 53 5212 21'722014 53 483-0 2'683968 541579-5 2'763028 54 521-4 2-121351:54 482-3 2-683351 55 581-5 2'762299 55 525-6 I 2-720691 55 481-1 2'68.2751 56 577'- 2'761572 56' 524-8 2-720019 56 481'0 2'682146 51 576-6 2'760845 51 524'- 2-719370 51 480'3 2'681542 58 575'6 2'763)120 58 523-2 -'7187111 479-1 2-683939 59 514-6 2'759398'59 522-5 2-718054 59 479'0 2'680337 5T4 -2159398 2-TI8054~~~~i'48' 60 5713'1 2'758674 60 5217 2-71739 60 418-3 2-679735 I, -. ____ __ _________________ _[_ _9 TABLE OF RPADIT AND THEIR LoGAr.ITHMS. 413 12 DEGREES. 13 DEGREES. 14 DEGREES. M. Radlius. Logaiiihm. M. 1arlu. Logal-ithlm. M.! Radills. Logarithlm. 0 4T8.3 2'6797835 0 441'7 2-645111 0 410'3 2.618075 1 477'7 2679135 1 4411 2,644557 1 409'8 2 612561 2 477-0 2'678535 21 440'5 2"644004 2 40(9'3 2-612048 3 4768 3 2.677936 3 1440'0 2'643451 8 408 8 2'611535 4 4756 67728 4 4894 2 64290' 4 468-83 20611023 a 475 0 2-6'7674,1 5 488-9 2 642348 5 4037'9 2-610511 6 474-4 2-676145 6 43838 2-641798 6 40G7'4' 2;610000 7 473-8 22675549 71 487 8 2'641248 T 40679 266C9490 8 473-1 2674954 8| 437-2 2-6 8 40699 8' 4064 2'608980 9 472 5 2'674360 9 436-7 2'640150 9 406-0 2'608471 10 471-8 2-673767 10 48361 2 639603 10 40'5 5 2-607962 11 4T712 2'673175 11 435-6 2'689056 I11 405'0 2-607454 12 470'5 2-672584 121 435'0 2'638510 12 404'5 2'606946 13 46'99 2-671993 13 484'5 2'6-7964. i 1'13 404'0: 2.'606439 142 43692 2'671403 14 41 438 9 2'637419 14 403'6 2'605933 15 468G6 2-670814 15 48334 2'636875 15 403'1 2'605428 16 46830 2'670226 16 482'8 266336882 16 402'6 2'604923 17 46373 2'669638 - 17 432'3 2"635789 T17 402:'2 2'604418 18 4667 2-669052 18 481 8 2-685247 18 401 7 2 6083914 19: 466'1 2'668466 19 48312 2:.347I5 1919 401-2 2'603411 20 465-5 2-667881 20 43807 2'6341:64 20| 400'8 2'662908: 21 4643- 2 667297 21 430,2'6o3624 211 400,3: 2-602406 22: 4642 2666713 22 429.'6 2,6331853 22,399'9 2'601905 23$ 463 6 2.666131 23 429'1 2,632546 23 899'4 2601404 24 463-0 2665;549 24 428-6 2*632008 24 898'9 2'60C904 25 462-3 2'664968 2 428'0 26314i 1 25 898'5' 260C404 26 4617 2 8664387 26 427'5 3'638:984 26 3 898'0 2-599985 27 461-1 2-668808 27 427'0 2-6308&8 27 8976 2.599406 28 460.'5 2-663229 28 426'4 2,6298638 28 I 897-1 2598908 29 | 4;599; 2:662651 29 4259 2629328 29 8967: 2'98411 30 459-2 2662074 380 425 4 2-628794 30 39B-2 2A597914 81 458-6 O 2-'61498 31 424 9 2,628261 31 895'7| 23597418 32 45380 2'660922 28 424'4 2'627728 91 32 1938 2'596922 83 457'4 2'660347 33 423'8 2627196 38 394'8 2596427 34 456-8 2-659773 34 423`3 2'626665 834 894'4 2'599833 35 4562 659200 35 422'8 2|626134 13.5 8939 3 2'5954839 36 4556 2-658628 36 422-8 2'625604 36 8938.5 2.594946 37 455-0 2'658056 37 4218 2'625074 387 393-0 2'594453 88 4,544 2'657485 88 421-3 23624546 88 8392-6 2598961 89 45308' 2'656915 1 89 420 7 2'624018 39 392'2 2593469 40Q 4S3-2 2.656345 4 4202 2-623490 840 391 7 2'592978 41 452 7 2:655776 41 419,7 2622968 1 41 891'2 | 2592487 42 452-1 2'655208 42 419'2 2-622487 1 42 890'8 2591997: 43 451'5 2-654641 48 418'7 2-621912 1 483 8904 2'591508 44 450.9 2.654075 44 418.2 2'i621387 /44 890.0 2.591o19 45 450'3 2,658509 I45 41T7.' 2-62086 45 889-5 25i90531 46 449-7 2'652944 46 4-17'2 2'620339 46 8891 2'90043 47 4491 2-6523880 4T 41671 2,619816 47 8886 2589556 48 448-6 2'651816 48 416'2 2-619294 48 388'2 1 2'589069 49 448'0' 2'651254 49 415'7 2'618772 m49 88S7'8 2:5885883 50 447-4 2'650691 - 50 415'2 2'618251 50 3873 2'588C97 51 446'8S 2'650130 51 414'7 2'617731 51 886'9 2'587612 52 446G2 2'649570 52 414'2 2'617211 52 886'5 2-587128 53 445'7 2'649010 53 413'7 2'616692 513 386'0 2'586644 54 45'1 2648451 54 413-2 2616173 i54 885'6 2'586161 55 4445 2'647893 1 55 4127 2.615655 55 385.2 2-585678 56 444-0 2'647335 56 412'2 2'615188 56 884'8 2585196 57 443-4 2646778 41157 241'614622 57 38438 2584714 58 442-8 2'646222 58 411'2 2'614106 58 883'9 2-584233 59 4422 2-645666 159 410-8 2-618590 59 38335 2583752 60 441'7 2'645111 60 410'3 2-618071 60 383-1 2-5832T2 3/* 414 TABLE OF CHORDS. TABLE Of Chords corresponding to every 100 feet on curve from 200 to 1000 feet, calculdated to every 15 minutes' rate of curvature, from 15 minutes to 8 degrees, radius of 1' being 5730 feet.' 200 feet. 300feet.400feet. 500feet. 600feet. 700 feet.!800feet. 900 feet.l10O ft. 15' 200-00 800-00 400'00 499 99 599-98 699-91 199 96 899 94 999 92 30' 200,00 299'99 899-98 499'96 599-93 699 89 799'84 899'77 999-69:45-' 1200 - 299'98 899'95 499'91 599'84 699'76 799-64 899.49 999-30 1 199'99 299-97' 899,92 499'85 599-78'699'5 a 799-36 899.09' 998-75 ~ 15a' 1199!99 299'95 399-88 499-76 599'58 699-83 799.00 898-57 998-05 1 830': 19998 299-93 8399-83 499-66 599'40 699-04 798-56 897'95' 997-18 1~ 45' 199'98 299'91' 899177 499-58 599-18 698-69 798'04 897-20 99615 2' 199-97 299-88 0 99701 499-89 598-94 698-30' 797144 896-351 994-98 20 15' 199'96 2998o5 399-61 499-23 598-65 697-84 196-76 895-38 993865 2~ 80' 199-95 299'81 899-52 499-05 598-84 697-34'196-01 894-30 992-17 20 45.119994 29977 399-42:498-85 597-99 69678' 1795-17 898-10' 990-52 8o 199-93 299-783 899-82 498-63 597-61 69617'1 79425 891-80' 988-73 8'15' -199.92 299-68 899-19 498-89 597119 69550' 798-26 890-38 986-718 18780'J 199-91 299-63 899071 498-14' 596-74 694-79 1792-18 88885' 984-68 8'45' 199-89 99'571 398-98' 497-86 596-26 694-02 791-08 887-21 982-42 4' 199-88 299-51 398-78: 497-57 595-74 693-20 789-80 885-45 97999 4b'15' 199-86 299-45 898-63 497-25 595-20 692'82 788-49 883-58 977-46 41 4'30' 199-85 299-38 898-46 496-92 594-62 691-40: 787-11 881-61i 974-75 4~ 45' 199'83 299'81 398-28 496-51 594-00 690-42 785-64 879-52 971-89 5' 199-81 299-24 398-10 496-20 59836 689-89-1 784-10 877-82 - 968-87 5~1 5' 199-791 299'16 897-90 495-81 892'68 688830:'782-48 875-02 965-72 5~ 80' 199-77 299-08 397-70 495-40 591-97 687'17' 780-79 872-61 - 962-42 5o 45' 199-75 299-00 897-49 1-494-98 591-22 685'98 779-01 870-08 958-96 6" -199783'298-90 -89726 1 494-58 590-45 684-75 771716 867-45 955-87 60 15' 19970 1 298-81 18971081 494071 589-64 683,46 771524 864-72 951-68 68vl0' 199 68 298-72 896-80 498-60 68881 682-13 7738-26 861-90 947-75 6 45' 19965 298-61 839654 493809 581-98 680-73 771-16 858-93 943-71 7 199'68' 298-51 8396-28 492'57 581 702 617929:769-01 855-87 939'54 7~ 15' 199-60 298'40 896-01 492-08 586-08 677179'176679 852-172 935-28 i 7o30' 19957 298'29 895-'73l 491-47 585-11 617625 764-49 849-45 980'78 7 o45' 1'99-54 1298-17: 49544 1 4909 1 584-12 6T7466 762-12 846-09. 926-20, 8 | 199-51 |298-05 895-14 490-31 58808 617801'1759-67 842-62 921-47 I 1'I, I l TABLES OF IR.tSED) SiNrsZ.Si) rEXTEN-SAL SECANTS. 415 TAB3LES OF NATURAL AND LOGARITHMIC VERSED SINES,I AND EXTERNAL SECANTS. Versed On the'Constructiowt of the hkables of Tersed Sines and External Secants. In the above figure it is required to find the value of versed sine F B = C G, of.arc B C = A B angie a, and, of external secant C D. in:terms of sine C F and tangent B D. i The chord'B C' 2 sine'' B C, and angle FC B is measured by arcAB = arc B C. Therefore making chord B C radius, B F will be the sine of angle F C B, and we have: Versed sine B F=.2 x sine F C B- 2 x (sine ~ a)2. That is, twice the square of sine of half given arc =- versed sine. I Making C F radius. B F. beco'mes tangent, and we hlave, versed, sine B F = CF x tangent FCB, or sine a x tangent ~ a. Now by similar triangles v. s. a': ex. sec. a:: cos. a: radius; and v.s. a: ex. sec. a:: sine a: tangent a; or, ex. sec. a = v. s. a radis tn. a x tangent cos]le a 1- tan. a x tangent 4, a. Then log. v. s. a = log. sine a + log. tan. -~ a - (10 = log. of R.), anld log. ex. sec. a = log. v. s. a + 10 - log. cos. a.; or, log. ex. see. a:= log. tan. a + log. tan. ~ a- 10. EXAMPLE, Log. sine 40~= 9 80S067 Log. tan. 40~=9'923513 Log. tan. 20~=- 9561066 Log. tan. 20= —9'561066 Log. v. s. 40~= 9-369133 Exsec. 40=-9'484879 416 VERSED SINES. 0DEGREE. I DEGREE. 2 DEGREES. Mlin. Nat. o I.ogalrillm.. in.' Nat. No. I.ogarititm 1 in. N at. No: sgalioln. 0 0'000000 0'000000 0 0-000152 6-182714 0 0000609 6-7'8440 1'000000 2621422 1 1'00015T'197071 1 i000619'791948 2'000000 08228482 2'000162 211194 2'000680'799097 8 *000000'580664 8 -000168'..225091 8'000640'806187 4'000000'830542 4 0000173 238770 4'000650 S81B219 5 *000(001 4,024362 - 5.000179.252227 5'000661'820194 6'000002 182,724 6'000184'265496 6'000672 -827114 7'000002 8316618 7'000190 -278557 71'000682.838980 8'000003 432602 8'000196'291426 8 *000693 *840792 9'00003 5384906 9'000201 -304106 9'000704'847551 10'000004'626422 10,000207 8316604 10 -000715 -854257 11'000305 -709206 11'000213'328925 11'000726'86(;912 12'300006'784784 12'000219 8341072 12'000787'867516 13'300007'854306 18'000225'853051 13'000748'874070 14'000008 918678 14'000232 8364868 14.000759'880577 15'000010 97s8662 15 000238 s376528 15'000771'887033 16 000011] 5'034662 16 000244'888082 16 1000782'893444 17'000012'088316 17 -000251'899887 17'000794 -899806 18'0O0014'136966 18'000257'410592 18 -'000805 -906123 19'000015'188924 1 19'000264'421657 19 /000817'912893 20'000017'228480 20'000271'432582 20'000829'918618 21'000018'270856 21'000278'443372 21'000841'924800 22'300020) 311266 22'000284'454030 22'000853 *930937 23'300022 3849877 23'000291 -464538 23.000865'937032 24'000024 *386842 24'000298'474960 24'000877'943084 25'000026'422302 25'000306'485238 25'000889'949093 26'000029'456366 26 -000313'495396 26'000902'955062 27'OG0081-'489140 27 *000320'505438 27 1000914'960991 28'000033'520736 28 00028 515364 28'000926'966879 29'000035'551216 29'000335'525179 29'000989'972726 80'0000838 ['580662 -30'000343 534882 80'000952'978536 31'000040'| 609143 831'000350,544480 81'000964 984305 82'000043'636720 32 0008581 553972 / 82 -000977 9900388 883'000046'663449 33'000366'563362 338'000990'995733 84'000049 689376' 34 0003874'572651 34'001013 7-001891 35'000052'714558 | 8'000382 -581841 35'001016 007013 36'000055 139024 36'000890 590937 1 36'001029 *012597 87'000058 762821 T 87' 000398'599986 1 8' 001043'01814 38'000061'785984 38 000406'608845 1 88'001056 |'023660 89'000064'808549 3 89'000415'617662 119'001069'029139 40'00006S' "830538 [ 40''000423'626393 40'001083'034584 41'000071'851985 41''000431'635034 41'001097'039995 42'000075 872916 1 42'000440'643591 42'001110'045372 43'300078'893353 48 ]'000449'652064 1 438'001124 050717 44'000082'913322 [ 44 000458q'660456 44'001188 056G27 45'000086''932845 45 ['000466'668768 ] 45'001152'061007 46'000090'951932 1 46'1:0004751 676999 46'001166'066554 47'000093''970611 47'000484'685156 47'001180 1071771 48'000097'988898 48 000493'693234 1 48'001194|'076955 49'0001021 6-006770 1 49'000503 T'701240 49'001208 -082119 50'000106'024354 1 50 I000512'709171 50'001222.'087232 51'000110' 041559 l 51' -000521'717032 51'001237'092325 52'000114'058420 52'000531 -724820 52'001251'097388 53'000119 074965 53'000540'732540 53 1001266,'102423 54'000123 |'091200 I 54' -000550'740192 54'001281'107428 55'000128 107146 1 55'000559'747778 55'001295'112405 56'000133'122788! 56''000569'755297 56'001810'117353 57'0001387-'138167 571 7-'000579'762752 57'6C01825:''122272 58 o000142'15326s 58 000589.'770144 58'001340'127165 59'000147'168116 59'000599'777472 59'001855'132031 60'000152'182714 60'000609'784740 60'001370 136868 EXTERNAL SECANTS. 417 0 DEGREE. 1 DEGREE. 2 DEGREES. oMn. Nat. No. Logarithm Alin. Nat No. Logalit. Mhin. Nat. No. T.oogarithn. O 0000000 000000 0 0 000152 6-182780 0 0000610 6-785005 1;000000 2-626422 1 000157 197189 1 *000620'792217 2 000000 38228482 2 $ 000106;211265 2 000680'799871 8 000000'580604 8;000168;225164 8'000640'806465 4'000000 &8930542 I 4 000170'288845 4 006051 818502 5 000000 4k024362 5 000179 252305 5'000661.820481 6'000001.182725 6 *000184;265576 6'000672'827406 7'000002 3816619 7 000190 278689 7'000683'884276 8;000003' 480260 8 8 000196'291511 8- 000094'841098 9 *000008:584907 9 000201;304193 9 00C0704 847857 10 000004:626424 10 000207 316694 10 000715'854568 11;000005'709209 I 11 000213 0329018 11'000726'861227 12'000006 *784787 112 *000219 8341167 12'000738'867886 18 *000007;854309 ii 13 000225;853149 18'000749 -874895 14. 000008'918682 11 14 -000232: 3;64969 14'000760'88G907 15'000010 97T8606 15;000238'3766,1 15 000772'887868 16;000011 5-08467 I 16'000244:1'888188 16'000783'893784 000012 17 000012 081 79941 17,'00021795'900157 18,.000014'136974 i 18;OOC257 ]410704 18'000800'906473 19.000015'1889338 19 1:00264'421772 19'000818 *912748 20 *000017 *228487 20;0271 482700 20'000830 918978 21'000018 270864'21 000278 F443498 21'000842'925165 22;000020 0811275 22 0002985:454154 22 000854'9818C8 28'000022'349882 28;0(00292;464685 28'00086;987408'24i oo00024'1886853 24 000299!475090 24'000878'943465 25: 000026'422314 25 000306'485371 25'000890'949479 26 /000029'456378 26 |0003138 495532 26'00090'955454 27'000001 489153 27;000820'505577 27'000915'961888 28:'000033'520750 28'000328 1 515506 28'000927'967282 29;000036'551280 29';000885' 525825 29'000940 1'97813480 000088'5806T9 80 /;0008431'535081 80 -;00c953'978950 31'000040'619151 8 31 00050'544632 81'006965'984724 82'000043 -636739 82'0008581 554128 32 000978'990468 8831 000046'668469 083'00036601 568521 883'000991'996863 84 ];000049;689089T 84'000378457281 84'001014 7'001827 85;000052'714581 1851'000082'.582007 85:'001017'007455 06 [000055'739048 86'000890' 591106 86'001081'018044 87' 0 0005$:762846 87 000898'600109 37'001044'018600 88'1 000061 -7860138 8 1;000406 6 09022 88 *001057'024119 89 *000064 -808577 89;000415'617842 89'001071;029604 401;000068 -830567 i 40['000428 1 626577 40'001084'085055 41'000071'852016 41'000432'635222 41'001098'040471 42'000075'872948 42'000440' 643:.82 42 001111 045854 43;060078'898887 43;000449'652259 43'001125|'051205 441'000082'918858 441'000458'660655 44 -'001139 "056521 45;000086' 932882 l45 1'000466'668771 45'001153 "0,61807.461 000090 /'951971 46'000476'677206 46 "001167'067061 47'1;000094 *'970652 47 1'000485 1 685866 47 "001181 "072284 48 1 000097 988940 48 1 000494 1'693448 48 "001195 "077474 491 000102:6'006814 49' 000503'701458 49 "001210 "0j&82644 50 000106'''024400 50'000512''709893 50 "001224'087768 51 000110 041607 51'000522'717258 51'001238 "(92862 |52'0001i4 058470 1 52 -000531 725051 52 "001253 "( 97932 583'000119' 075017 53'000540'782775 58 "001268'1C0297 54'000123' 91254 54'000550 |'740481 54 "001282 "107985 551'000128-!'107202 55 1'0005601'748021 55 1"001297 "112968 56'000133'122846 1 56'000570'755544 56'001312 "117922 57''00018T7 128227 67'000579 I'768004 57'001327'122848 5.8'000142'158380 58'000589'770400 58 "001342 "127747 59'000147'1 68180 59'000599'777782 59 "001357'132620 60'o'.000152' 182780 60 -,000610'785005 60 0013872 1387464 418 VERSED SINES. 2 001440 140464 2.'0024-,77'890824 2 -003856 586156 8 _ 001417 -151225 8'002497 0897455 38!,000882 89026 4 0014302 -155955 O4 O002518 1-3S401019 4:00809:7 b9'1886 5 001448 *160661 5 *002638.404572 5 1000983 594785 G6 0*C140083 105 0] l42 6'0020559 4008108 6' 0003,959 597578 I7 3101479 1609909 7 *002580 *41;1029 7:003985 o00410 8 *00149i5 1 s74629 8 1002601 41513T 8 004010'.600234 9 *00151 1 *17 9206(; 9'002622 418832 9 o00403 6(0604o 8 10 001527 1S8Sl19 10 *0C2643 *422111 10 *304063 i 608852 0I 0 OG4LI3 188077'11 *002664 *425T577 11'C04089 0611647 12 1 0,~55 9!1920912 12 OC2685 *429029:12`'004116 0614434 10 0315T7' 197422 13 -C02707 *4352468 13' 004142'617210 14 0 *0o15)92,201910 14 02C72T28 40358912 14 004169 ]6199SO 1 01(08 *20637G 20037 *15 0027B50 *409303 15 004195 * 622740 16 -001025' *210817 16'002771'442702 16.1'004222 0 625490 17 *00 1G41,2152.37 1 7'OC2793 *440087 1 17'04248 *628234 131 0010S58 219002 1 8 *OC2815.449458 18'004275 638(966 191 00107G.5 224018 19 |'0602837 -452817 | 191, 004302 63308692 20 |-0010G92 228:360 0 20 "-0028.59 *456162 1 20 004829 6864C9 211 -001709 *232092 Ij 21'OC2881 4594949' 21'OC4356 0,G9117 22 |001726- *237000 I1 22 | *002903'462815 | 22'00C4382 641816 23 0*01743'2412S9 23'002925 *466121 23'004410,644506 24 ~00~1760 *24B550 55 24'O'002947'469417 24'004438'647170 25 0 00177T7 *249802 Il 25'002970 *472099 25'004466'649864 26 0 017095!254,26 50 26'002992.475909 261.004493.052532 27 1 U018103 258232 ii 27 1'008015 *479227 1 27'004521' -655190 28 001830 262416 28 003037 482472 2'004548'657840 29,001847'266582 29'003060 *4857C5 29'004576'660482 80,001865 270725 30'003083'488926 80'004604 6068116 31,001883,274852 31'003106 492187 31 *'004682 0665743 82 001901 278957 821 003129'495304 082 004660' 668360 383'001919'283043 3'003152'498523 33'004688 *67(,972 34 1001937 1287109 4'003175'501C94 -84'004716 * 678574 35 1001955,291156 35'003198 504857 85'04744'67T6168 36'OC1973,295187 36 003221'508008 86 3 *004773 678759 37'001992'299196 87'008244'311147 T 37;004801:81884 88 1002010'303190'88 003268'514275 88 1'004830'68906 89'002028 3807162 39 1003291'517391 89'004858 *686470 40 *002047' 311119 40 003315'520498 40'004887 689C26 41 002066'315056 41'003389'523593 41'004916'691574 42 |002085 |318977 42 |03362'526677 42'0049-44'694116 43'002103 322879 43'003386'529750 43'004973 069G6649 441 002122 3826764 44'003410'532812 44'005002 699176 45 002141 33064 45'003434'535863 45'005031 *701696 46' 002160'334483 46'003459'538904 46'C005061; *704208 47 00G2179'338316 47'003488 |541933 47'005090 706713 48'002198'342138 48'003507.544953 48'005119 709210 49 002218 1 45932 49'008531'547961 49'005149 711700 50 1002237 1 49716'50'003556'550961 50-'005178 [714184 511 002257 853482 51'003581'553948 51'005208'716658 521 002276 857233 152'003605'556927 52'005238 719128 53'00,2296 368967 531 003630'559895 53 *'005267 ]721589 54 0(C2316 -364687 4'003655'562852 54'005297'-724044 55'002336 6890 55 003680'565800 55'0053271'726492 56'002355 8372076 56'003705'568737 156'005357'728934 57 01'02375 375746 5'003730'571665 57.'0538T 78318367 58 ~C20396 379403 58'00375 574582 i 58'00541:.733796 -9W1 oC002416 1 383048 591 003780'577492 59.'005448 736217 60 1 02436 1386669 60'003805'580889 60'005478 1738630 EXTERNAL SECANTS. 419 3 DEGRlE E. 4 DEGREES. l -DEGREES. Mi\. NO Nto l o thm. Alil. Na. N. N grt Ali.j Na.N. Log;ir (-.' 0 0'0013T2 7-137464 1 0 0002442 71387728 I 0 0 0008820 7T82845 1'01358'142281 1 002462'391346 1 1 0088845'584946 2'001408'147078 I 2 0024S83 3949 1 2 008871 879304 3 001419:151,1 l 3'002503 S39So4l 8'008897 690715 4'001484 156577 I 4'002524 4)2114 4 003923 8593586 5,001450'161290 5'002545'405676 5'003949 -596446 6:'0014685'165978 [6'002566'409221 1 86'00397s5 99301 7,001481 170642 71 7 002587:412751 7 004001'602144 8'001497 175279 8 *00260()8 416268 8'004027 604979 9'001513'179893 9 002629'419772 9'004053'60780U 10 "001529 184488: 10,002650'428261 10 004080 8610620 11'001545'189048 11'002671,426736 11!004107 8610427 12'001562'198390' 12'002698 430197;12'004133 i'816225 13'001578'198107 13'002714'433645 13'004159'619(118 14'001594'202602 14 -002786'487079 I 14'004186'621794 15'001611'207075 15 002757'440499 15'004218 1' 624566 16'001628'211528 16'002779'443907 16'004240 627327 17'001644'215951 17'002801 4478302 1 004267'80183 18'001661'220058 181'0028238'450682 18 304294'682827 19'001678'224736 19 002845'454051 [ 19'004821'635564 20,001695 229095 20'002867'457405 1 20 1004348 6388293 21'001712 128348 i 21':002889.'4607471 21'004:3875'641013 22'001729'237750 22'002911'464077 22'0044083'643724 23'001746'242047 1 23'002934'46793 2.1 0044830 646428 24,001763 24G620 I 24'002956 4T0699 24 004438 649102 2t5 f'0017811'2508575 25 |002978'478991 i 258'004485'8651808 26'001798 1.254S06, 261 003001'477210 1 26 1 -004513.6544S8 27'001816'259020 27'0038024 {480538 1 27'004541'657158 28'0018338 263211 28'003046'48793 8i 28'004569 1 8659820 29'001851'267385 29'003069'48T7086 I 29'004597'662474 80'001869 1 271586 10 2 003092'4902687 30 -004625 665120 81'001887'278571 31 1 003115:4938488 31 8004658 667759 82'001905'279783 1 32 008188 496694 32;004681'670388 33'001923.'283877! 33'003161'499894 383'004710'673018 34;'001941' 287951 34'303185,503075 34'0047388'675627 85'001959'292006! 35'008208'506248 85 004767 1'G78288 368'001977T1'29604;V'i 36'0032382 509409 8J 86'004T96 -'680887 37 001996'300062 8i 37'0032955 1'512555 i 37'004S24'683424 88'002014 ['304064 88 1 00327'9:5159697 88'004858 1'686009 89'0020821'808044 39 1.003302'O518283 8 39 -004Ss2 /'688585 40'002051'12009 40 0382 1'521940 401'004911'6911,54 41'082070'315954 }t 41'008830 9' 525049' 41;0049-10' 0698714 42./'002089'81938883 42 1,008374 1 "528140 42 1004969 6 %982G9 43'002108.'328798 4.3 0083398 5381228 43 i004993'693814:'44'002127'327688 44'3030422 5342986 i 44'005028'701854 45'002146 83381565 i 45'0(08446 58T7857 45 |0050857 703887 461;002165 ]385422 1 46 003471 1540409 1 46'005086'706411 47'002184 8839263 47 | 003495'543448 47 005116 I'708929 48'002203'343089 ji 48'003519'5468479 il48'1 305146'711489 t49'002223'846896 1 49'003844'549497 49'005175'710942 50'002242' 350689 1 0' 008509 {'552508 50'0058025'716489 51'002262' 854463 51 0035938.555506 51'0052085'718926'52'002281'358223 52 008618 "'558496 52'005265 I'7214C9 53'002301'361965 53'0086438 I'561474 8 53 0052985'7238888 54'002321'865694 54'008668'564442 54 00,8325'723851 55,5 002841'369406 {l 55'0036983' 567401,1 55'0058356'728812 561 0023861 { 78100 || 56,008718 1 570349 1 56'005386'731267 -57 1'002881 1 876779 |I 57' 008744t'578288 |i 5.7 -005416 I'738713 58'002401 1 80445 58'1 008769'576216 1'58'005447'7836155 59.-.i02422 00 s1 983 11 59 1 0087'94 1 57913T 11 519. 005-I78 1'788589 60 { 002442'387728 11 60 i.003820',582045'17 60 005508l,7410160 420 V~1ESED SINES. 6 -DEGREES. 7 DEGREES. 1 8 DEGREES. Mill. | is: No. i Logarithm, Mi | Nat.. N. | Logarithm,. tMin. 1 Nat No. L.oparithli. 0 0105478 i 7'7886(30 0 0'017454a 7872880) 0 00932 1 938199 1'005509 *741038 1 300T4S09 8174444 1'009772'990(080 2'00539'748488 2 00551 5 876(0502 2'009S03'991804 83 005570'745831 3'007561 *8 878555 3'009854 993601 4'005600'748220 4'007596 SS06038 4 V009S94 99.539t 5:005631 /,5061 5 { T001682.882647 5.009935.99s185 6:005662'752974 6'007668 *884686 6 009976'9989712 7:005693 755342 71;0077)4'836719 i7'010017 8-000754 8 005724 5757704 8'007740'-.88749 I 8 010058'002532 9:005755 760057 1 9'007776: 890773 9'010099'004307 10):051786 762406 10; )07813 S92793 i 10'010141'006079 11:005818 1 64749 11 0078;49 *894S08 11 010181'007841 12'005S49 8767084 12 *007885:'896818 12 010223'0C9611 15 |'005880'769413 13 007922'S9S8824 138'010265'011371 14 005912'771738 14 007958 900825 14'010307'013128 15 | 005944 774055 1 1[;007995 *902321 15 *010348 014883 16'005975'776364 16 008032'934813 16'010390'016632 1 7:006007 1 778671 17'088069'906800 17,010482'018379 18S'006039 1780968 Ij 18 008106 *908788 18'010474'020121 19 *:0061071 783261 19 [008143'910761 i 19 *010516'021861 2):006103 7855471 20 I:008180 9127834 1 20 -'010558'023597 21:006135 1787829 i 21'008217'914704 21'010600,025329 22:00616;7:7931.02 22 [008254 916170 22'010648'027058 23':006230 792839 283 008291'9)186283 28'010685'028783 2':006232 7 946833 24:0083829 920584 11 24'010728'030505' 006W5 T196891 25'008866 -9225386 25 010710'032223'26 006297 1799140 26'008404'9244883 26'010813'033989 2:006380'801385 217 008442'*926425 27'010856'035651 28: 0068362'803621 28 -008479 -928868 1 28'010898'037359 29:006095'805859 29'008517'930297 29'010941'039064 38:006428;808086 30:008555'932227 30'010934'040766 31'006461'810307 31'008593 9834152 31'011027'042465 32'006494'812524 32:008631 -936074 8 32'011070'044159 83'00652T7 814734 1 33'008669'937993 33'011113'045850 34 006560'816989 34'08087'939903 84'011151'041589 35 D06594'819139 35'008746'941811 35'011200'049225 36 306627'821332 8~6 008784'9431115 386 011248'050936 387 306661'828521 317 008828'945615 37 011287'052584 |38 18394'825104 38'008862'941511 38'011831 o054260 39 D06728'827s1 809 008910' 944903 39'011874 (539811 40 308762'880052 40'008939'951290 40'011418'057601 41 306795 8832218 41'0089T8'9531713 41'011462'059266 42 )065129'834879 42:0090117'195 52 42 0115036'06C92S 43 D1083'836535 43:009056 956927 4:3'011550 062588 D44 06897' 88685 |44 009895'958799 44'011594'064248 45 006982'840S80 45'009134 960666 45'011638'065896 46 006966'842969 46'009173;962529 46'011682'0617546 47 DO00100'845115 41 *009213;964388 47'011721'069192 48 |07004'8472832 48'0C9252 966243 48'011772'070886 49 017069'849356 49'009292'968094 49'011816'072476 50 00104 8514715 i 50'009331;969342 50'011860'074113 51 107138'853589 1'009311;971784 51'011905'075747 52 1 307173'8556917 52 009411'983624 52'011950'077878 58 307208'857800 1 53'009450 975459 1 53'011995'079007 54 1 072- 1'859898 54'009491;971291 54;012040'080631 55 3 007278 -861991 55'0019531';99i18 55'012085'082253 56 i 007313'864079 56'OC309552:980942 j 56'012130 -088872 5T 007848'866162 i 51'0(19611'932762 5T'012175'085488 58 007383'868240' 58 0( 9351;981578 58'012220'087100 59 007418'870312 59'0oc951 1'3393) 59'012266'088710 60 I 007454'872880 60 009128'|98199 60'012811'090816 EXTERNAL SECANTS. 421 6 DEGREES. i 7 DEGREES. 8 DEGREES. Iin. | N'at.No N. Logarithm. Alin Nat. No. |,ogalithln. lm. | Nat No. | Logarithln. 0 0.505508 7'741016 0 0-0.007510 7-875628 0 0!009828 7'992447 1'0;539'743437 1'007546'8777t;8 1'009873'994268 2 o00Sz70'745850 1 2 00781'879782 2'009910'996087 3 00560l1 74257 1 3 007618.881851 8'069952 997902 4.005632'750659 i' 4 007054'883915 4'0(o9998 999715 5'005663 7'53054 5'0(7691'885974 5'010U385 8-001521 6'005694 -755440 i6 007727 888029 6'010077'003326'7 005726'757822 i 7'007764'890078 7'01119'005126 8'005757'760197 i8 007801 892124 8'010160'006921 9'005788'762564 9'00787 894163 9'010203'008716 10.005820'764926 10 0u07874,896199 10 6,01;245'010505 11.005852'767283 11 0079 1,898230 11'010287'012292 12'005883'769632 1.2 007948,90256 12 0108329'014074 18 -005915 -771974 18'007985'90278 13'010372:'015852 14'005947'774313 14'008022'904295 14'010414'0176927 15.005979'776644 15 008059'906307 15'010457'019401 16'006011'778967 16.008097'908315 16 ]010499'021148 17'006043'781288 1 7'0(8134'910818 17'010542'022983 18'006076'783599 1 8 -008172 912318 18'010585'024694 19'006108'785906 i 19'08S209 914312 19 010628'026452 20'006141'788206 2-) 008247'916001 20,010671'C28207 21 -006173'790502 21.008285'91828T1 21 01G714,029957 22 -006206'792789 1 22 008823 92U270 22'010757'031705 23'006238 -795070 28 008361 *922244 23 010800 0833449 24'006271'797348 2i 4 0C8399:924216 24'01(844 -C835189 25'006304 799620 i 20 1 C8487 926185 25'61(887'036926 26'006337'801883 i 26 1008475'928148 26'01C931'088661 27'006370'804143 2T 2 008513'930107 7'01C975'403891 28'006403'806396 28 1'(8552'9892061 28'011018'C42118 29'006436'808645 29.0C8590'I 934012 29 -011062'042842 30'006470'810887 30.008629'985958 30'C11106['041563 31'006503'818122 31'0(8668'937900 31'011150'0-47281 32'006537 -815354 1 82.0c8706'939839 1 2'011194'C48994 33'006570'8i7578 83 008745 -94171 71 33'011288,c56iTC4 34'006604'819798 84.008T84'943701 84'011282'C52412 35'006638: 822012 35'008823 1945626 35'011326'054117 8z6' 006671'824220 Ij 86 -00U8862'947547 136'011371'055817 37''006705'826428 1 -37.008901'949464 81 7'011416'057514 38'006789'828621 3 008941'951877 388'011461'0592C9 89'006778'8308138 39'008980'953286 9 39'011506'06(899 40'006808'832998 1I 40.009020,955190 40 O'011550 062588 41'006842'835179 41 009059 957090 41'011595'064278 42'006876'837355 42 1069099 1'958986 142'011640'065954 43'006911'839526 1 43 009189 -960878 143' 011685'067683 44'006945'841691 1 44 009178'962767 44'011730'0C983C8 45 006980'843851 45'009218'964651 45'011776'07(980 46 -007015'846005 46 009258'9665831 46'011821'072650 471 007049'848155 4T 009298'968408 11 47 011866'074315 48 *007084 -850298 481 009339'970280 i 48'011912'075979 49'007119 852437 49 1 009379'972148 49'011957'0776S88 50 i 007154'854571 50 009419 -974013 50'012003'079295 51 -007189'856700 i 51 009460'975873 51'012049 G'8C949 52'007225'858823 52 009500'977730 i52'012095'C82599 53'007260'8609492 -i 5 009541'979583 53'012140'084248 54'007295'863055 54'009581'981432 f54'012187'C85892 55 1'007331'865163 -55 009622'983277 55'012232,087584 56 1 00767'867267 i6'009663'985119 56'012279 C-89172 571 007402'869365 571 009704'986956 j 57'012325 -'9(808 58 008 007488 871458 58 009745 988T90 58'012372'C92440 59'007474'873546 19'009786 1990619 59'012418':094060 60 -007010'875628.60 009828 1992446 60'.012465 -95696 36 422 VrEn, sE SIN Es. 9 DEGREES. 10 DEGREES. _- 11 DEGREES. AIin. I Nat No I I -9-itllln At rlil" Nat. NTo.' Logalithm. Ai lill. N aitt. No L,,itllm. 0 0-012311 8-091816. 0 0 15192 8-181622 0 10-018780 8-2641B76 1 012357 091920 1-'015242 -183065 1 -018428 -265486 2'012408 0931521 2'015293 184505 2 018-184 266796 3-8 012448 0'35119'8 015849 1894 3 018541 12806184'01249 (;93T714 4 015895'187878 4 018 596 269407 5'0121540 (C98306 5'015446 188811 5 018G51 270711 6'012586] 099891 6 015497 "190242 6'0187T7.272012 7 0126832 10148 ) 7'01548 -191671 7 6 0187(2 273298 8'012078 103964 8'015599'193097 8'018819 2741 S8 9 -012724 104644 9'0O15650 194520 9 018876 2719 4 10 -012770 106221t 10'015701'195942 10 -018932 277197 11 -01`2817T 107796 1 015752 *197T61 11 -018988 278487 12'012864'109%367 12'015834 -198778 12 -019045 279777 18 -012910 /110086 138 015856'200192 13 019101 281065 14'012957 I'11251 14''.0159 8.201604 14 -019153'2828,50 15 i'13003 i 11465T 1 15'015959 208014 15 019215 -238684 16 -018050 -1156025 16 016011 -204421 16 019272 -284915 17 -01309T7 117182 I T17;016068 -205826 17 -0193238 2861'94 18 -013144 -118787 18 016115;207229 1S *019385 -287474 19 - 183191 -1231238 19 0161 67 -208689 19 *019442 -288749 20 -018288 1218 3 20,016219 {210028 20 019499 2900C28 21'013286 -1210381 21'016271;-211424 21 *019557 -291296 22'0183338 -124927 2 1'016823;21282-7 22 7019614 -292566 23'018380;126463 23'01.6l76 -214209 28 -019671 -2938'3 24 *018428 It 128)06 24;016428 -1 215598 24 -019729 -2951C1 25.018475' -129542 25 016481;216986 25 -019786.2909666 26'0185238 -181074 i 26'016,80 -218871 2 0198.44 -291629 27'018570 -103260 27 1 01(6586 219753 27'019902 -298889 28'010618 -104131 28 1016609 -221184 28 019959'030149 29 -018666:- 13563 29'0166921 -222522 5 1 29 -020017 S00C406 30'018714 -17176! }0'016745 -223887 0 -02007T5 -802661 81'013762 - 13869o5 i.016798;225261 31 0201380 — 800916 32'018810 140212 82 8'016851 -226683 32 -020191 -1'85167 388 -113889 -1 41721 0 3 - 016934|;2280G2 8 33 020250 - 30(417 84'013907 -1432836 84'016958;229370 84 020808 -307666 85'013955!'144745 I 5 "'017011 -280784 5 020366.83C8912 86'014003 -146i21 36i'917065' 2822097 86 020425 - 10156 87'014052'147754 31 017118 238458 7 020483'3113099 38'014101 - 149255 3 1i171 -234817 88 0,20541 -8012600 89'0141491 15075-2 j 9: 117225 I -23617 39 020600 - 81-88) 40 -014198 -152248 1 017279 - 2871528 40 -20(659 -815117 41'014247 -'158741 41 017333:238880 41;020718 -8165o3 42 *01429, 6 -15281 ll 42 01788 7;240230 142 020777:817587 4348 014345 1'15719 0 43 017441 241578 j 43 0218086 1 818818 44 0141 1'15820 i 44: 017496 24292: 4JA4 -:0208S95 -320(C49 450'014448 -159686 45 017550 1;244267 451 -02954 0321278 46 4.014498 -161165 46:017604 | 245609 46 021014 822565'| 47 -014542 1 -162640 471 0176581 -21G949 471 03210703 - 82780 4S'014592 -16411 i 4S18;017712 1;248286 48'21108 3 -324953 4-9 -014641 -165589 49 |017767 -249621 49 -021192 - 926174 5.'014691 -16706') 50 -'017822 1 250955 53 1;-21252 -82795'51'014741 ]-16852T 51 |'017877 -| 252286 51'-021811 5;828618 52 - 014791 - 169992. 521 -017931 -208615 52 0;21071 - 39821 58 014841: 171454 50 ]'017986;24942 5 9 1 -021431 - 0331044 41 -014891 |:172914'54'018041 2562067 5 021491 - 8322 56 565 *014941 -17437-2 55 0'1809o6| 257591 55 021551 3334089 56'014991 175827 1' 6 -018151' 258911 56- 21(i11 -3 04678 57 T'015141'177279 57 1'1S82C061 -260280 T7 021671 - 808851 58 -015091 -178709 58 -018262;201848 58V -021782 -27W 91 591'01514 1 -18)171 59'.}18013i7i1;2G6282 59 -219791 |- -800293 60.015192 -.181622 60 -01837 -2G64176 60 ~ 721852 894-199 II 55'OI~~~~iC2~I 02j~sIl;3 ~21852.3g9~oo:3 EXTERNAL SECANTS. 423 9 DEGREES. 10 DEGREES. 11 DEGREES. Min. Nat. No. Log...ilth. Mlil. Not No. [I,garil. Mlil. Nat. No. |.ogaritll.. 0 0-012465 8.095696 I 0 0-015426 8-188271 0 0-018717 80272229 1'012512 9'C97320. 1'015478'1809 2 1'018774'2703564 2'012559 *C 93941! 2 v'015530 -191193 2'018032 *274899 8.012605i: 100559 3 015583'192658 3 *018S91 276260 4'012652'102174 814'01536 1.904116 4 018948'277559 5'0129,9'103787 15:015GS8 -195571 5'019006'278808 6'012746 105395 6'015740' 197025 6 019064 280218 7 012794 I107001 1 7 C01579311 193046 7'019122 281521 8 012841.10007 8'015046 19925 8 -019180 2282859 9 0(12889 1102 6 0 I 9 0'015899'201371 9 019239'284180 10'0129.36'111804 10'015952'202015 10.019297 285494 11 -0129804 11390 9 1 1 016005 204257 11'0190056'286818 12'013081 114990 12'016058'205697 12.0194t5'288128 18i3 013079'116579 13'016111, 2071308 13 019473 -289441 141 013127'118165 1 14/ -016164 1 208568 14.019532 /290751 15'.01175:'119749 1 15 016218.210001 15'0139591'292060 16.0132230 *121830 A 16 016271 211431 10'019650.'293866 17 -0103271 122989 i 17 1 O'016825 212859 17'019709 294670 18 01819.124483 j 18'01680-79.214285 18'019769 29,976 19 -010367'126055 19'0164003 215718 19'019828 *297276 20 -013416 I'127626 20]'0164861'217130 20'019807'298575 21.018464'129191 1 21'016540I -218549 21 -019947 -299874 22.013513 1300756 i 22'016594 -219965 22 -020006 -3011C9 23 0)180561 182318 128 O'016649 221380 23'020066 3024683 24'011610 133877 i 24 O'016708'222793 24 -02012G'3-8755 25 10186059 1085484 25'01 6757T 224230 25'020186 0835045 26 01.370i8 1 136987 126 I'016811 [225612 26'020246'068084 27 -013757 188587 1 27'01866 22T7017 27'020306 03T07619 28.01.8316 140086 28'01692), -2284291 28'02C566 -308905 29.0138506 141631!~ 29 -0169i75 229922 29'020426 310188 80 01095 9 1431T4 i 80'01703)'281221 80 -020486 -311469 811.010394 I 144714! 1'01 7085 282618 81:0(20547' -312749 82 0 -14004 I,146232 32'017140.284i014 82 o020608'314026 3.3.014054:'1477 7 38 r0171951'285406 8 3'C62668 -8315802 84 1 o-41 —10 I 149318 I 34 I'0172590'236793 04'02-',729 316D57T 85 014153 150819 85 3 0178051'288185 85 022790) 31S7848 86' 0142,03,152876 86 8 017860 2893572 6'02. 851'319118 07 | 0Q14253,158900 8 7 017416 -24-57 0 7'02912 I'820887 08. 143,2 -15 42:3 I 017472' 242089 00 8' 02- 9" 21644 39',14833 156941 309'01 75T7'24-719 89'021034-'822920 40 01.14403 158-058 40'017582 -245798 40'021C95'83241830 41 -014453 3 159973' 41' 40176038' 246474 411 021150'825444 42 ].014503'161485) i 421'017T95'2 I47848 42,'02121S8 326705 4361'014554'16 29911 i48 4 0177'51 7'2492191 40 1 -02120 1'0827963 44'314605 1645o00 44'01787 t'2 51580 44'021141'829220 45 i.01469 166004 i 45 017860:3 251956 45 021403 0330475 461.014706 1 167505 46'017919' 253822 2 46:021465 831728 47'0147571 1693)05 47'0179075'25466 47'021527 1 332979 |48'01408 |0 17'!5 l 40 |'018 32 1'2516147 | 4-.021589 -884229 49 -014859 [1719o 49'018089'257407 49'021651'8805477 50 014939 1'173438 i50 018145-'25S7G65 50' 021718'386724 51'014961'174977 51'0182102'261120 51'021776'337969 52'0150183 176464 5i2'018258'261478 52'021SS 8890211 53 015064'179-S 153'018315 262025 50 C021900'840453 54'0151159i:179480 541 0183721 -264174 54 0219603'841691 55'015167 183910 55, 018430 265522 55'0220!26'842931 561'015218 1'1S87 56I'0S184871 266867 56'0229080'-844167 51 1'015270 ['18336t 57'018544 1 268210 57 022151'845400 58''0153822'1853040 58,'01860 I,269552 58 I022214'846634 59,15'74'.1801 5. 9'018f59:' 9270891 59,022277'8470865 60 0.015T426' 18300271 60'017171T1 272229 60 0'0528041.1349095 424 - VERSED SINES. 12'DEGREES. 18 DEGREES. 1! 14 DEGREE S. Millt.a. No. |.,glii lnl. iin. Nhat N I.rifllm I Mi. Nat No. L,,garitlm 0 0-021852:8-18a9499 0 0(Ci25630 8 4087 47 0 0-029704 8-472819 1 021.913' 340700 1'025695'4(C9856 1'029775'473848 2'821974 8341900 2 *C25761'41(962 2 029845 *474874 3 022034 -3430C97 8 *025827 1412067 38 02991'6 475899 4'022095 *844293 4'025892 418171 )4 4 299G86 476925 5'022916 *845488 5'0G2v958 *414273 5 080(57 477948 6.022217 846681 6 026024 41a574 6 030127 478970 7'0229278 0847877 7'02690'41644j 7 *030199'479991 -8 *029238 0349062 - 8,026156 )417573 S 080270 481011 9'022400 *850249 9'026222. 4198669 i 9'030C41 482029 10 [-022461 85143 5 10'020288 4191-G764 10 0)80412 *488046 11'0C22523 -52620 I 11'0263-05 4201858 11 *030483 -484062 12 *022584'355802 12-'0269421 421951 12,030555 *485078 183 022646'354984 j 183 (22649f8 *4238,42 13 1 080626 4S86091 14 -022707 3856163 14;026564'4241181 14'08CG97 -487108 15 -0227 69 3857342 15'026621 *4.25219 15'03(769. 488115 16'022831 ['35S51s8 16i'(23687 [4268(9 16'C83 841 489125 17 5022892'359;93 17I 026754'427393 17'C3(912 *490183 18 -022954.360867 18,026821'428477 18'03(988,491141 19 023016'862089 19'026888 429563 19 O'01(56 -492148 208'023079'3033208 20 ('26955 4'430641 i 20'031128 *493153 21 -023141: 364376 21'027(022'4-31722 21 *031200'494157, 229'023203'365543 22'027089/'43280 I 22'031272 4-95160 23'023266'366719 23'(27157'4-387i3 8 7 23'081345 496162 24'-023328 *367872 24 (21224'434954 1 24'031417 497162 25 028393 ['3690835 25'0272991 43(6,29 25'031489;498162 26'023s403'370195 26'02785 9'40371(2 26 C'31562'499160 27 023515'371854 1 27'027427 1'438174 27'031634'500158 28'023578'372511 28'027494'49244 28'031707 501153 29'023641'378667 i 29'027562 I 4408314 29'081780'*502148 80'028704.74822 80'027630'441382 80'081852'508142'81-'023767'875974 381'027698 442449 81'031925'504134 82'028330'327125 82'027766'44814 82'031998' 505125 83 023893'378275 88'027834 444578 88'032071'506115 84 023956 379423 i 84]'027902 445(641 34'032144 -507105 85 024020'380569 85'027971'4467(2 85'0322181 508093 36'0243,83 3881715 86'028)309:447762 86'032291'509079 37'024147'882858 317'028107'448821 37'032364 -510065 38 -024210' 884001 88'028176'449878 88'032488 -511049, 89'024279'385141 89'028245'45(0935 39'032511'012082 40'024338'386279 40'028318'451990' 40'082585 5'13014 41'024402'887417 41'0283882 453043 41'032659'518990' 42'024465'888553 42'028451'454096 42'032732-'514976 438 -024529 388687 43 028520' 455147 43'082806'515955 44: 024594'390821 44'028589 1 456196 44'032880 516933 45.024658'391952 45i'028658 1 457244! 45'082954 517909' 46'024722'893082 1 46'028727 4582911 46'033028'518884 47 -024786.894210 47'028796' 45988| 47'038102'519858 48! i024051 1395338 48'028866'4600382 9 48'0388177'520832 49 024915'896463 49'0298935 [461425 49'0883251'521804 59'024930 89758T 50'029005'462468 50'083325'522775 51'025044 3 89S710 51'029074, 463508 51'083400 52374 5 52 *0251(9 399881 52'02'9144'1 464547 52'033475 524714 53'025174'400951 53'029214 465586 53'038549'-525681 54'025489 -402069 154'029238'466623 54'033624'526648 551'025304 *408185 55'0293583 *467659 55'033699 527614 1 56'025369 4043800 56'029423'46,8693 1 56'033774 1528578 057'0254384'405414 57'0294983'469726 57' 033849 1'529541 58 1 0254991 406527 581'029564'470759 58'033924'530504 59'025564'407637 59'029684'471789, 59'033999 5831465 6.0' l 025630' *408747 1 160 029704'472819 60 1034074 - 532425 EXTERNAL SjheCANTS. 425 - 12 DEGREES. 13 DEGREES. i 14 DEGREES.. Mill NM. NO. i: g. N'i. Nat. No Ni N 0t Nn. Logaritlm. 0 0022:341 8-49J995 i 0 0-020-94 i 8-42480:20 1 0 0-090614 8-485915 1 02249,4'859822 1'02i8. 7,8 I 421161 1 -080688'486976 2'022467' 351549 i 2 02442, -422293 21'0807683 -48033 8'022581 5277 8 0251 4234,1''0808:38 t 4895190 4 ~022594 i -358996 4 02051'I.424541 4 80913 -491147 5 *022658!85218 5' 026650' 425695 5'080988'491202'0227221 856488 60 02672)1'426826 6.031064'492256 7 022786j.3576602 7'026789 I'427955 7'031139'493308 ~'022319 i -858$74 8'026859,429084 8 08312165 -419460 9'02291. I 860088 i 9 026928 4300 -9' 1 031293 495410 10'022977'861301 8 10,026998'431394 10[ 0'818606 -406459 11'023042 8362518' i 11 -027068 432457 1l'0314421'497507 12.023106.86072:3 12.027158 i.480585 12'031518 498555 18.'028170'36490'2 1:.8'027218'484700 3 0815941'499600 14 02:'}235 8366138 }14'027278 481 9 14 0031670 5 10648 15 -023299'08367345 15 127849.4937 15 031746 -501683 16'023364 -36S548 ii 16'027419,4380550 16 -031822 -5027T30 17, 020429.369751 [ 17'027490'489170 17 I'081899 -5I 053770 18'023991 3870952 18'027560'440284 18'031975'504810 19'-023559 8372152 19'027631 [4418097 19'082052'50584'0 21'023624 -83780-48s 22.027702'442508 209 032128.50C8S7 21'028689.87454i4 21'027773;448619 21'082205'55T7923 22 -028754 3753789 22'02784,444727 221 032282 i,508951 23'028829 8376942 23'027915 445884 23.032859'59999 241'028885 83781293I 24.'027980 446950 24 -0328436 511025 25: 0203950 379314 25 0280157'44804 25 032513'51205 2(,'024016 -880502 26 -028129'449149 26 -0302599 -51088'024082' 881089 27'02820'450252, 27 052668'514119 23'02414S ~382874 24 23 -028272.451852 23 032745 -515140 29 - 024214'884058 29'023843'452452 29'032828'516174 2809'02429'385240 3 i3 9 023415,453551 30 082990 -.5i7200 S1.'0~214 80 Gfl I l. 385 1'023460 -886421 i 81i'028487 /454648 81'082978.518225 2'24412 87690 32'023559 455743 82 i'0833056'519249 80 02 -4 47T 8'888778 803'0286318'450888 88 -088104 -52027' 8 024514. -389954 i 54 028703'457931 84'083212'521294 85'02411 -391128 I 85 028775 -4593023 5 3' 80312 -522815 85 5 -08-329~~35 ~0823 5i81 36'02507867 892012 5 -1'028248 -4600115 80'0308003 -52,3330 7'024740 -893474 i 87'028920 -461203 87'083447,'524653 88'024811 891645 ii83'029993'462290 i 88 0838525 -525870 409'02491'39518 89'029905'460878 819'033604 i520638 49'02494 -890699 49'029183'464464 40 0Q836882 527401 41'092512'898146 i 41 -'29211'55 41'0807031 528416 42'025379 -8993010 42 -029284'46660 1. 42'088840'529429 48'025146'400478 i 48'02957' 467713 48[ "083919 {'580441 44'025214 -4016035 i 44['029430'4608790i 44'088998/'581458 45:'025281'41.2795 I 45'029508'469872 45'0834077'582468 46:'025348'403954 46'029577,'470950 46'084156'-508470 47:'025416'405110! 47'029650;47-2028 47'084286 -'584478 48'025484'406267 i 48'029724' 478108 i{ 48'084515'585485 49'025552'407421 i 49'029797.'4741i7 7 49'084895 -'536490 50'025602'40857, 5'0209871'475251 50'034474 -537495 51'02568S'409725 51'029915'476322 51'034554 -588498 52'0257560'410875 5-2'030019 -477392 52. 0434 634.839501 53'025824'412021 4 583' 030098'478462 53 - 034714 -540501 541'025892 418171 51'030167 -479081 1 54'084794'541502 55'025961'414016 55'030241'480593 55'084874'542501 56'026029'415460 9j 56' 0303151'481663 l 56 - 034954 -543499 57'026093 -416608 i 57'030890 -48278 57'0859035'544496 53'026166 -0,046 ASO,? 58'026010'417745: 53'0804604:488792 58'035115 -545498 59'026285'418884 { 59'8030539'484853 59'0355195'546487 69'0263004'42002 0 60'080614'485915 609'035276'547481 343* 426 VERSED SINES. 15 DEGREES. II 16 DEGREES. 17 DEGREES. 7 2I EGin E | E;t. N Nt N. I Log.ritl. |Mi. Nat No. Logarithm. M, t.,. _. 0 O 0 034074 ) 9'582425 O) 0088738 1 8,588140 0 0-048695 8'640434 1'0841a50'5888S4'038818 599388 1 048'783'641279 2 1'08422'584842 2 08899 5999036 2 043865'642123 8 0384300'535299 3,088979 5'90888 8 043951'642966 4 034876 5836255 4'9060 0'591728 1 4 044036'64809 5 08452 587210 089140 592623 5 044121 644650'6,084527'588168 6 0899221 59:3517 6 B 044207'645491 7 *084608 *589116 7, 08901:'59449'*044292'646830 8 *084679 o540068 8'089882-.595801 8'044378 647169 9 084755'541018: 9 089463 596193 9'044464'648008 10 0834881 -541968 I 1o 089544 1597088 10 0445508'648845 11'084907 *542916 i 11 089625 597971 11'044686'649682 12'084988 *5438868 12'089706 598859 12'044722 650517 183.'085060 -544839 1 18'089787 599747 13 044808'6518352 14 0351386 54575. i 14 *089869 *6006388 14 04489'652187 15 035218 546699 15'089950 60 118 15'044980'658020 16'085289'547642 16'040082'602403 16 04;5066'653852 17 035366 548584 17 040118 6083286 17'045158'654688 18 085448 549525 118 040195'604169 13'045289'655514 19'085520'550466 19 040276 6551 1 19 045826'656345 20.035596 {551405 2).040858 *605981 2'045412 {657174 21'035678 52342 21'40440 {606811 21'045-499'658002 22'085750 558279 22 040522 60769) 22'045586'658880 28 085S27 554215 23'040604 085G 2.3'045678'659657 24'035905'555150 24'040fi80 609445 24 04576 660485 215 |0835982 55603S4 25 *040768 G,10322 -6 25'045847 *661808 26'086059.557017 26'040850'611196 2'045934'662182 27 1 036137'*557948 27 0409088'612071 27,046021'662956 28 0386214'558880 1 28,041015 612945 28.046109'663779 293'086292 2 5039S9 29'041C93' {61381t 29'046196'664601 3 8 03608870,560738 30' 0411890 614689 80'046288'665422 81 -086447'561665 1 31'041268'615560 81'046370 666242 82 086525' 562592 82 041846'6164:30 32'046458'667061 88'086608'563518 98'041428'617299 88'046546 -667881 34'086681'564448 84'041511'61S167 34'046633'668698 85',086759 -565366 85'041594'619385 85'046721'669516 86'086837'566289 1 6'041678'6i9901 36'046809'670388 87''086916'67211 1 87'041761'620766 87'046897'671148 38 1 036994'568132 88 {'041844:'621681 388'046985'671968 89'087072'569052 39'041927'622495 89'047078'672776 40'087151'569971 i 40'042010'628857 40'047162'6738590 41'037230'570888 41'042094 -624220 41'047250.'674403 428'037808'571805 42'042178'6251,81 42'047389'675215 48'037887'572721 43'042261'625941 48'047427'676025 441'087466'53685 44'0422345'66800 44'047516'676836 45'037545'574550 45'042429'627659 45'047604'677646 46' 0376241'575462 46 -042518;' 628517 46'047693'678454 47'087708'576874 47'042597 629378 1 47'047782'679262 48'03778291'577285 48'042681'6802-380 48'0473871 680069 49'037861'578194 49'042765'681085' 49'047960': 680875 50' 087940 1 579108 I 50 1 042049 ~631989 50'048049'681681 51'088020'580012 51'042933'632792 51'048188'682485 52'088100'580919 52'043017 688645 52'04822T 683291 538 1'088179 1 581825 58'043102 1 6384496 58'048316'684093.54 1'038259'582780 54'043186'635847 54'048406'684896 55 -0388888'588634 55 048271 636197 55'048495'685698 56' 088418 1 - 845 56. 043856'687046 586 048584 -686498 57'088498'58549 57'048441'6878905 57'048674'687298 258'088578'58684i 581'048525'6 88742 t58 1049764 688098 59'088658'587241 59 04361 *639588 59'0485 84 1'688896 60'0 03388788 588140 60! 040695'640434 60'048944'689694 EXTERNAL SECANTS. 4271 15 DEGREES. 16 DEGREEDS. Rr.s,,,m ~ ~ hL N~. 17 DEGEES m Ntiogari.. Nt i n. a Nt. No.,ogaithm. m. I Nat No. I Logarithm. 0 0.035276 8 547481 0 o0o040299 8-605299 I 0 0o'045692 8-659838 1 0835357'548473 1 ] 040386,606283 1'045785'660721 2'035438'549466 2'040473'607167 21'645878'661604 8'085519'550457 8'040560'608100 8'045971'662486 4 - 085000 i551447 4'040647'6(9382 4'046065'668367 5'085681,552486!5'04o0735'6C9968 5'046158'664247 6'035762'558428 6'040822'610898 6'0462.52'665127 7'085843{'554400, 7'040909'611822 7'046845'666005 8'085925,555396 8''040997'612750 8'046489'666083 9'086006'556381 9'041085'618679 9'046583'667761 10[ 0088'557364,{ 10,041172'614605 10'046627'668637 I 1 086170'558847' 11'041260'615580 11i'046721'669513 12'036252'559828. 12| 041848'616455 12'046815 -670887 13'08633884'560309 I 138'0414837'617880 18'046910'671261 14 -086416'561289 1j 14{'041524]'618802 14'047004'672185 15'086498'562267 1 15}'041618'619224 I15'047C99'6780C8 16'036580'568245 16'041701'620146 16'047193'673879 17'086662'564221 17'041789'621066 117 1'047288'674749 18 086745'565197 i 18'041878'621982 18'047883'675619 191'086828'566172 119'041967'622905 19'047478'676490 20'036910'567146 20!'042055'623822 20'0475783'677858 21'036993'568118 21['042144'624789 21'047668'678226 22'037076'569089 22'042233'625655 22'047763'679C,098 23'087159'570063 23'042322'626570 23'047859'679960 24'037242'571030 24!'042412'627484 24'047954'68C827 25;'037825'571999 25!'042501'628898 25'048050'681690 26'037408'572967 I 26'042590'629310 26'048145'682553 27.037492'578937 27'042680'680222 27'048241'688417 28'037575'574899 23'042770'681183 28'048887'684280 29'0.37658'575863 29'042859'682048 29'0484831'685142 30'037742'576827 30'042949'632952 80'048529'686002 81' 087826'57779 1 81'043039'633861 81'048625'686862 82' 037910'578752 82'048129'634768 82'048722'687721 83'037994'579718 88.'048219'i635674 83'048818'688581 84' 038078'580673 84':.043309'636580 a4.048915'.689488 85'038162'5816831 85'048400'637486 13 5'049011'690296 861'038246'582589 86'043491'638389 86 ]'049108'691158 87'0388331'583547 i! 87'043580'689292 8 87'049205|'692008 88' 088415'584503 81 8'048671'640195 88'049302'692863 89' 088500'585458 39'043762'641096 91 9'049899'698717 40'038585'586412 40'043853'641997 40'049296'69457i1 41'088669'587365 41'04943'642893 41'049593'695424 41~~~~~~~~~~~{ 41{5 8 0909 052 42 038754',588318 42'044035'6438796 42'049691'696276 43'038839'589269 48.'044126'644694 1 43'049788'697127 44'038924'590219 44'044217'645591 44'049886'697978 45' 089009'591169 I 45'044309'646488 i 45'049983'698829 46'039095{'592117 46'044400'647884 1 46'050081'699677 47'089181'593075 47'044491'648272 47' 050179'700526 48'039266'594012 48 -044583'649173 1 48'50277'701373 49'039351'594958 49'044676'650076 49'050876' 702220 50.039487{'595932 50'044767'650958 50'040474'708066 51'039523'596846 51'044859'651850 51'05C572'708911 52 -339608'597789 52'044951'652741 52'050671'704757 53'039695{'598781 538 -045048'653630 53'056769'705600 54'039781'599672 54I'045136'654520 54'050868'706444 55'039867'600612 55'045228'655408 55'050967'707287 56'039958'601551 56'045321'656296 I 56'051066'708128.57'040040.'602489 571'045418'657188 57'051165'08969 58W'043126'608427T 58' 045506'658069 58'051264'709810 59'048203'604263 59' 045599'658953 59'0518638'710649 60'041299'605299 60 {'045692'659838 160'051462'711489 428 VERSED SINES. 18 DEGREES. 19 DEGREES. 20 DEGREES. Mi,. N-,,t,,~. 1..ga~lim. Mi. Nat. No. Logarithm. Min..... { N.t. No. Logai,i;i. 0 0-04944 8-689694 0 0'054481 88 2486M 0 006008 81780871 I 049}38 690492 1'054.76 -'817003 1.060407.781087 2; 049128'691288 2 -054671 7 7758 2'060507 7818`2 31 I049213 I692084 3'051766'738510 3'060606'782517 4 049304'692880 4'054861'789263 4.060706 7 083232 5 0498941'693674 5'0549561 740014 5.060806'783945 6'0494134 694467 6 05051.-:'740766 [6'060906 ]784659 7.04957)5 695260 7'055146 -741517i 7 061006'7835871 8 ~049665'696052 8'.42 742:c66 8 4061106 -786082 9 3049756'696843 i 9'0553377 v748015 9,061206 T'876794 10'049841'69T6338 18 0554430 (,43764 10'061306 787505 11 -049937 3 698423 11'055527 [ 144512 i 11' 061407 788215 12'05002:1'699213 12'055623I 745259 i 12 06100507'788924 13 -050119 700001 18 -'055719 746005 13.061608 7. 9634 1.! -050210 -'700788 14'05531'5:746752 14.061708.790341 15 1051301'7015T6 15'055911 I74T491: 15.061839 791049 16. [3,5039-'702861 16'056017 748241 16. 061909 /791756 17,05M4s8 -70-;147 11 113.78S94) 17.062010.792468 18, 050574.70981 1s'056199.749793 18 -062111.1793169 19 050666 T04(16 19,056295 750472 19[ -062212, 938s74 20 {050'757'705498 0i'0560691.'51218 20 1-062813 -794579 21.050849 6. G22 21'056483:751955 2t1.062414,795283 22.050941 T707063 22 056583 0752696 22 -062515 [;795987' 23'051033:'707845 23,056681 (0854 23 032017 796191 24 } -l051121i i''0862o 4 214'050677 i754175 24'062718 7 97893 25.051216 tC9404 25'05687'4 754913{ 25.0628202:2.798094 26 /05130'8 ('01183 26'056971. -755652 26 062921 7'98S795 27'051400 710961 27'0570C8'756890 [ 27'0600238,799496 28 051492 (11 1739 8 -'705(164 7 -57126 28 068124'800196: 29'051581 (712516 29'0572,61!757862 29 06l9,2268 -800895 30 -051676 713291 30'057853, 758597 30,0683298 80159.4: 31 -051769''714067 81'057456'759330 81 [063430;902298 82 "051881 j.714842 F 2.'055wo8.760067 82 1.0C32 802990 83 -051953 715615 F 33'057650 ('60830 33' -0,66364 -'808688 34 052046 1389 843 0774 8' 761534 61584 i 34 0. 368786 0438 35 -0521389 7171692 35'057845 762266 35'0603888 [805080 6'052232 F 717933 36'057942'762992: 36'068940'805775 37 052324'718704 37'058040 3763728 37'064043'806470F 38s /05241 1719474 38.058138 i z64459 38 I064145 - 807165 39 -052510'720244 39.'0582836 -765189 89 I064248'807858F 40.05`2603 (721013 40.e58334. T765918 40 064350,808551 41'052697 0'782 F 41 058432 (66647 41 134453 8192,441 421 052793'1 22549 42'058530.767875 42 1064556 8' 09937 43 8'052883 -'73'316. 43'058628 (768101 43'064659'810628 144 052976 (724082 i 44 /058726 (68829 i 44 064762 811819 45-.053070.72448. 1 45'058824 79I'9555 45 ]064865 819319C 46, 05381683 2725613. 46'058922 770279 46 64968 S19S 47T.05bo257 T7263'77 ij 47'059021 7171005 47 - 0650,71 -8184387 48:['05Q351: T'771401 4-4 1059119 771720 48 065174 814076 49'053444'727903 49'0592181 772458 49,065278 8'14765 5).0533588 (278666 501.059816 773175 50. 065381 8[ 4,0'2 51'053632'72942 i 51!'059415'773897 51 0654853 { 816189 52.0537.26 730183 52'059514 -774619 5 12,.0655S 8. 16825 53'053820 T'733094 5F'05961:3.775840 F 53'J656092 3817511 54 3 058015 (431(06 54.059712 1. 776060 / 54.6a79O 60.818198i 55,054009 732465 55 - 059811.776780 55 865899:818851 56'054103 7383224 56'059910 V777500 56 {066003'819565 57' 054198 [ Tf39816 57'060009.1778218 57 8066 1 27 32024 58 305499) 3'734787 58',60109.778936 [ 5 61669211'82C931 59 0448 43493 59 0602088,7796534 59 I 066315 821614 60 {05 1 60 8066420'5222~9G 6 40581'736248 2 )6'069308 78371 EXTERNAL SECANTS. 429 18 DEGREES. 19 DEGIREES. 20 DEGREES. M i.. | Laf. i N NILahMo.. | in. Nat. No. lgaritbill. 0I 0051462 S7114S9 0 0'057621 8-760578 0 0064178 8 8073S5 1 1 051562 712327 1 *057T727 - -7613876 1. 06429)0'808147 1 05161 - 713164 2 *05783.3 *762174 2 *064403 *808908 831 -051761 1714001 8- *057939 1 762971 8! -064511 *809669 4'051861'714S38 4 058045'176767 4 06469 810430 5 *-051960 715673 1 5 058152 *764562 5'064743;811190 6- 0520607'716508 6 *058258'765358 6 064856 811950 7 *05216(1.717342j 7 -058365.766152 7.6064969 81i208 8.052261 -718175 I 8'058472.766945 8 [.06J83 [813465 9,052361'71913(8 9' 058579 I,767738 9 C.06197 814224 10 052461 719839 11 1. 058686'768531 10 -065310 81498. 11.052562.7267 1 11.058793 -769323 11 -.065424 *8i5737 12 /*052663 721502 I1 12 -058900 770114 12 *065538 [816493 18 i.052763.722332 l 13 -059007 -77C904 13.065652 3817249 14 -052864'723160 14 059115'771695 14.065766 6818]i003 15 -052965'723993 t 15 059222 772484 15.065881 T818758 16'053166 *72481 16'059330 *773272 16 [.065995 [819511 17:,-053167 -725644 i 17 * 05938 774060 17.066110 [820265 18 -053268 -726470 i1 18 059545'774848 18 -066224 t821018 19 053370'727297 19 -059654:775636 19 0663o9 21760 20 -053471'728122 20'059762'776421 20.066454 822521 21 0535783 -28947 21 6059870'-777207 21.066569 S28272 t22!,58675' 729770 06 22 6059978 *777993 22 066684 *824023 23 053777 7T30594 28 /069087 *778777 23 [066800 /824773 24 -053879 1731415 i 24'060195 -779561 241 066915 /825528 25[.053981 *732237 25 060304'780343 25' 067030 826271 26 -054083~ *7.335-s 1 26 *060412 -781127 236.067146 827019 27 -054185.733878 | 27 *060521.78190f9 27 * 0672:62'S27767 28 [054287 7834698 I 28'060630'782699 28 [ G067877 828514 29 054390'735517 29'060740 7T83471 29.067493 829260 80 /-054492'7836835 30'060849'784252 30 067609. 830007 81 G054595 7837158 381 060958'785031 31 *'067726'8307T53 32 054698 -*77970 32.061068'785810 82 *067842'8381497 838 1054801 738785 883 061177,786588 33'OG67958 832242 34'054934'784602 84 G061287 787367 34 068075 832986 I85:.055007'740417 85.061397'78S1-1 85 I068191'888729 36 055110'741231 86 G061506'788915 836.GS06838 834472 87 3 055218'742044 I 87 061616 789G96 37 1068425'835214 88 055817 *742857 88'061726 7904T2 88 068542'835957 89:1,055420 743670 89 061837 1 791247 39,068669 836697 40 1 C055524 744482 40 -061947'792021 40 ]1-068775'837439 41]'055628 1745293 41 062158 792795' 41 I.068893'838178 42 1.055732 746108 42 062168'793568 42 C009311I 888919 481'055836 746912 43.062279'794340 43 069129'839658 44 1 055940 747721 44.062390 1795113 44 1 069247 I 40397 45 ['056044'748530 ii 45 062501'795884 45.069364' 41185 46,.056148'749338 46 06212'79654 46 069482 841871 47 -056253'750145 47.062728 797424 47.069600 842608 48 -056357'750951 48.062834 1798197 48 069718' 848345 491 i056462'751757 49.062945'798964 49 0,69836 844082 50I'056567 -752568 50.063057'799731 50 O069955'844817 51 -056672'753367 51:.06C168 *800499 B51'070073'845553 52'056777'754171 52 -063280 1801267 52'070192 846287 53 056882 1-54973 53.0638921'802033 53 070811'847021 5B41 -056987'755776 54 *063504'802799 54 1'0704301'847754 55 - -057092;756578 55 1363616'833565 55 070549 848487 56 057198' 757380 56 5 063728'804331 56 | -070668'849220 57 1'057304 1 758181 57'063840'895094 7 1'070787.'849952 58 0 -057409 *758980 58'063953'805858 58'070906'850682 I59I:'057515 T759779 59 -064065'806621 59 071025 851414 |60- 057621 7005T8 603.064178 807385 60 071145.852144 43 0 VERSED SINES. 21 DEGREES. 22 DEGREES. 23 DEGREES. Mlin. Na. No.. 11n.? Nat. No oajr NtNo..garoithm. 0 0-066420 8 822290 0 007T2816 8-862227 0 0-079498 8 900340 1 066524,822977 1 -072925 862877 1 07969 90(962 2 [ 066628 823658 2'078084 868526 2'079728'901582 3 06667188) 824688 6' 0873148'864175 8'.079837 902201 14 066837 T 82518 4 -0783258'864828 4'079951'902821 5'066942 825697 5 073362:865471 5 088664 9383402 6 606747 [826376 6 -073471'866118'6 *080178 ~'934057 7 6067151 /827(054 7 0783581'866765 7'080293'904675 S':067256 827731 8 -0873690'867411 8 080407.,905293 9,067861 0284C9 9'0788200 868('56 9'C8521'90,910 101;067466'82s985 10 073910'868701 10'080636'906527 11 i067571 8297630 11,G74(20 7'2869846 11 080750'907148 12' 1067676 830486 192'074180'80991 12'08865 907758 18 -06775S81 8 31110 13 %074239'870684 18'085C979'9~8374 14'067887 831785 i14'0743849 871277 i 14'081( 94 9'898 15'067992. 832459 15 T07460'871920 15'0812('9 i'909683 16''C68C97 8883131 16 074570'872562 l' 16'0813241'910216 17'0682u8.83380 1'37qI 684') 373953 17 081489'91C830 1 9 8 0689.074790 873845 18.'81554'.911443 1 0.68415 85148 19'074931 *874486 19i'081609'912056 2) o06ss20 s 835819 20 0o750Ol1 -875126 20'081784'912668 21.068626.83 6489 21.075122.875766 21'081899.913279 22;068829'837159 i92'075282'870405 4 22'C82014'9138890 28 068888'83829 28'0758343.8770441 28'082130'914501 24''068944 468497 24'075454 *877682' 24'082245'915111 2i5 {'9 0 0 839165 2- 0755656' 87825 25'682361 5915721 26'09157'8893, 26'075673 I 878957 26'082476'916831 27 0369263:840501' 27 075787'87K694 27 082592'9169`9 28 -0$696' 81167 23'075898'888230 28'082708'917548 29 %6C9476'8 8118,14 89'0760(9 )'880866 29'082824'918156 8) ~-698-9'842,199 33'0U-78121'8815(2 83)'062940'918764 1 -068489'80'6O5 31'(7 620252' 882137 30I'(83056'919371 2 0069 79'81P829 S82'076"843'832770 82'083172 -919977 3 5(914 -84'4493 8 3'1076455 88845 05'083288.920588 7010 4'8451T57 834'176566'884:88 8 34'(88404'921189,75 117.i 4 5GI 35.07667s8 884670 85'083521'921795 36 -i'2 j-846481 86 C'7379 0 885308 36'088687'922400 IS - ~ Ti1991 I 84714oD 37'7091G2 -'885935 37'083754'9283004'8'0886' 84 78"5' 038'07 886567:8'08087,, 923609 o9 010o8 j o848467 6'877125'887197 39'C88987'924212 40'070653'849127 40'77287'8788 T28 40'084104'924815 41 Oi'G I 036 941 4i0';077551'883458 41'084220'925418 41'070760'849787 4 O7705 831 41'8-2 42'0577867.830446 42,'1077562' 885.s 8 42'(084387'926620 975'851116 4'077574'889717 4'084454'926628 4'I 17128)'851764 44'377ST'894346 44'C84572'927224 45 S 3119 S) 45'077799''890974 45'084689'927824 46.o1293 85019 46'077912 -891602 46'084806'928425 47 71406 -8735.8 47'078024'892229 47'084923'929625 48. 971514 85491 48'078187 1'892056 48'085040'929625 9 71622 -55048 49'0782530.89342 49'085158'96C224 50'173')'8m503 50 -078363'894108 50'085275 -980828 51 -371839 -86358 51'078476'894784 51'085893 -931422 52'21 947 T'857012 I 52'078589'895358 52'085510.'932019 53'072055 85766 53'078702' -895986 53'085628'932617 54'07216 4 853819 54'078815i'896607 54'085746'938214 55'907227 2'35892 55'078928'897230 55'C85864'933811 56'0723881,83962 56'079141 -'897853 56'085982'9384407 57.'072149) 880 27 57'079154 898475' 57'086100' 95008 58'072393 86 927 58'010268'899097 58'086218'935598 596'7277 881578 59'079182' -899719 59'08633886'986198 3'172816,862227 61'079495'930340 6'086454 1'986787 |o c;,c it c. C -i, ot t;, "-, l..I CA L'Or-',- H 1-v..- rlp O C..co c t c 03, co cO,CO00 ~ _ o d co | L,: ~ 0 0o | l') 0 0o _ t0....: -4 = Of 4- CO K) k-1 1-1 VA cc w I- C=-. C, I -IIzl:,?L CD C=' y C. V( H., CA tp cc- t:: -oo co I4 ct: >tP O P co to O.o COC.... Gc, O O' V C o u I ".. C 0 4 D I GO i C i C. CG --'..0 000 Q, co QGCo. C, CO V4000iO0 0 CO CA to C LO 0,C- CO0.Q400 ) C -4 Q0 0C C0 ) 000 CA CO 00000 V W- -1 K) CO00.00CO000 0D0 CO -4 9 O 4 L cn 1,, l -D C::) 0- c;O OH Li |VT CO0 6L 1- - 0 0 0 0 GO - - - 00 G, C0 V ( C- O GO ^ G o-D o03 CL =0C=00 00 4 VT OV $C, GC L 0 0o V0 0b) C) CO, 0 _ CO COor 4. C- I (= d,;W CO (: CZ C L O Gt W. COO o C o O 0 —0.C0, 0. 0 W 00 0 0 0 0 0 cg cs H9 e co OC co co C9 CQ C: e CL c C 0 co t C3 Of Cx C,:" K0 ) Co e c D c9 to _C-;,;) e> C"D LI:) w L,;. =1 (J C9) co l- toC3 01. LID Z C:, 0,'D C- ( ot olr,~ COT cut c;z VI VT CO X CO.. mp l4D CO to co D; w o wr s tF C CD Oc M C u C.;: C C>'. t. e Cl CO O CD CD C s 0L C: to t. LO LO >o Lo l0 0 000 o C 00 C; O CO 0 0 00. C O O C 0 00 C 00 1 CO 0 C) O C ) 0 C C) C CO'- 0 | -4D43 - -, o;.'.; O0'; co lm (. Ot oD L<.-' O' c=: P-n 0-: tO o o O-D C r c; VT O cco OD ~n C, c, C L-; O O C)( CO V1 OCLIZ CT 0 to C CZD OOD c 01 COC CD O C) cD0 CD 00 C C CO Co c0.0 CC O tco S c: CD GO 9 o OX 9D C= C 0 OD C OD -o;( Oo 9D CZD C` G D D O CO 9D 9D GO OD | 9 cn CC tDc3 O5 qD c9r PD c tO P3 co 0D t co CID tc co to'O t CO CO = CO CC C ^ t C O cD CS O CZ) = k " C CO C = 1cD C3 C C' j W C-)O W CO C) CI) td G;D Gb C(D r CO Vt Gr O 0 C0- r 0C: C. GO 0 4 C O C O40.D CO bO G- CO C:0 GO 000 — 0C CO O l-0C1 C-' C.O Ooo C- C. o o,.,4 C-, C= C OD. G- C - 4 l CID CIO l3 (D Oz a 0 c C- O - Z) -I 9- C't O t>. ( C D G-CI) L) CD CD w l.D CO 4# n -O CIDO G; OD L.:) COD x C) F 0 CZO eo Cz ) t; C; G3: CD 4'. C;( C), COT 0C (n c3 CO, ca rA: ooooo Do Z~Doo o0o oo o co ao.o E6 - 00000.cc) CD 1c-0.Cc,)o CID c 0 c I 00 C/0G CY0 G O 00000CIO 0. GO 0. 00 00 OD 00 050 0 GO 000 0GO0r.0 0.0C O w;i C-5 65 c~ -S i w C C-) C- C,:,- 0 00000 0 0000 000 -1 c00->-co -'0co m C".5 Z%. C-) (50 C-, Cz) czI''-I — - C -1 4 C C> 0 0 0,: 0 L 00,:0 0 I_0 -00 CZ)0 0 C'0 0 VC00 -0 _ 432 V\RSE) SINES. 24 DEGREES. 25 DEGRIEES. 26 DEGREES, Miii. Nat.No. I Logarithm. Mih Nat. o. Logarithm., lin INat. No. Logarithm. 0-086454 8,936787 0 0-090692 8:971703 0 0101206 9-005206 1'0865738 9373882 1 -09815'972273 1;113 005753 2'086695 -937975 2 09939 I972840 2 101461 006300'086819'938509 8 C94061'973411 3 "101588;006843 4 -086929 i O'9162 4'694184 9703079 4:101716 I 007392 5.'087047 9399154 5'09408 -'974547 5'101845 007938 6 -087166 -940F346 96 094391 975115 101978,008485 7'087286'94988 1 7 C-5.975683 7 102100 00912 8 0.87404'941529 8'094678:976250:8 -1 2259 009572 9:os087523 -94212 9'C94829.976816 9 1 2:5'010116 10 -.'87642'942711 10'C941925.'977880 10:1,248 5 -010660. 11 0761 -.. 94300 11:095 49:977948 11:1,231 -011204 12 17'8880 9 403889 12) 095i73:978514 12 102740 -011746 18t'087999 -944478 1 095297 979578 1 1028 t012289 14 oss088li9 94507 14 095421:979643 14'102939'.012882 15'088288.945856 15'095545:983:207 15 108128'018373 16'088858'9462438 16'095669:984771 16;103256'018915 17 088177 946833 17 C957903:9810304 17 -10885 014456 19'C9859T i.4TAIS 1 8 -D939 1S 9S191 8 -108514 1 ~014998 ~ ~ 13::981898 58 ~ 014998 19.'C83716'918004' 0960"42:952-460 29::103:64) t'015538 2' -08886'948595 2) -09166:983i23 25 1 03772'-016078 21'088956'949175 21 -09291:913585 21:103951 -016618 22 -.089076 -949761 22 -090415:984146 22'104030 -017157 23 -089196'953346 23 -096540:984707:23 -104159 017696 24 -089316 -950931 24 -096665:985268 24 -104288 -018235 25 - 089437 -951515i 25 -'096793 -985829 25 -104417 -018773 26 - 089557 9529S8 26 -096914:986888 I26,104547'019311 27'089677 -952681 2'C97040:986948 27 -104676 019847 2'0879C 8 -953235 28 -097164:987507 i 28 104806 -020387 29 -]089918 -953848 29.097290 -988006 29:104936 -020924 830 -094039.-954429 80'097415 988624 80'105066'021460 31 -090159 -955011 i 81 -097548 -989182 31 -105196'021997 382 -090280 -955592 1 32 -097666 -989741 32 -105326 -022533 33' 090401:956173 833'097791'991298 33:105456'023069 34'099522'956753 34'097916:990854 34 ['105586'028603 35'090648 -957333 35 [-098042:991411 835 -105716 -0241839 36'090764 -957913 36'098168 -991968 6 -10 5846 i -024673 87 -095885 -958492 3,7 -'098293:992524 37 -105977'-025210 88 - C91006 -959071 38 - 098419 -993079 88:106107 -025742 89 -091127 -959649 89'098545:993684 89 -106237 -026275 40 -091249 -961228 40 -'098671 -994189 40 -'106867 -026808 41'091370 -960805 41 -098797 -994743 ~41 -106498'027342 42| -091492 -9613829 42 [-098923 -995297 42 -106629 -027874 43/ -091613'961939 48 -'099,)49:995851 43 -106760 - 028406 44 -091735 -962535 44'099175 -996404 44 -106891'-028938 45,-0'91857 -963111 45 -099302 -996957:45 -107021I -029470 46 - 091979.963687 46.'099128 -9975C9'46 -107152 -030000 47 -092101.-964962 47 -099555 -998061 47 -107283 -030531 48 - 092223 1 -964836 48 - 099681 -998613 48 -107414.031061 49.-092345'965412 49 -099898 -993164 49 -1075453 -031592 50 -092467 -965985 50'099934:999715 50 -107677 -032121 51 -092589.966539 51 -100061 9000266 51.107808 -032651 52 -'092711 -967181 52 -100188 -000887 52 -107940 -033180:53 092834 -967705 533 ~-100315 -001366 53 -108071 -003709 54 -092956 -968277 54 - 100442'001916:54 -108232 084237 55 -093078'968849 55 -100570 -002465 55 -a 108334 034765 56 -093231 -939421 56 -100697 -003014 56 -108466 -035293 57 -193324 -9869991 57 -100824 -003563 57 -108598[ -035820:8~'-C93447.970563 58 -100951 -004111 58 -S108730. 08648:59 -093569 -971130 59 -101079 -004660 59 -108862'036874:6 -193692 971703 60 -'11206'005206 60 -108994 -037401 EXTERNAL SECANTS. 433 24 DEGREES. 25 DEGREES. 1 26'OEtGEES. Mill. Nhat.No. Ltogarithm ill.. t. Nr 1 1N1i.' NIa Nat. I'Logarifl on,'o 0-0.9436 8-97o605 0-103318 901412 0.0-112602 9-051546 I ]'194178'9176708 i' 103528'015056 1 -112160'052154: 2' -094920 9785T i 2.103678'015685 2'112918'052?13 8'895062 91908 i 8.103828 016812 8'113076 053310 41'095204'978657 4.108911'016989 4.118235 T058909 5 109538417 979306 i5 / 104128'017566'5.1138893 0'54586 0 -CO954;89 919954 6.104219'018194 6: 118552 ['55195 71 095632'980603 7.104429,018821 1 118710'(;55799 8'095775'981250 8'104580 01917 8:113869 056406 9 -095918'981898 9.104780'020072 9.114028'(57012 10'-096061'982546 10i 104881.'020696 10.114181;051618 11'096204::'983191 10'10503?9-'021823 11.114847'058224 12[ 1C.96347'988337 i 12.105184'021948 12'114506'058828 13 -096490 984483 13 8[ 1050385I'022572 138'114666'059434 14 [',96634 985129 14:105486'023196 i 14 ]114826 (160(09 15'0967771'9851774 15 -105638'028820 15'114985'060642 16 -'196921''986117 16'105790 024444 16'115145'061246 11 -'97065' 987062 1I.105942:'025066 17 1 115306'061850 18'1097209 9377071 18'.106C94 025690 18'115466'062454 19 i'097353' 988050 19 -106246'026812 1 1 115626]'C63057 20'-09149$'988994 20'106398!'026934 20 -115787'036660 21 -917642,, 989636 21'106551 02556 21 11 2"'0977137'9932179 i 22''106103'028117 22 1118'1064863 23 4:097931'999210 23'106856;028798 28 -116269'0165465 24'-098016'991563 24'107009''029419 24'116481'066061 25 -'098221 9922:)5 25'.10716.2,'030040 25.116092 0666671 6'0o983366'992845 26'107015'030659. 26.'116753'0617268 217'098511:993486 271'1017468'I'081279 27 11915'067809 28'1986017 -994121 28'107621'031898 28'1170771'068410 29'988082'994766 29'1077715'032518 29'1172899'CC9010 30 498948'995406 30'107929'0331.36 30 -111400'C696E9 81'99091'996046 31'108082 083715-4 I 801 -117562'070269 32'199240'9966'84 32'108236'034373 32'17725'070868 83'699386'997823 33'1088390'0384991 338'117888'071461 34'1r"99532' 991961 34'108544'0356071 04'118049 0172064 35'199678'998599 35'1108699'0386225 35 -118212'012660 86-;'099824['99923 6 6'108854'036842r 36 -'18875'073261 37'0999711'999817 37.109008'037459 87.118509 06713861 88'100118 [9:000510 38'109163'038014 38'118702'074456 89:'100264'001146 39 00931.8'08690 39'118865'075053 40 "'100411'001183 40 109473'0809306 40 -119028'075649 41:'100558'002418 41 109628'039920 41'19192'076246 42'1007106'003053 42 109783'040535 42'119355['0716842 43'100858;003688 43 109989'041150 43'1i19519 ['077438 44'101000]'004822 1 44 110094 0417614 44'119688'078083 45'101148'004951 45 110250'042878 45 1i19848.'078629 46;'101296'005591 46 110406'042991 46'120012'179222 471 101444'006224 41 110561'048604 41'120116'.79817 48:'101591'006857 48 11017'044211 48'120340.'80412 49 ['101740'001491 49 110874'044829 49'120505.'081006 50 1 -.101885. [-008122 50 267 ~859 59'1018835'008122 50 111030 I'045441 50 206 d81599 51'102171,008155 51 i 111181'046053 5'120835 / 082193 52'102085'009385 521 111844'04665 52'121000'(882186 53'102334'010018 53 111500'047216 53'121166'888379 54'1020482'010649 54 111651.0417881 54'121331'0 83971 55'10o2631[. 011279 55 111814'048491 i5'121496'084568 56'102780'2011910 56 111912'049108 56'121662'(85155 5~1:'102930',012589'1 112129'-04971.8 5T'121828'085146 58''1030719 0181170 58 J112281T / 508328 58'121994.'086388 59 103228.'013798 59 112445 1.050938 59'122160'086929 60'103318:'01.4421 60 112602'051546 60'1223271'087520 37 4 34 VERSED SINES. 27 DEGREES. 28 DEGREES. 29 DEGREES. M- Ne t' lngaIlithm Ili. 1 Nat No j Logaritlm. | Min. Nat. No Logarithm. 0 0-108994 9-03740t I 0 0117052 9068880 0 0-125380 9C9s8229 1 109126 037927 1 1 17189 068OSS7 1 125522 C-98717 2 1092.58'0388452 2 i'117326'069893 2'125663 (C99206 8 109390.088978 3 3'117462'069899 3 125SC4 C99693 4'109522'039502 4 1 117599'070404 4'125945'100191 5- 109655'040027 5 *117736 07(;910 5! 126C86 100668 6'109787'040551 6 -117873'071415 6:1262$28 101155 7'109920:041076 7'118010'071919 7'126370 -101642 8 11005.8'041600 8'118147'072424 8 126512 102129 9 1101805'04212-3 9'I 118285 072928 9 -126653 102614 10'110318'042645 10'118422'078482 10'126791 1C3100 11 j 110451 1104 043168 t 11 S18559 -078935 11 12936 103585 12 -110584 04B3690) 12 i 118696'074487 12 -127078 [104070 13 11071T7 044213 13 118884 -074941 13 127220 ]104555 14.110850;044785 14'118972'075453 14 127862 [105040 15'110983'045256 15'119110 -075946 15 127504 105523 16 -111116'045777 16'119247'076448 16 127646 n106008 17.111249 I046297'1 119885'076948 17 -127789.1(16491 18'111883 040818 18'119523 077450 18 -127931 -106974 19'111516 04888 19'119661'077951 19 12873 107457 20'111650 047857 20 119799 *078452 20 -128216'107940 21'111783 048377 21'119937'078952 21 -128358'1C8423 22 111917 048396 22 120075'079452 22 -1285C1'1089(16 23 -112051 049415 23 120213'079951 23'128643 10 9387 24,112185 204933 2-'120851 080451 24 128786 109863 25'112319 *050451 25'120490 080951 25 -128929'110350 26'1124538 05096 26'120628'081449 26 -129072 1108.31 27'112587 0G514 27'120767'081947 27'129215 *111812 28 112T21 *052004 28 *120905 -082445 28'129858'111793 29'112855'052520 29'121044'082943 29 129501 -1122T8 30'112989 -0580371 0'121188 -083441 30 129644 1 12754 31 1138124 805355 81'121322 1088988 81 129788'113233 32 -118258'054009 332 -121461 -(84485 32'129931 -113712 83 1183893 05458 i 83'121600'084932 833 180074 -114192 84'113527 055099 34'121739 -085428 84'180218 -114671 85 -113662 -055614 85 121878 085924 85 130862 115149 86 113797 056129 86 3 122017 -086420 36'130505 -'115627 87'113931 056642 i 3'122157'086916 87 1380649 -116105 88 |114066 -057157 88'122296'087411 88 180793 -116583 89 -114201 057670 89'122485 -087905 89 180986 -117060 40 -114-336 058183 40'122575;088401 40'181080 -117587 41 -114471 -058696 41'122714'C88895 41 131224 -118015 42 -114606 -059208 42'122854 -089388 42'118368 -118491 48 114742'059721 48 -122994 -089882 43.181513 -118968 44'114877 -060232 44'123184 090376 44 16517 119443 45 -115013'060745 45'123273 090869 1 45 181801 -119919 46'115148'061256 46'123413 1 091861 1 4 1'131945 -120894 47 -115283 -061766 47 -128553 091854 4 1 132090 -120870 48 -115419 062277 11 23693 -092346 48 -1382285 -121845 491 115555 j062788 49 -128834 -09288 49 1 132379 - 121819 50 115691'063298 50'123974 -093329 50 i-132524 -122294 51'115827 1 063807 Ij 51'124114 -G(93821 51'182669 -122768 52 -115962 064316 52'124255 - (94312 52 -182814 -128242 53 -116093'064826 53 -124895 94802 53 -132958 -123715 54 16285 -065885 54 -124535 095293 541 183108 -124188 55 j 116371 -0065843 55'124676 095783 55 -183248 -124661 56'116507 -066351 56 -124817 -096372 56 - 138894 125185 571 -116643 066859 57 -124958 -096763 157 183589 -125607 58 -116779,067866 58'125098 -097251 1 58 -133684 126079 59'116916'067874 59 125289 -097740 59! 1338800 -126551 o60'117052 068880 60'125880 -098229 60. 1388975 -127022 EXTERNAL SECANTS. 435 27 DEGREES. 28 DEGREES. 29 DEGREES. Min Nat. No. Logarithm. Jinl. Nat No. Lognrii:hm. Min. N'at. No I.ogaritlbm. 0 0-122327 9.087520 0i 0 0-132570 9122445 0 0-148354 9-156410 1 *122493 0 88111 I 1 182745 1i28019 1 143,538 1569S8 2 *1226(30 C88700 2 -13)2920'123593 2'143723 15T7527 3 i 122826 (C8990 3 - 1833C96' 124166 8 3 143908'158084 4 *122993'089879 4 1833272'124738 4'144C93,'158642 5 *123160 090469 5.183448'125312 5 144278 *15919 6 *1238327 081057 6 -13624'125834 6 1444603 159757 7 123495'091647 * 7 180 12643 7 1449 160314 8 123662'C92236 8 -13397'127028 8.144884 16C0869 9 123829'(92823 9 Ii 1- -1o3 127600 9'145020 *161427 10 128997 G98410 I! 10.134330.128171 10 -145206'161983 11.124165.098998 1i1 -184506.128742 111 145891 -13625839 12 -1248338'94.585 li 12'-134683'129312 12'145578 163C95 18'124501'095173 1 13.14800'129883 183 145764'163650 14 -124669'C95760 9 14. 138(37.180453 14.145950'164206 15 124838 0C96346 1i 5 138215 1381024 15 146137.164760 16 -12500(6 0969382 13.13M592'180194 186 146824 -1635315 17.125175 -C97579 i 17 1185570 -1182162 17 -146511'165869 18.125344'C98103 18.135748 1827832 18 -146C98 -166423 19 125513'098688 19 -135926 18833C1 19 146S85 -166977 20 125682 (C99273 20'131404'130870 20'147072'137581 21.125851'099858 21 -13282'1344(8 21'147260'16808.5 22 -126021'100442 22 1806460'1050:6 -22'147448'168639 23'126191 -101C27 23 -186689'185574 23'147686'1069191 24'126860'101610 24 1836818'136142 24'147825 1lC9749 25'126530'102194 25 1 18C997 1 36710 25.148(12 I170296 26'126700'102776 26.137176'187277 26'148200'170849 27'126871'108361 27 1318o5.1837843 27 -148839'171401 28'1270)41 108944 1 28 -137584'1384(9 28 -148577'171953 29.127211 104525 29 1 -313 77 1 188976 29 -148766 172505 30.127882'105108 80.137893'189542 80 *148956'178C57 81'127553'105690 31 138073 1 1401C8 31'149145 1 73608 82 1127724 1106272 82 1138253 -140674 82 149884'174158 831'12T895'106853 83.188483 1412(9 83'149524.174T10 34 1283066'107434 84.188613'14180,4 84 149714'175261 85~'128237'108014 85 3 138T93'142369 85 3'149903'175810.36'1284C9'108596 36 10389T4 -142904 86'150C(93'176360 37 1128581 -1C9175 j37 189155 1 143499 87'150283'176910 88,128753.-lC9756 88 189336 |,144063 88'150474 -'177460 89'128925'1103885 89 1 83917'144626; 839 150G664 -178008 40'129697, 11C914 l 40,139698'145191 40'150854'178557 41'129269'1114938 41 1389880'145754 41'151045'179107 421 129441'112072 ]1 42.140061'146316 42 151286'179655 48'129614'112651 j 43'1402421 146879 438'15142T'180205 44'129787' 113228 44'140424'147442 44 1' 151619'180752 45 129960'113808 45'140606 148005 45'151810'181300 46'130132'114885 46 -140788'148566 46'152001'181847 47'130305'114962 47'14C970'149128 47'152193'182895 48'130479'115539 i 48.'1411538 149690 48'152885 182943 49'130652'116117 49'1418836'15(251 49}'152577'183489.50 3(18;s826 116694 50'141518'150812 50'152769''184086 51 13(3999'11:7269 i 51'141701'151378 51 *152962 184588 521 181173'117845 ii 52'1418841'151984 52'153154'185129 58'131847'118422 0 53'142067'15'2494 538.158347'185675 54'131522'118998 54'142250'158054 54'153540'186221 855'131696 119578 | 55'142434 158614 55.'158733 186766 56'131811 120148 536' 142G18'154173 56'158926 1878318 57 -132043 1120723 57'142802'154784 57'154120'187858 581 132220'121297 11 58 142986'155292 58'154313.'188403 59.'182895 121872 59'1483170'|155851'59 1'154507'188947 60'182570' 122445 |6'D; |14488i4. 1 6410 |60 154700['189491 433 VtoE)S1) 13SIE. 80 DEGREES. 1 DEGREES. 02 DEGREES. Till Nt No, L oga itll. lill. Nt No | g if, | Not I.go. O 013:3975 9 127C22 0 0-142838 9104827 O ~C151902 9-181706 1 184120'127494 { 1'1429883,10528 1 152106!182147 2 1384266 127964 2,148180 155R78 2'152261 -182587 83 104412 -1248i6 3 *148282 166192 9 8 152415 180Mi27 4'104558'128906 4'148482 156647 4'152569'188466 5 134703 1 129876 5'148588 1571(2 5'1527241'188906 6 134849 1298-1i(. G 1403738 15705,6 0 *152878'184045 7'134995'130016 0i 7'148883 158f010 7 -158088 1 184784 8 1.85141 18078S 6 8'144004 1,846'4 8 16581881 185223 9'185287'131254 9'144184 158917 9'158342'1850061 10 135438'181724! 10 q1440853 159370 10'158497i 18100 11 105579 -10.12192 11 1444885 159823 11.158652 -186538 12'185725 182660 12'144686 16C275 12 1583807'186975 13 1305872'100128 18'144787 160G728 183 158962' 187418 14.136018.138596 14'14490,37 16118o 14'154117 -187850 15'136165'184064, 15.145088' 1(16082 15'154272 -188287 16'100011'184551? 160'1452i09 162C83 10 15.4427'1S8724,17 136458 134998 17'145390'162535 17 -154(83 189161 18'186605 ]'135465!18'S 140-54A1 16298 18'154788 189598 19'1806751'135981 19'145092.'1G6486 19'154894'.190033 20'186898'186897 i 20-'145844'1.1688SS7 20 1155C49 -190 469 21 187(145''186863 08 21'149095 11587! 21.155205 2'19(;905 22 137192'187829 1 22 146146' 1G464788 22 158GO6 i 191340 9238 187839'107794 i 23 -1406298 9:165288 231 i15 516' 191775 24'137486'130259 i: 24 14(450:165687 24 i155672 192210'25'18376='1380724 il 2, -146CG1 16(;6187 25 1o5S28!192645 26'188771'124189 26' 146752;'16658,5 26.150984 198080 27'137928'139653 27 - 14C94 I 167C034 l27 156140.193514 28'138076i'140117 28'147056 -167480 285 1590 19948 29i'188228'140581 29'1472'8 167981 29 1I 564B2;194382 80'188871'141045 50'10 47300'168879 80'15669'O 194815 31'138518'141507 31'147512 108827 81 -156765' 195249 82'188666 141971 5 82 i'147604 1924'92 2 15(921 195682 8833'188814 142484 83 1478S17;1(09722:83'157078 1'196115 84'188962 1'142-96,' 84 i'1479C9'17019 34 157284'196547 5 1389110'148358 8 5'148121'1706115 1 31 157891'190;979 086'189258 | -148820 8i 6 1482738 1710638 86 157548 i 197412 87'189406 1 144282,i 7'158426I'1715(9 87;1577t5 197844 88'139554'144743 i 8'148578 171954 88'1578Fil 198275 89'189708 1 145204.;39'148780 1 172400 89 1158818'198707 40 1'189851:'145665 40'1488881 172846 40 158175'199188 41I'189999'146125 i 41'149C806 178292 41 1588032'199569 42'140147 146585 I 42'149189'17878(6 432 158490 1 200000 48'140296'147046 i 43 1498429'174181 48'158617 I 200480 44'140445'147506 i 44'149495 | -1741260 44;158S04 |'200860 45'140594 147905 45 149648 i'175070 45'158962'201290 46'140742'148424 46'149801'175514 46'159119'201728 471'140891'148888 47'149954 175958 47'159276 202149 48'141040 1'149842 48'150107 I'176401 48'169483 202579 49.1411891'149801 49 1 50261 176845 49;'15959!1'2030C8 50'-141888'150259 I 50'150414'177288 50 10159749 1-208407 51'141487'150707: 51'150568'177781 5:1 5'1,9907 208066 52'141686'151175 I 52'150721 1'178174 52 160005'204295 51 51 41780'1151082 53'150875' 178616 0 53 16C220'3 -204723 354'141985'152089 5 54'151028'179058 54'1660080'205151 55'142085'152546 55'151182'179500 55'160(508'205579 -56'142284' 158000 56'151836'17994.2 i 56'160697 l206006 57' 142884:158460 57''151490'180884 57'160855'2(06438:58 1 142508 158916 58'151644'180824 58 1161018''206860 591 -142683'.154872 59'151798'1 81265 59'161171'.207287'60- -.142888331 154827 60'.151952'181706 60'118330 207714 EXTERNAL SECANTS. 437 80 DEGREES. 31 DEGREES. 382 DEGREES. Min. I Isat. N.i JLogari:. m i. Ain. | Nat. No. Logaritllm. Min, Nat. No. L,a-id0 154700 9-189491 0 0166633 9-221T61 i1 0 0.1T9179 92.258286 1 I154894.190086 1.1668837 222291 1 119393 1 258805 2 155089 -190t59 2 161041 *222824 2 179608 25E4824 8 1*13288 191124 3 8 167245 *223834 3 *119S22 254844 4 1554s8 *191668 4 16740 -223885 4 -180037 255362 5'155573 i192211 i 5 167655 2244171 5 18025 5881 G 155867 *192754 6 1 lG7860 224941! 6 *183467 *256399 5I *6362 19329T7 7 168065 1'2254717! 7.180683 256917 8 -156257 193880 8 1 68270'226007 8'180899'257436 9 |156452 *19538 i 9 1684i6 |226587 9'181115 257951 1) *156643i 194925 10 168681 -227066 10 *1S1331 *258471 5I *1SGS44 195467 11 *16SS *227595 11 *181547 25S989 19 *151043 j196008 1 2'16939.3 228124 12 1817S63 i 2259505 13 15723.6 i 196550 13 169299 22S653 13 *182080 1 260023 14 -151412,.197091 14 -169505'229182 14 -182197 i260540 15 15a76(83 191633 15'1 69711.22911 1S235 261056 16 ~157824 1 93 1714 1 16918 23288 16 1826.0 2615T7 17'153021 193714 i17 -170125'230767 117' 182848 -262190 18 -158S218 199255 18 -170832' 231295 18 -18066' 262601 19 -159-15 -199T95 19 -170539 231822 19'188383 2683122 23 I -158612' 20035 3 20'170746 2832350 I 20'183501'263638 21.1538J9 -250875 21 -170953 232877i 21'183719 -264154 22 1,5937 21 2)1415 22'171161'233405 22 -183937 264669 23 *15921-4'2019144 23 -171369 2338932 23'184156'265184 24 i 159402 2 249 1 24'171577 234458 24'184374 1 265693 25'159690'203032 25'171785 2.34985 25'18t593' 266214 26'159198'2035711 26'11993'235510 26'184812'266129 27. 159997'204110 27'172201'236386 27'185031'267244 23 -160195'204643 23'172410 1236562 23'185250 -267758 29 -165393'205186 29'172619'237088 29 -185469'268272 80'160592 2057257 30'172828'237613 30 -185689'268786 i 31'130791'206261 31'1738037 288139 31'185909'269300 82'160991'206800 382 173246 2838663 8 32'186129'269814 33'161190' 207337 33'173456 1 239189 1 33 186349'270827 34'161890 1'207874 84 1 13665 2-3971 11 384 1865701 270840 85'161589 I'20840 35'178875'240237 35'186790 2711853 6 1'1617189 i'2)8947 36 1- 174085'2-40763 36'187011'271867 87 1 161989' 2-394841 37 174295'2412S6 37 1 187232'272319 88 j162189'210020 3 38 114585'241809 38'187453'212891 39'162689'210555 39 1 714716'242383 39 l187674'273404 40 *162589'211091 40i'174927'242857 40'187896'278916 41'162789'211626 41'175038'248881 41 188117'274428 42'162990 i'212161 42'175349'243903 42'188339'274940 43'163191'212697 1 43 i175560'244426 i 43'188561'275451 44 1168392'213232 44 175772 "244949 44 *188783'275963 45!163594 187366 45 |115983'2454711 45 *189006'276475 46 163395' 214801 1 46 17T6195' 245993 1 46 189228'2769S5 47'1 63997 *214865 i47'1 764071 246516 47'189-450'277495 48'164198'215868 1 48 116619 - 247037 48'1 89678'278006 49'164400' 21a590 1 49 176882'247559 I 49'1 893961 278517 50' 164603'21647 1 50' 1711044' 248081 50 S 193120'279028 51'1648051'216970 51'1T725T1'248602 51'190344'2795,38 52'-165008 217504 52 -177,5T0'249124 52'190568'280049 1 53 i 165210' 218036 5| 56'|117723' 249644 58'1907192 280559 54'1654113'218 569 51 -177896 250165.54'191615'281068 55 1165616 21910 1 55' 17i819'250685 5'191240'281578 56'165819'219634 56 178822 251206 56'191464'1 282187 51'166028'23016T 51 1718536'251121 51 19169S'22596 58 j'166226'220699'58 187.550'252246 58' 191914'28!3105 59'166430 i 221231 I 59 1 18964 252766 591'192169'283614 60 -166633'2217614 60 179179 233286 60'192364'284123 37* 438 VERSED SINES. 33 DEGREES. 34 DEGREES. 35 DEGREES. Min. Nat. No. Logarithm_ Nin. Nat. No. Log-arit!m. i. i n Nat. No. Logariin. 0 161830 9207714 017C963 923291 018848 92513 1'161488 -20S140 1'171120 233314 1'181015'257714 i 9~21714 ) 0 I.1?C63 i ~2~3201' 0' 0180848' j9211 2'161647'208533 2'171289'233727 2'181182'258114 3 -1618050t 2908~992 3.171452.234139 3 181049'258514 4,161964 2109418 4'171614'234552 4'181516 *258914 5 -162123'209843 5 171777'234964 5 181683'259314 6 132832 1 ~210269 6'171940'235378 6'181850'259714 7 216230 210693 7 172103 25788 7 182018'260114 8 I162599'211118 8'1792661'236199 8 9182185 260513 9:'169758'211543 9 172430'2386611 9'823 63 260912 10.162917 21197 10 12593 237022 10'182520.261310 11 163076'212391 11'172756'238743 11'182688'261709 12 163235'2128124 12.172920 2'37844 12 182855'262107 183 -163395'213238 13. 173083'238255 13'183023'262505 14 163555 218662 14'1783247'288665 14 183191'262903 15 163714'214085 15'173410 239075 15 183359 i 263301 16'1638874 214509 16'173574'239485 16 -188527 263699 17 164033'214931 17'17338'239895 17'183695;264096 18 164193 215353 18 ~T1739912 240394 18 -183863 -234493 19 16453 2157 19 174066'240713 19 184031 i'264890 20 164512 -216198 20'174230'241122 i20'184199'265286 21:'164072'216620 21'1748094 241531 21'184367'265683 22 -16,4832'f217042 22 174558'241949 22'184535 266389 2.3'164992'217462 2'174723.242349 23 184704'266476 24 165152'21784 24 I'174887'242757 24 1848721'266871 25'165312;'218805 25'175051'243165 25'185041'267267'2 1'65478'218723 26'1752161 243572 i 26.185210'267663 27 6568 3'6 19146 27 175880'248980 27'185378'268058' 116594'219987 29,175544'244387 28,185547'268453 ~9 2 65954'21992T 4 9'175709, 24479J4. 29'185716 I'268848 30' 166114 1.200 I ~75874 1 245201' 30'185885i'269248 31 166275'221826 31'1789 245608 31'186054'269637 32 166436 221246 32' 176204 246014 32'186223'270032:: 166597'221665 8'17669 246491 33 186392'270426 ~ 166157'2,28 34'-176534'246827 84'186561'270820 35'166918'22502 35 176699 247233 35 100 271214'267079~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~3'176699-4i'42,38 36'167079 ~292992 36'176864'247639 86'1806900 271608 37 -1067240. 28343 37'177029'248044 87'187069 -272001 38 -167401 1 223758 i 38'177194'248449 38'1872388'2728394 39'1-675369 -224175 39'477360'248855 39'187408'272787 40'167723 224593 40'177525'249259 40'187577'273180 41'167885'225011 41'177690'249664 41'187747'273573 42'168046 2295427 1 42 l'77856'250069 42'187917'273965 43'168208 25845 43 -178022'250473 43'188087'274357 44'168369:22662 44 -'178187'250876 44'188256'247449 45'168530 206678 45'178353'251280 45'188426'275140'4168692'229794 46'178519'251684 46'188596' 275532 47'654'29' 7511'T -17T8685'252087 47'188766'275924 48'1691316'227927 48'178851'252491 48 188936' -276315 49'169178 22842 i 49'179117'252894 49 189166'276706 50'169340 228758 50'1791803'253297 50'189277'277097 51.169502 1'9229173 15'179349'253699 51'189447 277487 52'1690.63'i299587 62'179515'254101 52 1'189617'277878 53'169S26:'230003 53'179682'254504 53'189788'278268 54'16998S'2380418 54'179848'354906 541 -189958'278658 55'170150'230882 55 -180015'255308 55'190129'279048 56'170812'231246 56'180182'255710 56'190300 279488 57'170475'231660 57'180348'256111 57'190471'279828 58, 170638'282074 58'180515'256512 58'190641'280217 59'170800'202487 59'180681'256913 59'190819'280606 60'170963'232901 60'180848'257313 60'190983'280995 EXSTERN AL'SEC ANTS.:439 838 D)EGREES. 4 DEGIIEES. 35 DEGGREES. | Mlli. Nat No. Logarihin.. 0 01.92364 9284123 0 0 206218 9 O14:26 0 02207115 98348949 1'192589'284681 1'206455] 3814825 1'.221024'844438 2 -192814'285189 2'206692 3 815823 2 I 221272 3844927 38'193040'285347 3'203929'315821 3 221521' 345415 4'193266'2861.55 4'2071617'816819 4'221771 845904 5'193492'286662'5'20404'316817 5 222020'346892 6'1937188'287171 6'207642 83171315 6'222270'846881 7 I'198945: "287677 7'207880 8317812 7'222520'347870 8'-1941172 28818 8'2081181: 818309 8'22771'847858 9 *194399'288692 9 ['208856'81880(6 9'223C21 3848846 10'94625' 289199 l 10['2085941'319302 10. 228272 3848833 11'194852'289t05 11'288833'319 99 11 - 223523 I349321 12 1 95C080 293211 l 12'209072'320296 12.2238714 -349808 13.1953071'290718 1 -18/'2T9311'320791 18 3 2240255 3850295 14'195586'290224 j 14';2C9150 3212SS 14.224276 I'350782 15 /195768'291730 15f'209790''821785 15 224528'851270 16 195992'292287! 16'21080'( 322280 16 -.225780 1 85175 117 [196220'292742 I 171'210270 3'22777 17.225031'852243 18.196448: 293247 i:'18[;21C510'-32822 18.225284'852730 19.19'6677'29:3753 19.210750 323671 19'225536 353216 20 "196906'294258 1 20) 210991'324263 20 225789 537C2 21'197135;`294763 1m 21;21121 3'824758 21.226042'3854189 22 "1978364 295268 i 22'211472'8252a53 i 22.226295'354675 28'19759';295771 I 238 211714 8 325749 2 23'226548'855161 24'19T823;296277 24''2119551 326248 24.226801 385546 25'-198053'296781, 25'212197'32738 25'227055',56131 26'198283'297285 iI 26 212438'321232 26'227310'856617 27'-198513'297789! 217 212680'3277G 27'227564'357102 28 198744 98 8293 28'212922'32820 28'227818'857587 29'198974:298797 29;218164'328713 29'228072'858072 80.199235'299299 | 30'2184517 83292 i 30'228326'358551 31.199486'29988' 311 "213650'321901.31'22S581'359141 82'1996671'303307 82 218892'330194 0 82'2238837'359526 833 199899 8300809 1! 83'2141385'33088 883 229092'8360011 84'230130'301312 3j 84'214389 8331181 4'229348'860495 85'200362'-301815 835'214622;83316 4 35'229604'86G979 86'20594 302317 1 86 214866;382167 1 6 229860'861463 87'2083326'-302829 37'215110'332059 87;230116'361947 88'211058'308822 08'215854 833315,2 88.230338'3(62431 89 -231290' 3303823 39'215598''33344 89'230630'362914 40 201523;304325 40'215842;334136 40 -20S0886'868898 41'201756'304827 41'21608T |334629 41'231148'363881 42'201990'305828 I 42'2168832'35121 42'281400'364864 43 -232223 38058830 43 216577' 35618 1 43'231658'364847 44'202456'3806331 44'216822'8136104 I 44'231916'385330 45'202690'3066832 45'217108'833395 415'2'32173'3365812 46'202924 "307383 i 46;'21781 337086 46'282431'8366295 471 203158 30T3884 i47 217559 83777 47''232690'866778 48 2083392 38083384 48'21136 3380 69 48'232949'367260 49 2313626'380884 49:218052| 388560 49'2833207'367742 50'213G861'3809'334 I50 218299 33950 i 50'2338466 3868224 51'204096'839834 51 218545' 339541 51'2338726'3G8706 521'20381'810334 52'218792 340031:52'233985 369188 53'204568'810834 5Ii'2190 340522 53'234245'369670 54 -20481 3O'8338 54 219287 341012 54 234505'370151 551'20508T'311882 1 55'219535 34l502 i 55 284764 8310632 561'21J5273'312831 56 219782 341991 56'.235025 8371114 571'25509 3812830 1 57'220030 342481 571 23285'871595 581'215745 i'318329 58 2202781 342970 i 58'235546'372076 59 2159A81'81382S 59-/ 220596 04 6 843460 59 |23580. 872557 60'206218 3143826 660 2207 1894 1 9' 235831 33031 440 VERSED SINES. 86 DEGREES. 87 DEGREE3. 38 DEGREES. Mlin, Nat Ni L|o gar.itlm liin.Nat. No. |Logarithm. rin.[ N at. o. L i,ga-i:,m. O 0 1911983 9 280995 0 021365 9030983 0 0-211990 9'326314 1'191154'281384 1 -231540'304361 1 -212169'326681 2'191832'281772 2'201715'304787 2'212348 3827047 8 -191496.282160 3 -20190. 305115 3 *212527.327414 4:191667 282548 4 202065'305492 41 212706 -827780 5 *191889'282936 5 202240'3058.68 5 *212886.328146 6 19C2010 2838324 61 202416'306245 6 *2130655 328512 7 *192181'283711 7'2022591'306621 7 *218244 3828877 8'1928353'28IC99 8 202767, 306998 8 -213424 3829243 9~1 192525'28443( 9'202913 *307374 9,.213603.329608 10'192X796'28483' 10 213118 *307749 10''213783 *329974 11 *:192868'285260 11'2383294'308125 11 *218963 8033089 12 -193040 *285947 1 2'203840.308501 12'214143 3830704 13 1938212'281033 13 203646'308876 13 214-323 331069 1-1 19:3304 * 286419 14 203822'3C9251 14'214503 3381433 1 5'193555 *286805 1 5 *208998'309625 15 *214683'33179S 16 193727'28719 16'204174'310000 16'214863'332162 17'193900. 2875 76 17'204350 310375 17 215043'33882526 18'194072 -287962 18 2t4527'310749 18 215224'332890 19.194244.2358347 19.204703'311124 19 *215404.333254 20 194416 2887322 2)'2048380 *311498 20 *215585 *333618 321'19415883 289117 21'205356]'311871 21,215765'338980 22 *194761'28953)2 22 -205237'312246 223 -215945'384344 23.194934'289887 23'205-09 *312619 238 216126.334706 24. *195106 *291271 24 205580-'312993 24-'216307.335070 25.195279'290655 25'205762'813366 25 *216487'335432 26.195452'291039 23'205989 3137-38 26'216668 335795 27 195362'291423 27'26115'314111 27'216849'336157 1 28 195797'291806 28 206293'314484 28'217030'386520 29 *1 95970'292193 29. 206470'314856 29 217211'336882 80.196143 *292573 30,20664T'315228 30'217392 3387244 31 193G316 -2929056 031 *206824,'315601 31'21,7578 337605 32 1949,0 1 32938809 3 82 237001'315992 32'2177M'337966 33.8 196663'293721 33'207178'3163844 33'217985 3388328 84'196836 i291104 1 84,237356'316716 84'218117 338689 85'197009 294486 85'207533 3817087 35'218299'3389050 86.197182 *294168 36'2077iO'317458 36 1'218480 -339411 37.197356 *295250 37'207880'317829 37'218661 3389171 38'197530'295332 38,208066 818200 38 [218843 -340132 89.197(03 29o013 39.208248 -318571 89.219024'340492 40.197877 296395 40 208421'318942 40'219206 3840802 41.198051 293776 41'208599'319311 41,219388 3841212 42'193225'297157 4'2 208777'819682 42'219570 3415783 48'1983938 29T537 43'208954'320051 438 219752 8341932 44'198572 297918 1 44 209182,320421 44 -219934 *842292 45 *198746 1'29329S 45'29809 320791 45'220116'342651 46.193920 293679 46,209489'321161 46'220298 -8343010 47 -19913;91'299359 47 209667 3821530 47'220480 3843369 48 1: 99269'299489 48'20'9845 821899 48 -220662 3843728 49'192443'299819 49'210023 322267 49'220844 344086 50 199617 3800193 50'210202 3822636 50'221027'14445 lo51 1 1)997-92 3 300577 51.-'210380'323005 51 221209 *344803 52 |199966 |300956 52'210559. 3'823373 52:1'221311 8345161 53 200141 0188335 53'210738 8238742 58 1'221574 8345519 54.'200315 8301714 54,2109161 -324110 54'221757T'845877'55. 205s90.302093 55,211095'324477 55'221940'346285 561'2;)0665'302471 56.2112783'824844 B56'222122 1346592 57'200840'302850 57'211452' -325212 571'2223051'346950 58'201015'308228 58 -211631'325579 58'222488 -347830 59.201190 30)3605 59 -211810.825947 59'222671'347664.60'201365'303983d 60'211990 326814 601'222854'348021 EXTERNsAL SECANTS. 441 86 DEGREES. 87 DEGREES. 88 DEGREES. Ailln. N. Nt.o. Logalitlim, Ntl. | Not. No. Logarill | lil. N at. No. I,ogaitlam 0 0-236068 9-8730837 0 0-252136 9.401634 0 0.269019 9-429782 1'236380'3783518 1 1'252410'402107 1'269307 -430248 2'236591 8373998 2'252685'402579 2'269595'4380712 8'2368538 8 74478 8 -252960'403052 3'269884'431178 4'237115 3749o58 41.253285 403525 4'270174 1431643 5 *237377 83754438 5 253511. 403997 5'270463'4832108 6'237640 3875918 1 6.253T87'404469 6'270753 432a573 7'237972 8376397 7'25406'3 404941 7 2711042 4833037 8'28816,5'376877 I 8 254-389'4085413 8' 271332'433502 9'238428'877857 1 9 254615'405884 9 2716823 483967 10 238691 -37786 1 10] 254892 -406855 10' -2171914'434482 11 *238954 *378315 11 I'2 5169'406827 11 1272205'434896 12 289218'878794 I 12 254416 -407299 12 *272496'485361 13 2239482 *879273 1 18'25528l3'407770 13'722788 4835825 ~14 - 239747'379752 1 1 4 2n5603'4.8241 14 -273081)'436289 15 9240011'380231. 15'2562718'40S711 15''273382 -486753 16.. 240276 8380709 1 16 256o56'409182 16 278664'487217 17~'240540'381187 1 17'256884'4096358 17'278956('437680 18 240805 3881666 i 18 2571181'410124 18'274249]'488144. 19 *241070'382143 19 257392'410595 19'274542 438608 20~ 24183 5'382621 120'257671'411065 20'274835'439072 21 -241601'388099 i 21'2o57950'411534 21'275128'439534 22 -24186T'883577 22,'2562823'412006 22 275421'439998 23~'242183'884055 23'25858 9 /'412475 23'275715'440460 24 -242400 38845382 24'25ST89'412946 24'276010'440924 25~'242567'3885010 25'259169'413415 25'276804'441386 26 -242933'385487 26::'259349'416884 26'276598'441849 27'2438200'385964 27'259630'414354 27'276893'442311 28'243467'3S86440 28'259910'414824 28'277188'442774 29~ 243735'386918 29'260191'415292 29`277584'443237 830 *244002 ~'387394 3 80'260472'415761 80'277780'443700 31'2-44270'387871 31o9'26(0754 416231 81'278075'444161 82 244539 3888347 1 32'261035'416699 82 -278370'444623 83 244807.'388823 o1 o3 *261817'417168 33'27S867 4445085 34/ 245075'389299 34 261600 1-'417638 8 4'27S963'445547 35 24,5344 -389775 85'261828'41S106 85 279260'446009 836(; 24513'390251 861.26916.'418374 8i6 279557'44647-1 87 -245882 390727 7'262448'419043 387 279855'446932 88/'246152'391203 i 88 -26278I1'419511 83 289152'447393 89 1'246422'891678 39 268015'1 419980 39 288450'447854 40 1 246691'392154 1 40'263298' 420447 40'280748 1448316 141'246961'392629 411'263581'420914 41'281046 |448777 421'247273'893104 42'268865' 421382 42'281845'449239 43;1'247502'393579 1 4 264150'421819 43'281648 449699 44 247773'394054 44''26443 1 422318 44'2S1912'450160 45 1 244044 94528 1 45 -264720' 422785 45['282242'450621 46['248815'395003 1 46'265005'423253 46'282541'451081 47/| 248587'8395478 47'265290 423720 47'282341'451542 48 1.24S859 1 95952 48'265575'424187 48 283140'452002 49 9'249031'396427 49'2658601'424653 49'283440'452462 50'249403''396901 50'2661461 425120 50'283741'452922 51 2496751 39734 51'266482'425587 51;284042'4533882 52:/'249948 8397848 52'2667191'426053 52'2A84343'4538842 58 - 250220'8398321 53'267006''426520 53'284644-'454302 54.'2504938'898793 54 -267293: 1'426987 54'284946''454762 55 -'259766 399269 55-'267589)'427452 55'285248'455222 56 | -21040'8399742 56 -267867:'4279148 56;2855350'455681 1571 251314'400216 57 -26815t-'429884 57'285852'456141 58 -251,588'40689 58'268'-42'428850 58'286154'456600 -59'251862'401161 D9 -'268 80'429316 59;286457'457059 60 292136'401634 60;'269019 |'429782'560 -286760'457518 i ss- ~~~ilsa~l 4011~ 69):~(is;0: ) 42936 1159 1;2S651'4571518 442 VERSED SITNES. 89 DEGREES. 40 DEGREES. 41 DEGREES. in N t No, | gaiti | li|. Min. Na INo. ooali lim A llin. Nat. N | Legalitlml. 0 0.222854 9.848021 0 0,288955 98369133 0 "0245291 9.389681.1 22.3037.48377 1 234143 369480 1 -245481 390018 2 228220'348734 2 284330'369827 2 -245672'39C856 3 223404'349090 3 234517 370174 28.245864'890694 4 22358.349446 4 2834704 870520 4.246055.891031 5'228770'349802 5'234891'870867 5.246246'891368 0 223954 8350158 6'235079 3712138 2464387'91705 7}'22413871 830514 7'285266 3871559 7 -246629 892042 8'224320 3,5G89 8.235454 371905 8 -2468201.-92379 9'224504'351224 9'235641'372251 9.247010 8392719 10'224688' 8351579 101'235829'372597 10 2472( 2'93052 11'224871 3851984 11'236016 372942 11 247894 893388 12'225055'8352289 12'2836204'873287 12 *247586'938T25 13 *225239 8352644 13'23 6392'3873682 13 247777'394060 14'2254231'352999 14'236580'378978 14 -247968 *3'894396 15'22567 353358853 15'236768 3748322 15 "'248160 *394732 16 225791 8538707 16'236956 1874667 16 - 248852 3-895067 17'225976'354062 17'237144'375011 17 ]'248544 ] 8395408 18'226160'354415 18'2387882! 875855 18 248786'895738 19 226344'854769 19'237520'375700 19 *248928 396074 20'226528 3855122 20'237708 3876044 20'249120 [ 396408 211'226723'8550i76 21'287897'763888 21'2498312'396748 22'226898 8355880 22'288085'3767831 22'249504 9397078 23O'227082'856183 23'288273'377075 23'249696 397412 24'227266'856535 24'288462'877418 24'2498891'97747 25'227451 8356888 25'288650'377762 25'25 082 83980S2 26'227636'357241 26'238889'878105 26['256275'398416 27'227821'85759-3 27'289028'3878448 1 27'250466 3898749 28'2289035'85T945 28'289216'878790 28'250659'899C83 29'228190 8'35897 29'289405'379133 29 250852'099417 80'228375'858650 80'289594.'379476 30 1251045 899751 81'228560'35900 1 81'289783'879818 81'251237'400083 32'228746'859853 82'239972:.80160 82'251430'40C417 83'228981'3897043 83'240162'380503 83'251623'400750 84'229116 -360056 34'240351' *388845 84'251816'401C383 85'229301'860407 85'240540'881186 1 85'252009'401415 36 8 229486'360757 86'240729' 881528 86'252202'4(01748 87'229672 -8361108 387'240919 881870 11'252395'402C81 38'229857'861459 88 i 41108'882211 88'252588'402413 89'230043 83618039 9'241297'382552 839'252782'402745 40'230229 8'362160 40'241486'3882892 40'252975'408077 41'230415,362510 41'241676'38328833 41'253168 -4034(9 42.230600.862860 42'241866 883574 42'2583362 -403741 43'230786 3638210 43'242056 388915 43'258566'44073 44'230972'363559 44'242245'884255 44'2538749'404404 45 - 231158 3639909 45'242435'8 84595 45'253942 4C4785 46 231344'364259 46'242625'384935 46'254186 405067 47 1 231530'364608 471 242815 1 85275 47'254330 405898 48['231716]'864957 48'243005'385615 48 |'254524 405729 49[1'2319031'65306 49'248195' 85955 49 i'254718'406060 50'232089'365654 50'243385 1 3886294 50'254912'406890 51'232275'366003 51'243575'886683 51'255106'406721 52'232461 8663851 52'248766 886973 152'255300'407051 53'232649'366699 53'248956'887812 53'255494'407881 54'232835'367048 54'244146 387650 54'255689 -407711 55'233021'867896 55 -24433887 87989 55'255883'4C8041 56'233209'3867744 56 -'2445281'888828 56'256077'408871 57 2388395'869991 57'244719'388667 57'256272'408C01 58'233582'368439 58'2449(9'889004 58'256466'4C9030 59'233769'368786 59'245100 18893483 59'256660'400859 60'233955'3691833 60'245291 8389681 60'256855'409688 EXTEnNAL SIECANTS., 443 89 DEGREES.' 40 DEGREES. 41 -DEGRTEES. Min. I Nat. No. Logarithm, in. N at. No. Logaritlhm Min. Nat. No. I.,gritlm. 0 0286760 9'457518 0 005407 9484879 0 00325013 9'511901 1'287063'457977 1 8305726'485332 1'3825348'512348 2 2878366 458486 2 8306045'48578D' 2 3256S4'512796 3'267m0.458895, 3 306364'486238 3 i326020.513244 4'287974'459358 4.306684'486691 4'326355'518691 5'288279'459812 5 3p307004'487144 5'326692'514138; 2388583'460270 6'307824 487597 6'3827029'514585 7 288813'460729.7 307644'4188049 7'3827365'150832 8'289193'461187 8.307965'488501 8 - 3'27702'51548) 9'289498'461645 9 302833'488954!9 *32840'515926 10 28988 462102 10.308607'489406 10'328378'516374 11' 2901(09 462560 11.303928,489858 11'328716'516820 12'290415'463018 12.309250'493810 12'329055:,517268 13 290721 *463476 13.809572'493761 13'329393'517714 14 291028'463934 14 8;309894 491214 14'*329781'518160 15 291335'464392 15.310217'491665 15'3G301070'518607 16'291641'464849 16.3i10540'492117 16'33 88)410'5190353 17 -291949'465307 17 83108603'492568 17'33807513'59499 18'292257,465764 18 8311186'493019 18 33881093 519945 13'292564 -466221 19'-311510'493471 19'331430'520392 21'292872'466678 20 3811894''490923 20'331770'580837 211'293181'467135 21 -312158 4948374 21'332111 521284 22'2993491)'467593 22'312082',494824 22'332452'5217,30 23'23798'468149 283 -12817'495276 23'3327941 522175 24'-294107'46.8505 216 *8313132'495726 24 8333136 522621 25 291-41T'468902 25 -3138457'496178 25'8333478'5238067 23 -~97297' 469419 7 26 3i'3782'49G628 26'' 33823)'523513 27/.2950U36'4698S5 27.314108'497079 27'334163'523958 28'291346'470831 23 831484'497529 28'384506'524403 29.295656'470787 29 8314760'497980 29 83348491'524849 830 295967'471244 30.315086 -4984j0 380'335193'521295 31.296278'471699 311 8315413'498O88 31'335536'525739 82.296589'472155 32 8315740 1'499330 32'3883 5'526185 883.293900'472611 33 8316068'499781 383'336324'52'6629 84 97312'48067 34'316396'500231 34',36569' 527074 85 -297524'478522 35 38167284 |500681 35'336914 1'527519 86'.297383'4713977 86'317052'501181 36'387260 1'527914 87:298148.474482 837'317881.501581 3T.833605'528-409 88 -293461'47188 8 8 817711 502831 38'33'790 5288583 89'298r74'475343 39'31838 9 1 502483 1 39'3382961'529297 40 -29938 3'475798 40 3818368'502929 1 40 /'338643G8'529742 41'299401'476258 41;318G'697'503873 i 41 I'338990'580186 42 1 299715'476708 4 42 8,319327'503828 42 1 339337'530631 43'300029 477168 3 48'319356'504277 43'839GS1'531075 44 1 300343'477617 44 31936871 504726 44'840031 531519 45'300658'478072 45'320018 |'505175 45 |'340379'5319683 46'300973'478527 46'320350 -50624 46'3840723 1'532408 47 301288'478981 47 -320681'506073 I 47'341077'532551 48 801603'479485 48'3210131 506522 481'3414251 5338295 49 801918 |4798S9 49 321845 |' 506971 { 49 31 341774 |'533739 50 302234'480344 50 8321677'507419 50 1'42128'5384182 51 8302550'48798:551 322009' 507867 i 51'3842473'584626 52'302366'481251 52'322342'508317 5 2' *342823 |'565870 53.808183'481705 5I 3'322675'508765 i 53-'3438173 53g5518 54' 393500'482,159 54 323008'509212 1 54 3435283 5-5956 55 303818.482613 55 32338841' 509661 55 343874'536400 56'304185'483066 56. 3283675/'510109 561.344226'56843 571'304452'483519 57['3249091'510,558 i 57 8344578'5 87287 58 18304770; 1'483973811 581'3'8248431'511005 58 4-4929 537729 59'305089'484426 59'38246781 -511453 59,'345281 I'538172 60'305407'4848479 60)'82508 13'511901 60'345683- 588615 444 TEIISIED SINES. 42 DEGREES. 403 DEGREES. 44 I)EGREES. Mill | Nat. No. Lognali hr. h in! Nat L'c. | Logaaritllm. | hlin. | Nat,o | Logarithm. | 0 0.256855 9.4C9381.0 02 s8040 9.429181 0 02808660 9-448181 2050 t 1 22 181001 28 4292 1: 2808628'448493 2.257245 410G46 2 2090'43'429822 2.281064 *448806 3'257439'410674 8i 269242 4380142 281266'449118 4'257634: 411003 4 269440'480463 4'281469'449431 5 237829'411332 5 2696839'430783 5'281671 449748 6 2358024.411668 6 2G9S8.481103 6 2818741.450055 7 -258219'411988 T 270037 431423 7 2S2076 /4350366 8 258414'412317 8 "2702306'481743 8 282279'450878 9 258609 5 412644 9'270434 }482061 9'282482'450990 10 258805'4129 72 10'2706331'482381 10/ 282684'451301' 11.239000'413299 11 270883'482701 11'282887'451613 12'259196'4138628 12 271082 I' 433020 12,283090'451924 13'259091'413955 i 13 i2712.31'483339 13'283293'452235 14.259586 414282,1 14'27140 4'3G8657' 14'283495'452545 15'259782 414609 ii 15 27,1629'483976 15 283698'452856 16'2599771 414935 1 16'27.1828/!434295 16'283901'453167 17'260173'415262!17'27.2028'4484613 17'284104'4584-77 18'260369'415889 18'272227'484931 18 2S84307'453788 19 260565 415916 19'272427'4835250 19,284510'454098 20'260760'416242 [ 2'272626'435568' i 20'284713'454408 21'260956|'416568 21'272826'485886 21'284917'454719' 22'261132 416894 22'273026'486204 22'285120 -'4,55028 23'261348 417220 28'27. 225'4236521 28'2853024'435388 24'261544'417546 24'273425'48889 24'285527'455647 25 -261740'417871 23'273625'487156 25 285731 1' 455957 26'261937'418197 I 26'273825'437473 26'285934'456266 27'262188.'418522 27'274025 487790 27`'286138 456576 28'262830'418847 28'274223'481'071 28''-286842'456885 29 2623526'419172 29'274425 i408424 29 28 286546 -457194 30'262722'419497 - 80'274625'488741 80,''286750'457508 31'262919'419822 3 81'274825 }4390957 31 286953'457811 32'263116'420147 32'275026 4309374 82' 287157 4F58120 83'268313'421472 8 33'275226'439690 88'287361'458428 34'263510'420796 84'275427 440006 34' 2875658'458736 35. 263706'421120 1 85'275628'440323 35'287770 459045 36'263903'421444 36'275828 144306'39 36.287974 4359353 37'264100'421768, 37['276029'4410954 1 37'28S178' -459661 38'264297'422092 9 / 38'276280 441270 38'2880838'459969 89'264494'422416 8 39'2764301'441585 391 2SS587'46C277 40'264691'422739 i 40'27(6631 441901 40'28S792'460585 41'264888'423062 1 41'276832 1'442216!4 41'288996 460892 42'265086'4233868 42'2770383'44251 42 -289200'461199' 43,'265283'423879 43'277234'442846. 43'289405'461506 44''265480'424032 il 44'277435'1438161 441'289610'461813 45' 265677'4243 54 45'277636 4i8347a' 45'2S9814 462120 46''263875 /'424677 11 46'277837'443790' 46'290019 462427 47'266072'424999 47'278039 444105 47'290224' 4627'34 48 -266270'425822 48'278240/ 1444419 481'2904-29 -468041 49'266468'425644 49'278441- 444784:/ 49 2906834''46'8347 50'266666'425967 |i 50'278642' 445047T / 50'290839 -'463653 51'266863'426289 51.278844'445361 51'291044 1 4609359 52'267061 -426611 52'279045 445675' 52'291249'464265 53'267259'426932 53'379247'445989 53'29R14541'464571 54'267457'427254 54'279448'446302 54 -291660 464S77 55'267655'427576 1 55'279650 1'446615 155'. 291865 465182 56I'267853'427897 56'279852'446929 56 2'292070 465488 57'268052'428219 57'280054 447242 11 57'292276'465794 58'268250'428539 58'280256'44,7555' 58,'292482 466099: 59'268448'428860 59'280458'4478,68 [ 59: 2926881'46644 60'268646'429181' 60'280660' 44-81:1 60'292893 466709 EXTErENL SECANNTS. 445 42 DEGREES. 43 DEGREES. 44 DEGREES. Min. Nat. No. Logarithin. M ill. N at. No. Imall. ithm. Mll. Nat. No. Logariflim. 0 08456833 95388615 0 0'0367328 9'565C54 0 0390164 9501247 1'349085'539(57 1'367699 565492 1 -'9554 -591681 2 343938'53950 2'868070'565930'2'39(945'592116 3'846691 539942 3 368441'566368 3 991336'592550 4 947344'540885 i 4'368813'566897 4 891728'592985 5 92478T99'540828 1 5'369186'567245 5 392121'593420'384T773'541270 6 369559 567684 6'892513'593854 7 848107'541712 7 369932'568122 7 3'92905'594288 8'848462'542156 8'70805'568560 8'393298'594722 9 948817'542597 9'370678'568997 9 839892'595157 10 349172'54393 10'371052'5694385 10'94085'595590 11 949528 3543482 11'371427'569674 11 83904479 3596(24 12'849884 *543924 12 371801 570311 1 2'694874 596459 13'350240'544366 13 *3O72176 5070749 1 13'3095209.596893 14.35o397 544808 14 9872551 5711.86 3 14 5895664'597626 15.330934'545249 15'872926 *571623 15 396305 9 597760 16 935110-.545699 16 -8783(32 572061 16'396555 *598194 17 T 331667'546332 1 7 -373679'572498 17 396851',598627 18'352025'546374 18'37433 5 579985 18 839724T7 3599061 19.852384 1547016 19.374432'578373 19'3N97644 *599495 20 52742'47457 20 7489'578810 20 98041 99928 21 -33100'547898 21 375187'574248 21 398439 60862 22 853459'548899 22 3795565'574685 22'0898887'600795 23'835818'518780 23' 379-13'5759122 23 ~99245.601229 24'354178'549222 24'3T3321'575559 94 I 99933'601661 25 354538'549663 25'3816700'57599 25'400082'62C59 26'354898'550104 26'977079 -'57 64382 0 26'40(481.'602528 27'.35258 559544 27'77478'56808 7 27'4 0481'.602962 28'355619'530985 28'877898 5706 2 421'60290 29'3989'51425'29'978918'5777421 29'40101'679828 80'353641'51866 30'38290'58179 30'402C082'604261 81'936703'552307 81 378979'578615 81'402489'604093 92'937065'552748 32'979360'579832 32'462834'665123 383 35742'553189 83'819741'59488 08'403235'605539 34'857790'553629 84'948123'579924' -14 408637'603991 35'386153'554069 85'38305'580361' 5'4C4C40'606425 36'385816'554509 96'380888'5835797 6'4C4449'606851 37'358883'554949 87 851270'581253 1 7'404846'607290 8 35924 55389 38'981633'981609 38'456249'6(107722 49'359608'555829 99'382033'382105 89 405653'6(155 40'859972'556269 40 382420'582541 40 4016058'608588 41'360337'5561709 41'382804'582977 41 4042'60.9020 42'836702'557149 42 38188'3583.412 42'4C6867 609432 43'361068'557589 43 383573'583848'407272 -69884 44'361433'5589280 44'383958'584284 44'407678'610316 45 -'361799'558467 45 384343'584719 45'408084'610748 46'862166'538917 46 984729'.585153 46'408490'611181 47'362522'589346 i 47'8985113'585591 4'48996 611613 48'362899'559783 48 985501'586028 48'4C9463'612045 49'363266'560225 49'385888'586462 49'409710'612477 50'363634'560665 50 386275'586893 50'410117'612908 51'364902'561104 51'886662'587332 151'410525'619940 32'364-370'561543 52 387950'587767 52'410984'613772 33'3647T39'561982 53 387498'588293 53'4110342'614203 54'365108'562421 154' 387826'588687 54'411751'614695 55'365477'562860 55'388215'589702 51 5'412160'615306 56'365847'563299 56'388604'5895071I 56'412570'615498 57'366217'563738 57'388993'389942 1/ 57'412981'615930 58'366587'564176 58'389383'593377 58'413391'610361 59'366957'564615 59'389773'590812 59'418802'616793 60'367328'565034 60'90164 391247 60'41.4213'617224 38 4-16 ns) S E. 45 DEGREES. 46' DEG-EEGS. 47 DEGnREES. TPi-i. I Nat. No. ILoga ithm N at. o. Logaliti m. Blin. Nat. No I Logalithm. 02923993 9466709 o 0'05342 9'484786 0 0 318001 9'502429 1 2993099'467014 1 05551'485083 1'18914'502720 2 293305 467819 2 615760 485081 2 018427'503010 3'293511 467624 3'3050970'48s5678 3 818641'5503800 4 -2963717'467928 4 -06180'4S5976'4 818858' 50850 5 298923'468233 1 0 1036089'4862763' 80190166 0503883 6'290129 46S537 6'306598 486570 6 319279'504170 7'294135'465842 7'306808;486866 7'319492' 5044 60 8 994541'469146 8 o37017'487166 8'319705 514750 9 291747 469149 9'307227'487460 9'319918' 05089 10'29495'4697583 10'30747'T 487757 10'20132 5005829 11'295119'470(356 1'307647'488056 1'320345 505618 12'295365'470360 12'307857'489849 12'320558'505907 13 2915572'470663 13'308067'488645 13 -020772'506196 14'295778'470936 14'808277'48S941 14'320986'5(6486 15'2909805 4712i0 15 688487'489237 15'321200 506775 16'293192 4715736 16 -308697'489582 16 321414 0507064 17 296393'471876 IT17 38937 489828 17 321627'5073852 18'291615'472179'18'391l8 490124 18'21841 5197641 19'296812.472481 12'8309328'490419 19'322054 007929 20'29:019'472784 23'39038 490714 20'322238' 508217 21 297226'473087 21 3G9748'4910C9 21'32242 508505 22 2974383'473389 22 609959'491305 22'322396'508793 23'297640'473691 23'310170 I'491600'23 322910 569081 24'297847'476993 i 24'310381'491895 24'323124 0519369 25'290304'474295 25'310591'492189'25'2308 5(;9657 26'293261'474597 26 310801'4924836 26'32552 05C9944 27'298468'474898 27'311012'492778 27'320766 510281 28 293676'475200 28'311223'4963072 28'323981'510519 29'29833'475502 29'311434'493866 29'324196'510807 30 299490'47538)3 830 311645'493660 30 824410'511094 31 299298'476104 31 311856'493950 31'324624'511381 32 4299506'476400 32'812068'4942419 82'324839'511668 83 299703'476706 383'312279'49454: 33'3250303 011955 34'299921'47970 314 3812490'494836 34'320260 512242 35'3003129'47700 30'5'831972'1491300 35'32433'512529 36'310367'477M69 3 6'8192913'490542 36'325698'012815 37'300545 F 4T9 37'315124'498716 37'325912'503101 38'300 1 02' 847829'33 313355'496309 38 -326127 5138 39'3105960'478519 39'831354'4963ID12 39'3268342'51336 40' 41168 5:T 40'8158779'49596 40'826517'51I894 41 -3013S76 -47~9109 1 41 -31.8970 1 ~196888 I 41 -M,6772'514245 42'301084'479-119 42'314182 497183 42'326987'014031 43 301793 479709 43'314893'497474 43'327203'514817i 44'302301'484)08 44'31460'5497766 44 -327418'510102 45'802210'4819608 45'314817'54958 8 45'3274388'5138 46'382418'4837 46' 2'315029 -3495830 46 7S'5105(7 47.312626 543 37,9O 47'310241 4986-42 -47'28060'5101)05 48' 332835'481236 11 48'83164043'498934 I 48'312S2SO'5(69-1244 49~'303043'4815510 -9'310665'419226 49'328495'510(029 53 3C39,52 481854:i 50'810877T 499018 00'328711'0168145',A - 6 ~920 51'3134'19'49989 51'823926 -'519 7 8 12'31370 482111 02'3816801'500131'52'829142'0517383 03'D33S78'482699 653'316513'53 1G0l2 53'32935S 0617068 54'334387'482998 04'31672726'5 08684 54'329673'0517912 05'314296'433296 Ir 55'316939'509976 0 5 3'020739'0 1836 56'304505'483090 560'17102' 081967 056'330005'518121 57 3804714'483893 07'31736- 31'001007 50'330221'018 KS 08'319203'464191. 58'317076'031848 I 8'330437' IN0,191 09.'310132'4 8461 8 8 59'3177089 -'5 2139 59'3313653'019373 6061805342'484786 60' 318901'032429 60'338,869'019606 EXTERNAL SECANTS. 44' 45 DEGREES. 46 DEGREES. 47 DEGREES. Mro. Nit. No. Logaritblin Aili. Not. No. Looo)itboo Min. INt, No. LTogaroitn. 0 0414213 9.617224 Ii0 0439557 9.643,15 0 100466279 9'668646 1'414625 -617655 1 439991'643444 1 466737 669072 2,415i37' 618087 2.4450425. 64872 2.467195'669493 8 415450'61S518 2, 440859'644300 3 467653 669924 4'4158663'618950 i 4.441294'644728 4'458112 -670350 5'416276'6193S1 5'441729'645156 j5 -465T1 -673T776 6'456689'619311 6 442164 645534 6 -469330 671221 7'417162'625242 7.4412650 -646012 7 -469490'716M27 8 417516 -623673 8 4-143057','3 646440 8 469951 672(.53 9 -417920'621(694 9 o 44347 5 -646869 9 -47041,2 672478:10 -413345 621535 10 -444912 *6472 97 10 -470873'672914 1'41370 621965 II 444350 *647725 11.471535 675560 12 419176 6229 i 192 444788 643150 12.471797'6753755 13 -419592 -622323 10 -4 45226'643581 13 -472260 -674181 14 ~421,038'6262o7 14 5665 64919 14 -4T27T23'674657 15 423425 -623688 15 -4146105 -649437 15 47137 -675066 16 -42t1842 62-4119 1 6.446544, -649364 16 -473650 -675458 17 -421259 -6246149 17 -4-16981 655292 17 474114 -675883 18 -421677'624980 18 -447125 650726 I8 474579 -676619 19 -422095'6254lu 19 -4,47865 -6,3,1147 19 4-75544 676764 20 -422513 -625840 90 -443836 -651574 90'47552.9 677159 21'422932'626271 21 4144748 3652062 21'47,5975 -677584 22 423351'626701 1 22 449190 "o5248f. 22 -476442 -678010 263 423771 -627131 I 23 449632 -652857 2.3 -4T6968 -678465 24'424191 -627561 25 -45)075' 653285 24 -4T753w'678860 25 424611 627991 25 450518 6553712 25 -477S463 679285 26'425161 -62849 1 26 -451961'6541 89 26 -478311'679710 27 j'425452'628851 27 -4951455 654,567 27 -478TT9 683165 28'425874'629281 1 28 451850 -65,1994 28 -479248'680560 29 *426296 -629711 29 -4522)94 655421 29 -4T971S'68936 60:426718:630141 j0 % 4"32529 655848 80, -483166'681411 81.'427141 1'630571 816 -4915'656276 81 -48,698'681836 32'427-563'631000 69 453631'6567063 32'481129 6U82261 86'427936'663 043 D 66'454,77'657160 63'481600'682686 34'428110'651860 64 454524 657557 65T 4'4S22)71 6813111 o5'42S335 -6322905 8 56 494971 657 1 5 4325-43'6835.36 66'429160'6311 12 36 4955419 65641 1 36'4335,1 5 -6889'60 oo7''65314 67 45583(7'6533SS38 67'4S3347 6434335 63'430129'633578 88 456315'659265 83'485950'60433 9 43345634',664008 689 456764'659392 09'484433 685235 41'444970'661437 40'452163'662 119 40 484937'635658.41'4315 86'664366 41 49176662'66554.5 41'435631'686283 48'-5563 4 1'6365295 4-49'4,0- 8112 -6 69 72 42'48356'68)658 46.'4322.39'685724 43'45S562'661593 43'4S36362'86 933 44'43))2367:'666154 44 459318'661825 44 -436316lS'687058 45.'-133:95'6665844 45 -4b9464 662251 45'4S3728'687782 46'455526'617512, 46'459915'662678 46'487760'688206 47'435'951'637441 47 460367'665104 47'433237'688531 48'4634530'67874 48 -464820'663531 48'433714'6893)55 49:'434310'658299 49'4612763'63958 49'439192'689479 50 4365239'633728 50'461726'664181'435U49675'689954 51'44,5669'619157 If 51'462179'664310 51'490149'6926328 52'45'6100'683956 I5952'462632'665236 52'491623'691752 53'4465(30'640014 3 5'463087'665663 53'4911S'691177 54'456961'649446 54'466542'6663389 54'491583'691601 55' 44 -7"93 1'642872, 55'463-1998' 6 6 6 51 6 55'492369'692326 56'437,3'641301 56'464,453'666942 56'49250 692450 57.' 4;28 64170 5'T 464906'667368 57 495061'692874 58 -4313869 1'642158 53'465365'667794 58' 4 9 355 1'696293 59'439123'642536 59.4622'332 1:59'496595-.693722 460 9557'645015 60'4662'i9'66346 61 444'694146 r -.t-.t-. 0. 0 F CO C.1....... 0l r.. CD CO.o X 0 0 X 0 tCO 0 - o F-... CO0 0. COCI _ COCOCtC~~oOCOCOCOCO COO 000000000z~~~ CO son0t-Co o o o c0000 x o o o o o o o cloO C' G - F cLa 0O CO | O -e i ~ * vo G > > U3 tz ) = > > ~ ~ ~8 e + co - 1 S t} ~, CtI CC G CS C) C C- C,4 C- N CO CO C 3 C3 O1 C1,14 * C: OQ t1 $- L t- X COD 0t 0 0 CO in AC..* I C " _ I O C. C C CO C;) C O C I _ C1OI C aA, ~.. C;.; C.. tI Q ~ C: I ~.., ~,........C' C | O H C H e C C COl COc CO CO cCO C O CO CO CO CO t- 0 o GO o o F.l-t —tCCICir;0000oooo0, o 0O0 CO0F-O Uo o3 CO -o I' 0 CO.0,F - 0..C Z,. _- __ COC___OC.)C I ~4 CID CO CO CO CO C,.) O-) C 0O C,.) CO C,:) CO. COF CO CO C, 2COCO22O COF - 0022- C ZCO0 C. C.o CO C CO 0C. O -C CcC COFCO 0-00' O CO CO Cc CO CO C C1~ r C~u~-cnl-c c 2200 00 2 20022 2 2e cr~rC; u~cl000 0c GA00 r-q- 0 CO -4 m t 0 c0 Co - w =`c H C:- C,0 EXTE rN'AL S ECANTS. 449 48 DEGREES. 49 DEGREES. 50 DEGREES. Ain. Nat.,No.. Logarilm n NN:,t. obg a MSin. Nat No LgitC 0 0-4914477 9-93140 0 j 024253 1 9,719341 00555724 9-74459 1 I 49190'691570 1'524763'719963 1 -556268'745230 2'495143'694994 2'525274'720880 2'556814'745710, 3'493927'6935418 38'525785'72' 819 3'557344 746123 4'491412: 695542 4' 752297'721231 4'557885'746545 5 ]'493390'0696266 5'526089 721605 51'5584-27'74C966 6 -497381 0 696689 6:'52722'722076 6'558909'747383 7]'49i7807 0 697113 7.527885'722498 1 7'559511'747,:9'8 493353 ]'697537 8 -528848'722920 8'560054'748233 9.493840 0 697961 9. 528863'723340 9'560598'748652,10' -499327 06983S5 10'52938 7287660 10 561142 74970 11'499314 0693808! 1'5298993 -724188 11'561687T -74949.1 12 |'5DM3 2 -699232 12'530448 724610 12'562232'749916 13 -5030790'699656 13 -53C9 92'725832 13'562778 -7503837 14 501279'703079 14.531440'725454 14 6'563324 /'75f758 15 -511708'700503 115 l 531 7T'725877 15 563371' 751180 16i. 5022'58 -700927 10.532175'726299 16 564418 -751601 17 5,'i2i749'701351. 17 -532999'726721 17'564966 -752122 18'51323 9 ]-791774t 18 533510'727148 18 -565514'752443 19!'5A,3730'702198 19' 53429'727565 19 |'566163'752805 20.534221 702621 20.534548'727987 20'566612'753286 2t1.534713'703015 21'5359 68'728409 21'56071,q2'758707 22 i'513-2.5 70340 22 9583589 T72882 22 567712 754123 -515093' 70389s1 20'586110'729254 9823'568268'754549 2,4 -'596191'7049: 5 21'53M1 729676 24'568815'754971 25'33i35685'704738 2 5371538'730C93 25'569367 ]'755892 26 i'5:i7130'705162 206'537675'730520 26 569919'755813 27'507674I'75585 27'538198 I'730942 27 570472'756214 28' 508169'706905 23.538721'7313064 28'571025 756655 29'5380664'706431 29:509245'731786 29 ]571579'757076 3)80 -519160'706854 308 539769,'32208 80,'572104 -757493 81 -599657'707278 81 543294'732630 i31'572689 {'757919 82'510154 -707701 82'540819'738052 82'5738244 - 753840 88 -I 510651'708124 33 8 58 145'784 74 8 4 3'578300'758761 84 51148'708547 34'541871 738896 84 574357'779182 85I'511646 708970 85'542998'734318 85'574914'759303 806 -512145'70939 806'542925'734740 68'575472'760024 87 [512645 709317 87'543452' 785168 1 87.'576030'760445 88 [ 513145 ['710240 88'543980 785583 88'576589'760,866 89'513645'710663 899 *544508;7836004 89 577148'761287 40' 51-146'711087 40'545087'786426 40'577738'76178S 41': 514647 -711539 41.545567'786848 41'578268 -762129 42 -515148'711932 5460397'78720 342'57&829'762550 43:i'515650.712355 i 43'5466028'73792 43'5795D90'762971 44 -'510152 712778 644'547159'738114 44'579952'768392 45'516(355'713200 45'547690'788535 45'580514'76381.$ 4;6'517153'7103623:46' 548222'738957 46'581077 -76423-4 47l,517662 714046 47' 548755'739379 471'581641'764655 48 ['518166'714469 48'549288'739800 48'582235'765',170 49 -518670' 714892 49'549821'740221 49'582770'765497 501'5191751.71531[4 50'550855'740643 50'583385'765918 511 519681'-715787:511 550890'741005 51'583900'766039 52,'5231188'716160 52?[ -551426'741487 52 i'584466'766760 531'523695[' 716588 53',5519061';741908 1 53'585033 -767181 54'521232'717006 54'552497' 742330 54'-585600 -767602 551'521709:'717428 55'5530933'742751 55'586c168 768022 506'522216'717850:560'558571'7431783 506'586737'768448 3 57)'522725'718-273 57'554109'743595 57'58T7306'768864 58'52282.4'718696 58'554647'744017 58'587875'769285 59'52:374,,'719118 1 591'55185'744434I8 59 ]'588445'7069736 60'524233'719.541 60'555724'744859 60 5895106'770127 38* *450 VERSED SINES. 51 DEGREES. 52 DEGREES. 58 DEGREES. Mit Nat. No. Logaritlhm Min | Nat No Logarithm. iu. Nat No. Logarithm. 0' 0870680 9'568999 0'8843389 9584714 0 00398185 600085 1 87C906.'569264 1'884568 84973 1 398417'60C038 2 3871182, 569528 2.384797 258522 2 398650 600GG91 8,371358 |'569798.3 88527:585491 8 898882:600845 4'871584 570C57 4 -'385256:585749 4 899115'601C98 5 871810'570822 5'885485:586008 5'899347:601851 6'872087:.570586 6'385715:586266 G'899580 G601608 7 372263 570850 7'3885944:586525 7'399812'601856 8 372490 1 571i14 8 8386174:586780 8'400045'6021C9 9 3872716'571878 9 -886404 587041 9'400278 -602862 10 8372943 571642 1 0'38668338:587299 10'400510'602614 11 373170 1'571906 11'386863'587557 11 -400743 *602867 12 *87396 1 572170 12 887093'587815 12 *40G976'608119 18'3736238 57244 1 18 88728'588078 183 4.012C9 6088371 14 *878850'.572697 14'887558'588881 14'401442'608623 15 374076 572960 15'387788'588588 15'401675'608875 16 *874303'573224 16'388018'588846 16'401908'604127 1~ -374458)0 573487! 17'888248'589103 17'402142'604379 ~18 *374757'578750 18 388473'589361 18'402375'604631 19 *374984 574018 19 *3888703'589618 19 1 *42608 -604582 20 875211,574276 20!'388933:589875 20'402841'605134'21 |875489'574539 21 *889164'590182 21'408074'605355 22 1 875GGG'574802 22!889394'590389 22'403308.605637 23 875893'575064 23 0889624'590646 23'403541 G605888 24 18761220'575327 T 24'889855 *'5990 3 24'403775'606189 251 3768348'575589 25 890085'591160 25'*4040C9'606891 26'87657.5' 575852 26'890316'591416 26 - *404242'606642 271 876803'576114 1 27 *890547 591673 27 -404476'606893 28 8377030'576376 28 3890777' 591929 28'404710'607144 29 0877258 1576688 29 *891008'592186 29 404948 607894 80 377485'576900 80[ 891289 592442 30'405177'607645 81 8377713.'577162 31!891469 -592698 31'405410'607896 82'83779411 -577424 32 8'391700 592954 982 405645'608146'83 8378168'577685 83'891981 -593210 83 1405879'608397 84. 0378396'577947 84'3892162'598466 84'406113'608647 5' 378624'578208 85}'3928.93'593721 35B 406847,608897 86'78852'5 78470 86 892624'593977 86'406581'609147 37 1 39080'578731 81 7 *892855'594233 87|'406815'609397 38 8379308'm578992 88 8'393086'594488 881 407049'609647 89'379536'579258 89'898817'594743 839 1407284'609897 40 1 79764'579514 40'8935491 594999 40'407518'610147 41 8379992'579775 41,8993780'595254 411 407753'610397 1 842 ~O2*1'583036 42'394012'5955C9, 421 407987'610646 43'380449'580297 4 43 8394243'595764! 43 1408221'610896 44 380677'580557 44'394474'596019 44|'408456'611145 45'388906'580818 45'394706'596274 45'408690'611894 46'1 81134'581078 46'894938'596528 46'408925'6116441 47'831363'581339 47 8395169'596783 47'409160'611893 48 |'81592'581599 48'895401'597038 48'409394'612142 49'881820' -581859 49 895682 1'597292 49'409629'612891 50 38828'49'582119 50'395864'597546 50'409864'612640 51'882278'582879 51'396096'597801 51'410099'612888 52'882506'5826389 52'896328'598155 52'410384'618187 53'8S2785'582899 * 53'396560 1 598309 58'410569'613886 54 1 082964'583158 54'396792 1598563 541'410804 6138684 55 3881193'588418 55'897024 1598817 55'411039'618883 56 388422'588677 1 56'397256'599071 56'411274'614181 57 380865T'588937 57 397488 599324 57'411509'614379 58 i 383880'584196 58 83977201 599578 58'411744'614627 59'384110 -584455 59'897953'599882 |59' 411979;614875,60 3884339'584714 1 60'398155:600085 60''412215'615124 EXTERSATL SECANTS. 451 51 DEGREES. 52 DEGREES. 53 DEGREES. M in. N it. N.. Logaritllm Mill Nat No. I.l -galitllm. Al 1! Nt N.! Io garifilt - 0589016 9770127 0 0-624269 9-795372 0 0?661640 9828622 1'589587 1770548 1.624875'795793 1.662282'8218(4 2'590159'778.969 2'625480,796214 2.662924 521464 1' 590731 771889 3'626086 I'796634 3.663567'821885 4' 591303 — 771810 4.626693 797055 4.664211 8223806 5'-5.91876 *772281 5'627300 -797476 5..664855 8i22727'6 592450'772652 6'6279038'79789B6 6 665500.'823148 7 *593025'773073 I 7'628517 798317 7 666145'82338569 8 593600'773494 8 *629126'798738 8 -.666791.-823990 9'.594175'773914 9'6'9736.799158 9..667489.824411 10'594751'774335 10 668C346'799579 10.668086 824883 11 "595327'774756 11'680957 800COO0 11 668784'825254 12'595904'775177 12'631569 I800421 I12 * 669383'25675 513 596482'775598 13'632181'808841 13 670032'826(96 14 - 597060 -776018 14 -6832794 [ 801262 14 * 67C682 ['826517 15 -597689'776439 1 15'6338407 801683 i5:'671833'S29.88 16 1 "598219 1'776860 I 16'634021 -'802104 1 16 -671985'827&GO 17'5987991 77T281 17 684634685 1802524 17 672637'827781 18'599380'777702 I18'6385251 802945 18'673290'8282(2 19'599960'778122 1 19 1 3685867 8C8866 19 678943 8'828C23 20 1'600542 1'778543 1I 20'636483 803887 20'674597'829C44 21'601124 778964 21 637100 804207 Ij 21. 675252'829466 22 6'601706 1'779385 || 22' 637717'804628 22'675907'829887 23'602289'779805 23 1 638335'805049 23.676563'3(8g8 i24/'602878 *'780226 // 24 "6839.54'805470 24'677220'830729 253'608458|'780647 | 25'639574 |'805891 25 *'677877'881T51 I 26'1 604043 *'781068 26' 6401941 800311 I 26 I678584'831572 27'604628'781488 27'6G40814 -806732 27'679193'831993 28 605214'781909 28'64143 -807153 3I 281 -'679852' 882415 29'; 60800 T'782330 1 29'642057'807574 29'C6880512'832886 80 606387'1 782750 30 "642680 ]'807995 80'681173' 838257 31'606975'788171 31 648830Q3 808415 81' 881884 888679 32 "607564'7885392 3 82'643926 -808836 82 682496'88341600 83'608153 " 784013! 33'644550 /'889257 83'. 6831539' 884t,22 841'608742'784433 I 34 64517'8C9678 834.-688822' 884943 3853 "669382'784854 85'6435801:'810(99 35 1 684486 8835864 86'.6099238 785275 36 646427'818C520 86 685150'.85786 37 610514' 785696 T3 87' 647C54 |'81940 87' 685815'838627 88'611106 1 786116 Ii 838 1 647681 |'811361 I 88'686481'886C629 39 ['611698'7865837 i0 89'1 6488(9'811782 89''687148'887(50 40 "'612291'786958 40'648938'812203 40.687815' 887472 41' 612884 " 787378 11 41'649567'812624-l 41 688483'887893 42'618478 1 787799 42 650197'813045 421'6891,2'8388815 43 "614073 "788220 ]I 43' 65(0827' 818466 43 689821'8388786 44''614668''788640 }l44 651458 1 813887 44'69C4911 89158 45 |'615264'789061 45'652090'814807 i.45 691161'889579 46' 615860'789482 46'652722'814728 1 46 | 691832'840001 47'1 615457'789903 47 -658855 1'815149 47 1 6926504 1 840423 48 1'617054'790323 48'6539891'815570 48' 698177'848844 49'1 617652'790744 49'654623'1 381991 i 49 698850'841266 50'618251'791165 50'655258'816412 50 |694524'841688 51'6188501'791586 51'655893.'8168883 51 693519'8421(9 52'619450'792006 1 52'656529 817254 l 52 1 95874 842581 583'620050 -'792427 Ti 58'657166'817675 i 53 696550'84298 54'620651'792848 54'657803 8180i96 54' 697227T'8433874 55'621253 1793268 55'658441'818517 55'697904'843796 56!'621855'793689 56'659080''818988 56 698582'844218 57 622458 1'794110 57'659719' 819809 9 57' 699261'84469 58'623CG1'794581 58 660859 819780 58' 699941'845061 -59'623665'794951 59'660999'820201 159 1 700621'845488 60'624269'795372 1 60'1 661640'820622 60 -701302'84t9C5 452 VERSED SINES. 54 DEGREES. 55 DEGREES. 56 DEGrEES. Min I Na No L-agrlritm. mhin | Nat. No. I Logaritlin. | Iil. N Nat Logarifhn. 0 1 0-412215 9615124 O 0'426428 9'629841 0 0'44i807 9'644249.1'412453.615871 1 426662 -630C84 1.441048.644486 2 -412685'615619 2'426900'630326 2.441289 644724 3 *412921'615867 8 427189'630569 3 *4415l'644961 4'4138156 616 114 4 4 4277:7 630811 4'441772 645198 5 *418892 -616862 5 427616'681054 5'442013 645485 6'4183628 616610 6 -427854[ -'681296 6 *442255'645678 7 41386:3 1616857 7 *4280938 631'38 7 *442496 *645910 8 414099('617104 8 428331'631780 8'442788'646147 9'414885 -617351 9 428570'682022 9'442980'646884 10.414571 0617599 i 10 *428809'632264 10.443221'646620 11.414807'617846! 11 429047:'632.505 11 *448463'646857 12. 415042'618092 i 12 *429286-'682747 12 *448704'647194 13 415278'6183889 18 429525'682989 138 44,946 6473380 14 415514. -618586 5 14.429764'68230 14 -444188'647567 15 415750 618888 1 15 480008 688472 15.444430'6478038 16'415986'619079 16.430042'638713 16.444672'648040 17: 416228'619326 i 17'430481 633954 17.444914'648276 18'416459 1'619572 I 18 -4830720-'634195 18 -445156a'648512 19'416695': 619818 19'430960'634437 19'445398'648748 20'416981'620065 2) *431199'634678 20'445640'648984 121'417168 1620311 21'481488'634919 21 445852'649220 22'4174041'62055711 22 431677 -685159 22 446224- 649456 23'417641'620803 2'3 431917 6835400 283 446366 649691 24'417877'621049 124 [ 482156'635641 24'4466(,8'64992T 25'418114'621294 25'432396' -635881 25'446851'650162 26'418350'621540 26'482685'686122 26. 447093'650398 27'418587'621786 27'432875''86862 27 447385'650688 28'4188283'622081 28 ~433114'636603 28'447578'650869 29'419060' "622276 i:29'438354 "636843 29'|447820'651104 80'419297'622522 81: 80'438594 "687088 801 "448063'651389 81'41954 "622'767 1 1'"48383 8 "6878238 81 "448306.'651574 32'419771'6230i2 1 2'4840783'637563 82'448548'651809 83'420008'628327 1188'484318 637803 83'448S91'652044 84'420245'628502 i/ 84 484553 688043' 34 449034'652279 35'420482'623747 85' 4:84793' 638283 85'4492760'652514 36'4213719 |623992 86'435033'638522 8 6'449519'652748 87'420956 624237 1 7 485273'638762 871 449762'652983 388' 421193'1 624481 8 483551.31'689001 38 450005'65321T 89'4214801'624726 3 89'-4358753'689241 89'450248 B658452 40'421668'624970 40 435993'689480 40'450491'653686 41'421905'625215 41'486284'689719 41.'450734'658920 42 *422148'625459'1 42 436474'689958 11 42-t 450977 — 654155 43 1 422380 1'625703 43' 486714'640197 1 43 B.451220'654889 44'422617 625947 44'-436955'640436 44'451463.'654623 4 422855'626191 45'437195'640675 45'451707'654857 46'423092'626435 46'437435'640914 46'451950'655090 47.'423300'626679 47'437676'641153 471 452193'655324 48'423568' 026923 48'487916'641891 48'4524371 655558 49'423805'627166 49 488157' 641630 49'4526GS0 655791 50'424043'627410 ~50'48898 "641868 50'4529241 656025 1' I424281'6276,4 51 *4886.39 642107 51'4538167'656258 52 1 424519 1 627897 52 ~4388879' *.642845 52 453411 656492 58'424757'628140 53'439120'642583 58'453654'656725 54 1424995'- 628:84 5 4'489361' 642821 541 43898 1 656958 55'4252381'628627 55'4839602.'643060 55''454142 -657191 56'425471 628870 i 56 4898438'6438298 56 -454885'657424 57'425709 1'629118 57 440084'648535 57 1-454629 1 65657 ~58'425947'1 629356 i 58'4430825'64T73 1 58'454873 657890 59'426185'629i598 59'440,566 644011 59'455117'658123 60'426423''6284|1 60'440807' *644249 60'455361'658356'~/~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ EXTERNAL SECANTS. 153 54 DEGREES. 55 DEGREES. 56 DEGREES. Din. I Nat No. Iogaitm. in. Nt. N. g N ioi. LilLm. M Iti. INat. No. | LOgarifhlbl. 0 10701802'9845905 0 0-74447 9871250 0 07SS291 9 896687 1 T701983 -846327 1 -744172 8716t378 1 7 89063 8S97112 2 *702665 *846749 2 -744897 *872096 2 *789886'S97587 3 1 703348 847170 8. 745623 872519 3 790609 897962 4'704032 |847592 4.746350'872942 4 791388'898387 5 -704716'848014 5.747078'873066 5'792158 -89S812 G -705401 -848436 6 *747806 *8707S9 6'792904 899287 7 -706087 *848858 -7 748585'874212 7 793710'899662 8 706773 1-849280 8 749265'874605 8 794488'900087 9'707460 *849702 9 *749996'875059 9 795266 900512 10'708148'850124 10.750727'875482 10'796045'91938 11 -708836'850546 11'751459'875905 11 *796825 901868 12'709525 *850968 12.752192'876829 12'797606'9017S8 13'710215 851890 13 *752926'876752 18'798387 902213 14'710906'851812 14'753661'877176 14 *799169 902639 15'711597 *852284 15 *754896'877599 1 15 *799952 903064 16 *712289 852656 16 *7a5182'8780238 16 *8007-37'9330-9) 17 *712982 *858078 17'755869 *878447 17 *801521 908915 18 -718675'858500 18:756606'878870 i 18 I 802307'904341 19 -714869 *853923 19 *75T84a'879294 19 *803894 *934766 20 -715064'854345 20.758084'879717 1 20'808881'905192 21 -715760'854767 21.758824'880141 21 *804669'905617 22'716456'855189 22 *759564'880564 1 22 S805458'906043 23'717153'855612 23.760f305'880983 28 -'806248'906469 24 -717850'856084 24 761048'881412 1 24 *807089'906894 25 -718548'856456 25.761791'881836 25 ]807830'907820 26 719247'856878 26 762535'882260 26.'808623'90774t 27 -719947 *857801 27 -7683279 *882688 27 *809416 908172 28 720648'857720 28 764024'883107 1 28 -'810210 90898, 29'721349'858145 29'64770'8885311 29!811005'909124 80 -722051'858568 80.765517'883955 1 80.-811801'99450 81 *722758'858990 81 *766265 884879 81 1 812598'909876 82 *728457'859412 82'767013 884808 1 82 8183895 9103802 83'724161'859885 383 767762 -885227 833 *814193'910728 84'724866'860257 84'768512 *885651 84'814993'911154 85'725571'860680 85 -769263'886075 85'815793'911580 86 726277'861103 86'770014'886499 1386'816594'912006 87'726984.861525 87'770767'886923 87'8t17896 912482 88 -727692|'861947 88'771520'887847 /8'818199'912859 89'728101'862870 89'772274'887712 89'819002'918285 40'729110'862793 40-'778029 1 888196 1 40'819806'918711 41'729320'868216 41' 778784''888620 41'820611'914188 42 -780580'8683638 42'774540'889144 42'821418'914564 48'781241'864061 43 775298t'889469 I[ 43'822225'914991 44'781958'864484 44'776056'889898 44'82308303'915417 45'782666'864906 45'776815'890817 45'828842'915846 46'*783880'865829 46'777574'893742 46'824651'916270 47'734694'865752 47'77883841'891166 47 825462'916697 48'734809'1 866175 48'779095'891591 48'826278'917124 49;'785525'866597 49'779857' 892015 49'827085'917550 50'7862411'867020 50'780620'892440 150 827898'917977 51'786958'1 867448 51'781384'892864 1 51 82812'918404 52'737676'867866 52'782148'893289 52'829527 91831. 53 7388395'868289 58'782913 1 893714 53'880843'919258 54'789115'868712 54'788679 894138 1 54'83110 919685 55'739835'869185 I 55 784446 1 894563 5'831977'920112 56' 740556'869558 56'785213 1'894988 56'S32796G'920509 57'7412771'869981 57 785981 /'895412 5 7 88615'92C966 58'742000'870404 58'786750'895837 1 58'834485'921398 59'7427238'87C827 59'787520'896262 59'885256 21 8201S 60 -7484471'871250 60'788291'8966S7 160 886078'9222-17 454 VERSED SINElS. 57 DEGnREES. 58 DEGREES. j IDEGREES. Ali,,. Nat No, I 1,ga~ibly, Miu, I N~t. N.. i Logar i,,.! ill.,,.t!N,),,,'),ii....' 0 455861 9658856 0 0'470081 9672178 0 0484962 9'('S57CS 1'455685'658588 1'47T827'672480 1'485211'(85981 2'45 5849'658821 2'47( 574'672628'2'48546'686154 8'456(893'659~54 8'47(821'672856'485710'(86877 4'45687 6'659286 4'471468'6781 4'485960'68(680 5'456581'59518 5'471815'69811 5'41862(9'(86823'456825'659750 6'471562'673589 6'486459'(87('46'457070'659982 7 4718&9'678766 7'48671O8'687269 8'4577 14 -'660215 8'472i56'6789989 8'4C98(98 6'87492 9'457558'660446 9'472808'6142221 I'4872(7:.687714 10 -4578(3'60678 10'472550'64448 I0'487457'(87917 11'458847'66(910 I 11'472797 674675 11'487767'68159 12.458292'661142 12 478(744 674 92 12'487957'88852 13 l 458586 861874 18'473291 6-75129 18'4882(:7,'6586,'4 14'458781'661605 14 478559 675856 14'488457'688Qs26 15'459825'661887 15 478786 675582'548877'C$9(:48 16 459270'662068 I 16'47488'6758(9 16I'8957 689270 17 } 459515'6628-00 17 474281'67636 17'4892(7 ('89492 18'459760'662581 1 47458'676262 18'489457'C89714 19'460004'662762 19 i'474776 [676489 19'489767 (C89986 20 -46(249'662048'2) *44759'23 676715 20.4899.57 (9,81551 21'46(494'668224 21'47521'670941 21'49C2C7'690880 22'4607839'663455 22'475518'677167 22'49(458'6906(2 23'46(984'668686 23'475766'6794' 26 4907(8'690682 24 -4612o29'66891 24'476(14 6620 24 49C958|'691045 26'461474.'664147 25'476262 677 846 25'4912(9 691266 26.'461719'664378 26'476510'678{72 26'491459'(91488 27'461965'6646(9 27'476758'678298 27'491710'C.917(9 28'462210'664889 28'477(f 5 67852 28'491960'(919830 29'462455']'665070 29'477258 678749'29'492211'(92151 80.462700'665300 80'477501'678975 80'492462'(92372 81'462946'665580 81'477749'67920 81'492712'692593 382'468191'665760 82'477997'679426 82'4929068'C92814 8 -'468436'665990 83'478246'679651 83'498214' 98(685 84'463682'666220 84'478494'679876 84'498465'C98256 85 468927'666450 85 478742'680182 835'498716'(91477 86'464173'666680 86'48991'68627 86'49(966'693697 87'464419'666910 817 4792(9'68(552 8'494217'(98918 88'464664{'667140 88'479487'18777 i 388'494468|'(9-1138 89'464910'667869 69'4797385'(818(2 899'494719'(9489 40'465156'667599 40'479984'681227 40'494970'(94579 41'4654t?2'667628 41 480282'81451 41 I'495221'(947)9 42.'465648''668058 42'480481'68167(6 42'495472 695819 43' 465894'668287 ]43 480170 681981 48'495724'695240 44'466140 1668516 44 *48(978'682125 44'495975'695460 45'466386'668745 45'481227'682850 i45'496226'695680 46'466632'668974 46'481475'682574 46'496477'695899 47'466878'669208 47'481724'682799 47'496729'696119 48'467124'669482 48'481978T'688628 48'496980'696869 49'467370'669661 49'482222'688247 49'4972382'696559 -50'467616'669889 50 482471'683471 50'497488'696778 51'467862'670118 51'482720'688695 51'467784'696998 52'468109'670347 52'482969'688919 52'497986'697217 53'468355}'670575 58'488218'684148 53.'498237'697436 54'46S(61'676804 54'488467'684867 54'498489'697656 55'6 8848'671082 55''488716'684590 55'498741'697875 56'4(C97:994'671260 56'488965'684814 56'498993'698094 57'4698341'671488 57'484214'6850(87 57'499244'698818 58' 48G:87'671716 58 484463 ['685261 58'499496 1'6985832 59.'469834'671945 59'484718'685484 59'499748'698751 60'470881'6721:73 60 484962'685708 60''500000 j'698970 EXTERNAL SECANTS. 455 51 DEGREES. i 58 DEGREES. I:,t Ni I Logarithm. I Nt, Noor\t N,. IM.,gih.m. im. Nat Ni. Logaritlihn. 0 03378 I9-92221' 0 083 004 9973868 1'8(19411'9522674 1 8819o9'949393 1 942544'974382'92-3101 2 888339'948823 942'943486'9141o 3'8335 5) i'923529 8'88972)'949253 I'944429'95169 4 8:39:315'9]-3956 4 89431'949683 4.945378'973503 5'81)2)2'924'384 5.891481'950114 5 946317'976037 ~'811029.921111 6 8 89268'9504.954 6'94726:3'9:6471 78'811851'92421:38. 893253'9530915 7'948210'97693:5 8' 81V288 6'925666 8 893139'19951405 8 949158' 967739'81351 9 9 9 814 2'92518336' 99501107'971773 10'84118'923:521 10.s95914'959,26 10'91101 9 2830 1 811) 9299"11 38 958I)2 392919 1 89 2'95697 20.95(9'978041 12.816S)112 /9213177 12 831192'9503121 2. 952961' 919075 1 4.81683'928)4 1 8 98353'953558 18.952915'.99510 14'801331 92,82:3-2 14 899175'953149 14 -948130'9 799-14, 15 488516'928360 15.9036S'954420 15'951526'981379 16.8,193D52'9292 88 1 93 1262'9*5451 130.956782 8'98f813 1 85)19)'929516 17 939156'955282 17 95774)'931248 18 *351I02' 929944 18 9 )352'955713 I18.958699 91 C82 19'851831'930312 19.933949'956144 19-.959659'932117 29'852111'9338)98 2.,).9 248i7'956575 20 9606321'932552 21'85'381'931228 i 21 I9.157468'957006 24 9 615S3'932897 22 854:393'931656 ] 292 93)1645'957437 221 962546'938422 23'8552334'932)85 9 3 -9)1375'951989 23'93311'983851 24'856:)71'932513 24 9 (8148'9583)0 2'93447-'984292 25'856921'932011 25) t993514'958732 25'965344'984172 26'853177'9133'69 36 -91;255'959163 26, 966411'985162 21'85S314'933198 27 9111'959595 27 96738))'985591 28'859161'93-1226 23 9129,66'963)026 28.'963850'160833 29 8Y)31.3)'931655 I29 2,97913'960458 29'969322'933463 30'831159'935)83 31 4 913SS1'936890 80'91(0294'986903 31'8 12 )09'913512 31 91497'931:J21 81.971238'987389 ~'81231)'93591 I 302 915700'961753 032.972242'977115 ~ 3'88112'.91963;39 3 916611'962185 83'97218'988210 34'831535'936798 31 911523'962617 34'914195'98343 35 835 1 2)'931221 3 928136'963049 5'9175173'9891182 36'883215 8'9317630 36 919350'963481 36'91615,2'989518 81'861131'938)85 87 932065'963913. 3 977113'939954 33'86793T'9385314'3 91132'964345 38'971815'99f;9) 39'833185'938912, 39 L 22399'964711 i 9.9791971'993-823 89'133'939371 i 1 9 23'915211 4 9, 0'.980081'991262 41'81451 j'9393', 1i41 9- 4 3937'965642 41'9S106) 01 991693 42'S8114-)'91)293) 42'. 9213517'93110759 42.932052'9!72l34 43'-81223 i'91)659 -,3'912718'966507.43 98303189'992571 44'81:3113'9111083 4'1'9213'936690 44'984:27'9930107 43'87412'911517 4'927624'96783 2 45'985017'998-1444 45'814431'9119 iT 4'923549'938567805 4 38608' 8) 4T1'815742-2 I'912313 45 ~ 929415'933238 41 98t101) )9131 348'866133)3 9'9123)3 1 4 93'986141 08'983310 48 9S199 )91175.8 49'871475'9 13235 49.9:31329'969103 49 9889 87'13191 50'878344'9148365 50.9 258'969536 50 989982'90953621 51'879213'91191 I9 51 93188'99939 51'99;919'9!)1(i 64 52 883)3 3'91 1524 52. 931119'9104G-2 52'9919171'f)5,1 5 8'8)95'9814953 533.9353050'910835 53'992975 i' -999)38 51'83821'945383. 54.9:359833'971268 154'991:39751'99376 55'898219)'915813 55 93691.' 97110 55'99496'991813 5'83:35714'9 13243 5 I 93183'912135 56'995978'9)8250 51'8344519'943313 57 -9T789'912568 51'996982'998 5 88353925'9417103 58 93908'9107001 58'991981'9991257 59'8882)2 2'941533 59'9406684'913445 59'998993'999562 60'887089'94796.3 60'941604'97888 60 1000000 1'0000000 NATURAL SINES AND TANG'ENTS TO EVERY DEO'REE AND MINUTE OF THE iUADRANT, E.XTENDED TO.SI-E N:PLA'CE$ OF DECIM'ALSi, 453 NAvt RAt, SiN-V' j lZb i ~ 3~ 4~ 5 _ _ 6~' 7, 0 —: I_ I __ —U —l C~ I~0 000 0000i017 4524'034 8995 052 88601069 7.565 087 1557 104 5285'121 8695 60 1 29,9 14 742'005 197:2 6264 070 0467 4455 8178122 1581 9 2 5818 018 00841 48:0, 91.69 8368 7050 105 1070 446s 18 0 8727 8249 7716 050 2i74 6270 08S 6251 1963 7355:,57 4 001 1636 6158]006 0620 4979T 9171 3148 6856. 123 c241 56 5 4544 9366 85300 78S83 071 2070 6046 97481 1128 55 6 74530019 1974 6437054 0788! 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12 49 014 2530I0 7015 049 1400 5641 9677 8411 692816 C((S 11 54089 9922 4008 854084 2576 6851 9813Q5 93I 10,-I 8048'0028 28.30 7214067 1446 6474 9245119 27044 58(1' 9 2 015 1256 5187 051 0119 4919 8373 102 2118 55911 86 17 8 41651 86441 8024 7251C85 1271 50S2 881'I7 1564! 7 10 7073030 1552 M929N068 0158 4169 7925120 1868 O 45 6 [5 99821 4459 8805 0055 1061100 0819 42561 78T 5 /6 016 2891 7366'651 1740 5957 9966 87112 5 7144188 (2918 4 5T 5799,004 0214 464-5 8859086 2864 663,5 121 0081 3089 3 / 87071 1181 75501069 1761 5762 9499 2991 5940 59 017 1616 6088052 0455 4660 8660104 21-92 58 6,1 1s5 60 4524- 8995 88600 7565,087 1557 5285 80(93 189 17 21 89 88~ - I; Soo~ 8s-sS1 s"~0 NATURAL COSINE!S. --— ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~o ----— l~o CC Co~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~O C G H> oHGo o o b-I _ —Coo'Iii n'1 - c,,,-i o cot o IO;-,'-C ~..oC)Co1o-o(oH - oHIo c.-.4 k- o 1 C'H C t C C -.D 4) oo AC.AC In 10 1-, AL. i.,: -rd, —-I,I, ",,I I c;,.. 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Co C H H( H o O~ ~ o Cii -SS.~ S S S S' Co C Ci A O H C a CO I- Go C C o H C:,-Co Hc C o C,, - IC cm -1 z An C -I) T l ( XC - I- I- I- C- C — L- co Co CO CO CO COD GO CoID ___ Co Co C C Co Co Co Co Co-1 C~ Co C C C Co 0 uPD AC (.11 Cf- CZ) Vo) — H W CO - -C CoC -I ) O Co C= CCCOCt Ctok-fI — cto) o..C-i c( In o COco-CO m c oCI a f-C In. 0co Co~~oC-CCCH~oH-CCCOI —C~o~HC~i H~~"CC-f iCDoC — 01 00o Iof t- oHCC".,ZIr- c o~1111~C-oooi.o H CC C CC 0 ) Co, CG C I o 1114 CoV.) j CI -- Co, n I Co -4 C, z co H- S to OC Co H Co Co —:z (D1 oI'' I- C cCo Co —,dI Ci C. I Co Co Co Co Co Co Co Co Co Co Co Co Co CoC Co CoC,-, t — C,,41~ O i"tk C, o3 a dqt-Z~ 0 c.4:z o:: c n 0-44'I- O Z C. ___ C CoN Co Co t Co Co C CCo(:: 1 co~l Co~ Co C to Co~ CO' C C IO )-oo Z 1o0o 11ooH-rCOH- IL-.)oC o,-.o C oCI tH o o CZC.I)CoI-.0I Ci ooo1Hoo~ -0C~~Cfoo~H~ 3~~;=0c~gc~O~C~~$~~C~~NT~ ~ -;~C~~Ol~j~)O-JI-O C~~Ind~T~~ cQ C~~ 01 T-( a O 0~~ O~ CD ~ C3 O ~~(5 l3 ~ tr3 ~~ ~~ Ti O 33 a C~~ V~~~-coIi - a C1 1 (5 r.) C., tD q or in cri - 4 UD'I CoO-Co l. Co C o -- Co CCI Co C lo C- UC Co Co CC H C — Co Co Co Co Cl U) Co r- H C- Co Co-'I C- lo H G C Co Co H H C- C Co Co C Co Co Co H H - Co. Co C COD C Co) Hw- Cf CC i Co CoG O C o H Co Co CD C= CCO C C ) H H H: H H H H H H H Co C Co Co C C:o Co Co Co Co Co Co Co C Co C C Co Co rC Co C. o I-<>Co C3CZ,,-C I M 1 OX' l CT l- c t - C oGo k ) OC-,C 1 0 LPC C) c C GO T+ it-c G Zco I C>CO c Co C o Co H (I Co CCo Co C — CO c HH -C C Co I-'.'I c cq o Cf Cf Co CCI CI Co CO Co ) CC C " i-C~ Co Co Co C Co Co Co Co Co Co Co Co C Co____ C,, Go I Q C-1;c4 C,; CO tI t,3 j Z -t (.1 CO O~, 0 0 3 aQ c,- t- tc) (. c4-4 => to I C-,-) -l i,!,I - — i CT( 7z GO 0, t3 I C:D 1 AC' CZM - I ( CO Oz Cq ('n a iq 0q - 4 (.,I C-1:4 1 oo U D C', - C-, - 11 1-j ilAC) S I- GO - c 16 --' —I 0 c3 - O In t- C/D C G1 -4 cn in t:1 l I- t a oo, c7 -4 1.3 t Z 1 DI.- GO -Z l Z- n1,i i C- 3 GO I- t — c An -,31 C~ ("I I I 0:, 0 Q Cio 1 — n -I C.),- - -r C3 c~l -e -r I Eo 03 H C-5 5 -~t CS3 ci, ot, rn i -0 I.-.) -P C.-,I H < t: 5 m r O Z i 4;C3 Z C /D o 01 c, U -) C 1 c~z CIO r -1 (Z;, O 1 a CO H tI. C> V.) I'D cl a C/D L'q Vu Z.) C C-1 03 ) 0 i 1~ t -C- z O C~ O - T L ( O C-,) m -,I.0 t.- C-,: CD 1-0 C-. 0 1 3 CO t( C,- (:-4 Lo (:C -,I CC H Co Co Co. Co Co HC,,C) C Co Co H Co Co C; T ~3 3 c I oo ~ I Cfo a 30 -Co n~e~~ I ~ Co Co_ co H3 ~ CCI Cor 1 ~ (~~ 9 O~~~~~ ~~~ H~ H~ H HO H3 H5 H~ H~ H1 CoC 3OC~L3C - CoI Co Co- Co( Cf Co3 CoC Co Co CC~ Co CC C CC C CC Co CC H H~ 3 4 H~ H3 HO H+ H( H Co Co Co CoCo Co Co Co Co Co Co 40.0 NATURAL SINES.,~ 9, 100 110 12 130 1.4 15 1' 0 1'39 1T:381156 4845 173 6482190 8890207- 9117 224 9511241 9219258 8190 60 I: 46121 7218 9846 191 69452C8 1962 225 345 242 2041 -259 1000 59 2 749215-7 0091 174 22111 3801 4807 5179 486$8. 3810 58 8 140 0372 29631 5075 6656 7652 80131 7685 6619] 57 4 3'252 586!: 7939 9510 209' 0497 226 0846243 0507 942S (56 5. 61-32 878 1:75 0803192 2365: 3341 36801 5329 260 223' 55 6 9012 158 1581 3667:' 5220 6186 6513 6150 5045'54 47 141 1892" 4458 65311 807.4 90830 93461 q971 TSs 8 a53 8 4172 7325 9395 193: 0928210 1874 227 2179244 1792-261 0662 52 9 651 159 01971176 2258. 3782 4718 5812 46103 8469 51 10 142 0581' 8069' 5121 6636 7561 7844 743388 6277 50 11: 3410 5940 79841 9490 211- 0405 228 OG77-245. 02541 9G85 49 1.2 6289 8812r17 0847194 2344 3248' 8(:9' 8074 262 1892 48 13 1 9168 160 1688 3710'5197 G691 6841 5894 4699 47 14 143,2047 4555 65731 -8050 89834 9172 87131 7506.6 15 [ 4926z 7426 9435195 09083212 17771229 2004 246 158833263 0812 45 16 7805161. 0297178' 2298i 3756- 4619 48851' 4852 3118 4q 17 144 0684' 3167[ 5160 6639 7462 76661 7171 925 43 18 3562 60381 80,22 9461 213 0804 230 80497 9990 87308 42 i.9 6440 8909179 0884 196 2314 8146 3828 247 28('9264 15836 41 20 96'19 162 1779 3746i 5166 5988 6159! 5627i 4342 40 21 145 2197 4650 8607. 8018 8829 89891 84451 7147 89 22''5751 7520 9469197T 870 214 1B671 231 1819'248 12681 9952 88 23 79513168 0390180 2330 3722 4512[' 4649 4,8i1 205 2757 37 24 146 0830' 3260i 5191 6573- 7853 7479 68991 55611 86 25 8708 6129 8052 94251215 0194 282~ C839 97161 8866 385 26 6585 8999 181 (,913 198 2276 83c5 3188 249 2588266 1170 834 27 94683 164 1868 3774 5127 5876 5967 5350 3978 33 28 147 23403 4738 6635 7978 8716 8796 8167 6777'' 32 29 52171 7607 9495 199 0829 216 1556 283 1625250 C(984l 90581 31 30 8094165'04761182 2835 36791 496[' 44541 38800:.267 28841 10 31 1|48 0971:. 83345 5215 65380 72861 7282 6616' 5187 29.32 83848'6214 8075 938801217 0076 234 0110 94321 7)89 28 83 67241 9082183. 0935s200 2230 29151 938 251 2248268 0192 27 a84 9601 166 1951' 3795' 5080[ 5754 5766l 50681 3594 26 35 [149'24777 4819l 6654 7930, 8593 8594 7879 60896 25 36 5,353 7687! 9514.201 0779218 1432 8235 1421 252 0694 919S 24 37 82801167 0556 184 23738 3629 4271 4248 85(;8269 28CO 23 38 150 1:1061 3428 5221 G478 7110 7075 68231 4801 22 89 )981 6291 8691 9327 9948' 99G2 9187 76(.2 21 40 6857' 91591185 0949'202- 21761219 2786 236 27291253 1952;270 04083 20 41.97.8331168 2826 3808 5o241 5624 55a5 47661 3204[ 19 42 1151 2608l 4894 6666 7873 8462 88811 75791 60C4 18 143 54841'7761 95241203 07211220 1800237'12071254 0893 88051 17 144 83591169 0628 186 2382 85691'4187 40813 8206'271 1605 16 45 152 12341 -8495 5240[ 6418 C974 6859 6019: 4404 15 46 - 41'C91 6362 8098 9265l 9811 96841 88821 7204 1 4 47 6984' -92281187 C956 204 21131221 2648 238 25101255 1645 272 0003 18 48 98581170 2095 3813'49611 54851'~ 53851 44581' 28C2 12 49 1533 7133 4961 66701 7808| 8821| 81599 7220| 5601 11 50 5607 7828 9528 205 06551222 1158 2839 C984256 0082 8400 10 51 84821171 0694 188 23851 85021 3994 38C8s 2894 273 1198 9 52 154 1:356'' 8560 5241 6349 68380 66883 5705 8997.8 58 42380 6425 8098 9195 9666 9457 85171 6794 7 -54 7104| 9291- 189' 954 206 20421223 2501,240 22801257 1328 9592 6 55 9978.172 2156' 8811 4888 M' 587 1 041- 4139274 2890 5 561155 2851|" 5022 66671 7734 8172 79271 69501 5187 4 1 57 57251 7887 9523'207 0580 224 1007241 0751 9760 6984 3 58. 8598'173 0'7521190 23791 3426 8842 3574158 252570275 0781. 2 9 1156 14721'6171 5234 62721 66761 68961 53811 85771:. 60. 48451 6482 8090 91171 9511{ 9219 81901. 63874| 0 31~ \ 80' I N9AT'URALd! 7a~ S E.4~0 |I~~~ - ~~~~NATURAL COSINES. NATUTR1tA TANGENTS. 46(31 80 Io 1 Io 11 | 120 13` 1 140 0 1L40 54081158 884 176 3270'194 83;3 212 5566 2;30 8682 249 8280 267 9432 60 1 8:375 6826 62ii9 6882 8606281 17401 6870)268 2610~ 59 2 141 1342 98k,9 92691 9S41 213 1647] 48111 0 94300 5728 58 8 408159 2791177 22869195 28611 4688 7876 250 2551 8847: 57 4' 7276 5774 52170 58811 7780)232 C941 5642269 1967I 56 5 142 02481 8757i 8270 891i'214 0772 4007 87341 5087i 55 61 3211 160 1740 178 1271196 1922 3814 7073 251 18261 8207i 54 7 6179 4724 427831 49431 6857 238 0140 4919270 1328 53 8 1. 9147] 77:.r8 72741 7964 9900 3207 8012 4449 52 9 148 2115 161 0692179 0276 197 (986'215 2944 6274'252 1106 75711 51 10 5384 3677 8279 408s 59B'88 90342 4200 271 0694' 50 1i | 8053 6(;62 6281] 7tj31 9:082 234 2410 7294 8817 49 12. 144 1('22, 9647 9284 198 O0053216 2077 5479 253 0389 6949 48 13 3991-1162 26432183 2287 3076 51221 8548 3484272 0064. 47 14 693;11 5618 5291 6100) 8167 285 1617 6580 38183 4615 99311 8693 8295 9124 217 1213 4687 96761 681: 45i 16 145 29311-163 1591 181 1299199:214S 4239' 7758 254 27T31 948.' 44I7'} 5872 4576 413J 5172 738(6-286 0829 1870 273s 2564 430 18 42 7563 78 8197218 03538 3900 89681 5690]5 42' 19'146 1813 164 0550 182 0313 200 1222 3400 69,71 255 20661 88171 41 20. 47841- 387. 8319 4248 6448 237 00441 5165 274 1945 40 21 1 7756' 6.25' 68241 7274 9491 8116 82641. 5072139 22 147 0727O 9951 9380231 3800219 25441 6189 256 13631 8201i 388 23 3699 165 25)3 2)1 18 3327 5598 9262 4463 275 1330 3724 66721 5489 5343 6854'. 8643 23'8 236.7564 4459 36 25 9.3441 8478 8351) 9381 229 1692! 5410)257 0664 7589. 35 26.148 2617 166 1467 184 1358 202 -2409' 47421 8485 8766 276 0719 84 27 1 5593j. 4456 4365( 5437T 7793 289 1560 68681 8850 3 28- 85638 74.46 7;3731 84651221 0844] 4635 99701 69811 32 29 149:1586167 0436185 0382'208 1494 8951 7711258 3073277 0113 831 80 45101 3426 33931 45231 6947 240 C781 6176 8245 80: 381: 1 74841 6417T 63991 7552J 99991 3864 98I) 6878! 29' 82- 71507 0458 94071 9409 234 0582 222 8051 6942 259 2384 95121 28 8 134:383168 2393 186 24181 36121 6104241 0019' 5488 278 2646' 27 64081 5393 54281 66431 9157 38971 8593 5780i 26 36: 93831 83811 88391 9674'223- 2211 6176,260 16991 8915. 25 36:. 151 2358 169 138783187 1449,205 27051 5265 9255J 48-05279 2050] 24 7 5333 - 43866 44601 5787} 8319 242 28334 7911 5186, 23 38 | 831.)9 7858' 7471 8769 224 1374 5414261 1018. 8b22 22 89'-:152 1285170 0851188 0483 206 1801 4429] 84941 4126280 1459 21 40 I 42629 3344' 3495 4804 7485248 15751' 7234 4591 20 41-. 72:8. 6388 65'071 7867 225 0541 4656-262 08421 7735, 19 42. 153 0215} 9031t 952 )297 0930 3597} 77873 8451 281 C8783 18 483| 319217~120325A189 25,831 3931,4 6654 244 0819 6560 4012 17 44"] 6170 532:) 55461 0968 9,711 198121 9670 7152 16B 45:1 9147 8314. 8559 298 0003 226 2769 6984 268 2780 282 0292, 15 46. 154 2125 172 18309190 157031 38108 5827 245 0068 5891 3482 14 47 5103 404 4587 607 888 81511 9002 6573 138 48: 8082 7800 7 6':2 91(9 227 1944 6236 264 2114 97151 12 49 155 1061.173 0296191 C617 209 2145 5003 93201 5226283 2857T 11 50 4040: 8292 8682 51811 8063 246 240,51 8339 59991 10 51 7019 628S' 6648 - 8218'228 1123 5491.265 1452 9144 9 52 9998 9285' 9664210 1255 4184 85771 4866284 2286 8 53 156 2978174 2232192 26801 4293 7244;247 16631 7680 5430 7 54 5958. 5279 5696 7831229 03061 47150 26 07941 8575 G6 55 I 89391. 8277 8713 211 0369 83671 78871 399 285 1720 5 56 1157 1919175 1275193 17831 8407 6429248 1925' 7025a 4866.4 57 4900 42738 4748 6446. 6492 401.3 267 0141 8012 3; 58 78811 T272 7766 9486 230 25551 7102 3257 286 1159 21 59 158 0863176 0271194 0784212 25251 5618'249 0191 6374 4806 1 60 1 88-41 3270 3803 55661 8682 8280 94921 74541 0 s8~ 8o~ 7g9~ 78~ 77~ ~ 7 75~ j 74. NATURAL COTANGENTS... _ _ _ _. _, _. 30 46 2. NATURAL SINES.' 160 I P7~:01 180I 190 20~ 210 22- 23~ O. 2'- 0683-.4' 292 37i7 389 0170:.325. 5682 342 0201 358 3679 374 6066 390 7311 60 i' 9170''.6499' 29361 8432 2935 5 6395 8763 G09S9 59 2. 276.1965 9280. 5T702326. 11S2 5668 9110837,5 1459391 2666 583.47,61.298 2061 8468 3932' 3400 359 1825: 4156 -53431 57 7556 4842 310 1234 66813438111833 4540, 6852 8019 56 5 277 6352 7623 8999 94301 865 7254 9547392 0695 55 6:1847L294. 0403 6764827 2179'6597 99968:376 2243 3371 54. q 5941 3183 9529 4928' 9329.360 2682 493S 6047; 53. 8 8736 59.638311 22941 7676 344 2060 5895 7.632 8722 52 9 278 1580. 8743 5058328 0424;1 4791- 8108 77 0327 893 1397 5 1 1,0' 4324!295 1522 7822 3172 7521361 0821 3021 4071 50 1. 7i18 4302312. 056 5919845 0252.534 5714 67451 49 12[ 9911 7081'3349 866'61 9829 6246 8408 9419! 48 13 279: 2704 9859 6112 829 141' 5712 8958 387 1101394 26C93 47 14'549 296 2638'8875 41601 8441362 1-669 3794 4766 46 15 8290 5416.313'1638 6906346 1171 4380 6486 7439 45 1.6 280 1083 81941 4400. 96531 8900 7091' 9178895 0111 44:1.7 3875297.0971 71(63330 2398 6628 9802379 1870 2783 43 18'' 6667 6'749 9925' 51441 9357363 2512i 4562 5455 42 1'9' 9459 6526314 2686 7889347 2085 5222 7253 8127 41 0 28 1 9322519803i 5448381 06341 4812 79321 9944 396 0798 40 2| 5"042 298 2079 8209 33791 7540 364 0641380 2634 3468 39 22 78388' 4856331'5 0069 6123348'0267[ 3351 5324.6139 38. 283 282 06241 7632 730' 8867 2994 6059 8014 88091 37T 415 299 0408 6490 3882.; 1611 57201 8768381 0704 397 1479 386 25.-,62051. 31841 9250 4355' 84470365 14761 8393 4148 35 26' 8995 5959 816 2010 7098849 1173 4184 6082 6818 34. |27 283 785 8734 184770 9841 8981 G68911 8770 9486 383 28: 45,75300.1509' 75293833 2584 6624 9599382 1459398 2155 32 29 364.42848317 0288 ".'5326 9349-,866-206' 414' 482331 30 284 0153 17058 30471 069 350 21,74 5012 6834 74911 30 8i 29421 98832 5805 384'08101 4798' 7719 95223899 0158 29 62 5731 8301. 2606 8,563 83552; 7523 867 0425 388 2209 2825 28 33' 8520 a5880880318..1321 6293 351' 0246 31830'48951 5492 27 384 285.1308 81531 4079 9)34 2970 5836 7582 8158 26 835 4096 302 0926 6836 3385 1775 5693 8541 384 0268'400 0825 25 86] 6884:'83699, 9593 4516 8416368 1246 2953" 34901 24 87 9671 6471819 2350 7256352 1119 3950 5639 6156 23 88 286 2458 9244 5106 9996 3862 6654 8324 8821 22 39/ 52461305 2016],7863386 27351 65841 9358385 1i0081401 1486 -21 40.8032'iS4788820 06'19 5475 93061369 2061 36983 4150 20 41 287 0819 7559 3374 8214:353 2027 4765 6377 68141 19: 42 "' 3605[8304 0331 6130137 0953 4748 7468 9060' 9478 18 I483 63911 8102' 8885 8691' 7469 870 0170386 17441402 21411 17,44 9177 5872 321 1640 6429!354 0190 28721 4427[ 48041 1'6 4.5 288 1963 86-1' 4895 91671 2910 557.41 7110] 74671 15 46 4748 305 1413 71498,38 19'5 5630 8276 97921403 01291 1A4 4T. 7530: 4183 9903j 46421 83501371 6977387 2474 2791 18 48 289 0318 6953822 2657 7879;355 1Q70 3678 5156 5431 12 49 8103 9723 541113'89 0116.38789 63791 78'71 8114 11 50.5887806'2492 8164 2852 6508.9079 388 05181404 0775 10 1 i 8671 5261'323 09171. 5589.9226172 1780 3199 3406:9 52 293 1455 8030 3670 88925 356 19441 44791 58801 6696.8 58 4239:T07 0798 6422[340 1060 4662 7179 85601 8756 7 54 7022 8566 91741 3796 7880 9878389 12401405 1416 6 55 9805 6334824- 1926 65811857 0097 373 2577 8919 4075 5 56 291. 2588'9102 4678 9265 2814 5275 6598' 6784 4 57.5371 308 1869 7429 341 20001 5531} 7973 9277 9393 3 -58 8153 4686825 0180 4741 824813-74 067190 1955406 2051 2 59 292 0935'7408 2981 7461358 964 3369 46339 479 1 Q0 3717 309 0170 5682342 0201 8679 6066 7311 7366 0 / 3~ 70 ~ 7 1~ 7 ~0 70 690 68~8 670 66~ / NATURAL COSINES. NATURAL TA-NGFNTS.4 6-3; 16; 170 1 190 20~ 21 122 23~ 286 74541805 73071824. 91973%.44 32763863 97302-3.83 8640'404. 0262424 4748 60 1 287 06023'Q 0488,825 24181 65303.64 29971884', 1978 386461. 8182 59. 2'8751'670 5601 9785 6292 5817 7031 425 1616 58 3 6900 6852 8848'8345 8040. 9588 86566405 04171 5051 57 4 288 0050 307 0038426 2066| 6296 65.2885 385 1996'38041 8487 56 5 8201 3218' 52845 9553 6182 5887}'7191'426 1924 55 6: 6352'6402 8504,846.2810' 9480' 8679406 0579i' 5861 54'7 9503 958627 21724! 60688366. 2779.18689 2021 3968} 8800 58 8 1289. 2655308,2771 49.441,9327 6079 53864 7358 427 2289 52 9' 5898]' 59~57 8165'84i7 286 -9379i 8708407 0.748' 5680 51 10 89611 91438 8 8 1887 5846 6.7 42680 887.2058 41389 9121 50 11 290 2114;309 2880''4610, 9107 59811 5898'7531 48 2563 49 12 5269'551.7833348 2368 9284! 8744 408 (J924 60051 48 18 84238 8705 29 10561 56868 2587 388 2091 48318 9449 47 14 291-1588310' 18983 42811 85981 5890 5439 7713,429 2894 46 1t 4784i' 5083 7505'349: 2156J 9195 8787 4C9. 1108 63889 45 16 - 891 8272 880 0781 542008369. 25003889 2136 4504 9785 44: 17 292 10478311 1462 8957 86.85 " 5806 5486 7901 430 3282 43 18'. "4205 4653 7184 350 1950 9112'888741012991 6680 42 19 7863 78458381 04111 5216 370'.24208390 2189 4697 431 0129 41 20 293 0521 312 1031 863.91 848318:57281 5541 8G9.7 3579 40 21 3680.42291 6868-851. 17501,' 90861 8894411. 1497 70301.9 22 6839.74 2 00971' 5018 871 2346 39,1 22471 4898432 0481 88 23: 9999318:3 0616] 83271 8287T 56561 5602 8800 3983 87 24: 294 81601' 3 88-01 65573.852'15561 89671' 0957'412. 170 7386 36 2 6321 7005 - 9788 4826 872 2278 892 2313. 5106488 0840 8'5 26.94338814 0200 383 80201 80961'55901 5670 8510 4295 84 2i 295 2641j 33,96 62523853 1368 890381 9027413 1915' 7751 883 28 5808 66593 94851 464038783 221T898 2886 53211484 1208 82 29 89711 979338.4 271.9 799121 55321 5745. 87281.465 31 30 296 21358315 2988 5958354 11861 88471 9105414 21861 8124 80' 31 5299' 6186| 9188, 4460374 2163 394 24651 5544435 1583 29 82'84641 9888 335 24241 7734 5479, 5827 89531 5'043 28 88: 29T 1680 316 2585 5660355 10101.87971 9189415 29368 8504 27 84 ~ 4T961 5785 8896'4286875 2115'895 2552' 5774486.'1966 26 85 7962 8986.336 2134' 75621 4883 5916 9186. 5429 25 86 298 11293817 2187 5372 356 0840 87581 9280416.2598 8893 24 37 42971 5389. 8610b 4118876 2078'896 2645L 6012437 2357 28 88 7465!' 8591887 1850o 7397 5394 60111 9426 i 58231 22 39 299 0634318 17941 5090 57 0676'8716 9378417 2841 9289 21 40' 88031 4998''8.330 3956'377 2038 97 27461 62571438 2756 20 41. 69-73 82028338 1571''72371 5361 6114!' 96783 62246 19 42 1300 01448319 1407' 4813858 05181 8685[ 9483'418 3091 9693 18 43 88315' 4618 8056.'8018378 2010 398 28531'659489 31638 17 44 6486 7819889 12991 7083: 5335 622.4! 9928 6634 1-6 45 9658820 1025 4543359 03671 86611 9595419 3348 440 0105' 15' 46 301 2881'. 4232 77871 8651379 1988 3899 2968 6769| 8578 14 4;'.60041 7440840 1032 6936. 5315. 6341'420 0190 7051 13 48 91783821 0649' 4278.360 0222 8644' 9715! 8613441 C526 12 49 302 2352 88581 7524 3508 880 1978400 3089, 7086 4001 11 50' 55271 7067841 0771' 6795 02 6465421 0460 7477 10 51 8703 322 0278. 40193861 00821 8638 98411 88851442 (.954 9 52 03 1879' 8489'7267 833713.81 1964401 3218! 7811 4432 8 5'3 50551 6700842 0516 6660 5296 6596 422 07388 7910 7 54 8282' 9912 38765 9949 8629 99741 4165 443 1890 6 55 304 1410823 31251 70151362 3240882 1962402 3354 73941 "4871 5 56 45881 6338 343 0266 6531 52961 6784 423 102381 8352 4 57 77(67 95,521 8518 9828 8631 403.80115' 44538444 1834 3 58 8050946'824. 266 6770386. 31153883 1967.3496 7884 5313 2 59 4126i 59811844 00231 64081 5308 6879424 1816 8802 1 60 73071 9197' 8276 97021' 8640 404,0262; 4748!445 2287 0 21 0 3 |1 7 2 | 71- | T9b | A 66 8 f /W NATURAL COTANGENTS. 4-t4,NATURnAL S:NES. I 24~1 25~1 26~ 1 27~ I 8 ~ I 3~ I 31~' " 4)6 78366 422 61883438 8711458 995 469.47-i6 484- 8(!96500 00008:515 0381!i 6BO 1'.47 0 424 8819): 6326'454'24971 7928;4'485:6640': 2519 2874. 2 2G31428 1'4551 8940' 5088 9852' 31841' 5871: 536758 3 53'37 49031439 155-3' 7679T470 2419 57271 7556! 78591 57 4 7993 (72.1' 41663455 0269j 49861 8270501 0073516'0351: 56 5 4)8 06419 9360' 6779 2859]1 755'48G 0812' 2591' 2842" 55 6 3305 424 19941 9392' 5449 471: 01191 33541 5107 53-33!.54 7 5960 4628j449 2604 888' 2685 5S95! 7624' 7824 53 8 8615'7262 4615 456 0627 5250 8436 502 0140517 05141 52 9 4:,9 12691 939.5 72271 3216 7815 487 09771 2655! 2804' 5i 1) 8938'425 25283 9838 5804472 0380 38517, 51701 5293' 50 11 65.77 5161i441 2448' 8392 2944 6057F 7685 7782 49 12 9230 77T9:3) 5059 457 0979 5508 8597 5030199 518 02701 481;:') 410 1833426 0425' 7668 38566 8071488 11836 27181 2758' 47 11 45386 80531;4422 0278' 6153 473 0634 386741 5227 5246' 46 1' 71 8 9 5687 2ss8371 8739' 3197 62121 77401 7733 4516. 9341 8318 5496458 1325 5759 8750:504 (252519 0219::44 17 411 2492 427 09419 8io04 8910 832149:1288' 2765 27(15! 43 1s: 5144 385791443 0712. 6496474 C8823 8251 5276 51911 42 19 7T95 621-,) 3319' 9380 8443 63611 7788 7676' 41 2 -'412 0445 8838, 592T459- 1665( 6004 8897 505 0298 520 01611 40 21 3896 428 1467) 8534i 4248 85644953 14833 2809 26461 389 22 5745 4`'951444 1140 6832.475 1124" 39681'5819 51301 38 23 8395 6723' 3743 9415 36831 65031 78281 761:3 3724. 413 1044 9351 6352 460 1998 6242 9038'506- 00338521- 06961 36 25' 86983 429 1979' 8957' 458:) 88801 491 1572 2846,' 2579' 35 26 63412 4606445 1562 7162 476 1359' 4105 5355 5061 34: 2'7: 8993'. 72331 4167 9744' 3917T 6638 7863 754'36 83 28 414 163' 9359! 6771 461 2825 64i741 9171 507 0870 522 002-1 32. 29 4285,480 24531 9375'496 931 49 17041 2877 2505 31 69321 51111446'1978 7486477 1588 4236' 58384 4986 30 31 9579' 7736 4581 462 00661' 4144 6767' 7890' 74661 29 32 415 2226!431 0861 718M- 26461 6700 9298 508 0396 9945! 28 3:31 4872! 29861 - 9786 5225. 9255493 1829'' 2901..52 24241 27 384 7517 56101447:2388 8041478 1810''4359 540:6. 49:8! 26.38; 416' 01631 8234 499464C 0382 43641 6889! 7910 73811 25: 361 2808'432 08571 71591 29601 69191 9439509 0414 98)9 24 87 3 ~ 45181 8481 448 0192 5581 38 9472494 1948' 29'18524 288361 23 |8 8 899 ] 610' 2792 815 479 20261 4476 5421) 4813' 22 39 417 0741', 8726 5392 46.1 -0692 4579 70'05.79241 7290 21. 40 33885483 13481 7992 3269 713il 953251 0426 ( 97661 203 41. 6028 3970 449 0591. 5845 9683.4953 20601' 928 525 22411 19 42: 8671 65911 8193 842048) 22835 4587 5429- 4717 18 43: 418 13313! 9212- 5789 465 0996 4786 7113 799,:' 7191 17 44 8956'484 18832' 8387 3571 73371 9639,511 0431(. 9665 16 45. 65971 4453 45 0934 6145 9S88 496 21651 2931 526 2189 15 46. 928391 70)72, 3582 8i19481 2188s 4690' 5431 4613 14 47 419 188), 93921' 6179466 12981 4987r1'7215' 79301 7(85 13 48: 4521435 28111 87751 8866 7537 9740'512 0429 9558: 12 49: 7161 4930 451 13872 6439482'0086 497 2264': 2927 527 2080 11 50 98)11 7548 3907, 9:)12| 26834 4787 5425 452 1) 51 420'244-i4836 0166 6568 467 1.584 5182 7-310 79231 6971 99 52 5083 274'9158i 4156'. 770'9833 518 04201 9443 8 53 7719~ 5101452 1753. 6727 483 0277,498'2355i 2916 528 191-4 7 5 1421 0358!: 8018 4347,. 9298 28241 48771 54131 4:883 61 55 2996437 -064' 6941'468 1869 5870[ 7:3991 7908 6853 5 56 56341 32511 95:5 4439 7916]' 9920 514 o04041 9822 4 57 8272i 5866458 2128' 70091484 0462,499 24411 2899 529 1790[ 3 |58 422 0909) 8682 47251! 9578 30071 4931 539.8 4258 2 59 38546438'10971 7313 469 2147 5552 "74811 7887 67261 1 60 6183j 37111 993 -4716 8096.500 0000 15.0381 91931' 65 64~ 63 6 ) ~ o 61~ 6 () 59~| 5s NATURAL COSINES. NATURAL TANGENTS. 465 24~ j 25~0 26~ 27~ I 28~ 29 i 300 31~ _. 445- 2287 466- 3077 487 7326 509. 5254 531 7094 554: 091577 85038008.6'61 60 1 5778 G66184488 C9271 8919.532 08261 68941' 7882601 2566 59 2 9260467 01611 4830510 — 25851 4559 5550o698578. 1262' 6527T 58 3. 4462747! 37051 81331 6252 8293 4504 5144 602 0490 57. 5 9726468 0796 5343 511 8588j 5765 556 2119 579 2912 8419 55 6 447: 3216i 4342 8949 7259 9503 59291 6797603 2386! 54 7- 6708! 789l)490 2557 512 0930 534.8242 9739 580.684!1 68354 53 S. 448;020469 1489 61661. 46021 6981 557 8551 4573 604 0323 52 9 39 -86938 49881 97751 8275535 07231 7364 84621 4294 51 10 1871 8539.491 3386513 1950 446555S 1179,581 2:-538 8266 50; 11 449 0682470,2G90| 69971 5625 8248 49941 6245 605 22401 49 12 41781 5643492 0610 93(2 586 1953..8811582 01391 6215'. 48 13 76751 91961 4224514 29801. 5699 559 26291 41t34,606 6192 47:14 450 1173471 2751! 7838!i 6658! 944(-; 6449 7980 4170! 46 15 4672 6306 493 1454 515 0388, 57 8194 560 0269 586 1828 8149 4,5. 16 8171! 9868 5071 4019, 6943 4(91 5726 607 21801 44 17 451 16725472 3420 8689 7702 588 0694 7914 9627! 6112 43 i8 5173 6978 494 2308 516 13S51 4445 531 178S 584 8528 60S OC:953 42 19 8676'473 0538' 5928 50691 8198j 55641 7431. 4(o8 ) 41 20 452 2179. 4C098 9549 8755.589 19521 9391585 1335 8367! 40 56838' 7659 495 8171 517 2441 570T7 62 8219 5241 669 2054i 389 91S8474 1222' 67941 61'29'94641 70481 91481 6(43 88 23 453 2694! 4785'496 0418! 9818540'221563 0879,586 8(156G610 0031 37 24 62011 8349 4043 518 3508!: 6980 4710 6965 41268 36 1i 9709475 1914 7669! 7199 541 07401 8543587 (876 8o199 35 1' 2 454 32181 5481 497 1297 519 C8911 45011564 2378 4783 611 23-14, 34 27.6728,'9048 4925 458t4 826! 3 6213i 8702 60Bll 33, 28 455 0238 476 2616 8554 8278542 2',27 565- 005058. 2616 612.-00(8 832 29:3750'6185 498 2185.520 1974 57.911 3888 65833 40C7 31 3,0 726'3! 9755 5816| 5671 95571 7728 59 04450, 80G8 30 81 456 0776 477 8326 949 9368'533 82324.566 1563 4369 613 2010 29 32 4290! 6899 499.3082 521 80671 7092 5411 82S9 6013 28 3'3'7806'478 04721 67171 6767544 0862! 9254,590 2211 614 00181 27 34 457 13221 4046'500 03521522 C463 461325617 8193' 6184 4C241 26 83,5 4889,:'7621 83989i 41701 841!4 694-1591 01(58 8082 25 8.6 8357.479 1197 7627] 7874 545 21771568 0791i 8984 615G 2041 24 387, 458 18771 4774 501 1266523 1578 5951 4639 7910 6052 23 388. 5971 8352 49061 528-8 9727 84881591 1839616 0064 22 89 8918480' 1932 8547 8990.546,351031569 23391 5768 4077 21 40. 459 2489/ 512 5.02-.2189 524 2698 7281' 6191! 9699 8(92 20 41 5962 9093 5882 6407 547 1060.570 (0045,593 3682 617 2108. 19 42 9486481 2675 94765525 0117 48401 38991 7565 6126 18 43' 460 3011 6.258-503 8121 8829] 86211 7755594- 1501 618 01455 17 44. 6587 9842 0768 7541;548 2404571 16121 5417 4166 16 45 461 0063482 8 3271504 04155526 1255 6188 5.4711 91375, 81881- 15 46. 3591. 701i4'4063 4969 99738 9831;595 8314 619 2211 14 47 7119 483 0601 771 8 868.5 549 8759572 3.192 7255 6236 1 48. 462 0649 41895905 18635827 2402.75471 7054,596 1196 620 0263 12 49 4179 7778 5015 6121) 550 1335!573-6918 5140. 4291 11 50 1.7710484.13681'8668 9' 9839 5125 47851 9)84 882) 10 51 463 12438'4959 506 2322528 856 0 8916 8649 597 86830 621 2851 9 52 47-76 8552 5977 7281.551 27(;8574 2516 6978 6383 8 53 8810 485 2145 9633 529 10i4,t1 65'32 6385 598 0926 622 0417 7 54 464 1845 5789507 32903 4727552 0297'575 02551 4877i 4452 6 55- 5382 9 341 69431. 84521 40931 4126 8828' 8i88 5 56.8919486.2981'508 0607T530 2178 78931 7999599 2781628 2527 4 57 465 2457' 6528! 4267 1 5b,6'553 31688576 1878 -6785 6566 3 58 35996 487 0126 7929t 9684 5488 5748 600 0691 624. 0607 2 59 9536 3726 509 1591!531 3364' 92881 96251 4648. 4650 1 60 466 3771' 7326 52a.41 71.94 54 3091!57 85031 8606~ 8694 0 | 5 | 4 630' | i GO2 | o 59~ I s | NATURAL COTANGENTS. 4,06 NATURAL SINES, l **o o,oo 7o6 1, 9 l~ 320 330 340 350 330 3 380 3 9'0 529 9193 544 6390 559 1929 573 5764'7' 7i853 601 8150 615 6615 629 02,4. 60 1 50 1659 8830 4340, 8147 588 266 6026 0478 892; 7] 5464' 9 2 -4125545 1269 6751574 0529 2'5;8 2795'616 11981 7724 58 O 6591 0707 9162 2911 46110 51171 0489 9983 57 4:9'587' 6145 560 15722 5292 7262 7459 5788, 60 2242' 56 5 51 15211 8583 9811 7672 6 9618 9760 8069' 45001 55 6 8 s 9 i986 546 1(720 6890 575 01 53 589 1964 663 26'80 617 059 6758' 54 7 6430 03456 8798 24321 4814i. 4400' 2648 90;15 58 8 8913 5892 561 12i;6 4811' 66631 6719, 4936 681 1272., 02 9 532 1376 8328 86141 71901 9!;12' 9088 72241 528 51 810 889 547 0'763 60211 9568 590 1361 6'04' 1856': 95111 5784 50 11 631a' 3198 828 5,6 1946' 37(9 3674618 1798: 80C9' 49 I2 8763 56;82.562 2(834 4323 6C57,: 5991i 4(846622 CM93' 48 1 33 1224 8863 89'i' 67001 8464 83(;8 6370 2547 47 14 3685 548 8499 5645 9076 591 60750 6C5 0624' 8655 4800 46 15 6145 2902 149 57 1452 3.96 2940 619'C.9 76 1253 45 13,. 86351 50365563 8C453 0827. 54421 52,555 8224 93(6, 44 1 5834 o5' 7797 ~ 2857 62'2:. 7871 7570 5507 633 15571 43 18 3523'549 (.228 5260 8573 592, 0182 9884 7790 38(9 49 19 5982' 2859 7663578 C950 2476 W66 2198 620 0673 60<9 41 20 8440 5' 911564 (1066 0823 4819 4511 2855 8310 40 21 535 0(898 7528 2467 5696 7163. 6824- 4636 634 059 922 31355 9950 4889 8069, 950C.'9186 6917 2881 8 23 6812 550 2,879 727) 579 C440 583 1847 607 14471 9198 506571 57 24 8268 487 9673 28121- 4189 3758 621 1478 73.5 06 25 536 0724 7286 565 2017( 5183 6580m 6069 3757. 95%58 35 26 179' 96638 4469{ 7558 8871 88797 6006 685 1800 84 27 5634 551 2918'1 68(68 9923 594 1211 6C8 0689' 8314 4046 33 28 889 451S 9267589 2292 8550'1 2 29028 622 C692 6292 32 29 37 ( 6944 566 1665 461 58ss9 5306 2870T 8587 1 0 2996, 970' 4062 7(00 228 7614 5146'66 6782 80 1 54419 5520 1795 6450 99897 595 C566 09221 742 81 6026 29 32 7902 4220 8856 581 1785 2904 69 2229', 998 52761 28 33 588 0354 6645567 1252 41132 52411 4585 623 1974 751,31 27 0-4 28{:; 1.(9 8-'648 649s 7;5771 6841 42481 9750, 26 35 525 5 58-1492 6043 8864 99131, 9147 6522607 1998 25, (76 8 0915' 8487582 12`0.596 224.9 610 1152 8796 4240 24 68-3 89 (1158 6~S 5 68 CS-3 4 95 487 015T 6: 86S (82 0595 4584i 3756 624 1(C9 6481 23 S b268 87 60 $225 5959 6918 6860 3042, 8721 22' 558 554 1182 5619 823 92521 83631 5614 638 8961 21 40 751 363 8811'1582- C687597 1586 611 66661.7885 82(1 20 41 9955 6.24 569 C4(8 38(:50 89191 2969 625 6156 544C 19 49 540 21(3 844-4 2795 5412 C251 5270 2427 7678 18 43 4851 555 8864 5187 T 7774 8588 7572 4896 9916 i1 44 7298 s 2s8 7577584 o116 598 c915 9873 636 9 21 16 45 9T415 570.2 99Fs23 2397 (`246 612 2178 92385 48901 5 46 541 2191 8121 530 2357 48571 5577 4473 626 15'8 6621 14 47 46871556 059 4741-7 7217 7906 6772 37,71 8862 13 48 7G082' 2956 7186I' 9577,599 02866i 9071 6(08'40 1(97 12 49: 95 27 583 9524 585 1936 253 613 1669, 8005 839 t42 1971 1912 4291 489 866 627C571 C 556 l 0'4 4415 557 (c2.6 4299 6652 7221 5964 2807 77069 9 )2 0859 2621 6686 9l01( 9549 8260 5162 641 OC82 8 93'2 58036 9178 586 1367 60Q 1876 614 (556 7366 2264 7 54 543 1744 7451 572 1409 0724 42''2 2852 9631 4496 6 65 4187 9815 84 680 6528 5147 328 1894 628 5 56 662855 22798 6229 84851 8854'7442 4157 8958 4 07 90)69 4692 86141587 07901601 1179 9736 6420 642 1189 3 8 544 1510' 7105 573 89908 61451 350(3 615 2029 8682 3418 2 59 0951 9517 3881 5499 5827 4322629 6943 5647 1.4826994, 564T21 60 390559 1929 576-14 7853 8150 6615 r 204 7876 0 5i.580 505" 540 53 590 510 (500 N-ATURAL COSINES. NATURAL TANGENTS. 467 | 2 0 33 | 34 3a | 3(30 30;. 3s |39~.2: 8'691 649,4')76 674 5J81- 700 230'51726 5425,1753 5-4t 781t 2856'309 7840 60 162 5 27-9 82121.9318 6411] 9871'i51 401302. 74542'810 2658 59 2 6 G83653) 2a350675:355:31701 0749.727 4.318, 4(66.S2 2229' 7478' 58 3 626 081i'.:6'99 779,) 5089 - 87i'G:' 9~:32, 69.19,811 2800 57 1 43831'651 -.06'3L 676 2'23[ 9:30);72 18 91,55 879T 76-3 16111 7124: 56:8935a 47741 ~6268 702 37731 7671J 83Ost9 6305 812 1951]. 55 6 627-2933'8918 677 0591 8118 729 2125 756 2941 784 10021 6780' 54 7 "7042 652:8o6411 4752703 24(641 6582i 7G514 5700'813 16111 53 8 62S:1098' 7211 899 1 6813730 1041 757 U9;) 785 04001 6444.! 52 9 I 5155653 133 678 3243 704 116l1 o501o 6668 5108814 12801 51 10 92141 5511 7492. 55151 9963 758 12481. 9S808 61138 50 11.629 3274 9363679 1741 1 9869 4423' 5829 78.6 4515815 0958 49 12 73836 654 317| 5993 705 4~224 88941 79 0413 9224 58o1' 48 183 6803-1899' 79T2 68)10216 8581 7382 33621 4999 787 8935 816 0646! 47 14 5.64('655 2129 4'591'7006 941 78'321 9587! 8649 54938 46 15 9533 628'is 8758 730)1783 23i3,76o9 4177 788 33646817 03431 45 16 631-3593 656 0417'681 3)16707 16611 67771 8169 80821 5195: 44 17T 7667. 46:)9 7276 6023834 1253761 8363 789 2802'818 0049 43 18 632 17:3' 8772 62 158, 708 08393, 57301 7959 7524 4905' 42 19 5310 657 29371 5311 4 4763735 30210 762 2557 790 2248 9764' 41 2) 933.:I:7103 683.0066 91331 46911 7157 6975819 4625' 4) 21 6.3838959 658 1271:433'.3709 35041 9174 763 1759 791 17031 9483 89 22 8)35 5441 86)1j 7878 736 3660.1 6363 6-1484823 43541 33 23 1634 2113 9612 63428710 287110 22531 147764 0969792 1167 922.2 37 24 6193 659 37385:'7143 663:) 737 2:6361 5577 5902821 41(9'' 36 25 6.35' 027i41 795)65 141S16 711 109' 7127.765 0188793 0640| 8965 35 26 4-357 66)3 2Ls6:691 539)C788 16720 4800', 5379 822 3401') 84 27 8I111 63'.31 9969; 9772; 6115. 9414794 01211 8718 03 62 G636 25T7 661 0492 66 4247 4712 41 739 -011 7 46 40331 4365823 3597 32 29 6B14 4673 8528' 851.3 5110 86. 149 96111 8479 81 830 637 0703.835 687.2810713. 2931 96111767 3270 795 4359 24 333-1 380 31'4793 662'8949.093' 7321740 41131 789:3, 9110 82511 29 32 883351 7225 6'33 1379 714 17121 8618 768 21517 796 3862 828 3141) 2S'33 638 2978 663 14831 566B6; 6106'741 31211.7144 86171 80881' 27 7073:: 5)11 9955.150 os7ui 7 61;3769 1773797 3374 826 2925 26 35 6839'1169.9792 639 4211 -' 4.393742 21431 640-4 81341 7821 25.36 5237 664 39341 88331 92971 6655770 1037798 2895,827 2719 24 37 936.3 8178 69) 28:32 716 8693 743 11701 5672 76259. 762 23 38 643 3487 665 2373' 712 81)0 5636 771 0339 791 2125 828 2523 22 89 1 759'..670 69L 142,5'17: 2.5')5 741 02o41 494- 7193' 7429 21 43) 641:1673 663. 0769' 57251 69911 472 1 9589 800 1963 829 28377 2.) 41 5i79'9'94939692 0026718..13191 9246'772 4233 6736 - 7247, 19 43.933'S3 9171 4:32,8 57229 745 3? 701 8 3S78801 1511 830 216; 18 43 642 39) Gti- 3371 863:3 719 01411 - 8293 773 8526 62S38 7075' 1.7 41 6 305 75s'693 693239 45514746 282.1 8176.802 10671831 1992' 16 45 64'-3'221 665 31783. 721T 89701o 73)-1, 774 23271 5849 6912 15 46 6329 595 69-4 1557 723 3833S7747 18831 7481803 0683282 1834' 14 47 644 0-11'669 02.)5' 5868! 733i861 642 775 2137 5418 6759 113 48 453!' 4417 695 0181;721 227 748 09561 6795 834 C206 8833 1686t, 12 49[ -8'g38.863301 48 963 6 653 5494776 1455a 4997 66151 1 53 645 2797 670 2315 8813 722 1075 749 00331 6118' 9790 834 154 7 1.0 51 6918' 7i61 696 13131 55032 4575777. 07S S05 48 81.64 S1 9 52 6-16 1041671 1235)' 74511::.9933) 9119 54-48 9: 382 835 14181 8 )3 | t5165 5599 697 1773.723 43Gli753 3665 77 0117 806 41811 6357 7 51 9291.9721',6971 7931 8s211. 4788 8983 308 1298 6 53 617 3117,672 3941-698'0422T272 32277751 27621 9460 807 378S7 62-12 5 56 |.7:61 81691 4749 76631 7314 779 4135 85938837 11SS 4 57 618 1676673 2:393 9378'725 2101,752 18671 8812808 349!11 613 18 3 8 -5S)S 6624699' D09 654)0 6423780 3492; 8212-888 1(8I 2,59 31 67T4 0851'i 7741 726.982, 753 ~ C9311.' 817 809 302f (6041'1 6) 19 4?76 5)s700 207.52. -42 5541781 2356 7840S839 0996: 0O. 7". _ Gi" _ -~~" 5-il"- | 53" S | 5 | 1 | L I 37 5' 54' 53 I 2 I sof |ATURAL- COTANGENTS.. *40S JNATURAL SlNES. 406 41: 42 | 4 | 44 | 40 4' 47 |' 0 ~ 642 7876; 656 0.590 669'1306 681 9934 694 6584:707 - 1068 19 8898 731 85387 60 1'643 01041 2785 3468682 2111 - 8676 81-24 541;8 5521 59 2 23382 4983 5628 42387 95 0767 5180.:7438 7503 58 3 4559' 7174 7789 633 28581 72386:9457 9486 57 4 6785f 9367 99481 8489 4949; 9291 720 1476 782 1467 56 5 9011657 1560 670 2108 683 o013 7039'708 1345 3494 449 55 6 644 1:236 8752 4266' 2738'9128 898 5511 54291 54 7 3461 59441 64241 4861 696:12171 5451: 7528 7409 53 8 5685 - 8185 85821 6984 -051 7504 9544 9388 52 9 7909 658 0826 671 0739' 9107 53921 9556721 1559 733:1367 51 10 6415' 0182 25161 2S95'684 1229'7479.709 1607 3574:3345 50 11 2355 4706 5051 83350 95651 3657 5589 5322 49 12 4577 6S951 7206 5471697 16511 5707 7602 7299 48 13 6798 9083' 9361 7591 3736' 1757 9615 9275 47 14, 9019 659 1271-672 1515. 9711 5821 9806 722 16287834"1250 46 15'646 1240 8458: 3668 685 1830 7905710"18541 3640 3225 45 16 3 84601 56451 5821 3948[ 9988 3901 5651:5199 44 17 5679'78311 79731 6066 698 2071-:'5948 7661'7178 43 18 7898660 001'7 673 0125, 8184 4153 7995 9671 9146 42 19 647 0116 -22021 2276 686 0300 6234 711: 00411723 16817385 1118 41 20 2334 4886 4427 2416 8315~ 2086 3690 3090 40 21 4551 6570 6577 4532 99 0396 4130 5693 5061 39:22 6 6767 4 8 74 07271 66471 2476] 6174 7T105 7032 38 28 8984 661 0936-674 0876 8761 4555 8218 9712 9002 87:24 1648 11991- 8119 024687 03751 6688712 0260724 1719786 0971 386 25 8414 5300:51721 298: 87111 28038 87241 2940 35.26 628 74821 7319 5101 700 0789 4844 57291 4908 34 27 78421: 96621 9466i 7213 - 2866 6885 77841 6875 33 28.1649 0056662 1826T75:16121 932,5 49421 8426 9738 8842 32 29 228 4022 8757688 1t435 701871'8 0465,725 17417T37 0808 131 30 4480 6200 5902 85463 9093: 2504 3744[ 2773 80 81 6692 8379. 046 5655701 1167 45481' 5746 4788 29 32 89033663 0557 67`6 0190 7765 3241 6581 77471 67081 28:83 1650 1114. 2784{ 22333'98731 5314 8618 9748 8666 207 34 8824 4910 44760689 19811 73877114 0655726 17483788 0629 26 85 5583 7087 6618! 40891 94591- 2691 3748 2592 25 36 7742 9262 87T60 6196 072 1531 4127 5747 4S53 24 87 9951 664 1487 677 0901' 8802 36011 6762 7745. 6515'283 38'1 651 2158 3612 8041'695 040o'71 721. 8796 97438 8475 22 389 4866 -578 5181 2512 77411715 08380127 174017839 0451 21 40: 6572 7959 7320 46f71 9811 87361 2394 20 41 0 8778665- 0131 94591 621 703 1879 5 43531 19 42 652 0984 2304'678 1597 88241 047 6927 7728 6311 18.43' 8189 4475 38784691 09271 60141 8959 9722 8268 17'44 ] 53941 66461 5871 3029 80811716 0989728 1716'740 0225 16 I45:1 7598 8817 80071 513104 0147: 38019 3710 2181 15 46 9801.666 0987679 01483 7232 2218:5049 517031 41871 14 47 1653 2004 31.56 2278 98.32. 4278 70781 7695 6092 18 48 4206 5325 4413692 1432 6342 9106 9686 80461 12 49 6408 74938 6547 8531 8406717 11841729 1677 741 0000 11 50 8609 9661 8681 5630 705 0469 3161 3668 1953 10 51 1654 0810 667 1828 680 0818 7728 2532 5187 5657 89051 9 52 8010 8994 2946 9825 4594 7213 7646 5857'*8 53. 5209 6160 5078:693 1922 6655 9238 9685 7808 7 54 7408 8326 7209 4018 8716718 12681730 1623 9758'6 55 1 96071668 0490 93391 611470 07161 83287 36101742 1708:5 56 1655 1804 2655 681 1469 8209 2838 5310'5597 8658 4 57 4002 4818 3599 694 03804 4894 73881 7588 5606 3 58 6198 6981 57,281 2398 695381 55 9 5681 754 -2 59 9 8895'91441 856 4491 9011719 13771781 1558 9502 1 60 656 0590669 1306 99841 6584 07 1068:88981:8537741448 0 1:49~ 1 -48 47~:46~ 45~ -4~ 43~ 44 3 42~ NATURAL CGOSINES. NATURA L TANGENTS. 469 400 41_ _ 42 | 430 440 450 460 47 O 839 0996 869 28671900 4040982 5161 965 6888 100 000001-08 55303 107323687 60 1 5955- 79761 9309933'0591 966 2511 05819 61333 29943 59 2 840 0915870 3087 901 45801 6034 8137 11642 678367 86203 58 8 5B781 82001 9854934 14W9 967 3767 17469 73404 42467 57 4 841 0844 871 3016 902 51311 6928 9399 23298 79445 48734) 6 5 5812 8435 938 0411 935 2380 968 5035 29131 85489 55006 55 6 842 0782 872 35561 56931 7834 969 0674 34968 91538 61282 54 7 5755' 8680,994 0979936 8292 6316 40807 97589 67561 53 8 813 0780873 38061 62671 8753 970 1962 46651 104 08645 73845 52 9 5708' 8935'905 1557 937 4216 7610 52497 09T04 80132 51 108 44 0688 874 40671 68511 9683 971 8262 58348 15767 86423 50 11| 56701 9201 906 2147 938 5153 8917 64201 21833 92718 4, 12 845 0655 875 4838' 7446939 0625 972 4575 70058 27904 99018 48 13,56431 9178907 2748' 6101 978 0236 75918 83977 108 05321 47 14 816 0683 876 46201 8053940 1579 5901 81782 40055 11628 46 151 56251 9765'938 38860! 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6846 62638.550.6959 5241 6208 89212 55128 34912 11 50 884 1926 895 1506 927 0914 960 0829 994 1991. 95203 61841 41865 10 51! 7009 6747! 6324 -6421 77771 103 01196 67558 47823 9 52865 259 896 1991 928 1738 961' 2016 995 8566 07194. 78779 54284 8 53 7181| 72381 71541 7634 9D58 18195 80004 60750 7 54 866 2272 897 -2487 929 2573 962 8215 996 515 19199 86233 67219 6 55 786 5 77391 7996;1 8819 997 095 25208 92466 783693 5 56 867 2460 898 2994 933 84-21 963 4427 6756 81220 98702 80171 4 57. 755 8 82511 8849964 0037 998 2562 37235 1 07 0494 86653 3 58 868 2659 899 8512 931 4280! 5651 8371 43254 1118 93140 59. 7762 87T51 9714965 1268 999 4184 492T7 1748 99680 60 869 2867 900 4040 932 5151 68881000 l 000.55803 286871106125 0 490 480 | 47 | 40040 44 430 42.| NAT.VTRIAL CQTANGENTS. 40 470 N~ATURAL SINES. 480 490 500 510 520 530 540 0-74 1448 1754 7096'76 0444 777-1460 788 0108 798 6855'809 0170 61, 1 3894 I. 9004 2314 3290 1898 8105 18709 59 2 5340 755 0911 4183 5120 3688 9855 3588 58 8 7285 2818 6051 6949 5477 799 1604 5296 57 4 9229 4724 7918 8777 7266 3352 7004 56 5 7441 1173 6683, 9785 778 0604 9054 5100 8710 55 6 3115 8535,767 1652 2481 789.0841 6847 810 0416 54 7 5358 756 0439 3517 4258 2827 8593 2122 53 8 6999 2843 5882 6084 4413 800 0338 8826 52 91.' 8941. 446 7246 7909 6198 2083 5580.51 10 745 0881 6148 9110 97833 7988 3827 7234 50 11 221' 805. 768 0973.779 1557 9767 5571 8988 49 12 4760 9951 2885 380 j793 1550 7814 8110638 48 18 6699 757 1851 4697 52D2 3838 9056 289 47 14 8686 6751 6558 7024 51.15 801 0797 4041 48 15 746 0574 5651 8418 8'8845 6896 2588 5740 145 16 2510 7548 769 0278 783 0665 8676 4278 7489 44 17 44436 9443 2187. 2485 791 0456 6018 9187 48 18 6382 T758 1843 3996 4304 2235 7756 812 0835 42 19 8317 3240 5853 6123 4014 9495 2532 41 20 747 0251 5136 7710 I 7940 5792 802 1282 4229 40 21 2184 7081 9567 9757 756 9 5925 39 22 4117 8926 770 1423 781 1574 9345 4705 720 38 23 68849 759 9820 3278 3390 792 1121 8440 9314 87 24 1 7981 2713 5182 5205 2896 8175 813 1008 36 25 9912 4606 6986 7019 4671 9909 2701 35 26 i748 1842 6493 8840 8883 6445 8083 1642 4893 34 27 8772 8:389 771 0692 782 064 8218 i 338875 6084 33 28 5701 1760 0283 2544 2459 I 9990 5107 7'775 32 29 7629 2170 4395 4270 793 1762 6888 9466 31 30 9557 4060 6246 6082 83533 8569 814 1155 80 81 749 1484 5949 8096 7892 5804 804 0299 2844 29 321' 3411 7887 9945 9702 7074 2028 4532 28 38 5887 9724 T772 1794 1783 1511.8843 3756 6220 27 34 7262 761 1611 83642 8820, 794 0611 5484 7906 26 35 9187 3497 5489 5127 2379 7211 9593 25 36 750 1111 5088 7836 6935 4146 j 8938 815 1278 24 87 83034 7288 9182 -8741 5913 805 0664 2983 23 381 4957 9152 773 1027 784 0547 681 7 2389 4647 22 39 6879 762 10386 2872 2852 9444 4113 6830.21 40 8800. 2919 4716 4157 t795'1208 58371 ~ 8013 20 41 751 0721 4802 6559 5961 2972 75601 9695 19 42 2641 6683 8402 7764 4735 9283 816 1876 18 4.3 4561 8564 774 0244 9566 6497 806 1005 3056 17 44 6480 763 0445 2086 785 1368 8259 2726 4736 16 45 ~ 8398 2325 3926 8169 796 0020 4446 6416 15 46 752 0816 4204 5767 4970 1780 6166 8C94 14 47 i 2233 6082 76068 6770 8 3540 7885 9772 13 48 4149 7960 9445 8569.5299 9803 817 1449 12 491 6065 9888 775 1288 786 0867 7058 807 1321 38125 11 50 7980 784 1714 8121 2165 8815 8 3038 4801 10 51 98941 3590 4957'1 8963 797. 0572 4754 6476 9 52'1753 1808 5465 6794 5759 2829 6470 8151 8 58 8721 7840 8629 7555.4084 8185.9824 7 54 5634 9214 776 0464 9850 5839 9899 818 1497 8 55 7 546 1765 1087 2298.787 1145 7594 808 1612 3109 5 56 9457 2960 4132 2939. 9847 3825 4841 4 57 1754 1368 4832 5965 4732 798 1100 5037.6512 3 58 3278 6704 7797 6524 2858 6749 8182 2 59 5187 8574 9629 8816'4604 8460 9852 1 60 7 8096 T66 0444 777 1460 788 0108 1 355 809 0170 19 1520 0 41 oI 40~ ] 39~_ 38~ 370 36~,-, NATURAL COSINES. 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NATURAL TANGENTS. 4 ~62"630 1 64 66 6 CS 0 1'88 07265 196 26105 2'05 03038 2-14451069 2'24 60368 2'35 58524 2'47 50869 60) 1 20410 40221 181851 61366i 77962 77591, 71612 s9 2 33693 54864 338149 77683 95580 96683o 9386 58 3' 46924 68518 485311 94021:2'2513221 2'36 15801 2'4813190 57 4 60172 82688 63732 2'15 10378' 30885 34946 84023 56 5 73436 96874 78950 26757 48572 54118 54T87 55 6 83713 197 11077 94187 43156 66283 7831(3 75781 54 }1.8 jls 00o066`7 iii~l 6 7 22~ 0' 8 I 3 9 ~ 40 5 3 27 189 00006 25296 2.06 69442 59575 84016 92540! 96706 53 8 13313 39531 24706 760152'2601778 287117912'4917660 52 9 26635 53782 40008 92476 19554 81068 38645 51 10 39971 68350 55318 2.16 08958 87857 50372 59661 50 11 53322 82334 70646 25460 55184 69703 80707 49 12 666881 96635 859941 419883 73035 89060 2'50 01784 48 13 80068 1'98109522'07 01359 585271 90909 2'3808444 22891 47 14 / 93464 25286 16743 75091 2'208807 1 27855 4402 46 15 1'9906874 39636 32146 91677 26729 472931 65198 45 16 20299 54003 47567 2'17 08283 44674 66758 86398 44 17 33738 68387 63007 24011 62643 86250 2'51 07629 40 18 47193 82787 78465 41559 80636 2039 05769 28890 42 19 60663 97204 93942 58229 98653 25316 50183 41 920 74147 1'99 11637 27i208 09438 74920 2'28 16693 44889: 71507 40 21 87647 26087 249583 911631 84758 644901 92863 39 22 119101162 40554 40487 2'18 08364 52846 84118 25214249 38 23 14691 55038 56039 25119 70959 24003774O 35667 37 24 28236 69539 71610 41894 89096 234571 57117 36 25 41795 83056 87200 58691 2'29 O7257 43168 78598 35 26 55370 938590 209 02809 75510 25442 62936 2'53 00111 34 27 6896012'0013142 184371 92349 43651 82672 21655 33 28 32565 27710 84085 219 09210 61885 2'41 02465 43231 32 29 96186 42295 49751 2'6090 81143 22286 64889 31 30 1'9209821 56897 65436 42997 98425 421061 86479 30 01 23472 71516 81140 59923 2 00 16732 62013 2.54 08151 29 82 07138 86153 96864 76871 85064 81918 29855 28 33 50819 201 008062'I 1126071 9:3'840 534201242 018511 51591 27 34 64516 10474T 28369 2-21 10831 71801 21812i 70059: 26 85 78228 00164! 44150) 27840 93206 418)11 951301 25 36 91906 44869 59951 44878 2031 08667 61819 2-5516992 24 07 1-93 05699 59092 75771 61934 27192 81864 88858' 20 88 194575 74001 91611 79312 451/T 2-43 019081 60756j 22 39 93201 890882 11o0747o 96112 64176 220411 82686 21 40 47023 2'0200862 20348 221 10234, 82606 42172 25604649 2) 41 60825 18654 39246 80079 2'32 011601 62381! 26645 19 42 74645 33462i 55164 47545 19740, 82519: 48674 18 43 88481 48289 71101 64703 88345 244 02736 707835 17 44 1.94 0233 60133 87057 81944 5'6975 22982 92810 16 45 16200 7799012'12 03034 99177 75630 48256 257 14957 15 46/ 800883 92870 19030 222 164032 94311 63559; 87118 14 47] 43981 2,00 07769' 359461 00709 2033 10017 838911 59312 13 48 57896 22683 51082 51009 31748 2'45 04252 815912 49 71826 37615 67107 68331 50505 24642 2'58 03800 11 50 85772 52565 882138 85676 692871 45061! 26094 10 51 99733 67532 99308 228 00043 88095 65510 48421 9 52 195137111 82517.2.1315423 20433 203406928 85987 707827 8 53 27704 97519 1559 784 25787246 06494. 112TS45i 25T8T 22-46 06494 98177 54!41713 2'0412540 47714 55280 44672 27030 2'59 15606 6 55 557391 275781 63890 72738 635821 47596 30868 5 56 69780 426 84 8 3085 91,218 82519 681911 60564 4 57 88837 577081 93301 2.24 0772112035 01481 88816 83895 3 58 97910 72800 2'14 12537 25247 204692'47 09470 2'60 05659 2 59 1'9612000 87910 28793 42796 09483 30155' 28258 1 60 26105 2'0508038 45069 60368 58524 50869 50891 0 271 266 250 240 23~ 22 I 210 NATURAL COTANGENTS. 476 NATURAL SINES. 69~ I 70~ 1~ ( 7-2~ 7.3 74~ 5 0 98 5804 1939 6926 945 5186 191 0565 956 3048 961 2617 965 9258 60 1 6846 7921 6182 1464 3898 8418 966 0011 59 2 7888 8914 7078 2361 4747 4219 0762 58 8/- 8928 99)7 802:3 8258 5595 5019 1513 57 4 9968 940 0899 8908 4154 6443 5818 2263 56 5 934 1007 1891 9911 5050 7290 6616 3012 55 6 2045 2881 946 0854 5944 8136 7413 8761 54 %..I 8082 871 1795 6838 8981 8210 4508 53 8. 4119 4860 2736 7731 9925 9005 5255 52 9'5154 5848 8677 8623'957 0669 9800 6001 51 1 0 6189 68435 4616 9514 1512 962 0594 6746 50 i 722:8 78922 5555 952 0404 2354 1387 7490 49 12 8257 8808 6493 i 1294 8195 2180 8284 48 18 9289 9 79093 7430 2188' 4085 2972 8977 47 14 9. 0321 941 07.77 8366 8071 4875 83762 9718 46 15 13529 1760 9001 3958 5714 4552 967 0459 45 16 2382 2743 947 0236 4844 G552 5642 1200 44 17 3412 8724 1170 5730 7889 6130 1989 43 18 4440 4705 2103 6615 8225 6917 2678 42 19 5408 5608 3035 7499 9)61 7704 3415 4L 20 6495 6665 3966 8882 9895 8490 4152 40 2i 7521 7644 4897.9264 958 0729 9275 4888 89 221 8547 8021 5827 953 0146 1.562 963 0060 5624 38 208 9571 9598 6756 1027 2394 0843 6358 87 24 986 0595.942 0575 7684 190)7 226 1626 7092 36 251 1618 1550 8612 2786 4056 2408 7825 85 26 26411 2525 9588 3664 4886 3189 8557 31 27 8662 3493 948 0464 4542 5715 3969 9283 88 28 4683 4471 1889 5418 6543 4748' 968 0018 82 29 57030 5444 2;313 6294 7871 5527 1 0748 4 30 6722 6415 3237 7170 8197 6305 1476 30 77 1 T40- 786 4159 8044 9023 7081 2204 -29 2 8758 8055 5081 8917 9848 7858 2931 28 833 9774 4 90924 6002 9790 959 06721 8633 3658 27 84 937 0790 943 0293 6922 954 0662 1496 - 9407 43883 26 1806 1260' 7842 1533 2818 964 0181 5108 25 86 2820 2227 876.0 2403 8140 1 6954 5882 24 87 883 381'92 9678 8278 3961 1726 6555 23 08 4846 4157 949 0595 4141 4781 1 2497 7277 22 89 5858 5122 1 511 509 5600 826S 799S 21 40 68 G9 6085 2426 1 876 6418 4037 8719. 20 41 7880 7048 0841 6743 7236 4806 9488 19 42 8889 8010 4255 7608 8058 5574 969 0157 18 48 9898 8971 5168 8473 8869 6841 0875 17 44' 938 (:9:}6 9931 608) 9886 9684 7108 1593 16 45 1913, 944 0890 ( C991 955 0199 960 0499 | 7873 2309 15 46 1 2920 i1819 79012 1062 1: 12 8638 8025 14 47] 3925 2807 8812 1923 2125 9402 3740 13 48 4930 0764 9721 2784 2987 965 0165 44503 12 49 1 5984 4720 950 0629 364:3 8748 0927 5167 11 50 6988 5675 1586 4502 4558 - 1689 5879 10 51' 7940 6630 2448 0 5861 5368 2449 6591 9 52 8942 7584 8848 1 6218 6177 8209 7801 8 58 9943 8537 4253 7074 6984 3968 8011 7 54 909 09483 9489 5157 7980 7792 1 4726 8720 1 55 1942 945 0441 6061 8785 8598 5484 9428 5 26 2940 1891 6968 9689 9403 1 6240 970 0186 4 57 3988 2341 7865 956 0492 961 0208. 6996 842 0 3 58 1 4935 3290 8766 1845 1012 7751 1548 2 59 5981 4238 9'66 2197 18151 8505 2258 1 60 6926 5186 951 0565 8048 2617 9258 2957 0 20' 1 1 17~ 16~ 15~' 14 NATURAL COSINES. INATURnATL TANGINTS. 417 G9~ 70~'|71~ | 2 | 73~ 7 274~0 75 0 2-6050891 2'4 74 74774 29042109 8-07 76835 3-27 08526 834874144 3-'78 20508 60 1 735581 996611 69576 308 07825 42588 8349 124701 63980, 59 2 96259 275 24588 97s9 38 79 76715 5087437407546 58 8 2'61 18995 49554 291 216491 684688 88 1(9071 893;56 51207 71 4 41766 74561 52256 99122 451643'5027916- 94963 56 5 64571 996081 799-;98'39298311 79487, 66555,83'753 88815 55 6 87411 2-76 24695 2-92 07610 60596 8-29 13876 8-51 052781 82768 54 7 2-62 10286 49822 853858 91416 4888030 44070 3876 26807 58 8 38196 7499(0 63152 8310 222911 828511 829461 7{;947 52 9 56141 2'77 00199 90995! 532238 383017438 3-52 21902 377 15185 51 10 79121 25448 293 18885' 84210; 52091' 6(79881. 59519 50 11 2-63 02136 507881 46822 8311 15254! 86811 8353 000548'78 039511 49 12 25186 76069 74807i 46853 831 21598'9251 484811 48 18 4827112'78 01440 2 94 02840 775(9 56452' 78528 931089 47 14 71392 26853 80921 8312 08722 91373 8354 17886 8379 37885' 46 15 94549.52307 5919501 89991 303226362 57089251 82661! 45 16 2'6417741 77802: 872271 -73171 61419 96846,8.80 27585 44 17 40969 2'79 08389 2'95 15453 3'13 (02701 96543 3855 86449' 726(9 43 18 64232 289171 48727 84141 38'1 81786: 7618 3'381 17738' 42 19 87T531 545871 72050 G65689 66997 356 156900 62957, 41 20 26 6910867 80198 2'96 00422 97194 3034 02326 55749 882 08281i 40 21 84238 2'80 05901 28842 3.14 28807 T 37724 95(81 53707 89 22 57645 0 316463 5;7812 (60478 73191 357 356961 99233 88 23 810891 57483 858831 92207 8-35 C8728 75794 388 44861 37 24 2'66 045691 83263 297 14899 3 815 23994.4433888 358 15T9751 95911 86 25 28085 2'81 091341 43016' 558401 80008 5G241 3.84 36424 35 26 51638 35048 716838 87744 o-36 i5753' 9f;5981 82358 34 27 75227 61004 298 00400 316 19706( 51568 359 87024 38528~,96 33 28 988531 87003 291671 51728 87453 77541 745T37 82 29 2-67 22516 2'82 13045 57988 8f3 8 3837 234(:8 8360 18146 386 20782 31 80 46215 89129 86850 3 817 159481 59484 588351 671311 380 31 69951 65256 2'99 15766 481471 955311 996f.9 3887 13584 29 32 937251 91426 44734! S0406 8388 31699 3'61404691 60142 28 33 268 17535 2883 17639 73751 3-1S 127241 67988 81415. 388 06805 27 84 413881 48896 3800 (2820 45102 8'89 04249 3'62 224471 58574' 26 85 652671 701961 31909 77.540 403681 63566 3'89 0C448 25 86 891901 96589] 611(99 3'10 089 77085 3'68 04771 47429 24 387 269 13149 284 229261 9038301 42598 8340 13612 46064 94516 23 38 871471 49356 3801 19603 752171 52101 87444 890 4171(0 22 89 61181] 758311 48926 3,20 ()7971i 86882 364 289111 89011 21 40 85254 2s85 02849 783301 40638 3'41 23(626 70467 8 91 86420, 20 41 2'70 093641 28911 3802 07728 734401 60443 3865121111 839378 19 42 335131 55517 87207 3821 06304 973833 538448392 31560 18 48 576991 82168 6673871 9228 3842 342971 956651 79297 17 44 81923 286 08863 968320 72215 71384 386687575 898 271411 16 45 2-71 06186 85602 303 25954 3-22 05263 8-48 08446 795751 7,5094 15 46 380487 62386 55641 388373 456381 367 2166538'94 231571 14 47 54826 89215 85381! 715461 82891 638451 713831 13 48 79234 287 16388 -04 15173 3283 0478803'44 28f226 3868 06115 395 19615 12 49 2-72 036201 43007 45018 88378i 57685 484751 68011 11 50 280763 699170 T4915 71488 95120 90927;896163518 10 51 525691 96979 3'05 04866 3'24 04860 3-45 32679 3869 834691 651871 9 52 77102 2888 24383 34870 38346! 70315 76104 397 18S68 8 53 2783 01674 51132 64923' 71895 3'46 08026 3870 188301 62712 7 54 26284 78277 95038 3825 05508 45813 61648 3'98 11669 6 55 50934 2'89 05467 3'06 25203 39184' 88676 3871 04558; 60739, 5 56 756283 827041 554211 72924 8-47 21616 47561 3899 09924 4 57 2'74 008521 599841 85694 32G 06728 596382 90658' 59223 3 58 251201 8s3143071602 41596 97726 372 38847 40008686 2 59 49927 2-90 14688 464001 74529 3-48 85896 771311 58165 1 60 74774 42109 76835 827 085263 74144 3-7320508 40107809 0 20~ 19~ 18~ I 17~ 16~ I 15~ I 140~ NATURAL COTANGENTS. 478. NATURAL SINES. I 6 79- SO 81 ( 8-2- / 0 970 2957 974 701 978s 1476 981 6272 9848 078 9876 888S 9902 681 60 1 8661 4855 2080 6826'582 9877 838 9903 085 59 2 4363 3008 2684 7380 9849 086! 792 489 58 3 65065 5660 3287 7933 589 9878 245 891 57 4 5766 6311 3889 8485 9850 091 697.9904 293 53 5 6466 6902 4493 903 8 593 9879 148 694 5 6 7165' 76(12 5090 9587 9851 093 599 9905 095 54 7 783 8231 568 9 9S2 0137 598 9889 048 494 53 ~I 9880 048:~~~~~~~~I 494 53 8 8561 8939 6288 0686 98 52;92 497 893 52 9 9258 95,56 6886 1234 59,9) 945 9'.906 290 51 10 9953 975 0203 7483 1781 9853 087 9881 8392 687 50 11 971 0649 0849 8'i79 2327 583: 838 9907 083 49 12 1343 i494 8674 2873 9854 079 9882 284 478 48 13 20336 2138 9268 3417 574 728.873 47 14 2729 2781 98(012 3961 9855 068 9883 172 9908 266 46 15 3421 3423 3J79 0455 4504 561 6 15 619 45 16 4112 4965 1047 5046 9856 053 9884 057 9909 051 44 17 48032 4706 1638 5587 544 498 442 43 18 i 5491 5345 2228 6128 9857 035 939 802 42 19 6180 5985 2818 6668 524 9885 878' 9910 221 41 20 6867 662, 3406 706 9858 018 817 610 40 21 7554 720' 3994 77?44 501 9886 255 997 39 22 8240 7897 4581 8282 " 988 692 9911 384 38 28 8926 85338 5167 8818 9859 475 9887 128 70 387 24 9610 9168 5752 9353 960 564 9912 155 36 25 i92 0294 9802 6337 9888 9860 445 998 540 85 261 0976 976 0435 6921 983 3422 2 929 9888 432 92 34 27: 1658 1068 7504 0955 9861 412 865 9913 306 33 28 28339 1699 8086 1487 894 9889 297 688 32 291 83320 2330 8668 2019 198G2 375 728 9914 069 831 30 3699 2. 963 9247 2549 856 9890 159 449 30 31 4378 3589 9827 3079 9863 886 588 828 29 32 5056 1 4218 1980 0405 3608 815 9891 017 i9915 203 28 833 5733. 4845 0983 4136 9864'293 4.451 504 27 34 6409 5472 1560 4663 770 872 [ 911 26 35. 7084 60983 2136 5189 9865 246 9892 298 9916 337 25 36 7759 6723 2712 5715 I 722 723 712- 24 37 8482 71347 2386 - 6239 9366 193 9893 14S 9917 006 23 38 913.5 7970 3860 6763 670 572 459 22 89 9777 8593 4433 7286 9867 143 994 832 21 40 973 04549 9215 5005 7808 615 9894 416 9918 204 20 41 1119 9886 556 830 9868 087 88 574 19 42 1789 977 0-456 6147 8850 ~ 557 9895 258 944 18 43 24581 1075 6716 9370 9869 027 677 99919 314 17 44 3125 1693 7285 9830.496 9896 096 I 63 16 45: 3793 2311 7853 1984 0407 964 514 9920 049 15 46' 4459: j 2928 8420 0(924 9870 431 931 416 14 47 5124i - 8544 8986 1441 897 9897347 782 13 48'5789. 4159 9552 1956 9871 363 762 9921 147 12 49 6453 4773 981 0116 2471 827 9898 177 511 1t 50'7116 5387 0680 2985 9872 291 590 874 10 51 7778 5999 1243 3498 754 9899 003 9922 237 9 52 8439, 6611 1805 4310 9873 216 415 599 8 S3 9100 7222 2366 4521 678 826 99 T 54 9760 7832 2927 5032 9874 138 9909 237 9923 319 6 55 9T4 0419 8142 3486 5542 598 646 6T9 5 56 1077 9050 4045 6050 9875 057 9901 055 9924 037 4 57 1734 9658 4603 6558. 514 462 394 3 58 2390 978 0265 5160 7066 972 869 751 2 59 3046 0871 5716 7572 9876 428 9902 275 9925 107 1 60 8701 1476 6272 8078 88 681 462 08 13' 12 111 l 90 1 _ NATURAL COSINES, NATURAL TANGENTS. 479 i 76 7 78~ 79 80~ S1 82o 0 401 078091488 147594-70 46301 514 46540 56 712818 683137515T7 153697 60 1 J 575701 72316 4-71 18686 515 2-557 8094461 256601 804190 59 2 4'02 07446 4'34 30018 81256 5'16 05813 906394 376126 455308 58 8 1 574401 87866 472 49012 86811 57 008663 496092 6070T56 57 4 4-038 07550 4385 45861 478316954 517 67051 1012,56 616502 759437 56 5 57779 4'36 04008 850883 518 48085 199173 737859 912456 55 6 4'04 08125B 62293!474 53401 519 29264 297416: 8586651 72 063116 54 7 58590 437 20781 4-75 21907 520 10738 895988 989422 220422 58 8 405 09174 79817 90608 92459 494S89 6 410263388 375878 52 9 59877 4 38 38054476 5949015'21 74428 594122 225301 15,30987 51 10!4'06 10700 969401477 28568 5'22 56647 693688 8484[28 687255 50 11 61648 4'89 55977 978837 523889116 793588 472017 844184 49 12 407 127074'40 1516414-78 67300 524 21886 898825 596070 173 001780 48 18 63892 74504 4'79 86957 5'25 04809 994400 720591 160047 47 14 Q08 15199 4'41 38996 4'80 06808 8803515'8 095315 845581 318989 46 15 666271 986411 76854 52671517 196572 971048 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968799 606423 15 46 1425 2392314.61 21908 5-03 499-85 92740 51508356.9 11C859 789489 14 47 79501 86788 5.04 26T001 554 850521 6282172 252489 973s96 13 48 1426 3521814'62 51832 5'05 08690 5.55 776631 741865 895192 7'9 158151 12 49 91072 4.68 17056 80907 5.56 70574 855867 538847 843758 11 50 14.2747066 8245T7 506583525.57 638786 9T0279 6082385 530224 10 51 1428 03199 1464 48034 5.07 36025 558 57802)62 82 8106 826781 717555 9 52 59472 465 13788 508 1892315-5951121' 200847 971806 905756 8 58 429 15885 797211 92061 5'60 45247 816007 7'0 117411:830 0948385 7 54 T72440 46645832 509 704261561 39680G 482086 268662 2847906 6 55 1480 2918614671212415'10 49024]5-62 3442.1 548388 4104829 475647 5 56 85974 785951511 278555.-63 29474] 665515 557905 667394 4 57 4'31 42955-163 452481512 0692115'64 24888 782868 7059384' 860042 8 58 432 00079 4'69 12088 8622415 65 20516| 900651'85457788,1 058599 2 59 57347 ~ 79100/513 657 66163156 1650961S 8 66.003826 248071 I 60 43388 14759 4,70 4680115 14 45540 5-67 12818 137515 1536971 448464 0 I 12 1 1 1)~ oi 8 ~ t NATURAL COTANGENT'. 4S0 iNATURAL 5NIES. 830 84 0 850 860 870 880 8 90 019925 462 9945 219 9961 647 9975 641 9986 295 9993908 999S 4T7 60 1 816-1 521 9962 200 843 447 9994 009 527 59 9926 169 825 452 9976 045 598 110 1 577 58 3 521 9946 127 704 245 748 209 625 57 4 873 428 954. 445 8 981 308 673 56 519927 224 729 9563 204 645:1 9987 (46 405 720 55 6 573'9947 028 45,3 843 194 502 766 54 7 922 8627 701 9977 040 840 598 812 58 8 9928 271 6 25 948 I 287 486 693 856 52 9 618 921 9964 195 433 601 788 900 51 10 965 9948 217 440 627 775 881 942 50 11 9929 310 513 685 821 919 974 984 49 12 655 857:-929 9978 015 9988 061 9995 066 9999 025 48 18 | 999 -.9949 C11.9965 172 207 203 157 065 47 14 19930 342 893: 414. 899 1 44 24 105 43 15 685 685 655 589 484 36 143 45 16 9931 026 976 895. 779 62~ 424 181 44 17 367 9950 266 9966 185 968 761 512 218 43 18 706 556 374 9979 156I- 89 599 264 42 19 6932 045l 844 612 343 9989 0135 684 289 41 o20' 384 9951 132 849 30 171 770 823 40 21 721 419 9967 0 5 716 016 85,4 357 39 22 9933 057 705 821 930' 440 937 389 88 23 8 390.990 555 9980 c84 573 9996 020 421 87 24 728 9952 274 789 267 706 101 452 36 25-99-34 062/ 557 9968 022 450 887 182 ~ 8 285 26 I 395 840 254 631 968 262 511 34 27 727 9950 122 485.811 9990 t98 841 5389 88 28 9935 058 403 715 991 227 419 567 82 29 8389 683. 945 9981 1.70( 355 49 590 81 30 719 962 9969 173 348 4892 573 619 30 31 9936 047 9954 240 401 525 6 9 6,9 644 29 32 375 518 628 701 734 724 668 28 38 703 795 854 877 859 7 662 27 34 9937 029 9955 070 9970 080 9982 052 983 8H 714 26 385 355 345 304 225 9991 106 94:3 7086 25 36 679 6219 528 398 228 9997 (o1I 756 24' 37 9938 003 893 750 570 350 C8H 776 23 38 3826.9956 165 972 742 470. 156I 795 22 139 648 437 997-1 193 912:590 224 813 21 40 960 7u8 1 413 9980 082 709 292 831 20 41 9939 290 978 6338 250 827 300 847 19 428 610 9957 247 851 418 944 426 868 13 48 928 515 99729 069 485 9992 060 492 878' 17 44 9940 246 783 286 751 176 556 892 1 45 568 9958 049 502 917 2' 90 620.9005 15 46 880 3815 717 9984 081 404 683 917 14 47 9941 195 580. 981 245 517 745 928 13 4 510 844 99708 145 408.629 4 807 939 12 49 823.9959 10(7 857 570 740 867 949 11 50 9942 1816 370 569 781 8/ 927 9 58 10 51 448 681 780 891 960:986.966 9 52 760 892 990 9985 050 9993 069 9998 044 9783 8 53 9943 070 9960 152 9974 199 209 177 101 979 7 54 379 411 408 367 284 157 985 6 55 688 669 615 524 3 891 213 989 5 56 996 926!.822 680 495 267 99 4 67 9944 380 19961 183 9975 028 885 600 382 996 3 58 609 438 2833 9,89 704 8374 998 2 59 914 693 437 8986 1483 806. 426.00O0 000 1 60 9945 219 94 641 295 918 T 000 0 60. o 40 20 0 0 NATURAL COSINES. NATURAT, TANGENTS. 481 11880 840 "8K 8 S 7 880 890 0'1 44346495143645 11'430052 14'300666 19'081137 28'636253 57'289962 60 1 6297861 410613 468474 360696I' 187930' 877089 58'261174 59 2 8379111 679068: 507154 4212301 295922 29'122005 59'265872 58 8'2 035289 949022: 546o93 482273 405133 371106 60'305820 57 4 23438 9'6 220486' 585294 543833, 515584! 624499 61'382905 56 5 484485 493-1475' 624761' 6065916 627296' 882299 62'499154 55 6 635547 7680001 664495 668529 740t291 80'144619 63'656741 54 7 8875799'7 044075 7045001 7316791 8545911 411580 64-858008 53 8 8-3 040586 321713 744779, 795372 970219, 6833071 66-105473 52 9 244577 600927 785333, 859616 20-087199 959928 67'401854 51 10 449558 881732 826167i 924417 205553 31-241577 68-750087 50 11 655536 9-8 1641491 867282 989784 325308' 528392 70'153346 49 12 8625191 4481661 908652 15'055723 446486 820516 71-615070 48 13 84 0705151 7383823 950370 122242 569115 82.118099 73.138991 47 14 279531 99 021125 992349. 1893491 6982201 421295 74-729165 46 15 489573 310088 1203-1622' 257652 818828 730264 761890009 45 16 700651 600724 0771929' 325358 945966 33'045173 78-126342 44 17 912772 8930501 123062 390276 21'074664' 366194 79-943430 43 18 8'5 125943 10-018708 163286 463814 204949 693509 81-847041 42 19 340172 048283 2)6716 533901 336851' 3-14-027303 83-843507 41 20 555468 0780311 250505 604784 470401i 367771 85'939791 40 21 771838 1079541 294609 676233 605630 715115 88'143572 89 22 989290 1380541 339028 748337 742569 351669546 80-460336 38 23 8'6 207803 168332' 38376 821105 8812511 431282 82-908487 87 24 427475 1907891 423831 894545 22-021710 800553 85-489475 36 25 648223 229428 474221 968667 163980 36'1759 88-2179-43 35 26 870088 201249 519942 16 043482 838097, 562659 101-10690 34 27 8'7093077 291255 565997 118998 454096 956001 104-17094 4 28 317198 322447 6120993 195225 602015 87-'35789 10742648 82 29 [542461 358827 6591251 272174 751392 7686131 110'89205 31 30 768874 385397 706235 849855 923766 388-188459 114-58865 30 31 993446 417158 7536384 428279 28'-057677i 617738 118'54018 29 328'8 225186 449112 811417 507456 213666 89-056771 122'77096 28 33 455108 481261 8495571 587896 3717771 505895 127-32184 27 34 686206 513607 893058 668112 532052 965460 13221851 26 35 918505 546151 946924 749614 691537 40'435837 137-50745 25 36 89 152009 578895 996160 831915 8592771 917412 143'23712 24 37 386726 611841 13'045769 915025 24'C2289320 41'410588 149'465G,2 23 88 622668 644992 0957571 993957 195714 9157903 156-25908 22 39 859843 678348 146127 I7'083724 867509 42-43846 16370019 21 40 90 098261 711913 196883 169-037 541758 964077 171-88540 20 41 3807933 745687 248031 255809 718512 4-5308122 1811-93220 19 42 578867 779373 299574 343155 897826 44-0661121 190'98419 18 483 821074 813872 351518 431385 25-079757 638590 202-21875 17 44 9-1 064564 848283 403867 520516 264061 45-226141 214-85762 16 45 309308 882921 456625 6105591 451700 829351 229-1816 15 46 555406 917775 509799 701529 641802 46-448862 245-55198 14 47 83283S 952851 563891 793442 834823 47-085343 264'44080 13 48 9-2 051564 988150 617409 886010 28'030736 739501 286-47773 12 49 301627 11023676 671856 980150 229638 48'412084 312-52137 11 50 553035 059431 726738 18-074977 431600 49.103881 343'77371 10 51 805802 095416 782060 170807 6866901 815726 881'97099 9 52 9'3 059986 131635 887827 267654 844984 50548506 429-'7175 8 53 815450 168089 894045 365537 27'0565571 51-303157 491-10600 7 54 5723551 204780 950719 464471 271486 52-080673 572'95721 6 55 830663 241712 14.007856 564473 489853 882109 687-54887 5 56 9-4090384 278885 065459 665562 711740 53-'708587 859-43680 4 57 851531 316304 123536 767754 937233 54-561300 1145-9153 3 58 614116 353970 182092 871068 28-166422 55'441517 1718'8732 2 59 878149 891885 241134 975523 3993971 56-350590 3487-7467 1 60 9'5143645 430052 300666 19-081137T 636253 57-289962 Infinite. 0 60 5 0 4 31 20 I 1 0 I NATURAL COTANGENTS. 41 LOGARITHMS OF NUMBERS FROIO 1 TO 10,000. N. N. L og. N. Log. N. Log. 1 0o000000'26 1414973 51 1'707570 i 76 1 880814 2 0'301030 27 1.481364 52 1 716003 77 1 886491 3 0 -477121 28 1-447158 53 1 7 24276 78 1P892095 4 0 602060 29 1 46.2398 54 1'732394 79 1'897627 5 O'698970 30 1 477121 55 1 *740363 80 1*903090 6 0-778151 31 1 491362 51 6 1 748188 81 1 908485 7 08S450)98 32 15 05150 57 1 755875 82 1 913814 8 01903090 33 1a518514 5 a8 1 763428 83 1 91907 8 9 0O954243 34 1 531479 59 1P770852 S4 1-924279 10 1l000000 35 1'544068 60 171'78151 85 1 929419 11 I 1 041393 3(6 1 55630(3 (61 1'785330 86 10934498 12 o079181 37 1568202 62 1' 92,392 17 1 99519 9 13 1* 113943 38 1 579784 63, 1 799341 88 1 944483 14 1 1463128 39 1 P591065a 64 1 806180 89 1- 949390 15 1176091 40 1'602060) 65 1'812913 1 90 1 954243 16 i 20412( 41 161278'1 66 18'1 1819544 j 91 1'959041 17 1 230449 42 1'6B23249| 67 1 826075 92 1'963788 18 1 255273 43 1 6334681 68 1832509 93 1 968483 19 1' 178754 4-1 1 (43-53.69 I s8388 94 1 9 972 128 20 1'301030 45 1 653213 70 1845098 95 1 977724 21 1'3220219 46 662758 1 1 851258 96 198221 22 1 3')12423 47- 1'6723098 72 1 857 332 5 97 1 986772 23 136728 48 1'681241 73 1'868323 1 98 11'991226 24 380211 49 1 690196 74 1.8692 2 9.) 1 995635 25 11397940 50 1'698970 7 1:875061 1100 2 000000 4-4 LcIUGrTlrMor OF-:NUM.-BERS. No. 2 3 4 5 7 8 9 Diff. 100 000000'000434 000868 001801 001734'002166 002598 ot80,'29 008461.0(OC.891 482 1 4321 47511 5181 560.9 6038i 64661 689'! 78211 77481 8174 428 2 8600 9;126 9451 9876 010300'010724 011147 0115706C11993 012415 424 8 0128370138259013680 0141007 45211 49401.53G6 57791 6191 66160 420 4 70331 7451: 7868 8284' 87001 9116 95802 9947;620861020775 416 / 5 021189 021608 022016 1022428 022841 023822 C0236640240175 44861 4896' 412 6 63061 5715 61251 6533 69421 73503 77571 8164 85714 8978! 4(8 7 93841 9789 030195 038600 081004'031408 0318121032216 C32619 C83021 404 8 030424 038826 4227 4628 50291 54801 8380 6230) 6629: 702S: 400 9 74261'7825: 8223' 8620. 90171 9114. 9811 046267 04062C4C998'897 110 041393 0C41787 042182 6 42o6 042969 043362 043755 C44148 044540 C44932 893 1 5323 57114 6105 6495 6885! 7275 76641 8053 84421 8880'90 2 9218 9606 9993105088) 050766i051153 05158s C51924 0528C9 CB2694 86 3 (03078 053468 058846 4230 46131 49961 58378 5760 6142i 6524 383 4 69)51 7283 76661 8046'. 8416 88051 91850 9063 9042 C0C3201 079 5 106698 061075 061452 061829 062206 C62582 062958 06383 0687( 9 40883 376 6 44581 4832 5206l 59583 59531 6326B 6C(91 7071| 74431; 7815 3783 7 81861 85571 8928; 9298 9668 C7C03807C4071C70776071145071514' 370 8- 0718821072250072617107298507338521 3718 4('85 4451 4816 b5182 866 9 5547 5912:6276 66401 70041 73681 7781 8694 8457i 8819. 863 120 079181 079543' 09934i 080663C0626 08 C937 C81347 681707 82C67 (824:26' 860 1 C82785 088144 0885031 88.31 4219 4516, 4934 5291 56471 60((14 357 2 6360 6716 7071. 7426; 77Sli 8136' 8490 8845 91981 95521 855 3 9905 090258 090611 09.983 C091815'6901667 C92018 C92870 (92721 c930711; 52 4 0934221 87721 41222 4471i 48201 5169 5518 5866 6215 6562349 5: 6910' 72511 76041 951 8298 84-41 8990 985,. 9681160C0i26 346 6 100371 1007154101059 101403 101747'1 2091 1(,234 162777 108119 8462; 343 7 8804 4146 44871 489281 1699 5510 58510. 61911 6181 68711 341 8 7210 7549! 78881 822711 85651 8908 9241 9579 9916110231 3.838 9 1105901 i10926 1112638111599 111934 112270 112605o112940118275} 869 835 130 118943 114277 114611 114944 115278 115611 11 943 11627611668, 116940 833 1 7211 7603 7934 82651' 85951 8926i 92561 9,5861t 915'12(245 830' 2 120574 120908 121231 121560 1 21888 12221 61 22544 122871 1231981 8525 328 3 3852 41781 4504 48801 5156i 5481,5806 61311 6456 6781 825 4 7105 7429 753 801S6 8199 8S22 9004 93681 69013800 12 323 5 130334 13065138091 1f3129 131319 131939 132260 132580 1329000 3219 321 6 38391 38-58 4177 4491 48141 51331 0451 5769 6(86 64031 318 7 67211 7037 7354 76711 179871 830;3 S618 89341 2249, 9564 316 8 9879 140194 140508 143822 141136 141460'41763 142076 142889 1427021 314 9 14801513 8321 3689' 39511 42681 45741 4885 51961 b5071 5818 311 140 146128 146438 146748 147058 147386 147676 147 985148294 148603148911 8(9 1 92191 95271 9883515014215044091507561510C8151370151676151982 807 2 152288'152594'152900 8.205!' 3510) 38151 41201- 44241 4728 5032 305 3 5836i 56401 5943: 62461! 6549 680521 7154 74571 7759 80611 303 4 88621 86641 89651 92661 95671 9868s160168160469 160769161068[ 301 5 6168616716196 22616256416281'16166 841616162266 1254 16268 8161 3460 37581 4055 299 6 453 465 49471 52-141 55411 5838 6184: 6430 6726 7022 297 71 7317i 76131 79198 82381 8497! 8792 908 9380 96741 9968 295 8 170262 170055 170848 171141 171484 171726 1721019 1728111726038172895 298 9 3186 34781 3769 406q01 43511 46411 49832 52221 55121 5862) 291 150 176091 176881 176670 176959 177248 177586 177825 178113 1784r;.1786.89 289 1 891T, 926' 9552 9839 18(1 2' 180413 180699 18C986 181272 181 58 2871 2 1181844182129 182415 1827001 29985 82701 85550 88899 4123 44071 285 3 4691 4975j 5259 0542 582951 61(8. 63911 6674 6956 92891 283 4 175211 7803, 8084 83661 86471 89281 92091 9490 97711190051 281 5 190382 190612 190892 191171 191451191730 192010 1922S9 1925671 2846 279 6 3125i- 403 3681; 89 4237i 4514 47921 5069' 461 6281 278 7 59001 6176 64583 67291 70051 7281 75561 788321 81071 8382 276 8 | 86571 89821 9206 94811 9755 200029 2003038 200577200850 201124 274 9 1201397 201670 201943 202216 2024881 2761 [ 80881 883051 857T1 848 272 No. 1 2 3 1 4 51 7 I 9 uf. 6.... LoGARITAMS OF'NUnBERS. 485 NO. 0 1 2 | 4 6 7 8 9-|Dif I 160 204120 204891 2046631204934 205204 205475 2C5746 206016 206286 206556! 271 1 6826i 70961 73651 7684-: 79041- 8170 84411. 8710. 8979'. 9247.269 2 9515! 9783 2100511210319 210586 210853 211121 21188S 211'654211921 267 3 212188 212454! 27201. 2986: 82521 8t518 3783 4049 43141 4579 266 4 48441 5109 53783 5638 59102 6166 6480 6694' 69571 7221| 264 51 7484! 7747 8010 82711 85361 8798, 9060 94~ 3: 95S5! 9846 262' 6 220108 220370 220631 220892 2211531221414 221675 221936 222196 222456 261 7 2i716 2976 8326 3496' 875518 4015 4274 438i 47,92 5051 259 8 5319- 5568 5826 6084!6' 6342 660(0 6858 1 711Sf 7872 7680 2582 9 7887| 8144 8400 86571 89183 9170T 9426 9082. 9988230193 256 170 230449 230704 230'960 231215 231470 281724 231979 232234 232488 232742 255 1' 29961 3230 3504 37T371 4011 4264 40171, 47701 50123 5276 253 2 5328 5781 6033 6285 65,7 6789 7041 7292 7544 7795 252 38 8)46! 82'97 8548 8799 9049, 9299 9550 9800 2400530240300 250; 4 240549 240799 241048 241297 241546 24179 242044 2422938 2541 2790/ 249. 5 8038 8286 835341 8782 4o30 4277 4525 4772, 5019 5266 248 6 [55183 5759 6006 6252' 64991 6745 6991 7287 7482 7728 246 7- 79731 8.219 84641 8709 89,4! 919s 9443 93871 9 9320250176 245: 8 250420 250664 250908'251151 251895 251638 2518811252125 252868 2610 243 91- - 28583 8039618838 85301 8S221 4064 4306, 4548, 4791) 5031 242 183' 2t527.3 255514 255755 255996 256237 256477 256718 256958 257198 257409 241 1 7679' 791S8 81581 8893 86871 8877 9116t: 98551 9594. 9833 239 2:1 260071 260810 260518 260787 261025 261268 261501 261789 261976 262214 288 3 2411 2688 2925 8162:. 3399 86306 8873/ 4109! 4346 4582 2837 4 4818 5054 5290 5525' 5761 5996' 6232 64671 6702 69371 285 5; 71721 7406t 7641 7875! 8110 83441 8578{ 8812! 90461. 9279 234 6 951.3 9746 9930 270218 27(1446'27(679 27(;912271144 2718]77 271609 283 7 271842 272074 272806) 2588! 27701 8001 82381 84641 3696j 8927T 232 8 41581 4389 46201 4850 50811 5311 5542 600)21 6232 280 9 - 64621 6692, 6921. 7151i1 788.0 760j9 73S8' 88 61: 8296 8025 229 190 278754 278982 279201 279439-279667 279895 280123 2303.51 280578 280806 228 1 281033 281231 281488 28231715 281942 282169 2396' 2622| 2849'8075 227 2 88301 8527 8753. 8979 4205!' 44311 46356 48S2- 5107 5332 226 3 55571 5782 60071 62321 6456 6681 69051, 7180| 7854 7578 225 | 4 7832 8026 1 8249 847.3, 86961 8920 9140. 9366 95689 9812 2293 5 290035 290257 290480 290732 290925 291147 291339 291591,291818 292834 222 6 2256 2478 2699 29201- 3141 83631: 8584' 3804 40251 4246 221 7' 44G6 4687 4907 5127 5347 5567 57871 6007 6226 6446 220 8 6665 6884 7104 7323 7542 7761 7979 81981 8416 86351 219 9' 88531 9071 92891 95071 9725 9943 300131 300378 800595 8008131 218 200 301030 301247 301464'301681 331898 302114 302331 83(2547 30274 802980 217 1 8196 8412 3628 3844 4059, 42751 4491! 4706 49211 5186 216 2 5351 5566 5781 5996 6211 64251 6689 68541 70681 7282 215 3 >7496 7710 7924 8137 8351| 8564' 8778 -8991 9204 94171 213 4 9630 9843 10056 310268 310481 310693 310906 311118 3113830 311142 212 5 311754 311966 2177j 2389 2600 2812 38023 32834 8445 86561 211 6 38671 40T78 4289 4499 4710 49201 5180' 5340 551| 5760 210 7,5970 6180 6390 6599 6809 7018 72271 7436 76461 7854 209 8 8063 8272 8481 8689 8898' 91061 9314 9522 97301 9988 20-8 9- 820146 320354 320562 320769 820977 321184 321391 321598 321805 822812 207 210 322219 322426 3226333228.39 323046323252 823458 323665 323871824077 206 1 42821 4488 4694 4S99' 51051 531()01 5516, 5721 59261 61311 205 2' 6336 6541 674560 G) 71551: 78359 7363 7767 7972 81761 204 3 i'8380' 8583 8787 8991. 9194' 98.398 9G6011 985 830008 330211| 203 4 380414 330617 330819 331022 8331225 331427 331630 8318821 20841 2236 202 5;'2488 2640 2842 8044 8246 834471, 3,649 88501 401 4253 202 6'4464 46515 4856 50371 521 5458- 565 8 368 58591 6059 6260 201 7 6460 6660 6860 7060 7260 7459 7659, 7858 8058 8257 200 8 8456 8656| 8855 9054 9250 94511 9650' 98493400478340246| 199 9 340444340642 34084141089 341237 34148S 341682 341830| 2028 22251 1938 No o 4 21 1 4 I 5 617 8 9 Dif 41* 486 LOGARIThMSS.OF. NUMBERS. N.o | i O|: 2 3 - 4 5 |6 7 8 |9 9 if, 220 8424283 342620 842817 34-3014 843212 3434C9 3436061343802 834399991344196 197 1 4892: 4589: 47851 49811, 5178 5374 5570 5766. 5962. 6157 196 2. 6.858 6549 67441 69391 7135i 7830 7525 7720 7915 8110 195 8:{1 83051 85001 86941- 8889 908883 9278 9472 9666 98603580054 194: 4 850248 850442 350636 350829 351023 851216 851410 351603 851796 1989 193 5 2183: 2375 2568 2761 29541 3147 38839 8532 3724 8916 1938 6: 4io81 4301 4493 46851 48761 5068 5260 5452 5643 5834 192: 7: 60261 6217 6408 65991 67901- 6981 7172 7363 7554 7744 191: 8: 79835 8125. 83161 8506' 86968 8886: 9076 9266 9456 9646 190. 9' 9835 360025 860215 360404.860693 860783 860972 361161 861350 3615389 189 2830' 861728 8361917 862105 362294 862482 362671 862859 863048 363286 363424 188,.1 8612 8800! 8988 4176! 436831 4551 4739 4926{. 5113 5301 188. 8 7856 7542 7729 7915!: 8101 8287 84783 8659; 8845 9030 186:3 4:[ 9216 94011 9587 9772 995883701431370328 37051383706981370883 185 5i 371068 371253871437 3716-22371806 1991 2175a 2360 25441. 2728 1846"- 2912 3096. 8280 8464I 8647 8831 4015 4198 4882 4565 184' 7 4748 4982! 5115 5298 5481 5664 5846 6029 6212 6394 188. 8. 6577{ 67,59' 6942 7124 7306 7488 7670 7852 8034 8216 182. 9- 8898 85801 87611 8943 9124 9306 9487 9668 9849 380030 181 240. 380211 8380892 880578 880754 880934 8811153 881296 8814761 881656 881837 181 1- 20171: 2197i 23771 2557 27371 2917 30971 82771 845.6 38636 180 2: 8815 8995 41741 4853. 4583 4712 4891 5070 5249 5428 179 5066 57851 5964 6142 63821 6499 6677 68561 7084 7212 178 4:1 7390 75681 7746. 7923 8101 8279 8456 86341 8811 8989 178. 5. 9666 9348 9201 9698 987539005113902288390451390582 390759, 177 6 890938 8911123912888914648916411 1817 1993 2169. 2345 2521i 176 7 2697 2871i: 8048 83224 8400 8575 8751 8926 4101 4277 176 8' 4452 46271 48021 4977 5152 5326 5501 56761. 5850 6025 175 9 6199 68741 65481 6722 6896 7071 7245. 74191 7T592 7766 174 250 897940 898114 898287 398461 898634 898808 898981 8991854899828 899501 178 1 9674 9847 400020'400192 400365 400588400711 4008831401056 401228 173 2 4014014015781 1745 1917 2089 2261 2488 26051. 2777 29491 172. 8: 8121 82921 3464 8635 880o1 8978{ 4149 4320 44921 46631 17 4 4884 5005 5176 5346 5517 5688 5858 6029[: 6199 6370 171 5; 16540l 6710[ 6881{ 70511 72211: 7891 7561 1 77311 7901 8070T 70. 6-:: 8240[ 8410( 85791- 8749{ 89181 9087 92571 9426 9595[. 9764 169., 7 9988 4101029410271 410440 410609 410777 410946 411114 4112883411451 169: 81411620 1788' 1956 2124 229318 2461 2699 296 2964 3132 168 | 9 8800 8467 3635 8808 8970 4187 485 44721 46391 4806 167 - 260 4149783415140 415307 415474 415641 415808 415974'416141 416308 416474 167 1 6641 68071 69783 7189 7306 7472 76388 7804. 7970. 8135 16.66 2 88301 84671 86883 87981 8964 9129{ 9295; 9460 9625 9791 165 31 9956 420121 420286 420451 420616 420781 42094542111014212751421439.165 4 421604 1768 19331 209T7 2261{ 24261 25901 27541 2918 8082 164 5 8246 83410 3574 3787 8901 4065 4228 4392 4555 4718 1646 4882 5045 5208 5871 5534 56971 5860 6028. 6186 63491 163 7' 6511 66741 6886 6999, 7161[ 7324 7486 7648 7811 79781 162 8 8135 8297 8459 8621! 8788 8944 9106 9268 9429 95911 162 { 91 97521 9914430075430236 40398 480559 480720 43088114104214812031 161 270 431864 431525 481685 431846 482007 482167 432328 424884826491 432809 161 1 2969 3810 38290 8450 3610 3770 3980 4090 4249 4409 160 2 45691 4729 4888 504S] 5207 58671 5526 56851 5844}. 6004 159 38 6168 6322 6481 66401 6799 69571 7116 7275 7433 7592 159. 4'{ 7751 7909 80671 8226 8884 8542 8701 8859 9017 9175 158- 5. 933 9491 9648 98061 9964440122440279440437[4405941440752 158. 6 440909441066441224441381 441588 16951 1852 2009 2166 2828 157 7 { 2480 2637 27931 29501 81061 826 83419 8576 8732 38891 157 8 40451 4201 4857 45181 4669 48251 4981 5187 5293 5449 156 9 5604 5760 5915 6071 6226 6882 6587 6692 6848 7008 155 No. I O 1 2 3 4 15 I 6 9 7 Di8 LOGARITHMS OF NUI.BEnRS.. 487 No. 0 1 |2 3 4 5 6 7 8 9 D;ff. o 1o 44 l 1 446 1 ~ 280 447158 4478183447468144628447778 447933 448088 448242 4483971448552 155. 8706. 8861[ 9015 91701 9824 9478 963388 9871 9941 4b5951 154 2 450249 450403 450557 450711 450865 451018 451172 451826 451479 168833- 154! 8 1786. 19401 2093 2247 24001. 2553 2706.. 2859 8012 8165' 153 4 83318 8471 8624 87772 89301 4082 4285' 4887 4540 4692: 153 5 4845 4997 5150 5302 5454 5606 5758 5910 6062. 614 152: 6 63661 6518 6670 6821 6978 7125 7276 7428 7579 7731' 152 7 7T88~2 8038 8184 8336 8,487 8688 8789 8940 9091 9242 151 8. 9892 9543 9694 9845, 9905 460146 460296460447.460597 460748 151 9 460898 461048 461198 4613484614.19 1649. 1799, 1948. 20981 2248 150 290 0.62398 462548 462697 462847 462997 463146 46329.6 463445 468594 468344 150 1 3898 4042 4191 48401 4490 46891 4788 4986: 5085] 5284' 149 2 53838 5582 5680 5829' 5977 6126 6274. 64283 65711 6719 149 8 6S681 7016 7164 7812 7460 76(;8 7756 79C4 8052 82001 148 4 88471 84951 8648 87901 8988 91i85 9238 98801 9527 9675 148 5 9322! 9969 470116 4702638 470410 470557 470704 47(851'47(998 471145' 147 6 471292 471488 15851 1732 1878 2025 2171. 2318, 2464'2610i 146 7 27561 2908 38049 31951 3341 8487 8688 8779 8925 40711 146 8 4216 43621 4508 46538' 4799 4944 5b(901 52851 581 5526' 146 9 5671 5816 5962 6107 6252 63.9 6542, 66871 6832 6976 145 300 477121 477266 477411 477555 477700 477844 477989 478183 478278 478422 145 1 85661 8T111 8855 8999! 9143 9287i 9481 95751 97191 986 144 2. 480007 480151 480294 480438 480582 480{725 X8869 481012 481156 481299 1448 1448 1586 1729'18721 2016 21591 282 2445 2588 273811 143 4 2874 3016 8159 38802' 8445 8587 780 871 415 4157i 143 5 4800 4442 4585 47271 4869 5011 5158' 5295 5487 5579 142 6 5721 5868 6005 6147 6289 6480 65721 6714 6855 6997i 142 7 7138 72801 7421 7563 7704 7845 79861 8127 8269 8410i 141 8 8551 8692 88388 8974 91141 9255 98961 95871 9677 9818 141 9 9958 490099i490289 490880 490520'49066 44980 4949 1081 491222 140 | 310 491362J491502.491642 491782 491922 I49 214922014492841!492481 4926211 14i0 1 27601 2900. 8040 8171 819 1 3858 3591 87i 4015 139 2 4155 4294 44881 4572 4711 4850 4989 5128 52671 5406 189 3 5544 5688 5822 5960 6099 6288 6087 6515 6658 6791' 189 4 690 7068 7206 744 748 7621 7759 797 885 8178 138 - S 8311 8441 85865 8724 8862 8999 9187 9275 94121 955C 138 6 9687 981 99625000)991500286'5008T4 5C9511 500648500785500922 138 501059'501196.501338 1470 1607 17441 1880 2017 2154 22911 137 8 2427 25641 27001 2871 2978 3109 8246 8882 8518 3651 186 9 379T 89271 40681 4199 4335 4471 4607 4748 48781 5014 186 8320 1505150 505286 505421 505557 505698 505828 50C964 506099 506234 5063870 186 1 6505 6640 6776 6911 7046 7181 7816 7451''75'861 7721 185 2 78561 7991 8126 8260 8895 8580 8664 8799 8984 9068 185. 92081 9887 9471 9606 97401 9874 5100C9510143510277T510411 134 4 510545510679 5108181510947 511081'511215 1349 1482 1616 1750 134 I 5 18883 2017 2151 2284 2418, 251 2.684 2818 2951 3084 18 6 82181 38351 3484 86171 8750 8883 4016 4149 4282 4415 133 7 4548 4681 4818 4946 5079 5211] 5844 5476 56609 5741 138 8 5874! 6006 6139 627 6403 65851 6.6681 68001 6982 7064 132,9 87146 78281 7460 7592 724 7855 79 8119 8251 8882 182 1830 518514 518646'51877518909 519040'151917115198038519434 519566 519697 181 1 9828.9959520090520221 520358 520484 520615 520745 520876 5210071 181 2 1521188521269i 1400 1580 16611 1792 1922 2058 2188o 2314 181 8 2444 2515 2705 28851 2966 3896 38226 8856 8486 86161 180 4 87461 8816 4006 4186 4266 4898 4526 4656 4785 4915 18(0 5 5045 5174 5804 5484 5563 598 5822 5951 6081 6210 129 6 6339 6469 6598 6727 68561 698 7114 7248 7872 7501. 129'7 76800 7759 7888 80161 8145 8274 8402 8531 8660 87889 129 8 8917 9045 9174 9802 91481 9559 9687 9815 9948530072 128'9 530200580828580456580584580712 580840 80968 581096 538'228 1351 128 No. I I 1 3 4 15 6 1 8 9 Diff. 488 LOGARITHMS OF NUMBERS. qNo. 0 1 21 32 43 5| 6 9 Diff 5.4) 1514'9' 50316i37 531734 531862'5319930532117 53224.5 532372 532500 5326271 128 1 2751!. 232 309 31386. 32641 83891i 3518 3645 38772 38991 127 2 4)2' 415: 1 42131 4 4407 45341 4661 4787 4914' 50411 5167' 12T 3 5291 54211 5547 5674. 5800. 5927 6053 61801 6306 6432' 126 4 6553 6685 611 6987 7063[ 7189[: 7815 74411 7567 76931 126 5 7819 7915i1 8)71 8197 8221 8448S 8574 8699 8825 8911! 126 6. 9)786. 92321 9:327 9452 9578 97031 9829 9954 540079 54t,234' 125 7 51.3 3 5401435 540530 540705 540830 540C55 541080 541205, 18330, 14' 125,8 11j79 178041 18291 1931'20781 22081 2327' 2452; 25T76 2T71 125 9 2325 2953) 83J74 8199 382:3 3482 J 8571 3696 8820 89.14 124 3.) 54)G63 54192, 5:44316 51,443 544564 544688 544312'544936 545060 545183 124. 1 53)71T 5131 535551 5678-1 5802] 5925i 6049i 6172 6296 6419 124 2 6.38. 66G6 6789 6918 78036 7159' 7282 7405 7.529: 7652 123 3 7771 7893 8 )21 8141 8267 83891 8512 8685' 8758 8851 123 4 9);3 9L231 9289i 93fl 91949 96161i 97389 9861 99S45518i06 123 5 5.)223' 5 t3)81 )473 550593 550717 550840'5509682 551084 551206 1382 122 6 1.13) 1572 1:691 1316 19388 2060 21811 230i3 2425' 2547 122 7 26331 2993) 2'11 3083 8155 327j6 38898 8519 386401 3'62 121 8 33Q33 4004 412.3 42,17 48368 4489 46180 43711 48.52 49783 121 9 5J98 5215, 53:61 5457 5578 56991 5820 5940i 60611 6182'.121 033 5 333)3 556120 5383584556681 556785 556905 557026 557146 557267-557887! 120 1 75, )7 7627' 7.zl18 78631 7.9881 8108! 82281 884-9 8469, 8589 120 2 8T7)9 8329 89-8' 933J68 91881 9308! 9428! 9548' 9667 9787 120 8 9) i7 560026 53J146 5602653563388515680504 561i24 560743 560863 5609892 119' 4 5611 1 1221 1134) 1459' 157T8 16981 1S171 1936. 21055 21741 119 5 23)0 2 112 25-31t1 265) 2769 2887 3006 81251 8244' 3862 119 6 131 8630' 3718 83371 8955 4074 4192 4311 4429 454S8 119 7 4j33 47841 49,)38 5021 5189 5257 5376 5494 5612: 57801 118 8 50813 596' 60)814 62021 63201 6437T 65551 6673! 67911 6939 118 9 702i 7141 72621. 7379 749T 7614 77732 78491 79671 80841 118 370 5682,21 568319 5684.3 636S554 568671 568788 5'689.5 569023 569140 5692571 117 1 974 991 991' 93'81 9725 98i21 9959.570u76.57u1935703.-;9570426 1.17 2 570513 57066,;57To776,570893571010:571126 1243 1359 1476 15921 117 83 1.7T9 18251 1912 2358 2174| 2291 2407 25231 26391 27?55 116 41 28721 29338 3104 32231 33836 3452 35681 3684! 38001 3915o 116 5 4)381 4147 426'31 43791 44941 4610 4726 48411 49571 5072 116 61 5188 53.338 5419 5504 5650 5765 5880 59961 6111 62261 115 7 6311 6157 6157 6687 6302 6917 7032 71471 72621 7877 115 81 7492| 7607 7722 7836 79511 8066 8181 82951 8410i 8525 115 9 8809 87511 8868 898 9'097 9212 9826 94411 9555 96691 114 380 3579781 579898 530012 580126 580241 580353 580469 580588 580697 580811 114 1 58)9235;5810389 11531 1267 1381 1495 1 608 1722 18836 1950 114 2 20Mo1 21771 2291 24)4 2518 2631 2745 2858 29'72 3085 114 3 3199 3 312 8426 8539 8652 3765 38791 39921 4105 4218 113 4 4:331 4444i 4557 4670 47831 4896 5009 5122 5 5348 113 5 5861 5574T 5686 5799 5912 6024 6137 6250 6362 6475 113 665387 67i 67001 6812 6925 70387 7149 72621 7374 74861 7599[ 112 7 7711 78231 7985 8047 816I0 8272 8884, 8496{ 8608' 8720) 112 8 8312, 89-11 9J56, 9167 9279, 9891 9M503l 9615, 9726' 98881 112 9 995) 590061 599173 590284 590396 590507!590619 590730 59)'842 590953 112 890 591.06g 5911T6 591287 591399 591510 591621 591732 591848 591955 592066 111 1. 2177, 2288 2399 25101 2(621 27832 28483 2954 8064 3175 111 2 38283 9 350 88 8618 8729 88401 8950 4061 4171| 4282 111 83 4393 4503 4614 4721 48384 4945i 5055 5165 5276 5886 110 4 5193 56' 5717 55827 5937 60471 6157.626.7 68771 64871 11. 5 65971 6707 68t71 6927j 7087T 71461 7256 7866 7476 7586 110 6 7695 7805' 79141 8024 8134! 82431 88531 8462 8572| 8681 110 71 87911 89001 9091 9119 92281 93871 9446 955 665 96 74 109 8 1 9383 9992.600101,600210 600819 600428'600537 600646 600755 600864 109 9 601)97T 631082' 1191 12991 14081 15171 16251 -1784[ 18481 1951 169 No. u ili 1 3 4 51 69: I 18l 9 Diff. LoG.AITI-IMS OiF NuMBERS.: 489 No. | ~ I 2 3 |4 5 | 6| 8 9, 9 |i t 400 602060 602169 602277 602386 602494 6026038602T11 602819602928 6030836 108 1 3144 3253 3361 3469 3577 8686 8794 8902 4010 4118 1(8 2 4226 4384 4442 4550 4658 4766 4874 4982 5089 5197 108 3. 5305 5413 5521 5628 57361 844 5951 6059 6166 6274 108 4. 6.881 6489 6596 67041 6811 6919 7026. 7138 72411 T848 107 5 7455 7562: 7669 7777 7884 7991 8098 8205 8312 8419 107 6 8526 8633 8740 8847 89541 9i;61 9167Z 9274 93811 9488 107 7: 9594 9701 9808, 9914 610021 610128 610234 610341 610447 610554 107 8 6106680610767610870 610979i 10861 1192 1298 1405 15111 1617 106 9 17231 18291 19361 2042. 2148 2254 2360, 2466 2572 7 2678 106 410 612784 612890 o612996 613102613207 613313 613419 613525 613630 613736. 106. 1 3842 3947 40531 4159 4264 4370 447514581 4686. 4792 106 2 4897 50038 5108!. 5213 5319 5424 5529 5634 5740' 58451 105 3' 5950 6055 6160' 6265 6370 6476 65811' 66861 6790 6895! 105 4 7000 7105 72101 7315 7420 7525 7629, 7734 T839 7943! 105 5 8048 8153 82571. 8362 8466 8571 8676 878 88.4 88841 89, 10C5 6' 9093 9198 9302 94061 9511 9615 9T719 9824. 9928 6200082 104' 7 620136 620240 620844 620448 620552 620656 620760 620864 62968' 1072 104 8: 11761, 12801 18841 14881. 1592[: 16951. 1799 19031 20071 2110 104. 9- 2214 2318 2421 2525 2628 2732 2835 29391 3042 3146 104 420 623249 623383 623456 623559 62836638 628766 28869 628973 624076 624179 103 1 42982: 4385 44881 45911 4695 4798 4901 5004 5107 5210 103 2 5812: 5415 5518 56211 5724 5827T 5929' 6082 6185 62883 103 3: 63401 64438 6546 66481- 6751 6858 6956 70581 7161 7263 103 4 7366 7468 7571 7673. 7775 7878 7980 8082 8185 82871 102 I 8:889' 8491 8593 8695ai 8797[1 8900' 90(12 9104 206 93C8. 102 6 9-110 95121 96131: 9715'. 9817 9919 630021t630123 630224 680826! 102 7 630428 630530 663106630733 630838563C9361 1038 1139 124 1 102 8 14441 1545 16471' 1748' 1849. 1951l: 2052 21531O 22a5 28561 101 9 24571 2559 2661 27611 28i2. 2963: 3064 3165 3266 3367 101 430- 633468 633569 633670 6338771 633872 633973 634074 684175 634276 634376, 101 1 4477 45781 4679 4779; 48801 4981 5081 5182 5:83 53831 101 2 5484 55841 5685! 5785! 5886 5986i 6087 6187 6287 6388 100! 8 6488 6588 6688 6789. 68891 6989 7089 7189 7290 73901 100 4 74900 7590 7690 77901 78901 7991 8(090 8190 8290: 8889 100 5 8489 8589 8689 8789 8888 8988 9088 9188 9287' 9387' 100 6 9486 95861 9686 9185N 9885 9984.6400846401836402836400821 99 7 640481 640581 640680 640779 640879.64C908 10771 11771 1276 18751 99 8 14T4 15731 1672 17711 1871 19.01 2069' 2168 2267 28361 99: 9 2465 25631 2662 2761 2860 2959 3058 3156 3255 338854 9 440. 6433 643551 643650 643749-643847 643946 644044 6441438644242 644340 98 1 44891 4537 4636| 4734 4832 49312 5029;' 5127 5226' 524 98. 2 5422 5521 5619 57171 5815 5913, 6011 6110 6208 6306 98, 8 6404 6502, 6600 6698: 6796 68941 6992- 7089 7187 7285 98: 4 3883 7481I 75.791 767.61 7774 7 9692 8067S 8165 82621 98.' 8360 84581 8555 8653. 8750 8848. 8945' 9043' 9140 92871 97': 6 9335. 94321 9530 96271 9.724 9821, 991:9.650016.650113 6502101 971 1 7 650308 650405!650502650599.650696 650793 6508901 09871 1084 1181 97 8 i20781 1375 1472 1569 16661 1762. 1859 195'6 2053 2150 97 91 2246 2343 24401 2as6R 2633 27301 2 2923 3019' 3116 907 450 1653213 653309 653405 653502 653598 653695 6538791 653888 653984 6540890 96. 1 4177 42731 4369 4465 4562 4658 4754- 48501 4946. 5042 96 2 5138 5235 5331' 5427 55283 5619; 57151 5810 5906 6002. 96 3 6198 61941 6290 6886' 6482 65771 66783 6769 6864 69601 96 4 7056 71521 7247 7343 7438 7534, 7629 7.72 7820 7916 96 5 8;)111 81071 82021 8298 8893 8488 8584 8679 8774 8870 95' 6 8965 90601 9155 92501 9846 94411 95386 9631 9726' 9821 95 7 9916 660011 6601.06 660201 660296.660891 660486 660581660676'660771 95 8'6608651 960 1055 11501 1245 1339 1434 15o29 1623 1718 95 9 1813 1907 2002 2096. 2191 2286' 2880. 2475 2569 2663 95',No.' 02 1 - 3 14 5 7 8 1 9 )iff. 490 LO'GARITIIMS OF NUMBERS. 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6767 6880 6894 68.7 69571 7020 7083 7146 72101 7278 78861 7899 7462 7525 68 8 7588 7652 77151 7778 7841 79041 79671 8080 89 8156 68 9' } 8219 8282 8,845 8408 8471 8584 8597 8660 8728 8786 68 690. 888849 838912 838975 889038 839101 889164 889227 889289 889352 8S9415 63 1 9478 9541! 9604! 96671 9729 9792 9855 99181 99811840048 68 2- 840106 840169.840282 840294 8408571840420 840482 840545 840608 0671 63 8 07.38i 0796 0859 09211 C984 1046 1109 1172 1284 1297 68 4 1859 14221 1485 1547 1610 16721 1785 1797 1860 1922 68 /5 ~19851 20471 2110 -2172 2285 2297 2360 2422 2484 2547 62.61 2609 2672 2784 2796 2859:2921.2983! 3046.8108 8170 62 7. 82338 3295 885T1 420 8482 8544 86066 0669 37811 3793 62 8 88551 8918 8980o 4042 41041 4166 4229 4291 48583 4415 62 I 921 44771 489 4601 4664 4726 4788 4850 4912 49741 5086 62 So 1 2 3 4 6 6 I 1 8 1 9 Diff 42 494 LOGARITHIaIS OF NUmIFBES. No. 1 213 14 5 1 6 1 7 8 9 Diff. 700 f845098 885160 815222'845281 845346845408 845470 8455-32 845594 84565651 692 1I 5718 57S'O 58-21 5904 5966 6028! 6090 6151 6218 62751 62 2 6337 6399 6461 65283 65851 6646' 6708 6770 6882 68941 62 3 6955~ 70171 7079 7141! 72802 72641 7826 7388 7449'7511 62 4 7573 7684 7696 7758' 7819' 7881! 7943 8104 8066 8128 62 5 8189' 82511 8812 8374 84:835 8497] 8559 8623 8682 87431 62 6 83)5 83668 8928 8989' 9511 91121 9174 9285' 9297 9358 61 7 9419 9t131 9542! 968)4 9665B 9726' 9783 9849 9911 9972' 61 8 859083 851)095 850156 850217 850279 85034( 850401 8583462 850524 850585 61. 9 0648 T 0707 069 08381 08911 0952i 1014'1075 1186 1197 61 710 8512958S51820 851381 851442 851508 851564 851625 851686 851747 851809; 61. 1 1 1870 1931 1992 08 053 2114 2175i 2236 2297 2358 2419i 61 2 218) 2541 2602 2663 27241 2785 28461 2907 2968 8 29 61 8 31)91 81591 3211 382721 383981 3891 3455' 3516 3577; 8687 61 4 3698 83759 3820 3331 3941 4002i 4063 4124 4185 42-5 61 5 438)6 4867 4423 4133 4549' 4610 4670 4731 4792 4852 61 6 4913 4974 5034 5095 5156 5216 52]77 5337 5398 5459 61 7 55191 55801 5610 5701 5761 5822 5882 5948 6003 6064 61 8 6124 6185 624t5 6306 63668 6427 6487 6548 6608 6668 60 9 67291 6G89 G68591 6910 6970 7081 7091 7152 7212 7272| 60 720 857832855793 8.57458 3 5751 857574 857634 857694 857755 857815 857875 60 1 7935! 7995i 8056a 8116i 8176 82861 82971 8857 8417 8-177 60 2 85871 859 1 8.O57 8T71, 8773 838 8898 8958 93118 90781 60 83 91383 9193! 92531 9318 9379 9439 9499' 9559 9619' 9619 60 4 9329i 9799 93091 9918 997 8600388 60093 8698 158 86218 8602781 60 5 86)3338'6)393 86' 48 8681518'860578 0687' 0697 0757 0817 0877 60 6 0937i 096 1056 1116i 1176 36 1236 19,51 1355 1415 1475 60 7 15384 159: 1651 17141 1773 18833 1893' 1952 2012 21721 60 8 2131 21911 2251 23101 2370 24301 2489' 2549 26308 2668'. 60 9 27128 2787| 2817 29)61 2966 30251 3085 3144 3234 3263' 60 733 8'388238868382 8380412 863501 863561 883628 868689 868789 863799 863858 59 1 3917 397T1 4)36 49631 4155 42141 42741 4338 4392 44521 59 2 4511 45701 4630 4689 474S 4808 4S 67 4928 4985 50451 59 3 5104 51683 5222 52821 58341 54001 5459 5519 5578' 5687i 59 4 5696 5755! 5314 58741 59:33 5992 6051.6110 6169 6228' 59 5 6287 684t 6101 64851 6524 65838 6642 6701 6760! 6819, 59 6 6378 69871 6993' 7055' 7114 71731 7232 7291 7850 7439: 59 7 7467 7523 7535 7614 7703' 7762 7821 7880 79389 7998 59 8 8356 8115" 8174 8233 8292 85901 8409 8468 8527 8586; 59 9 86441 8703 8762 8821 887'9 8933 89971 9056 9114' 9173 59 7408 889293 869349 869408 889456 869525 869584 869642 869701 869760 59 1 91183 9377T 9935[ 9994 870053 870111 870170 8T0228 870287 870345. 59 2 870404 879432 870521 870579 0688 0698 0755. 0813 0872. (19'30 58 38 1 89 917 1106 1161 12281 1281 13389 1398 1456 1515; 58 4 i573 161' 16991 1748 18386 1865 1923 19381 28)40 2098 58 5 21561 2215 22T78 283831 2389 2448 2506' 2564' 2622 2681i 58 6 2739 2797j 23551 2918 2972' 8300 8088! 3146 8204 8262 58 7 8821[ 8879 84:371 8495 8553 8611 8669 387271 3785 38 — 58 8 39932 3909 4018 4076 4134' 4192 42501 43088 4866 442-3 58 9 4182 45401 4593! 4656 4714' 4772 4880 4888 4945i 5008 58 I I I I I 75) 875061'875119 875177 875285 875298 875851 875409 875466 875524 875582' 58 1 56491 5398 57561 58138 5871! 5929 59871 6045' 6102 6160i.58 2- 6218 6276 6383 6391 6449 6507 6564 6622 66811 6737i 58 38 67951 68583 6910| 6968 7026 70838 7141! 71991 7256 7814' 58'4 7371t;7429 74871:75448 7602 7659 7717i 7774! 78832 788T9 58 5 7947 8004 8062'8119 8177j 82 823 8292. 8349' 8407 8464' 57 6 8522 8579 8637 8 8694 8752 8809, 88661 8924! 8981 9089,57 7 9096 9158 9211 9268 9325' 93838 9440'9497! 9555 9612 57 8 96691 9726 978,! 9811 9898. 9956 880018 880070 889127 889185! 57 9 889242 88299 880856'880411 830471850528i'0585, 0642:0699:0756 57 1 No. I ( I0 1 1 2 I 3 1 4 5 D 7 8 9 IDff. L]OGARITHMS OF NUMBERS. 495 No. 0 | 1 2 2 3 4 5 0 7 8 9 IDiff. 760 880814 880C811880928 88,9S 5 881042 881099 8811.56 8812138881271'881328 57 1 1885 1442' 1499 1556 1613 1670 17271 17841 1841 1898 57. 2 1955 2012 2069 2126' 2188 2240 22971 2354. 2411 2468 57 8 2525 2581 2688 2695' 2752 28099 2866 2923! 2983 307 57 4 89o 3 150 8207 8264 83821 8877 8434 34911 8548 3605 57 5 8661 3718 8775 88821 3888 3945 4002. 4059 41151 4172 57 6 4229 42851 4342 48991 4455 4512 4569 46251 46821 4789 57 7 4795 4852' 4969 4965 5022 5078 51835 5192, 52481 5805 57 8 5361 54181 54741 55811 5587 5644 5700( 5757i 58181 5870 57 9 5926i 5983 60391 6096 6152 6209 6265 6821 6 484 56 770 886491 886547 886604 886660 886716 886773 886829 886885 886942 8869981 56 1 70541 7111 7167 72238 7280I 7836, 7892 7449 75f5! 7561; 56 2 7617 76741 7730 77861 7842 7898' 7955 80111 80671 8123 56 8 8179 8286 8292 8348: 84041 8460 8516 8578 86291 8685 56 4 -8741 87971 8853 8949 8965o 9021 9077 9184 9190 9246 56 B 9'302, 9858! 9414 94701 95261 9582 9638 96941 97501 9806 56 6' 9862, 9918' 9974 890030 890086 89ul41 890197 89t,258 S903(9 890865 56 7 s890421 891477,895833 0589 0645 07001 0756 0812 08681 0924 56 8 09808 10851 1091 11471 1203.12591 1814 18701 1426 1482 56 9 lb87 1593i 1649 1705: 1760 1816i 182 1928 1983 2089 56 780' 892095 892150 892206 892262 892817 892873 892429 892484 892540 892595 56 1 2651 27071 27621 2818 28701 2929; 29851 8040i 3096 8151 56 2 8207 8262 83181 38313 8429 84841 8540 8595 8651 3706 56 38: 8762 3817 3878 89281 8984 4039 4(994! 41501 4205 4261 55 4 4816 4871 4427 4482 4 58[ 45983 4648 47041 4759 4814 55 5 4870 4925, 49801 50361 50911 1464' 5201 5257 5812 5367 55 6 5428 5478; 55881 5588 5644 5699 57541 580.9 5864 5920 55 7'59751 60801 6085 61401 6195 62511 6806 68611 6416 6471 55 8 6526 65811 66386 6692 67471 6802 68571 69121 6967 7022 55 79- 7077 T1382 7187 7242; 7297 T7852 74071 74621 7517 7572 55 790 897627 897682 897T87 8977929897847 897902 897957 898012 8981067 898122 55 1 8176i 8231i 8986! 8'41 8P961 84".1 s 85061 8561 8615 8670 55 2 8725 87801 88853 8890, 944 8999 9054! 91(9 9164 9218 55 8 92731 9828i 988:3 i 94 92 9547{ 96f21 9656 9711 9766t 55 4' 98211 98751 9980j 99890(089 900,94 901149 900208 900258 908312 55 5 "900867 900422 90476 900531i 0586 C0640 (069 0749 0804 08591 55 6 19131 C96(98 1022! 1077j 1131 1186 1240! 1295! 1349 1404; 55 7 14581 15181 15671 1629 1676 1781 17851 18401 1S94 1948 54 8 2003 20571 2112; 2166 2221 2,75 2,829! 2884! 2488 24921 54 9'I 254T7 26011 2655 2710; 0,764 2818 2878 29272 2981 80868 54 800 903090 908144 9038199 908258 9088307 9088361 9834-16 908470 908524 903578 54 l 1 8683 8687: 87411 879: 8849 8904 3958 401121 4066 4120 54 2 41741 4229'4280 488l;491 444 4 499 455-3 4607 -4661:54 8 47106 47701 4824 48785 49382 4986' 50401 59-41 5148 5202} 54 4' 52561 53101 5864 5418 5472! 5526 55801 5684 5688 5742 54 I' 5796! 5850 5904 5958! 6(121 6066 6119, 6178 627 6281 54 6 6385i 6889; 6448 6497; (551 6604 66158 6712 6766 68208 54 7 6874 6927i 6981. 7085 7U891 7143:7196' 7250 7804 7358 54 1 8 74111 7465 7519! 7578 7(626 7680: 7T84 -7787 7T841 7895 54 9 7949 8002, 8060 8110! 816Si8 82171 M2M 8894 8378 8431 54 810 908485 908589 908592 908646 908699 908758 91880i7 968860 908914 908967 54 1 921 91 9)74 91281 9181i 39285! 9289 9342 9:396 94419 9508 54 2 9556! 96101 96681 916' 97701 9828 9877' 99301 998491003871 3 8 910091 910144 910197 910251 910804910058 910411 910464 910518'0571 53 4 0624 0678! 0781 0784 8 0888 08911 C944 0998 1051 11041 53 I 1158 1211i 1264 1817 1371 1424! 1477T 1530 1584 1637 53 1690 17483 1797 1850 190381 1956! 2009 2068 2116:2169 58 27 2222 2275; 2328'2881 2485 2488! 2541 2594:2647 2700 53 8 2753 28060 2859 2918; 22966 80191 8072 81251:178 82831 53'9 2841 8887 88901 8448. 84961 8549' 3602 86551 870.8'8761 58 No. O' 1 2131 14 15 6 7 1 8 1 9 Diff. 496 LOGArmTl[MS OF NUMBERS,;No'. O 1 1 2 4 6 7. 8 9 Dff 820 918814'918867 913920 918973 914026 914079 9141812. 914184 914287 914290, 5. 1 4848 4896 4449; 4502 45551 4608 46.6(0i 4713' 476( 48191 58 2 4872 4925' 4977 508 5083 5106' 5189 5241 5294 5847- 5g 81 5400 5453' 5505 5558; 5611 5664' 5716 5769 5822 5875 53 4 5927 5980 6083 6085 6183 61911 6248 6296 6349 6401 13 5 6454 6507 6559 66121 6664 6717i 6770[ 6822 8075 6927 -18 6 6980 7083, 7085 7138 7190 7243, 7295 7848 7460 741,38.8 7 7506 75t58 7611 7663 7716 7768 7820 7878 7925i 798 152 8 8080 8083; 8185 81881 8240 t6293 8845 8897 8450 s5,.2 2 9 8555 86071 8059 s7121 8764 t816i.8s69 8921 8973 9.26 52 830 919078 919180 919188 919285 919287 919140 919892 919444 91.9496 91:549 52 1 9601[ 96538 97061 97518 98101 9:62 99141 9967 92(019 92C(: 71 52 2 920128 920176 920228 920281) 92(8032 920( 84 9~0436 920489 05411 C1983 52 8 0645 06971 0749 c801 0881 096C6 698 1010, 162.1114, 52 4 1166 1218 1270 1022 1874 1426 14781 1580 1512 1684' 52 5 1686 1738 17090 1842 1894 1946 1998 -2500 21(2 29154 52 6- 2206 2258 281(0 2862' 2414 2466 25181 2570! 2622 2674 52 7 2725 27771 2829 2881 2988 2985 -8(87 3(89 81401 81092 1 2 8 8244 8296' 8848 899 8451 b15(3 1555 80667 80658 871'01 52 9 8762 8814' 865 8917 89G9 4(21: 4072i 124 41761 42281 52 840 924279 924881 924388 924484 924486..924088 924589 924641 924(98 9247441 52 1. 4796 4848 4699 4951 5C068 5C54 51t66 5157 522(;9 5261' 12 2 5312 53641 1415 1467 551.8 55701 5621 5678 5725 57760 52 8 53828 5879 5981 ~982 6084 6(:85 6137 6188 6240 62911i 5.1 4 6842'6894 6445 6497 654:8 6600' 6651 6762 6714 68 51i 5 68571 6908' 6959 7011 70621 7114' 7165 7216' 7268 7819 51:6 7870 7422 7473 724 7576 7627, 7678 7701 7781 7882, 5i 7 7888 79835 7986 8C87 8S88] 8140 8191 8242 8298 8845 51 8 8896 8447 8498 8549 8601 862 8781.87154 8865 88571 51 9 8908 8959 9010 91 ul 91121 9163 S2151 9266 9817 9068 51 850 929419 929470 929521 929o72 929628 929674 929725 929776 9298271922V.879 1:1 99801 9981 980082 980188 930184 980185 98:286 980287 93C808 908('889 51 2 983440 9804911 c5421 c921 0648 C94 07451 0796, C847 68998 51 3 0949 i0001 1051 11C2 1158 12104 1541 ( 16(:5, 14(i71 51 4 1458 1509; 1560 1610 1661 1712 1763 1814' 18651 1915! 51 5 1966.20171 2068 2118 21691 2220 22711. 222 28721 24223 5i 6 2474 25241 2575 2626 26771 27227i.2.778 2829 28'79 29801 5 7 2981 8811 8682 81881 8188 828.4' 82851 8885 386 4873'1 51 8 84871 8588 385891 3689 86901 8740 87911.88411 8892. 8943 51 9 8993 4044i 4094 4145 4195i 4246, 4290' 48471 4897 4448 51 860 934498 984549 984599 934650 934700 9804751 9 9848152 9349(.2193495 50.1 5008 50 54' 51041 5154 52(5' 5215 5806. 5856 54C6 5457 50 2 5507 5558 5608 5658 57(19 571t9 68691 1860 5916 5960 50:8 6G11 6061 61114 61621 6212 6262 68138 6868,.6418 64638 50 4 65141 6564 6614! 6665 6715 6765 6815 6865' C916 6966 50 l5 70161 7066 71171 7167 72171 7267. 7817 7867i 74181 7468, 50.6 75118 7568 7618 7668 77181 7769' 7819 7869 7919 79691 50 71 8019 8069 8119 8169 82191 8269 88201 8870 84201 8470 50 88 8520 8570 8620 8670 87201 87701 88201 8870 89201 8970 50 9 9020 90To 9120 9170 92201 9270 98201 9869 9419 9469 50 870 98951.9 989569 919619 989669 9 897719 9 89869 98 998989 98991:8 99968 50 I1 940018 9400(68 940118 940168 940218 94C267 94C817 94;3867 940417 940467 50 I2$ 0516; 0566 0616i 0666 071'6 07651 0815 (,865 C9151 C964 50 3 101i4 1064 1114 11'68 1218 128 1181 1862 14i21 1462 50 I.4:| 1511 1561i 1611 1660 1710 1760j 1869 1859 1969 1958 50 5 2008 2058 -2107 2'157.2207T 2256 280.6 2855 24(5 245M 50.61 241 2554 2608 2658 2702 27521 28012 2851; 2901 2950 50 7:.:|80Q00 8049:8099 81481 ~8198 8247 3297 8846 889'6 8445 49' 8:8 6:495[ 83544 3598 364-3 8692 38742! -8791 8841i 8890 69991 9 989 40s 4088 413 4186 4286 4285 48851 4384 4433 4 No. 1: 3 4 1 5 i 6 1 8 9 IDi. LOGArITHIIMS OF -NUMBERS. 497 No. o 1 213 41 5 1 9 Diff. 880 914483 944532 944581 944631 944680 944729 944779 944823 944877 944927 49 1 49t6 5025i 5074 5124' 5173 5222 52721 5321 5370 5419 49 2 5469 5518' 5567 5616 56651 5715 5764 5813 5862 5912 49 3 5961 60101 6,59 6108 6157j 62071 6256 6305! 6354 6403 49 4 6452' 65031 6551 66001 6649 6698 6747 6796 6845 6894 49 5 6943 6992 7041 7090 7140 7189 7238 7287 7336 7388 49 6 7434" 7483' 7532 7581 7163!'6 7679 7728 7777 7826 7875 49 7 79241 79731 8022 80T70 8119 81683 8217 8266 8315 8364 49 8 841:3 8462' 3511 8560! 836t9 86571 8706 8755 8804 8853 49 9 8902 8931' 8999 9348 9v97 9146 9195 9244 9292 9341 49 893a 949390 949439 949483 949536 949535 949634 949683 949731 949780 949S29 49 1 98378 9923 99 75 933J)24 95)073 950121 953170)950219 95026T 9530816 49 2 95036586953414 95'3462: 0511 0560 06U8 0657 0706 0754' 0808 49 3 0851i 0900i 0949 099Ti 1046 1J95' 1143 1192 1240 1289 49 4 13.8 1386 1435 1483 1532 15801 1629 1677 1726 1775' 49 5 1823 1872' 1920' 1969' 2017 20663 2114 2161 2211 22601 48 6 2308 23536 2105 2453' 2532 2550. 2599 2647 26961 2744 48 7 2792 2841! 2889 2938' 29386 30341 3083 3131 3180 3228 48 8 3276 8325 3873 3421 83470 35181 3566 3615 3663 3711 48 9 3736i 3808' 3856 3905i 8933 4001' 4049 4098 4146 4194 48 930 -[954243 954291 954339 954387 954435 954484 954532 954580 954628 954677 48 1 4725 47783 4821 4869 4918 4966 5014 5062! 5110 5158 48 2 5207; 52551 58033 53511 5399 54471 5495 55431 5592 5640 48 3 5683 5733 5784 5382 5830 59281 5976 6024 6072 6120 48 4 6168 6213' 26 6313 6361 6409 6457 6505 6553 6601 48 5 6649 669't 6745 6793' 6840 68881 69-6 6984 7032 7080 48 6 7123 7176 2 722 7272 7320 7368, 7416 7464 7512 7559 48 7 6371 7655' 7703 77511 7799 7847 7894 7942 7990 8038 48 8 8)86 8134 8181' 8229 8277 8325 837 1 8421 8468 8516 48 9 8564' 8612 86359 8707 8755 88031 8850 8898 8946 8994 48 910 959341 959389 959137 95919185 99232 959280 959328 959375 959423 959471 48 1 9518 95663 9614! 9661, 97091 97571 93041 98521 9900! 9947 48 2 9995 96Y042 930093,9601389 60185 960233 960280 960328 963376 960423 48 3 960471 0518 0566 61 0613 0661 0709j 0756 (0804 0851 0899 48 4 0946 09941 10411 1089 1136[ 1184 1231 1279 1326 1374 48 5 1421 1469. 15161 15631 16111 1658 1706 1753 1801 1848 47 6 1895 1943 199.) 2033 2085 2132' 2180 22271 2275 2:322 47 7 2369 2417: 2464 2511 2559 2606 2653 2701 2748 2795 47 8 2843| 28901 29:37 29851 30832 3079' 8126 8174 3221 3268 47 9] 3316 3363 3413 3457 3504| 3552 3599 3646 33693 3741 47 920. 963788 963835 963832 963929 9639771964024 964071 964118 964165 964212 47 1 4261 439)7 4354 44011 4448 4495 4542 4590 4637 4684 47 2' 4731 4778 48251 4872. 4919 4966 5013 5061 5108 5155 47 3 52o2 5249 5296 5343 53901 5437, 54841 55311 5578 5625 47 4 5672 5719 57661 58183 5860 59071 5954 6001 6048 6095 47 5. 6142 6189 6236! 62S3 63291 6376 6423 6470 6517 6564 47 6. 6611 6658 6705 6752 67991 6841 6892 6939 6986 7033 47 7 7083s 7127 71783 7220 72671 73141 7361 74081 74541 7501 47 81 7548 7595 T642 7688 7735 I 7782 7829 7875 7922 7969 47 9 80161 8062 8109 8156 8200' 8249 8296 834 8390 8486 47 930 968483 963530 968576.963623 968670 968716 968763 968810 968856 968903 47 1.8950 8996 9343 9930J 9136; 9183 92291 9276i 93281 9369 47 2 9416! 9463 9509 9o556 9802' 9649 96951 97421 9789 9838 47 3! 93821 9928 9975 970'021 970068 970114 970161970207 970254 9703001 47 4.970347 970393 970440 0486 0533i 0579 0626 0672 0719 07651 46 5 08121 0858 0904 0951 09971 1044 1090 1187 1183 1229 46 6 1276 1322 1369 1415 1461 1508' 1554 1601 1647 1693 46 7 1740 1786 1832 1879 1925 19711 2018 2064 2110 21571 46 8 2203 2249 2295 2342 23881 2434! 2481 2527 2573 2619 46 9 2666 2T12 2758 2804 2851' 2897i 2943 2989 3035 3082 46 I No. 0 1 1 2 1 3 1 4! 5 1 8 9 Diff. 42 498 LOGARlITilMS OF NUMABERS. No.' 0 1 2_L 3 4 | 5 6 |7 8 _9 Diff. 9401975128 973174 973220 978266 978318 973389 973405 973451 978497 973543 46 1 8590 $636 8'682 3728 8774' 88s20' 3866 89131 8959 49.05 46 12. 4051i 4097j 4148 41'89 42835 4281| 4827i 48741 442() 4466 46 8, 4512 4558! 46044 4653' 4696 4742] 4788 48341 4880 4926 46:4.4972 50181 5064 51101 51561 5202' 5248' 5294 5840 5886. 46.5 5482 54781 55 70.5616' 5662 5707i 5753 5799 5845' 46;6. 5891. 5987 5953 6029 60751 6121 6167! 6212 6258 6304 46 7 6850 6896 6442 6488 6533 6579 6625 6671 6717 6763 46 8 -6808 6854 690 6946 6992 7037 7ut 3 7129 7175 7220 40 9' 7266 7312 7358 7403 7449i 74951 7541 7586 7632 7678 46 |950 977724 977769 977815 977861 977906 977952 977998 978043 978089 978185 46 I 1 8181 82261 8272 8817 88638 84019 84541 8500 8546 85911 46 2 8637 8688. 8728/ 8774 88'219 88651 891l 8956 9)02 90471 46 8 9093 9188i 9184 9230 92751 9321l 9866 9412 9457, 9503 46 4 9548 95941 9639 96851 97301 9776, 9821 9867 9912 9958 46 5 980003 980049 981094 980140 980185 9802381 980276 98u322 980867 980412 45 6 0458 /)5038 0549 0594 0640 06851 0780 07761 0821 0867 45 7 0912 6957, 1003 1048' 1093 1139 1184 1229' 1275 1820 45 8 1366 1411 145'6 1501i 1547 1592 1687 16838 1728 1773 45 9 1819 1864 190o9 1954' 2000 2045 2C90 2185i 2181 2226 45 960 982271 982316 982862 982407 982452 982497 982543 982588 9826383 982678 45 i 2723 27691 2814 2859 29J04 29491 29941 830140 3()85 810 45 317 82201 3265 883310' 856'. 3401 83446 3491 8536 3881 45 3 8626 3671 8716 37621 3807 3852i 8897 8942 83987 4032 45 4 4077 4122 4167 4212 4257T 4302 4347 4892 448T 4482 45 5 4527 4572i 4617 4662 47071 472 4797 4842 4887 4982 45 6 4977 5022 5067 5112.5157 520i21 5247.5292 5837 5882 4$ 7 5426 54711 5516 5561 56061 56511 5696 5741 578:6 5830 45 8 5875 5920 5905 6010 60551 6100 (,144 6189 6284 6279 45 9 6324 6369 6413 6458 65031 6548 G598 6637 6682 6727 45: 970 986772 9868171986861 98690698695)1 986996 987040 987085 987138098!7175 45 1 72191 72641 7309 78538 7898 74438 7488 75821 7577 7622 45 2 7666G 7711 7756 7800 7845 7890 79341 7979 8024 8068 45 31 811i 81571 8202 8247 8291 8336 8.811 8425 8470 8541 45 41 85591 8604 86481 8693 8737 8782 8826 8871 8916 89601 45 5 9005 9049 9094 9188 9183 9227 9272 9:316 9861 9405 45 6 94501 9494 939 95831 96281 9.672 97171 97611 9806 98501 44 7 9895 9939 9983 0(990028 990072 990117 990161 990206 9902501990294! 44 8 99033999088319904281 0472 05161 05611 06105 0650 06941 07388 44 9 0788j 0827T 0871 0916 09601 1004 1049 16938 1187 1182 44 980 991226 991270 991315 991359 9914-03 991448 991492 991536 991580 991625 44 1 1669 17113 1758 18021 1846 1890 1935' 1979' 2023 20671 44 2 21111 2156 2200 22441 2288 2883 2377 2421 2465 2509 44 8 25541 2598 2642 26861 2730 2774 2819 2863 2907 29511 44 4 299531 039 80831 8127| i'72 82161 8260 804 8348 8892 44 3436' 3480 3524 3568 3613 8657 3701 3745 37891 3833 44 6 88771 89211 89651 4009i 40,531 4097 4141 4185 4229 4278 44 7 48317 4361 44051'4449 4493! 4587 4581. 4625 4669 4713 44 8 4757 4801 4845 48891 49831 4977 5021 50651 5108 5152 44 51961 5240 5284 5328i 5372 54161 5460 5504 47 5591 44 990 995635 995679 995723 995767 995811 995854 99589S 995942 995986 996080 44 1'6074 6117 6161' 6235 6249 6293 68371 6880 6424 6468 44.2 65129 6555 6599 66i483 6687 67311 6774 6818' 6862, 6906 44 8 6949 6998 70871 7080 7124 7168.7212' 72551 7299 78.483 44 4 786. 7480 7474 7517 75611 7605 7648 7692 77861 7779 44 5 78238 7867 7910 79541 7998 80411 8085' 8129 8172 8216 44 6 8259 8303 88471 8890 8434! 84771 85211 85641 8608 "8652 44 71 8695 8789 8782 8826 8869' 8913 8956 9000 90431 9087. 44 913 914 9218 92611 9385 9848 9392 9435 9479 9522 44 9 9565 9609 9652 9696 9T79 9783 08261 97 9913li 9957i i4 No 0 1 2 3 6 8 Diff LENGTHS OF CIRCULAR ARCS. 499 TABLE of the Lengths of:Circular Arcs, radius being unity. Se.e Length of arc Mmn. Length of arc.: De. Length of are.l Deg. Lent!h ofarc. 1 0-0000048 1 0-0002909 1 0017458833 61 1'06i6509 2 o0000097 2.0005818 2 08349;06 62 -0821042 8 0000145 3 0008727 0 0523599 63 ['0995575 4,0000194 4 *0011686 4'0698182 64 1170108 5 *0000242 5 1 0014514 i 5' 0872665 65 1344641 6'0000291 6, 0017458 6 1047198 66'1519174 7'00003.39 7' 0020362 7 11221730 67 |. 1693707 8 *0000388 8'0023271 8 11396268 68:1868240.9'00004. 1 9 1 0026181 9 1570796 69'2042773 10 0000485 i 10'00290i89 10- 1745829 70' 2217805 1-1 0000583 1 11'0031998 11.1919862 71'2391839 12; ~'0000582 12 0034907 12. 2094395 72'2566372 13'0000680 1 83 -0037816 183 2268928 78'2740905 14 0000679 14;0040725 14 I 2443461 74.2915438 15,000072T 15,0043684 15 261799 1 75 3089970 16 0000776 16 - 0046543 16 2792527 76 - -8264502 1,7 0000824 17 I 0049452 17 2967060! 7'8439084 1''00008783 18 0052861 18 3141593 78 83613567 19 0000921 1 19'0055270 i 19 83316136 79'87S8100 2) 0000970 1 20'0058178 i 20 3490659 80 8962634 1 21i 0001018 I! 21 0061087 21. 3665192 81 4187167 22 0001067 22 1 006399 22 3883972 1 82 14311700 28 0001115 1 283.0066935 23 -4014258 83 1 4486233 24 [0001164 1 24.0069814 24.4188791 84,4660766 21 0001212 25'0072728 25'48368824 85 1'4S85299 26 - 0001261 26 0075682 26'4587857 86'5009882 27 0001809 27'0078540 27 -4712390 87 -5184365 28'0001858 I 28'0081449 28'4886923 88'5358898 29 000ooolo406 29 1 084358 29'5061456 89 a55.33431 830 0001454 390 1 0087266 3 80 5285988 90 {5707968 81'0001502 31'0093175 31 -5410521 91 -5882496 32'0001551 32'0093084 82'5585054 92'6057029 83.0001599 838 0095993 38 5759587 93'6231562 84,0001648 84'0098902 34' 5934120 94 I 6406095 85'0001698 35 010l811 85 1 6108653 95 -6580628 86'0001745'0 8 3 0104720 36 1 6288186 96 6755561 87 *0001793 37'0107629 3 87 6457719 1 97'6929694 88 ]0001842 88'0110588 38 -6632252 98'-7104227 39 0001890 389 | 0118447 89'6806785 99 -7278760 40'000i939 40'0116835 40'6981317 100'7453293 41'0001987 41'0119264 41.7155850 1'7627826 42'0002036 42'01221738 42 78830388 2 7882859 48,0002081 43'0125082 43'7504916 3 7976892 44'0002133 1 44'0127991 44'7679449 4 8154125 45'0002151. 45'0130900 45' 7858982 5' 8825958 46'0002280 46'01388809 46'8028515 6 18500491 47'0002278 47'0136718 47'8208048 7'8675024 48'0002327 48'0189627 48'8377581 8'8499557 49 10002875 49 014256 49 8552113 9 1'9024090 50'0002424 50'0145444 50.8726646 110 9198622 51 10002472 5i 0148858 51.8901179 11'9378155 52 -0002521 52'0151262 52'9075712! 12 1 9547688 58 1000259 58'0154171 6 58 1 9250245 13 1 9722221 64'0002618 54'0157080 1 54 9424778 14 9896754 05' *0002666 | 55 09989 9599311 15 20071287 56 i0002715 56'0162898 56,9773844 1 6 -0245820 57'0002763 57'0165807 I 57 9948877 17 10420353 53.0002812 58 016816718 58 1.0122910 18'0594888 59.0002280 1 59'0171625 159'0297488 19.0769419 f60 0002909 60'0174583 i 60'0471978 1120'094,951 500 LENG'TIIS OFI CIRCULT,AR ics.C. TABLE of the Lezgthls of circ ular Arcs, radius beiny unity. Deg.!Length ofarc Deg.!Length ctflrc. Deg Length ofarc De. Legtofar. 121 21118484 11181 8 3'1590460 i| 241 42062485 8301 I 5-2584411 2 1293017 2'1764993, 2 *2236968 2'2708944, 3o 14t6750 3!1939526 3 2411501 3 2 883477 4 j 11G42083' 4 2114059 4 *2236984 4 2868010 5 1810616 5 - 2288592 5 *2760567 5 2 * 32.2542 6 *1991149 6 i246312 51 2935100 6 -3407075! 7 2165S2 l 7 2687.658 7: 10968 7'358 I 608 1 8 *234021.5 1 1 8 1 2812191 l 8 1 *3284166'8 87.56141 19.2514748 0 1 9 - 29~6724 9 3458699 9' 8930674 130 1268928 190 6316126 2 32 101 410520T 1 *2868813! 1 i -335789 1 i 3807764 1 4279740 2'3038346 2 3510322 2'38982297 2 4454273 3 3212879 311 3 | 684855 3 1 4156830 3'4628806 4 3387412 I 4 3589388 1 4 1 4831363 1 4 *4808339 ~5 3861945 5'4083921 4505896 5'4977872 6 83736478 6 42s08454 6 4680429 6 36152405 7 3911011! I7 882937 7 4854962 7'5326938 8 *4085544 8 4557520! 8 5029405 F 8 5501471 9 *4260077 F 9 I 4732053 j 9'5204028 9 I'56766004 140 *4434609 200'4906585 260'5378560 8323 5850536 1'46091443 F 1'5081118 1'5553093 1' *6025069 2 -4T83676 2 5255651 2 *5727626 l 2 *6199202 3 *49582C9 3' -5430184 3 *5902160 3 6374135 4 5132742 4 5604717 4 *6076693 4 6548668 5 0 *5307274 5 i'5779250 | 5 6251225 5'6723201 6 5481807 6 5953783 61 G 64257581 6 6897744 7'5656340 7 1'628016 7 *6600291 7 1 7072267 8 36880873! 8 16323849 8 -6T41824 8 7246800!9 |60005406 1 9 64477'382 9 *6949357 9 1'7421332 1 60,6179989 210'6651914 270'7123890 j 30 7595865 1 i j635447T2!1 1 6826447T 1 -'7298423 1 i'7770898' 2'6529005 i 2 700C80 2'7472956 2 17944931 3 1 6703538 V 1 38'7175513' 3 *7647489 3'8119464 4' *6878071 TI 4'17350046 4 17822022 4 i'8293997 5 7052604 I 5'7524579 5'7996554 5'8468530 6 7227187 6 7699112 6 8171087 6 8643063 7 7401670 1 7878645 7'8345620 7'8817596 8 T1,75T6203,1 8'8048178 8 i 8520153 8'8992129 9'7750786 9'8222711 9'8694686 9'9166661 F 160'7925268 220'8397243 280' 8869219. 1 340 9341194 1 8099801' 1'8571776' 1 9043752 1'9515727 2'8274884 2'8746309 2 9218285' 2 1'9690260 | 3 8448867 i 8920842 3'992818'1 6 9864793!4 -S623400 1 4'9095375 4 *93B7351 i 4 60039326 87979.33 926990 5 59741888 218859 6'8972466 6 9444441 6 9916416 6 08892 17 9146999 7'9618974 7 5-0090949 7 0562955'8 9321532 8 9793507'08 264582 8 0737458 9 |9496065 9 ~ 9968040 9 9 0440015 i 9'0911990 i 170 *9670597 T 230 4 01425792 290 60614548.8350'1086523 1'9845130' 1 0317103 1'0789081 1 1261056 2 38001966T 2'04916,39 2 60968614 ]2 1435589 3 019419i 8 3 0666172 3 1188147 38 1610122 4 |0868729 I 4 0840705 4 |1812680 4 |1784655 5 05438262 i F'1015287 5'14872183 5'1959188 6 0707o795 6 1189770 6 1661746 6 2183721 7'0892828 7 1364303 7 186.s279 7'2308254 8 |1066861 8 158886 1 8 2010812 i 8 2482787 9 1241394 9'1713869 9 2185845'9 2657820 180'1415927 Ti 240 1887902 800 28359878 1 60 2831853 I 1i I 9 F! 2 8T LONG:CUOoDS. 601 IEXPLANATION OF THE USES AND APPLICATIONS OF THE TABLE OF LONG CHIORDS, PROBLEM. Required to find the distances or abscissas on tlie chord from7?twhich, if ordinbates or perpendiculars be drawn, they will pass through the station points on the curve. EXAMrPLE. —Let the given curve be 1000 ft. long of 5~ curvature, i or 1146 ft. radius. For the first station from the beginning we have chord for 1000 ft. - chord for 800 ft. 2 ~~ — 1st distance, chord 800 - chord 600 2nd distance, etc. 2 Then by table we have, Intermediate 968687 - 784.1~0 Distance. Distance. Total. 2 -- 92'385 Ist dist. - 92385 78410- 593'36 2 —- -95'370 2nd " = 187'755 593-36 - 398'10 -97,630 3rd' 285-385 89810 - 198'81 99-645 4th " = 385'030 198'81 0000 - 99405 5- th "' = 484'435 96S887S 484435 = half length = 2 Thus for any given station we take from the length of the whole chord the length of a chord of twice as many stations less than the one under consideration; that is, 1st station from beginning 2 less; 2 from beginning, 4 less, etc., and take half the difference. If the chord had been for 900 ft. of curve, we should have, 877'32 - 68939 -- = 93965 = 1st distance. 689'39 - 496'20 96'595 - 2nd " 2 496'20 - 299'24 -- 98'480 = 3rd " 29924 — 100 - = 99-620 = 4th L' 2 388-660 Add 5io- 877'32 438'66 - =half length of chord. 5082 MIDDLE OnnINATES. In like manner we may find the ordinates connecting these, abscissas with their points on the curve. Let the length of chord and radius be as already given. Then lwe have, M lid. ordinate 1000 ft.- mid. ordinate 800 ft. curve = ordinate at 1st station. 3Mid. ordinate 1000 ft. - mid. ordinate 600 ft., = ordinate at 2nd station. For this purpose we have calculated a table of middle ordinates corresponding to that of long chords. From this we have, 10739 - 69'13 = 38'62 = 1st ordinate. 107'39- 39'06= (i8'33= 2nd " 107'39- 17'41 - 89 98= 3rd " 10739 - 4'36 = 103'03 - 4th " 107'39 -- 000 = 107'39 = 5th or middle ordinate. Were the chord for 900 ft. of curve we should have by tables, 8725 -- 53'05 = 34'20 = 1st ordinate. 87-25 -27'1.7 = 60'08 = 2nd " 87 25 - 981i = 77-44 =3rd 87 25- 109 = 86'16 = 4th " 87 25 - 000 = 872 -2 middle This will sufficiently demonstrate how the ordinates can be obtained for any other length of chord or curve. The same principle obtains in regard to any other rate of curvature. After passing the middle ordinate, their lengths will be repeated inversely; as will also be the intermediate lengths of abscissas. Then from end of first abscissa erect first ordinate, and so on in regular rotation. MIDDLr ORDINATES. 503 TABLE Of Middle Ordinates fromt Cihords subtending Curves of from 100 to 1000 feet in lenqgth; calculated to every 15' of Curvature from 15' to 8~. cRadius of 1~ being 5730 feet. LENGTIIS OF ARCS. 100 1 200 800 1 400 1 0 1 600 1 700 800 900 1 000 MIDDLE ORDINATES. Curvature. 0" 15' 0-06 0 22 0o49 0o87 1.36 1 96 2 67 3-49 4.42 5 45 80 0-11 044 0-98 1 75 2-78 8398 5-34 698 8-88 10-90 45 016 06'5 1147 2'62 4'09 589 8-01 10-47 13825 I1-85 1~ 00 I 0 22 0'87 1-96 0349 5b45 7-85 10069 18 96 17-67 21-80 15 027 1 09 2'45 4'86 6-82 9'81 18-36 1744 22(107 27'24 30 0 33 1'31 2'94 5'283 8-18 11-77 16803 20(98 26-48 32 68 45 0 88 1'53 0 43 6'11 9'54 1807 18 70 24-41 30-88 38811 15 0:49 1'96 4'41 7'85 j12'26 17'64 24-02 8318 39-G6 48'93 2300 0 2-18 4-91 8 72 113 1682 19-60 268 444 5 483. 45 0-60 2-40 540 959 114-98 216 29 8829 4841 59371 8~ 00' 0 65 2'62 5'89 10-46 16'34 2852 38198! 4174 52'78 65' (8 15 0'71 2-84 60-8 11-831 17'70 25-47 83463 45-19 57'13 70'44 80 0876 8'05 6-87 12'20 19'06 27-42 87'28 480G3 61.47 75'78 45 0 82 3827 738 13'07 20'41 29 36 39 92'520(7 0580 81'10 4' 00' 087T 49 7T85 13 94 12177 311 1426 550 70-12 86-40 15 0893 3 71 8 84 148t1 23812 83325 45 19 58 92 74'43 91'68 30 1 0398 3098 882 15'68 24-47 8,5'19 47-S2 62-84 78-72 90694 45 1 1-04 4-14 9'32 16'55!25-82 7-183 5044 65-74 82'99 1(;218 51 00' 1 09 4'36 9481 17 41 127-17 390C6 1 5305 ('913 82. 107-89 15 ii`~15 4'58 10'30 18'28 -2852 0'99 90'50 i12'58 -15 l 1-15 4 980 1741 2 1 55-67 72 90*50 15S 1 30!120 4'80 10'79 19415 29S7 42'92 5827 T 7588 95783117'75 45 1 25 501 11-27 20'01 13121 44-84 60-86 7925 9994 192 89 6~ 00' 11 5-23 11-76 20'88 38255 46-76 16845 82-60 lC413 l12799 15 136 545 12'25 21'74 33 89. 4867 66 04 85,98 1(0830 11388307 30 11 1-42 5'67 12-74 22-60 385283 5059 268.625 920 11245 138S12 1 45 11 147 5'89 13'23 2304 7 86'52'50 11.8 9257 1S1416 31 7~ 00' 1530 6'10 13-71 24-33 3791 54'40 7084 9587'120( 9!148 12 15 15 6-82 14-20 25'19, 89'24 56'80 76380 9915 124'78 1.3 07 30 16 4 6-54 14'69 26005 40'57 58'19 78'84 1Cj242 128'84 157 98 45 1 69 676 15-18 26'91 41'90 60'08 81387 10C68 1132i8 16286 8~ 00' 1-7 6'98 15'66 2;77T 443 3 16197 8390 108 92 183689 j167 70 On the principles by wlhich the following tables are calculatecl. Let n = linear opening of switch rail, s = angular opening of rail, f = angle of frog, g gauge of track. Let x = length of chord from opening of switch rail to point of frog. Then will the amount of curvature between the opening of rail where curve commences and point of fiog- f -- s; therefore' the instrument setting over the open end of switch rail with a backsight on the fixed end of it, the instrumental deflection to the point 604 LENGTHS OF TURNOUTS. of frog will be But if the backsight be taken on a point (say 5 inches distant) parallel with the main track, the deflection will'f-s' f +s then be - 2 - s f * Making the value of x, radius, g- m 2 2 he e s will be homologous to the sine of f~. Then we have, Sin (f~ s N R:: g~- 1~: ~i R(g-m n)'2 ) ": g- x sin f s8 EXAMPLE: Calling s 1~ 15', f 6~ 45' 4'70, m 0 and g —m 4'28, we have sin. 40: R:: 4-28: x 6136ft. When a double openinrg of a switch ranil fori a double turnoout occurs, we have, sin. ( _ 2 ) R: -2 x 0: x = distance to nearest frog. The linear and angular opening of rail being the same, this table may be adapted to aily other gauge by increasing the value of r as given in this table, and thie length of radius of turinout 12 per cent. for every additional inch in the gauge. This is a little too much; the correction for a 6 ft. gauge being about 30 per cent. Thus 100 ft. chord of turnout on this track will give 130 ft. on 6 ft. gauge, and 1000 ft. radius will give 1300 ft. This is for a straight lin-e. When on a curve goingf the same way as turnout, it is sufficiently accurate for practice to add rate of curve of main track to that of the table; but when going in opposite direction, subtract it; thus making relative departure from main track the same as on a straight line. EXAMPLE: Tlius a 5~ frog for a 4ft. 8~ inch gauge gives a distance of 78-5 ft. curvature 4~ 46'. If the main track wTere a 4~ curve and going the same way, distance being the same, the rate of curvatiure would be 40 46'+ 4.= 8~ 46', radius 653 ft.; but going the other way 4~046' - 4~ = 0~ 46', radius 7473 ft. TABLE FOR LOCATING T-URNOUTS. 505 T AB:LE Of distanes:on chord from opening of swioitch giail where the icurve commences, to point of frog,:radius of curvature manzd rate per 100 ft., calculated to every 15 minutes of frog angle, from 30 to 150. Constant data: opening, of switch rail 5 inches = 42 ft., average angular opening say 10 1.5', rails being from 18 to 20 ft. long. Vlariable data gauges of road. Gauge 4ft. 8} inches = 470 ft. Rate of Rate of Angle of Length of curvre per Angle ot Length of curve per frog. Distances. raius. ft. frog. Distances. radius. 100 ft. 30 11543 37793 1 31' I 90 47:99. 3550 160,09' 15' 109 02 3023 3 1 50 15' 46 78 335'3 117 07 30' 103'28 261i32 2 11 30' 45'69 1 317'6 18 04 45' 98'12 2249'0 2 33 45' 44 66 301`3 19 02 4 93'45 1947'2 2 56~ 10~ 43'67 286-2 20 -03 1i5' 89.21 1704.0 3 22 1:15' 42-72I 27-2'2 21. 05 30'- 8533 15080- 3 48 8 30' 4180 2593 22 08 45' 81-78 1339'0 4 17 1 45' 40195 247'2 *23 13 5~ 7 8-51 1199-8 4 461 110 40 11 236 0 24 -20 15' 75'50 1081-6 5 18 I 15' 3936 2254 25 28' 30' 72'70 9803 5 51 30' 38'55 215'8 26 37 45' 70 01 892'9;6 25. 45'`37'81 20'6'6 27 48 60 67'69 81i68 7 01;127 ~ 37-10 198'0 29 - 15' 65'44 715'1 7 39 / 615'/ 641 91 1895 30 13 30' 63 33 69004 8 18 30'' 35 15 1824 31 27 45' 6136: 639,4 8 58 45': 35`12 17532 32 43 7O 59 50 593 0 9 40 13~, 3451- 1Q866 34 02 15' 577.5 550/8 10 24 5 15' 33391 162.2 35 23 30' 5601 514o6 11 09 30', -333.34 156 3 36 45 45' 54{55 481'1 11 56 45' 83279 150.6 38 08 8 53:08 415.8 12 44 14~ 32'26 14'5i3 39 32 15' 51'69 42833 13 35 I 15' 31574' 140'2 40' 58:30' 5036 398-3 14 25. -30' 31-24 13-5,4 42 26 45' 4911 34 15 17 45' 30 7 130 —8 43 -56:15 30 28 126 5 415 26 48 506 TABLE FOR LOCATING TURNOuJTS. TA:BLE Of distances on chord fro)m? opening of:swiitchl rail to point f/ frog, radius of curvature and rate per 100 ft. Gauge 4ft. 10 inches. Rate of Ratfe of Angle of Leneth of curve per Angle of Length of curve per tro ". Distances. radil s. soo ft. fi'og. Distances. radius. it. 3~ 118'89 3892' 1I 28' 9~ 49'42 365'7 150 41 15' 11-229 3217-0 1 47 15' 418 345 3 16 36 30' 106.37 2709- 2 07 - 30' 47 06 327 1 17 32 45' 101'06 2316' 2 28 45' 46 00' 310 3 18 294~5 96.25 2006. 2 51i I10 44.98 294-7 19 28 15' 91 88 1755' 16 15' 44.00 280'3 20 27 30' 8788 1553 3 41~ 30'1 4306 267- 21 28i 45' 84'23 1379' 4 09+ 45' 42.17 254 6 22 31~ 5~ 80-86 1235- 4 38V 11~ i4131 243' 23 36 15' 7776 1134' 5 03 15'` 40'48 2323 24 42 30' 1488 1009' 5 404l 30' 39'70 222'2 25 49 45' 72"21 919' 6 14 4: 5' 38:94 212'7 26 58 6 69-72 841' 6 49 112 3821 2039 28 09 15' 67-40 772' 7 251 15'1 3750 195'5 29 21 30' 65-22 712. 8 03o 30' 36 82 187 8 30 3 45' 63 20 658' 8 42~ 45' 36-17 1805 31 46 q7 61 28 610 9 231 113~ 35'54 173 6 33 00 15' 59-48 568' 10 06 15' 34-92 167' 34 18 30' 57 79 530' 10 50 30' 34'34- 160'9 35 39 45' 56-18 49.5 11 35 45'1 33-77 155 37 00 8 54-67 4643 12 21 14~ 33'22 149 6 38 20 15' 5324 436' 13 09 15' 3269 144-4 39 44 30' 5187 4102 13 -59 30' 32-17 139-4 41 10 45' 50-58 386-6 14 ~50 45'- 31'67 134-7 42 36 15~ 3118 1302 44 04 TABLE FOR LOCATING TURNOUTS. B07' TA:BLE; Of distances on chord from opening of switch rail to point offrog, radius -of curvature and rate per 100 ft. Gauge 5 feet. ~'' Rate of I Rate of Angle of Length of curve per e f Length of curve per frog. iDistances. radius. lOo ft.: rog. Distances. radius. 100 ft; 3~ 12351 4036' 1~254' 9 5124 379.9 15O 05' 15' 116'65 34836' 1 40 15' 50'00 3587 15 58 30' 110-50 2810' 2 02''30 48-88 339-8 16 52 45' 104-98 2403 2 23' 45' 47 78 322-3 17' 48 4g 100-00 I2080] 2 45 10i~ 4672 3062. 18 44 15' 95345 1820' 3 08'i 15' 45 71 291-2 19' 42 30' 9130 1611. 3 33 30' 4473 277 4 20 40 45'' 8750 1430' I 4 00 45' 43l81: 2645 21 40 50 84' 1281' 4 28 11~ 4291 2625' 22 42 15' 80/7 8 1156O 4 57 IS' 4200 2412 23 46 30' 7778 - 1047' 5 27'30' 41024 2309 24 52 45' 71500- 965 5 58 45' 4045 221 0 26 01 6, 72 32 873' 6 33 120 3969 211 8 27 10 15' 7000 802' -7 09 15' 38;95 2027 28 20 30' 67761 739o 7 46. 30' 3825 195 1 29 30 45' 65O65 684' 8 23 I 45' 37 57 187s-5 30 40 7~'S 6366-1 634 9 02 1130 36 92 180 2 31 50'15' 6178 S90' 9 43 "15' I 36 28 1738 33 02 30' 60-00 550- 10 25 30' 3567 1672 34 17 45'.58'36 514 11 09 I 45' 34508 161 1 35' 35 8 66'79 482' 11 54 114 3451 1554 36 55 15' 5530 I 459 12 40 15 3396 150 0 38 16 30'' 5'488 426 13 27 I 30' 33'421 1448. 8 39 38 45' 5254 |401- 14 17, 4D5' 32'90 1399 41 00 -15~ 32'39 135O3 1 42 23 508 TABLE:. FOR LOCA.TING TURNOU.TS.. TABLE Of distances on chord from. opening of -sitch rail to point of frog, radiu.s of curvature and rate- per 100 feet. Gauge 5 feet 6 inches. Rate of Rate of Angle of. Length of curve per!Angle of Length of curve per frog. Distatces, radius. 100 f.:frog. istances, radius. 1oot. 3~ 136.8 4478- 1 17' 99. 5687 420'7 130 39/ 15'. 1291 9. 370 1 32. 15. 5' SS40 3974 14 27 30' 112238. 3116 1 50 30' 54 14 3764 15 14 45' 116,27. 2.664' 2 09 45' 52'92 3'57 0 16 04 40 1-10075 2307' 2 29 10u 51i 74 339 1 16 65 1-5' 10571; 2019' 2 50 15' 50 62 322 5 17 47 30' 101 11 1786' 3 12 30' 49054 307 2 18 40 45' 96'90 1586' 3 37 45:' 48.52 292-9 19 35 50.93-03 14210 4 02 11 47 52 280 0 20 30 15' 89:46 1281. 4 28 15! 46 52 267 2. 21 28 30' 86014. 1161. 4 B 30' 45'68 2.55 7 22 26 45' 83-15- 1062' 5. 24 45.' 44'80 244'8 23 26 6~: 80,16 967, 5i 56 12~ 43'96 234'2 24 30 15' 7753: 888;8 6 27 15' 1 4314' 224'7 25 33 30' 75 04 819' 7 00 3.0' 42 36 215 9 26 3.6 45' 72'71 757'6 7 34 45' 4161 2077 27 40 7o 70:50- 702.8 8 10 13~ 4089 199'7 28 46 15' 6843 65.3 —8: 8 46 1 15' 4018: 1922 29 54 30' 66 47 609 8. I1 24 30' 39'50 18.5;:2 31 02 45' 64.564 57Q0 101 04 45' 885 178-4 3.2 11 8 62 8:91 53.4 -10 45 I14 382-2 1721 33 21 15'' 61:25 5016 11 27 15' 37461 166-1 34 33 30' 59 67 | 471-9 12 10 30'! 37|01 1604 35 47 45' 5819: 4448 12 54 45' 3644 154'9 37 0.3.1150 38187 1500 38 18,........... 81'8 TABLE FOR LOCATING TURNOUTS. 509 T AB L E Of distances on chord from openilg of switch rail to point offrog, radius of curvature and rate per- 100 ft. Gauge 6 feet. Rate of Rate of Angle of Length of curve: per Angle of Length of curve per firog. Distances. radi s. 100 ft. flrog. Distances. radius. 100 ft. 830 15006 49131 1 10'J 90 62-40 4616 120 26' 15' 141173 4060'3 1 24j 15' 60 81 435 9 13 10 30' 134 26 3419 3 1 40j 30' 59 40 412-9 13 55 45' 127 56 2923 7 1 57k 45' 58-06 3917 14 40 40 121 50 2531 4 2 16 100 56'77 3721 15 25 1 5 115-97 2215-2 2 3-5 15' 56554 353 9 16 12 - 30' 110 93 19604 2 556 30' 54 35 337 1 17 00 45' 106 31 1740-7 3 17i: 45' 5324 3214 17 50 5~ 10206 15600 3 40i 110 5214 3068 18 42 15' 98-15 1406 1 4..04k 15' 5104 -293 2 19 34 30' 9451 12744 4: 30 30' 5012 2805 20 27 45' 91-14 11608 4 56 45' 491 5 2686 21 22 6~ 88'00 1061'8 5 24 12~ 48:23 257 4 22 18 15' 8507 9750 -5 53 15' 47 33.246 0 23 15 30' 82 33 898 8 6. 23 30' 46 47 237 1 24 12 45' 79177 8312 6 54 45' 45 66 227 9 25 12 7" 1 71735. 17170 7 26 130 4486 219?2 26 12 15' 15 08 1711i-3 8 00 15' 44 08 210 9 27 14 30' 72 94 669 0 8 34 30' 43 3-4 203 2 28 17 45' 70'92 625 4 9 10- 45' 42 63 195 8 29 20 8~ 69-00 586-0 9 471 14~ 4194 188 9 30 23 15 617 20. 550.3 10 251 15' 41'26 1823 391 28 30' 65 47 517 8 11 05 30' 40:61 176-0 32 36 45' 63 84 4880 1 1.4.6 45'- 39 98 1700 93 45 150 39 36 164 5 33 54 431* 51 0 MISCELLANEOUS N"OTES AND EXAMPLES. MISCELLANEOUS NOTES AND EXAMPLES. SUPPOSE a curve contain 57.' 24' curvature, distance between centres of inner and outer track 5ft. Required difference between main and outside track. By table of circular arcs;: 57~ gives 0'99483771 240-" _00069814 1-0018191Multiply 5 5I0090955 Ans. 5 ft. To find the length of any circular are,, multiply tabular are of given number of degrees by the radius. Half of this. tabular length gives the tabular area of a section of some number of degrees, and this tabular area multiplied by the square of radius, gives the required area of sector; or this tabular area, multiplied by the difference of the squares of the two radii, gives the area of a ring. Thus if inner radius — 3 ft., outer - 4, thickness being 1, we have 42.- 32 = 7, which multiplied by tabular area gives area required. Suppose the radius of the intrados of an arch containing 134~ 46' is 6'3 ft., the thickness of voussoirs. — 1'5. Then 82- 652 - 21'75. 134~ gives 2'3387412 46' " 0-0133809 1340 46' " 2'3521221 x.2175 = 51 16 nearly, 5116 and 2 — =625'08 = area.: When the span and rise are given to find: the curvature of arc, rise make - nat. tang. ~ curvature. half span EXAMPLE.-Suppose span = 18 ft., rise _ 6 ft., then V = 0:666667 = nat. tang. 330 411', and 330 41~' x 4 = 1340 46' of curvature. Let it be required to find radius, we would then have, ( span)2 + (rise)'2 9a + 6 radius.. Thus =- 9756 =radius of are. 2 x rise 2 x 6 Had it been a 12 ft. span and 4 ft. rise, radius would' have teen 6'5 feet. Analogous to this last example, and derived from the same proposition of geometry, is an easy method of determining the distance across a river or ravine. Let the instrument be at. B with a foresight; upon C across river; from B lay off a right angle to D. Set the instrument over D and AMISCELLANEOUS NOTES AND EXAMPLES. 511 lay off from D C a right angle D A meeting C B produced in A. Then by similar triangles, B DV AB: BD:: BD:BC; orAB -BC. Suppose that B D 2500 50 ft. and A B = 3 ft., then — = 833'3 ft. To Triangulate round an Obstruction o0 a Curve. EXAMPLE.-Suppose in running a 3~ curve, I find the point for sta. 2645 to be occupied by a house; I find, however, that 2644 + 75 and 2645 + 25 are clear of the house; also, that I have sufficient room for an equilateral triangle whose sides are 50 ft. each. Establish 2644 + 75 and set the instrument over it. Now suppose the last reliable point on curve to be at sta. 2640. The instrumental deflection from 2640 to 2645 + 25 = 525 ft. is 7~ 52i'. Set the vernier to this reading, and clamp the instrument with a backsight on 2640, so that, when the vernier is at 0, the telescope may point towards 2645 + 25. Unclamp vernier, set the reading at 60~, and measure 50 ft. in line of telescope. Set instrument over this point, and turn the interior angle = 60~, measuring 50 ft. as before. Set the transit over this last point, sta. 2645 + 25, with the vernier at 60~ so that the zero line shall coincide with the chord from 2644 + 75 to 2045 + 25. Clamp the instrument with a sight on the second point or vertex of triangle. Then set the vernier at 1~ 52-', the instrumental deflection for 125 ft., and the telescope will point in direction of sta. 2646, from whence continue the curve, if required, as before. This was an expedient applied to advantage by a former associate in making the final location of the Ohio and Mississippi R. R., Ripley County, Indiana. Similar examples and corollaries to previous propositions might be added indefinitely, but this would transcend the proper limits of the work. To an adept practitioner possessing ordinary faculties of generalization, it is believed the rules and formulas already given will be suggestive of the means of solving most of the other problems which may occur in practice. INDEX. Accumulated Work, 56, Boiling point, height of, at different Adhesion, 176. heights of the bairometer, 287. Air, weight of given bulks of, 63; ex- Bolts and nuts, sizes of, 288. pansion of, by heat, 231; properties of Bookbinders' recipes, 352-355. atmospheric, 277. Boyle's law of elastic force, 177. Alabaster, to harden and polish, 67. Brass, weight of superficial foot of, 144. Algebraic symbols, 15. Bronze powder, 159. Alloys, 286; recipes for various, 1t5159. Bronze, recipes for various kinds of, Alto-relievo, engraving in, 222. 347, 348. Amalgams, 89. Bronzing liquids, 108; for gun. barrels, Amonton's law of elastic force, 177. 179; for tin castings, ib. Analysis of various organic substances, Bricklaying, 297. 269. Building, terms used in, 881-841. Anchors, size of, proportioned to the tonnage of vessels, 272. Cables, strength of, 271; size of, accordAnimals, work of, 55. ing to the tonnage of vessels, 272. Approximations, decimal, 41. Capillary attraction, 224. Arches, hemispherical, how to construct, Carbon, 173. 295. Carpentry, 305. Architectural terms, vocabulary of, 831- Cast-iron, dimensions of cylindrical 841. columns of, 142; compression of, 209; Arcs, circular, table of the lengths of, bars, breaking weight of, 207; bars, 499. transverse strength of, 211; beam, Arithmetic, decimal, 24; instrumental, pressure it will sustain, 140; pillars, 42. table of ultimate breaking- wveight in Arithmetical signs, 18. tons, 119-122; pillars, strength of, 216, Artificial stones, recipes for making 217; pipes, 143, 149; pipes, table of different kinds of, 362-865. Z weight of, 131; plates, weiliht of, 245; Atmosphere, temperature and weight shafts, table of strenlgts of, 128, 124; of the, at various heights, 279. wheel, to find the horse-power it is Attraction, centre of, 175; capillary, 224. capable of transmitting, 246. Axle grease, 283. Cement, 300, 302. Cements, recipes for various, 262-269. Balls, cast-iron, weight of, 129. Centre of attraction, equilibrium, fricBeam, cast-iron, pressure it will sustain, tion, gravity, and gyration, 175, 176. 146. Centre of gyration, 58; of percussion, ib. Beams, equilibrium and pressure of, 59; Centrifugal force, 50. transverse strength of, 209-211, 247;: Chain rigging, scale of proofs for, 291. deflection of, 212. Chains and ropes, comparative scale of, Bending and gluini-up, method of join- 247. ing woodwork by, 828, 829. Chalk drawings, how to render permaBinary compounds, 250-255. nent, 858. Black, for miniature painters, 203. Chords, table of, 414; uses and applicaBlacking, recipes for, 202; 208. tions of the table of long, 501. Blacklead pencils, 2013. Circles, table of the diameters, circumBoard measure, 146. ferences, and areas of, 160-166; cirBoiler tubes, table of surface of, 241. cumferences and areas of, from one to Boilers, engine, 47; the, of steam- fifty feet, 167-173. engines, and their proportions, 77. Cisterns, capacity of, in gallons, 274. 514 INDEX. Clothes balls, 159. Electrical machines, amalgam for, 89. Coal, number of cubic feet to a ton, 280. Electrotype manipulations, solutions Cocks, bore and weight of, 134. used in, 109. Cog wheels, pitch of, i68. Enamel, various recipes for making, 848, Cohesion, 176. 849. Coins, method of taking impressions Engine, horse-power of, 57; boilers, 47; from, 118. locomotive, general proportions of, 76. Combustion, spontaneous, to prevent, 79. Engraving in alto-relievo, 222. Composition, for covering buildings, 268; Engravings, transfer of, to plaster casts, for walls, 804; and resolution of forces, 129; how to take casts of, 133. 812.. Equilibrium, centre of, 175, of beams, Compounds, binary, 250-255. 59. Compression, 176. Equivalents of sixty-two simple subConducting powers of substances, 290. stances, 248. Copper, liquids for bronzing medals, Etching fluid for copper and steel, 214.. figures, &c., of, 108; tubing, weight of, Extension, 176. 134; weight of superficial foot of, 144;. bean and feather shot, 152; etching Factitious stones, recipes for maling fluidfor, 214; rods or bolts, weight of, different kinds of, 862-3865. 278; pipes, weight of, 274; sheet, Feeding properties of different vegeweight of a square foot of, 275; dimen- tables, 249. sions and fweight of, 183. Ferrules for tool handles, solder for, 133. Crane, to calculate the different parts of, Feet, table of solid, 245. as respects mechanical advantage, 58. Figures, copper, bronzing liquids for, Crayons, recipes for, 358. 108: Crystals, diamond prismatic, for win- Floors, the construction of, 818-320. doews, 82. Fluids, elastic, mechanical laws of, 177Cubes and cube roots of numbers, table 179.. of, 180-201. 7Formnlrl E for running lines, locating Cubic measure, 147. side-tracks, &c., 375. Cupolas, plan to take away the lateral Forces, parallelogram of, 59. thrust of, 295. Fresco painting, 132. Cylinders, cast metal, 148, Friction, centre of, 175. Fuel, power of various species of, 288. Dalton's law of elastic force, 177; ex- Furniture oil, 127; paste, 159. perinment on gases,.178. Fusing point and fusibility of metals, Dampness, how to preserve walls from, 282, 284. 158. I Decay of wood, means of preventing, Gases, illuminating, 178; propertieo af, 307. 278; pressure at which certain, are Decimal arithmetic, 24; approximations, liquified, 257. 41. Gauge-points for the engineer's and Decomposition, vegetable, circum- common slide rule, 45. stances favorable to, 305, 306. Gay-Lussac's law of elastic force, 177. Diamond crystals, prismatic, for win- Geometry, practical, 16. dows, 82. Gilding, recipes for different kinds of, Digging, measurement of, 301. 859-361. Distance, how to estimate, 82. Glass, perfectly black hard, 82; soluble, Domes, plan to take away. the lateral 222; recipes for making different kinds -thrust of, 295. of 242; how to stain various colors, Dovetailing, process of, 324. 861, 862; paper, how to make, 304. Drawings, chalk and pencil, how to ren- Glazes; 102. der permanent, 858. Globes, liquid amalgam for silvering, 89. Driving wheels, revolutions per mile of, Glue, 263, 264, 267; recipes for various 86. ~ kinds of, 842. Drums, velocity of, 68. Gluing joints,. process of, 323. Ductility of metals, 289, 290. Gold lustre for stone-ware, 70. Dyeing hats, recipe for, 256; recipes for Gradients, table of, and resistance per compound colors in dyeing, 350, 851. ton for each, 118. Gravities, specific, of simple substances, Earthenware, glaze for, 102. 248; specific, of bodies, 281. Elastic force of steam, 80; fluids, mocha- Gravity, 47; specific, 61; centre of, 175. nical laws of, 177-179, Grease, scouring drops for removing, Elasticity of torsion, 143. 103: for axles, 288. TABLE,, FOR CURVING RAILROAD IRON., BY CHARLES HASLETT, CIVIL ENGINEER. G[vFN to the nearest sixteenth of an inch, the f a IO feet r il beina one inches and 11 inches. The s ring ill be tested at each point by a suitab] spring,.0 a 2 p w fourth that of a 20 feet'ail the spring of a 9 feet rail bei: gone fourth that of an 18 Where the sign + occurs curve a little more; and where the sign - n feet rail. When the rail is properly curved, the spring at fourth the length from little less., the end of the rail will be three-fourths that at the middle. At 5 10 and 15 feet of a Opposite the rate of curvature and under the length of rail, will be 20 feet rail, the spring, for a.19 degree curve (301 -feet M'dius) would be 112- inches, 2 quired spring, in inche'sand -part,% of an inch. Defiection LENGTH OF RAIL IN FEET, per Ce P4 100 ft. 10 11 -12- 1'3 14 15 16 17 1 18 19 20 21 22 23 24 25 26 27 28 29 30 10 I IL + -a —- -fq6 —6 4 + I.1+ __ 00' 5730 1 6 8 8 8 1 6 IL 6'3-:!- T2 116 — 16 8 8 16 1 5 4584 L- _jL +:LP + IL ] — IL I+ 3:L A_ - 1P + 16.1 6 16 1 16 5+ 4 4 -41+ T 6__ 30 3820 + _jL + + 3 3 3 X I+ -A — _6 6 16 16 TW __ 4 4 16 IL6 16 8 I T6 8! 8 W 8 6 + 4 45 3274 + -I- Y+ V- I I+ 3 3 3.+ I I I+ 5 5 5+.;1- 3+ Tw W 4 -— T T-6- Fg-8 3 1. IL+ 5 87 7 2000 2865 I + _i_+ I IL 5 IA IL+ I+ 3 4 T Z_ 1W 2 TW_ __ 1 6 8 8 8, 8 1 6 1 6 15 2546 + I I-I- I+ 3 3+ J+ 5 5 7 7 JL _f 1 6 1 _6 5+ -1 + 1 6+ 2 9 I 1 6 4 1 6 1 6 -Fif _q 1Y I+ 5 3 3 W + % 87 87 30 2292 -T -A- + 4 1W TW 1 6 8 1 6 1 6 6 2 2 1 6 W 6 iff 1 6 8 8 1 9 5 + 45 2083 I+ 3 3+ IL- -1+ 5'I 7 - 1 6 + -A6- -817 8 T _6 4 4 1 6 1 6 8 1 6 Y__ -29 1 6 8 8 I V_ 5 5 3 3 7 + 1 + + 3000 1910 + "_ + + -- g 8 8+ 8 r6 1 6 1 6 8 -- T 2 29+ 5w+ 8 8. 1 6 3, 3+ I I I - _6. -I I -- 3L I+ + 15 1763 + I I + + 5 + -+ + -27 1- 6 8 8 T W -1-9- 4 T T-6 -1 6 8 8 6 29 1 6 1 6 4 30 1637 -L+ +!-+':P + + 5 5 7 1+ + IL3 - - 1 6 6 1.U + t__ - -- - _f W 8 1 6 1 6 4 1 6 8 T6 W+ IL 6 ff 16+ 3 3 3 L_ 16 1 IL + 13+ 7+ 1' 528 1- -1+ T-6 8 1 6 1 6 4 1 6 8 45 87 I-.1 6 fW -+ 366 87 2 + 7 1 4000 1432 I+ 3 3+ + 7+ 1 3C 4-5 5+ I _]6 + A4+ _6 8 T 8 1 6 8 4 5 1 Q___ 96+.1 5 3 7 + I I I 1 3 7 1348 _1 8 1 6 1W -1 6 4 16 IL 6 8 +.1 6 Y.1 6 61 6 3 0 1273 4- 3 + -1:11+ 5 5+ 3 7 1 3.+!_ 5 II_ A 1 3-_ i -_ fA- I 116 V6 4 -1 6 T _6 8 I 6 2'y 1 6 W 6 4 1 6 8 6 1 & 3 7 1 - 5+ I+ 3 + _f 3+ 7 + IL 45 1206 + 1W fW+ _I jL + 8 4 6 8T W 8 4 Y6 8 1 6 1 6 2 8 6 I I iw- 19P -3- 3 5. 3 7 1- I 1-1. I,+I-L 5000- 1146 8 16'176 4 4 2 29+ T -6+ 1 6 1W8 13-6:L IL I Zs __ 1I -1 16 1092 + -i- + IL + Tw- + IL+ I -6 _ff I+ 1 6 1W 4 3 5 5 _3 7 V +-13 + 7 + + 5 30 1041 + y + J- -6 8 1 6 -1- + J_+ -]L C,+ 4 16 _V6 8' 6 1 6 2 6 TW 8 3, 3+'L7 IL 5 + 1, _ 1 _,.L IL f __ I I+ 3 45 996 IL -Ji- A 8+ 6 4 8 86 4 8 9 5 3 5 3 + I+ V + + + 7 a W- A+ -L+ 1 6 4 8 4 1 6`00 iF-. 4+ TX 8 2+ 56: _fW 1 6 8 1- 1-L- +.15 918 -a- A- -J- 1+ + 3 7 + + IL 3__ 15 + IL 3 1 -A — 1-1 16 8 170 1 6 1 6 1 6 8 I W 1 6 8 2. + 5 + ij- _1 IL + + + ov 881 + 11-1 1-!k- + I I I 6 8 i _6 2 T 6 1 6 4 1 48 I 6- 2 3.3 + + + 3 + I I n+1jL 1 6:' 6 j- + X -+ -1 __ I 1W W. ]L W 4 -rw -45- 848- - + V6 1 6 8 1 __6 8 2 + + Il+ 87 1 3 I 7 7 00 818 1P _I IC + J- I + -f'6 IL 6 + -If _6 - 1 _A6___'A 4 + 1 6 3 I+ 5 3 IL, 9 5 1 1 8 3 15_ 1 6 1 5 I+ 792 f3 3,+ T 2'- 4 - - - 1 __P__ + _16 + 4 V6 W I 1 6 16 I W 6 1 6 8 2 1 6 764 -3 + 3+ 76 9P 5+ 11+ 1 3 5+ + 1 7-1 9- JA + 1 + 1-6 - T _ff 2 T 6 1W + IL 6 4 1 6 8 9 W 1 6 1 6 1 6 8 1 V6 4-5 739 -3+ 5 JL + 3+ + I+ -f W A4- 1 3 -L'+ 1 3 +Ill + I 1.6 I _6 8 T6 1 6 I IL+ 2 fW 1 6.16+ 8000 716 3 5 1 3+ 1 5 1 + 1.1 5 7 7 8 T 6 4 I -6 1.6 1 6 2 Il + in 1 6 8 1 6 + 3L W I 8+1 +'W' + 5+ JX__ 8 76 Li — I- 3+ _ I j__ 113- 1 J_,65 + W -1 6 4 8 4 -6 W 1 1 611 is 694 1 1-6' IL 6 +1X- 11 _6 I 6 -:+ 3 13- 7 + ]L 2 JA_ 10 674 J IL + 1-3 1 1+ J A -+ IL 1 A4 + 1W W I 6 V6 9 T6 8 6 4 8 8 1 1 5 IL 3+ 1 5 2 45.654 5 1 + I + + +1 1 6 1TZ6__ -9 81 1 6 2 T 6 1 6 1 I 6' 19+ IT6 1 6 0 6' + ]L -6 + A+'j- +.1 + W- II- 3 + + I IL. -3 + 1-f 00 5 7 8 8 91 63 W 8 J 6 1 6 4 8 1 62 2-1 7 1 & IL IL + 13 15 619 IT -1 -P'J I+ + I + I A+ 1 f+ 1 7+2-L — 2 —a- - 2 6 T 6 1 6 4 8 5- + IL 1JL_ 30 603 +.13 + I 3+ 5 - IL 3 1 1 5+2:1 + 21 - V6_ -:2, V W 1 6 1 6 W -u 1 6 1W 1 6 1 6 1 6 _f W 4 5 + 5'+ 3 IA+ 5 3 45 581 y 11_L + J;, I P- 2 21 + 2-5 — - -1 6 T __ T 6 6 8 2 1 6 8 1 6 L+ 4- A 7 1+ 9+ IL-1 3 5 10000 573 1 - _L__ 1 IL + J j 3+ 1A+ I 3. +11 + 2 2-;L + 2: 1 6 8 _6 2 1 6 1 6 1 6 8 4 8 2 4 T W 1 6 8 15. 669 1 - _ 1._ +I IL + I I IL 5+ 2-1 -- 2-1ILW -47 6 — Z — 1-_L- 1 6 4 6 2 W 1 6 1-6 + 7 9 1 6 1:1 I 6 1 6 8 L+ + 1 3 9P 5 30' 545 J + I 11 1A- + 1 + I -f-u + 1 1-:1 + I I 2 2-L + 2 2 _r 6 8 1 6 I W 2 53 1 9 5 1.2 3 7 -L.1 9. I - + - + 11 1.1 1A. it 1 2 1 2-! -+ 96 -16 1 6 2 TW IL 29 I 1.0 7 1 5+ 3 1 3 1 00 521 + IL 5 + IT + 3 IA- + 1116- 11-5- 2-L - 2 -I 2 -fZ6 —-24' 8 I 1 6 4 1 6 2 J. 6 1W 8 2 8 31 V _38f 8 + I I-+ 5. _f W 3 -9: 7 11.+ ji 3 7 I 7 111 + J -rIL -61 + 2 -- 21- + 2-&- 21 + V6 + 16 4 1 6 1 6 2 8 1 5 609 _:a -1 —6 - I I J- 1 8 16 1 6 1 6 1 6 8 7 IL + 3,0 498 + 1 5 i- 1-JL + I 9+ i 1W - 2 + 2 2:3 2:1 +2'J-+ 3- T W -6- T 6 3. 6 -1-6 4 2 1 6 I 2+ 8 1 6:L -8 T 6- 1 6 5 70 + 5 I 1 7+ IL 1 3 1 5 45 487 - - I a 3+ J1 2 -f!-6.+ 21 2-L — 2 Q — + 2A+ -8 8 8 8 2 -8- 1 6 4 176 1 6 4 + 6 6 4 I 1 7 1 1 2 040 477 + 1 13 + 1 1 + 1 1A. + 2-L 2 + 2A -0 _+ + + 1:1 2-2L- I 6 1 6 8 IL 6 Y 8 1 6 1 6 W 8 4 8 29 1 16 116 1 6 8 1 6 W + 8 213+ _jL+ 9 5 3 13 5 1:1 + 1 iL + 1 7 7 15 468 -5 1 2 2-13-,br-.- 2 5 + 2 yl 211 2 IL __ _8 4 T6 T -6 8 4 IL W-, I (3 1 6 8 iw 1 6 8 5 _96 5 11 7 9+ 7 30 458 - I W+ 3 1.3 + 1 5 1 1 1 3 2 2 j16- + 23- 2 9 23 21 5 6_ 1 -6 IL W T6 8 1 6 iw 5+ 3 7 + I 8 W, + 3+ 35. 449' 5' + 11L I 3 + 1.15- 2 1+ 1,2 IT 2 7 2A 2 13- 3 3 _f T+ W 1 6 I -6 2 - 1 fW I 1 6 8 1 TW 8 -13000 44 1 -A-+ 7 __9+ 11 1.1 JA + 1 3 27 3-L161 3 - i + 21. l2.y5X — 21 - I 1 6 2 _W 4 8 8 4 8 2 8 6 _f -6 81, 2 T -6- 8 - 1 5 432. 7 i9 3+ + I IL 2 5 4 21 + 2'-L + 2J-A — 31 11 13 + 1-1 + I'L +2 8 4 W 8 4 8 2 1 _6 16 T6 I W q,_1 6 -1 6 1A _1 I IL 1 3-. 424 7 7+ 1 30 4-1 1.1 -4-1 i- I 4-I. i -L-1 i I 1 -4- 3 IL 5 -4- i INsDE;x. 515 Grooving and lapping, method of join- Lateral pressure, resistance to, 186. ing woodwork by, 328. Laws of motion, 47. Gun barrels, bronzing liquids for, 179. Lead and. lead pipe, weight of, 184; Gunpowder, composition of different weight of thicknesses o0, 145; sheet, kinds of, 285. weight of a square foot of, 275. Gutta percha, solvents for, 179. Lever, the, 68. Gyration, centre of, 53, 176. Lime and hair, 301. Lines of railroad, formulae for running, Hardness of metals, ratio of the, 23. 75. Hardnesses of bodies, ratios of, 289. Linseed oil, for varnishing, 87. HIats, recipe for dyeing, 256. Liquids, table of properties of, 288; Heat, latent, in steam, 72; properties bronzing, 108.. and miscellaneous effects of; 228-281- Locomotive engines, general proporcapacities of bodies for transmitting tions of, 76. 2835; power of various substances to Logarithms, use of the tables of, 219transmit, 291. 222; tables of, from I to 10,000, 483. Heating power of various combustible Lustre, gold, forstone-ware, 70. - substances, 82. Lutes, fire, 266. Heel balls, 103. Horse-power, 55; of an engine, 57; of a Machinery, estimated power of man or steam-engine. rule for obtaining, 75; horse as applied to, 286. nominal, of low pressure engines, 83; Machines, work of, 56. of high pressure, 84: capable of being Malleability of metals, 289, 290. transmitted by a cast-iron wheel, to Maltha, or Greek imastic, 303. find the, 246; of-wheels, proportionate Man, units of work of a, 55; power, comstrength of, 258; comparison of, as ap- parison of, as applied to machinery, plied to machinery, 286. 286.. Horses, traction of, 56. Map colors, 143. Marble, plaster imitation of, 303; recipes Illuminating gases, 178. for staining and cleaning, 349. Inches, table of.solid, 245. Mariotte's law of elastic force, 177. Inclined plane, the, 66. Materials, strength of, 135; strain and Ink, recipes for making different kinds stress of, 204; of construction, weight of, 365-369. of, 234; conducting power of, 235. Instrumental arithmetic, 42. Masonry, different kinds of, 292, 293. Instruments, copper, bronzing liquids Mastic, Greek; 303. for, 108. Mechanical powers, the, and their appliIron, to preserve from rust, 153; cast- cation, 63. iron pillars, table of ultimate breaking Medals, copper, bronzingliquids for, 108. weight in tons, 119-122; cast, corn- Mensuration of superficies, 27, 44; of pression of, 209; cast and wrought iron solids, 36, 46; oftimber, 146; of circles, shafts, tables of strengths of, 123-126; 160. fiat and rolled, table of weight of, 130; Metals, ratio. of the hardness of, 283; malleable, weight of one foot length of, weight of, 97; paste for cleaning, 146; 182; sheet, weight of superficial foot and other substances, weight of, 285; of, 144; sheet and boiler plate, weight expansion of, by heat, 228; linear of a square foot of, 275; wrought, dilatation of, by heat, 230; fusing point square, round, and flat, tables of and fusibility of, 282, 284: ductility weight of, 97-117; wrought, tensile and malleability of, 289, 290. and compressive strength of, 208. Modelling wax, 86. Ivory, recipes for softening, hardening, Motion, laws of, 47. -dyeing, marking, &c., 174, 175. Mortar, 800. Mortice and tenon, 326-328. Joiners' work, finishing of, 330, 8331. Joints, method of gluing, 823. Numbers, properties of, 238, 239; table Journals-of shafts, strength of, 149. of. useful,. 240, 241; table of prime numbers to 5000, 243-245. Knot table, 259-261. Numeration, 42. Kyan, Mr., his method of seasoning tim- Nuts, sizes of, equal in strength to their ber, 310-312. bolts, 286. Lac, preparations of, 89. Land-surveying, 46. Oakum, weight of a cubic foot of, 180. Latent heat in steam, 72. Oil, linseed, for varnishing, 87. 5i 6 IKYNDEX. Oil, for furniture,.127; for watchmakers,. ladii, table of, and their logarithms, 409. 182; number ofgallons' in a cubic foot Railroads,'fornimnle for- running'lines, of. o280.. locating side-t-racks, &c. Or. ganic,substances, analysis.f.various,:Riedtrction,!24.'269. Refle'cting powers of arious substances, Oscillation, centre of, -16. 291. Resolution of forces, 312..Paint, old, solvent for, -102. lRigging, chain, scale df proofs for, 291. Paper glass and stone, how to make, 804. Roofs, the construction of, 314-317. Papers, recipes fbr preparing, for various Ropes, strength of, 218, 27-1;comparative purposes,.342-347, str'ength and weight of, 136;hempen, Parallel motion, table. to.determine dis- capacity of sustaining strain, 278; and tances of movable points in a, 22..chains, comparative scale of, 247. Parallelogram of forces, 59. Rule, slie, lidedesription o-f,-and manner Partitions, timber,-322, 323.'otusing, 42. Paste for cleaning metals, 146; for furni- list -to'preserve iron friom, 153. ture, 158; common, 265.; for laying cloth or leather on table tops, 267. Safety' alve,'rule to'ind'the prioper Pencil drawings, how to render perma- diameter of, 75.. nent, 358. Scouring balls, 159. Pencils, blacklead, 203. Scouiing drops:for Teinoving grease, 1;08. Pendulum, oscillations of,'49, 50. Screw,'the,'67. Pendulums, 249. Scre-w-cuWtting, change wheels for, 279. Percussion, centre of,:58. ScreWs, V-triead, niitmber of threads to I Petrifactioi of.wood, 70. an iich in, 27.' Pew's composition for covering build-: Sciibi-ng, the' op'eratidn of, 829. ings, 2'68.' Sealing-wax, recipesfor m'akingdifferent Photographs, paper for, 845-347. kinds of, 369;,370. Pile-driving, force in, 19.' Seasoning timber, 3078312. IPillars, cast-iron, table of ultim'aite bretk- Secants, external, table of, 415. ing weight in tons, 119-122.' Seed-lac, 89. Pipe, lead, weight of, 134. Shafts, cast and wronght-iron, tables of Pipes, weight of, of various metals, 128;.strengths_ of, 128-126; strength of cast-iron, tadble of weight. of,"131; -cast- journal sof, 149; daiameters of, 151; Teironi 148, 149. sistance of, to torsion, 1i50. Pitch of cog wheels,:68. Shears, tension of the guise:and shearPlane, the inclined, 66.: leg of,;;60. Plaster, in' in'mitation of'marble,'803. Sheet-iron, weight of superficial'foot of, Plastering, 800. 144. Plastic figures, amalgam for varnishing,:Shelllae, 89..89.... Side-ticaclks, oln'raiirooads, formiilie for Platina for spri'ngs, 145. locating, 372. IPlatina-mohr, -176. Signatures, metho'd of taking fae-similes Polishes, recipes for various,: 851, 352. of,- 182. Polishing wood, recipes for, 226; 227. Signs, ari'thmetical, 18. Peorcelin, glaze'for, 102.'Silica,'178. Poriphyry, to cut, with -iron, 1:46. Silver, an indelible black to' wri te on,?Power, steam, -1.'81; recipes for purifying and redwcing, Power of man or.horse as applied to' 201. machinery,'286.;Sfvering powder, 145. Powers, the mechanical,'. and their:ap- Sines, versed table: of, 415:;, ~attural, 457. plication, 63. Size', 17. Practical geometry, 16. Slide' iule, descript'ion o:f, ald -manner Pressure of beams; 59: lateral, 1'36'; ver- of using, 42. tical, 141: of steam at different te-ni- Soldterrf6i'tor'toise-shell,-128';fforfe-ruiiles.peratures, 78. oftool'hianle's, 833.'Prime numbers,.table of, to-5000, 243-245. Solders, iecirpes'for various;,235. Pulley, the, 65. Solid measure, 147....Pulleys, velocity of, 68. Solids, mnensurationi-of,'86, 46. Pump, common; pr-inc-iple of, 231.'S'otvents for gutta p.ercha, 179. Pumps, quantity of water'pekr -inertfoot, Sound, velocity of, 176. in, 276. Specific gravity, 61; gravities, table of, Putty, old, selvent-for~ 102. 62; of simplP`'sunstan'ces,g248 -specific'Radiating" powers of substances,-:290. gravities and otherr'iprope'rt'"'s, 281. IN IeSX. 517 Spontaneous combustion, to prevent. 79. DUnit of work, 54. Springs, platina, 145. Squares and square roots of numbers, Varnishes, 89-97. table of, 180-201. Vegetable decomposition. circumstances Staining woods, processes for, 218, 219. favorable to, 805, 806. Stains, method of removing, 179; vari- Vegetables, the feeding properties of' ous, for glass, 861, 862. different, 249. Steam, pressure of, at. different terl- Velocity of falling bodies, 47; of wheels, iperatures, 78; elastic force of, 80; drums, pulleys, &c., 68.'force and temperature of, in atmo- Versed silles, table of, 415. Ipheres, 81; pressure of, 87, 88. Vertical pressure, resistance of bodies Stoalm-engine, 71; table of surface of to, 141. b9ilel tubes of, 241; various problems Vocabulary of teclnical terms used in concerning, 227, 228. building, 381-341. Steam-engines, power of, 46; no:ninal hlose-power of low pressure, 838; of Walls, preservation of. fromn dampness, highl presslur~, 84; proportions of con- 158. densing, 85. Watchmalker's oil, 132. Steam power, 71. i Water, weight of civen bulks of, O8: exSteel, recipe to preserve articles made pansion of, byheat, 229; boiling pointsj of, 173; etching fluid for, 214; cast, of salt, 283; how to find the quantityi composition for welding, 268. and weight of, in pipes of any given I Stick-lae, 89. size, 280; discharge of, over weils.l Stne, methods of joining, 294-296; pa- 285. per, how to make, 304; ware, gold Wax, modelling, 86. lustre for, 70. Wax, sealing, recipes for making difStones, artificial, recipes for making dif- ferent kinds of: 369, 370. ferent kinds of, 862-365. Wedge, the. 66. Strain and stress of materials. 204. Weirs, discharge of water over, 285. Strength of materials, 135. Welding composition, 201. Stucco, fire-proof, for wood, 268; for Wheel and axle, the, 65. inside walls, 301; Higgins', 302. Wheel, cast-iron, to find the hiorse-power Superficies, mensuration of. 27', 4'1. it is capable of transmitting. 246. I Symbols, algebraic, 15. Wheels, velocity of, 8;: toothed, ib.:l strength of, 152; driving, revolutions Tallow, weight of a cubic foot of, 28). iper mile of, 86; table of the proporTangents. table of, 457. tions of 154: cast-iron, strength of Technical terms used inl building, 331- teeth of, 127; dimnensions of, in actuil il - 841.. | use. 246; proportionate strength of, in Tenaperature, gradations of, 236. 283. horse-power, 258; toothed. table for Terms used in building, 331-3l1. calculatinhg the pitch of. 27: chane, Terra cotta, 268. for screw-cutting, 270. Therimomnetrical scales, comparative ta- Wind, speed ind force of, at differentl ble of the degrees of the thlree. 2'2. velocities, 286. 238. Wood, strength of, 23; to petrify, 70; Tides, variations in, 118. fire-proof stucco for, 268: component; Timber, results of experiments on the parts of different varieties of', 278: elasticitv and strength of various specific gravities and other propelties species of, 136; mensuration of, 146: of, 2S2; iilesan s of preventing decay in, cause of the decay of, 805; methods of i 07. seasoning, 807-12; partitions, 822. 823. 1Woods, processes for sta.ining. 218. 219; i Timbers, framing of, 812; methods of how to polish and preserve, 226, 227. scarfing, 821. Woodwork, 305. different methods of Tin castings, bronzing liqulids for. 179. joining, 82.3-26. Toothed wheels, 68. i Work. units of, 54: of men and aUnilIaf'S. Torsion, elasticity of, 143: resistance to. 55; of machineos, 56; accumlnulated., by shafts, 150.. 56. Tortoise-shell, solder for, 128; ground \Vroght-irrn shafts, tables of strengths, for metbal, 189. of. 125, 126; tensile and c:)ipressivc Transverse action, resistance to, 186; strength of, 208; bar. transverse flexure strength of bodies, 247. of, by horizontal pressure. 218; experiTrusses, or timbers framied together for ments on transverse strength of the support of weight, 820. rectangular tubes of, 215.!~2-! l~~~~~~4 JUST PUBLISHED, TIIE AMERICAN COTTAGE BUILDER,'F SEIRIES Or; Designs, Plans, and Sypeifications, Fronl $200 to $,000'for Homes for the People. BY JOHN BULLOCKI9 AUTHOR OF THE RUDIMENTS OF ARCHITE!TCURE AND BUIIDING.?1 Crown Octavo. Price, $1,75. The object of this work is to furnish a practical manual of Designs for;' HIomeo fur the People." from a twenty-five dolla.r cottage to a twenty thousand dollan palace. giving estimates as to cost. furnishing llans and specifications, and treating upon thes rudiments of the arts called into exercise by the practice of cottage building. From its practical character it must become a desirable text-book to the builder and mechanic. as we'l as to those contemplating building; either in;rnamental or plain styles. This work is adapted to suplly the wants of all classes in every grade of living. from the humblest cottage to the stately mansion. who are interested in the erection of beautiful and convenient houses. The estimated cost is given to each respective plan. so that the expenses of each building may be accurately determined on. Every effort has been made to add to its intrinsic value-consistently with its practical character-by giving it variety, in collecting all new facts relating to recent improvements in the art of WARsIuNG and VENTILATION) GARnDe1)N-;, &c.. &c. l'he whole work will be found a' matter of fact" book on cottages and country residence,-a desideratum so long needed by all persons who contemplate building a. home. ~pitnitls of tU)e Learing JourItals. This volume is a valuable contribution to the art of rural architecture. It is illustrated with a. lmsber of excellent engravins representing v;rious styles of cottages,'and in the fiull details and clear descrip'tions of each style, which it gives, it will be found to be a safe guide botlh for taste andt econlomy. —-Courier anil Enquirer. Architectural treatises of more pretension than this have frequentlS passed under our notice, but we remember few volumes of tle kind that are likely to be so useful to thle public. Mr. Bullock furnishes designs of' cottages in every style, and at every cost. from $25 up to $25,000, and supplies ininute plans and specifications, estimates as to cost, &c. He also teaclies very fully and intelligently the rudiments of the arts employed in cottage building.. —-Conmercial Advertiser. This work contains twenty one chlapters of souud sense, twenty illustrations proper, and about sixty plans and illustrations of dlesigns. hIr. Bullock is an experienced writer upon matters of architecture and art in general, and his instructions may be firmly relied upon —drawn as they are from liard study and fruitful familh-arity..-.Buflalo Express. Mr. Bullock is fitvorably known as a clever writer ctn architectural subliects, and his books have always been popular. This handsome volume is profissely illustrated witil designs in every style for cottage residences. Thle letter.press descriptions and instructions are full, concise, and cleat, and renler thle treatise in our opinion useful not only to those about to erect cointry homes for them selves, but to professional builders whlio may be called upon to undertake suchi' works for otlers. —. N. Y. Express. This is a most useful book in this country. It consists of a few plain directions for building small ntod Imodlerate sized countrsy houses, witll a view to internal comlfbrlt, and external beauty and ptro -,riety. Everv one who tlblnks of erecting a hIouse out of town shlould read tli6 book. It is simple, aoil abounding with excellent llints, wvlicll may save many pounds after the house is erected. Fit for rich and poor. —Itontreal Herald. T'lihe scope of this work is compreliensive, embracing cottages of the lowest cost. as well as buildsrags wllich migllt more appropriately be called palaces. It firnishes plans and'specifications fob:' elclh, and estimates of'cost. Tle imliortant subject of warmino andl ventilating is duly considered, lso tliat of drainage* and tlhere is an instructive clhapter on lanlscape gardening. The plates (,f de -igcis, and illustrations of tle details of tle designs, are very numerous, antd thle subjects lucidly ex i!ibited. —-Journal of Conamerce. Te oblject ofthis book is to furnishl those ersons about to build witlh designs of houses costing trom two hunsdreed dollars to twenty tlhousaind, and with them valtuable intlrlmati on the subjects of ventilation and gardening. Among the plates are the drawings of Prince All,orL's Model Cottages, aind also full lans ibr Fowler's Octagonal House, both of tllem curioeities well vortll examination.-. Boston Journal. One object ofhllis truly excellent book is to encolurage tlhe spirit of ilmprovessmenlt —to aid the clll t;vation of agricultural tsate in the poor man's holme, iend to teacli all the irt of ceinbining the usefu I with the tasteful anti elegant. —-Pittsbrur g Token. Published by STRINGER & TOWNSEND, 222 Broadway, N. Y.,,7n:l for sale 1y all the prinwcipal Booksscllers. An Indispensible Woirk for the Mechanic, the Engineer & the Artist ale rlatlttat irangl tsman's BOOK OF INDUSTRIAL DESIGN FORMING A COMPLETE COURSE OF Mechanical, Engineering & Architectural Drawing. TRANSLATED FROlM THE FRENCH OF THE MESSRS. ARMENGAUD AND M. AMOUROUX, WITH ADDITIONS AND IMPROVEMENTS BY PItOFESSOR WILLIAMI JOHNSON. GENERAL CONTENTS: LINEAR DRAWING.-THE STUDY OF PROJECTION.-ON COLOURING SECTIONS WITP APPLICATIONS.-THE iNTERSECTION AND DEVELOPMENT OF SURFACES, WITH APPLICATIONS.-THE STUDY AND CONSTRUCTION OF TOOTHED GEAR.-ELEMENTARY PRINCIPLES OF SHADOWS.-APPLICATION OF SHADOWS TO TOOTHED GEAR.-THE CUTTING AND SHAPING OF MASONRY.-THE STUDY OF IMACHINERY AND SKETCHING.-OBLIQUE PROJECTIONS.-PARALLEL PERSPECTIVE.-TRUE PERSPECTIVE.-EXAMIPLES OF FIN' ISHED DRAWINGS OF 1MACHINERY.-DRAWING INSTRUMIENTS in one quarto volume, containing 105 pages of Steel Plates, 75 Wood Engravings, and 200 pages of Lettcrps-ess. Bound in cloth, $6.50; in half Turkey, $8.00. This superb work has been prepared by its distinguished Editor to meet the requirements so long felt ofa comprehensive treatise on INDUSTRIAL DESIGN. His aim has been to present the well-known French work in a much improved English form, and at a price Drought within the reach of all. Every Principal, Mechanic, Builder, Machinist, Engineer, and Artist, has felt the want of a complete course of Industrial Desigtl-not a mere collection of the first principles of Geometrical Drawing, but a work that wvould be useful as well in Schools and Colleges as in the Workshop and Drawing Offices of the Engineer and Architect. The special mission of the PIRACTICAL DIAUGHTSAIAN'S BOOK OF INDUSTRIAL Dr.SIGN may almost be gathered from its title-page. It is intended to furnish gradually de. veloeped lessons in Geometrical Drawing, applied directly to the various branches of tlh: Industrial Arts: comprehending LINEAR DESIGN proper ISOMETRICAL PERSPECTIYE, or the study of Projectons s; the Drawing of TOOTHED WHEELS and ECCENTRICS With SHADOWING and COLOURINsG OBLIQUE PROJECTIONS i and the study of PARALLEL and EXACT PERSPE:CTI'E: each division being accompanied by special applications to the extensive ranges of MECHANICS, ARCHITECTURE, FOUNDRY-WORKS, CARPENTRY, JOINERY, METAL MANUFACTURES generally, HYDRAULICS, the construction of STF. AM ENGINES, and MILL-WORK. In its compilation, the feeble attraction generally offered to students in elementary form has been carefully considered; and after every geometrical problem, a practical example of its application has been added, to facilitate its Comprehension and increase its value. A series of PLATES, marked,, B, &c., are also interspersed throughout the work, as examples of finished drawis)gs of machinery.The Letterpress relating to these Plates, together with an illustrated chapter on DRAWINO INSTRUAIENTS, forms an appropriate Appendix to the Volume. The general explanatory text embraces not only a description of the objects and their movements, but also tables as.d practical rules, more particularly those relating to the dimensions of the principal details of machinery, as facilitating actual construction. Fr-on WT'.,1. No-tonz, Esq.. Professor of Civil Engineering in Yale College. In the Practical Draughtsman we at last have a work on Industrial design, at the same time suflitiently comprehensive in its scope and minute in its detail to meet the wants of the student of engineering. Mly opinion of its merits is abundantly indicated by the fact that I have adopted it as a text-book in the Sclhool of Engineering attaclhed to Yale College. So far as I have made use of it, I fiLd that it meets my expectatios, and doubt not that it will give general satisfaction, if introduced nto our Engineering and Industrial Schools. W'. A. NORTON, Professor of Civil Engineering in Yale College. From the Practical Mechanic's Joits-al, (Glasgow.) Such a bcok in English was greatly needed. We possess for the first time on our iterature a etandard work, in many respects greatly superior to tie French volume. Every page is full of p!eas. ant instruction, and demonstrates competent intelligence and excessive care. The original work is a class-book in the Industrial Schools on the continent, and tile present English form of it will hold a high place wherever its subject is studied, either in our own lands or in the United States. It is frori Its cheapness one of the few good books which every principal shoukl possess as a work ofreference, and which every learner should have readily accessible for constant study. Published by STRINGER & TOWNSEND 222 Broadway, N, Y.,