TIlE UNIVERSAL MO)DERN CAMBIST, AND FOREIGN AND DOMESTIC COMMERCIAL CALCULATOR; A ])ICTIONARY OF NUMERICAL, ARITIIMETICAL, AND MATIl'MATICAL FACTS, TABLES, DATA, FORMULA;,S, AND PRA.CTICA.L RULES FOR. BUSINESS-1MEN, MERC-IANTS, BANKERS, BROKERS, AND ACCOUNTANTS. BY E. S. WI.NSLOW, Autlor of "Conprellenlsive Mathematics," " Computist' Manuall," " Machinists' and IMechanics' Practical Calculator and Guide," "Tin-plate and Sheet-iron Workers' Monitor." i3EVENTII EDITION, IPEVISED AND ENTLARGED. BOSTON: I - I. Y T:I D j BY J Ti IAUT I(. 1872. FOR LOCAL, AGENTS, SEP NEXT PAGIE. Single copies of Winslow's MathematicalWorks are delivered at the counter or sent by mail, post-paid, on receipt of price, as follows, viz: COMPREHENSIVE MATHEMATICS...... $2.50 THE UNIVERSAL MODERN CAMiBIST, AND FOREIGN AND DOMESTIC. COMMERCIAL CALCULATOR. 1.50 THE MERCHANTS' CALCTLATOR AND ACCOUNTANTS' DESK COMPANION 1.00 TIE MACHINISTS' AND MECHANICS' PRACTICAL CALCULATOR AND GUIDE.1.75 Sold by G. W. LINDSEY, Bookseller and Stationer, No. 1031 Washington St., Boston. Also by J. C. WINSLOW, Danville, Vermilion Co., Illinois. See ADVERTISEMENT, page iv. Entered, according to Act of Congress, in the year 1S67, by E. S. WINSLOW, in the Clerl's Office of the District Court of the District of Massachusetts. Entered, according to Act of Congress, in the year 1870, by E. S. WINSLOW, in the Clerk's Offlice of the District Court of the District of Massachusetts. Entered, according to Act of Congress, ill the year 1872, by E. S. WINSLOW. in the Office of the Librarian of Congress, at Washington. Stereotyped by C. J. Peters and Son, 5 Washington Street.,' PREFAC E TO THE COMPREHIENSIVE MATHEMATICS. ON presentin this work to the public, it may be proper to state that it has been designed and written mainly for the practical man.. It contains a vast array of Numerical, Arithmetical, and 1Matihemnatical facts, tables, data, formulas, and rules, pertaining to a gCreat variety of subjects, and applicable to a diversity of ends, as well as much information of a more general nature, valuable to the artisan, and commercial classes; thus meeting the wants, in an eminent degree, of the lovers of the exact sciences, and the practical wants of students in the mathematics. The facts and data alluded to have been gathered, with much care and patience, from a great variety of sources, or derived, often by toilsome investigations, from known and accredited truths. The care that has been taken in respect to these, it is thought, sh'ould secure for this particular department reliance and trust. The tables, which are numerous, have, with few exceptions, been composed and arranged expressly for the work, and a confidence is felt that they may be relied on for accuracy. From the valuable works of:Dr. Ure, Adcock, Gregory, Grier, Brunton; fiom the publications of the transactions of London, Edinburgh, and Dublin Philosophical Societies; and from the publications by the Smithsonian Institute, much valuable information has been (ained, relating mainly to machinery and the arts; aind to these sources the author feels indebted. The conciseness with which the work has been generally written would, perhaps, be found an objection, were it not that all the propositions and problems of intricacy are accompanied with examples and illustrations, and, in the matters of Geometry, additionally accompanied with diagrams. The whole, it is thought, will appear clear to him who consults it. A prominent feature in the design has been to produce a useful work, and one which in the way of price shall be readily accessible to all. 111. iv PREFACE. PREFACE TO TIllE UNIVERSAL MODERN CAM BYST, AND FOREIGN AND DOMESTIC COMMERCIAL CALCULATOR. This work is composed of the first five sections of the author's "COMPrnEIIENSIVE MIATIIEMATICS." It was thought advisable to publish this portion of that work in a separate form on account of' price; more especially as it contains all of a commercial nature treated of in that work. Indeed, the contents of that work were,arranged expressly to this end. The Table of Contents in both works is the same: the work being stereotyped, this could not well be avoided. The Table of Contents, therefore, in either work, is that of the, "CCOMPREHENSIVE MATHEMATICS; " and the first five sections thereof, that is, Section I., Section II., Section III., Section A., and Section B., is that of the " Universal Modern Cambyst, and Foreign and Domestic Commercial Calculator." P I E E TFA C E TO TIIE TIN-PLATE AND SHEET-IRON WORKERS' MONILTOR. TThis work is composed of Section VI. of the author's " CoMPREIIENSIVE MATAIEMATICS," with portions of other sections of that work. It embraces all that is contained in the last-mentionct4 work of special interest to the Tinsmith, as such. It may be relied on for accuracy in all particulars, and is believed to be thle first and only reliable work of the kind ever published. It is published in separate form bn account of price, and with the view of affording apprentices and students every possible facility of obtaining it. It contains over 100 pages, nearly 50 diagrams, and stl.pby-step directions for constructing, mechanlically, not less than 30 unlike and different patterns, embracing all of the more difficult and complicated in use, and several of new and beautiful designs. CONTENTS. S. E C T I 0 N A. WEIGQ TS AND MEASURES. PAGE Foreign Moncys of Account... al PAOE Foreign Linear and Surface Meas- LoaNG OR LINEAR EASURE.. o 25 lres.............. a8 Cloth Measure........ 25 Foreign Weights. a26 Land Measure.......... 25 Foreign Liquid Measures.... a37 Engineer's Chain.......... 25 Foreign Dry Measures... a44 Shoemaker's Measure......26 Memoranda, &c., relative to Foreign Miscellaneous Measures... 26 Moneys, &c........ a50 United-States Customs' Tares. a52 SQUARE OR SUPERFICIAL MEASURE.............26 Measure for Land........ 26 Circular Measure......... 27 CUBIC OR SOLID MEASURE... 27 GENERAL MEASURE OF WEIGIIT, 28 SECTION I. GrossWeight....... 28 Troy Weight.28 Apothecaries' Weight...... 28 MONEYS OF ACCOUNT, COINS, Diamonds, Measure of Value, &c., 28 WEIGIITS, AND MEASURES OF LIQUID MEASURE........ 28 TItE UNITED STATES; FOREIGN Imperial Liquid Measure..... 20 GO(LD COINS, &C. Ale Measure........... 29 DRY MEASURE........... 29 EXPYLANATIONS OF SIGNS... 12 Imperial Dry Measulre...... 30 Moneys of Account of the United States.............13 Comparative Value of Gold and Silver.........13 SECTION II. Gold, pure; value of, by weight. 15 Mint Gold, Standard of, &c.. 15 Gold Coins, their weights and val- MISCELTLANEOUS FACTS, CALCUnes..15 LATIONS, AND MATHEMATICAL Silver, pure; value of, by weight. 16 DATA. Mint Silver, Standard of, &c.... 16 Silver Coins, their weights and S G TE TBLE OF, 31 values.16 SPECIFIC GRAVITIES, TABLES OF, 31 Copper Coins, &c......... 16 Weight per Bushel of Articles. 35 Present Par Value of Silver Coins Weight per Barrel of Articles.. 35 issued prior to June, 1853.. 17 Weights of different Measures of Currencies of the different States various. Articles.35 of the Union....... 17 Weight of Coals, &c., TABLES. 35, 55 Tnie Metrical System of Weights Practical Approximate Weight in and Measures.........* 185 Pounds of Various Articles... 36 Foreign Gold Coins, TABLES of, &c. 19 Foreign Silver Coins, Values of, 25 ROPES AND CABLES....... 36 1 * v vi CONTENTS. PAGaS PAGE Weight and Strength of Iron To find the Dimensions of Vessels Chains........... 37 of different Forms, for holding Comparative Weight of Ietals, Given Quantities.. 62 TABLE 38 CASK GAUGING, all Forms of Weight of Rolled Iron, Square Bar, Casks. 63 TABLES............ 38 Casks..............63 TABLES.38 Weight of Various Metals, differ- To find the Contents of a Cask, the ent Forms of Bar. 39 same as would be given by the Weight of Round-rolled Iron, TA- Gauging Rod... 6 BL-....40 To find the Diagonal and Length Weight of Cast-iron Prisms of dif- of a Cask. GG. 6 ferent forms, &c......... 40 ULLAGF........ 67 Weight of Flat-rolled Iron, TABLE, 4 To find the Ullage of a Standing Weight of Different MIAetals,in Plate, 44 Csk......... 7 TIIE AmERICAN WIRE GAUGE. 45 To find the Ullage when the Cask is upon its Bilge....... 67 The Values of the Nos. American To find the Quantity of Liquor in a Wire Gauge and Birmingham Cask by its Weight...... 68 Wire Gauge, in the United States, Customary Rule by Freighting Merinch, TABLES Of........ 45 chants for finding the Cubic The Number of Linear Feet in a uIeasure.mnent of Casks..... 68 Pound of different kinds of Wire TONNAGE OF VESSELS, to Calcuof different Sizes, TABLE, of, &C., 46 late.... 69 Characteristics, &c., of Alloys of OFCONDUITS, PI..O...70 Copper and Zinc, — BRASS.... 47. 7 The Weight per Square Foot of dif- To find the requisite thickness of a ferent Rolled Metals of different Pipe to support a Given Head of thicknesses by the Wire Gauge, ~Water............. 70 TABLE.......48 To find the Velocity of Water passTIN PLATES, Sizes, &c., TABLE - t9 ing through a Pipe....... 71 Sheet Iron, Sheet Zinc, Copper To find the Head of Water requiSheathing, Yellow Metal, Weight site to a Required Velocity of, &c-.4. 6;.. 49 ththrough a Pipe. 71 Capacity in Gallons of Cylindrical To find the Quantity of Water DisCans, &c., TABLE -..-..... s.. 50'charged by a Pipe in a Given Weight of Pipes......... Time 71 Weight of Pipes, TABLE. - 53 To find the Specific Gravity of a Weight of Cast-iron and Lead Body heavier than Water.... 72 Balls 54 To find the-Specific Gravity of a Weight of Hollow Baills or Shells, 54 Body lighter than Water.... 72 Analysis of Coals. 55 To find the Specific Gravity of a Weight, Heating Power, &c., of Fluid.-....f the 72 Coals and other kinds of Fuel, find the Q TABLE........... 55 several Metals composing an Alloy...............72 MENSURATION OF LUMBER... 56 To find the Lifting-power of a Bal-x loon 73 Board Measure - 56........... C)To find the Diameter of a Balloon To Measure Sqlua~re Timber -..... 56; ~equal to the Raising of a Given To Measure itound Timber.. - -5eight 73 TAB LE relative to the Measurement To find the Tllickness of a ollow of Rolund Timber-.......... - 57 i Metallic Globe that shall have a TO find the Solidity of the greatest Given Buoyancy in a Given Rectangular Stick that can be cut Li uid fiom a Log of GivenDiniensions, 58 To Cut a Squar-e Sheet of Metal so To find the Solidity of the greatest as to form a Vessel of the GreatSquare Stick that can be cut from est Capacity the Sheet admits of. 73 a Round Stick of Given Dimen- Comparative Cohesive Forces of sions........... 59 Substances, TABLV........ 7 7 SlOBS-9 6 Substances, TABLE. -.. 7! - To find the Contents of a Log in Alloys having a Tenacity greater Alloys having a Tenacity greater Board Measure.......... 9 than the Su m of their ConGAUGING........... 60 stituenlts............74 CONTENTS. vii PAGE PAGE Alloys having a Density greater Chemical and other Properties of. than the Mean of their -Con- Various Substances....... 90 stituents........... 75 Alloys having a Density less than the Mean of their Constituents. 75 Relative Powers of different Metals E C T N III. to Conduct Electricity...... 75 Dilations of Solids by Heat, TABLE 75 Melting Points of Metals and other PRACTICAL ARITII.IETIC. Substances, TABLE....... 76 Relative Powers of Substances to Radiate Heat, TABLE... 76 Boiling Points of Fluids.. 76 Reduction of Vulgar Fractions. 95 Freezing Points of Fluids. 77 Addition of Vulgar Fractions.. 99 Expansion of Fluids byIIHeat.. 77 Subtraction of Vulgar Fractions. 99 Relative Powers of Substances to Division of Vulgar Fractions... 100 Conduct Heat....... 77 Multiplication of Vulgar Fractions 100 Ductility and Maleability of Metals, 77 Multiplication and Division of Quantity per cent. of Nutritious Fractions Combined...... 101 Matter contained in different Ar- CANCELLATION...... 96, 9)7, 102 ticles of Food......... 78 Standard, &c., of Alcohol.... 78 To Reduce a Fraction in a higher, Quantity per cent. of Absolute Al an equivalent in a gi en lowcohol contained in different Pure er denomination 102 Liquors, Wines, &c., TABLE.. 78 To Reduce a Fraction in a lower, Proof of Spirituous Liquors... 78 to an equivalent in a given highComparative Weight of Timber in er denomination 102 a Green and Seasoned State, TA o To Reduce a Fraction to Whole BLEo, &C. X. 7-/ Numbers in lower given denomRelative Power of different kinds of inations...........103 Fuel to Produce Heat, TABLEC, 79 To Reduce Fractions in lower deRelative Illuminating Power of dif- nominations to given hi.her d.ferent Materials, Table and Re- nominations.o 103 marks.....,.. 80 To work Vulgar Fractions by tile.rM M..T.S.... dr s Rule of Three, or Proportion.. 104 THERMOMsETERS, different kinds, to Reduce one to another, &c.,. 82DECIMAL FRACTIONS.. 10 tHORSue-POWER oan.other, 83 Addition of Decimals...... 105 Subtraction of Decimals..105 Animal Power.......... 83 Multiplication of Decimals.... 10 STEAM, TABLES in relation to, Division of Decimals... 106 &c. 8:3, 308 Reduction of Decimals...... 107 Velocity and Force of Wind, TA- To work Decimals by the Rule of BLE. 84 Three.......... 108 Curvature of the Earth....84, 213 Proportion, or Rule of Three.. 109 Degrees of Longitude, Lengths of, Compound Proportion..110 &c............ 84 Conjoined Proportion, or Chain TIMIi, with respect to Longitude, 84 Rule.............. 112 Velocity of Sound..... 84 PECENTAGE..........11 Velocity of Light........ 85 GRAVITATIO N......... 85, 302 INTEREST 120 Compound Interest.......122 Area of the Earth, its Density, &c., 85 C o nere12 AChemical Elements........86Bank Interest, or Bank Discount. 127 Elementary Constituents ofBodies, DISCOUNT...........129 TABLE............87 Compound Discount.129 Combinations by Weight of the Profit and Loss.......... 30 Gases in forming Compounds, Equation of Payments......1:32 TABLE... 87 General Average.... 134 Combinations by Volume of the Assessment of Taxes....... 136 Gases, their Condensation, &c., Insurance............ 6 in formingf Compounds.... 89 Life Insurance..136 Atomic WVeight......... 89 Fellowship.......138 Hii CONTENTh. PAOBE PAG Alligation......... 139 To draw a Triangle equal in Area Involution.... 141 to two given Triangles..... 183 Evolution........... 141 To describe a Circle equal in Area To Extract the Square Root... 142 to two given Circles...... 183 To Extract the Cube Root... 143 To construct a Tothed, or CogTo Extract any Root........145 Wheel.. 183 Arithmetical Progression.....146 Geometrical Progression.O.... 1150 ANNUITIES..........154 ANNUITIES15 MENSURATION OF LINES AND SUPEROf Installments generally....164 FICIES..Pf',ERMUTATION..........166 TRPIANGLES........... 185 COMBINATION......... 167 Of Right-Angled Triangles.. ~ 186 PROBLEMS............160 Of Oblique-Angled Triangles. 187 To find the Area of a Triangle. 188 To find the Hypotenuse of a Triangle...189 S E C TI o N V. To find the Base, or Perpendicular, of a Triangle....... 188, 189 To find the Height of an inaccesG EOzMETRY. sible Object........... 189 To find the Distance of an inacDTiEFINITIONS, CONSTRUCTIQON OF cessible Object....1... 190 FIGURlES, &C.......... 172 To find the Area of a Square, To Bisect a Line.........17 Rectangle, Rhombus, or RhomTo 1Erect a Perpendicular... 76 boid.............. 190 To Let Fall a Perpendicular.. 176 To find the Airea of a Trapezoid. 191 To Erect a Perpendicular on the To find the Arlea of a Trapezium. 191 end of a Line........ 177 OF POLYGONS, TABLE, &C..... 194 To draw a Circle through any three To find the Perpendicular of a points not in a straight line, and Rhombus, Rhomboid, or Trapeto find the Centre of a Circle, or zoid...... 192'Arc................ 17.7 To find the Diagonal of a RhomTo find the Length of an Arc of bus, Rhomboid, or Trapezoid. 192 a Circle approximately by DMe- To find the Area of a regular or chanics... 177 irregular Polygon....... 195 From a' given Point to draw a Tangent to a Circle....... 177 CIRCLE............ 196 To draw from or to the Circumfer- The Circle and its Sections.... 197 ence of a Circle, lines tending To find the Diameter, Circumferto the Centre, when the latter is ence, and Area of a Circle... 198 inaccessible........ 177 To find the Length of an Arc of a To describe an Oval Archi on a Circle......... 199 given Conjugate l)iameter... 178 To find the Area of a Sector of a To describe an Oval of a given Circle............. 201 Length and Breadth..178 To find the Area of a Segment of To describe an Arc or Segment of a Circle............ 201 a Circle of Large Radius.... 179 To find the Area of a Zone. 202 To describe an Oval Arch, the To find the Diameter of a Circle of Span and Rise being given.. 179 which a given Zone is a part.. 202 Gothic Arches, to draw... 180 To find the Arca of.a Crescent. 202 Polygons, to construct... 181 To find the Side of a Square that Polygons, to inscribe in- a given shall contain an Area equal to Circle..............181 that of a given Circle 202 Polygons, to circumscribe about a To find the Diameter of a Circle given Circle......... 181 that shall have an Area equal to To produce a Square of the same that of a given Square 202 Area as a given Triangle....181 To find the Diameters of three To construct a Parabola... 182,355 equal circles the greatest that To Construct a Hyperbola.. 182, 349 can be inscribed in a given CirTo bisect any given Triangle... 182 cle............202 PAGE PAGE To find the Diameters offour equal Of Spherical Segments..... 219 circles the greatest that can be Of Spherical Zones 220 inscribed in a given Circle...202'To find the greatest Cube that can To find the Side of a Square inscribed in a given Circle.... 203 To find the Diameter of a Circle OF SPEROIDS......... 21 that will circumscribe a given Of Segments of Spheroids. 221 Triangle............ 203 Of the Middle Frustum ofa SpherTo find the Diameter of the great- old............. 05, 221 est Circle that call be inscribed OF SPINDLES.222 in a given Triangle... 203 To divide a Circle into any num- Of the Middle Frustum of a Paraber of Concentric Circles of bolic Spindle.65, 222 equal Areas.......... 204 OF PARABOLIC CONOIDS 65, 2:23 To find the Area of the space con- OF HYPERBOLOIDS....... 223 taised between two Concentric Circles. 205 To find the Surface of a Cylindrical Ring............ 224 ELLIPSE.................. 205 To find the Solidity of a CylindriTo find the Area of an Ellipse.. 207 cal Ring............ 224 To find the Length of the Circum- OF THE REGULAR BODIES.. 225 ference of an Ellipse.... 207 To find the Area of an Elliptic Seg- PROMISCUOUS EXAMPLES IN merit........... 207 GEOMETRY.6..........226 PARABOLA.......... 209 TRIGONOMETRY........ 231 To find the Area of a Parabola. 210 TABLES OF SINES, COSINES, TANTo find the Area of a Zone of a GENTS, &C.241 Parabola........... 210 TABLES OF SQUARES, CUBES,'o find the Altitude of a Parabola, 210 SQUARE AND CUBE RooTrs, &c. 245 To find the Length of a Semi-parabola.. 210 HYPIEBOLA........ 211 SECTION V. To find the Length of a Semihyperbola... 212 IMEICHANICAL POWERS, MECHANITo finld the Area of a Hyperbola. 212 CAL CENTRES, CIRCULAR MO-'(CYCLOID,..212 TION, STRENGTH OF MATERIElPICYCLOID, 213 ALS; STEAM, THE STEAM ENCYCLOIDAL ELLIPSE,.... 213 GINE, ETC. CATENARY,........... 379 THE LEVER.... 271 To find the Distance of Objects atXLE 272 Sea, &c............ 213 STIROWEY ORTIIE PULLEY........... 273 STEREOMETRY, OR MENSURATION T L OF SOLIDS. TIIE INCLINED PLANE..... 274 OF PRISMS........... 214 THE WEDGE......... 275 Of Right Prisms or Cubes.... 215 THE SCREW.......... * 275 OF PARALLELOPIPEDONS.... 215 Transverse Strength of Bodies. 279 OF PYrAMIDS.......... 21- Deflections of Shafts, &c.. 28 Resistance of Bodies to Tortion. 287 OF FRUSTUMS OF PYRAMIDS. 216 Resistance of Bodies to CompresOF PRIS3IOIDS.......... 216 sion.............. 289 OF THE WEDGE......... 217 CENTRES OF SURFACES..... 291 OF CYLINDERS......... 217 CENTRES OF SOLIDS....... 293 To find the Length of a Helix. 217 CENTRES OF OSCILLATION AND OF CONES.....S....... 218 PERCUSSION.......... 294 OF FRUSTUMS OF CONES.... 65, 218 CENTRE OF GYRATION..~,. 298 OF SPHERES OR GLOBES.... 219 CENTRAL FORCES........ 300 PAGE PAOX FixY AVIIlE5;S....... 301 To construct a Patt.crn lfor the Tl111 (-1'N 1)x01......... 301 Lateral Portion of a vessel in the O GIIAVITY...... 302 form of a Frustum of a Cone of I (11 GRAVrITY.......:~30 given diameters and depth... 335 To find( thle [Icight of a Stlream To construct a Pattern for the pliojectled vertically from a Pipe, 303 Body of a vessel in the form of a,To illd the Power requisite to Frustum of a Cone of given clilroject a Stream to any given mensions, without plotting the HIeight......... 303 dimensions....... 338 OF PENDULUMS. 304 To constrnlct a Pattern for thle SC IEW-CUTTING IN A LATHE.. 303 Lateral Portion of a Flaring Vessel of given symmetry of outline Table of Change Whleels for andgiven capacity....... 339 Screw-Cutting in a Lathe... 30S given capacity.39 TABLE OF RELTIVE' PROPOROFt S'rriA-. 1 ND THEI STE Am\ E-xGINE...........308 TIONS, CIHIORDS, &C....... 339 The special tabulalr figure, the diVelocity of Projectiles, &c.... 313 ameter of one end, and the Cubic Steam, acting expansively.. - 313 Capacity of the vessel being Of the Eccentric in a Steam En- given, to find the diameter of Cine.. 314 the other end........ 342,OF CONTINUOUS CIRCULAR MO- To construct a Pattern for the body TION.......314 of a Flaring Vessel of givel To find the number of Revo- tabular outline, and given dlmentions madte nby te last, to one sions, without plotting the ditiOllS made by the lsst to oane mensions..... 344 revolution of the first, ii a train The Capacity in gallons of a vessel of Wheels and Pinions.... 315 in the form of a Frulstum of a lhe distance from Centre to Ccn- Cone being given, and any two tre of two Wheels to work in nd al tw tie of two Wheels to work in of its dimensions to filnd the contact given, and thle ratio of other dimesion......... 346 Velocity between them, to find To cotstruct Patterns for fl-ring thleir Rcequisite Diametels... 317 oval vessels of diferent eccentrlTo find the Velocity of a LBelt. 317 cities d given densions, Nos Tro find the D)raft on a Malchine. 317 cities and given dimensions, Nos. To find the Rtevolutions of tue 1,2,3. bae.;.3 348' Throstle Spindle-........ 318' To describe the bases for Nos.1,2, 3, 349'rllrost~le Sp~indlle.......... 318, To find the Twist given to the OF CYLINDRICAL ELBOWS... 354 Yarn by the Throstle.... 318 To construct a Pattern for a RightTIEr orF WVIIEELS, &C... 31S angled Cylindrical Elbow.... 35 To construct a Tooth, &c... 319 To construct Oblique-angled El-'1, ostle Tol,&c.......... 1)bows.... 35f( To find the Horse-Power of a bows-358 Tooth... a319 To construct tight-angled Elliptic'1'ootll.. *,39]los......~...3 5.' ) ti Elbows..359 J)IJILNALS OF SIHAFTS.. 320 To construct Oblique-angled EllipIIYDROSTATICS........... 2() tic Elbows...... 359 IIYI)RAULICS........... 3 To construct Itight Semi-hyperboWATEIB-W1ES......... 32.3 las by intersecting lines.. 355, 359'''o lill(l t.le Powxvel of' a Strealll. 324 To construct the Quadrant of a CirTo construct a Water-Whleel to a cle by intersecting lines... 360 (ivenl Power anlld Falll..... 325 To construct the Quadrant of a )YNAMICs............ 326 given Ellipse by intersecting Il IYio Sr ATLC PRESS.3211.. 326 lines.. 3G0 Ucuevi-al Aplplications of Principles To construct the Quadrant of a( Cyinl Dynamics...... 327 cloidal Ellipse by intersecting 11EAT,.SCenisilc, Latent, etc... 329 lines.. 360 To describe an Ellipse of given diSECTION VI. mensions by means of two Posts, a Pencil, and a String.... 360 COVERINGS OF SOLIDS, OR PROB- To find the length of the circumLEMNS IN PATTE RN CUTTING. ference of a given Ellipse.. 207, 361 To construlct a Semi-parabola by RE)IARKS AND DElFINITIONS... 333, intersecting lines........ 361 Page. OVALS, to describe 178, 350, 352, 353 OF CIRCULAR ELBOWS.... 361SECTION VII TABLE applicable to Circular Elbows......... 362 Page. To construct a Right-angled Circu- CATENAEY 3.'3 lar Elbow of 3, 4, 5, 6, 7, or 8 DCIvING BELTS AND PUILLEYS, pieces, &c........ 361 THEIR RELATIONS TO ONE ANTo construct a Collar for a Cyliln- OTHER,&C.......382 drical Pipe of the same diameter To calculate the necessary Leu'gth as the receiving pipe.... 365 of an Open Belt Tb construct a Cylindrical Collar To calculate the necessary Leth of a given Diameter to fit a Re- of a Cross Belt....... 384 ceivilg-pipe of a greater given Of the relations of the forces at Diameter..... 366 work on belted machines, tile reTo construct a Cylindrical Collar brendths of tll dlivi to fit an Elliptic-cylinder at either belts, the alstract sizes oI the right section of the Ellipse.. 367 pulleys, &c....385 To construct a Cylindrical Collar To calculate the necessary stress of a given Diameter, to fit a Cyl- of a belt......... 38 inider of the same Diameter, at To calculate the necessary breadth any given Angle to thle side of of a belt. 389 the ylinder..... MILL HOPPERS, to construct, c. 392 To construct a Cylindrical Collar, SOLDERS, AILOYS, AND COMPOSIor Spout, of a given Diameter, to TONS...3 fit a Cylinder of a greater given Diameter, at a given Angle to the side of the Cylinder..... 368 OF SPOUTS FOR VESSELS.... 369 Of Pitched or Bevelled Covers.. 370 To construct a Bevelled Circular Cover of a given R4se and given Diameter....... 371 SECTION B. To construct a Pattern for a Bevelled Elliptical Cover of a Oiven Rise to fit an Elliptical Boiler of REDUCTIONS, EXCHANGE, INVESTgiven Diameters...... 371 ENTS, MIXED NEGOTIATIONS, To, construct a Bevelled Cover of a ETO, ETC......... b 1 given Rise, to fit a False-Oval Boiler of given length and width 371 OF CAN-ToPS........ 372 To construct a Can-Top of' a given Depth and given Diameters.. 372 To construct a Can-Top of a given Pitch, and given Diameters.. 373 OF LIPS FOR MEASURES.... 374 To construct a Lip for a Measure, the Diameter of the Top of the Measure beingr given.... 375 OF SHEET PANS........ 375 To cut the Corners for a Perpendicular-sided Sheet Pan... 376 To cut the Corners for an Obliquesided Sheet Pan.376 To construct a Heart, or Ieartshaped Cake-Cutter..... 376 To construct a Mouth-piece for a Speaking-Tube...... 376 To construct a Pattern for the Body of a Circular-bottonmed Flarinl Coal-Hod, all the curves to be arcs of circles....... 377 DEFI NITION S OF THIE SIGNS USED IN THIE FOLLOWING WORK. Equal to. The sign of equality; as 16 oz. = I lb. + Plus, or More. The sign of addition; as 8 + 12 = 20. - Minus, or Less. The sign of subtraction; as 12 - 8 - 4. X Mulbiplied by. The sign of multiplication; as 12 X 8 = 9i. Divided by. The sign of division; as 12 +- 4 = 3. DiYfference betuween the given numbers or quantities; thus, 12. 8, o0 8 p 12, shows that the less number is to be subtracted from the greater, and the difference, or remainder, only, is'to be used; so, too, height w breadth, shows that the difference between the height and breadth is to be taken.:::Pro'porlion; as2:4::3:6; that is, as2isto4, sois3to 6. V/ Sign of the square root; prefixed to any number indicates that the square root of that number is to be taken, or employed; as /611 = 8. 4v/ Sign of the-cube root; and indicates that the cube root of the nlnlber to which it is prefixed is to be employed, instead of the number itself; as 4/64 = 4.'To be squared, or -the square qf'; shows that the square of the number to which it is affixed is the quantity to be employed; as 122 * 6 24 4; that is, that the square of.12, or 144 - 6 = 24. Indicates that the cube of the number to which it is sul)joined is to to be used; as 43 = 64.. ecimal point, or separatrix. See IDECIMAL FRACTIONS. Vinculum. Signifies that the two or more quantities over wlhich it is drawn, are to be taken collectively, or as forming one quantity; thus, 4 + t6 X 4 - 40; whereas, without the vinculum, 4 + 6 X 4 = 28; also, 12 - 2 X 3 + 4- 2; and / 5 -2 32 4 So, also, V (52 - 3" ) = 4, and (4 + 6) X 4 -=40. 42 halffof42 or half of the square of 4 - (42 )2(the square of half the square of 4) = 64. Asb or J(b)2 (half the square of b.) (Ab)2 (the square of half b ) (2b)' (the square of twice b.) SECTION I. MONEYS, WEIGHTS AND MEASURES, OF TIlE UNITED STATES; —TIIEtR DENOMINATIONS, VALUES, COMPARATIVE VALUES, MAGNITUDES, &c. MONEYS OF ACCOUNT OF TIIE UNITED STATES. These are the mill, the cent, the dime, and the dollar. 10 mills = 1 cent, 10 cents = 1 dime, 10 dimes = 1 dollar. The dollar is the unit or ultimate money of account of the United States, or of what is sometimes called Federal money. In practice, the dime, as a denomination of value, is rejected. Thus, 10 mills - 1 cent, and 100 cents = 1 dollar. This mark, $, is equivalent to the word dollar, or dollars, in this money. COINS OF THE UNITED STATES. Until June, 1834, the government of the United States estimated gold in comparison with silver as 15 to 1, and in comparison with copper as 850 to 1. From June, 1834, until February, 1853, the same government estimated gold in comparison with silver as 16 to 1, and in comparison with copper as 720 to 1. For all time since February, 1853, this government has estimated gold in comparison with silver as 14-1m2o1 to 1, and in comparison with copper as 720 to 1. The standard for mint gold with this government until 1834. was 11 parts pure gold and I part alloy, the alloy to consist of silver and copper mixed, not exceeding one half copper. The gold coins, therefore, struck at the United States mint prior to 1834, are 22 carats fine. 2 14 CURRENCY OF TIlE UNITED STATES. In what, until 1834, constituted a dollar of gold coin of United States mintago, there were put 24.75 grains of pure gold; and 27 grains of the standard mint gold of that day were at that time worth $1. Twenty-seven grains of that gold, or gold of that standard, are now, by the present government standard of valuation, worth $1.0652. The standard for mint silver with this government until 1834, was 1485 parts pure silver and 179 parts pure copper,- 8-9 parts pure silver and 1 part pure copper. The silver coins, therefore, struck at the United States mint prior to 1834, are 10f-9 C ounces fine. In that which, until 1834, constituted a dollar of silver coin of this government's mintage, there were put 3714 grains of pure silver; and 416 grains of the standard mint silver of that day were at that time of the value of $1. Four hundred and sixteen grains of that silver, or silver of that standard, are now, by the. present government standard of valuation, worth $1.0744. The cent, until 1834, was of pure copper, and weighed 208 grains; since 1834, pure copper, weight 168 grains. The standard for mint gold with this government is now, and for all time since June, 1834, has been, 9 parts pure gold and one part alloy, the alloy to consist of silver and copper mixed, not exceeding one half silver. The gold coins, therefore, struck at the United States mint and dated subsequent to 1834, are 214 carats fine. The standard weight for these coins is 254 grains to the dollar; and in every 254 grains of these coins there are 23 22 grains of pure gold. The standard for mint silver with this government is now, and for all time since June, 1834, has been, 9 parts pure silver and 1 Iart pure copper. The silver coins, therefore, struck at the United States mint and dated subsequent to 1834, are 104 ounces fine. In what, from June, 1834, until February, 1853, constituted a dollar of silver coin of this government's mintage, there were put 371i grains of pure silver; and 4124 grains of the standard mint silver of that day (the present standard) were worth, from June, 1834, until February, 1853, $1. Four hundred twelve and one half grains of this standard of silver are now worth, by the present standard of valuation, $1.0742. The standard weight for silver coins with this government at present is 384 grains to the dollar. (See page a51.) The foregoing is not applicable to the silver three-cent pieces, so called, authorized by the Congress of 1850-51. These pieces are composed of 3 parts silver and 1 part copper; and their staldard weiflght is 121 grains each. They are worth, at full weight, macas CURRENCY OF] TIIE UNITED STATES. 15 ured by the present standard of 3415.6 grains of fine silver to the dollar, 2.685 cents each. The-nickel and bronze tokens or coins, authorized by Congress at diffirent times between the years 1856 and 1867, are as followss: — Nickel 1-cent token: -88 parts copper and 12 parts nickel; weight, 72 grains. Nickel 3-cent token: copper and nickel mixed, not exceeding one-fourth nickel; weight, 32 grains. Nickel 5-cent token: copper and nickel mixed, not exceeding one-fourth nickel; weight, 77.16 grains, or 5 grammes; diameter, 2 centimetres. Bronze 1-cent token: 95 parts copper, an(l 5 parts tin and zinc; weight, 48 grains. Bronze 2-cent token: 95 parts copper, and 5 parts tin and zinc; weight 96 grains. NOTE. -In the preceding calculations of values, and in the following, as is now the common custom everywhere, no value has been assigned to the alloy, either in the silver coins or the gold coins. In general, it is simply copper, and will not net more, it is assumed, than the cost of recovering. In the United States, the law relating to coinage, previous to 1834, required that the alloy for gold coins should consist of silver and copper mixed, not exceeding one-half copper; and the present law provides that it shall consist of silver and copper mixed, not exceeding one-half silver. BOSTON, July, 1869. GOLD, -PURE. 24 carats fine _ pure gold. 1 grain $0.043066. 23.22 grains - $1.00. 1 dwt. -- $1.033592. 1 ounce - $20.671834. MINT GOLD, - U. S. Alloy, practically.all colpper. Nine parts of pure gold and one part of alloy, or 213 carats fine - standard coin. 1 grain - $0.03876. 25- grains = $1.00. 1 dwt. - $0.93023. 1 ounce -- $18.60465. GOLD COINS,- U. S. Denominations. Weight in Standard grains. value. Double Eagle, -516 $20.00 Eagle, - - 258 10.00 Half-Eagle, - 129 j 5.00 Quarter-Eagle, - 64 2.50 Triple Gold Dollar, 772 3.00 Gold Dollar, - 255 1.00 Eagle, prior to 183-1 (com. value $10.62), 270 10.909 Half do, " " - (conl. value = $5.31), 135 5.454 l{J CURRENCY OF T'IE UNITED STATES. Private and Uncurtren. ri Sa.les. A. Bechtler, N. C., $5 pi'ce, - - $4.75 " " 2.4". - 2.37 1..93 T. Reed, Georgia, 5'" - - 4.75 " " 2" - - -" 2.37 " " 1 ".....93 1Moffit, Californin, 5 " - - 1'29 5.00 SILVER, — PURE. 12 ounces fine - Pure Silver. I dwt. - $0.06944. 345.6 grailns = $1. 1 ounce = $1.38889. MINT SILVER.- U. S. Alloy, all copper. Nine parts pure silver and one part alloy; or 10 oz. 16 dwts. fine = Standard Coin. 1 dwt. = $0.0625. 384 grains = $1.00. 1 ounce = $1.25. SILVER COINS.- U. S. Weight il Standard Grains. Value. Dollar, - 384 $1.00 Ialf Dollar, - - - - 192.50 Quarter Dollar, - 96.25 Dime,. 3825.10 IIalf Dime, - 9 1.05'Three-Cent Piece,, silver and g colper, 12,.03 rtle copper coins of the United States are the CENT and n..hTF CEN'T; tlhey are of' pure copper. The weight of the former is MIS0, grains. alnd that of the latter, 84 grains. N\TE.- -T!le silver coins of the Ulnited( States, sIssiid since Feblruary, 1S53, are not legal tender in the United States in sums exceedinllvgre dollars. CURRENCY OF TIIE UNITED STATES. 17 TABLE, Exhibiting the standard weight and present par value of the silver coins of the United States, of dates subsequent to 1834, and prior to 1853. Weight In Present Gratins. par Value. Dollar, - 412. $1.0742 IIalf Dollar, -. - - 206k.5371 Quarter Dollar, - - - - 103.2685 Dime, -. 41k.1074 Jlalf Dimenc, - - - - 20i.0537 Three-Cent Piece, - - - 12.03 CURRENCIES O-F TIHE DIFFERENT STATES OF TIIE UNION. I Farthings = 1 Penny, 12 Pence = 1 Shilling, 20 Shillings =, 1 Pound. In Massachusetts, Connecticut, Rhode Island, New Hampshire, Vermont, Maine, Kentucky, Indiana, Illinois, Missouri, Virginia, Tennessee, Mississippi, Texas and Florida, 6 shillings = 1 dollar; $1 _. 3y ~. In New York, Ohio and Michigan, 8 shillings = 1 dollar; $1l ~. In New Jersey, Pennsylvania, Delaware and Maryland, 7 shlillings and 6 pence _ 1 dollar; 1 dollar B= In North Carolina, 10 shillings = 1 dollar; $1 =. in South Carolina and Georgia, 4 shillings and 8 pence - I dolilr; $1 = 7 ~. NOTE.- These currencies, so called, are nominal at present in a great mneasu'e. The denominations serve in the different States as verbal expressions of value. But they are neither the names of the moneys of ac:count in any of the States, nor are they the nationa! names of any of the real moneys in circulation. All values in money in the United States are legally expressed in dollars, cents, and mills. 2* 18 METRICAL SYSTEM OF WEIGHTS AND MEASURES. TIlE METRICAL SYSTEM OF WEIGHITS AND MEASURES. In this system, the METRE is the basis, and is one forty-millionth of' the polar circumference of the earth. The METRE is the principal unit measure of length; the ARE of surface; the STERE of solidity; the LITRE of capacity; and the GRAM of weight. The gram is the weight, in a vacuum, of one cubic centimetre of pure water at its maximum density. The Metre, almost exactly. - 39.37 U. S. inches. The Are (100 square metres) = 3.95367 " square rods. The Stere (a cubic metre). = 35.31445 " cubic feet. The Litre (a cubic decimetre) i 61.023 " inches. 1.05668 " wine quarts. The Gram... = 15.43235 " grains. The divisions by 10, 100, 1,000, of each of these units, are expressed by the same prefixes, viz., deci, centi, milli; and the multiples by 10, 100, 1,000, 10,000, of each, by deca, hecto, kilo, mnyria. The former series were derived from the Latin language, the latter from the Greek. To illustrate with the metre:10 millimetres - 1 centimetre, 10 centimetres - 1 decimetre, i 0 decimetres — 1 METRE, 10 METRES = 1 decametre, 10 decametres 1 hectometre, 10 hectometres = 1 kilometre, 10 kilometres -1 myriametre. In commerce, the ordinary weight is the kilogram, and 100 kilograms (usually called kilos) = 1 quintal; 10 quintals = 1 millier, or tonneau. The kilogram 15,432.35 *. 7000 = 2.20462 avoirdupois pounds. In practice, the terms milliare, declare, decare, kiliare, and myriare are usually dropped, and 100 centare = 1 are; 100 ares _ 1 hectare. Also the terms millistere, hectostere, kilistere, and myriastere, are usually rejected, and 100 centisteres = 1 decistere; 10 decisteres -1 stere; 10 steres 1 decastere = 353.1445 cubic feet. 1 centiare (square metre) 1.19598526 square yards. 1 kilometre.. 0.62137 statute miles. 1 hectare.... -- 2.471 U. S. acres. 1 kilolitre... - 1 stere = 61,023.377953 cubic in. A hectolitre - 26.41748 wine gallons -- 2.83 774 Winchester bush. NOTE. - The system is the one recommended by the Statistical Congress of 1865 as a general system of weights and measures to be adopted by all nations. FOREIGN GOLD COINS. 19 FOREIGN -GOLD COINS. NOTE. —' The coins of any country, both -gold and silver, circulating as foreign in any other, particularly those of the smaller denominations, are usually held at an estimate below their standard -par value, compared with the money standard of the country in which they circulate as foreign. Many of them, more particularly the silver, having circulation in the United States, are much worn and otherwise depreciated. In some instances, owing to frequent changes made both with regard to weight and purity, certain of them, having the same name and general appearance, bear a premium at home; others, a discount. Others, again, can hardly be said to have a definable value anywhere. The par value of the old pistole of Geneva, for instance, weighing 1031 grains, is $3.985, while that of the new, weighing 879 grains, would, at the same degree of purity, be worth but $3.386; whereas, owing to its higher standard of fineness, its par value is $3.443. The ducat of Austria, coined in 1831, weighs 532 grains, - its purity is 23.64, and its par value $2.269; while the half sovereign, closelyresembling the ducat, coined in 1835, and weighing 87 grains, has a purity only of 21.64, and a par value, consequently, of but $3.378. The circulating value of the ducat in the' United States, in general, is $2.20, and that of the half sovereign of Austria, $3.25. Stttanl.rd Standalrd Par value Circulatinl Par ailARGENTINE REPUBLIC. of ovejyt in volno in no per purity in in Federal Federal grann. carltts. raits. money. money. cti. Doubloon to 1832, 19.56 418 $14.671 $ 3.5-0 " to " 20.83 415 15.512 3.73 AUSTRIA. Sovereign, half in proportion, to 1785, 22.00 170 6.711 6.50 3.94 Sovereign, half in proportion, since 1785, 21.64 174 6.756 6.50 3.88 Ducat, double in proportion, 23.64 53t 2.269 2.20 4.24 BELGIUM. Sovereign, half in pro., 22.00 17 0 6.711 3.94 20 FOREIGN GOLD COINS. Standard Stanwldard Par value Circulating Par valof weight in value in ue per purity in in Federal Federal grain. carats. [rains. iorney. money. cts.'rwenty Franc, more in pro. 21.50 99A $3.840 $3.83 3.85 I)ucat, 2.20 BOLIVIA, COLOMBIA, CHILI, ECUADOR, PERU, NEW GRtENADA, and- MEXICO. For the modern coins, &c., of these States, see Foreign MAoneys of Account, SEC. A. Doubloon, (8 E) 20.86 417 15.620 15.60 3.74 Half do. I " 208s 7.810 7.50 Quarter do. " 104[ 3.905 3.75 " Eighth do. " 52 1.952 1.75 1, Sixteenth do. ": 26.976.90 C" Pistole, half in pro., 3.75 BRAZIL. For the modern coinage of this Empire, see Foreign Mloneys of Account and Coins, SEC. A. I)obraon, 22.00 828 32.719 32.00 3.95 l)obra, " 438 17.306 17.00 4 Joannes, (standard variable) " 432 17.064 $i3to$17 Half (lo. do. do. t" 216 8.532 $6 to8.50 - Mloidore, (BBBB) half in pro., (stand(ard variable) 21.79 165 6.451 6.00 3.90 Crus;ado, do do. 164.?635 4 DENSMARIf K. Christia,, d'or 21.74 103 4.018 3.90 I)tcat, species, 23.48 53 | 2.254 2.20 4.21 " current, 21.03 48 1.R I 3.77 FRANCE. Thelc are but few goldc coins of France now in circulation other than multiples of the sttandard franc, napoleons, fractional and double, in pro. Chr. d'or, d(itl,)le in pro.,'21.60 101 3.914 3.90 3.87 FOREIGN GOLD COINS. 21;Standard S tanidtrd Pur value Circullttin Par valof. weight in value in ue per purity in in Federal Federal grain. curats. grains. mon)ev. monev. ro'n. Franc d'or, double in pro., 21.60 101 $3.914 $3.90 3.87 Louis d'or, " " " to 1786, 21.49i 1'25A 4.840 3.85 Louis d'or, double in pro., since 1786, 21.68 118 4.573 4.50 3.87 Napole)n (20 F.) double &c. 21.60 99. 3.856 3.83 " GERMANY. BADEN. Ziehnl Gulden, 5 in pro.. 21.60 1o05 4.088 4.00 3.87 BAVARIA. Carolin, 18.49 149i 4.952 3.32 Ducat, double in pro., 23.58 531 2.275 2.20 4.23.IMaximilian, 18.49 100 3.317 3.31 BRUNSWICK.! Ducat, 23.22 53,4 2.220 4.16 Pistole, double in pro., 21.60 1174 4.548 3.87'I'en Thaler, 5 in pro., to 1813,' 21.55 202 7.811 7.80 3.86 Ten Thaler, less in pro., si;ce 1813, 21.50 204 7.873 7.80 3.85 HANOVER. nelc;at, (d11)le ill pro., 23.83 53- 2.287 2.20 4.t27 (CeoW re'or, "4 " " 21.67 102. 3.987 3.88 Zehllnlhaler, 5 "" 21.36 204A 7.838 7.80 3.83 HESSE.'Fen Thaler, 5 in pro., to 1785, 21.36 202 7.742 Ten Thaler, 5 in pro., since 1785, -21.41 203 7.799 3.84 SAXONY. T)lucat, 23.49 53.4 2.256 2.20 4.21 Augustus d'or, double in pro., since 1784. 21.55 1024 3.964 3.86 WURTEMBURG. I'Carolinn 18.51 1474 4.899 3.32 j)uc:at, 23.28 534. 2.235 4.17 22 FOREIGN GOLD COINS. Standaird 6Stlandr Par vaIVo C(ircrllatlang Par valt of weight in v;alue in ue per rpa'ty in Federal Federal grain. c;e rI ts. ns. money. n1 oney. cts. GREAT BRITAIN. (Alloy, since 1826, all copper.) The modern gold coins of this Kingcom. are the sovereign, fractional, double, &c. Guinea, half in pro., to 1785, 22.00 127 $5.016 3.95 Guinea, half in pro., since 1785, " 1294 5.111 $5.00, Sovereign, half in pro., " 12341 4.866 4.83 " Five do. i " 6164G 24.332 24.20 i Sovereign, (dragon) half in pro., " 122.4 4.838 4.80 Double Sovereign (dragon) " 246 9.7 17 0.67 " GREECE. Twenty Drachm, more in pro., 21.60 89 3.441 3.40 3.87 HOLLAND. I)uclat, 23.58 534 2.263 2.20 4.23 Ryder, 22.00 153 6.043 3.95 Double do. S,. 309 12.205; Ten Gulden, 5 in pro., 21.60 103 3.988 3.98 3.87 INDIA. Pagoda, star, 19.00 524 1.798 3.40 Mohur, (E. I. Co.) 1835. 22.00 180 7.106 6. 45 3.95 Half Sovereign, do.. 2.41 BOMBAY. Rupee, 22.09 179.095 3.96 MADRAS. Ripee, | 22.00 180 7.106 3.95 ITALY. ETURIA, Ruspone, 23.97 1614 6.935 4.30 GENOA, Sequin, 23.86 534 2.291 4.28 MILAN, Pistole, 21.76 97. 3.807 3.90 " Sequin, 23.76 53 2.281 4. -~~~~~.f FOREIGN GOLD COINS. 23 Standard. Stalndrd Pnr vilite (cill tingr Par vailof wevilht tn -le intl ue per purity hi in Federal IFelerul grain. carats. grains. mnnev. money. ctis. MILAN, Twenty Lire, more in proportion, 21.58 99; $3.853 $3.83 3.86 NAPLES, Ducat, multiples in pro., 21.43 22..865 3.84 NAPLES, Oncetta, 23.88 58 2.485 4.28 PAIMAIA, Doppia, to 1786, 21.24 110 4.192 3.81 " Pistole, since 1796, 20.95 110 4.135 3.75 Twenty Lire, 21.60 399 3.859 3.83 3.87 PI n1DMONT, Carlino, half in pro., since 1785, 21.69 702 27.321 3.89 I)IEDMONT, Pistole, half in pro., since 1785, 21.54 14(0 5.411 3.86 PIEDMONT, Sequin, half in pro., since 1785, 23.64 531 2.280 4.23 PIEDMONT, Twenty Lire, more in pro., 20.00 99,1 3.563 3.50 3.59 iROME, Ten Scudi, 5 in pro. 21.60 267.[ 10.368 3.87 -" -Sequin, since 1760, 23.90 52-A 2.251 4.28 SARDINIA, Carlino, A in pro., 21.31 247. 9.465 3.82 TUSCANY, Zechino, double in pro., 23.86 534 2.302 4:30 VT:Nilr,:, Zechino, double in pro., 23.8.1 54 2.310 MALTA. Sequin, 23.70 53A- 2.275 4.25 Louis d'or, double and demi in pro., 20.25 128 4.651 3.63 NETHEI LANDS. Ducat, 23.52 53A 2.257 4.21 Zehn Gulden, 5 in pro., 21.55 103i 4.013 4.00 3.86 POLAND. I)ucat, 23.58 53. 2.264 4.23 PORTUGAL. The modern Portuguese gold coins are the coroat of 5000 reis, parts andl nmultiples in proportion. See Si.:C. A. 24 FOREIGN GOLD COINS. Standard Standard Par value Circulating Par val.of weight in value in ue per purity in in Federal Federal grain. carats. grti ns. money. money. cta. Dobraon, 24,000 reis, 22.00 828 $32.706 $32.00 3.95 Dobra, "4 438 17.301 17.00 " Joannes, (standard variable) " 432 17.064 $13 to $17 " Half " " " 4 4 216 8.532 $6 to 8.5( "' Moidore, 4000 reis, " [4 to 4-i Coroa, 5000 " " 147.J 5.83 5.75 cc Milrea, 22.00 194.780 3.95 PRUSSIA.. Ducat, 23.49 53A 2.255 2.20 4.21 Frederick d'or, double in pro., 21.60 102A 3.973 3.87 RUSSIA. Ducat, 23.64 54 2.291 4.24 Imperial, (10 R.) half inl pro., 1801, 23.55 1854i 7.828 4.22 Imperial, (10 R.) half in pro., since 1818, 22.00 201~ 7.949 7.90 3.95 SICILY. Oncia, double in pro., 20.39,68 2.495 3.64 Twenty Lire, more in pro., 21.60 994 3.856 3.83 3.87 SPAIN. For the new standard of coinage, denominations, &c., of' this Kinidom, see Foreign M:oneys of Account and Coins, SEC. A. Doubloon (8 S) parts in pro. 21.45 416A 16.031 16.00 3.84 " (8 E) parts as Bolivian, &c. 20.86 417 15.620 15.60 13.74 Pistole, to 1782, 21.48 103 3.970 3.85 sifnce " 20.93 104 3.906 3.75 Escudo, to 1788, 5 20.98 52 1.957 3.76 " since " 20.42 52 1.'05 3.66 Coronilla " 1800, 20.29 27 983 3.64 SWEDEN. Ducat 23.45 53 2.230 4.20 -1 _ _ _ _ _ _ _ 1 LONG OR LINEAR BIEASUT:E. 25 Staindard Standairdl Plr Valle CIi'tlatit i Par val of weigrlht in vtlle in tue per prity in in Federal Federtl gral Co ra ts. zgrins. moneyv. mioiey. cts. SWITZER LANI). BEIINE, DUCat, douilel in pro., 23.53 47 $1.984 4.22 BIERNE, Pistole, 21.62 117 4.558. 3.88 GENEVA, Pistole, 21.87 87. 3.443 3.P92 " ", (,/li) 21.51 1031 3.1985:3.85 ZURIC1h, Ducat, doutll in pro., 23.50 53A 2.256 4 2 TURKEY. Misseir, half in p)ro. 1820, 15.88 36,1 1.040 52.8.1 Sequin fonducli. 19.25 53 1.830 3.15 Yeermeeblekblekl, 22.88 73~ 3.027 4.10 NOTE. —For futll and particular specificartions regarding modern foreign gold and( silver coills, sce lr'reirl Alone1ys of Account and coins, SEc. A. The standard silver 5-firanc piece of France is worth $1.00471 in tlle silver coilns of the United States; hut 5 francs in the standard gold coills of Fran:ce.are worth but $0.96472i in the gold coins of the United States. LONG OR LINEAR MEASURE. - U; S. STANDARD.- A brass rod, the length of which, at 620 Fahrenhleit, is 36 o - ooo that of a pendulum beating seconds in vacuo, at the level of the sea, at the latitude of London 3.oo —o 0 at 32~ Fall., at the gravitation at New York, - the Yard. 6 points -I line. 1 5. yards (l16 ft.)_ I rod. I2 lines (72 points) -- 1 inch. 40 rods (220 yds.) 1 furlong. 12 inches 1 foot. I 8 fur. (5280 feet) = 1 stat. mile. 3 feet (36 inches) = 1 yard. SPECIAL, FOR CLOTH. 24t inches - 1 nail. 4 quarters (36 inches) =1 yard. 4 tiails (9 inches) -1 quarter. SPECIAL, FOR LAND. 7 92% inches = 1 link. 100 links (66 feet) - I chain. 95 links =1 rod. 80 chains (320 rods) 1 s. mile. ENGINEER? S CHAIN. 10 inches 1 link.120 links (100 feet) 1 chain. 3 26 SQUARE OR SUPERFICIAL 3lEASURE. SlHOEMAKER'S MEASURE. No. 1 is 41i incles in length, and each succeeding number is an addition of - of an inch. No. 1 man's size - 811 inchcs. MISCELLANEOUS. HIair's breadth =-1 inch. Fathom -=6 fect. Digit - 10 lines. Knot = 473 feet. Palm = 3 inches. Cable's lengti = 120 fatlolm,,. HIand = 4 " Geometrical pace — 4.4 feet. Span. 9 12 particular things = 1 dozen. 12 dozen (144) = 1 gross. 12 gross (1728) = 1 great gross. 20 particular things = 1 score. 24 sheets of paper = 1 quire. 20 quires = 1 ream. SQUARE OR SUPERFICIAL MEAS U RE. (Length X brceadlth.) 144 square inches 1 square foot. 9 " feet - 1 " yard. 304 " yards - 1 " rod. 40 " rods - 1 rood. 4 roods -- 1 acre. SPECIAL, FOR LAND. 62~4-4 square inches - 1 square link. 10000 " links -- 1 " chain. 10 " chains — 1 acre. Square rod - 2-72- square feet. Rood 1210 " yards. 10890 " feet. Acre (160 (qulare rods)= = 4840 " yardse 43560 " feet. 640 acres. Square nmile 102400 sq. rods. 220 X 198 square feet The square of 12.649 "' rods " t of 69.5701 " yrds ace. " " of 208.710321 " feet CUBIC OR SOLID L.EASURE. 27 CIIIUCLAIR IMEASURIE. M~inute, or 1 G-Creat Circle 360 dleglees. Geogra- 1.151 s. miles. Elquatorial ciI- 1llical m. 08 et. cuimfercee -( 24897 s. m. (60") of the earth ( leagrue _3 miles. Equatorial diam.= 7925 = S60 geo. miles. Polar diam. - 7899 " e 69.158 s. ms. Mean radius - 3955.92 Signrlu(Lj zod.)- 30 degrees. Nore. - In tile expressions, square feet and feet square, there is this difference; viz.. the iolller expresses an area il which tllere are as many square feet as the number 1isieul, atdl the latter an area in which there are as many square feet as the square of the inumlller lalied. Thle flrmer particularizes no fol' of area, the latter asserts a square firn. CUBIC OR SOLID MEASURE. -U. S. (Leng' th X breadth X depth.) 1.273 cylindrical feet. Cubic foot, _ J2200 " inches. 1728 cu. inches -- 3300 spherical " 6600 conical " [0.785398 cubic feet. Cyiindrical foot I - 1357.2 " inches. 1728 " inches t -- 2592 spherical L5184 conical 27 cubic feet = 1 cubic yard. 40 " of round timnber 1 ton. 42 " of shipping " -1 ton. 50 " of hewn " I ton. 1~2i8 ((" _ 1 cord. Cubic foot of pure water,1 at the maximum density at the level of the sea, - 2A avoirdupois pounds. 1000'; ounces. (39~.83, barometer 30 1000 inches) { 49.1 " pounds. Cylindrical foot 785.4 " ouns. 785.4 "6 ounces. (0.036169" pounds Cubic inch - 0.5787 " ounces. 253.1829 grains. Cylindrical in 0.028415 avd. pounds. 0.4546 " ounces Pound 27.648 cubic ildches. " distilled = 27.7015 " " Cubic inch" 252.6934 grains. Pound at 62~, distilled - 27.7274 cub. inchies. Cubic inch at 62~, " - 252.458 grainls. 6 "( 390.83, in vacuo =- 253.0864'~ Cubic fiot of salt water (sea) weighs 64.3 pounds. 28 GENERAL MEASURE OF WEIGHIT. GENERAL, MEASURE OF WEIGHIT.- U. S. AVOIRDUPOIS. SPECIAL - TROY. S'IANDARD. - The l)pound is the (Exclusively for gold and silb weight, taken in air, of 27.7015 ver Blllion, precious stoines, and cubic inches of distilled water at gold, silver and copler coins, and its maximum density, (39~.83 F., ith reference to their monetary the barometer being at 30 inches) value only.) =27.7274 cubic inches of distilled 24 grains I pennyw't. water at 62~ = 7000 Troy grains. 20dwts. (480 grs.)= 1 ounce. 12 oz. (5760 grs.) = 1 pound. 271 1 g l-ns - dranm. 16 drains (4371 grs.) 1 ounce. SPECIAL AP CAIES. 16 ounces (7000 grs.)= 1 pound. (Exclusively for compounding medicines, for recipes and preSPECIAL - GROSS. scriptions.) 28 pounds = 1 quarter. 20 grains = I scrtlile, )o 4 quarters 1 quintal. 3 scruples = Idram, 75 112 pounds 1 cwt. 8 drams(480 g.)= 1 ounce, 5. 20 cwt. 1 ton. 12 oz. (5760 g.) 1 polllnl, lb SYIECIAL- DIAMOND. 1 lb. avoir. _ 1'13 lbs. troy. 6ll i)arts = 1 grain= 0.8 troy gr. 1 lb. troy _ 144- lbs. avoir. 4 grs. = I carat = 3.2 "' 1 oz. avoir. troy. 1 oz. troy -- 1i,-75 oz. avoir. NOTE. - The comlparative value of diamonds of the sa;me quality is as Ilhe s(iquare of h.lleir respective weights. A diamond of fair quality, weighing I carat in the roughl state, is estimated worth about $9 T5i0.; and it will require one of twice that weight to inake one when worked down equal to I carat in weight. Hence, to dleterilue the value of a wrought diamond of any given number of carats: - Rtolc. - Dobille the weiglit inl carats and multiply the square by 9.50. Thus, the value of a wroughlit diamlond, weighing 2 carats, is 2-+2= 4 X 4 = 16 X 9.50$ 152. LIQUID MEASURE. IJ. S.'rhe " Wine" or " Winchester" Gallon, of 231 cllbic inches capacity, is the Government or Customs gallon of the United States for all liquids, and the legal gallon of each state in which no law exists fixing a state or statute gallon of its own. It contains 583.'784 grains of distilled water at 39'3.83, the barometer being at 30 ilches. 4 gills = 1 pint, 2 pints = 1 quart. 4 quarts, or 231 cubic in. gallon. 0.13368 cub. ft., 294.1176 cyl. in. 8.355 av'd. lbs. purc water. 'ORY V EASURE. 29 0.128 cubic foot. Liquid gallon of the) 221.184" in. State of New ~ork, — 8 avoid. Ilbs. pure water 281.62 cylindric in. t 3)30 b 30 in. at 39..3, b. 30 in. Barrel - 311 g:llonis. ilum(llc, = 84 gallons. Tierce = 42 I'' ile (1r l3ttt = 126 i" Hogshead = 63 u" t 252 " Imperial gallon, 10 av'd lbs. distilled water 277.274 cub. in. - at 62~ F., b. 30 in. Ale gallon, _ 11 av'd lbs. pure water 282 cub. in. S at 390.83, b. 30 in. (0.8331 Imperial gallon. I Wine gallon= 0.8191 Ale (0.10742 W. bushel. 1 Imperial gallon - 1.2 Wine gallons. DRY MEASURE. U. S. The "Winchester Bushel," so called, of 2150 4 2 cubic inches c:lpaclity, is the Government bushel of the United States, and the legal hbishlel of each state having no special or statute bushel of its owln.'l'he standard Winchester bushel measure is a cylindrical vessel havinlg an outside diameter of 19A inches, an inside diameter of 18,; inc!hes, and all inside depth of 8 inches. The standard " heaped " or cal " bushel of England was this measure heaped to a true cone 6 inches high, the base being 19A inches, or equal to the outside diameter of the measure. Its ratio to the even bushel was, therefore, as 1.98, nearly, to 1. The present " Imperial " measure of England has the same outside diameter and the same depth as the Winchester, and all internal diameter of 18.8 inches, and the same height of cone is retawined for forming the heaped bushel. Its ratio, therefore, to tihe even bushel is a trifle less than was that of the Winchester. In the Iliiited States the "heaped bushel " is usually estimated at 5 even peeks, or as 1.25 to 1 of the standard even bushel, which, if takei, as * By enactment of the Legislature of the State of New York, this gallon ceased to he the legal gallon of that State, April 11, 1852; and the United States Government gallon, of 2:31 cuIbic inches capacity, was adopted in its stead. 3* IDRY MEASURB. the rule, requires a cone on tle Winchester measure of 5.4 inches to equal the heaped W\inchester bushel. 4 gills... = 1 pint. 2 pints.. = 1 quart. 4 quarts - 1 gallon uarts or half peck. 8 quarts.. 1 peck. 4 pecks F 1 bushel. 2150.42 cubic in. _ 2738 cyl. in. 1.244456 " ft. - 77.7785 av'd lbs. 1.5844 cyl. " J pure water. Bushel of the )1.28 cubic feet. State of New York,* 2211.84 " in. 2816.1955 cyl. in. ) 80 av'd lbs. pure water. 1.272 cubic feet. Bushel of Connecticut, = 2198 " in. 79.50 av'd lbs. pure water. Heaped Win. bushel 2747.7 cubic in. 1.28 — even" " = " 1.59 cubic ft. Imperial bushel = 2218.192 " in. Chaldron - 36 Winch. heaped bushels, 1 Winchester bushel = 0.9694 Imperial bushel. 9.3092 Wine gallons. 1 Imperial bushel = 1.0315 Winchester bushels. NOTE. - The Imperial bushel, mentioned above, is the present legal bushel of Great Britain; and the Imperial gallon, mentioned on the preceding page, is the presenlt leg-al gallon of Great Britain, for all liquids. The gallon for liquids is the samne as the gallo, for dry measure. Eight Imperial gallons make one bushel. The sulbdivisions of the gaLllon and the bushel, and their denominations, are the same as in the Winchester measures. In Great Britain, in a(ddition to the denominations of dry measure used in the Unitedl States, the Strike,........- 2 bushels. Last......... = 80 bushels. Coomb,...... = 4 " I Sack of corn..... = 3 Quarter,.....= 8 " B ole of corn,. = 6 " Wey or loadtl..=. —40 " Last of gunpowder,.. = 42 balrrels. * This bushel ceased to be the legal bushel of this State April 11, 1852, and the United States Government bushel, of 2150 42- cubic inches capacity, was aloptd( as the legal bushel in its stead. i This bushel is now, January, 1852, no longer the legal bushel of this State, and the standar(l Winchester bushel is adopted in its stead. SECTION II. MISCELLANEOUS FACTS, CALCULATIONS, AND PRACTICAL MATIIEMATICAL DATA. SPECIFIC GRAVITIES. The specific gravity of a body is its weight relative to the weight of an equal bulk of pure water at the maximum density, (39~.83, b. 30 in.) tile water being taken as 1., a cubic foot of which weigths 1000 avoirdupois ounces, or 62t lbs. The specific gravity, therefore, of any body multiplied by 1000, or, which is the same thing, the decimal being carried to three. places of figures, or thousands, as in the following TABLES, the whole taken as an integer equals the number of ounces in a cubic foot of the material: multiplied by 62.5, or considered an integer and divided by 16, it equals the number of pounds in a cubic foot; and multiplied by.036169, or taken as an integer and divided by 27648, it equals the decimal fraction of a pound per cubic inch; by which, it is readily seen, the specific gravity of a commodity being known, its weight per any given bulk is easily and accurately ascertained; as, also, its specific gravity, the weight and bulk being known. The weight of any one article relative to that of any other, is as its respective specific gravity to the specific gravity of the other. Specific Specific MIETALS. gravity. grvl. Antimony,... 6.712 Gold, pure, hammered, 19.546 Arsenic,... 5.810 Iridium,... 15.363 Bismuth,... 9.823 Iron, cast,. 7.209 Bronze,... 8.700 " wrought,.. 7.787 Brass, best,... 8.504 Lead,.. 11.352 Copper, cast,.. 8.788 Mercury, 32~,.. 13.598 " wire-drawn,. 8.878 " 60~,.. 13.580 Cadmium,... 8.604 " -39~,.. 15.000 Cobalt,.., 7.700 Manganese,. 8.013 Chromium,.. 5.900 Molybdenum,.. 8.611 (Clucinium,.. 3.000 Nickel,.. 8.9280 Gold, pure, cast,. 19.258 Osmium,. 10.000 32 SPECIFIC GRAVITIES. Specific Specific gravity. gravity Platinum, cast,.. 19.500 Granite, red,.. 2.625 hammered,. 20.337 Lockport,. 2.655 " rolled,. 22.069 " Quincy,. 2.652 Potassium, 600,.. 0.865 " Susquehanna, 2.704 Palladium,.. 11.870 Grindstone,... 2.143 Rhodium,. 11.000 Gypsum, opaque, 2.168 Silver, pure, cast,. 10.474 Hone, white,.. 2.876 " hammered,. 10.511 Hornblende,.. 3.600 Sodium,... 0.970 Ivory,... 1.822 Steel, soft,. 7.836 Jasper,... 2.690 " tempered,. 7.818 Limestone, green,. 3.180 T'in, cast,.. 7.291 " white,. 3.156 Tellurium,.. 6.115 Lime, compact,.. 2.720 Tungsten,.. 17.600 " foliated,.. 2.837 Titanium,. 4.200 " quick,.. 0.804 Uranium,.. 9.000 Loadstone,... 4.930 Zinc, cast,.. 6.861 Magnesia, hyd.,.. 2.333 Marble, common,. 2.686 STONES AND EARTHS. " white Ital.. 2.708 Alabaster, white,. 2.730 " Rutland, Vt.,. 2.708 " yellow,. 2.699 " Parian,.. 2.838 Amber,... 1.078 Nitre, crude,.. 1.900 Asbestos, starry,. 3.073 Pearl, oriental,.. 2.650 Borax,... 1.714 Peat, hard,.. 1.329 Bone, ox,... 1.656 Porcelain, China,. 2.385 Brick,... 1.900 Porphyra, red,.. 2.766 Chalk, white, 2.782 " green,. 2.675 Charcoal,....441 Quartz,... 2.647 " triturated,. 1.380 Rock Crystal,.. 2.654 Cinnabar,... 7.786 Ruby,... 4.283 Clay,... 1.934 Stone, common,.. 2.520 Coal, bitum. avg.,. 1.270 " paving,. 2.416 " anth. ". 1.520 " pumice,.. 0.915 Coral, red,.. 2.700 " rotten,.. 1.981 Earth, loose,.. 1.500 Salt, common, solid,. 2.130 Emery,... 4.000 Saltpetre, refined,. 2.090 Feldspar,.. 2.500 Sand, dry,... 1.800 Flint, white,.. 2.594 Serpentine,.. 2.430 " black,.. 2.582 Shale,... 2.600 Garnet,... 4.085 Slate,... 2.672 Glass, flint, 2.933 Spar, fluor,.. 3.156 " white,.. 2.892 Stalactite,... 2.324 plate,.. 2.710 Talc, black,.. 2.900' green,.. 2.642 Topaz,... 4.011 SPECIFIC GRAVITIES. o3 Specific Specific gravity. gravity SIxIPLE SUBSTANCES, Pine, yellow,...568 neilther meclallic nor -gaseous. Poplar, white,...383 P'oron,... 1.96;8 Plum,.....785 Biomine,... 2.970 Quince,...705 Carbon,.. 3.521 Spruce, white,.551 Iodine,... 4.943 Sassafras,...482 Phosphorus,. 1.770 Sycamore,...604 Selenium,... 4.320 Walnut,....671 Silicon,... 1.184 Willow,....585 Sulphur,... 1.990 Yew, Spanish,...807 " Dutch,..788 WOOI)S, (dry.) Apple,.. 0.793 Highly seasoned Antm. Alder,....800 Ash, white...722 Ash,...760 Beech,..624 Beech,....696 Birch,....526 Birch,....720 Cedar,.....452 B3ox, French,.. 1.328 Cherry,....606 " Dutch,...912 Cypress,....441 Cedar,....561 Elm,.....600 Cherry,....715 Fir,.....491 Chestnut,....610 Hickory, red,...838 Cocoa, ~.. 1.040 Maple, hard,..560 Cork,.240 Oak, white, upland,..687 Cypress,....644 " James River,..759 Ebony, American,. 1.331 Pine, yellow,..541 " foreign,.. 1.290 " pitch,....536'Elm,.....671 " white,...473 Fir, yellow,...657 Poplar, (tulip,)...587' white,...569 Spruce, white,...465 IIacmetac,....592 IHickory, red,...900 GUIS, FATS, &C. lignum vitr,. 1.333 Asphaltum,.905 Larch,...544 Asphaltum, 1.650 Logwood,....913 Beeswax,....965 Mahogany, Spanish, best, 1.065 Butter,....942 " comrn.,.800 Camphor,....988 L" St. Domingo,.720 Gamboge,.. 1.222 Maple, red,...750 Gunpowder,..900 Mulberry,....897 " shaken,. 1.000 Oak, live,... 1.120 solid 1.550 " white,...785, 1.800 Orange,..705 Gum, Arabic,.. 1.454 Pear,....661 " Caoutchouc...933 Pine, white,...554 " Mastic,. 1.074 034., SPECIFIC GlAVITl-ES;. S pl)t!~ 8 I Specifie -'rgtvity. gtavity. I[onev,. 1.450 Wine, champagne,..997 Ice,.....930 " claret,...994 Indigco,... 1.0009 port,...997 Lard,....941 " sherry,...992 Pitch,... 1.253 Rosin,... 1.100 Spermaceti,.943 ELASTIC FLUIDS: Starch, 1.530 The measure of which is atmospheric air, Sngar, dry,.. 1.606 at 600, b. 30 in., its assumedl nravity I; ole Tallow,....938 cubic foot of which wei-ghs 527.0.1 grains, _ Tar, 1.257.305 of a grain per cubic inch. It is, at this temperature and density, to pure water at the maximunl density, as.00120(16 to 1, LIQUIDS. or as I to 83().1. Acid, acetic,.. 1.062 SIMPLE OR ELEMICNTARY GASfS. citric,. 1.034 Hydrogen,.0689 fluoric,.1.0648 Oxygen,.1025 " nitric, 1.485 n. " nitrous, 1.420 Nitrogen,..9720 64 sulphtric, 1.846 Fluorine, Chlorine, 2.470'6' muriatic,. 1.200 it sili'c5 2.660 Calrbon, vaporXf; t.422 Alcohol, anhyd. 794 (teoreicl,) " 90 /....834 Bleer,... 1.034 COMPOUND GASES. ilood, human,.. 1.054 Ammoniacal,...591 Camphene, pure,...863 Carhbonic acid,.. 1.525 Cider, whole,.. 1.018 " oxide,...973 Ether, sulph.,...715 Carbureted hydrogen,..55!9 " nitric,...908 Chloro-carbonic,. 3.38!) Milk, cow's,.. 1.032 Cyanogen,... 1.818 Mlolasses, 75 /.. 1.400 Muriatic acid gas,. 1.24.7 ()ils, linseed,...934 Nitrous acid gas,. 3.176." olive,...917 Nitrous oxide gas, 1.040 "' rapeseed,.927 Olefiant, gas.978 " sassafras,.. 1.090 Phosphureted hydrogen, 1.185 " turpentine, coin...875 Sulphureted ". 1.177 sperm, pure,..874 Steam, 2120.484 " whale, p'f'l,.923 Smoke, of wood,..900 Proof spirits,...925 " of coal,..102 Viliegar,... 1.025 Vapor, of water,..623 Water, pure,.. 1.000 " of alcohol,. 1.613 sea,.. 1.026 " of spirits turpentine, 5.013 " )ead sea,. 1.240 WE1OIrT Pi' mt flU 1- DA1TEUL - CALLON, &C. 35 Weight per Bushel (corn Ex. tariff) of diffcrcnt Grains, Seeds, 4c. Articles. lbs. Articles. lbs Barley, (N. E. 47 lbs.). 48 Hemp seed,... 40 Beans,.... 60 Oats,.... 32 Buckwheat,.. 46 Peas,.... 62 Blue-grass seed.. 14 Rye,.... 56 Corn, (N. Y. 53 lbs.). 56 Salt, T. I.,. 80 Cranberries,... " boiled,.. 56 Clover seed,... 601 Timothy seedl,. 45 Dried Apples,.. 22 Wheat,.... 60 " Peaches, 33 Potatoes, h'p'd,.. 60 F;lax seed, (N. E. 52 lbs.) 56 Mlalt,.... 38 Weighlt per Barrel (Legal or by Usage) of different Articles. Flour,... 196 lbs. ICider, in Mass.,. 32 gals Boiled Salt,.. 280" ISoap,... 256 lbs. Beef;... 200 " Raisins,... 112 ork,.. 200 " Anchovies,.. 30 " Pickled Fish,. 200 " Lime,.220 " ",'' "L i:n I Ground Plastcr, Massachusetts, - 30 gls. Hydraulic Cement,. 300' A Gallon of Oil weighs.... 7 lbs. A " " Molasses, standard, (75 per cent.,) 11 " A " " Linseed Oil (usagfe, 71 lbs.) 7.788 " A Firkin of Butter, (legal,)... 56 " A Keg of powder,... 25 A Ilogshead of Sallt is... 8 bush A Perch of Stone - 243 cubic feet. A Gallon of Alcohol, 90 per cent., weighs. 6.965 lbs. A " " Proof Spirits, ". 7.732 " A " " AVine, (avc.rage,) ". 8.3 " A " " Spern Oil, ". 7.33 " A " " Whale " p'f d, ". 7.71 " A " " Olive " ". 7.66 " A " " Spirits Turpentine, ". 7.31 " A " " Catmphene, pure,. 7.21 " Wc'ight of Coals, 4c}., broken to the medium size, per Mfeasure oJ CQapacity. The average weilght of Bitulminous Coals, broken as above, is about 62 per cent. that of a bulk of equal dimensions in the solid mass, or 36 ROPES AND CABLES. of the specific gravity of the article.; that of Anthracite is about 57 per cent. Average weight I Averageo weight per. pereubio foot. lbs. W. Coal bushel. } Ibs. Anthracite,... 54 Anthracite,... 86 Bituminous,... 50 Bituminous,... 80 Charcoal, of pine,.. 18.6 Charcoal, hard wood,. 30 " of hard wood,. 19.021 Coke, best,... 32 Practical Approximate Weight in Pounds of Various Articles. Sand, dry, per cubic foot,... 95 Clay, compact, per cubic foot,... 135 Granite, " " "... 165 Lime, quick, " " "... 50 Marble, " " "... 169 Slate,,,, " "... 167 Peat, hard, " " "... 83 Seasoned Beech Wood, per cord,... 5616 " Yellow Birch Wood, per cord,. 4736 " Red Maple Wood, " ".. 5040 "," Oak Wood, ".. 6200 " White Pine Wood, " ".. 4264 " Hickory Wood, " ".. 6960 " Chestnut Wood, " 4880 Meadow Hay, well settled, per cubic foot, 4} lbs., or 445 cubic feet - 2000 lbs., or 498-0- cubic feet i long ton Meadow Hlay, in large old stacks, per cubic foot, 5 Clover Hay, in settled bulk, 4 " a 4" Corn on Cob, in crib, " " " 22 " shelled, in bin, " " " 45 Wheat, in bin, " 4 " 48 Oats, in bin, " " -" 251 Potatoes, in bin, C" "" 38j Common Brick, 7 X 3 X 2in. "M,.. 4500 Front " 8X 4} X 2- in. ".. 6185 ROPES AND CABLES. i The STRENGTII of cords depends somewhat upon the fineness of the strands; — damp cordage is stronger than dry, and untarred stonger than tarred; but the latter is impervious to water and less elastic. SILK cords have three times the strength of those of flax of e(ltial circumference, and MANILLA has about half that of hemp. WEIGHT AND STRENGTH OF IRON CHAINS. 37 Ropes made of IRON WIRE are full three times stronger than those of hemp of equal circumference. White ropes are found to be most durable. The best qualities of hemp are —1. pearl gray; 2. greenish; 3. yellow. A brown color has less strength. THE iBREAKING WEIGHT of a good hemp rope -is 6400 lbs. per square inlch, but no cordage may be conllted on with safety as capable of susta.ining a weight or strain above lhalf that required to break it, and htle weight of the rope itself should!)e included in the estimate.'1IlHE RELIABLE STRENGTH of a good hemp cable, in pounds, is usually estimated' as equal to the square of its circumference in inches X by 120. That of rope X 200. Thus, a cable of 9 inches in circumference may be relied on as having a sustaining power X 9 X 120 7 — 920 lbs.'THE WEIGHT, in pounds, of a cable laid rope, per linear foot = the sllrare of its circumference in inches X.036, very nearly. The weight, in pounds, of a linear foot of manilla rope = the square of its circumference in inches X.03, very nearly. Thus, a allnilla rope of three inches circumference weighs per linear foot 3 X 3 X..03 -- 1,7 ibs.,=3 7 feet per lb. A good helmp rope stretches about -6, and its diameter is diminished about - before brcaking. WEIGHT AND STRENGTII OF IRON CIIAINS. Diameter of Weight of Brealking Diameter of t eight of Breaking Wire 1 Foot Weight Wire 1 Foot Weighlt in Inches. of Chain. of Chtain. in Inches. of Chain. of Chain. lbs. lbs. lbs. lbs. 3 0.325 2240 s 4.217 26880 4 ] ~0.G5 4256 1 4 4.833 32704 -1 G 0.967 6720 5.75 38752 1.383 9634 13 6.G67 45696 7 - 71 1.767 1321G - 7.5 51744 1 2.633 17248 1 5 9.333 58464 I,9, 13.333 21728 1 10.817 65632 38 COihMPARATIVE WEIGIIT OF \IETALS. Comparalice IlViighlt (of Metals, I1'ciglht per Measure of Solidity, c)-c Specific Ratio of Poonlils in a CulIic (iraviLt. C(mlparison Eotl. 1 ll'. Iron, w-rouglt or rolled. 7.787 1. 486.65.28163 Cast Iron,... 7.20.9258 450.55.26073 Steel, soft, rolled,. 7.836 1.0064 489.75.283421 Copper, pIure, ". 8.878 1.1401 554.83.32110 3rass, best, ".. 8.604 1.1050 537.75.3112 Bronze, gun metal,.. 8,700 1.1173 543.75.31461 Lead.. 11.352 1.4579 709.50.4106 TABLE, Eahibitinlg the Weight in pounds qf One Foot in Length of WlVroug:it or Rollcd Iron of any size, (cross section,) from A inch to 12 inches, SQIUARE BnR. Size Weighll Size Wei-lit Size Weight Size Weigllt in is in in ill in i, i i INlols. Poellds. Inches. POrindIs. ilhwes-. Pouinds. Inihcs. PoesldIs. 3.053 24 1!9.066'1,4 72.305 7- 203.024 4.211 24 21.120' 49 76.264 8 216.336 {.475 24 23.2921 4 i 80.333 81 230.068 4.845 24 25.560 5 84.480 84 244.220 / 1.320 24 27.9391 51 88.784 84 258.800 4 1.901 3 30.416 54'93.168 9 273.792, 2.588 3: 33.010j 54 97.657 94 289.220 1 3.380 34 35.704 54 102.240 94 305.056 1 4.278 34 38.503i 54 106.953 91 321.332 14 5.280 31 41.4081 54 111.756 10 337.920 1 6.390 34 44.4181 5J 116.671 104 355.136 14 7.604 3i 47.534 6 121.664 104 372.672 1 8.926 3 50.756] 6./ 132.040 104 390.628 1. 10.352 4 54.084 (61 1412.816'11 408.9;0 14 11.883 41 57.517 64 154.012 11i 427.812 2 13.520 44 61.055 7 165.632 114 447.02,1 24 15.263 44 64.700~ 74 17-7.672 11 466.6841 2i 17.1121 4 68.448 74 190.136 12 486.656 COjILPAR1ATIVE WEGliTi 0F iMLETALS. 39 To determine the weight, in pounds, of one 0oot in leiigthl, or of ally lellgthl, of a bar of any of the following metals of form prescribed, of any size, multiply the weight in pounds, of an equal length of square rolled iron of the same size, (see table of square rolled iron,) if the weight ble sought of Iron, Round rolled, by......7854 Steel, Square " "... 1.0064 i" Round " ".7904 Cast Iron, Square bar, "......9258 " " Round " "'..... 271 Copper, Square rolled,".... 1.1401:' Round " ".8....8954 13 rass, Square " ".... 1.105 (" Round "' "...8679 lBronze, Square bar, ".1.1173 " Round " "......8775 Lead, Square " "... 1.4579'3 cc" Round " " 1.145 The weight of a bar of any metal, or other substance, of any given length, of a Jlat forim, (and any other formn may be included in the rule,) is readily obtained by multiplying its cubic contenits (feet or inches) by the weight (pounds, ounces, or grains) of a cubic foot or inch of the article sought to be weighed; that is - Icngthl X breadth X thickness X weigh-ltl r twit r it f measure. For the weight in pounds of a cubic foot or illch of differelnt metals, seo " TABLE of weights of metals per measure of solidity, &c." OR, F01R FLAT OR SQUARE BARS, Multiply the sectional area in inches by the lengtil in feet, and that product, if the metal be Wrought Iron, by...... 3.3795 Cast "...... 3.1287 Steel, ".3.4 EXAMPLE. - Reqtuired the weight of a bar of steel, whose lenagth is 7 teect, breadth 21 inclhes, and thickness I of all inch. 2.5 X.75 X 7 X 3./1 -- 41.625 lbs. An7s. EXAMrPLE.- Required thle weigllt of a east iron beam, whose length is 14 feet, breadth 9 inclies, and thickness 1, inch. 14 X 9 X 1.5 X 3.1287 =591.32 l1b. Ans. 40 WEIGHT OF' 1OUND IROLLED IRON. TABLE, Exh ibiting the weight in pountls of One Foot in Length oj' Rotund Rolled Iron of any diamneteler,f Jrot inch to 12 inches. Diameter Weigliht iDiam. in Weighllt Diam. in Weight'Diam. il Weight in inches. in lbs. inches. in lbs. inches. ill lbs. inches. irl l)s. 4i a.041 24 14.975 4[ 56.788 7 159).4564.165 2' 16.688 43 59.900; 8.;1.85);.373 24 18.293 44 63.0941 t 189.(;); /.663 24L 20.076 5 66.752 84t 191.8108 4 1.043 24 21.944 54 69.731 83 203.260 1.493 3 23.888 5i 73.172 9 215.040 / 2.032 3 25.926 5 i 76.700 94 227.152 1 2.654 3 t 28.040 54 80.304 94 239.600 14 3.360 3 30.240 5 / 84.001 943 252.376 14 4.172 31 32.512 5;} 87.776 10 266.288 1 5.019 34 34.886 54- 191.634 104 278.924 14 5.972 34 37.332 6 95.552 10 292.688 1 7.010 34 39.864 64 103.704 104 306.800 1i 8.128 4 42.464 G6 112.160 11 321.216 1 4 9.333 4 45.174 6`4 120.960 114 336.004 2 10.616 44t 47.952 7 130.048 114 351.104 24 11.988 44 50.815 7A. 139.544 11 366.536 24t 13.4401 44 53.760 74 1149.328 12 382.208'To find the weiglit of an equilateral three-sided cast iron prism. width of side in inches2 X 1. 354 X length in feet - weight in lbs. EXAMPLE. - A three-sided cast iron prism is 14 feet in length, and the width of each side is 6 inches; required the weight of the prisnm. 62 X 1.354 X 14 = 682.4 lbs. Ans. To find thle weight of an equilateral rectang ular cast iron prism. width of side in inches2 X 3.128 X length in feet = weight in lbs. To find the weight of an equilateralfive-sided cast iron prism. width of side in inches2 X 5.381 X length in feet_ weight in lbs. To find the weight qf aA equilateral six-sided cast iron prism. width of side in inches2 X 8.128 X length in feet = weight in lbs.'To find the weight of an equilateral eight-sided cast iron prism width of side in inches2 X 15.1 X length in feet - weight in lbs. To find the weight of a cast iron cylinder. diameter in inches2 X 2.457 X length in feet = weight in lbs. In a quantity of cast iron weighing 125 lbs., how many cubic inches? - By tabular weight per cubic inch - 125 - -.26073 = 479.4 cubic inches. Ans. RELATING TO CAST IRON. 41 Or, by tabular weight per cubic foot - 450.55:'1728:: 125: 479.4 cubic inches. Ans. IHow many cubic inches of copper will weigh as much as 479.4 cubic inches of cast iron? By tabular weight per cubic inch -.3211:.26073: 479.4 389.27 cul;ic inches. Ans. Or, by specific gravities - 8.878: 7.209:: 479.4: 389.27 cul]ic inches. Ants. Or, by tabular ratio of weight -.9258 479.4 X 1.1401 -389.2x. A cast ircn rectangular weight is to be constructed hlavillg breadth of 4 inchles and a thickness of 2 inches, and its w\eigllt is to be 18 lbs.; what must be its length? 1S 4X2X 6 38.63 inches. Ans. 4X2X.26073 A cast iron cylinder is to be 2 inches in diameter, and is to weighl 6 lbs.; what must be its length?.26073 X.7854 =.2047 lb. = weight of 1 cyl. inch, then 6 2X. 2047 - 7.327 inchIes. Ains. A cast iron cylinder is to weigh 6 lbs., and its length is to be 7.327 inches; what must be its diameter?.? 6 \ (7.327 X.2047 )-2 inches. Ans. A cast iron weight, in the form of a prismoid, or the fruslrmln of a pyramid, or the frusirum of a cone, is to be constructed tllat x ill weigh 14 lbs., and the area of one of the bases is to be 16 inchies. and that of the other 4 inches; what must be the length of the weight? 14 V16 X 4 =8 and 8 - 16 + 4 3 9.33, and 9.a3 X'-(;4.-: 5.75 inches. Ans. NOTE.- For Rules in detail pertaining to the foregoing, see GOieoMlml:, Menfl rtii of superficies - of solids. A model for a piece of casting, made of dry white pine, woil,is 7 lbs.; what will the casting weigh, if made of' conmmon brass I By specific gravities -.554 8.604:: 7: 108.71 lbs. Ans. Norz. - As the specific gravity of the substance of which the model is conlllszeul l.eluf grtverally remainn to somine extent uncertain, calculations of this kind can oily he rt:li,.; a..pp roxima te. 4 TABLE E1.;hibiting the Weight of One Foot in. Length of Flat, Rolled Iron; Breadth and Thickness in Inches, Weight in Pounds. r.,and Th. Wci't. Br., and Th. Wei't. Br. and Th. Wei't. Br. and Th. Wei't. inch. Ibs. inch. Ibs. inch. lbs. inch. lbs., by -.211 4l by 4l 3.696 14 by 4 2.-957 24 by 3.5911.422 1 4'224 H 3.696 8 4.488i 0.634 11 4.7522 4 4.435 4 5.386' by.264 14 by.581 6/ 5.175 4 6.284; 4.528 41.161 1 5.914 1 7.1811 4.792 4[1.742 1A 6.653 14 8.079] 4 1.056 I 2.323 1,1 7.393 14 8.977 4 by.316 / 2.904 14 8.132 14 9.874 4.633 4 3.485 14 8.871 1 10.772 I.950 -14.066 14 9.610 /21 by.950 a1.267 1 4.647 14 by.792 4 1.901 I 1.584 14 5.228 4 1.584 1 2.8511 1 by 4.369 14 5.808 41 2.376 hA 3.8021 A.739 12 by R.634 42 3.168 [ 4.752 I 4g 1.108 4 1.267 ] 3.960: 5.703 2 1.478 1 1.901 4.752 6 i 6 653! 1.848 4 2.534 ] 5.544 1 7.604. l/2.218 } 3.168 1 6.336 14 8.554 1 by.422 4 3.802 1 7.129 14-. 9.505.845 4 4.435 [1 4 7.921 14 1.455 1.267 1 5.069 1' - 8.713 1 11.406; 41.690 11 5.703 1 9.505 1 12.8356 | 2.112 14 6.337 1[ 10.297 14 13.307 42.534 ~ by-16.970 143 11.089 24 by 1.003 |g2.957//l~ by [.686 2 by 4.845 4 2.006 14 by.475 4 t1.373 4] 1.690 4 3.010.950 4! 2.059 I 4 2.534 I 4.013 1.425 A 12.746 4 3.379 g 5.016 4 1.901 13.432 4..224 [ 6.019 4 2.376 4.119 4; 5.069 4 7.023 4 2.851 14.805 4 5.914 1 8.026 1 3.326 1 5.492 1 6.759 14A 9.029i 1 3.802 1 6.178 14 7.604 14 10.032 1 by 4.528 14i 6.864 141 8.449 1 11.036 t 1.056 1 7.551 14 - 9.294 14 12.039 41.584 1418.237 1 1110.138 1- 13.042 4 2.112 1 by 1.783 2J b9y 1.898 [ 11 14.046 4 2.640,1l.478 4 1.795 2 16.052 | 3 8.16:8 4 2.218!1 2.693 24 by [ 1.0561 WEiaIIT OF FLAT, ItOLrLED IRON. 43 TABLE. - Continucd. Br. anr. and Th. WWight. Br. and Th.lWeight. Br. an d T h. Weight. ht. Br. n. Weij inch. lbs. inch. lbs. inch. lbs. inch. lbs. 24 by 4 2.112 24 by 13 16.264 34 by 4 6.865 34 by 1t 20.594 4 3.168 1417.426 4 8.238 11 22.178 A 4.224 2 18.587 4 9.610 1-j 23.762 4 5.280 24119.749 1 10.983 2 25.347 4 6.336 24 2*0.911 1 12.356 2 28.515 4 7.393 24 by 4 1.214 1i 18.729 - 24 31.683 1 8.449 2.429 14 15.102 24 34.851 la 9.505 4 3.644 14 16.475 4 by 4 1.690 14 10.561 4 4.858 14 17.848 4 3.379 14 11.617 g 6.073.14 19.221 4 6.759 14 12.673 4 7.287 14 — 20.594 4 10.139 14 13.729 4 8.502 2 21.967 1 13.518 14 14.785 1 9.716 24 24.713 14 16.898 1ti- 15.841 1 10.931 24 27.459 14 20.277 2 16.898 14[12.145 3.4 by _ 1.478 14 23.657 24 by A 1.109 14 13.360 4 2.957 2 27.036 4 2.218 11 14.574 4 4.436 24 30.416 I 3.327 14 15.789 4 5.914 24 33.795 4aj 4.436 14 17.003 I 4 7.393 24 37.175 4 5.545 1I 18.218 4 8.871 3 40.555 4 6.653 - 2 19.432 4 10.350 34 43.934 4 7.762 24 20.647 1 11.828 44 by a 1.795 1 8.871 2i 21.861 14 13.307 4 3.591 1 A 9.980 3 by 4 1.267 14 14.785j 7.181 14 11.089 4 2.535 14 16.264 4 10.772: 14 12.198 4 3.802 1 17.743 1 14.363 14 13.307 4 5.069 1g 19.221 14 17.9541 14 14.416 n 6.837 14 20.700 1a 21.5441 14 15.525 4 7.604 j 122.178 14125.1351 14 16.634 i 8.871 2 23.657 2 28.726 2 17.742 110.139 24 26.614 24 32.317 24 18.851 14 11.406 24 29.571 24 35.908 24 by 4 1.162 1 112.673 2iL 32.528 24 39.498 1 2.323 1:113.941 3: by 4 1.584 3 43.089 4 3.485 14 15.208 4 3.168 3 46.680 / 4.647 14 16.4785 4 4.752 3 50.271 g 5.808 14 17.743 4 6.337| 44 by 4[ 3.80'2 4 6.970 14 19.010.] 7.921 4 7.604 4l 8.132 24120.277 4 9.505 4 11.406 1 9.294 2 22.812 H 11.089 1 15.208 14 10.455 24 25.345 1 12.673 1 14 19.010 14 11.617 34 by A 1.373 14 14.257 1 1 22.812 1. 12.779 4 2.746 141 15.842 14 26.614 14 13.940 4l.119 14 17.426(i 2 30.416 14 15.102 A 5.492 1 119.010 I 24134.218.~~ _. 1,,_9_0_0 44 WEIGHT. OF FLAT, ROLLED IRON. TABLE.- Continuecd. Br. and Th. Weight. IBr. and Th. WVeight. Br. and Th. Weiht. Br. and Th. Weight. inch. lbs. inch. lWs. inch. lbs. inch. lbs. 4. by 2- 38-020 4-4 by 3 48.158 54 by 4 13.307 5.4 by 2 37.175 244 1.822 3 52.172 1 17.743 24 46.469 3 45.624 34156.185 14122.178 3 55.762 34 49.426 5 by 4 4.224 14!26.614 54 by 4 4.858 34 53.22& A 8.449 14- 31.049 4 9.716 44 by i 4.013 4 12.673 2 35.485 4 14.574 A 8.026 1 16.898 24 39.921 1 19.432 4[12.040 1 21.122 24 44.356 14 24.2901 1 16.053 14 25.347 3 53.228 14 29.1461 14- 20.066 1 29.571 54k by 4 4.647 14 34.0071 14 24.079 2 33.795 A 9.294 2 38.865 14 *28.092 2 38.020 4 13.941 24 43.723! 2 32.106 2142.244 1 18.587 2. 48.581 24 36.119 3 46.469 14 23.234 3 58.297 24 40.132 54 by 4 4.436 1, 27.881 6 by 4 5.069 24 44.145 4 8.871 13132.528 WEIGIIT OF METALS IN PLATE. The weight of a SQUARE FOOT one inch thick of Malleable Iron.. -40.554 lbs. Corn. plate ".. 37.761 " Cast Iron... — 37.546 " Copper, wrought.. -46.240 " " corn. pllte. _45.312 " Brass, plate, com..-. - 42.812 " Zinc, cast, pure.. 35.734 " " sheet... -- 37.448 "C Lead, cast... = 59.125 And for any other thickness, greater or less, it is the same in proportion; thus,.a square foot of sheet copper -r- of an inch thick =46.24 - 16 2.89 lbs. And 5 square feet at that thickness - 2.89 X 5 - 14.45 lbs., &c. So, too, 5 square feet at 2j inches thickness = 46.24 X 2.5 X 5 - 578 lbs. AMERICAN WIRE GAUGE. 45 THE AMERICAN WIRE GAUGE. The American Wire Gauge was prepared by Messrs. Brown and; Sharp, manufacturers of machinists' tools, Providence, R. I. It iQ graded upon geomtrical principles, is rapidly becoming the standlard gauge -with manulftclurers of' wire and plate in the United States, and cannot fhil to supersede the use of the Birmingham Gauge in this country. TABLE Showing the Linear Measures represented by NAos. American Wire Gauge and Birmingham Wire Gauge, or the values of the Nos. in the United-States Slandard Inch. Ameriean American Birm. American Birm. American Birm. No. Gauge. Gaugoe No. Uauge. Gauge No. Gauge. Gauge. No Gauge. Giauge. IncJ&. bech. _ Inch. Inch. Inch. Inch. Inch. Inch. 0000.46000.45 8.1284.165 19.03589.042 30.01003.0 12 000.40964142.5l 9.11443.148 20.031961.0351 31.00893.010 00.36480.3801 10 1.10189.134 21.02846.039 32.00795.009! 0.324861.340 11.09074.120 22.02535.028 33.00708.008 1.280301.3001 121.08081.109 23.02257.025 34.00630.007 2 225763!.22841 13:07196.095 24.02010.021 35.00561.005 31.229421.2591 14.06408.083 25.01790[.020 36.00500.004 4'.204311.238:; 15.05707 07[2 26.01594.0181 37.00445 51.181941.220 16.05082.065 27.01419[.016! 38.00396 61.16202.2031] 17.04526.058 28.01264.0141 39.00353 7.144281.180i1 18.04030i.049 29.01126.013; 40.00314 Thus the DIAMETER or size of No. 4 wire, American gauge, is 0.20431 of an inch; Birmingham gauge, 0.238 of an inch: so the'ruT'ICKrNESS of No. 4 plate, American gauge, is 0.20431 of' an inch; Birmingham gauge, 0.238 of' an inch; and so Ibr the other Nos. on the gauges respectively. TABLE Showing the Number of Linear Feet in One Pound, Avoirdluois, oJ Different IKinds of Wire; Sizes or Diamneters corresponding to Nos. American Wire-gauge. No. Iron. Copper. Brass. Iron, Copper. Brass. Feet. Feet. Feet. Feet. Feet. sFeet. 0000 1.7834 1.5616 1.6552 19 293.00 256.57 271.94 000 2.2488 1.9692 2.0872 20 396.41 347.12. 367.92 00 2.8356 2.4830 2.6318 21 465.83 407.91 432.35 0 3.5757 3.1311 3.3187 22 587.35 514.32 545.13 1 4.5088 3.9482 4.1847 23 740.74 648.63 687.50 2 5.6854 4.9785 5.2768 24 934.03 817.89 866.90 3 7.1695 6.2780 6.65421 25 1177.7 1031.3 1093.0 4 9.0403 7.9162 8.3906 26 1485.0 1300.4 1378.3 5 11.400 9.9825 10.581 2 7 1872.7 1639.8 1738.1 6 14.375 12.588 13.342 28 2361.4 2067.8 2191.7 7 18.127 15.873 16.824 29 2977.9 2607.6 2763.8 8 22.857 20.015 21.214 30 3754.8 3287.9 3484.9 9 28.819 25.235 26.748 31 4734.2 4145.5 4394.0 10 -36.348 31.828 33.735 32 5970.6 5221.2 5541.4 11 45.829 40.131 42.535 33 7528.1 6592.0 6987.0 12 57.790 50.604 53.636 34 9495.G 8314.9 8813.1 13 72.949 63.878 67.706 35 11972 10483 11111 14 91.861 80.439 85.258 3G 15094 13217 14009 15 115.86 100.75 107.53 37 19030 16664 17662 16 146.10 127.94 135.GO 38 24003 21018 22278 1 17 184.26 168.35 171.02 39 30266 26503 28091 18 232.34 203.45 215.64 40 38176 33342 35432 NOTE. -In this TABLE the iron and copper employed are supposed to be nearly pure. The specific&gravity of the former was taken at 7.774; that of the latter, at 8.878. The specific gravity of the brass was taken at 8.376. WIRE AND WIRE GAUGES. 47 To find the number of feet in a pound of wire of any material not given in the TABLE, of an size, Amnzerican gauge, its specific gravity being known. RULE. - Multiply the number of feet in a pound of iron wire of the same size by 7.774, and divide the product by the specific gravl ity of the wire whose length is sought; or ordinarily, for steel wire, multiply the number of feet in a pound of iron wire of the same size by 0.991. To find t~he number of feet in a pound of wire of any given No., Birmingham gauge. RULE. Multiply the number of feet in a pound of the same kind of wire, same No., American gauge, by the size, Anmerican gauge, and divide the product by the size, Birmingham gauge. EXAMPLE. - In a pound of copper wire No. 16, American gauge, there are 127.94 feet: how many feet are there of the same kind of wire, same No., Birmingham gauge? (127.94 X.05082).065 = 100.03. Ans. To find the weight of any given length of wire of any given No. or size, A nerican gauge, or the length in any given weight, by help of the foregoing TABLE. EXAMPLE.- Required the weight of 600 feet of No. 18 iron wire. 600 9- 232.34 - 2.5822 lbs. 2 lbs. 9! oz., nearly. Ans. EXAMPLE.- Required the length in feet of 2. lbs. of No. 31 brass wire. 4394 X 2.5 = 10985. Ans. Characteristics of Alloys of Copper and Zinc - Brass. Parts by Weilght. Specific Color, Denomination. Copper. Zinc. Gravity. 83 17 8.415 Yellowish Red. Bath Metal. 80 20 8.448 " " I)utch Brass. 74- 25} 8.397 Pale yellow.. Rolle Sheet Brass. 66 341 8.299 Full " English Sheet Brass. 491 501 8.230 " " German Sheet Brass. 33 67 8.284 Deep " Watchmaker's Brass. NOTE.-To alloys of copper and zinc, generally, there is added a, small quantity of lead, which renders them the better adapted for turning, plan inir, or filing; and, for the same reason, to alloys of copper and tin, there is ust.ually added a snmall quantity of zinc (see ALLOYS AND CoIPOSl'rIToSs). TABLE Showving the Weight of One Square Foot of Rolled Metals, tlickness corresponding to Nos., American WTlire-gauge. Tlhickness. Iron. Steel. Copper. Brass. Lead. Zinc. No. Pounds. Pounds. Pounds. Pounds. Pounds. Pounds. 1 10.849 10.999 13.109 12.401 17.102 110.833 2 9.6611 9.7953 11.674 11.043.15.228 9.6466 3 8.6032 8.7227 10.396'9.8340 13.562 8.5903 4 7.6616 7.7680 9.2578 8.7576 12.078 7.6501 5 6.8228 6.9175 8.2442 7.7988 10.755 6.81'26 6 6.0758 6.1601 7.3416 6.9450 9.5779 6.0667 7 5.4105 5.4856 6.5377 6.1845 8.5291 5.4024 8 4.8184 4.8853 5.8222 5.5077 7.5957 4.8112 9 4.2911 4.3507 5.1851 4.9050 6.7645 1 4.2847 10 3.8209 3.8740- 4.6169 4.3675 6.0233 3.8151 11 3.4028 3.4501 4.1117 3.8896 5.3642 3.3977 12 3.0303 3.0720 3.6616 3.4638 4.7770 3.0257 13 2.6985 2.7360 3.2607 3.0845 4.2539 2.6934 14 2.4035 2.4365 2.9042 2.7473 3.7889 2.3999. 15 2.1401 2.1698 2.5829 2.4463 3.3737 2.1369 1G 1.9058 1.9322 2.3028 2.1784 3.0043 1.9029 17 1.6971 1.7207 2.0506 1.9399 2.6753 1.6945 18 1.5114 1.5324 1.8263 1.7276 2.3826 1.5091 19 1.3459 1.3646 1.6263 1.5384 2.1217 1.3439 20 1.1985 1.2152 1.4482 1.3700 1.8893 1.1967 21 1.0673 1.0821 1.2897 1.2300 1.6768 1.0657 22.95051.96371 1.1485 1.0865 1.4984.94908 23'.84641.85815 1.0227.96749 1.3343.84514 24.75375.764221.91078.86158 1.1882.75262 25.67125.68057.81109.76728 1.0582.670226.59775.60605.72228.683261.94229.59685 27.53231.53970.64345.608461.83913.53151 28.47404.48062.57280.54185.74728.473333 29.42214.42800.51009.48242.66546.421511 30.375941.38116.45426.42972,.592(;3.375381 NoT'. -In calculating the foregoing TABLET,, the specific gravities were taken as follows: viz., iron, 7.200; steel, 7.300; copper, 8.7'00; brass, 8.230; lead, 11.350; Zine, 7.189. TIN PLATES. 49 TIN PLATES. F B |Size of |o. of Nt Sizc of No. of N|et Brand Sheets in Shects Wleight Brand Marks. Sheets in Sheets Weightlt Marks. Inches. in Box. in lbs. Inches. in Box. il ls. 10C 14 X 1-1 ) 10 iS)XX 105 X 11 200 21 IC'. 14X10 225 1Il2 SDXXX 15 X 11 200 2 11-C 14 X 10 22) t119 SDXXXX 15 X 11 200,) I!IX 14 X 10 225 117 TT' 14 X 10 225 11i 14 X 10 225 140 " ICI12 X 1 225 1!2 1XX 14 X10 22 161 " IX'12X 12 225 147 IXXXX 14 10 22 182 4 IXXj12 X 12 225' 1(; IXX XX 14 X 1 0 9 )5 203 I XXX 12 X 1 2 225 18 IX 1 X 14 200 174 " IXXXX!12 X 12 225 210 1lXX 14 11 " 00 200 1 IC'20 X 141 112 11' I )C0 17X12&I 100 10o I X;20 X 14 112 140 i DX, 17 X 12 100 16 112 161 I DXX 17X121 100 147 1 IXXX20X 14 112 182 DXXX 17 X 12 100 168 1"IXXXXi20 X 14 112 203 I)XXXXI17 12L 100 189 liTernes IC.20X< 14 112 1!2 18l_)0 15X 11 200 168 " IX20X14 112 140;SD X 15 X 11 200 189]. __ _ I NO'iTE. -The above'TAr1iE includes all the regular sizes and qualities of' tiln pliltes, except " wasters." Other sizes, such as 10 X 10, 11 X 11, 13 X 13, &c., of the dilercnt; bra.nds, are often imported into the United Statcs to order. Common English Sheet Iron, Nos. 10 to 28, Birmingham gauge, widths from 24 to 36 inches. R1. G. Sheet Iron, Nos. 10 to 30, Birimigliam gauge, widths from 21 to 36 inches. American Puddled Sheet Iron, Nos. 22 to 28, Birmingham gauge, widths from 24 to. 36 inches. Russia Sheet Iron, Nos. 16 to 8 inclusive, Russia gau,e, sheets 28 X 56 inches. Sheet Zinc, Nos. 16 to 8, Liege giuge, widths from 24 to 40 inches; length 84 inches. Copper Sheathing, 14 X 48 inches, 14 to 3'2 oz. (even numbers), I"'r square foot. Yellow Metal, in sheets, 48 X 14 inches, 14 to 32 oz. (even nullh1el:rs), per square foot..5 TABLE Shlowvisg the Capacity, in Wline Gallons, of Cylindrical Cans, ej different diameters, at Otne Inch depth. Diameter in Inchcs. |Dian'r. Gallons. Dianl'r. Gallons. Dialn'lr. Gallons. Diasn'r. Gallols. inches, inches. inches. inches. 6.1224 124.5102 184 1.164 241- 2.083 6i.1328 12..5313 18I 1.195 25 2.12.5 6.1437 1 2I.5527 19 1.227 254 2.167 6:1.1549 13.5746 1914 1.260 25. l 2.2-11 7.1666 134.5969 19A 1.293 25 2.25 | 74 1.1787 134.6197 1951} 1.326 26 2.298 74i.1913 131.6428 20 1.360. 26I 1 2.343 4 V 7i.2042 14.6664 201 1.394 264 2.388 8.2176 1 14.690 204 1.429'3 261 2.433 84.2314 144.7149 20:1 1.4'64 27 2.479 84.2457 14,.7397 21 1.499 274 2.524 8~.2603. 15.7650 214 1.535 274 2.571 9.2754 154.7907 211 1.572 271 2.518 9:.2909 154.8169 211 1.608 28 2.666 O9.3069 151.8434 22 1.646 284 2.713 9'.3233 16.8 04 224 1. 83 28A 2.762 10.3400 164.,8978 224 1.721 281 2.810 104.3572 164.9257 221 1.760 29 2.859 104.3749 161.9539 23 1.799 291 2.909 101J.3929 17.9826 234L 1.837 294 3.009 11.4114 i 174 1.0120 234 1.877 30 3.060 114.4303 174 1.0410 231 1.918 304 3.163 11l..4497 171 1.0710 2- 1.958 31 3.264 114.4694 18 1.1020 244 1.999 314-A 3.374 12.4896 184 1 1.1320 244 1 2.0-41 32 3.482 Applications of theforeyoing TABLE. EXAMPLE. - A eylindrical can is 11- inches in diameter, and its dlpth is 18 3 inches; required its capacity..4303 X 18-7 = 8 gallons. Ans. EXAMPLE. - The diameter of a can containing oil is 261 inches, and thl1e oil is 141 inches in depth. IIow many gallons are there of the oil? 2.388 X 14A =34.6 gallons. Ans. l:x.MILE. - A can is to be constructed that will hold just 36 gal-. *o',.s.;-ad its diameter is to be 18 inches; what must be its depthl? 36 - 1.102 = 32. inches. Ans. CAPACITY OF CYLINDRICAL CANS. 51 EXAMPLEU,. -A cylinldrical can is to b:) constructed that shall have a deptl of 15 inches ald a caIpacity of just 5 gallons; what must be its diatmetcr? 5 + 15-.3333 = capacity of can in gallouln for each inch of depth; antd against.3333 gallon in the table, or the quantity in gallons nearest thereto, is 10 inches, the required, or nearest tabular diameter. Ans. No'rpE. - The table is not iktended to meet demands of the nature of the one contained in the last example, with accuracy, unless the fractional plart of the diameter, if there be a flictional part, is ~, ~ or j inch. As, however, the diameter opposite the tabular gallon nearest the one sought, even at its greatest possible remove, can be but about L inch from tle diameter required, we can, by inspection, determine the. diameter to be taken, or true answerI' to the inquiry, sufficiently near for practical purposes, be the fraction what it may. Or, to throw thle demand into a mathematical formula: As the tabular gallon nearest the tone sought is to the diameter opposite, so is the tabular gallon requirecd to the required diameter, nearly. Thus, in answer to the last query,.3400: 10:: 3333: 9.8 inches, the required or true diameter, nearly. F:)r a mathematical formula strictly applicable to this questfion, see GAUcINu Or, for a formula more strictly geometrical, we have Capacity X 231 Depth X.7854 The true diameter, therefore, for the supposed can, is 231 X ches.,%/l' i5.7m — = 9.9 — inches. 52 VWEIGIIT OFI rlPES. WAIlGIITr OF PIPES. The weight of ONE FOOT IN LENrJG'H of a pipe, of any diameter and thickness, may be ascertained by multiplying the square of its exterior diameter, in inches, by the weight of 12 cylindrical inches of the material of which the pipe is composed, and by multiplying the square of its interior diameter, in inches, by the same factor and subtracting the product of the latter from that of the former, - the remainder or difference will be the weight. This is evident from the ftact that the process obtains the weight of two solid cylinders of equal length, (one foot,) the diameter of one being that of the pipe, and the other that of the vacancy, or bore. For very large pipes, the dimensions m'ayI be taken in feet, and the weight of a cylindrical foot of the material usedl as the factor, or multiplier, if desired. T're weight of' 12 cylindrical inches (length 1 foot, diameter I inch) of Malleable Iron = 2.6543 lbs. Cast Iron = 2.4573 " Copper, wrought, = 3.0317 " Lead " = 3.8697 " Cast Iron —I cyl. foot- = 353.86 " T'heretfre — E'XAMPLE.- Required the weight of a copper pipe whose length is 5 feet, exterior diameter 31 inches, and interior diameter 3 inches. 3 - X - X 10.5625 X 3.0317 = 32.022 + 3 X 3 = 9 X 3.0317 -27.285 + Ans. 4.737 X 5 = 23.685 lbs. ExAMPLE.- Required the weight of a cast iron pipe, whose lenglth is 10 feet, exterior diameter 38 inches, and interior diameter 3 feet. 382 X 2.4573 - 36'2 X 2.4573 = 363.68 X 10 = 3636.8 lbs. Ans. Or, 38' - 362 = 148 X 2.4573 = 363.68 X 10 - 3636.8 lbs. Ans. gEXAMPLE. - Required the weight of a lead pipe, whose length is 1200 feet, exterior diameter - of an inch, and interior diameter -1of an inch.; X7 4- 9.765625, and 9 X 9- 28!!=.316406, aid.765625 -.316406 -.449219 X 3.8697 X 1200 = 2086 lbs. Ans. EXAMPLE. -The length of a cast-iron cylinder is 1 foot, its exterior diameter is 12 inches, and its interior diameter 10 inc('hes: required its weight. 122-_ 10'2 — 44 X 2.4573 = 108.12 lbs. Ans. Or, 144: 353.86: 44: 108.12 lbs. Ans. WEIGIHT OF PIPES. 53 Thefollowing TrABLE exhibits the coefficients of weight, in pounds, oJ onefoot in length, of various thicknesses, of diffJerent kinds of pipe, (J any diameter whatever. Thickikess Wrought in Inches. Iron. Copper. Lead. 1.332.379.484 r.664.758.9675.995 1.137 1.451 1.327 1.516 1.935 1.658 1.894 2.417 3 1.99 2.274 2.901 7~! 2.323 2.653 3.386 1 i 2.654 3.032 3.87 5T5 3.318 3.79 4.837 3.981 4.548 5.805 CAST IRON.'Thickness. Factor. Thickness. Pactor. Thickness. Factor. 13 1.842 6.143 14 12.287 2.457 7.372 1- 14.744 ~4.13.686 8.C 14 17.201 1 4.901 1 9.829 2 19.659 To obtain the -weight of pipes by means of the above TABLE -:ULl. - llMultiply the diameter of the pipe, taken from the interior stirfiLce of lhe metal on the one side to the exterior surface on the o!p)posite, (interior diameter + thickness,) in inches, by the numbler in the table under the respective metal's name, and opposite the thickness corresponding to that of the pipe -the product will be the vweight, in pounds, of ONE foot in length of the pipe, and that product unltiplied by the length of the pipe, in feet, will give the weight for -nly length required. EXAMPLE. - Required the weight of a copper pipe whose length is 5 feet, interior diameter and thickness 3} inches, and thickness A of an inch. 3- _ - 3.125 X 1.516 X 5 = 23.687 lbs. Ans. EIXAMPLE. — Required the weight of a cast iron pipe, 10 feet in Itergth, whose interior diameter is 3 feet, and whose thickness is I inch. 36 + 1 -- 37 X 9.829 X 10 3636.73 lbs. Ans. 5* 54 WEIGHT OF BALLS AND SHELLS. WEIGHT OF CAST IRON AND LEAD BALLS. To finld the weight of a slphere or globe of any materialRULE. - Multiply the cube of the diameter, in inches, or feet, by the weight of a spherical inch or foot of the material. The weight of a spherical inch of Cast Iron. -.1365 lbs. Lead.. -=.215 " Therefore -EXAMPLE. — Required the weight of a leaden ball whlose diameter is 4 of an inch. 1 X 4 X 4- 1 =.015625 X.215.00336 lb. Ans. EXAMIPL'E.- Required the weight of a cast iron ball whose diameter is 8 inches. 8' X.1365 = 69.888 lbs. Ans. EXAMPLE. - ow many leaden balls, having a diameter 4 of an inch e;ceh, are there in a pound? I -1-.00336 - - 10_00 =00 298. Ans. EXAMPLE. - What must be the diameter of a cast iron ball, to weigh 69.888 lbs? 69.888-.1365 - //512 = 8 inches. Ans. EXAMIPLE. -What (must be the diameter of a leaden ball to equal in weight that of a cast iron ball, whose diameter is 8 inchles? [Lead is to cast iron as.215 to.1365, as 1.575 to 1.] 83_ 512 + 1.575 - =4325 -= 6.875 inches. Ans. WEIGHT OF HOLLOW BALLS OR SHELLS. The weight of a hollow ball is the weight of a solid ball of the same diameter, less the weight of a solid ball whose diameter is that of the interior diameter of the shell. EXAMPLE. — Required the weight of a cast iron shell whose exterior diameter is 6.1 inches, and interior diameter 44 inches. 61 _ 2 _ X 5 X 25 -='244.14 X.1365- 33.33 4, =4 253 X.1365 - 10.48 22.85 lbs. Ans. Or, If we multiply the difference of the cubes, in inches, of the.two diameters - the exterior and interior —by the weight of a spherlical inch, we shall obtain the same result. ExAMPLE. - Required the weight of a cast iron shell \.whose ex terior diameter is 10 inches and interior diameter 8 inches. 10( -83 X.1365 -- 66.612 lbs. Ans. ANALYSIS 01? t0ALA1 ANALYSIS OF COALS. Description. Volatile Matter. Carbon. Ash. Breckinridge, Ky., 62.25 29.10 8.65' Albert," N. B., 61.74 32.14 6.12 Chippenville, Pa., 49.80 Kanawha, " 41.85 Pittsburg, " 32.95 Cannel, sp. gr. 1.4 35.28 64.72 Newcastle, 24.72 75.28 Cumberland, 18.40 80. 1.60 Anthracite, a'v'g., 3.43 89.46 7.11 Woods of most descriptions vary little from 80 per cent. volatile matter, and n20 per cent. charcoal. TABLE. — Exhibiting the Weights, Evaporative Powers, 4c., of Fuels, from Report of Professor Walter R. Johnson. Lbs. of Water Weishat tit 212 derees Lbs. of Water Weigwht of prfconverte itor at 212 degrees Clinkers Designaition of Ft'le l. |c c itovertce i ntvo lonkers Ft i Chic Steam by I converted into fromin 100 Ibs. Ity. Foot. Cubic i Foot of Steam by I lb. of Coal. Fltel. of Fuel. ANTHRACITE COALS. Beaver Meadow, No. 3 1.610 54.93 526.5 9.21 1.01 Beaver Meadow, No. 5 1.554 56.19 572.9 9.88.60 Forest Improvement 1.477 53.66 577.3 10.06.81 Lackawanna _ 1.421 48.89 493.0 9.7.9 1.24 Lehigh 1.590 55.32 515.4 8.93 1.08 Peach Mountain 1.464 53.79 581.3 10.11 3.03 BrITUMINOUvs COALS. Blossburgh 1.324 53.05 522.6 9.72 3.40 Cannelton; Ia. 1.273 47.65 360.0 7.34 1.64 Clover H1ill 1.285 45.49 359.3 7.67 3.86 ICumberland, average, 1.325 53.60 552.8 10.07 3.33 -Liverpoo1 1.262 47.88 411.2 7.84 1.86 Midlothian 1.294 54.04 461.6 8.29 8.82 Newcastle 1.257 50.82 453.9 8.66 3.14 Pictou 1.318 49.25 478.7 8.41 6.13 Pittsburmh 1.252 46.81 384.1 8.20.94 Scotch 1.519 51.09 369.1 6.95 5.63 Sydney 1.338 47.44 386.1 7.99 2.25 Cor,6. Cutnberland 31,57 284.0 8.99 3.55 Midlothian 32.70 282.5 8.63 1().5 Na.tural Virginia 1.323 46.64 407.9 8.47 5.31 Wood. irv P'ine Wood 21.01 98.6 4.69 56 M:ENSUtntATION OF' LUMlLPEFLo MENSURATION OF LUMBER. To find the contents of a board. RULE.- Multiply the length in feet by the width in inches, and divide the product by 12; the quotient will be the contents in square feet. EXAMPLE. -A board is 16 feet long and 10 inches wide; how many square feet does it contain. 16.X 10 = 160 12 = 13-4. Ans. To find the contents of a plank, joist, or stick of square timber. RULE. - Multiply the product of the depth an d width in inches b}y the length in feet, and divide the last product by 12; the quotient is the contents in feet, board measure. EXAMPLE.-A joist is 16 feet long,, 5 inches wide, and 2A inches thick; how many feet does it contain, board measure? 5 X 2.5 X 16 - 12 =16 —. Ans. To find the solidity of a plank, joist, or stick of square timber. RULE. -Multiply the product of the depth and width in inches by the length in feet, and divide the last product by 144; the quotient will be the contents in cubic feet. EXAMPLE. - A stick of timber is 10 by 6 inches, and 14 feet in length; what is its solidity? 10 X 6 - 60 X 14= 840 * 1.14 =5, feet. Ans. NOTE. — If aboard, plank, or joist is narrower at one end than the other, add the two ends together and divide the sum by 2; the quotient will be the mean width. And if a stick of squared timber, whose solidity is required, is narrower at one end than the other (A + a + — A n) — 3 = mean area. A and' a being the areas of the ends. To measure round timber. RULE (IN GENERAL PRACTICE.) -- Multiply the length, in feet, aby lhe square of i the girt, in inches, taken about ~ the distance from the larger end, and divide the product by 144; the quotient is considered the contents in cubic feet. For a strictly correct rule for neasuring round timber, see A;lENSURATION OF SOLDs- Frustum ofa Cone. EXAMPLE.- A stick of round timber is 40 feet in length, and girts 88 inches; what is its solidity? 88 2 X 4=22 X 22 484X40=193G60+ 144 =134.44 cub. ft. Ans. MENSURATION OF LUNBER. 57 TIe following TABLE is intended to facilitate the mleasuring of Round Timber, and is predicated upon theforegoing RULsE. r |i Crilt in Area in Girt iln Area illn iGirtin Area in I Girt inl Area in Inc hes. Feet. Inches. F'eet. Inches. I'eet. Inches. Feet. 6.25 12 1. 18 2.25 24 4. 6,.272 124 1.042 1841- 2.313 244 4.084 6.A.294 12I 1.085 18.4 2.376 24& 4.168 6!:1.317 12i 1.129 184 2.442 244 4.254 7.34 13 1.174 19 2.506 25 4.34 74.364 13. 1.219 194 2.574 25:- 4.428 7.39 134 1.265 194 2.64 25& 4.516 7:4.417 134 1.313 194 I 2.709 254 4.605 8.444 14 1.361 20 2.777 26 4.694 8.472. 144 1.41 204 / 2.898 26i 4.785 8.501 144 1.46 20. 2.917 26. 4.876 8.531 144 1.511 204 2.99 264 4.969 9.562 15 1.562 21 3.062 27 5.062 94.594 154, 1.615 214 3.136 27& 5.158 94.626 15. 1.668 21 3.209 274 5.252 94 659 154 1.722 214 3.285 27 65.348 10.694 16 1.777 22 3.362 28 5.444 104.73 164t 1.833 224 3.438 284 5.542 10.766 164 1.89 224 3.516 28t1 5.64 - 10.803 164 1.948 224 3.598 284 5.74 ] 1.84 17 2.006 23 3.673 29 5.84 114l.878 174 2.066 23.1 3.754 294 5.941 11.918s 174 2.126 234 3.835 294 6.044 114.909 174 2.187 231 3.917 30 - 6.25 To find the solidity of a log by help of the preceding TABLE. RULE. - Multiply the tabular area opposite the corresponding { girt, by the length of the log in feet, anad the product will be the solidity in feet. EXAMPLtE. -The u girt of a log is 22 inches, and the length of the log is 40 feet; required the solidity of the log. 3.362 X 40 = 134.48 cubic feet. Ans. NOTE. - Though custom has establishedl, in a very general way, the preceding method as tha:t whereby to measure round timber, and holds, in most instances, the solidity to be that which the method will give,.thstre seems, if the object sought be the real solidity of the stick, neither accuracy, justice, nor certainty, in the practice. Thus, in the l)prcediilg exarnple, the stick was supposed to be 40 feet in length, and 88 inches in cirlculference at I the distarne from the larger end, an(l was found, I)3y the method, to contain 1:4.44- culbic feet: now 88 -- 3.1416 = 28 inches, - the diameter at - the distance from tile greater base, anti retaining this diameter ond the lengtht, we may 58 ME8fNStflATiON 01F LtUIBEIt. Slluppo)')se, withl sliifiient lilerallity, ani; without h)ein fila from the general run of such sticks, the diat!ll etcr at the greater base to be 30. inches, and tha't of the less to be 24 i tches, aIl -I Bly a correct rule the stick contains - 30 X 2=-720- 12=732 X.7854 X 40=22996- 144-=159.7 cubic feet, or 19 per cent. more than given bIy the method under consideration a anld we need hardly ad(ld that the nearer the stick approaches to the figure of a cylinder, the wider will be the difference bletween the truth and the result obtained by the method referred to. Thus, supl)ose the stick a cylinder, 28 inches in diameter, and 40 feet in length; and we halve, by tile fallal: cious rule, as above, 134.44 cubic feet and - By a correct method, we have - 282 X.7854 X 40 = 24630 * 144= 171 cubic feet, or over 27 per cent. more than fur. nished by-the erroneous mode of practice. Again: suppose the stick in the form of a cone, 30 inches at the base, and tapering to a point at 150 feet in length; and we have, by a correct rule - 30 - 3=300 X.7854 X 150 =33534:3 - 144 — 245.44 cubic feet; and by the ordinary method of gauging, or the aforementioned practice, we have - 20 X 3.1416- 62.832' 4= 15.7082 X 150 = 37011.19 -- 144= 257 cubic feet, or nearly 4: per cent. more than the stick actually contains. In short, without taking into account anything for the thickness of the bark, that may be supposed to be on the stick, the method is correct only when the stick tapers at tile rate of 5.j inches diameter per each 10 feet in length, or over ~ inch diameter to each foot ill length of tile stick. If, however, we suppose the stick as before, (30 inches at the greater base, 24 inches at the smaller, and 40 feet in length,) and suppose the bark upon it to be 1 inch thick, we s1alll have, by the usual method, 134.44 cubic feet, as before. And, exclusive of the bark, b)y a correct method, we shall have. 30 - 2 X 24 - 2 - 616 + 12 = 628 X.7851 X 40 = 19729 - 144 = 137 cubic feet,. or only about 2 per cent. more than that furnished us by the usual practice. The following simple rule for measuring round timber is sufficiently correct for most practical purposes:RULE. - Multiply the square of one-fifth of the mean girt, (exclusive of bark,) in inches, by twice the length of the stick in feet, and divide the product by 144; the quotient will be the solidity in feet. To find the solidity of the greatest rectangular stick that may be cut from a given log, or from a stick of round timiber of given ditnenszons. RULE.- Multiply the square of the mean diameter of the log, in inches, by half the length of the log, in feet, and divide the product by 144. EXAMPLE. - The diameter (exclusive of bark) of the greater base of a stick of round timber is 30 inches, and that of the less base is 24 inches, and the stick is 40 feet in length; required the solidity of the greatest rectangular stick that may be cut from it. 30 X 24 + - (30 - 24)2 - 732 = square of mean diameter,* and 732 X 20 = 14640 144= 101. cubic feet. Ans. E ixcept in the case of a cylinder, there is a difference betwixt the mean diameter of a s:,lil haviing circulhtr bases, and the middle diameter of that solidl. The mean diameter reittlues the solitl to a cylilnder; the middle diameter is the di:meter midway between hlh, ttai ba:ses. MUNfSIATION OI L,1iN t:Oil1. 59 NoT. - The foregoing stilk will make 14640 -. 16 915 feet of square-edged boar(Is I i:h1 tlliel; Or, 101! X 9 —915. To find the solidity of lhe greatest square stick that may be cutl from a given log, or from a slick of round limber of given dimensions. RuLE. M- ultiply the square of the diameter of the less end of tile log, in inclhes, by half the length of the log, in feet, and divide the product by 144. EXAMPLE.- The preceding supposed log will make a square stick containing - 242 X 4o-_1152 - 144 =80 cubic feet. Diamncter multiplied by.7071 = side of inscribed square. To find the contents, in Board Measure, of a log, no allowance being made for wane or saw-chip. RILE. - Multiply the square of the mean diameter, in inches, by the length in feet, and divide the product by 15.28. Or, Multiply the square of the mean diameter in inches, by the length in feet, and that product by.7854, and divide the last product by:12. The cub)ic contents of a log multiplied by 12, equal the contents of the log, board measure. The convex surface of a Frustum of a Cone - (C + c) X A slant length; C being the circumference of the greater base, and c the circumference of the less. tj60.GAUGING. GAUGING.'RULES for finding the capacity in gallons or bushels of different shaped Cisterns, Bins, Caslcs, c4c., and also, by way of examples, for constructing them to given capacitics. RULE - 1. When the vessel is rectangular. Multiply the interior length, breadth, and depth, in feet together, and the product by the capacity of a cubic foot, in gallons or bushels, as desired for its ca pacity. RULE - 2. When the vessel is cylindrical. Multiply the square of its interior diameter in feet, by its interior depth in feet, and the product by the capacity of a cylindrical foot in gallons or bushels, as desired for its capacity. RULE -3. When the vessel is a rhombus or rhomboid. Multiply its interior length, in feet, its right-angular breath in feet, and its depth in feet together, and the product by the capacity of a cubic foot in the special measure desired for its capacity. RULE - 4. When the vessel is afrustum of a cone - a round vessel larger at one end than the other, whose bases are planes. Multiply the interior diameter of the two ends together, in feet, add B the square of their difference in feet to the product, multiply the sum by the perpendicular depth of the vessel in feet, and that product by the capacity of a cylindrical foot in the unit of measure desired for its capacity. RULE- 5. TVhen the vessel is a prismoid or the frustum of any r egular pyramid. To the square root of the product of the areas of its euds in feet, add the areas of its ends in feet, multiply the sum by - its perpendicular depth in feet, and that product by the capacity of a cubic foot in gallons or bushels, as desired for its capacity. If it is found more convenient to take the dimensions in inches, do so; proceed as directed for feet, divide the product by 1728, and multil)ly the quotient by the capacity of the respective foot as directed. Or, multiply the capacity in inches by the capacity of the respective inch in gallons or bushels; -by the quotient obtained by dividing the capacity of the respective foot in gallons or bushels by 1728 - for the contents. RULE -6. When the vessel is a barrel- hogshead, pipe, 4'c. Multiply the difference in inches between the bung diameter and head diameter, (interior,) if the staves be much curved, by.7 medium curved,. by.65 straighter than medium, by.6f See page 63. nearly straight,. by.55 J and add the product to the head diameter, taken in inches; then multiply the square of the sum by the length of the cask in inches, and divide the product by the capacity in cylindrical inches of a gallon or GAUGING. 61 bushel las desired for the contents. Or, divide the contents in cylindrical inches, as above found, by 1728, and multiply the quotient by the capacity of a cylindrical foot in gallons or bushels as desired for its contents. Or, multiply the capacity in cylindrical inches by the capacity of a cylindrical inch, in gallons or bushels, as desired,that is, by the quotient obtained by dividing the capacity of a cylindricall foot in gallons or bushels, by 1728, for the contents. Tlhe capacity of a CUBIC FOOT- CYLINDRICAL FOOT 7.4805 Winchester wine gallons. 5.8751 Winchester wine gallons. 6. 1276 Ale 4.8126 Ale 6.2321 Imnperial " 4.8947 Inlperihal.80356 Winchester bushel..63111 Winchester buslhel..62888 " heaped ".49391' heaped' ".64285 " - 1 even.5089 " 1 even ".779 Impcrial " ".61183 Ilnperial EXAMPLE. — Rcquire'd the capacity in Winchester bushels of a rectangular bin, whose interior length is 12 feet, breadth 6 feet, and depth 5 feet. 12 X 6 X 5 X.8035 = 289.26 bushels. Ans. EXAMPLE. -Required the ca.pacity in Winchester wine gallons of a cylindrical can, whose interior diameter is 18 inches, and depth 3 feet. 18 X 18 X 36X 5.875 1723 = 39.66 gallons. Ans. Or, 1.5 X 1.5 X 3 X 5.875 = 39.66 gallons. Ans. Or, 18 X 18 X 36 X.0034 = 39.66 gallons. Ans. EXaMPLE. -IIow many Winchester bushels in 39.66 wine gallons. 39.66 X.10742 = 4.26 bushels. Ans. EXASMPLE. - Iow many wine gallons in 4.26 Winchester bushels? 4.26 X 9.3092 = 39.66 gallons, Ans. EAMPLE. - IIOW mnany wine g;allons will a cistern in the form of a frustum of a cone hold, having the interior diameter of one of its ends 6 feet, and that at the other 8 feet, and its perpendicular depth 9 feet? 8 - 6 = 2, and 22 3 = 1.333 = ~ square of dif. of diameters, and 6 X 8 + 1.333 = 49.333 X 9 X 5.8751 = 2608.55 gals. Ans. Or, 6 X 8 - S 8 — 6' = 148 X 9 X 5.8751 = 2608.55 gals. Ans. Or, (8' - 6:) ~ (8 -6) = 148 X 9 X 5.8751= 2(;08.55 gals. l nAs Or, 96 - 72 = 2 and (242 -- 3) = 192, and 96 X 72 + 1`92 = 7104 X 108 X.0034 = 2608.55 gals. Ans. G 62 GAUGING. EXAMPlE. - WVIhat is tlhe cappacity in Winchester bushels of a cistern whose form is prismoid, the dimensions (interior) of one end blcing'8 bly 6 ibet, of the other 4 by 3 feet, and its perpendicular depth 12.f tbt 8 X = 48 = area; of one end, and 4 X 3- 12 area of the other end; tithen 48 X24 48 12 = 58 6 4 +X 4 1 X.80356 270 bush els. A ns. O(r, (8 + 4) 2 = 6(, and (G -- 3) 2 2=4.5 = m-ean sectional areas of onids, an 6 X 4.5 X 4 -=4 area of mean perimeter, then 8 X; + 4 X 3 -- X 4.5 X 4=168 X J2- X.80356 =270 bus. Ans. ExAml'L. - What must be the depth of a rectangular bin whose length is 12 feet, and breadth 6 feet, to hold 289.26 bushels I 289.26 +- (12 X 6 X.80356)- 5 feet. Ans. EXAMPLIE. -A cylindrical can, whose depth is to be 36 inches, is required to be made that will hold 40 gallons; what must be the dialneter of the can? 40 -- (3 X 5.8751) - /2.27 = 1.506 feet. Ans. Or, 40 -+ (36 X.0034) - /323.8 = 18.07 inches. Ans. ExAm'ILE. - A cylindrical can, whllose interior diameter is to be 18 inches, is required that will hold 40 gallons; what must be the interior depth of the can? 40 -(182 X.0034) = 36.31 inches. Ans. Or, 40. (1.52 X 5.8751) = 3.026 feet. Ans. ExAM'LE. - A cistern is to be built in the form of a frustum of a cone, thant will hold 1800 gallons, and tbe diameter of one of its ends is to be 5 feet, and that of the other 7.A feet; what must be the depth? 7.5 - 5 = 2.5, and 2.5 + 3 2.0833 a= square of difference of diameter, and 1800 - (7.5 X 5 + 2.0833) X 5.8751 = 7.74 feet. Ans. /7.-'5 X 5 7.52 + 52 _ Or, 1800- 3 X 5.8751 7.74 feet. Ants. EXAS,.\LE. -The form, capacity, depth, and diameter of one end being determined on, and being as above, what must be the dialleter -1 tihe other end? -- h —3 = y, c being the solidity in cylindrical mcasuxetcmlt. /h GAUGING. G3 the depth, d thile diameter of the given end or baLse, and y a quantity tie square root of which is the sum of the required baso and half the given base; then 1800 + 5.8751 = 306.378 = solidity in cylindrical feet, and 306.378 +- = = 118.75- (52' - ) -- /100- 10 - 7.5 feet. Ans. iEXamiLE.- A measure is to be built in the form of a frustum of a cone, that will hold exactly 1 wine gallon, and the diameter of' one of its ends is to be 4 inches, and that of the other 6 inches; what must be its depth? 1 - (6 X 4 + 11i) X.0034 = 11.61 inches. Aits. 231 6 X4-+-62-+-42 Or,.7854 + 11.61 inches. Ans EXAMPLE.- A measure in the formn of a frustum of a cone holds 1 wine gallon the diameter of one of its-ends is 6 inches, and its depth is 1.1.61 inches; what is the diameter of the other end? 3 - 941 1 1.61L.= 76 (62 ) 49 7 _ 64. —! 8 5 z -— 2,'t.1176 --'-. 2 - 4 inches. Ans. CASK GAUGING. CASKC-GAUGING, in a general sense, is a practical art, rather than a scicntific achievement or problem, and makes no pretensions to strict accuracy with regard to the conclusions arrived at. The aimn is, by mneans of a few satisfactory measurements taken of the outside, and an estimate of the probable mean thickness of the material of which the cask is composed (of' which there must always remain some doubt), or by means of a few measurements taken of' the inside, to detcermine, 1st, the capacity of the cask, and, 2d, the ullage, or capacity of the occupied or unoccupied space in a cask but partly full. And the Rule (RULE 6, page 60), which reduces the supposed cask, or cask oft' supposed curvature, to a cylinder, is as practically correct for the capacity of ordinary casks, as any rule, or set of rules, that can be offlrcd for general purposes. Casks have no fixed fbrm of their own, to which they severally and collectively correspond, nor are they in any considerable degree in conformity with any regular geometrical fi(gure. Some casks a few - those having their staves much curved througalout their entire length, Tare nearest in keeping with the middle frustum of a spheroid; others, slightly less curved' than the preceding, correspond in a considerable degree to the middle 64 GAUGING. fiustum or' a pdrabolic spindle; others, again —those having very little lonritudinal curvature of stave to their semi-lengths —are nearly in'~eeping with the c7nidl rustums of a paraboloid; and others - a very few - those whose staves are straight from the bung dia.m. eter to the heads, or equal to t!iat form, are in accordance w ith the equalfrustums of a cone. The gauging rod, which is intended to be correct fol caisks Of. thie most common form, gives for all casks, as may be seen in one coi' the following EXAM PLES, a solidity slightly greater (about 2 l per Cent.) than would, be olbtained by-supposing the cask in conformity witli the third figure above alluded to. The RULE for finding the contents of a cask, byfour dimcnsions, hereafter to be given, is intended as a general Rule for all casks, and, when the diameter midway between the bung and head can be accurately ascertained, will lead to a-very close approach to the truth. From the length of a cask; taken fronm outside to outside of the heads, with callipers, it is usual to deduct from 1 to 2 inches, to correspond with the thickness of the heads, according to the size of the cask, and-the remainder is taken as the lengtl of the interior. To the diameter of each head, taken externally, from i inch to J-th inch should be added for common-sized barrels,,-1 inch for 40 gal laon casks, and from j inch to T-6 inch for larger casks, to correspond with the interior diameters of the heads. If the staves are of uniform thickness, any sectional diameter of a cask may be nearly or quite ascertained, by dividing the circumference at that place by 3.1416, and subtracting twice the thickness of the stave from the quotient. For obtaining the diagonal of a cask by mathematical process,the interior length, &c. &c. -see Rules, below. In the following formulas D denotes the bung diameter, d the head diameter, and I the length of the cask. T'ile solidity of any cask is equal to its length mlultipliedby the square of its mean diameter multiplied by'854. To calculate the contents of a cask frontm bur dimensions. RULE. - To the square of the bung diameter add the square of' the head diameter, and the square of double the diameter midway between the bung and head, and multiply the sum by ~- the length of the cask, for its cyvindi'ical contents; the product multiplied by.0034 expresses tie contents in wine gallons. EXAMPLE. -The length of the cask is 40 inches, its bung diameter 28 inches, head diameter 20 inches, and the diameter midway be dAIJOINO. 65 tween the hung and head is 25.6 inches; how many gallons' caipacity has the cask? 20+)- 282 + 25.06 X 2 =3805.44X 4 0-X.0034 =86.26 gals. An.s, (D- + dp -4-'2 m) X' 1 X.7854 = cubic contents. D2 + d2 +d- 2 6 GIIC = square of mean (dialueter. By RULE 6, p. 6S, this cask will hold28 - 20= 8 X.65- 5.2 - 20- 25.2-X 25.2 X 40 X.0034 s,-;.T gallons. hI/lten the cask is in lthe form of tlhe middle Jfustum of a sp)/l c,'id - D2 +- d2 square of mean diameter. And a cask of this form, hlaving the same head diameter, lhunm diameter, and length as the preceding, will hold - 2 X 282+ 20 X 40 X.0034 89.216 gallons. 3 Whllen the cask is in the form of the middlle frustum of' a paradbolic spindle. - D2 -F + d — 2- (Dw d)2 = square of mean diameter. And a cask of this form, having the same head'diamneter, bungl diameter, and length'as the preceding, will hold - 522l -t 133= 656 - 8.533 = 64764G7 X 40 X.0034 = 88.055 gal1s. VIWhen the cask is in the form of tiwlo cqualfrustums of a plaraboloid. A DO + A d ~ = square of mean diameter. And a eask of this form, having the same lead diam.lleter, In diameter, and length as tle preceding, will hold28 + 20-. 2 X 40 X.0034 80.51 gallons. When the cask is in the form of the equalfrustums of a cone. A D2" + d2 - i (D w d)2 square of mean diameter. Or, lD2+ -td2+ Dd- " " " Or, DXd+l(D-d)~2=",,,, And a cask of this form, having the same. head diamnletr, Ihlng diameter, and length as the preceding, will hold28 X 20r + 21~ X40X.0034-: 79.06 g.als C*. To find the coktents of a cask the same as would be given by the gauging rod. The gauging rod is constructed upon the principle that the cubo of the diagonal of a cask, in inches, multiplied by -4Y, equals tlhe contents of the cask, in Imperial gallons. The contents in wine gallons of either of the aforementioned casks, therefore, by the gauging rod, would be31.2-1 X.0027 = 821 gals. The decimal coefficient to take the place of.0027, for finding the contents of a cask in the form of the middle frustum of a spheroid =.002926; and for finding the contents of a cask in the form of the equal frustums of a cone =.002593. And between these extremle s lies the decimal for other casks, or casks of intervening figures. To find the diagonal of a cask,: wwhen the interior is inaccessible. RULE. -From the bung diameter subtract half the difference of the bung and head diameters, and to the square of the remainder add the square of half the length of the cask, and the square root of the sum will be the diagonal. EXAMPLE. - What is the diagonal of a cask whose bung diameter is 98 inches, head diameter 20 inches, and length 40 inches? 28- 20=8 - 2=4, and 28- 4 = 4, then V/(242 - 20'). = 31.241 inches. Ans. To find the leng, th of a cask, the head diameter, bung diameter and diagonal being given../.diagonal2 ----- - 1. And the interior length of a cask, whose interior head diameter, bung diameter and diagonal, are as the preceding, will be -/ (31.2412 242) = 20 X 2 = 40 inches. To find the solidity of a sphere. D2 X: D X.7854 = cubic contents, D being the diameter. To find the solidity of a sphcricalf rustum. ( h h - b+ 2) X h X.7854- cubic contents, b and d being the bases, and h the height. NOTE. -For Rules in detail peirtaining to the foregoing figures, and for ether figures, See AIENSURATIOY OF S)O[lDS. t;,LAGE. 67 ULLAGE. The ullage or -walntage of a cask is the quantity the cask lacks of being full.. 7' find the ullage ~(f a standing cask, when lthe cask is Ialf full or mor e. ROLE. -To the square of the head diameter, add the square of the diameter at the surflce of the liquor, and the square of twice the diameter midwav between the surface of the liquor and the upper head, and divide the sum by 6; the'quotient, multiplied by the distance from the surface of the liquor to the upper head, multiplied 1by..0034, will give the ullage in wine gallons. ExA.IMPLE. The diameters are as follows- at the upper head, 20 inches; at the surface of the liquor, 22 inches; andat a point midway between these, 214 inches; and the distance from the upper head to the surface of the liquor is 5 inches; required the ullage. (202 + 22-{- 21.25 X 2 ) 6 — 448.37 X 5 X.0034 7.62 gal}lons. Ans. When the cask is standing, and less than hayffull, to find the ullage. RULriE. Make use of the hung diameter in place of the head diameter, and proceed in all respects as directed in the last Rile, and add the quantity found to half the capacity of the cask; the sum will be the ullage. EXAMPTE. — The bung (diameter is 28 inches; the diameter at tle surftace of the liquor, below the -bung, is 26 inches; the diameter midway between the bung and the surface of the liquor is 27.3 inches; and the distance ftom thle surface of the liquor to the bung diameter is 5 inchlles; required the quantity the cask lacks of being half full; and also the ullage of tlhe cask, its capacity being 86.26ti galllons. (2s82 -262 + 27.3 X 2) 6- = 740.2 X 5 X.0034 = 12.58 gallons less than a full. Ans. And, 86.26+ 2 43.13 + 12.58 = 55.73 gallons ullage. Ans. W4hen the cask is upon its bilge, anlld /halfull or more, to find the ullag'e. RULE.Z Divide the distance firon the bung to the surfalce of tlle liquor- (the height of the empty segment) — by the whole ullinn diameter, and take the quotient as the height of the segment of a; circle whose diameter is 1, and find the area of tle segelllnt; nultiply the are;, by the capacity of the ca-sk, in g:allons, and tllt produc't by 1.25; the last product will be the ilage, iin gallons, a's 68 UeLLiAdE. found by the aid of the wantanre-rod; and will be correct for casks of the most common form. NOTE. - The area of the segment of a circle = (ch'd 4 are q- ~ ch'd + arc + ch'd seg.) X height seg. X?4*, very nearly. And, having the diameter of the circle and the height of the segment given, the chord of half the arc, and the chord of the segment may be found, thus - radius - height = cosine; radius2 - cosine2 = sine2; V(sine2) X 2 = ch'd of seg. sine2 + heitght seg.2 -O c'd 1 arc2, and V (ce'd ar c') - ch'd 1 arc. EXAMPLE. — The hung diameter is 28 inches, the height of the empty segment 5.6 inches, and the capacity of the cask 86.26 galions; required the ullage of the cask, in gallons. 5.6 - 28 =.2 = height of seg., diameter as 1. 1 2 2.5 = radius..5 --.2 =.3 = cosine..52 -.32 =.16 = sine2, or square of half the balse of the segment. V/.16 =.4 X 2 --.8 = chord of segment, or base of segmnent..4' +.22 =.2 square of chord of half the are. V.2 -.4472 s chord of half the are, then -.4472 +- 3 =.1491, and.1491 +.4472 +:8 X.2 X ~4 =.1117, area of segment, and.1117 X 86.26 X 1.25 =12 gallons. Ans. W/ten tIhe cask is upon its bilge, and less than halJ full, to find the ullage. RULE. Divide the depth of the liquor by the hung, diameter, and proceed in all respects as directed in the last Rtule; then subtract thle quantity found from the capacity of the cask, and the difference will be the ullagre of the cask. To find the quantity of liquor in a cask by its wei gSlt. EXAMPLE. -The weigrht of a cask of proof spilits is 300 lbs., and the wei(ht of the empty cask (tare) is 32 lbg. lHow many gallons are there of the liquor' 300 - 32 268 7.732 = 34? gallons. Ans. Customary Rule by Frcightingr 3erchants, for findlin the cubic measuremsent of casks. Bung diameter' X 4 length of cask = cubic mneasuremlent. NOTE. - One culbi foot contains 7. 4805 wine gallons. * For several Rules in dletail, for finding the area of the seC".mnllt of a circle, see Gioi.tETlT - Mcn.sI ration o/' Superficies. TOSNNAE. 69 TONNAGE. GOVERNMENT MEASUREMENT.* length - a breadth X b)readth X depth 2= tonna.ge In a double-decked vessel, the lenoth is reckoned from the fore part of the main stemr to the aftelr side of the sternpost above the ulpper deck; the bre-adth is taken at the broadest partLabove thle main wales, and half this breadthl is.taken for the depth. In a single-decked vessel the lengith and breadth are taken as for a double-decked vessel, and the distance between the ceiling of the hold and the under side of the deck- plank is taken as the depth. EXAMPLE. - The length of -a double-decked vessel is 260 feet, and the breadth is GO0 feet; required the tonnage. 0(;0- o6-X- - 224 X 60 X G - 403200 -- 95 = 4244.2 tons. Ans. ExsnmErE. The length of a single-decked vessel is 180 feet, thle breadth 34 feet, and depth 18 feet; required the tonnage. 180 - - of 34 = 159.6 X 34 X 18 - 95 = 1028.16 tons. Ans. CARPENTER S MEASUREMENT. IFor a doulbledecked — length of keel X blreadth main bearm X A breadth 95 = — -tonni(-e. For a single-decked length of keel X breadth main beam X depth of hold - - -, = tornage *'The set of ruiles legalized-by the Congress of 183. for ascertaining the nominal or (lovernment, tonnage of vessels are not inserted in this work, partly bec:Alse of their great lengths in detail, and the multiplicity of the mechanical adm.e:surelcents required, and partly because they can be of use to but a few individu:ls; and to those they are furnished by the Government. CON'DMMTS on L't14S~. OF CONDUITS OR PIPES. Pressure of;VVatcr in Vertical Pipes, c'c. h height of column in inches; o = circumference of column in inches; t - thickness of pipe in inches equal in strength to lateral pressure at base of column; w - weight of a cubic inch of water in pounds; C = cohesive strength in pounds per inch area of transverse section of the materiall of which the pipe is composed — TABLE, p. 74. ho -= area of interior of pipe in inches; hw = pressure in pounds per squtare inch at the base of the column, or maximum lateral pressure in ituinds per squire inch on the pipe tending to burst it; how _= Inmaximum lateral pressure in pounds on the. pipe, tending to burst it at the bottom; and how - 2 == mean lateral pressure in pounds on the pipe, or pressure in pounds on the pipe tending to burst it at half the height of the colululri. /how+ C = t; how-. t-C; Ct + W = h; Ct:h?. o.0 NaoRT. -The reliable cohesion of a material is not above I its ultimate foirce,.s giveti ill t.lle'rable of Cohesive Forces. By experiment, it has been foiund that a cast irlll Ipilt 15 inches in diameter and [ of an inch thick, will su pport a head of water of 600 feetit andI lthat one oftihe same diameter made ofoak, and two inches thick. will support a head of!1,30 feet: 12000 lbs. per sqtare inch for cast iron, 1200 for oak, 7,50 for lead, are counlte:afe estilmalte. The ultimate cohesion of an alloy, composed of lead 8 parts and zinle I part, is 3010) pounds per square inch. Concerning the Discharge of Pipes, 4dc. Snmall pipes, whether vertical, horizontal, or inclined, under equal helads, discharge proportionally less water than large ones. Tl'hat form of pipe, therefore, which presents the least perimeter to its afcia, other things being equal, will give the greatest discharge. A round pipe, consequently, will discharge more water in a given time than a pilpe of any other form, of equal area. * The greater the length of a pipe discharging vertically, the greater the discharge. Becanase the friction of the particles against its sidels, and consequent retardation, is more than overcome by the gravity of tihe fluid.'I'he greater thle length of a pipe discharging horizontally, thle less p)roaportionally will be the discharge. The proportion compared' with a1 less length is in the inverse ratio of the square root of the tuwo lengths, nearly. Other things being equal, rectilinear pipes give a greater discharfe than curvilinear, and curvilinear greater than angtular. The heal, the diameters and the lengths being the same, the time occupied in. pa:ssing an equal quantity of vwater through a straight pipe is 9; ~through one curved to a semicircle 10, and thromugh one(n Ihavinor one righlt'angle, otherwise straight, 11. All interior inequalities aind romgllhness should be avoided. It has been ascertained that a velocity of 60 li(el a minute (I to;ol a second) through a hIorizomntal pipe, 4 inches ill!inltitnter all(.{!0) ficl COQDUTTS (R PP1ES. 71t in length, is produced by a head 2# inches, only T of an inch above the tipper surface of the orifice; and that, to maintain an equal velocity through a pipe similarly situated, of equal length, having a diameter of i inch only, a head of 1-5 feet is required. To increase the velocity through the last mentioned pipe to 2 feet a second, requires a liead 4{-j feet; to 3 feet, a head of 10 —'; to 4 feet, a head of 17T, &c. From the foregoing, the following, it is believed, reliable rules, are dedluced.. 7o fincd the velocity of water passing through a straight horizontal pipe oJ any'iegh and diameter, the head, or height o( the fluid above the centre of the orifice, being known. RULE. - Multiply the head, in feet, by 2500, and divide the product by the length of the pipe, in feet, multiplied by 13.9, divided by tlhe interior dialneter of the pipe in inches; the square root of the quotient will be the'velocity in feet per second. EXAMPLE. - The head is 6 feet, the length of the pipe 1340 feet, and its diameter 5 inches; required the velocity of the water passing throughl it. 2500 X 6 = 15000 *- (i34o-5; 3.9): = 4.03 = 2 feet per second. Ans. To find the head necessary to produce a required velocity through a lipe of given length and diameter. RULE. - Multiply the square of the required velocity, in feet, per second, by the length of the pipe multiplied by the quotient obtained by dividing 13.9 by the diameter oft the pipe ill inches, and divide the product thus obtained by 2500; the quotient will be tile head in fteet. EXAMPLE. - The length of a pipe lying horizontal and straight is 1340 feet, and its diameter is 5 inches; what head is necessary to cause the water to flow through it at the rate of 2 feet a second? 22 X 1340 X l:39 -- 2500 = 6 feet. Ans. To find-the quantity of water flowing throulgh a pipe of any length aend diameter. RULE. - Multiply the velocity in feet per second by the area of tile discharging orifice, in feet, and the product is the quantity in cubic feet discharged per second. EXAMPLE. - Tle velocity is 2 feet a second, and the diameter of tlle pipe 5 inches; what quantity of water is discharged in e:wchl second i)f time? 5 - 12 =.4166, and.41662 X.7854 X 2 =.273 cubic foot. An,;. MISCtLLANOUJS PUO3LEtIS. MISCELLANEOUS PROBLEMS. To Jind tilc specific gravity of a body -heavier than water. RULE. -Weilgh the body in water and out of water, and divide thile weight out of water by the difference of the two weights. EXAMPLE. -A piece of metal weighs 10 lbs. in atmosphere, and but 84 in water; required its specific gravity. 10- 8.25 = 1.75, and 10 - 1.75 - 5.-714. Ans. T'o find the specific gravity of a body lighter than water. -RUJLE. -Weigh the body in air; then connect it with a piece oft inetal whose weight, both in and out of water, is known, and of silt ficient weight that the two will sink in water; and find their comlnbiied weight in water; then divide the weight of the body in air by the weight of the two substances in air, less the sum of the diifbrence of the:vWeight of the metal in air and water and the comrnbied weiglit of the:-two substances in water, and the quotient will lbe the sliecitic gravity sought. E]XAMPLE.- The combined weight, in water, of a piece of wood, and piece of metal, is 4 lbs.; the wood weighs in atmosi)here. 10 lls.; anid the metal in atmosphere:12, and in water 11 lbs.; require(l Ilie specific gravity of the wood. 10 (. (10- 12-12 11.+ 4) —.588. Ans. To find the specific gravity of a fluid. RULE. - Multiply the known specific gravity of a body bly the (lifference of its weight in and out of the fluid, and divide the product by its weight out of the fluid; the quotient will lie the specific gravIlity of ihe fluid in which the body is weighed. EXMPLE.- Tile specific gravity of a brass hall is 8:6; its wei.ght in atmosphere is 8 oz., and in a certain fluid 74 oz.; required the specific gravity of the fluid. 8 - 7.25 -=.75, and 8.6 X.75 - 6.45, and 6.45 8. 806. Ais. To find, the proportion of one to the other of twlo simples formingr a compound, or the extent to which a metal is dcbascd, (the nietal adnd the alloy used being known.) The Rule strictly bears upon that of Alligation Alternate, which see. EXAMPLE. —The specific gravity of gold is 19.258, and that of copper, 8.788; an article composed of the two metals, has a specific gravity of 18; in what proportion are the metals mixed? 18., 19.258 X 8.788 =-11.055 18 r 8.788 X 19.258- 177.4, then MISCELLANEOUS PROBLEMIS. 73 11.055 + 177.4: 11.055: 18 - 1.056 copper, As 11.055 -+ 177.4 177.4:: 18 = 16.944 gold. - Or, 18 - 1.056 = 16.914 gold. Copper to gold as 1 to 16.04 -- To fin;d the lifting power of a balloon. RULE. -Multiply the capacity of the balloon, in feet, by the diiftrence of weight between a cubic foot of atmosphere and a cubic foot of the gas used to inflate the balloon, and the product is the weight the balloon will raise. EXAMPLE.- A balloon, whose diameter is 24 feet, is inflated wit'l hydrogen; what weight will it raise 1 Specific gravity of air is 1, weight of a cubic foot 527.04 grains; specific gravity of hydrogen is.0689. 527.04 X.0689 - 36.31 grains - weight of 1 cubic foot of hydlrogen.. 5927'.04 - 36.31- 490.73 grs. - dif. of weight of air and hydrogen.'143 X.5236 = 7238.2.1 - capacity in cubic feet of balloon. Then, 7238.24 X 490.73 = 3552021 grs. 3 -i50 —507 Ans. To find the diameter of a balloon that shall be equal to the raising of a given weight. The weight to be raised is 507 4 lbs. 507.4 X 7000- 490.73 -= 7238.24, and 7238.24 +-.5236 = 4/ 13824 -24 feet. Ans. To find the thickness of a concave or hollow metallic ball or globe, that shall have a given buoyancy in a given liquid. EXAMPLE. - A concave globe is to be made of brass, specific gravity 8.6, and its diameter is to be 12 itlches; what must be its thick-:ess that it may sink exactly to its centre in pure water I Weight of a cubic inch of water.036169 lb.; of the brass.3112 lb. Then, 12; X.5236 X.036169 2 = 16.3625 cubic inches of water to bhe displaced. 16.3625 +..3112 = 52.5787 cubic inches of metal in the ball. 122 X 3.1416 - 452.39 square inches of surface of the ball. Aod, 52.5787 - 452.39 --.1162 + - inch thick, full. Ans.'1) cut a square sheet of copper, tin, etc., so as to form a vessel of the greatest cubical capacity the sheet admits of. IULE. - From each corner of the sheet,- at right angles to the side,,ut 1 part of the length of the side, and turn up the sides till the (orters meet. 7 74 COMPl'ARATIVE COIIESLVE FORCE. Comparative Cohesive Florce of AMetals, Woods, and other substances, lWirought Iron (medium quality) being the unit of comparison, or 1; the cohesive force of which is 60000 lbs. per inch. transverse area. Wrought iron,... 1.00 Ash, white,...23 "'" wire,. 1.71 " red,... 30 Copper, cast,....40 Beech,.....19 " wire,....76 Birch,.....25 (-old, cast,... 34 Box,...33'" wire,....51 Cedar,.... 19 Iron, cast, (average)...38 Chestnut, sweet,...17 lead, "....015 Cypress,.....10 m. " illed,... 055 Elm,.....22 Platinum, wire,...88 Locust,.....34 Silver, cast,....66 Mahogany, best,.36( " wire,....68 Maple,...18 Steel, soft,... 2.00 Oak, Amer., white,. 19 " fine,... 2.25 Pine, pitch,...20 Tin, cast block,...083 Sycamore,...2'2 Zinc, "...043 Walnut,.....30 " sheet,.27 Willow,...2.2 B1rass, cast....7 5 Ivory,.....27 Gun metal,...50 Whalebone,...13 Gold 5, copper 1,...83 Marble,.....15 Silver 5, " 1,...80 Glass, plate,..16 Brick,.....05 Hemp fibres, glued,.. 1.53 Slate,.....20 The strength of white oak to cast iron, is as 2 to 9. The stiffness of " " " " is as 1 to 13. To leterminze the weight, or force, in pounds, necessary to tear asunder a bar, rod, or piece of any of the above named substances, of any (riven transverse area: RULE. - Multiply the comparative cohesive force of the substance, as given in the table, by the cohesive force per square inch, area of cross section (60000 lbs.) of wrought iron, which gives the cohesive force of 1 square inch area of cross section of the substance whose power is sought to be ascertained, and the product of 1 square inch thus found, multiplied by area of cross section, in inches, of the rod, piece, or bar itself, gives the cohesive force thereof. Alloys having a tenacity greater than the sum of their constituents. Swedish copper 6 pts., Malacca tin 1; tenacity per sq. inch, 64000 lbs (htili copper 6 pts., Malacca tin 1; " " " 60000 ".;l!;,:n copper 5 pts., Banca tin 1; " " " 57000 ()f A. itrlesea copper 6 pts., Cornish tin 1; " " " 41000 LINEAR DILATION OF SOLIDS BY IE AT. 75 Common block-tin 4 pts., lead 1, zine 1'; tenacity per sq. in., 13000 lbs. Malacca tin 4 pts., re~grlus of antimony 1;" " " 12000 " Block-tin 3 pts., lead 1 part; " " " 10000," }Block-tin 8 pts., zinc I part; " " " 10200 " Zinc I part, lead 1 part; " " " 4500 "' Alloys having a density greater than the mean of their constituents. GOLD with antimony, bismuth, cobalt, tin, or zinc. SILVER with antimony, bismuth, lead, tin, or zinc. COPPER with bismuth, palladium, tin, or zinc. 1,EnD with antimony. PLATINUM with molybdinutlm.'PALLADIUM with bismuth. Alloys having a density less than the mean of their constituents. GOLD with copper, iron, iridium, lead, nickel, or silver. SILVER with copper or cobalt. IRON with antimony, bismuth, or lead. TIN with antimony, Icad, or palladium. NIcKEL with arsenic. ZINC with antimony. IREL.ATIVE POWER OF DIFFERENT METALS TO CONDUCT ELECTHICITY, (the mass of each being equal.) Copper,.... 1000 Platinum,... 188 Gold,.... 936 1 Iron,.... 158 Silver,.... 736 Tin,.... 155 Zinc,.... 2851 Lead,.... 83 LINEAR DILATION OF SOLIDS BY lIEAT. Iengthl which a bar heated to 2120 has greater than when at the temperature of 320. f)rass, cast,...0018671 Iron, wrought,...0012575 Copper,..0017674 Lead,....0028568 Fire brick,...0004928 Marble,....0011016 Glass,....0008545 Platinum,....0009342 Gold,....0014880 Silver,..0020205 Granite,...0007894 Steel,....0011898 ironl, cast,..0011111 Zinc,....0029420 NoTB. -To find the surface dilation of any particular article. double its linear dilation,. al to finld thedilation in volzme, trilile it. -'o find the elongation in lilear inches per liinear lt(t, of ary particular article, mulltiply its respective linear dilation, as given in the YAnmis. by 12. 76 EFFECTS OF HEAT. iELTING POINT OF METrALS AND OTHER BODIES. Lime, palladium, platinum, porcelain, rhodium, silex, may be melted by means of strong lenses, or by the hydro-oxygen blowpipe. Cobalt, mangranese, plaster of Paris, pottery, iron, nickel, &c., at from 23000 to 3250~ Fahrenheit; others as follows:Degrees Fah. De.rees F,,h. Antimony,.. 809 Nitre, 660 Beeswax, bleached,.. 155 Silver,.1873 Bismuth,.... 506 Solder, common,. 475 Brass,..1900 " plumbers',. 360 Copper,.... 1996 Sugar,. 400 Glass, flint,... 1178 Sulphur,.. 226 Gold,. 2216 Tallow,. 127 Lead,.... 612 Tin,.... 442 Mercury,.... -39 Zinc,.... 680 Cast iron thorouglly melts at.... 2386 Greatest heat of a smith's forge, (com.). 2346 Welding heat of iron,.... 1892 Iron red hot in twilight,...... 884 Lead 1, tin 1, bismuth 4,melts at...... 201 ILead 2, tin 3, bismuth 5,' "...... 212 RELATIVE POWER OF DIFFERENT BODIES TO RADIATE HEAT. Water,.... 100l Lead, bright,. 19 Copper,.... 12 Mercury,... 20 Glass,.... 90 Paper, white, 100 Ice,... 85 Silver.... 12 India inkl,.. 88 Tin, blackened,.. 100 Iron, polished, 15 " clean,... 12 Lampblack,.. 1001 " scraped,... 16 NOTE. - The power of a body to reflect heat is inverse to its power of radiation. BOILING POINT OF LIQUIDS. Barometer at 30 in. Acid, nitric,.. 253~ Oils, essential, avg.,. 318~ " sulphuric,.. 6000 " turpentine,.. 316; Alcohol, anhyd.,.. 168.5o " linseed,.. 6410 " 90 per cent.,. 174o Phosphorus,.. 554o Ether, sulph.,.. 970 Sulphur,... 560~ Mercury,... 656~ Water,... 212~ NOTE. - Barometer at 31 inches, water boils at 2130.57; at 29, it boils at 21f(o.:3;- at 28, it boils at 20SO.69; at 27, it boils at 2060.85, and in vacuo it boils at 880. No li(lit. under pressure of the atmosphere alone, can be heated above its boilnlog point.. At t,:l poiat the steam emitted sustains the weight of the atmosphere. EFFECTS OF IIEAT, ETC. 77 FREEZING POINT OF LIQUIDS. Acid, nitric,... — 5501 Oil, linseed, avg.,. -11~ " sulphuric,. 1~ Proof spirits,.-7~ wEther,.. — 470 Spirits turpentine,. 16 Mercury,.. -3~ Vinegar,. 28~ Milk,.... 30~ Water,.. 32~ ()il, cinnamon,.. 30" Wine, strong,. 200 " fennel,... 14~l Rapesced Oil,.. 250 " olive,... 360 NOTE. - Water expands in freezing.ll, or ~ its bulk. EXI'ANSION OF FLUIDS BY BEING H[EATED FROM 320 TO 2i25 F'. Atmospheric air,;- per each degree,.375 Gases, all kinds, a-" ". Mercury, exposed,....018 Muriatic acid, (sp. gr. 1.137,)..060 Nitric acid, (sp. gr. 1.40,)..110 Sulphuric acid, (sp. gr. 1.85,)..060 " ether, - to its boiling point,.070 Alcohol, (90 per cent.,) " "...110 Oils, fixed,.080 " turpentine,.....070 Water,........046 RELATIVE POWER OF SUBSTANCES TQ CONDUCT HEAT. G(Cold,... 1000 Zinc,.... 363 Silver, 9. 973 Tin,.... 301 Copper,... 898 Lead,.... 180 Ilatinum,. 381 Porcelain,... 12 ron,.... 374 Fire brick,.. 11 NOTE. - Different woods have a conducting power in ratio to each other, as is their respective specific gravities, the more dense having the greater. METALS IN ORDER OF DUCTILITY AND MALLEABILITY. Ductility. - VMilleability. 1. Platinum. 1. Gold. 2. Gold. 2. Silver. 3. Silver. 3. Copper. 4. Iron. 4. Tin. 5. Copper. 5. Platinum. 6. Zinc. 6. Lead. 7. Tin. 7. Zinc. 8. Lead. 8. Iron. 7* 78 NUTRITIVE AND ALCOIHOLIC PROPERTIES OF BODIES. Quantity per cent. by weight of Nutritious Matter contained in- different articles of Food. Articles. per ct. Articles. per ct. Lentils,.... 94 Oats,... 74 Peas,.... 93 Meats, avg., 35 Beans,.... 92 Potatoes,. 25 Corn, (maize,).. 89 Beets,... 14 Wheat,.. 85 Carrots,. 10 Barley,.... 83 Cabbage,. 7 Rice,.... 88 Greens, 6. i Rye,.... 79 Turnips, white, 4 2)ecific gravity, and quantity per cent., by volume, of Absolute Alcohol contained, necessary to constitute the following named unadulterated articles. Sp. rrav. Per cent. Articles. 60~, b. 30. of Alcl,hol. Absolute Alcohol, (anhydrous,)...7939 100 Alcohol, highest by distillation,....825 92.6 " commercial standard,....8335 90 Proof Spirits, - standard,....9254 54 Quantity per cent., by volume, (general average) of Absolute Alcohol contained in different pure or unadulterated liquors, Vrines, 4-c. Liquors, &c. per cent. Wines. per cent. Rum, 50 Port,.. 22 Brandy,.... 50 Madeira,.. 20 in, Holland,... 4.8 Sherry,... 18 Whiskey, Scotch,.. 50 Lisbon,... 17 " -Irish,.. 50 Claret.... 10 Cider, whole,.. 9 Malaga,.... 16 Ale,... 8 Champagne,... 14 Porter,... 7 Burgundy,... 12 Brown Stout,... 6 Muscat,... 17 Perry,.... 9 Currant,... 1' Proof of Spirituous Liquors. The weight, in air, of a cubic inch of Proof Spirits, at 60~ F., is 233 grains; therefore, an inch cube of any heavy body, at that temperature, weighing 233 grains less in spirits than in air, shows the spirits in which it is weighed to be proof. If the body lose less of its weight, the spirit is above proof, - if more, it is below. COMPARATIVE WEIGHT OF TIMBER. 79 Comparative Weight of different kinds of Timber in a green and perfectly seasoned state. Assuming the weight of each kind destitute of water to be 100, that of the same kind green is as follows: — Ash,.. 153 Cedar,. 148 Maple, red,. 149 Beech,.. 174 Elm, swamp.. 198 Oak, Am.,. 151 Birch,.. 169 I Fir, Amer..,. 171 Pine, white,. 152 NOTE. - Woods which have been felled, cleft and housed for 12 months, still retain from 20 to 25 per cent. of water. They therefore contain but from 75 to 80 per cent. of heating matter; and it will require from 23 to 29 per cent. the weight of such woods to dispel the water they contain. They are, therefore, less valuable by weight, as fuel, by this per cent., than woods perfectly free from moisture. They never, however, contain, exposed to an ordinary atmosphere, less than 10 per cent. of water, however long kept; antl even though rendered anhydrous byr a strong heat, they again imbibe, on exposure to the atmosphere, from 10 to 12 per cent, of dampness. Relative power of different seasoned Woods, Coals, T4c., as fuel, to pro. duce heat, -the Woods supposed to be seasoned to mean dryness, (77J per cent.,) and the other articles to contain but their usual quantity of moisture. Ratio of Henting Power per eqiata Bulk. Weight. Hickory, shell-bark,.1.00 1.00 " red-heart,....81.99 Walnut, com...,95 -.98 Beech, red,... ~.74.99 Chestnut, -..49.98 Elm, white,......58.98 Maple, hard,....66.98 Oak, white,......81.99 " red,.. 69.91) Pine, white,....42 1.01 " yellow,....48 1.03 Birch, black,.....63.99 " white,...48.91) Coal, Cumberland, (bit.)....56 2.28' Lackawanna (anth.)...2.28 2.22 c" Lehigh, ". 2.39 2.03 Newcastle, (bit.). 2.10 1.96 Pictou, (bit.)... 2.21 1.91 " Pittsburgh, (bit.). 1.78 1.82 " Peach Mountain, (anth.) 2.69 2.29 Charcoal,... 1.14 2.53 Coke, Virginia, natural,. 1.89 2.1o2 " Cumberland, 1.31.25 Peat, ordinary,... 6'2 Alcohol, common,...0 2 13eeswax, yellow,. 2.90( Tallow,.... 3 10 E80'iLLITMINATION. NOTE. - By help ot the preceding table, the price of either one article being known, tile relative or par value of either other, as fuel, may be readily ascertained: — EAMPLE: Maple (66): $5.00: Pine (42): $3.18. ILLUMINATION ARTIFICIAL. The following TABLE shows:1. The materials and methods of using - column Materials. 2. The comparative maximum intensity of light afforded by each material, used or consumed as indicated, - column Intensities. 3. The weight, in grains, of material consumed per hour, by ench method respectively, in producing its respective light, or light of intensity ascribed - column. Weight. 4. The ratio of weight required of each material, under each special method of consumption, for the production of equal lights inl equal times - column Ratios. Materials. Intens. Weirht. Ratios. Camphene, Paragon Lamp,... 16. 853 1. Sperm Oil. Parker's heating Lamp,.. 11. 696 1.19 " Mech. or Carcel ".. 10. 815 1.53 " " French annular ".. 5. 543 2.04 " " Common hand ". 1. 112 2.10 Whale " p'fd., P's heating".. 9. 780 1.63 Wl1ax Candles, 3's or 4's, 15 in. or 12 in.,. 1. 125 2.35..". 6's, 9 in.,.....92 122 2.50 Spern " 4's, 13A in.,.. 1. 142 2.66 Stearine" 4's, 13A in.,.... 1 168 3.15 Tallow," dipped, 10's,....70 150 4.02 " " mould, 10's,.....66 132 3.75 " "' 8's,....57 132 4.35 i" "s it 6's,.79 163 3.87 "C it "4 4's, 133 in.,.. 186 3.49'Coal Gas," intensity being... 1 411 NOTE. - The consumption of 1.43 cubic feet of gas per hotir, gives a light equal to one wax candle, - the consumption of 1.96 cubic feet per hour, a light equal to four wax cantiles, and the consumption of 3 cubic feet per hour, a light eqtlal to ten wax candles. A cilic foot of gas weighs 286 grains.'rThe average yield of bi-carbureted hydrogen - Olefiant gas - Coal Gas, obtained from the following articles, is as annexed. 1 lb. Bituminous Coal, 41 cubic feet. 1 lb. Oil, or Oleine,.15 " " 1 lb. Tar,. 12. 1 lb. Rosin, or Pitch,.10 " A pound of good Lancashire cannel coal, or of good Scotch cannel, will yield, on an average, 5.95 cubic feet of good illuminating gas. From the English Boghead cannel, by White's hydro-carbon process, 17 cubic feet to the pound are commonly obtained. A pipe whose interior diameter is. inch, will supply gas equal in illuminating powver to 20 wax candles. The sp. gr. of coal-gas from cannel coal ranges from 0.6 0 to 0.370, while that from good working bituminous comal in generali ranges from 0.420 to 0.370. ILLUMINATI0N. 81 Results qf Experiments by M3r. Clegg of London, relative to te conveyance of coal-gas through pipes of different lengths, diamcters, A$c. Diameter of pipe Length of pipe Pressure in inches Specific gravity, Q!antity in cubic in inches. in yards. of water. air as 1. per disargd per hour. 0.5 10 1.25 0.4 120 0.5 59 1.25 0.4 60 0.62 41 1.38 0.559 99 0.62 62 1.34 0.559 - 83 0.62 93 1.34 0.559 74 0.62 19 1.34 0.559 57 0.62 138 1.34 0.559 53 2. 25 0.5 0.528 1630 10. 100 3. 0.4 120,000 10. 1760 3. 0.4 30,000 18. 1760 1. 0.4 66,000 26. 4300 0.475 0.42 80,000 26. 3130 0.8 0.42 103,000 26. 4300 2.25 0.42 175,000 From these experiments and others, Mr. Clegg derives the data for the following general approximate rules, viz.: Q _ 1350d\2 (hd); or, when the length of the pipe is above 400 or 500 times its diameter; or, in any case, if the pressure be measured in the pipe instead of in the gas-holder,Q - 1350d24jl.4 in which Q represents the quantity of gas in cubic feet passed per hour, I the length of the pipe in yards, d the diameter of the pipe in inclles, h the pressure in inches of water, and s the specific gravity of the gas. Therefore, ordinarily for short mains of large diameters, or when the quotient of 361l d is less than 400, 1 3 5 02d"h Q2s(lq- d) 13502~dYL1 13502 -d; h - s(l+d); and 1 +- d 1 Q Q.s l3502slnd ad d And for long pipes of small diameters, or when 1 and 1 + -d are practically equal. 13502d5h Qs;an I; and d dQs' 13502 d N 1350h' Now, in these formulas for Q, by Mr. Clegg,-the probable retardation of the flow of the gas due to the friction of its particles in the pipes has been taken into account; and the results by the fobrmulas agree as closely as coul( be expected with the given experilllents, although the former average about 7 per cent. more tian the latter. 826'~T.IICl3TZ.SIMETEF11. The pipi)eS a]re lhere supposed to be straighrt, and to lie horizontal, or e(quall ill (ffect to that condition, and the gas is supposed to be deflivered without force at the discharging end. EXAMPIE. - What pressure in inches of water is required to convey 852 cubic feet of gas per hour, of specific gravity 0.398, through a pipe 4 inches in diameter and 6 miles in length'? 8522 X 10560 X.398 102 5 1.6348 inchlies. Ans. 13502 X 45 NOTE. —The illuminating power of coal-gas is nearly as its specific gravity, the more dense being the better. THERMOMETERS. Boiling point. | Freezineg point. Fahrenheit's,... 2120 320 Reaumur's,.... 80 0 Centigrade,.... 100 00 To reduce Reaumur to Fahrenheit. When it is desired to reduce the +-o, (degrees above the zero):RULE. - Multiply thq degrees Reaumur, by 2.25, and add 320 to the product; the sum will be the degrees Fahrenheit. When it is desired to reduce the - o, (degrees below the zero):RULE. - Multiply the - ~ Reaumur by 2.25, and subtract the product from 320; the difference will be-the degrees Fahrenheit. EXAMPLE.- The degrees R. are 40; required the equivalent degrees F. 40 X 2.25 = 90 + 32 = 122~. Ans. EXAMPLE. - The degrees below 0, R., are 10; what are the corresponding degrees F.? 10 X 2.25 _ 22.5, and 32 - 22.5 = 9A~. Ans. EXAMPLE. -The degrees below 0, R., are 16; what point on the scale F. corresponds thereto? 16 X 2.25-=-36, and 32 - 36 - 4; 4 below0. Ans. To reduce the Centigrade to Fahrenheit. RULE. - Multiply the degrees C. by 1.8, and in all other respects proceed as directed for Reaumur, above. NoTr. -- The zero of Wedgewood's pyrometer is fixed at the temperature of iron red-hot ill daylight, - 10770 F., and each degree W. equals 130o F. The instrument is not consilered reliable, and is but little used HORSE PCOWER -A- MAiL ioWVtVt - 8TEAM. SEA tHORSE POW IEll. A HGRSE-POWER, in machinery, as a measure of force, is estimated equal to the raising of 33000 lbs. over a single pulley one foot a minute, = 550 lbs. raised one foot a second,= 1000 lbs. raised 33 feet a minute. ANIMAL POWER. A man of ordinary strength is supposed capable of exerting a force of 30 lbs. for 10 hours in a day, at a velocity of 2j feet a second, 75 lbs. raised 1 foot a second.'the ordinary working power of a horse is calculated at 750 lbs. for 8 hours in a day, at a velocity of 2 feet a second, = 375 lbs. raised I foot a second, = 5 times the effective power of a man during associated labor, and 4 times his power per day; and as machinery may be supposed to work continually, = a trifle less than 23 per cent. per day of a machine horse-power. STEAM.'Table exhlibiting, the expansive force and various conditions of steam under different degrees of temperature. Degrees of Pressutre in Density. Volume. Spec. gravity. Weight of;a heat. atmnospheres. VWater as 1. Water as 1. Air as 1. cubigc i,ot in grains. 212 1.00059 1694.484 254 250.5 2.00110 909.915 483 27G 6 3.00160 625 1.330 700 293.8 4.00210 476 1.728 910 308 5.00258 387 2.120 1110 359 10.00492 203 3.970 2100 418.5 -20.00973 106 7.440 3940 [An atmosphere is 14-Ti lbs. to the square inch.] NOTE. - By the above table it is seen that any given quantity of-steam having a temperature of 212~ F., occupies a space, under the ordinary pressure of the atmosphere, 1694 times greater than it occupied when as water in a natural state. It exerts a mechanical force, consequently, — = 1694 times the weight or force of the atmosphere resting on the bulk from- which it was generated, or resting on 1-1694th of the space it occupies. A, force, if we consider the volume as so many cubic inches, equal to the raising of 2087 Ibs -12 inches high, by a quantity of steam less than a cubic foot, heated only to the temnperature of boiling water, and weighing but 248 grains, and that, too, the product of a single cubic inch of water. The mean pressure of the a.tmosphere at the earth's surface is equal to the weight of a column of mercury 29.9 inches in height, or to a.olumn of water 33.87 feet in height, 2116.8 lbs. per square foot, or 84 VELOCITY AND FORCE OF WIND. 14.7 lbs. per square inch. Its density above the earth is uniformly less as its altitude is greater, and its extent is not above 50 miles - its mean altitude is about 45 miles; at 44 miles-it ceases to reflect light. Were it of uniform density throughout, and of that at the surface, its altitude would be but 5j miles. Its weight is to pure water of equal temperature and volume, as 1 to 829. It revolves with the earth, and its average humidity, at 40~ of latitude, is 4 grains per cubic foot. Its weight at 60~, b. 30, compared with an equal bulk of pure water at 40~, b. 30, is as 1 to 830.1. VELOCITY AND FORCE OF WIND. Mean velocity in Miles per Feet per Force in lbs. per Appellations. hour. second. square foot. Just perceptible,.. 2, 3..032 Gentle, pleasant wind,. 4A 6. -.101 Pleasant, brisk gale,.. 129 18.80 Very brisk, ".. 22 33 2.52 High wind,... 32A 47. 5.23 Very high wind,... 42A 62i 8.92 Storm, or tempest,.. 50 73-k 12.30 Great storm,... 60 88 17.71 _Hurricane,... 80 117 31.49 Tornado, moving buildings, &c., 100 146.7 49.20 The curvature of the earth is 6.99 incles (.5825 foot) in a single statute mile, or 8.05 inches in a geographical mile, and is as the sqtuare of the distance for any distance greater or less, or space between two levels; thus, for three statute miles it is 1: 32:: 6.99: 54 feet, nearly. The horizontal refraction is T. Degrees of longitude are to each other in length, as the cosines of their latitudes. At the equator a degree of longitude is 60 geographical miles in length, at 90~ of latitude it is 0; consequently, a degree of longitude at 5~.. -59.77 miles. 40~.. = 45.96 miles, 10~.. 59.09 " 500.. -38.57 " 20.. — 56.38 " I70~ -20.52 " 30~.. 51.96 " 185~ -- 5.23 " Time is to longitude 4 minutes to a degree, - faster, east of any given point; slower, west. The mean velocity of sound at the temperature of 330 is 1100 feet a second.. Its velocity is increased A a foot a second for every degree GRAVITATION. 85 above 330, and decreased a a foot a second for every degree below 33 In water, sound passes at the rate of 4,708 feet a second. Light travels at the rate of 192,000 miles per second. GRAVITATION. GRAVITY, or GRAVITATION, is a property of all bodies, by which they mutually attract each other proportionally to their masses, and inversely as the square of the distance of their centres apart. Practically, therefore, with reference to our Earth and the bodies Upon or near its surface, gravity is a constant force centred at the Earth's centre, and is there continually operating to draw all bodies with a uniformly accelerating velocity to that point, and through very nearly equal spaces, in equal intervals of time from rest, at all localities. Putting Rl. to represent the Equatorial radius of the earth, and r to represent the Polar, and making I? - 3962.5 sttatute miles, and r =3949.5, which is nearly in accordance with the mean of the most- reliable measurements of the arcs of a degree of latitude at different localities, we have ed- (R2- r) l2' _.006550751i,the square of the elliptic ity of the earth, and 1-' = 2R1' -(2 + el sin21), the radius at any given latitude 1. And since the initial velocity due to gravity at the level of the sea at the Equator is G 32.0741 feet per second, or, in other words, since a body fallingi in vacuo at the equator, at the level of the sea, describes a space of 16.03705 feet in the first second of time from rest, we have gy [Rl VG. R]i, the initial velocity at the level of the sea at any given radius 1R; or g- (22441.2 + R)2. And finally= g ( 2(41 )i (X -1 28R) at any given radius R, at any given altitude, h, inl bet, above the level of the sea. NOTE. -When 1, reckoned from the equtltor, is higher than 45~, sin l = cos2 (90 - 1). The Impulse, or force with whliclh a falling body strikes, is the product of its weight and velocity (the weight multiplied by the square root of the product of the space fallen through and 64.33, or 4 times 16&1); thus, 100 lbs., falling 50 fret, will strike with a force, 50 X 64.333 = / 3216.66 = 56.71 X 100 = 5671 lbs. An entire revolution of the earth, from wvest to east, is performed in 23 hours, 56 minutes, and 4 seconds. A solar year 365 days, 5 hours, 48 minutes, 57 seconds. Tile area of the earth is nearly 1 97,000,000 squarie miles. Its crust is supposed to be about 30 miles inl thickness, and its mean density 5 times that of water. About 1- of its area, or 150,000,000 square miles, is covered by water. The portions of land in the several 8 86 CHEMICAL ELEMENTS. divisions, in square miles, are, in round numbcrs, as follows, VIZ: Asia,. 16,300,000 Europe,.. 3,700,000 Africa,. 11,000,000 Australia,.. 3,000,000 A. merica,. 14,500,000 America is 9000 miles long, or 1%e60 the circumference of the earth. The!)pel)ulation of the globe is about 1,000,000,000, of which there are, ill Asia,.. 456,000,000 Africa,.62,000,000 Europe,.. 258,000,000 America,. 55,000,000 CHEMICAL ELEMENTS.''he chemnical elements -simple substances in nature- as far as has been determined, are 62 in number: 13 non-metallic and 49 metallic. Of the non-metallic, 5 - bromine, chlorine, fiuorine, iodine, and orygen, (formerly termed "supporters of combustion,") have an intense affinity for all the others, which they penetrate, corrode, and apparently consume, always with the production, to some extent, of light and heat. They are all nlon-conductors of electricity and negative electrics. The remaining 8 —hydrogen, nitrogen or azote, carbon, boron, silicon, phosphorus, selenium, and sulphur, are eminently susceptible of the impressions of the preceding five; when acted upon by either of themn to a certain extent, light and heat are manifestly evolved, aind they are thereby converted into inconlbustible compounds. Of the metals, 7 —potassium, sodium, calcium, barytium, lithium, strontium, and magnesia, by the action of oxygen, are converted into bodies possessed of alkaline properties. Seven of them - glucinum, erbium, terbium, yttrium, allumium, zirconium, and thorium, - by the action of oxygen, are converted into the earths proper. In short, all the metals are acted upon by oxygen, as also by most or all of the non-metallic family. The compounds thus formed are alkaline, saline, or acidulous, or an allkali, a salt, or an acid, according to the nature of the materials and the extent of combination. Metals combine with each other, forming alloys. If one of tihe metals in combination is mercury, the compound is called an almalgram. Silicon is the base of the mineral worid, and carbon of the organ.ized. For a very general list of the metals, see TABLE OF SIECIFR(; (iIA V ITIES. CONSTITUENTS OF 130UIES. 87 TABLE E.thibiling tIhe Elementary Constituents and per cctl. by wcight lf cach, in 100 parts of diffcrent compounds. Comxpounds. Constituents andl per cent.. H,. yroen. Oxygen. Nitrogen. Carbon. Atmospheric air,... a 19.96 79.84 Water, pure,... 11.1 88.9 Alcohol, anhydrous,.. 12.9 34.44 52.66 Olive oil,.,. 13.4 9.4 77.2 Sperm".... 10.97 10.13 78.!9 Castor".... 10.3 ]15.7 74.00 Stearine, (solid of fats,) 11.23 6.3 0.30 82.17 Oleine, (liquid of fats,). 11.54 12.07 0.35 76.03 Linseed oil,.... 11.35.12.64 76.01 Oil of turpentine,.. 11.74 3.66 84.6 "Camphlene," (pure spts. turp.) 11.5 88.5 Caoutchouc, (gum elastic,) 10. 90. Camphor,.... 11.14 11.48 77.38 Copal, resin,.. 9. 11.1 79.9 Guaiac, resin,... 7.05 25.07 67.88 Wax, yellow,... 11.37 7.94 80.69 Coals, cannel, 3.93 21.05 2.80 72.22 " Cumberland, 3.02 14.42 2.56 80. " Anthracite, b.. 93. Charcoal,.... 97. Diamond,.... 100. Oak wood, dry, c 5.69 41.78 52.53 Beech" "... 5.82 42.73 51.45 Acetic acid, dry,.. 5.82 46.64 47.54 Citric " crystals,.. 4.5 59.7 35.8 Oxalic " dry,.. 79.67 20.33 Malic, " crystals,.. 3.51 55.02 41.47 Tartaric" dry,. 3. 60.2 36.80 Formic ". 2.68 64.78 32.54 Tannin, tannic acid, solid, 4.20 44.24 51.56 Nitric acid, dry,...73.85 26.15 Nitrous" anhydrous, liquid, G1 32 30.68 Ammoniacal gas,.. 17.47 82.53 Carbonic acid "... 72.32 27.68 Carb. hydrogen gas,.. 24.51 75.49 Bi-carb. hyd., olefient gas,. 14.05 85.95 Cyanogen ". 53.8 46.2 Nitric oxyde'. 53. 47.00 Nitrous " ". 36.36 63.64 Ether, sulphuric,... 13.85 21.24 65.05 Creosote,... 7.8 16. 76.2 _3.366.64 CONSTITUENTS OF BODIES. Constituenti R!td per ceiit. Compounds. vdroen. Oxygen. Azoe. I Carbon. Morphia,.. 6.37 16.29 5. 72.34 Quina, - quinin,. 7.52 8.61 8.11 75.76 Veratrine,.... 8.55 19.61 5.05 66.79 Indigo, 4... 4.38 14.25 10. 71.37 Silk, pure white, 3.94 34.04 11.33 50.69 Starch, - farina, dextrine, 6.8 49.7 43.5 Sugar,.... 6.29 50.33 43.38 Gluten,.... 7.8 22. 14.5 55.7 Wheat,.. c 6. 44.4 2.4 47.02 Rye,..... 5.7 45.3 1.7 47.03 Oats,... 6.6 38.2 2.3 52.9 Potatoes,.... 6.1 46.4 1.06 45.9 Peas,..... 6.4 41.3 4.3 48. Beet root,... 6.2 46.3 1.8 45.7 Turnips, 6. 45.9 1.8 46.3 Fibrin,.... d 7.03 20.30 19.31 53.36 i Gelatin,.... d 7.91 27.21 17. 47.88 Albumen,.... d 7.54 23.88 15.70 52.88 Muriatic acid gas, - Hydrogen 5.53 + 94.47 chlorine. Sulphuric acid, dry, - Oxygen 79.67 +- 20.33 sulphur. Silicic acid - Silica, dry, - Oxygen 51.96 +- 48.04 silicon. Boracic acid-Borax, dry,- " 68.81 + 31.19 boron. a. The atmosphere, in addition to its constituents as given in the table, contains, besides a small quantity of vapor, from 1 to 3 parts in a tihouisand of carbonic acid gas, and a trace merely of ammoniacal gas. b. Anthracite coal, charcoal, plumbago, coke, &e., have no other constituent than carbon; they are combined, to a small extent, with foreign matters, such as iron, silica, sulphur, alumina, &c. c. The constituents of woods, grains, &c., are given per cent., witllout regard to the foreign matters (metallic) which they contain. In oak, chestnut, and Norway pine, the ashes amount to about -T of 1 per cent., and in ash and-maple to 7 of 1. In anthracite coals, at an average, they are about 7 per cent. d. Fibrin, Gelatin, Albumen-Proximate animal constituents — Nutritious properties of animal matter. Fi brin is the basis of the muscle (lean meat) of all animals, and is also a large constituent of the blood. Gelatin exists largely in the skin, cartilages, ligaments, tendons and bones of animals. It also exists in the muscles and the membranes. Albumen exist.s in the skin, glands and vessels, and in the serum of the blood. It constitutes nearly the whole of the white of an egg.. CONSTITUENTS OFl BODIES. 99 TIHE RELATIVE QUANTITIES BY VOLUME of the several gases going to constitute any particular compound, are readily ascertained by help. of their respective specific gravities, compared with their relative weights, as given per cent. in the preceding table:-thus, the sp. gr. of hydrogen is.0689, and that of oxygen 1.1025, and 1.1025 +-.0689 -= 16; showing the weight of the latter to be 16 times that of the former per equal volumes, or, relatively. as 16 to 1. -The per cent. by weight, as shown by the table, in wr'iich these two gases combine to form water, for instance, is 11.1 anldl 88.9; or 11.1,f' hydrogen and 88.9 of oxygen in 100 of the co mpiound; or as 88.9 - 11.1,-as 8 to 1: 16 - 8-=2: two volumes, therefore, of tllht lighter gas (hydrogen) combine with one of oxygen to form water.Water, consequently, is a Protoxide of Hydrogen. Upon the principle of ATOMIC WEIGHTS, - relative quantities, by weight, in which the elements combine in forming compounds, based upon the standard already shown,- we have, for water, ILt' + OS = Aq. 9. 9. That is, an atom of hydrogen is represented b\l 1, an atom of oxygen by 8, and an atom of water by 9. By the same rule as the preceding, the constituents of atmospheric air are found to be to each other, in volume, as 4 to 1; four vollllces of nitrogen and one volume of oxygen make one volume of atmospheric air. The weight of nitrogen to hydrogen, per equal volumes, is as.972 to.0689, as 14.11 to 1. Atomically, therefore, it is as 7.055 to 1; hence, we have N4 + 0 = 36.22, the atomic weight of atmospheric air. The vast condensation of the gases which takes place, in some instances, in forming compounds, may be conceived of, and the process for determining the same exhibited by a single illustration. We will take, foir exam ple,~ water. A single cubic inch of distilled water, at 60~, weighs 252.48 grains. Its weight is to that of dry atmosphere, at the same temperature, as 827.8 to 1. A cubic inch of dry atmosphere, therefore, at that density, weighs.305 of a grain. Hydrogern, we find by the table of Specific Gravities, weighs.0689 as much as atmosphere, and oxygen 1.1025 as much. A cubic inch of hydrogen, therefore, weighs.0689 X.305 -=.0210145 of a grain, and a cubic inch of oxygen 1.1025 X.305, =.3362625 of a grain. The constituents of water by volume are 2 of the first mentioned gnas to 1 of the latter; and.0210145 X 2 +.3362625 =.3782915 of a: grain, = weight of three cubic inches of the uncondensed compound,. of which,.1260972 of a grain, is the weight of a volume 1 cubic i ch. As the weight of a given volume of the uncondensed compound, is to the weight of an equal volume of the condensed compound, so are their respective volumes, inversely: then -.1260972: 252.48:: 1: 2002.26 the number of cubic inches of t ie two gases condliensed into 1 inch to form water; a condensation of 2001 tiines. Of this volume of gases, I, or 1334.84 cubic inches, is hydrogen; the remaining third, 667.42 cubic inches. is oxygen. 8 go fk0PRTits, ttEc., oP o1ttos. The foregoing method, though strictly correct, does not exhibit in a general way the most expeditious for solving questions of that nature, the condensation which takes place in the gases on being converted into solids, or dense compounds. It was resorted to, in part, as a means through which to exhibit principles and proportions pertaining thereto. As before; one cubic inch of water weighs 252.48 grains, 1 of which, or 28.05+ grains, is hydrogen, and -, or 224.43- grains, is oxygen. The volumefof 1 grain of oxygen is 2.97+ cubic inches. and tile volume of hydrogen is 16 times as much, or 47.58- cubic inches. Therefore, 28.05 X 47.58 = 1334.62, and 224.43 X 2.97 - 665.56, - 2001.18, condensation, as before. Properties of the SIMPLE SUBSTANCES, and some of their compounds, not given in theforegoing. BRomINE, - at common temperatures, a deep reddish-brown volatile liquid; taste caustic; odor rank; boils at 116"; congeals at 4c; exists in sea-water, in many salt and mineral springs, and in most marine plants; action upon the animal system very energetic and poisonous - a single drop placed upon the beak of a bird destroys the bird almost instantly. A lighted taper, enveloped in its fumes, burns with a flame green at the base and red at the top; powdered tin or antimony brought in contact is instantly inflamed; potash is exploded with violence. CHLORINE, -a greenish-yellow, dense gas; taste astringent; odor pungent and disagreeable; by a pressure of 60 lbs. to the square inch is reduced to a liquid, and thence, by a reduction of the temperature below 32~, into a solid. It exists largely in sea-water-is a constitulent of common salt, and forms compounds with many minerals; is deleterious, irritating to the lungs, and corrosive; has eminent bleaching properties, and is the greatest disinfecting agent known; a lighted taper immersed in it burns with a red flame; pulverized antimony is inflamed on coming in contact, so is linen saturated with oil of turpentine; phosphorus is ignited by it, and burns, while immersed, with a pale-green flame; with hydrogen, mixed measure for measure, it is highly explosive anid dangerous. FLUORINE, — a gas, similar to chlorine,- exists abundantly in fluor -spar. OXYGEN, a transparent, colorless, tasteless, inodorous, innoxions gas; supports respiration and combustion, buit will not sustain life for any length of tinie, if breatihed in a pure state. It is by far the most abundant substance in existence' constitutes 1 of the atmosphere; PROPtrTIES, ETc., OF IODIES. 91, of water; and nearly the whole crust of the earth is oxidized substances. For further combinations and properties, see tables of.Elementary Constituents and Chemical Elermcnts. IODINE, - at common temperatures, a soft, pliable, opaque, bluishblack solid; taste acrid; odor pungent and unpleasant; fuses at 2250; boils at 3470; its vapor is of a beautiful violet color; it inflames phosphorus, and is an energetic poison; exists mainly in sea-weeds and sponges. HYDROGEN, - a transparent, colorless, tasteless, inodorous, innoxious gas; if pure, will not support respiration; if mixed with oxy. gen, produces a profound sleep; exists largely in water; is the basis of most liquids, and is by far the lightest substance known; burns in the atmosphere with a pale, bluish light; mixed with common air, 1 measure to 3, it is explosive; mixed with oxygen, 2 measures to 1, it is violently so. NITROGEN, or Azote, -a transparent, colorless, tasteless, inodorous gas; will not support respiration or combustion, if pure; exists largely as a constituent of the atmosphere — in animars, and in fungous plants; is evolved from some hot springs; in connection with some bodies, appears combustible. CAnBON, - the diamond is the only pure carbon in existence; pure carbon cannot be formed by art; charcoal is 97 per cent. carbon; plumbago, 95; anthracite, 93. Carbon is supposed by some to be the hardest substance in nature. A piece of charcoal will scratch glass; but it is doubtful if tis is not due to the form of its crystals, rather than to the first mentioned quality. It is doubtless the most durable. For combinations, &c., see table. BORON, - a tasteless, inodorous, dark olive-colored solid. SILIcoN, a tasteless, inodorous solid, of a dark-brown color; exists largely in soils, quartz, flint, rock-crystal, &c.; burns readily in air-vividly in oxygen gas; explodes with soda, potassa, barryta. PnosPIionus, — a transparent, nearly colorless solid, of a waxlike texture; fuses at 108~, and at 5500 is converted into a vapor; exists mainly in bones -most abundant in those of man - is poison ous; at common temperatures it is luminous in the dark, and by friction is instantly ignited, burning with an intense, hot, white flame; must be kept immersed in water. SELENIUM, — a tasteless, inodorous, opau(le, brittle, lead-colored 92 PROPERTIES, ETC., OF BODIES. solid, in the mass; in powder, a deep-red color; becomes fluid at 216~, boils at 650~; vapor, a deep yellow; exists but sparingly, mainly il combination with volcanic matter; is found in small quantities combined with the ores of lead, silver, copper, mercury. Ammoniacal gas, - N + H3; transparent, colorless, highly pungent and stimulating; alkaline; is converted into a transparent liquid by a pressure of 6.5 atmospheres, at 50'; does not support respiration; is inflammable. Carbonic acid gas, -C + 0; transparent, colorless, inodorous, dense; is converted into a liquid by a pressure of 36 atmospheres; exists extensively in nature, in mines, deep wells, pits; is evolved from the earth, from ordinary combustion, especially from the combustion of charcoal, and from many mineral springs; is expired by man and animals; forms 44 per cent. of the carbonate qf lime called marble; the brisk, sparkling appearance of soda-water, and most mineral waters, is due to its presence. It is neither a combustible nor a sup. porter of combustion; and, when mixed with the atmosphere to an extent in which a candle will not burn, is destructive of life. Being heavier than atmosphere, it may be drawn up from wells in large open buckets; or it may be expelled by exploding gunpowder near the bottom. Large quantities of water thrown in will absorb it. The above gas is expired by man to the extent of 1632 cubic inches per hour; it is generated by the burning of a wax candle to the ex-,tent of 800 cubic inches per hour; and, by the burning of "CIamnphene," (in the production of light equal to that afforded by 1 wax candle,) to the extent of 875 cubic inches per hour. Two burning candles, therefore, vitiate the air to about the same extent as 1 plrSO!1. Carbonic osxide gas, C + 0; transparent, colorless, insipid; odor offensive; does not support combustion; an animal confined in it soon dies; is highly inflammable, burning with a pale blue flame; mixed with oxygen, 1 to 2, is explosive-with atmosphere, even in small quantity, is productive of giddiness and fainting. Carburetedhydrogcn gas, - C + H'2; transparent, colorless, tasteless, nearly inodorous; exists in marshes and stagnant pools —is there formed by the decomposition of vegetable matter; extinguishes all burning bodies, but at the same time is itself highly combustible, burning with a bright hbut yellowish flame; it is destructive to life, if resplired. Cyanogf en -- Bicarulmrt o Niilrogen - a gas, - N + C2; trans pa rent, colorless, hi,;hl y pungent and irritating'; lnder a pressure of PROPERTIES ETC., OF BODIES. 93 3.6 atui.oshtiles, becomes a limpid liquid; burns with a beautiful purple flame. Hydr'ochloric acid gas - Muriatic acid gas, - -II - C1. (chlorine); transparent, colorless, pungent, acrid, suffocating; strong acid taste. Nitrous oxide gas — Protoxide of iitrogen, "laughing gas," - N + O; transparent, colorless, ino(lorous; taste sweetish; powerful stimulant, when breathed, exciting both to mental and muscular action; can support respiration but from 3 to 4 minutes; is often pernicious in its effects. Nitric oxide gas - Binoxide of Nitrogen,- N + 0~; transparent, colorless; wholly irrespirable; lighted charcoal and phosphorus burn in it with increased brilliancy. Olefiant gas- Bicarburcted hiydrjr'cn gas -" coal gas," C1 +112; transparent, colorless, tasteless, nearly inodorous, when pure; does not support respiration or combustion; a lighted taper immersed in it is immediately extinguished. It burns with a strong, clear, white light; mixed with oxygen, in the proportion of I volume to 3, it is highly explosive and dangerous. Phosphureted hydrogen gas, -P + H3; colorless; odor highly offensive; taste bitter; exists in the vicinity of swamps, marshes, and grave-yards; is formed by the decomposition of bones, mainly; is highly inflammable; takes fire spontaneously on coming in contact with the atmosphere; mixed with pure oxygen, it explodes. It is the veritable "Will o' the wisp." Sulphurctcd hydrogen gas - Hydrosulphuric acid gas, - S +- II; transparent, colorless; taste exceedingly nauseous; odor offensive and disgusting; is fiurnished by the sulphurets of the metals in general- also by filthy sewers and putrescent eggs. It is very destructive to life; placed on the skin of animals, it proves fatal. It burns with a pale blue flame, and, mixed with pure oxygen, it is explosive. Hydrocyanic acid -- Prussic acid, - N - C2 + HI; a colorless, limpid, highly volatile liquid; odor strong, but agreeable - similar to that of peach-blossoms; it boils at 790 and congeals at 0; exists in laurel, the bitter almond, peach and peach kernel. It is a most virulent poison, - a drop placed upon a man's arm caused death in a few minutes. A cat, or dog, punctured in the tongue with a needle fresh dipped in it, is almost instantly deprived of life. Ilydroftuoric acid, - F + II; a colorless liquid, in well stopped lead or silver bottles, at any temperature between 32~ and 59~. It is obtained by the action of sulphuric acid on fluor-spar. It readily acts upon and is used for etching on glass. It is the most destructive to animal matter of any known substance. Nitrohydrochloric acid -" qua regia," - (1 part nitric acid and 4 parts muriatic acid, by measure;) - a solvent for gold. The best solvent for gold is a solution of sal ammoniac in nitric acid. Nitrosulphuric acid, — (1 part nitric acid and 10 parts sulphuric acid, by measure) - a solvent for silver; scarcely acts upon gold, iron, copper, or lead, unless diluted with water; is used for separating the silver from old plated ware, &c. The best solvent for silver, and one which will not act in the least upon gold, copper, iron, -or lead, is a solution of 1 part of nitre in 10 parts of concentrated siul-, phuric acid, by weight, heated to 1600. This mixture will dissolve about i its weight of silver. The silver may be recovered by adding common salt to the solution, and the chloride decomposed by the carbonate of soda. &elenic acid, - Se + 0; obtained by fusing nitrate of potassa with selenium - a solvent for gold, iron, copper, and zinc. Silicic acid, - (Silica - silex; base Silicon) — Si- 0-'; exists largely in sand. Common glass is fused sand and protoxide of potassium (carbonate of potassa- potash) in the proportion of 1 part by weight of the former to 3 of the latter. Manganese, compounded with oxygen, in different proportions, imparts the various colors and tints given to fancy glass ware, now so generally in vogue. Butylene,- C4+ H4; sp. gr. 1.98348; a gaseous hydro-carbon derived from the distillation of coal-tar; illuminating power, compared with that of olefient gas, as 2 to 1. Propylene, - C3 + H3; sp. gr. 1.4511; a gaseous hydro-carbon derived from the distillation of coal-tar; illunminating po wer, compared with that of olefient gas, as 1.5 to 1. Napthaline Vapor, - C +II4, the vapor of solidified olefient gas. Tturpentihne Vapor, - CI' + I18. SECTION III. PRACTICAL ARITHMETIC. VULGAR FRACTIONS. A fr'action is one or more parts of a UNIT. A vulgar fr'action consists of two terms, one written above tile other, with a line drawn between them. The term below the line is called the denominator, as showing tle denomination of-:the fraction, or number of parts into which the unit is broken. The term above the line is called the numerator, as numbering the parts employed. These together constitute the fraction and its value. A vulgar fraction always denotes division, of which the denotniniltor is the divisor and the numerator the dividend. Its value as a unit is the quotient arising therefrom. A simplefJ'action is either a proper or improper fraction. A proper fraction is one whose numerator is less than its dfenominator, as,-,, 2A, &c. An improperJiaction has its numerator equal to or greater than its denominator, as i, 4, 2, &c. A mixedifraction is a compounld of a whole number and a fraction, as I, 5a1, la, &c. A compound fraction is a fraction of a fraction, as a of ~; a of a of l, &c. A counplex fraction has a fraction for its numerator or denom4 4 2 5A inator, or both, as, I and is read 4 - 4; A_;; 5 4, &c. REDUCTION OF VULGAR FRACTIONS. To rceduce afiaction to its lowest terms. This consists in concentrating the expression without changing the value of the fraction or the relation of its parts. It supposes division, and, consequently, by a measure or me;,sures common to both terms. It is said to be accomplished when no number greater tha;:l 1 will divide both terms without a remainde;: - therefore. 96 VULGAR FRACTIONS. RULE. — Divide both terms by any number that will divide them ~ ithout a remainder, and the quotient again as before; continue so to do until no number greater than 1 will divide them, - or divide by thie greatest common measure at once. EX AMPLE. - Reduce,861T:. to its lowest terms. A8) 56_4y ~ 216. 2 -S= - 9=1 — 3 4. Ans. To rcduce an improper fraction to a mixed or whole number. RULE. — Divide the numerator by the denominator and to the whole number in the quotient annex the remainder, if any, in form of a fraction, making the divisor the denominator as before; then reduce the fraction to its lowest terms. EXAMPLE. 4 = 1 1- 1- -13; 2- =2. To reduce a mixedfraction to an equivalent improper fraction. RULE. - Multiply the whole number by the denominator of the fiactional part, and to the product add the numerator, and place their sun over the said denominator. EXAMPLE. -Reduce 3 and 12. to improper fractions. 3 X 4 12- 1-. Ans. 12 X 9 +8 16. Ans. To reduce a whole number to an equivalent fraction halving a given denominator.. RULE. - Multiply the whole number by the given denominator, and place the said denominator under the product. ExAvLLE. - How may 8 be converted into a fraction whose denominator is 12? 8X 12 =. Ans. To reduce a compoundfiraction to a simple one. RULE. Multiply all the numerators together for a numerator, and all the denominators together for a denominator; the fraction thuis formed will be an equivalent, but often not in its lowest terms. Or, concentrate the expression, when practicable, by reciprocally expunging', or writing out, such factors as exist or are attainable common to both terms, and then multiply the remaining terms as directed above. NOTE. - This last practice is called cancellation. or cancelling the terms. It consists, as has been stated, in reciprocally annulling, or casting out, eqlotl values fromi bothl termls, wvtlhreb)y the expression is concentrate(d, and the rel;tion of the parts kept undisturbed; a,.l it may always be carried to the extent of reduLcilig the fiactitm to its lowest torins, before any multiplication, as final, is resortod to; and often, therefore, to the extent that such multiplication is inadmissible, the terms haviiig been cancelled by (naci other until tbut a single nuilmler is left in each. VULGAR FRACTIONS. 97 EXAMPLI:.- Reduce L of j of L to a simple fr;action. Operation by multiplication, - X 3 X I 6 1. Ans. Operation by cancellation, - = =1. Ans. EXAMPLE. -Reduce 2 of of -ofg of of 2 to a siiniple fraction. By multiplication, X ) X -- X I X a X X 3X 2Q a A ns The last example stated 2 3 12 6 5 2 for cancellation, | 3 4 8 8 9 PROCESS OF CANCELLING THE ABOVE. 1. The 3 in nlm. equals the 3 in denom., therefore erase both. 2. The first 2 in num. equals or measures the 4 in denom. twzice, therefore illace a 2 under the 4, and erase the 4 and 2 which measured it- (as 4: 2:: 2: 1.) 3. The 2 (remailling factor of 4 and 2 erased) in denom., and the remaining 2 in nunl., will cancel each other, - erase them. 4. The 12 anld 6 in lmol. - 72, andl the 9 and 8 in denom. = 72; these, therefore, in their relations as factors equal each other, and may be erased. The remaining factors represent the true value of the compound fraction, and will be found -, as by multiplication. EXAMPLE. Reduce I a of 7-2 to a simple fraction. 3 3 1X'Or, [ X I~ (= 18+ 6, and 12. 6) =- U X fa 13X!1 3 X-l~ =a.~ Ans. 2 To reduce two or rmorefracCtions to a common denominator. RULE.-Multiply each numerator by all the denominators except its own, for the new numerators; and multiply all the denominators together, for a common denominator. NOTE.-Whole numbers and fractions other than simple, must first be reduced to simple fractions before they can be reduced to fractions having a common denominator. EXAMPLE.-Reduce j and - to fractions having a common denominator. 2 2x4 *8 3 3x3 9 ]% + ~; - = and -s; that is, 8 - 8 x 4 = 12' and 4 4 x -l; and 2, 3 3 4 12 —4and 3 -~j and?- are fractions having a common denominator. EXAMPLE,-Reduce e, a,', and -' to fractions having a common denominator: 1+~+~+~, = ~11 12 a II,: 10a -6 0 12 job n II + 8, -3 2 440~ 240 4, 4 244 - I.S, 120, I12t 120 6,.875, 4-. 9 98 VULGAR FRACTIONS. T1o 7 cduce a complex fraction to a simple one. 1 lULE — Multiply the numerator of the upper fraction by the denominator of the lower, for the new numerator; and the denominator of the upper by the numerator of the lower for the new denominator. EXAMPLES. Reduce; - and 5- each to a simple fraction. 3 a3' 4, _ * a 1. 4- *. 13; 2 - Ix4 4 X2,; 51: 2~iR' T T' ~- ~~L~-3 ) Z ~2 4 2'6 3' 2 &A, andri d 1 aL Ans. L-1, and 12 X -, = 1. As. To reduce Vulgar Fractions to equivalent Decimals. RULE.- Divide the numerator by the denominator; the quotient is the decimal, or the' whole number and decimal, as the case may be. ExAMPLE. -Reduce i, 43, 14, to decimals. 7 8 0.875; 4 _,= 4.6; 14 1 — =1.166 +. Ans. To find the greatest common measure of two or more given numbers. RULE. - Divide the given numbers by any measure common to them all, and set the quotients in a line beneath; then divide the quotients by any measure common to them, and set the quoticnts beneath; and so on until the quotients are no longer coinmon multiples of any one number greater tban unity; the product of all the divisors or common measures employed will be the greatest common- measure. EXAMPLE. -What is the greatest common measure of 84 and 36? Also of 32, 24, and 16? Also of 182, 104, and 52? 4)84.36 8)32.24.16 2)182.104.52 3)21.9 4.3.2.-8. Ans. 13) 91.52.26 7.3-4X3=-12. Ans. 7. 4.2.-26. Ans. NOTE. - When any number in the series is prime to either of the others, the numbers are collectively incommensurable; that is to say, their greatest common measure is unity, or 1. To find the least coemm on multilple of two or more given numbers. RULE. -Divide all the given numbers that are commensurable with each other by any measure that is common to them, and set the quotients, together with the undivided numbers, if any, in a line beneath; then divide the quantities in the second line as before, and so on until no two quantities in the last line are common multiples of any number greater than unity, or 1; the product of all the commnon measures employed into the product of all the numbers in the last line will be the least common multiple of the given numbers. VULGAR FRACTIONS. 99 EXAMPLE. - WVhat is the least common multiple of 27 and 36? Also of 182, 104, andl 52? Also of 24, 14, 12, and 7? 9)27.36 2)182. 104.52 3 4 —4 X 3 X 9 108. Ans. 13) 91. 52.26 2) 7.4.2 7. 2. 1.728. Ans. 7) 24. 14.12.7 2)24. 2.12.1 6) 12 1 6 2.1 1. 168. Ans. ADDITION OF VULGAR FRACTIONS. Sum of` thle Irolducts of each numerator with all the denominators except that of the numerator involved, fortms numerator of sum. Product of all the denominators firms denominator of sum. RULE.:-Arrange the several fractions to be added, one after another, in a line from left to right; then multiply the numerator of the first by the denominator of the second, and the denominator of the first by the numerator of. the second, and add the two products iogcelher for the numerator of the sum; then multiply the two denominators together for its denomninator; bring down the next fraction, and proceed in like manner as before, continuing so to do until all the friactions have been brought down and added. Or, reduce all to a common denominator, then add the numerators together for the numerator of the sunm, and write tile common denominator beneath. EXAM PLES. - Add together 1, ~,, and. I 7 +-.1 46 4 _ _ 234 = 3 31. A~ns. 2 3 -6 4 2 -r 3 - — 2 - I 2 + 3 - - 1 5 and 9 + 4 - 6- R and +r 2.4, Iz Z " 4 4, 2- 4h I TY 3 -f-a - U -12 12 la 12 3 AAAns. +=~-{"= I —,sand - =-J'- __24, and _t~ +.4 4 4 ==-?-34a. An~s. SUBTRACTION OF VULGAR FRACTIONS. P'ioduct of numerator of minuend and denominator of subtrahend, forms numerator of minuent, for common denominator. Product of numerator of subtrahend and denominator of minuend, forms numerator of subtrahend, for common denominator. Pr6duct of denominators forms common denominator. Difference of new found numerators forms the numerator, and common denominator lithe,ehmminator, of the difference, or remainder sought. RULE. — Write the subtrahend to the right of the minuend, with lhe sign (-) between them; then multiply the numerator of the minuend by the denominator of the subtrahend, and the denominiator of ilie minuend by the numerator of the subtrahend; subtract ihe latter!irdut from the former, aid to the remainder or difference affix the 100 VULGAR FRACTIONS. product of the two denominlators for a denominlator; the sum thus formed is the answer, or true difference. EXAMPLES. - Subtract I from', also a from -T. 4 = 684, = ~. An. 11 3 55-.5.84 Ans. ~ 5=-=& Ants. DIVISION OF VULGAR FRACTIONS. PrIoduct of numerators of dividend and denominators of divisor, fornls numerator of quotient. Product of denominators of dividend and numerators of divisor, forms denominator of quotient; therefore, RULJE. -Write the divisor to the right of the dividend with the sign (~) between them; then multiply the numerator of the dividend,by the denominator of the divisor, for the numerator of the quotient, and the denominator of the dividend by the numerator of the divisor, for the denominator of the quotient. Or, invert the divisor, and multiply as in multiplication of fractions. Or, proceed by cancellation, when practicable. EXAMPLES. - Divide 1 by -; 4 by 1; 4 by 1; and 2 of o 2 4 2 -; and 2 ~ f 8 of 4 by, of I of 3 of 2 I *-4 4.. 6. 4 2. X 23 =X,-. Ans 2'45m 4~ 24 I 3RU, 3 x IX1 1 X = X = 6 - 5 and 1 X 1 X xI. 2 6 1 iE 4 Y T- 4 -TT 4 2 3 -'2' and 5z. s_ I = 62. Ans. FORM FOR CANCELLATION. — EXAMPLE LAST GIVEN. 1 3 5 4 4 2 4 3 20 Ans., as above. 2 4 6 3, 1 3 2 3 Ans., NOTE.- The fitregoing example can be cancelled to the extent of leaving but a 4 and a 5 (= 20) numuerators, anti a 3 denominator. Units, or l's, in the expressions, are valueless, as a sum multiplied by I is not increased. MULTIPLICATION OF VULGAR FRACTIONS. Product of numerators of multiplier and multiplicand, forms numerator of produlct. Product of denominators of multiplier and multiplicand, forms denominator of product. RULE. - Multiply the numerators together for a numerator, and thle denominators together for the denominator. EXAMPLES. - Multiply I by [; - by 7; T - by 1J2;1 of 2 o(f' 4 tby 3 of I of 2. 1) 1 =1; 1 >lying by the denominator of a fractirln of e(lial valule whose nutnerator is l, or miultiplying by the denominator of a fi'acti(in of eiual value whose numerator is more than 1, and dividing the product by the nurmerator. D)ividing by a fraction is equivalent to multiplying by its. denominator andl dividilig the proltuct by its numerator, or dividing by its numerator and multiplying the quotient by its denominator. Thus,.5. ---, and.75 =-5 = I. Andi 12.24.5 — 2 41, alll 12.24 X 2 = 24.48. So, also, 12.24 -.75 16. 32, and 12.24 X 4 -- 48.96 - 3-:16.32. This method of accomplishing divisionl Iay often be resoried to with convemlience. REDUCTION OF DECIMALS. To reduce a decimnal in a higiher to whole numbers in successive lo7,our? idenominations. RULE. -M ultiply the decimal by that number in the next lower denomination that equals ONEt of the denomination of the decimal, and point off as many places for a remainder as the decimal so multiplied has places. Mulltiply tle reimainlder by the number in the next lower denomination that equals I of thle denomination of the remainder, and pioint off as befbre; so continue, until the reduction is carried to the lowest denomination required. EXAMPLE. - What is the value of.62525 of a dollar?.62525 100 Cents, 62.525()0 10 Mills, 5.25000 An.. 62 cents 5i mills. l08 DECIDMAL FRACTIONS. ExAMPLE. - What is the value of.46325 of a barrel?.46325 32 Gallons, 14.82400 4 Quarts, 3.296 2 Pints,.592 4 Gills, 2.368. Ans. 14 gals. 3 qts. 2 U6~- gills. EXAMPLE.- HOW many pence in-.875 of a pound?.875 X 240 = 210. Ans. To reduce decimals, or whole numbers and decimals, in lower denominations, to their value in a higher denomination. RULE.- Reduce all the given denominations to their value in the lowest denomination, then divide their sum by the number required of the lowest denomination to make ONE of the denomination to which the whole is to be reduced. EXAMPLE. - Reduce 14 gallons, 3 quarts, 2.368 gills, to the decimal of a barrel. 14 X 4 = 56 + 3 -59 X 8 = 472 + 2.368 = 474.368. 8 X 4 X 32 1024 ) 474.368 (.46325. Ans. To work decimals, or whole numbers and decimals, by the Rule of Three, or Proportion. RuLE. - State the question and work it as in whole numbers, taking care to point off as many places for decimals in the product to lie used as the dividend, as there are decimals in the two terms which form it, and to remove the decimal point therein as many places to the right as there are decimals in the term to be used as a divisor, before the division is had. EXAMPLE. — If.75 of a pound of copper is worthl.31 of a dollar how much is 3.75 lbs. worth?.75:.31::3.75.31 375 1125.75 ) 1.16,25 ($1.55. Ans. rROrORTION. 109 PROPORTION, OR RULE OF THREE. TIE RULE OF PROPORTION involves the employment of three terms -a divisor and two factors for forming a dividend —and seeks a quotient, which, when the proposition is written in ratio, bears the same relation to the third term that the second term bears to the first. Two of the terms given are of like name or nature, and the other is of the name or nature of the quotient or answer sought. That of the nature of the answer is always one of the factors for forming,the dividend, and, if the answer is to be greater than that term, the larger of' the remaining two is the other; but if the answer is to be less thlan that term, the less of the remaining two is the other- the relnaining term is the divisor. EXAMPLE.-If $12 buy 4 yards of cloth, how many yards will.$108 buy? 4 X 108 108 -af - = - = 36 yards. Ans. %3 3 -EXAMPLE. - It 4 yards of cloth cost $12, how many dollars will 36 yards cost? 12 X 36 -X 108 dollars. Ans. 4 EXAMPLE. - If 30 men can finish a piece of work in 12 days, how many men will be required to finish it in 8 days? 30 X 12 = 45 men. Ans. EXAMPLE. -If 45 men. require 8 days to finish a piece of work, how many men will finish the same work in 12 days? 45 X 8 12 -30 men. Ans. EXAMPLE. - If 8 days are required by 45 men to finish a piece of work, how many days will be required by 30 men to finish the same work?'8 X 45 30 = 12 days. Ans. 30 ENXAIMPLE. -If 12 days are required by 30 men to perform a piece of work, how many days will be required by 45 men to do the samle work? 12 X 30 12 X 30 8 days. Ans. 45 EXAMPLE.- I borrowed of my friend $150, which I kept 3 months, and, on returning it, lent himn $200; how long may he keep tile sum 10 110 COMPOUND PROPORTION. that the interest, at the same rate per cent., may amour. t to that which his own wolll have drawn? 150. X 3 + 200 = 2: months. Ans. I:XAAIll,:LE. -- A garrison of 250 men is provided with provisions for 30 days, how many: men must be sent out that the provisions may last those remaining 492 days: 250X< 30 42 - 179, and 250 - 179 = 71. Ans. IXARMPEI:. If to the short arm of a lever 2 inches from the fulcrum there be suspended a weight of 100 lbs., what power on tile long arm of the lever 20 inches from the fulcrum will be required to raise it? 20: 2 100 = 10 lbs. Ans. I',XAMPLE. - At what distance from the fulcrum on the long arm of a lever must I place a pound weight, to equipoise or weigh 20 lbs. suspended 2 inches from the fulcrum at the other end l 1: 2::20 40 inches. Ans. Nolr. I- f we examine the foregoing with reference to the fact, we shall see that every proiposition in simple proportion consists of a term and a half! or, in other words, of a co.poundcl terin consisting of two factors, andl a factor for which another factor is sought that together shall equial the coImpound. We have only to multiply the factors of tile conlipouiid together - a.nd a little observation will enable us to distinguish it - and divide by the remnaining factor, and the work is accompl)ished. See COMPOUND PROPORTION. COMPOUND PROPORTION, OR DOUBLE RULE OF THREE. COMPOUND PROPORTION, like single proportion, consists of THIREE terms given by which to find a fourth - a divisor and two factors for forming a dividend —but unlike single proportion, one or more of the terms is a compound, or consists of two or more factors; and sometimes a portion of the fourth term is given, which, however, is always a iart of the divisor. Of the given terms, two are suppositive, dissimilar in their natures, antd relate to each other, and to each other only; and upon their relat.iln the whole is made to depend; the remaining term is of the nature of one of the former, and relates to the fourth term, which is of the altlilre of the other.'!'he object sought is a number, which, multiplied into the factor or clttors of the fourth term.given, if any, and if not, which of itself', hIears the same proportion to the dissimilar term to which it relates,;I:s the suppositive term of like nature bears to the term to which it relates. RULE. — Observe the denomination in which the demand is m;ade,.and of the suppositive terms make that of like nature the secoind,;il, hIhe other the first; make the remaining term tile third term; a;ll, ii' COMPOUD i PROPORTION. 111 there are any factors pertaining to the fourth term, affis them to the first; multiply the second and third terms together and divide by the first, and the quotient is the answer, term, or portion of a term. sought. EXAMPLE. -If 12 horses in 6 days consume 36 bushels of oats how many bushels will suffice 21 horses 7 days 2 12 X 6: 36::21 X7:x. 36 X 21 X 7 147. - _ 73 3A bushels. Ans. 2 EXAMlPLE.- If 12 horses in 6 days consume 36 bushels of oats, lhow many horses will consume 733 bushels in 7 days? 36: 12 X 6:: 73a: 7 X x. 12 X 6 X 73. 147 -36 X 7 = 21 horses. Ans 36 X7 7 XAIMPLE. -If the interest on $1 is l.l cts. for 73 days, (exact interest at 7 per cent.,) what will be the interest on $150.42 for 146 days. 73: 1.4:: 150.42 X 146: x. 1.4 X 150.42 X 146 $4.21. Ans. 73 EXAMPLE. -If the interest on $1 is 1.2 cts. for 73 days, (exact interest at 6 per cent.,) whtat will be the interest on $1~t5 for 90 days? 73: 1.2 125 X 90: x $1.85. Ans. EXAMPLE. - If $100 at 7 per cent.. gain $1.75 in 3 months, how much at 6 per cent. will $170 gain in 11 months? 100 X 7 X 3: 1.75:: 170 X 6 X 11.5: x. 1.75 X 170 X 6 X 11.5 -- 100 X 7 X 3 = $9.77,5. Ans. EXAMPLE. - By working 10 hours a day 6 men laid 22 rods of wall ill 3 days; how many men at that rate, who work but 9 hours a day, will lay 40 rods of wall in 8 days? 22: 6 X3X 10:: 40: 9X8X x. 6 )' 3 X 10 X 40.S22 X 9 X 8- 4-T. Ans. EXAMPLE. -If it costs $112 to keep 16 horses 30 days, and it costs as much to keep 2 horses as it costs to keep 5 oxen, how much w:ill:it cost to keep 28 oxen 36 days? 112 CONJOINED PROPORTION, OR CIAIN RULE. 16 X 30: 112: 2 X 28 X 260: x. Or, - 16 X 30 X 5: 11'2:: 28 X 36 X 2:x. 7 12 I1 2s8 $' ~ 28X 12 X 7 -— 8$0' 8 X 12X7_ $94.08. Ans by 6 0 5 5X 5 5 EXAMPLE. -If 24 Inen, in 8 days of 10 hours each, can dig a trench 250 feet long, 8 feet wide, and 4 feet deep, how many nell, in 12 days of eight hours each, will be required to dig a trench 80 feet long, 6 feet wide, and 4 feet deep? 250X 8X4:2IX 8X 10::80X6X4: 12X8Xx=5-. Ans. EXAMPLE. -If 120 men in six months perform a given task, working 10 hours a day, how many men will be required to accomplish a like task in 5 months, working 9 hours a day? 120 X 6 X 10 = 5 X 9 X Tr. Or,-1 1'20 X X 10:: 1: 5 X 9 Xx. 160. Ans. EXAMPLE. - The weight of a bar of wrought iron, 1 foot in length, 1 inch in breadth, and 1 inch thick, being 3.38 lbs., (and it is so,) what will be the weight of that bar whose length is 124 feet, breadth 34 inches, and thickness S of an inch? 1:3.38:: 12.5 X 3.25 X.75: x. Or,- 1: 3.38 X X, and 3.38 X 25 X 13 X 3 = 10298+ lbs. Ans. X44 of a ar f wrught irn, EXAMPLE. - The weight of a bar of wrought iron, one foot in length and 1 inch square, being 3.38 lbs., what length shall I cut from a bar whose breadth is 2j inches, and thickness A inch, in order to obtain 10 lbs.? 3.38: 1:: 10': JL- X X I X 10 X 4 X 2 = 2 feet 1-4 inches. Ans. 3.38 X 11 X 1 CONJOINED PROPORTION, OR CHAIN RULE. THE CHAIN RULE is a process foir determining the value of a given quantity in one denomination of value, in some other given denomination of value; or the immediate relationship which exists between two denominations of value, by means of a chain of approximate steps, CONJOINED PROPORTION, OR CHAIN RULE. 113 circumstances, or equivalent values, known to exist, which connect them. In every instance at least five terms or values are employed in the process, and in all instances the number employed will be uneven. A proposition involving but three terms, of this nature, is a question in single proportion. The equivalent values employed are divided into antecedents and consequents, or caulses and effects~; and the value or quantity for which an equivalent is so,,ght, is called the odd term. RULE.- 1. When the value in the dcnomilaio(n of the first anlececlent is sought of a given quantity in the denonmination (f the last consequent. - Multiply all the antecedents and the udd term together for a dividend, and all the consequents together for a divisor; the quotient will be the answer or equivalent sought. RULE. -2. When the value in the denomination of the last consequent is sought of a given quantity in the denomination oJ the first antecedent. Multiply all the consequents and the odd term together for a dividend, and all the antecedents together for a divisor; the quotient will be the answer required. EXAMPLE. - I am required.to give the value, in Federal money, of 5 Canada shillings, and know no immediate connection or relationship between the two currencies -that of Canada and that of the United States. The nearest that I do know is that 20 Canada shillings have a value equal to 32 New York shillings, and that 12 New York shillings equal in value 9 New England shillings, and that 15 New England shillings equal $2.50; and with this knowledge will seek the value, in Federal money, of the 5 Canada shillings. 2.50 X 9 X 32 X 6'-X X 1. Ans. 15 X 12 X 20 EXAMPLE. - If $2. equal 15 New England shillings, and nine shillings in New England equal 12 shillings in New York, and 32 shillings in New York equal 20 shillings in Canada, how many shillings in Canada will equal $1. 15 - s0 1 at, (99,, 1j~ -- 5 shillings. Ans. 3 % EXAMPLE. -If 14 bushels of wheat weigh as much as 15 bushel, of fine salt, and 10 bushels of fine salt as much as 7 bushels of coarse, and 7 bushels of coarse salt as much as 4 bushels of sand, how many bushels of sand will weigh as much as 40 bushels of wheat? 1.5 X 7 X0 4 X 40 - 17+- bushels. Ans. 14 X 10 X 7 10 114 DPERCENTAGE. PERCENTAGE. Pure percentage, or PERCENTAGE, is a rate by the hundred of a part of a quantity or number denominated the principal, or basis. But percentage, considered as a means, and as commonly applied, is mixed and related in an eminent degree; and in this light may be regarded as divided into orders bearing different names. Thus Interest is percentage related to intervals of time in the past. Discount is percentage related to interest, and intervals of time in the.future. Profit and Loss is comparative percentage, or percentage felated to the positive and negative- interests in business, etc., etc. Pure percentage is commonly called BROKERAGE when paid to a broker for services in his line. It is called COMMISSION when paid to or received by a factor or commission merchant for buying or selling goods. It is called PREMIUM by an insurance company, when taken for insuring against loss. It is called PRIMAGE when it is a charge in addition to the freight of a vessel, etc. Comparative percentage relates to the differences of quantities, and is confined always to the idea of more or less. It implies ratio. Thllis description of. percentage, though much in practice, seems not to be well understood; and often a quantity is indirectly stated to be many times less than nothing, or many times greater than it is. The difference of two quantities cannot be as great as a hundred per cent. of the greater, however widely unequal the quantities may be, nor as small as no per cent. of the greater or lesser, however nearly equal they may be. No quantity or number can be as small as 1 time less than another quantity or number; and therefore cannot be as small as 100 per cent. less. But, since one quantity may be many by 1 time, or many times greater than another with which it is compared, it may be said to be many by 100 times, or many hundred per cent. greater. When one of two quantities in comparison is stated to be three times less, or three hnndred per cent. less, for instance, than the other, the expression is incorrect and absurd. The meaning evid(lntly is, that it is two-thirds less, or only one-third as large as the other,- that it is 662 per cent. less, or only 33~ per cent. as large as the other. In common comparison, 1 is the measuring unit. In percentage, 100 is the measuring unit. Let a principal. b percentage. s amount (sum of tlle principal and percentage). d =difference of the principal and percentage. r rate of the percentagc. p rate per cent. of tlle percentage. a - s b — b' r= 100b. p 100s. (100 +-), b s-a- ar cap + 100, p 100r =100b -- a 100(s - a) 4 a, r=p — 100-b a (s-a) *- a, s a+b =a(1 + r)- a(100 +p) - 100, d = a - b = 2a - s = s - 2b =- a(1 -r). To find the Percentage. EXAMPLES. What is 4 of 1 per cent. of $200? b-ar = ap * 100 $0.50. Ans. 8 of 2 per cent. of 50 is what part-of 50? 50 X 8 X 2 = Ans. 7 X 100 What is 8 of - of I of 24 per cent. of 150 lbs.? 150 X 12 - 100 18 lbs. Ans. What is 2*3 per cent. of 19 bushels? Jo X ~0~ —0.45125 bushels. Ans. Bought a job lot of merchandise for $850, and sold it the same day, brokerage, 21 per cent., fobr $975; what was the net gain? s -sr- a- s (sr- +a) = s(1l- r) - a - 975 -975 X.025 - 850 — $100.625. Ans. To find the Rate or Rate Per Cent. EXAMPLES. What per cent. of $20 is $2? r=-b.- a, lOOb -- a-10 per cent. Ans. 12 dozen is equal to what per. cent. of 2 dozen? 12 - 2 6, 600 per cent. Ans. 116 PERCENTAGE. What part of 5s lbs. is 4 of 2 lbs.? rX4 if.=61 - 0.27 1. Ans. 241 per cent. is what per cent. of 364 per cent.? 66 per cent. Ans. For an article that cost $4, $5 were received; what per cent of $4 was received? p -5 X 100 + 4 = 125 per cent. Ans. A farmer sowed 4 bushels of wheat, which produced 48 bushels; what per cent. was the increase? 48 is more than 4 by what per cent. of 4? The diffeirence of 48 and 4 is what per cent. of 4? a -b a 100(a-b) 48- 4 _ 48 4 - 1 _ b --,p p 4 100(48- 4) - 4- 1100 per cent. Ans. WVhat per cent. would have been the decrease, if he had sowe'd 48 bushels, and harvested only 4 bushels? 4 is less than 48 by what rate of 48? The difference of 48 and 4 is what per cent. of 48? r (a-b) *. - a1 _ - = 0.91i, or 912 per cent. Ans. a Since water is composed of 8 atoms of oxygen and 1 atom of hydrogen, what per cent. of it is oxygen'? 8 is what per cent. of the sum of' 8 and 1? a b 100a 8 r' - -,p1.8889-, a + b_1 a- b' at- +b 8 1 or 88.89-per cent. Ans. What per cent. of it is hydrogen? 1 is what per cent. of the,'im of 8 and 1? a b lOOb 1 r --. - - -1111-, or a + b a+b' P a +b=- 8+1- l+or 11.11 + per cent. -Ains. How many volumes of water must be added to 100 volumes of 90 per cent. alcohol to reduce it to 50 per cent. alcohol or common proof? 90 is more than 50 by what per cent. of 50? The differk-nce of 90 and 50 is what per cent. of 50? (a -) 100 (90 -50)100. Ans. b'50 PERCENTAGE. 117 How many volumes of 50 per cent. alcohol must be added to 100 volumes of 90 per cent. alcohol to produce 80 per cent. alcohol? 90 is more than 80 by what per cent. of the difference of 80 and 50? The difference of 90 and 80 is what per cent. of the difference of 80 and 50? (a-b)100 (90-80)1-00 b- b' 80 — 50 Ans. How many volumes of 90 per cent. alcohol must be added to 100 volumes of 50 per cent. alcohol to raise it to 80 per cent. alcohol? 50 is less than 80 by what per cent. of the difference of 90 and 80? The difference of 80 and 50 is what per cent. of the difference of 90 and 80? (b- b) 1.00 _ (80 - 50)100 00. An. -- 300. Ans. a —b 90 —80 If to 2 volumes of 95 per cent. alcohol, 1 volume of 50 per cent. alcohol be added, what per cent. alcohol will be the mixture? The sum of 50 and twice 95 is what per cent. of the sum of 2 and 1? 2a + b _ 2 X 95 + 50 80 per cent. Ans. 2 -1 - 2-+1 In a barrel of apples, the number of sound ones was 60 per cent. greater than the number that were damaged. What per cent. less was the number that were damaged than the number that were sound? 60 per cent. is what per cent. of the sum of 100 per cent. and 60 per cent.?.6 is what rate of 1 +.6? a 100 O.a 1 60 r- -1- -- 1 -- --.375, or 1 +-a 1 1. a 1. a 1a+ 60 371 per cent. Ans. Since the number of damaged apples was 37i per cent. less than the number that were sound, what per cent. greater was the number that were sound than the number that were damaged? r=a' (1 -a) =. (1- a) - 1 = 60 per cent. Ans. Since the number of sound ones was 60 per cent. greater than the number that were damaged, what per cent. of the whole were sound? a+a2 1 +.a 100+60__ r 2a - - 2 2 + - 100 - 80 per cent. Ans. 2a 2 - 2 What per cent. of the whole were damaged? (100- 60). 2 -20 per cent. Ans. Since 20 per cent. of the apples were damaged, what per cent. less was the number that were dainaged than the number that were sound? 1 — 2.a 1 100- 2a 4 100 r 20 —.a 2 - 2. a 2-2.a' 200- 2a 200 40 37j per cent. Ans. What per cent. greater was the number that were sound than the number that were damaged? r 2-(1 +2.a) 2 - 2.a I 60 per cent. Ans. Since 80 per cent. of the whole were sound, what per cent. less was the number that were damaged than the number that were sound? 2.a- 1 1 2X.80-1 r = — - - - -— 8 —- 37k per cent. Ans. 2.a 2.a 2 X.80 Since the number of damaged ones was 37j per cent. less than the number that were sound, what per cent. of the whole were sound? 1 _ 100 100 r2-2.a' P 2 —2 a 2 X 37 5 SOper cent. Ans. Since 80 per cent. of the whole were sound, what per cent. greater was the number that were sound than the number that were damaged? 2 -.a r 2 2.a - 1 2 X.80 1 60 per cent. Ans. 2 Lost 20 per cent. of a cargo of coal by jettison, and 5 per cent. of the remainder by screening, what per cent. of the coal was saved? a -bt- d } r- (1 —- rt) (1 -rl) - (1-.20) -(1 -.20) dt - bI - d"t X.05- (1 -.20)(1 —.05) - 76 per cent. Ans. d" - b"' - d"', &c. Yesterday drew 12 per cent. of my balance of $4,273 in the bank, and deposited $1,000; and to-day have drawn 31- per cent. of the balance left over, or as it stood( last night.:What per cent. of the sum of the first-mentioned balance and deposit of yesterday have I drawn? b'+- b" 512 +- 1487.575 r 4 7 1 37.9354 -- per cent. Ans. a —+m 42173 -+- 1000 PERCENTAGE. I1g tWhat per cent. of the said sum is remaining in the bank? 1-b + b"' a +- m — b'- + a - m -b) I I -- = 62.0646 - per cent. Ans. What per cent., predicating it upon the first-mentioned balance, have I drawn? bt + btt 512.76 -+ 1487.576 r..5.. — =46.8134- per cent. Ans. a 4273 What per cent. have I drawn, predicating it upon what I now have in the bank? b' +- b" b +i bi" a b +- m -bi" a -+ -m — (b' +,)-61.1225 + per cent. Ans. What amount of money must I deposit to make good 629 per cent. of the aforementioned sumn? d = r (a + m) + b' + b" - (a + m) = r (a - m)-0 1" = $22.96. Ans. To find t7he Principal or Basis. EXAMPLES. The percentage being 250, and the rate.06, what is the principal? a=b_ — r 100b *p p-250 --.OG =25,000 6 - 4,1661. Ans. A tax at the rate of A of 1 per cent. on the valuation was $27.50. WVhat was the valuation? a- = $3,300. Ans. Sold 120 barrels of flour, which amounted to 12 per cent. of a certain consignment. The consignment consisted of how many barrels? 120-0.12 1,000. Ans. 216 bushels is more by 8 per cent., or 8 per cent. more. than whllat number of bushels? 8 per cent. more than what number is equal to 216'? WVhat number, plus 8 per cent. of it, will make 216' a-=s -- (1+r) =216 - 1.08 200. Ans. 200 Ibs. is less by 8 per cent., or 8 per cent. less, than what ntlm 120 INTEREST. ber of lbs.? 8 per cent. less than what number is 200? What numlber, minus 8 per cent. of it, is equal to 200? a =d' (1-r) = 200 - (1-.08)-=2179. Ans.. 217 -- 2l79 X.08 - 200 = a,- b=- d a (1 - r). To a quantity of silver, a quantity of copper equal to 20 per cent. of the silver is to be added, and the mass is to weigh 22 ounces. What weight of silver is required? a = s+ (1 +-r)=- 22 1.2 = 18A ounces. Ans. What weight of copper is required? S S7' s- - - -r1+ 32 ounces. Ans. To a quantity of copper, a quantity of nickel equal to 62, per cent. of the copper, a quantity of zinc equal to 33~ per cent. of the copper, and a quantity of lead equal to 5 per cent. of the copper, are to be added; and the whole is to weigh 40k pounds. The weighlt of each constituent of the alloy is required. s 40k 1 +- rf- r' +- r1 -- 1 +.6,2:+. 33 +-. 05 = 20 lbs. of copper, 1 b = 20 r = 1 2 lbs. of nickel, A bf= 20 rf 6 2 lbs. of zinc, bt -- 20 r1- 1 lb. of lead. J INTEREST. Universal for any rate per cent. T = time in months and decimal parts of a month; t= time in days; P = principal; r = rate per cent., expressed decimally; i=interest. PXTXr PX t Xr 12 365 12 i 365 i 12 i 3G5 i 12i 3G5 i Tr tr Pr Pr PT Pt EXAMPLE. — A promissory note, made April 27, 1864, for INTE1REST. 121 $325- anld intceestat at 6 per cent., matured Oct. 6, 1865: what was the interest? Oct. is 10th mointh. Time from April 27 to Oct. 6 (one of Aprilis 4th mollt. the dates always included) - 162 days,: m. dt. which, added to the 365 days in the year 1865. 10 __'64. 4. 27 preceding - 527 days. NoTE. —One day's interest at least is gencr. ally lost by computing the time in years and Time 1I. 5. 9 months, or months, instead of days. 825.25 X 17.3 X.06 - 12- $71.38. Ans. 825.25 X 527 X.06 365 = $71.49. Ans. To find a constant divisor, k, for any given rate per cent. When the time is taken in months, k - 12 - r. When the time is taken in days, k = 365 - r; thus, PXt Tlen tlte RATE is 6 per cent. 60 n3 =Interest. PXt lWhen the RATE is 7 per cent. 5214 Interest, &c. EXAMPLE. - Required the interest on $750 for 93 days, at 7 per cent. 750 X 93 5214 = $13.38. Ans. EXAMPLE. - What is the rate per cent. when $450 gains $94i in 3 years? 450: 100 94.5 ~ 3x = 7 per cent. Ans. 94.5. 3 X 450=.07. Ans. EXAMPLE. - In what time will $125 at 6 per cent. gain $18:? 6: 100 18.75: 125 X x = 2 years. Ans. 18.75 125 X.06O= 22 years. Ans. EXAMPLE. - What principal at 5 per cent. interest will gain $167 in 18 months? 5: 100 16.875: 1.5 X x $225. Ans. 16.875 X 12' 18 X.05. $225. Ans. 11 122 COMPOUND INTEREST. IcVhn parltial paynments have been mnde. RULrE. - Find the amount (sum of the principal and interest) up to the time of the first payment, and deduct the payment therefrom; then find the interest on the remainder up to the next payment, add it to the remainder, or new principal, and from the sum subtract the next payment; and so on for all the payments; then find the amount up to the time of final payment for the final amount.' COMPOUND INTEREST. If we calculate the interest on a debt for one year, and then on the same debt for another year, and again on the same debt for still another year, the sum -will be the simpnlle interest on the debt fobr three years. But, on the contrary, if we calculate the interest on the debt for one year, and then on the amzount (sum of the principal and interest) fobr the next year, and then on the second amount for the third year, the sum of the interest so calculated will be the comlpound interest, or yearly compound interest, on the debt for three years; equal to the simple interest on the debt for thfree years, plus the yearly compound interest on the first year's interest for two years, plus the simple interest on the second year's interest for one year. So, if' we divide the time into shorter periods than a year, and proceed fbir the interest as last suggested, the interest will be compound. Thus we have half-yearly conmpound interest, or compound interest semi-annually, quarteryearly compound interest, or compound interest quarterly, &c. This method of computing interest is predicated upon the natural idea, that interest, when it becomes due by stipulation and is withheld, commences to draw interest, and continues at use to tile holder, at the same rate as the principal, until it is paid, like other over-due demands; and that the interest so made matures and becomes due as often, and at the same periods, as that on the principal. It will be perceived by the foregoing that the worlking-time in compound interest is the interval between the stipulated payments of the interest, or between one stipulated payment of the interest. and that of another; and that the working-rate is pro rata to the rate per annum. Thus the amount of $100 at semi-annual compound interest for 2 years, at 6 per cent. per annum, is COMIPOUND INTEIEST. 123 100 X (1.03)4 -$112.550881 - $112.55, or 100..03 3. 100. 103..03 3.09 103. \ 106.09.03 3.1827 106.09 109.2727.03 3.278181 109.2727 $112.550881, as before. if we let P principal or debt at interest, r - worklin-rate of interest, n -- number of intervals into which the whole time is divided for the payment of interest, or number of consecutive. intervals Por the payment of interest that have transpired without a payment havingg been made,' - compound interest, A = P +- i or amount, then A=-P(1-+r)"; P (1A),; r=, - — 1; - (I+r)D; i=A-P. EXAMPLE. - What is the compound interest, or yearly compound interest, on $100 for 1 years, at 6 per cent. a year? 100X 1.OGX1.03 —109.18- 100_ $9.18. Ans. EXAMIPLE. — What is the amount of $560.46, at 7 per cent. compound interest per year, fbr 6 years and 57 days? 560.46 X (1.07)0 X( 07X5 35 ) —$850.29. Ans. 124 COMPOUND INTEREST. EXA.MrLE. - The principal is $250, the rate 8 per cent. a year, the time 2 years, and the interest compound per quarter year: required the amount. 250 X (1.8) — $292.91. Ans. When Partial Payments have been made. RULE. - Find the amount up to the first payment, and deduct the payment therefrom; then find the amount up to the next payment, and therefrom deduct that payment; and so on for all the payments; then iind the amount up to the time of final payment, for the final amount. EXAMPLE. — A note of hand for $500 and interest from date, at 6 per cent. a year, has been paid in part as follows; viz., two years and four months from the date of the note, by an indorsement of $50; and three years from that indorsement, by an inlorsement of $150. It is now eight months since the last payment was made, and the demand is to be settled in full: required the amount at the present time, interest being compound per year. 500 X (1.06)2 X 1.02 - 50 =523.036 (1.06)3 622.944 150 472.944 1.04 $491.86. Ans. The following table shows (1+-r) raised to all the integer powers from 1 to 12 inclusive; r being taken at 4, 5, 6, 7, 8, and 10 per cent. If the numbers in the column headed years are taken to represent years, then 4 per cent., 5 per cent., &c., at the head of the columns of powers, will stand for per cent. per annum: if they are taken to represent half-years, then 4 per cent., 5 per cent., &c., will stand for per cent. per half-year, &c. The quantities in the columns are powers of (1 +-r), of which the numbers referr-el to and standing opposite, respectively, are the exponents. Thlus, 1.26248, in the 6 per cent. column, and against 4 in the column marked years, -(1.06)4; and so with the others. The powers or quantities in the columns are co-efficients in the calculations. COMPOUND INTEREST. 125 Years. 4 per cent. 5 per cent. 6 per cent. 7 per cent. 8 per cent. 10 per cent 1 1.04 1.05 1.06 1.07 1.08 1.10 2 1.0816 1.1025 1.1236 1.1449 1.1664 1.21 3 1.12486 1.15762 1.19102 1.22504 1.25971 1.331 4 1.16986 1.21551 1.26248 1.3108 1.36049 1.4641 5 1.21665 1.27628 1.33823 1.40255 1.46933 1.61051 6 1.26532 1.3401 1.41.852 1.50073 1.58687 1.77156 7 1.31593 1.4071 - 1.50363 1.60578 1.71382 1.94872 8 1.36857 1.47746 1.59385 1.71819 1.85093 2.14359 9 1.42331 1.55133 1.68948 1.83846 1.999 2.35795 10 1.48024 1.62889 1.79085 1.96715 2.15892 2.59374 I 11 1.53945 1.71034 1.8983 2.10485 2.33164 2.85312 L 12 1.6010311.79586 2.0122 2.25219 2.51817 3.13843 NOTE.- If a co-efficient is wanted for a greater number of years or intervals of time than is given in the table, square the tabular co-efficient opposite half that number of intervals, or cube the tabular co-efficient opposite one-third that number of intervals, &c., for the co-efficient required. Thus, 1.9992'- 1.9586873. = 1.081 X 1.08 1.083.996, the co-efficient for 18 years or intervals at 8 per cent. per interval, &c. If the compound interest alone is sought on a given principal, subtract I from the tabular power corresponding to the time and rate, and multiply the remainder by the given principal; the product will be the compound interest. Thus (1.26532-1) X 100 = $26.532, the yearly compound interest, at 4 per cent. per annum, on $100 for 6 years, or the hal:fyearly compound interest, at 8 per cent. per annum, on $100 for 3 years, or the halcyearly compound interest, at 4 per cent. per half year, on $100 for 6 half-years. EXAMPLE. -What is the amount of $125.54, at 5 per cent. compound interest, for 7 years, 21 days? 21 X.05 1 + - 1.00288, the co-efficient for the odd days; and, turning to the 5 per cent. column in the table, we find against 7, in the column of years, 1.4071, the co-efficient for 7 years: then 125.54 X 1.4071 X 1.00288 - $178.20. Ans. EXAMPLE. -In what time, at 7 per cent. compound interest per annum, will $1000 gain $462? A P - (1 +r): then 1462 1000-1.462, the co-efficient demanded. Turning now to the 7 per cent. column in the table, we find the nearest less co-efficient there (there being none that exactly corresponds) to be that for 5 years; viz., 1.40255. And 1 1 -11.0 7-.60553, the fraction of a year over 5 years to the answer..60553 X 365 - 221 days: 5 years, 221 days. Ans. 11* 126 COMIPOUND INTEREST. The followinc TABLE is of the same nature as the preceding, and is applicable when the interest becomes due at regular intervals short of a year, or when the working-rate in compound interest is less than 4 per cent. The quantities in the 1 3 per cent. column apply to quarter-yearly compound interest when the rate is 7 per cent. a year; and those in the 1 per cent. column, to quarterly compound interest when the rate is 5 per cent. a year; also the former are applicable to monthly compound interest at 21 per cent. per annum, and the hitter to monthly compound interest at 15 per cent. per annum; and so relatively, throughout the table. 1 1.035 1.03 1.025 1.02 1.0175 1.01- 1.0125 1.01 1.005 2 1 071123 1.009 1.050631.0404 1.0353111.03023 1.0251611.0201 1.01003 3 1.10872 1.09273 1.0768911.06121 1.05342J1.04568 1.0379711.0303 1.01508 4 1.14752 1.1255111.10381 1.08243 1.0718611.06136 1.05095 1.0406 1.02015 5 1.18769 1.1a59)2711.13141 1.10408 1.09062i1.07728 1.06408 1.05101 1.02525 6 1.29225 l.19405 1.1596911.12616 1.1077 1.09344 1.0774 1.0615211.03038 7 l.2284 1.22987 1.1 886911.14869 1.12709 1.10984 1.09087 1.0721411.03553 8 1.31681 1.2667711.2184 11.17166 1.14681 1.12649 1.10451!1.08986:1 04071 9 1.3629 1.3047711.24886,1.19509 1.16688 1.14339 1.1183111.0936911.04591 10 1.4106 1.343921.2800811.21899 1.1873 1.1605411.1322911.10462 1.05114 11 1.45997 1.3842311.31209 1.24337 1.20808 1.17795 1.14645 1.11567 1.0564 12 1.51107 1.4257611.34489 1.26824 1.22922 1.1956211.1607811.12683 1.06168 EXAMPLE. -What is the amount of $750 for 4 years and 40 days, allowing Dhalf-yearly compound interest, at 7 per cent. a year? In this case, the workingr-rate for the full periods of time is 3. per cent., and there are 8 such full periods; then, seeking the co-efficient in the 3- per cent. column, we find against 8, in the column of times, the quantity or co-efficient 1.31681; and 1 + 40 X.07 365 1.00767: therefore 750 X 1.31681 X 1.00767 $995.18. Ans. EXAMPLE. - What is the amount of $1000 at compound interest per quarter-year, at 1 I per cent. per quarter-year, for 4 years? 1000 X 1.126492 X 1.015=$1288.01. Ans. BANK INTEREST. 127 BANK INTEREST OR BANK DISCOUNT. A bank loans money on a. promissory note made payable with. out interest at a future period. The operation is called discounting the note at bank, and is as follows: The bank takes the note, finds the interest on it for three days more time than by its own tenor it has to rull, subtracts it from the principal, and hands the balance, called the avails of the note, in its own bills, to the party soliciting the loan, or offering the note for discount, as it is called; whereby the note becomes the property of the bank, and the maker and indorsers are held for its payment when it matures. The three days mentioned are called days of grace, and the note does not become due to the bank until three days after it becomes due by its own tenor. These proceedings are sanctioned by usage, and protected by law. Bank interest, then, is bank discount, and bank discount is bank interest. But bank discount is not disco unt, nor is it what is called legal interest on the money loaned. It is the interest on the money loaned, plus the interest on the interest of the loan, plus the interest on the difference of the sum taken and the interest on the loan for the time of' the loan! A kind of interest more onerous, if any description of interest be onerous, than compound interest, rate for rate and time for time, as may be readily perceived. Let P =principal or face of the note.,- working-rate of the interest for the time of the loan. a = avails of the note or sum borrowed. i = bank interest. t time of the loan. R: r:: T: t. R being the rate per cent. per annum, and T one year. P=a (1l-r). a-=P-Pr. i=-Pr. r-(P-a)-' P. If we let n represent the time of the note in months, Rn 3R r —2 - 365. But it is the practice with many banks to count the days of grace as so many 360ths of a year. Putting d to represent the the time of the note in days, -Rd 3 R r= I 3, true time and rate. 365 With some banks, it is the practice, in calculating interests to take the time, when it does not exceed 93 days, as so many 360ths of a year. A note haviing 3 months to run firom Aug. 10 for instance, will 128 n,.NK INTERIEST. fall (lne Nov. 10-13; but one having 90 days to run from Aug. 10 will fall Nov. 8-11. The time including grace of the former is 3 mo. 3 ds., and that of the latter 3 mo. 2 ds., mean time. Nevertheless, the former embraces 95 days, or one day more than mean time, and the latter but 93 days. The following table shows 1 - r, mean time, for the intervals of time set down in the left-hand column; R being taken at 4, 5, 6, 7, and( 8 per cent. per annum, as set down at the top of the columns. |'J'ime. 4 5 6 7 8 [ rob. ds. per cent. per cent. per cent. 1 3.996333.995417.9945.993583.992667 2 3.993.99125.9895.98775.986 3 3.989667.987083.9845.981917.979333 4 3.986333.982917.9795.976083.972667 5 3.983.97875.9745.9 7025.966 6 3.979667.974583.9695.96441 7.959333 7 3.976333.9 7041 7.9645.958583.952667 8 3.973.96625.9595.95275.946 9 3.969667.962083.9545.946917.939333 10 3.966333.957917.9495.941083.932667 11 3.963.95375.9445.93525.926 12 3.959667.949583.9395.92941 7.919333 Putting k to represent the tabular quantity 1 - r, a — Pk, P - a' k, i P - a-= P - Pkc. EXAMPLE. - What will be the avails of a note for $1,250 payable in 4 months if discounted at a bank, interest being 7 per cent. a year? The tabular constant 1 - r, in the 7 per cent. column, against 4 months and 3 days in the time column, is.976083, and $1,250 X.976083 - $1,220.10. Ans. E XAMPLE. - For what sum must I make a note having 6 months to run, in order that the avails at bank, if discounted on the day of the date of the note, may amount to $956.38, interest being 6 per cent. per annum? By the table, $956.38 ~.9695 $986.47. Ans. EXAMPLE. - What is the rate of bank interest when the nominal or legal rate is 7 per cent.?.07 (1-.07) =.07527 -7 7 + 27 per cent. Ans. NOTE.-A note having 5 months to run from Feb. 1 will fall due Jllyl 1-4; and the time, including grace, is 5 mo. 3 d. — 155 days, mean tilee. But the time in days fiom Feb. 1 to July 4, when Febrriuary has but 28 do(es, is 153 days only, or 2 days short of mean time. See SEC. B., 1). 3. DISCOUNT. - COMPOUND DISCOUNT. 129 DISCOUNT. DISCOUNT is a deduction of the interest on the present worth or availability of a debt not yet due, in consideration of its present payment. The prin cipal is the present nominal value of the debt, interest include(d, if' any interest has accrued. The time is tlhe interval from the present to the date at which the debt will become due. The rate is the legal rate of interest, if no other rate is specified; and the present worth is that sum of money, which, if put at interest at the same rate and for the same time as the discount, will amount to the principal. Let a represent the principal, d the discount, w the present worth, and i the interest on one dollar for the time and at the rate of the discount. w=a' (1 +i)= a -d. d ai- (1+-i) a-a w. a- d (1-+- i)' i d+- w. EXAMPLE. Required the discount on $250 for 8 months at 6 per cent. The interest on $1 for 8 months at 6 per cent. is.04 of a dollar, or 4 cts.; and 250 X.04' (1 +.04) = $9.6154. Ans. EXAMPLE.- Required the present worth of $1272.62 due 24 7 days hence, discount 7 per cent. The interest on $1 for 247 days at 7 per cent. = 247 X.0 7 ~ 365 0.04737, and 1272.62 - 1.04737 -$1215.06. Ans. NoTEI.-" Taking of, in common parlance, a certain per centum from the face of a demand, is equal to deducting the interest, at that rate per cent.tum, on the present worth for 1 year, plus the interest on the interest of the present worttl, at the same rate per ceatum for 1 year. Sewe Sc. B., p. 1. COMPOUND DISCOUNT. COMPOUND DISCOUNT is to compound interest what simple discount is to simple interest. In both cases of discount, the difference between the principal and the discount is that sum of money, which, if put at interest for the same length of time, at the same rate, and in the same general manner as the discount, will amount to the principal. RULE.-Add 1 to the rate per cent. of the discount for the 130 COSIrOC.g SC 0UT. working-timeh, and raise the sum to a power corresponding with the number ol' working-tilnes; divide the principal by the power, and the quotient twill be the present worth; subtract the present worth fiorn the principal, and the remainder will be the compound discount. NOTE'. -The TABLES of the powers of 1 + r, applicable to compound:interest, are equally applicable to compound discount. EXAMPLE. — Required the present worth of a debt of $250, allowing yearly compound discount, at 7 per cent. a year, for 3 years 84 days. 1 + —- — _ 7 1.01611, the working-rate for the 84 days, and 365 250 ~ (1.073 X 1.01611) = $200.84. Ans. EXAMPLE. - What is the present worth of a debt of $150.25, due 3 years, 3 months, and 10 days hence, without interest, allowing compound discount per qua/ter-year, at 1 per cent. per quarter-year? 150.25 (1.015 1.0X 1. 10) Ans. By table, 150.25' (1.19562 X 1.015 X 1.00164) $123.61. A ns. NOTE. - What is here denominated the debt, or principal, represents the debt at the close of the time of the discount; that is, if the debt be on interest, the interest must be included in what is here called the debt, or principal. PROFIT AND LOSS. The term "PROFIT AND Loss," as intimated in treating of PERCElNTAGE, relates to the positive and negative interests in business, and embraces the idea of both. Both profit and loss are absolute quantities, and are expressed by the (lifefrence of the cost price andt selling price that limit them. They are usually, however, estimated by percentage, predicated upon the first-mentioned price or prime cost. AXhen the selling price is greater than the cost price, or when the money obtained by the disposal of' property exceeds what the property cost, the difference is positive,, and denotes increase, profit, or gain. Conversely, when the cost price is greater than the sellinhg price, or when property is disposed of for less money than it cost, the difference is negative, and denotes decrease, loss, or tROFIT AND LOSS. 131 waste. So, the difference of the two prices, ~divided by the cost price, expresses the rate of gain on the cost when the selling price is the greater, — expresses the rate of loss on the cost when the cost price is the greater. Let c represent the cost price, purchase price, par value, or sum of money paid for the property; s, the selling price, trade price, premium price, or sum of money received in exchange for the property; r, the rate of the profit or loss; p, the rate per cent. of the profit or loss. To find the rate or rate per cent. of the profit or loss. r _ $ c. ( = ( c) 100. Moreover, when the difference is c C $ S positive, r - - 1; and, when it is negative, r= 1-. C C EXAMPLE. - Paid $4 for an article, and sold it for $5. What pecr cent. was gained? 5 is mrore than 4 by what per cent.. of 4? The ditffrence of 5 and 4 is what per cent. of 4'? 5 -4 = $1, gained; and.25 = - 1. 25 per cent. Ans. EXAMPLE. - Paid $5 for an article, and sold it for $4. Wvhat per cent. was lost? 4 is less than 5 by what per cent. of 5? The difference of 4 and 5 is what per cent. of 5? 4- 5 - - 1 $1, lost; and =.20 W1 —a. 20 per cent. Ans. 5 EXAMPLE. - A whistle that cost 3 cents was sold for 20 cents! The profit was how much pcr cent? (20. 3)' 3- 5 or 566'. per cent. Ans. EXAMPLE. - A fop paid $10 for a well-made and well-fitting pair of boots for his own wear, that were worth what they cost him; but, being told that they were unfashionably large, sold them for $4. HIis vanity cost him what per cent. of the purchase price? 1 -X 4 —.6 or 60 per cent. Ans. To find a price long a given per cent. of the cost, or to find a selling price that shall be the sum of the cost price and a given per cent. of it. s c + cr — c (1 +r) - c (100 +p) 100. EXAMPLE.- At what price must I sell an article that cost $2.35 to gain 25 per cent.? 2.35, more 25 per cent. of it, is how muchl? The sum of $2.35 and 25 per cent. of it is how much? 2.35 T- 2.35 X.25 — = 2.35 X 1.25 $ ~2.933. A1ns. 132 EQUATION O PAYMENTS. To find a price short a given per cent. of the cost, or to find a selling price that shall be the difference of the cost price and a given per cent. of it. s _ c-cr c (1-r) -c (100-p)' 100. EXAMPLE. - I have a damaged article of merchandise that cost $2.75, and I wish to mark it for sale at 30 per cent. below cost. At what price shall I mark it? 2.75 less 30 per cent. of it is how much? The difference of $2.75 and 30 per cent. of it is how much? 2.75 (1 -.30) = 2.75 X.7 $1.925. Ans. To find the cost price when the selling price and profit per cent. ai e given. -c + cr = (1+ r).'.c=s (1+- +r) = 100 S * (100 -)-). EXAMPLE. - What cost that article whose selling price, $4, is long 25 per cent. of the costs? What price, more 25 per cent. of it, is equal to $4? $4 is the sum of what price and 25 per cent. of it? 400 - 125 —$3.20. Ans. To find the cost price when the selling price and loss per cent, are given. s c-cr —c(1 —r).'. c=s - (1-r)- 100s'- (100-p) EXAMPLE. - What cost that article whose selling price, $375, is short 7 per cent. of the cost? What price less 7 per cent. of it is equal to $375? $375 is the difference of what price and 7 per cent. of it? 375 -.- (I -.07) = 375 *.93 = 375 X 100 (100 - 7)$403.226. Ains. EQUATION OF PAYMENTS. TiE EQUATION OF PAYMENTS, or Averaging of Accounts, as it is more frequently called, practically consists in finding the common time of maturity of two or more debts due at different times, and is either special or general; special when it is made in regar(l to a given interchangeable rate of interest and discount, in which the magnitude of the rate slightly affects the time, since discount consumes more time per dollar, rate for rate, than interest; and general, when it is made disrespectfal of rate, or common in the greatest possible degree to all rates. RULE, for common purposes. - Multiply each debt by the numnber of days between its own date of maturity and that of the debt earliest due, and divide the sum of the products by the sum of the debts; the quotient will express tlhe common time in days subsequent to the leading date. EQUATION OF PAYMENTS. 133 The fbllowing exhibits the face of an account in the ledger, and the time (date) at which it averages due is required. 1860, April 10 -$250.26 - 6 mo. Due Oct. 10. " June 25 - 320.56 6 " " Dec. 25. " July 10 - 50.62 - 3 " " Oct. 10. " Aug. 1 210.84 - 4 "' " Dec. 1. " " 18 -- 73.40- 5' " Jan. 18. " Oct. 15 100. - cash " Oct. 15. I!XAMNIPLE. - Practical method of stating and working. 1860. Due Oct. 10, $301 - " " Dec.25, 321 X 76 -24396. i' " " 1, 211 X 52 10972. " " Jan. 18, 73 X 100 = 7300. " Oct. 15, 100 X 5= 500. 1006 ) 43168 ( 43 days, =Nov. 22, 1860. Ans. COMPOUND AVERAGE. COMPOUND AVEIhAGe consists in finding the tirue at which the balance of an account or demand averages due, whose sides -the debit and the credit - average due at different dates. RULE. - Multiply the less sum or side by the difference in drays between the two dates -that at which the debit side averages due and that at which the credit side averages due - and divide the product by the difference of the sums or sides; the quotient will be the number of days that one of the dates must be set back, or the other forward, to mark the time sought; for which last, SPECIAL RULE. Earlier date with larger sum, set back from earlier. Later date with larger sum, set forward from later. EXAMPLE. - The debit side of an account in the ledger foots up $400, and averages due Oct. 12, 1860; the credit side of the same account foots $300, and averages due Nov. 16, 1860. At what d-ate does the balance or difference between the two sides average due. 400 300 300 35 100 ) 10500 ( 105 days earlier than Oct. 12, =June 29,1860. Ans. EXAMPLE. — The debit side of an obligation foots $250, and averazres due May 17, 1860; the credit side of the same obligation foots $i75, and averages due May 1, 1860. At what date does the differenice of the sides average due? Q50 175 175 16 75 ) 2800 ( 37/ days later thian May 17, - June 23, 1860. Ans. 12 134 GENERAL AVERAGE. GENERAL AVERAGE. -It is the established usage thlat whatever of either of the three commercial interests —the ship, the cargo, or the freight —is voluntarily sacrificed or destroyed for the general good, or with the viewv of saving' the most that may be saved when all is in imminent dangcer of being lost, is matter of general loss to the respective interests, and not more especially to the interest voluntarily abandoned than to the others. So, too, the losses and damages inci(lcnt to the voluntary sacrifice, and collateral therewith, together witl the expenditures which the master has been compelled to Inmake for the general good, in consequence of disaster, are matters of' general average, or are to be contributed for, pro rata, by the several interests. The contributory interests are the ship, the cargo, and the fiteight, at their net values, independent of charges, premiums paid for insurance, &c. The contributory value of the ship, generally, is her value at the port of departure at the time of leaving, less the premium paid fbr her insurance. The contributory value of the cargo is its net value, in a sound state, at the port of destination, if the voyage be completed; or its invoice value if the voyage be broken up and the cargo returned to the port whence it was shipped; or its market-value at any intermediate port, where of necessity it is discharged and disposed-of. The value of the goods jettisoned, and to be contributed for, is their value after the same manner; and that value is a part of the contributory value of the cargo, as well as a matter of general average. The contributory value of the freight, generally, is the gross amount or amount per freight-list, less one-third part thereof; in most of the States; but, in the State of New York, one-half thereof, for seamen's wages and other expenses. The loss of freight byje.ttison, when any freight is earned, is matter of general average. If the cargo is transshipped on board another vessel, and in that waiy sent to the port of destination, the contributory value of the frieight is the gross amount, less the sum paid the other vessel.'1Ti( voluntary damage to the ship, with a view to the general good, - suchll as throwing over her furniture, destroying her equir)nillts, cutting away her masts, breaking up her decks to get at the (argfo Ibr the purpose of throwing it over, &c., -is contributed for;at two-tllirlds the cost of repairing and restoring; the new articles beimng supposed one-half better, or worth one-half more, than the old. GENERAL AVERAGE. 135 If we let V contributory value of the vessel, C contributory value of the cargo, F - contributory value of the freight, d- aggregate amount of losses to be averasged, then d' (V + C + F) =- r, the per cent. of each interest that each must contribute, and V X - Vessel's share of the loss, C X r Cargo's share of the loss, F X r = Freigrht's share of the loss. When a contributory interest's share of the loss is to be distributed among, the several owners of that interest, the same pro rata method is to be observed: thus A X r = sum A must contribute, B X r = sum B must contribute, D X r - sum D must contribute; A, B, and D being A's, B's, and D's respective shares in that interest. 136 ASSESSMENT OF TAXES. -INSURANCE. ASSESSMENT OF TAXES. G amount of taxable property, real and personal, as per grand list. A= amount of money to be raised, including thlie whole poll-tax. T amount of money to be raised on property alone. n =number of ratable polls. h = poll-tax per head. r = rate per cent. to be raised on taxable property. P - an individual's taxable-property, as per grand list. b =P's poll-tax. T=A-An. r-=T~-, G. Pr+-b-=P's tax, including poll. INSURANCE. INSURANCE is a written contract of indemnity, called the policy, by which one party (the insurer or underwriter) engages, for a stipulated sum, called the premium (usually a per cent. on the value of the property insured), to insure another against a risk or loss to which he is exposed. Let P =Principal, or amount insured or, r = rate per cent. of insurance, a - premium for insurance. a — Pr. r- a - P; P=-a ar. EXAMPLE. — What is the premium for insuring on $4500 at 11 per cent.? 4500 X.015 - $67.50. Ans. LIFE-INSURANCE. Life-insurance is predicated upon the even chance in years, called the expectation of life, that an individual in general health at any given age appears by the rates of- mortality to have of living beyond that age. Thle Carlisle Tables of Expectation, column C in the following tables, are used almost or quite exclusively in England, and by some insurance-companies in the United States; while those by Dr). Wigglesworth, column WV, computed with special reference to the rates of mortality in this country, are used by others. The Supreme Court of Massachusetts has adopted the:Wigg-ic, LIFE INSURANCE. 137 iorth rates of expectation in estimating the value of life-annuities and life-estates. TABLE Of Ages and Expectations from Birth to 103 Years. ge. C. W ge. Cw. V. Age. C.. Age. C. W. 0 38.72 28.151 26 3 7.14 31.93 52 19.68i20.05 78 6.12 6.59 1 44.68 36.78i 27 30.41 31.50 53 18.97}19.46 79 5.80 6.21 2 47.55 38 74/ 28 35.69 31.08 5-4 18.28j18.92 80 5.51 5.85 3 49.82 40.01 29 35.0030.66 55 17.58 18.35 81 5.21 5.50 4 50.76 40.73 30 34:34.130.25 56 16.89 17.78 82 4.93 5.16 5 51.25 40.88 31 33.68 29.83 57 16.21 17.20 83 4.65 4.87 6 51.17 40.69 32 33.03 29.43 58 15.55 16.63 84 4.39 4.66 7 50.80 40.47 33 32.36( 29.02 59 14.92 16.04 85 4.12 4.57 8 50.24 40.14 34 31.6.662.6 G60 14.34 15.45 86 3.90 4.21 9 49.57 39.72 35 31.00 28.22 61 13.82 14.86 87 3.71 3.90 10 48.82 39.23 36 30.32 27.78 6(2 13.31 14.26 88 3.59 3.67 11 48.04 38.64 37 29.64 27.34 63 12.81 13.66 89 3.47 3.56 12 47.27 38.02 \38 28.9G 26.91 64 12.30 13.05 90 3.28 3.43 13 146.51 37.4 1{ 39 28.28 126.47 65 11.79 12.43 91 3.26 3.32 14 45.75 36.791 40 *27.61 26.04 66 11.27 11.96 92 3.37 3.12 15 45.00136.17 41 26.97 25.61 67 10.75 11.48 93 3.48 2.40 16 44.271 35.7 6 42 26.34 25.19 68 10.23 11.01 94 3.53 1.98 17 43.57 35.3 7 43 25.71 24.77 69 9.70 10.50 95 3.53 1.62 18 42.817 34.98 44 25.09 24.35 70 9.18 10.06 96 3.46 19 42.17 34.591 45 2 446 23.92 71 8.65 9.60 97 3.28 20 41.4(;6 34.221 46 23.82 23.357 72 8.16 9.14 98 3.07 21 40.75 33.84 47 23.17 22.831 73 7.72 8.69 99 2.77 22 40.04133.46 48 22.50 22.271 74 7.33 8.25 100 2.28 23 39.31133.0 8 49 21.81 21.721 75 7.01 7.83 101 1.79 24 38.59132.70 50 21.11 21.17 76 6.69 7.40 102 1.30 25 37.86 32.33 51 20.39120.61 77 6.40 6.99 103 0.83 Thus, by the tables, a man in general good health at 21 years of are has an even chance, by the Carlisle rate of mortality, of livingr 40.} years longer; by the Wigglesworth rate, of' living 33,8 4 years longer. So a man in general good health, at 60 years oft' age, has, by the Carlisle rate, an even chance of living 1,4.34 years longer; by the Wigglesworth rate, an even chance of living 15.45 years longcr, etc. 12* 138 FELLOWSHIP. FELLOWSHIP. FELLOWSHIP calls for the distribution of a given effect to eachl of the several causes associated in its production, proportional to their respective magnitudes one with another. It is a rule, therefore, adapted to the use of partners associated in business, in achieving a pro rala distribution among themselves as irdlividuals, of the profits or losses pertaining to the company. RULE.- Multiply each partner's investment or share of the capital stock, by the whole gain or loss, and divide the product by the suImt of all the shares, or gross capital. EXAMPLE. -Three men, A, B, and C, enter into partnership. A invests $500, B $700, and C $300. They trade and gain $400). What is each partner's share of the profits? A, $500 500 X 400 - 1500 - $133.331 = A's share. t, 700 700 X 400 — 1500- 186.66- = B's " C, 300 300 X 400 - 1500 = 80.00 = C's " $1500 = gross capital. $400.00 Proof. EXAMPLE.-D's investment of $600 has been employed eight nmorths; E's, of $500, five months; and F's, of $300, five months, the profits of the company are $500, and are to be (lividcd pro rant among the partners. What is each partner's share? D), $600 X 8 = 4800 X 500 + 8800 $272.73, D's share. E, 500 X 5 = 2500 X 500 +. 8800 = 142.05, E's " F', 300 X 5 = 1500 X 500 - 8800= 85.22, F's 8800 $500. Proof'. iEXANrPLE.- Of $120 distributed, there were given to A, A; to B, 4-; to C, -; and to D, I, and there was nothing remaining. NV lat sum did each receive? of 120 40 X 120 -- 114 = $421 = A's share. 4of 120 -30 X 120 — 114 - 311 B's " -of 120 24 X 120 -- 114- 25-5 -= C's " o- of l20 = 20 X 120 -114 = 211 = D's " 114 $120. Proof. EXAMPLE. -Divide thle nlumber 180 into 3 parts, which shall be to each other as 4, 4, 1. iof 180 90 X 180 195 — 83.08 4of 180 60 X 180 195) 55.38: of 180 45 X 180' 195 — 41.54 195 180.00 Proof. ALLIGATION. 139 EXAMPLE. — $400 arc to be divided between A, B, and C, in the ratio of 2 to A, I to B, and I to C; how much wvill each receive? i of 400 200, and 200 X 400' 500 $10G A's share. of 400 200, and 200 X 400 500- 160 = B's share. of 400 = 100, and 100 X 400 500- 80 C's share. 500 $400. Proof. ALLIGATION. ALLIGATION lMedial is a method by which to find the mean price of a mixture or compound, consisting of two or more articles or ingredients, tile quantity and price of each being given. RULE. - Multiply each quantity by its price, and divide the sum of the products by the sum of the quantities; the quotient will be the price per unity of measure of the mixture; and, having found the price of the given quantities as mixed, any quantities of the same materials, taken in like proportions, will be at the same price. EXAMPLE. -If 20 lbs. of sugar at 8 cents, 40 lbs. at 7 cents, and 80 lbs. at 5 cents per pound, be mixed together, what will be the mean price, or price per pound, of the mixture? 20 X 8- 160 40 X 7 = 280 80 X 5 -- 400 140 ) 840 ( 6 cents. Ans. The several kinds, then, at their respective prices, taken in the proportion of I at 8, 2 at 7, and 4 at 5 cts., will form a mixture worth 6 cts. a pound. EXAMPLE. -- If 10 lbs. of nickel are worth $2, and 24 lbs. of copper are worth $4A, and 8 lbs. of kinc are worth 40 cts., and, 1 lb. of lead is worth 5 cts., wheat are 5 lbs. of pretty good German silver worthi (2 O o + 4 5 _+_4 0 + 5Y_5 81 cents. Ans. ALLIGATION Alternate is a method by which to find what quantity of each of two or more articles or ingredients, whose prices or qualities are given, must be taken to form a mixture or compound that shall be at a given price or of a given quality between the two extremes. It also applies to the finding of relative quantities when the quantity of one or more of the articles is limited. RULE. - Connect the given prices or qualities - a less than the given mean with that one or either one that is greCtcr - and to the extent that all be thus connected; then place tlhe diiffrence between 140 ALLIGATION. each given and the given mean opposite, not the given, or the given mean, but the given with which it is alligated; the number standing opposite each price or quality will be the quantity that must be taken at that price, or of that quality, to form a mixture or compound at the price or of the quality desired. And, being proportions respectively to each other, they may be taken in ratio greater or less, as desired. See SEc. B, page 18 b. EXAMPLE. - In what proportions shall 1 mix teas at 48 cents a pound and 54 cents a pound, that the mean price may be 50 cents a poundl In the proportions 5048 ] 4 lbs at 48 cts ns. 541 2 lbs. at 54 cts. s. Or, as 2 at 48 to I at 54. 3 X 50 = 150. EXAMPLE. -In what proportions shall I mix teas at 48, 54, and 72 cents a pound, that the mixture may average 60 cents a pound? 48 12, 12 at 48 ) (2 at 48) ~ 60 5411 12, 12 at 54 - 2 at 54 Ans. (72-1 12- +6, 18 at 72) 3 at 72) FXAMPLE. - A wine dealer has received an order for a quantity of wine at 50 cts. a gallon. He has none ready ma'nufjcturcd at that price. He has it at 40 cts., at 56 cts., and at 80 cents a gallon, and lie lIhas water that cost him nothing. IIe wishes to fill the order withl a mixture composed of the four materials -the water and the three different priced wines. In what proportions may he mnix theml, that the mean or average price shall be 50 cents a gallon?. Ans. Ans. F00 6 00r — 30 - 30 50 4 0i30) Or, 50 G4h 6-30 36 ri50 56.I 10 10 so S10i 800 50- 10=60 -96 gals. = 136 gals. Ans. Ans. 5 0 30 Or, 50 40 6 30 630 5056 5 0 0 -10 80 — 50 ~ 10 80-2 10 = 176 gals. = 112 gals. If, now, having folld llie proportions desired, it is wished to limit onei of til areticles ill (lllllatity sy tile best win, tot 8 gallons in the INVOLUTION - EVOLUTION. 141 mixture -- the ploportions of the remaining articles thereto are found lhus: Instance, 1st example,10: 8 50- 40 10 8 * 30-4 24 And the mixture will consist of 10:8:: 6. 43 i 8 + 40 + 24 + 44 == 76~ gallons.;f, instead, it is desired to mix a given quantity, say 100 gallons, and proportioned, say as in first example, the quantity to he taken of each is ascertained by the following RULE. - As the sum of the relative quantities is to the quantity required, so is each relative quantity to the quantity required of it respectively. The sum of the relative quantities alluded to is 6 + 30 +- 50 + 10 =96; then, 96: 100:: 6 = 64 96: 100:: 30 314 96: 100 ~: 50 -52 1 96: 100:: 10 — 102 INVOLUTION. INVOLUTION consists in involving, that is, in multiplying a number one or more times into itself. The number so involved is called the root, and the product arising from such involution, its power. The second power, or square, of the root, is obtained by multiplying the root once into itself, as 4 X 4 16(; 4 being the root and 16 its square. The third power, or cube, of a number, is obtained by multiplying the number twice into itself, as 4 X 4 X 4 = 64; and so on for.anyi power whatever. When a number is to be involved into itself, a small figure calledi the index or exponent is placed at its right, indicating the number of times it is to be so involved, or the power to which it is to be raised,l. Thus, 34 =3 X 3 X 3 X 3=81; and 43 -4 X 4 X 4 =64. EVOLUTION. EVOLUTION is the opposite of Involution. It consists in finding ai root of a given number, instead of a power of a given root. Wilen the root of a number is required or indicated, the number is written with the N! before it: and the character or denomination of tihe root, if it be otler than the square root, is defined by an index 142 EVOLUTION. figcrre placed over the sign. When the square root of a number is required, the sign (/) is placed before the number, but the index (2) is usually omitted. Thus, V/25, shows that the square root of 25 is required, or to be taken; and,W/25 shows that the cube root is required. The operation is-usually called extracting the root. TO EXTRACT THE SQUARE ROOT. RULE - 1. Separate the given number into periods of two figures each, by placing a point over the first figure, third, fifth, &c., counting from right to left — the root will consist of as many figures as there are periods. 2. Find the greatest square in the left hand period, and place its root in the quotient; subtract the square of the root from the left hand period, and to the remainder bring down the next period for a (l i vidend. 3. Multiply the root so far found -- the figure in the quotient - by 2, for a divisor; see how many times the divisor is contained in the dividend, except the right hand figure, and place the result (the number of times it is contained) in the quotient, to the right of the figure' already there, and also to the right of the divisor; multiply the divisor, thus increase4, by the last figure in the quotient, and subtract the product from the dividend, and to the remainder bring down the next period for a dividend. 4. Multiply the quotient - the root so far found (now consisting of two figures) - by 2, as before, and take the product for a divisor; see how many times the divisor is contained in the dividend, except the right hand figure, and place the result in the quotient, and to the right of the divisor, as before; multiply the divisor, as it now stands, by the figure last placed in the quotient, and subtract the product from the dividend, and to the remainder bring down the next period for a dividend, as before. 5. Multiply the quotient (now consisting of 3 figures) by 2, as before, and take the product fpr a divisor, and in all respects proceed as when seeking for the last two figures in the quotient. The quotient, when all the periods have been brought down and divided, will be the root sought. NOTE. - 1. If there is a remainder after finding the integer of a root, annex periods of ciphers thereto, and proceed as when seeking for the integer. The quotient figures will be the decimal portion of the root. 2. If the given number is a decimal, or consists of a whole number and decimal, point off the decimal from left to right, by placing the point over the secon(l, fourth, sixth, &c., figures therein, and fill the last period, if incomplete, by annexing a cipher. 3. If the dividend does not contain the divisor, a cipher nmist be placed in the quotient, idal also at the right of the divisor, and the next period broughlt down; then the dividelnd nim1st be divided by the divisor as increased. 4. If the qllotient fisgure. obltained by dividing by the double of the root, is too large, as will sometimes be ithe case, (see 3d Example) it mlotst. ie droppedl, iand a less - one which is t.he true measnlre -- taken in its sitce. 'tVo LUTION, 1 43 EXA ^.lPI.l. -- Rlequired( the square root of 123456.432. 123456.4320 (351.3636+. Ans. 9 65) 334 325 70.) 956 701 7023) 25543 21069 70266 ) 447420 421596 702723) 2582400 2108169 7027266 ) 47423100 42163596 5259504 EXAMPLE.- Required the square root, of 10621. Also, of 28561 o162i( lo03.05. -. Ans. 28'561(169. Ans. 1. 1 203 ) 00621 26) 185 609 156 20605 ) 120000 329) 2961 103025 2961 16975 TO EXTRACT TIIE CUBE ROOT. RULE- 1. Separate the given number into periods of three figures each, by placing a point over the first, fourth, seventh, &c., counting from right to left — the root will consist of as many figures as there are periods. 2. Find the greatest cube in the left hand period, and place its root in the quotient; subtract the cube of the root from the left hand period, and to the remainder bring down the next period for a dividend. 3. Multiply the square of the quotient by 300, for a divisor; see how many times the divisor is contained in the dividend, and place the result (except that the remainder is large, diminished by one or two units) in the quotient. 4. Multiply the divisor by the figure last placed in the quotient, and to the product add the square of the same figure, multiplied by tlhia other figure, or figures, in thie quotient,:and by 20; andl add -also theret.; 144 EVOLUTION. the cube of the same figure, and take the sum for the subtrahend; sutl. tract the subtrahend from the dividend, and to the remainder bring down the next period for a dividend, with which proceed as with the preceding, so continuing until the whole is completed. NOTE -1. Decimals must be pointed from left to right, by placing a point over the third, si.xth, &c., figures in that direction. 2. if the divisor ia not contained by the dividend, place a cipher in the quotient, anl( annex two ciphers to the divisor, and bring down the next period for a dividend, aild use the divisor, as thus increased, for finding the next quotient figure. 3. If there is a remainder after finding the integer of the root, annex a period of three ciphers thereto, and proceed for the decimal of the -root as if seeking for the integer, aII. nexing.a period of three ciphers to each remainder until the decimal is carried to as maliiy places of figures as desired. EXAMPLE. -- Required the cube root of 47421875.6324. 4742i875.632400 ( 361.959+. 27 Ans. 32 X 300- = 2700 ) 20421 6 16200 62 X 3X 30 = 3240 C) - 216 — 19656 362 X 300 - 388800 ) 765875 388800 12 X.36 X 30 - 1080.1 13 - 389881 361'i X 300 = 39096300 ) 375994639 351866700 92 X 361 XX 30 - 877230 93 = 729 - 352744659 3619!' X 300 = 3929'3148300 ) 232-4997340) 5 19645741500 52'X 3619 X 30 = 2714250 53- 125 = 19648455875 361952 X 300 = 393023-1407500 ) 36015175250040 9.35372 10667500) 92 X 36195 X 30 - 87953850 93 7 7.!) - 3537298622079 61418902921 EVOLUTION. 145 EXAmI'LE.- -Required the cube root of 32768. Also, of 8489664. 3276 8(320. 8489664(204. 27 Ans. 8 Ans. 3'X 300 -2700 ) 5768 22 X 300 = 120000 )489664 2 4 5400 480000 X 3 X 30=360 42X 20 X 30 = 9600 23 = 8 5768 43 = G64 —489664 Generfal Rule for extracting the roots of all powers, or for finding aLy proposed root of a given number. 1. Point off the given number into periods of as many figures each, counting from right to left, as correspond with the denomination of the root required; that is, if the cube root be required, into periods of three figures, if the fourth root, into periods of four-figures, &c. 2. Find the first figure of the root by inspection or trial,~and place it at the right of the number, in the form of a quotient; raise this quotient figure to a power corresponding with the denomination of the robt sought, and subtract that power from the left hand period, mnd to the remainder bring down the first figure of the next period, for a dividend. 3. Raise the root thus far found (the quotient figure) to a power next inferior in denomination to that of the root required, multiply this power by the number or index figure of the root required, antl take the product for a, divisor; find the number of times the divisor is contained in the dividend, and place the result (except that the 1remainder is large, diminished by one or two units) in the quotient, for the second figure of the root. 4. Raise the root thus far found (now consisting of two figures) to a power corresponding in denomination with the root required, an(d sultract that power froln the two left hand periods, and to the remainder bring down the first figure of the third period, for a dividend; find a new divisor, as before, and so proceed until the whole root is extracted. EXA.PLE. -- Required the fifth root of 45435424. 45435424(34. Ans. 35 = 243. 34 X 5.) 2113 345- 45435424 13 146 ALUATIIMETICAL P'ROGRESSION. gx4AM'LE.- Required the fifth rOot of 432040.0O54. 432046.03546 (13.4 + Ans. 15 14X 5) 33 135 371293 134 X 5) 607470 13.4 -5= 43204003424...... 116 For instructions touching special cases, see NOTES relative to the extraction of the square root, and to the extraction of the cube root. The A/ of the &/ of any nulber =- 4 of that number ",/ of the 4 = /. " A/ of the 4 of the/ 1 4/. ",/ of the 4/- = /. 4' of the / 4/, &c. ARITHMETICAL PROGRESSION. A series of three or more numbers, increasing or decreasing by equal differences, is called an arithmeticalprogression. If the numbers progressively increase, the series is called an ascending arithmetical progression; and if they progressively decrease, the series is called a descending arithmetical progression. The numbers forming the series are called the terms of the progression, of which the first and the last are called the extremes, tand the others the means. The difference between the consecutive terms, or that quantity ty which the numbers respectively increase upon each other, or decrease from each other, is called the common difference. Thus, 3, 5, 7, 9, 11, &c., is an ascending arithmetical progression, and 11, 9, 7, 5, 3, is a descending arithmetical progression. In these progressions, in both instances, 11 and 3 are the extremes, of which 11 is the greater extreme, and 3 is the less extreme. The numbers between these, (9, 7, 5,) are the means. In every arithmetical progression, the sum of the extremes is equal to the sum of any two means that are equally distant from the extremes; and is, therefore, equal to twice the middle term, when the series consists of an odd number of terms. Thus, in the foregoing series, 3 - 11 = 5 + 9 =-7 X 2. The greater extreme, the less extreme, the number of trInms. tche ARITIIMIETICAL PROGRESSION. 147 common difference, and the sum of the Icrms, are called tlle five properties of an arithmetical progression, of whllich, any three being given, the other two may be found. Let s represent the sum of the terms. " E " the greater extreme. " e " the less extreme. " d " the common difference. " n " the number of terms. The caltremes of an arithmetical progression and the numnber of terms being given, to find the sum of thle terms. (E + e) X A:2 stun of the terms. NEXAM11PLE. - What is the sum of all the even numbers firom 2 to 100, inclusive? 102 X 50~2 2550. Ans. ExrxmL,. - IIow many times does the hammer of a common clock strike in 12 hours? (1 + 12) X 12. 2 = 78 times. Ans. (E-e + 1e X -2 - 3um of the terms. (E X 2-n-1 X d) X A n sum of the terms. (2 e +- n-1 X d) X ~ n = sum of the terms.''he grcaler extreme, lthe common difference, and the number of tcrms of an arithmetical progression being given, to find the less extreme. E - (d X n- 1) = less extreme. Ex^AMPLE.- A man travelled 18 days, and every day 3 miles fairther than on the preceding; on the last day he travelled 56 miles; hlow many miles did he travel the first day? 56- (18 1 X 3) = 5 miles. Ans. s /n I X d s_ _ 2 3 = less extreme. Xn 2xtr n X 2 E less extreme. 148 ARItTUJ1 ErICAL PROGRESSION. A/ (I )X< 2 + d)2-s X d X 8 4 — d less extremc, when 4/ (2E + d )2 8 s d is equal to, or greater than d. / (2 E + d)2 -- 8 s d - d = less extreme, when 2 / (2 E + d)- 8 s d is less than d. 4(2 e. d)2 + 8 sd d=;greater extreme 2 d X n - I + e = greater extreme. s n —IXd n + 2 -- = greater extreme. 2 s- n - e = greater extreme. Thie extrcmes of an arithmetical progression and the common (liffrence being given, to find the number of terms. E - e +. d 1 — number of terms. EXAMPLE. -As a heavy body, falling freely through space, descends G16TZ foet in the first second of its descent, 48 —3 feet in the next second, 80-5 in the third second, and so on; how many seconds had that body been falling, that descended 305k-f feet in the last second of its descent 305 -- 1G -- 289.- 321 9q + 1 = 10 seconds. - Alns. V (2 e, d)2 - 8 s d - d —e + d + 1 = number of terms. 2 2 s E -+ /(2 E + d)s - 8 s d + d number of terms when 2 V/ (2 E + d)2 - 8 s d is equal to, or greater thlan d. 2 s- E+ -- (2E - d) -8s d d numnber of terms;whlen 2 (2 E. d)2 8 s d is less than d. sX2 e number of terms. A:tRIII 1 I'C.A. PiOOltloSISON. 149 7'The t.rlremes of an arithmetical progrcssion, and the number oJ terms being given, to find the common difference. E- e common difference. EXAMPLE. - One of the extremes of an arithmetical progression is 28 and the other is 100, and there are 19d terms in the series; required the common difference. 100 w 28 1.w 19= -. Ans. E- e Ee- 1) = common difference. 2s n 2e 2n- 1 n.e-= common difference. 2E- (2s — n) - common difference. n- I EXAMPLE. - The less extreme of an arithmetical progression is 28, the sum of the terms 1216, and the number of terms 19; required the 7th term in the series, descending. 1216 X 2 - +19 - 128 - sum of the extremes. 128- 28 = 100 = greater extreme. 100 - 28 = 72 = difference of extremes. 72 n - 1 (18) - 4 - common difference. 100- (7 - 1 X 4) 76 7th term descendinlg. Ans. Required the 5th term from the less extreme, in an arithmtletical progression, whose greatest extreme is 100, common difference 4,.and numnber of terms 19. 100 - (19-5 X 4)= 44. Ans.'To find any assigned number of arithmetical means, between two g-iven numbers or extremes. RuLE. - Subtract the less extreme from the greater, divide the remainder by 1 more than the number of means required, and the quotient will be the common difference between the extremes; which, added to the less extreme, gives the least mean, and, added to that, gives the next greater, and so on. Or, E - e-rn 1 = d, E being the greater extreme, e the less extreme, m the numnber of means required, and d the common difference. And e - d, e + 2 d, e + 3 d, &c.; or, E - d, E-2 d, E-3 d, ec., will give the means required. 13 * 150 GEOMETRICAL PROGRESSION. EXAMrLE. - Required to find 5 arithmetical means between the numbers 18 and 3. 18 - 3 = 15 - G6-=2a, and 3 + = 5, + 2A =8 + 2A = 10A + 2A. = 13 + 2- = 15A. 5A, 8, 10t, 13, 15L, therefWre, are 5 arithmetical means, between the extremes, 3 and 18. NoTre. - The arithmetical mean between any two numbers may be found by dividing the sull of those numbers by 2 i thus, the arithmetical mean of 9 and 8 is (9+-8) 2- 8~. GEOMETRICAL PROGRESSION. A series of three or more numbers, increasing by a common multi plier, or decreasing by a common divisor, is called a geometrical /)'ogrtssion. If the greater numbers of the progression are to the right, the progression is called an ascending geometrical progression, but, on the contrary, if they are to the left, it is called a descending geometrical proression. The number by which the progression is fornled, that is, the common multiplier, or divisor, is called the rafio. The numbers forming the series are called the terms of the progression, of which the first-and the last are called the extremes, and the others the means. The greater of the extremes is called the greater extreme, and the less the less extreme. Thus, 3, 6, 12, 24, 48, is an ascending geometrical progression, biecause 48 is *as many times greater than 24, as 24 is greater than 12, &c.; and 250, 50, 10, 2, is a descending geometrical progrcssion, because 2 is as many times less than 10, as 10 is less than 50, &c. In the first mentioned series, (3, 6, 12, 24, 48,) 48 is the greater extreme, and 3 is the less extreme; the numbers 6, 12, 24 are the neans in that progression. So, too, of the progression 250, 50, 10, 2; 250 and 2 are the extremes, and 50 and 10 are the means. In the first mentioned progression, 2 is the ratio, and in the last, or in the progression 2, 10, 50, 250, 5 is the ratio. In a geometrical progression, tihe product of the two extremes is equal to the product of any two means that are equally distant from the extremes, and, also, equal to the square of the middle term, when the progression consists of an odd number of terms. Th'us, in the progression 2, 6, 18, 54, 162; 162 X 2 - 54 X 6 s18 X 18. When, a geometrical progression has but 3 terms, either of thie GEOMIETRICAL PROGRESSIoN. 151 extremes is called a third proportional to the other two; and the middle term, consequently, is a mean proportional between them. Thus, in the progression 48, 12, 3,3 is a third proportional to 48 and 12, because 48 divided by the ratio 12, and 12 divided by the ratio -- 3; or 3 X ratio 12, and 12 9 ratio 48: 12 is the mean proportional, because 12 X 12 - 48 Xg. Of the 5 properties of a geometrical progression, viz., the greater extreme, the less extreme, the number of terms, the ratio, and the sum of the terms, any three being given, the other two may be found Let s represent the sum of the terms, "E " the greater extreme. " e " the less extreme. " r " the ratio. n " the number ofterms. n when affixed as an index or exponent, represent that tho term, number, or quantity, to which it is affixed, is to be raised to a power equal to the number of terms in the respective progression, &e. Any three of lhe five parts of a geometrical progression being given, to find the remaining two parts. E e -_1 + E -sum of the terms. -E ><'- e = Sum of the terms. r- 1 rn X c - e r - 1 -= sum of the terms. r i E- (E+ rn-i) r -_ 1 q-E = sum of the terms. E-e - (E - e) 1 E sum of the terms. "- (E + e) — 1 1EXAMPLE. — The greater extreme of a geometrical progression is 162, the less extreme is 2, and there are 5 terms in the progression.' required the sum of the series. 162 -2 s8 -(-1612 )i G242. Ans. 1 (162 +2) - 1 s-X - l- +e - - ~- }- greater extrem-e. 152 G(EOMETRICAL PROtRENSION....... i.?.,. X e = greater extreme. n -l ) e = greater extreme. sX rn- Xr-1 r - 1 =s greater extreme. s - (s - E) X r = less extreme. E - r"-' less extreme. s X r-lX r- 1 rn — X 1 e rn-I =less extreme. s e - E -- ratio. s E * /- = ratio. sXr —i e + 1 = rn; n, tllerefore, is equal to the number of times sX — 1 tllat r must be multiplied into itself to equal e + 1 s X r —l1 s-(s-E) Xr EXAMPLE. — A farmer proposed to a drover that he would sell him 12 sheep and allow him to select them from his flock, provided the drover would pay 1 cent for the first selected, 3 cents for the second, 9 cents for the third, and so on; what sum of money would 12 sheep amnount to, at that rate rnX e —e r- -1 = s, then 312 X 1 — 1 3-1 - $2657.20. Ans. NOTE. - Ratio, cubed = ratio2; ratio, squared = ratiol, &c. When it is required to find a high power of a ratio, it is convenient to proceed as follows, viz.: write dozwn a few of the lower or leading powers of the ratio, soec-essively as they arise, in a line, one,fter another, and place thlcir rdspective inditcis over them; tlhe.n GEOMETRICAL PROGItESSION. 1 53 will the product of such of those powers as stand under such indices whose sum is equal to the index of the required power, equall the power required. EXAMPLE. - Required the 11th power of 3. 1 2 3 4 5 3 9 27 81 243 Icere 5 --- 4 + 2 _ 11, consequently, 243 X 81 X 9 = 11th power of 3, or 5 X 2 + 1 =11, consequently, 243 X 243 X 3 = 11th power of 3, or 4 X 2 + 3 = 11, consequently, 81 X 81 X 27 11th power of 3, or 3 X 3 + 2-= 11, consequently, 273 X 9 = r = 177147. Ans. To find any assigned n7umber of geometrical means, between two given numbers or extremes. RULE. - Divide the greater given number by the less, and from the quotient extract that root whose index is 1 more than the number of means required; that is, if 1 mean be required, extract the square root; if two, the cube root, &c., and the root will be the commnon ratio of all the terms; which, multiplied by the less given extreme, will give the least mean; and that, multiplied by the s;iil root, will give the next greater mean, and so on, for, all the meanls required. Or the greater extreme may be divided by the common ratio, for the greatest' mean; that by the same ratio, for the next less, and so on. EXAMPrLE. -Required to find 5 geometrical means between the numbers 3 and 2187. 2187 +. 3 = 729, and tV/729 = 3, then3X3=9X3=27X3-=81 X3-243 X 3=729, that is, tlhe numbers 9, 27, 81, 243, 729 are the 5 geometrical means between 3 and 2187. NOTE. —The geometrical mean between any two given numbers is equal to the square root of the product of those numbers. Thus the gcometrical- mean between 5 andl 2O, = %/(5 X 20) = 10. 154 ANNUITIES. ANNUITIES. AN annuity, strictly speaking and practically, is a certain sum of money by-the year; payable, usually, either in a single payment yearly, or in half, half-yearly, quarter, quarter-yearly, &c., and for a succession of years, greater or less, or forever. Pensions, awards, bequests, and the like, that are made payable in fixed sums for a succession of payments, are commonly rated by the year, and denominated annuities. A current annuity that has already commenced, or that is to commence after an interval of time not greater than that between the stipulated payments, is said to be in possession. One that is to commence or cease on the occurrence of an indeterminate event, as upon the death of an individual, is a reversionary, contingent, or life annuity. One that is to commence at a given period, and to continue for a given number of years or payments, is a certain annuity. One that is to continue from a given time, forever, is a perpetual annuity, or a perpetuity. Annuity payments do not exist fractionally: they mature, and exist only in that state, and are then due. A current annuity commences with a payment, and terminates with a payment. One current in the past is measured from a present included payment, closes with an included payment, and is said to be in arrears or forborne, from a supposed cancelled payment one regular interval or time, beyond. One current in the future is measured from the present to the first included payment of the series, and from thence is said to continue to the close; but if the interval from the present to the first included payment is equal to that between the successive payments, it is supposed to continue from the present. Annuities in negotiation are a(ljusted, with regard to time, by interest, or discount, or both. The TABLES applicable to compound interest and compound discount are applicable in adjusting annuities at compound rates. To fintl'the Amount of a Current Annuity in Arrears. LETMrA. - The amount of an annuity that has been forborne for a riven time is equal to the sum of the several payments that have become due in that time, plus the interest on each, from the time it became due, until the close of the time. ANNUITIES. 155 Then the amount of an annuity of $100, payabl)le in a single payment annually, but delayed of payment 4 years, allowing simpie interest at 6 per cent. on the payments, is 100 X 1.18 118 100-X 1.12- 112 100 X 1.06 -- 106 100 X = 100- $436. And at 6 per cent. compound interest on the payments, it is 100 X (1.OG)8 - 119.10 100 X (1.06)2 - 112.36 100 X (1.06)1 = 106.00 o00 X 1 100.00 - $437.46. At 6 per cent. simple interest, when payable in half, halfyearly, it is 50 X 1.21 = 60.50 50 X 1.18 -- 59.00 50>X 1.15 = 57.50 50 X 1.12 -56.00 50 X 1.09 = 54.50 50 X 1.06 - 53.00 50 X 1.03 -51.50 -50 X i -50.00 $442. And at G per cenlt. compound interest PER ANNUM, when payable in half-yearly instalments, it is 50 X (1.06)3 X 1.03 =61.34 50 X (1.06) 3 = 59.55 50 X, (1.06)2 X 1.03 = 57.86 50 X (1.OG)2 = 56.18 50 X (1.06)1 X 1.03 = 54.59 50 X 1.06 =53.00 50 X 1.03 = —51.50 50 X 1. =50.00= —$444.02. From the foregoing, we tlerive the following general RUwLES:Let P annuity or yearly sum, r =rate of interest per annum, a -rate of discount per annum, n or" -= nominal time of the annuity in full years, A = amount for the full years, D =present worth for the full years. 156 A NNU~ITIES. When the annuity is payable in a single payment yearly, A= Pn (l + r(n-P), Simple Interest. A P O +r), Compound Interest. When payable in equal half-yearly instalments, A = Pn (1 +r(nl+ 4 ), Simple Interest. A-P X (+r-1 X (1 +') Compound Interest. When payable in equal third-yearly instalments, A =Pn (i + r(n27) 3r ), Simple Interest. A —P (I+r)- (1 +), Compound Interest. When payable in quarter-yearly instalments, A = -Pn(1 (2 +- i-8), Simple Interest. A = p (+r(1 - 3- 1 ), Compound Interest. When there are odd payments, to find the amount, S. When 1 half-yearly, S = A(1 + r) + IP. 1 third-yearly, S = A(1 -+ r) + AP. 2 " S=A(1+ir)+P( l+r)+ P -A(1 +r) P(6 +r) 9. 1 quarter-yearly, S = A(1 -+ r) + 4P. 2 " S A(1 ~ r)+ P(8 +-r) + 16. 3 " S = A(1 + 3 r)-+ P(3 + 3 r) -4. For any number of equal and regular payments at compound interest per interval between the payments, S = Pt (l )-, and for any number of equal and regular payments at simple interest per interval between the payments, S P'ln (1 + r'(n-)); P' being a payment, nt or n' the number of payments, and rI the rate of interest per interyal between the payments. But this must not ble confounded with compound interest annually, on payments occurring semi-annually, quarterly, &c. EXAMPLE. - What is the amount of an annuity of $150, payable in half, half-yearly, but delayed of payment 2 years and 72 d(ays, allowing compound interest per annum at 7 per cent.? 150 X'(07' -1=$310.50, the amount for 2 years, if payable;n yearly payments, and ANNUITIES. 157 310.50 X (1i4 )-$31.5.93, the amount for 2 years, if payable in half-yearly payments, and 315.93 X ( ~1.h3- 2) =- $320.29, the amount for 2 years and 72 days, if payable in half-yearly payments. Ans. EXAMPLE. - What is the amount of an allowance, pension, or award, of $100 a year, payable quarterly, but forborne 3~ years, interest compound per annum at 6 per cent.? 100 (1.o6- 1 (1.o6x 3) $325.52, the amount for 3 years, and 325.52 (1 +.03) +- 100X 8.06 ~ 16= $385.66. Ans. EXAMPLE. - What is the amount of $100 a year, payable in quarterly payments, and in arrears 4 years, interest being compound per quarter-year, at 6 per cent. a year? 25 [(1 ~06- 1] I 4. By tabular powers of (1 + r), page 125, =$448.30. Ans. To find the Present lWorth of an Annuity Current. LE~IMA..- The present worth of an annuity that is to continue for a given time is equal to that sum of money, which, if put at interest from the present time to the close of the payments, will amount to the amount of the payments at that time; and therefore, the times being full, is equal to the sum of the several payments, discounted, respectively, at the rate of interest for their respective times. NOTE. -If the foregoing proposition is tenable, it follows, since simple interest is due and payable annually, that the true present worth of an annuity hiaving more than one year to run cannot be found by simple interest and dis. count. By simple interest and discount, at 6 per cent., predicating the rule upon the foregoing lemma, the anstountt of $100, payable annually, and in arrears for 4 years, is $436; and the lcesent worth, at 06 per cent., is se6 100 100 100 -+ fS~ + -- + = $349..124+i.18 1-.1121.06 But $340 at 6 per cent. interest for 4 years, with the payments of interest annually, will amount to $440.30; and at interest simply for 4 years it will amount to only $432.76. Then the present worth of an annuity of $100, payable in a silngle payment yearly, and to continue 4 3-ears, or to become due 1, 2, 3, and 4 years hlence, interest and discount being compoullnd per annum, and each at 6 per cent.14 158 ANNUITIES. P P P P (I + + (1 + + (1 + r + $346.51 = 100 x (1.06)3-=119.10 100 X (1.06)2 = 112.36 100 X (1.06) = 106.00 100 X 1 -100.00 -437.46' (1.06)'- $346.51. Andl interest at 6 per cent. and discount at 10, both compound, it is 100 X (1.06) - 119.10 100 X (1.06)2 112.36 100 X 1.06 106.00 100 X 1 -- 100.00 _ 437.46 -(1.10)4-$298.79. Therefore, when the annuity is payable in a single payment yearly from the present time, D =P (1 +a) =(l+ -)n when r and a are equal. When payable in half-yearly payments, D - P X(+r )- X (1+ 1 ) r(1 + a) When payable in third-yearly payments, P x [(l +rI -l]x ( +r) r(1 + a) When payable in quarter-yearly payments, P [(1 +rM - l] (1 +ir) r(1 + a) When there are odd payments, to find the present worth, S. Tlhere being a half-yearly, S 4 —- L 1 third-yearly, S — a + 1i +a " 2 " S. 1 +2P (l+lr) "' 1 quarter-yearl y, S D — I I+-a I+Ia " 2 " S 1 (+i 3 D ] PS (-r).!' 3 " S S — -=i-}-a1 + a + a For any number of equal payments, at equal intervals between the payments, S — P' X (1l+a')?c; P' being a payment,, n' tLc (1 + OUI bein ANNUITIES. 159 number of payments, and r' and a' the rates per interval between the payments. NOTE. - Since (r) -is the co-efficient of P, for its present worth, at r(l+a)n compound interest and discount, for the time n, at the rates r, a, it follows that tables of co-efficients of P for its present worth, at given rates, for any number of years, may be easily made. Thus (1.064 -1) 4-i.064 x.06 = 3.46511, the co-efficient of an annuity, P, for 4 years' continuance, interest and discount being compound per annum, at 6 per cent.; and (1.060 -1) (1.062 X.06)='1.8333), tile co-efficient for 2 years, &c. If the annuity is deferred, then the difference of two of these co-efficients (one of them that for the time deferred, and the other that for the sum of tile time deferred and the time of the annuity) will be the co-efficient of I' for its present worth. Thus 3.46511- 1.83339 = 1.63172, the co-efficient of an annuity, 1', for its present worth, when it is to commence two years hence, and to continue2 years, interest and discount being compound per annum, at 6 per cent. each; or D= 1.63172 P. Jin like manner, tables of other co-efficients, such as the formula suggest, may be made that will greatly assist in calculating annuities. EXAMPLE. - What is the present worth of an award of $500 a year, payable in half-yearly instalments, the 1st payment to mature 6 months hence, and the annuity to continue three years; interest and discount being 7 per cent., compounded yearly? 500 x [(1.07)"-1] (1. ). *.07 >X (1.07)-3 $1335.13. Ans. EXAMPLE. - What is the present worth of an annuity of $100, payable in half-yearly payments, and to continue 1' years; interest and discount being 6 per cent. per annum? 100 X [1.06 - 1] X 1. -0 D. X10 4 95.755, and.06 X 1.06 - 95.755 50 1.03 + -13 $141.51. Ans. EXAMPrLI.- What is the present worth of an annuity of $500, payable in semi-annual instalments, and to continue 10L years, interest and discount being compound per annum, the former at 6 per cent., and the latter at 89? 500 [(1.06)l- 1] (1..) 500.06 (1.08)10 (1. 2 2([1j A 250 1.O81 X 1.04 + 1.04- Ans. 160 ANNUITIES. By tabular powers of 1 + r, page 125:500 X.79085 2 $3052.64, the present worth for 10 years' con-.06 X 2.15892 tinuance, if payable in yearly payments, and 3052.64 X 1.015 = $3098.43, the present worth for 10 years' continuance, if payablc in hlaltyearly payments, and 3098.43- 1.04 +- 500 ~ 2 X 1.04 $3219.64. Ans. When the interval of time from the present to the 1st payi ment is shorter than that between the consecutive payments, and the annuity is payable in a single payment yearly, A,1 (1 + r) _-1] (1 + 3) d r A P[(l+r) —-1] (1') 3) D ( +-I'Y (n1 6 ))(~a (6)5 - d)(1 + ~a(36(5-d)' d being the time in days from the present to the 1st payment. So, if thle annuity is payable in half-yearly, third-yearly, or quarter-yearly instalments, multiply by 1 + - r, 1 + I r, or 1 + _ r, as befbre directed; and if there are odd payments proceed for the present worth, S, as already directed. EXAIMPLE. - Required the present worth of an annuity of $100, payable yearly, to commence 4 months hence, and to continue 4 years; interest and discount being 6 per cent. annually. 100 X (1.064 —1) X (1. l -).06 X 1.063 X (1.).0 x (12) $360,24. Ans. To find the Present Worth of a Dfe rred Current Annuily, or of an Annuity in Reversion. When the annuity is payable in a single payment yearly, and the deferred time embraces full years only, D=P'r (1 - a)( ) n' being the deferred time. If it is payable in half-yearly, third-yearly, or quarter-yearly instalments, multiply by 1 + 4 r, 1 + it r, or 1 + 8 r, as already ANNUITIES. 161 directed; and, if there are odd payments, find the present wortll, S, as already directed. EXAMPLE. - What is the present worth of an annuity of $150, payable yearly, to commence 2 years hence, and to continue 4 years; interest and discount being compound per annum, at 6 per cent.? 150 X (1.064 - 1) +.06 X 1.066 = $162.59. Ans. EXAMPLE. - Required the present worth of an annuity of $500,'payable in semi-annual instalments, to commence 21 years hence, and to continue 6 years; allowing compound interest and discount annually at 7 per cent. 500 X (1.076 - 1) X 1.-07 4 -$2046.44. Ans..07.07 X 1.078 X 1. - EXAMPLE. - Required the present worth of an allowance, pension, or award of $125 a year, payable in half-every half-year, to commence 7 months 24 days hence, and to continue 62 years; interest and discount being compound per annum at 5 per cent. 125 X (1.056 - 1) X 1.0125 125.05 X 1.056 X 1.03247 X 1.025 + 2 X 1.025 = $668 Ans. o, 125 x [(1.05)0 — 1] Or 125 X [(1.05)- 11= $634.47, the present worth for 6.05 > (1.05)0 years' continuance, if payable in yearly instalments; and 631.47 X 1.- =$642.40, the present worth for 6 years' continuance, if payable in half-yearly instalments; and 642.40 ~ (I +.05 x 237) the present worth for 6 years' continuance, if payable in llalf-yearly instalments, and to commence 7 months, 24 days hence; and 622.20 +2X $668. Ans. 622.20 + 2 (1 2 1.-25 To find the Present Worth of a Perpetuity. LEMmA. - The present worth of an annuity to commence one year hence, and to continue forever, is expressed by that sum of money whose interest for 1 year is equal to the amount of the 14* 162 ANNUITIES. annuity for 1 year; and so, pro rata, for perpetuities. otherwise regularly affected. Then when the annuity is to commence 1 year hence, and is payable in a single payment yearly... D) P * r. Payable in half-yearly instalments. D P(1 -- ). r Payable in third-yearly instalments.. D -P(1 + ). Payable in quarter-yearly instalments. D -=P(1 r) r _EXAMPLE. - What is the present worth of a perpetuity of $150 a year, payable in a single payment yearly from the present time; intcrest at 6 per cent? 150 ~.06 -- $2500. Ans. EXAMPLE. -- What is the present worth of a perpetuity of $150 a year, payable in semi-annual instalments, and to commence 4 months hence; interest 7 per cent? Pr(l- 4r).+ 07(12-4) $2187.36 Ans. r 12 EXAMTPLE. - Required the present worth of a perpetuity of $400 a year, payable in quarterly payments, and to commence 6 years hence; interest and discount being 5 per cent., compound per year. P(1 +-] r) 400 X I."' r(1 r)u.00X 8 =$G081.G5. Ans. r( 1 + a)".05 X 1.056 - The Amount, Tilze, and Rate given, to find the Annuity. When payable in a single payment yearly from the present time, Ar Ar P- = r)'-I; half-yearly, P- Ar [ +(l r-1](l r) Ar third-yearly, P = ( r) [(l quarterly, Ar (1 -+- r)[(1 r 11 r) 1 ANNUITIES. 163 and so, pro rata, for other fractional units of the integral unit. Ar Ar Ar Therefore (1 - r)"- 1 or (- r),or (1 ) or &c. or ( + 3 r)' EXAMPLE. -- What annuity, payable in quarterly payments from the present time, will amount to $3000 in 12 years; interest, being compound per annum, at 8 per cent.? 3000 X.08-+- [(10812- 1)X1. 3 XO 08] = $153.48. Ans. EXAMPLE. - What length of time must a current anhuity of $400, payable in quarterly payments~ remain unpaid, that it may amount to $2500; interest being 7 per cent. yearly? 25400X.07:.4263094-=5.-tyears, and 5 years by tabld of 400X 1.0 >.o7 /.420394 35 (1 +r) — 1 =.402552: therefore.40552 -1) 07 308 days, 5 years, 308 days. Ans. The Present 7Worth, Time, and Rate given, to find the Annuity. When payable in a single payminent yearly from the present time, pDr(1+-r); hallf-yearly, rP D)r(1-) thirdyearly, P -[(1 - 1 D ](.; quarter-yearly, P = Dr(1 - r-)a P Dr(1+ r I)n(-; r)' )&c. Therefore, (1 + )"; -p _ Dr P(l + r) 1]1 -}-P(1 - +r) P( 1 + ~,r)-Dr P(1 + Ir) Dr' EXAMPL.E.- What annuity, payable in half-yearly instalments, and to continue 3 years, is at present worth $1335.13; discount a(nd interest beingr compound per year, at 7 per cent? 1335.13X.07X 1.07 (1.07 —- 1) X1'-7 $500. Ans..4 164 ANNUITIES. OF INSTALMENTS GENERALLY. Any certain sum of money to be paid on a debt periodically until the debt is paid is called an instalment; and a debt so made payable is said to be payable by instalments. Let D principal or debt to be paid, n number of years in which the debt is to be paid, r rate of interest per annum, p = instalment or periodical payment. When the instalments are payable yearly, and the debt is at interest, Dr(+ 1 + r)n _. _; D p [(1 -+- r)_-_l(1 -+r)- -i (+ — - Dr' r(+ -. When payable half-yearly, Dr(l1 r)" P = 2 C(: q- 0" — 1](: q- 1r ]; (1S+")"=[ ]( -)+P-D,+; D 2p +( +O". When the debt is not on interest, and the instalments are payable yearly, Dr Dr-p D ( + r)"' — p -( +); ( 1 + r)= p; D p r EXAMPLE. What yearly instalment will pay a debt of $4000 in 4 years, the debt being on interest the while, at 6 per cent. annually? 4000 X.06 X 1.064'- (1.064 - 1) = — $1154.37. Ans. EXAMPLE. -What semi-annual instalment will pay a debt of $1500 in 3 years, the debt bearing interest at 7 per cent. yearly? 4500 X.07 X 1.07 $82 $842.62. Ans. 2X (1.07 —1) X 1.0175 When the debt is on interest, and is payable in equal yearly instalments, p D(1- + rn) n(l + -!(.), at simple interest; but simple interest is not strictly applicable to instalments. See NoTE, p. 157. ANNUITIES. 1G5 When a debt has been dininished at regular intervcals by the payment of a constant sum, to find the remaining debt at the close of the last payment. When the debt is on interest, and the payments have been made yearly from the date of the debt, d-p (p-Dr)( + ); r (p dr r p -Dri _ Dr(1 + r)- dr P - (1- r)- 1 When the payments have been made half-yearly, d =p +p(1 + r) - (1 -+ r)5 [p + P (1 + r) -Dr] -. r, &c. EXAMPLE.- On a debt of $1000, drawing interest the while at 8 per cent. a year, there has been paid yearly, from the date of the debt, $200 for 6 years: required the unpaid debt at the close of the last payment. [200 — 1.086 (200 -.08 X 1000)]..08 = $119.69. Ans. EXAMPLE. - On a note of hand for $1000, and interest from date, at 8 per cent. annually, the following payments have been made; viz., $100 at the close of every half-year from the date of the note, for 6 years. How much remained unpaid at the close,of the last payment? [204 - 1.086(204 -.08 X 1000)] --.08 $90.34. Ans. NOTE.-IIn the foregoing, I have treated the terms annual interest and interest payable annually as synonymous in meaning with the terms compound interest and compound interest per annum, and they are so in equity and in factl: besides, simple interest is inapplicable, in equity, to instalment payments. If the debtor stipulates to pay the interest annually on a debt, and abides his contlract, he will pay it when it becomes due, and it then becomes a principal in the hands of the creditor, to be let, it is fair to suppose, upon as favorable terms to himself as he'let the principal which grew it: whereby lihe realizes equal to compound interest per annum on the first principal: moreover, if tie debtor withholds it from the creditor, it is fair to suppose that lie considers it of as much worth to himself as a like part of the principal. 166 R PlLiUTATfON. PERMUTATION. PEIMtJTATION, in the mathem'atics, has reference to the greatest number-of unlike relative positions, that a given number of things, either wholly unlike, or unlike only in part, may bc placed in. It considers the number of changes, therefore, that may be made, in thle arrangement of the things, under different given circumstances. ro find the number of changes that can be made in the order of arralngcmenl of a given number of things, when the things are all differcnl. RULE.- Find the product of the natural series of numbers, fro:n I up to the given number of things, inclusive; and thait product will be the number of changes or permutations that may be imde. E;xAMPrLE. -In how many different relative positions may 12 persons be seated at a table. 1 X 2 3 X 4 5 X 7 X X9X lOX l 12 479,001,600. Ans. To find the number of changes that can be made in the older of arrang ement of a given number of things, when that number is conposcd oJ' several d~i[erent things, and of sevcral which are alilie. RULE. — Find the number of changes that could be made if the things were all unlike, as in first example. Then find the number of cha.nges that could be made with the several things of each kind, if they were unlike. Lastly, divide the number first found by thle prodlemt of the numbers last found, and the quotient will be tlle number of permutations or changes that the collection admits of. EXAMPLrE.-Required the number of permutations that can be made with the letters a, bb, ccc, dddd, = 10 letters. 1 X 2X3X4X5XGX7X8X9X10=3628800 1X2XGX24= 288 -12,G00 l.s To find the number qf permutations that can be made with a given number of different things, by taking an assi g7ned number of them at a time. RULE. -Take a series of numbers beginning with the number of thinfgs given, and decreasing by 1 continually, until the number of terms is equal to the number of things that are to be taken at.a time; then will the product of tlme series be the nuimmbger of ch:nges tlhat may be made. COMIrINATION. 167 EXAM\iPLE. - What number of changes can be malde with the numbers 1, 2, 3, 4, 5, 6, taking three of them at a time? 6 — 1 = 5, 5 — 1=4, then G X 5 X 4 120. Ans. What number, by taking 4 of them at a time? 6X5X4X3=360. Ans. EXAMPLE. - Arrange the three letters a, b, c, into the greatest number of permutations possible. ar1c, acb, bac, bca, cab, cba, =6 permutations. Ans. EXAMPLE..- Arrange the four letters a, b, a, b, into the great,'s; numllber of permutations possible. abab, albb, abba, bbaa, baba, baab, -. 6 permutations. - Ans. COMBINATION. COMBINATION, in the mathematics, has reference to the number of unlike groups, which may be formed from a given number of different things, by taking any assigned number of them, less than the whole at a time. It does not regard the relative positions of the things, one with another, in any of the collections or groups. Bult it exacts that each group, in all instances, shall have the assigned number of members in it, and that, in every group, in every instance, there shall be a like number of members. It exacts, therefore, that no two groups shall be. composed of precisely the same members. ro find the number of combinations that can be made from a gniten number of. different things, by taking any given nunmber of th/m at a time. RULE. — Take a series of numbers beginning with that which is equal to the number of things from which the combinations are to be made, and decreasing by 1, continually, until we number of terms is equal to the number of things that are to be taken at a time, and find the product of those numbers or terms. Then take the natural series, 1, 2, 3, 4, &c., up to the number of things that are to be taken at a time, and find the product of that series. Lastly, divide the product first found by the product last found, and the quotient will express the number of combinations that can be made. EXAMPLE. -What number of combinations can be made fiom 8 different things, by taking 4 of them at a time? 8X7 X 6X5 1680 1X2X3X4 70 Ans. 1C 8 co.B r,lNATION. Whlat number, by taking 5 of them at a time? 8X7X6X5X4 6720 X2X3X4X5-120 56 Ans. Whlat number, by taking 3 of them at a time? 8 X 7 X 6 336 1X2X3 — =656. Ans. EXAMPLE. - What number of combinations can be made from 5 different things, by taking three of them at a time I 5X4X3 60 - -X2X 3- ----- 10. Ans..1X2X 3 6 Wlbmit number, by taking 2 of them at a'time? 5X 4 20 10. Ans 1X2 2 EAMrPLE. -Form 5 letters, a, b, c, d, e, into 10 combinations of 2 letters each; that is, into 10 unlike groups of two letters eacb. ab, ac, ad, ae, be, bd, be, cd, ce, de. Ans. Form them into the greatest number of combinations possible, in collections of thlree each. abc, abid, abe, acd, ace, ade, bcd, bce, bde, cde. Adns. SECTION A.?OREIGN MONEYS OF ACCOUNT, COINS, WEIGHTS, AND MEASURES,,EDUCED TO THEIR VALUES IN THE MONEY, WEIGIITS, AND; M[EASURES OF THE UNITED STATES. THE many changes that have been made in the moneys of acount, coins, weights, and measures of different countries, by hieir respective governments, within the: last few years, chiefly, lough not in.all cases, by the adoption of the Metric System, or ystems bearing aliquot relations thereto, have compelled the auhor to re-write this section of the work, in a great measure, since hie first editionwas published; and it is the intention that this dition, and subsequent editions that may be published, shall conuin this section strictly correct in all particulars at the time of oing to the press. The Federal units of comparison in the following tables, unless therwise expressed, are as follows; viz., the dollar of 100 cents, i gold; the commercial or avoirdupois pound, of 7000 grains; the omnmercial yard, of 36 inches; the commercial or wine gallon, f 231 cubic inches; the commercial or Winchester bushel, of 150% cubic inches; the standard foot, of 12 inches; the statute tile, statute acre, &c. The value in Federal money, therefore, affixed to any particular enomination of a foreign money of account in the following:ibles, is the equivalent, or intrinsic par, of that denomination 1 United-States gold coins. It is predicated upon the standard reight and purity of the coins coined especially to represent that enomination, or conventionally held to be the measure of its alue, compared with the standard weight and purity of the gold oins of the United States, that represent the dollar or its multiles. Thus, in respect to those countries in which gold is made the,easure of value and chief legal tender, it is the intrinsic par, old for gold; and, in respect to those countries in which silver is 1 1 2a FOREIGN MiONEYS Or ACCOUNT. made the measure of value and chief legal tender, it is the par value of that denomination in United-States gold coins, based upon the almost constantly prevailing relative commercial values for many years past, of gold to silver, as 15] to 1, for equal weights. It is, therefore, in a commercial point of view, the intrinsic par of that denomination, in Federal gold coins, in all cases. The denomination itself, to which the Federal value is immediately affixed, is usually the integer, or ultimate money of account, of the country especially referred to. It is a money of account in that country always, but not always the name of a circulating coin. Occasionally, even, its value is not represented by any known single circulating coin. From the foregoing remarks, it will be perceived that, when the mintage relative values of gold to silver, in any particular country, are maintained at rates nearer to each other than 15] to 1 for equal weights, the gold coins of that country are commonly worth more, as commercial material, than its silver coins of the same denominations, or same prescribed values; and, conversely, that when the mintage relative values are limited to rates more remote from each other than 153 to 1, the gold coins are commonly worth less than the silver coins. Thus, the Federal dollar, in standard silver coins, is ordinarily worth, as commercial material, but 14.88372 + 15.375 - 961 cents in Federal gold coins. But, since most Governments make silver the chief measure of value, the mint value of gold is usually purposely placed above its commercial worth. Thus, twenty francs in French gold coins are ordinarily worth, as commercial material, but 15.375 X 20 15.5 = 19.8387 francs in French silver coins. It is true that the silver coins of the United States, in small sums, for immediate use, in limited localities at home, may occasionally sell in exchange for the gold coins at their nominal values, or even at a premium, according to the local demand and supply; and the same may happen with regard to the gold coins in exchange for the silver, in France, and those other countries where gold is purposely over-valued in the mintage; but these conditions do not affect the general commercial relations of the metals: they are due only to a slight derangement in the required distributions of the two kinds of coins. In Germany and Austria, the mint relative values of gold and silver for the Zollverein money, are as 151 to 1, for equal weights. FOREIGN MONEYS OF ACCOUNT. a3 FOREIGN MONEYS OF ACCOUNT AND COINS REDUCED TO THEIR VALUES IN FEDERAL MONEY..Foreign. U. States. ABYSSINIA, (E. AFRICA). - Massuah: The old Venetian zechino (sequin) is current here at 50 harfs; and 23 harfs - 1 pataka, or old Spanish dollar, - $1.01385 Austrian rix-dollars and Spanish dollars are current here at 1 pataka each. NOTE. -The old peso duro colonato, or Carolus silver dollar of Spain, contained, at mint usage, 415 grains of mint silver g4 fine = $1.041353; but it is no longer struck at the mint, and those in circulation are more or less abraded. It is now valued, throughout the British Possessions in North America, and generally, wherever it circulates by tale, or is made the integer of the moneys of account, at 50 pence sterling in gold = $1.0138512. The Austrian rix-dollar (tallaro), scudo, or crown, which, by the way, has not been coined since 1858, except on orders for foreign circulation, contains 1 Vienna mark of fine silver, or 361.11 grains = $1.0114911. This is often called the German dollar; and the Venetian dollar is of the same value. ALGERIA (N. AFRICA). - Algiers, Bona, &c.: 100 centimes — = 1 Franc -. - = 0.19452 ARABIA. -Muscat: 20 goz = 1 namooda, 20 m. 1 currant Spanish dollar - - - = 1.01385 NOTE. -1 goz =2 paras, and 1 mamooda -1 piastre of Egypt. See EGYPT. Mlocha, Hodeida: 2 crats — 1 commasse; 60 c. = 1 Mexican or Spanish dollar by tale = 360 grains of fine silver, 1.- 1.00838 160 crats = 1 wakega or troy ounce (gold and silver weight). Jidda: Same as at Alexandria, Egypt. Aden: 80 caveers = 1 piastre of account = current Spanish dollar, - - - - 0.84488 Also, as at Calcutta. Official, as in Great Britain. AUSTRALIA. - Sidney, Melbourne, Iobart Town, and A ustralasia generally: Standard of purity, denominations, values, and relative values, since 1855, same as in Great Britain. AUSTRIA. - Vienna, Prague, Trieste, Ragusa, 6c. Zollverein money: Standard for gold and silver coins = fine, each; relative values, gold to silver as 15.375 to 1. 4 a FOREIGN MIONETS OF ACCOUNT. Foreign. U. States. 4 pfenninge - 1 kreuzer; 60 k. 1 gulden, or florind g Zollverein pfund (11 grams) of fine silver, or 171.471 grains, - - — $0.4803 1 gulden = 1 Zollverein thaler; 1107 gulden 80 Zoll. krones; 81 gulden _ 200 francs; U. S. Customs value of gulden AZORE ISLANDS. - Fayal, Terceira, Corvo, St. Michael, c.: 1000 reis - 1 milreis of account= ~ old current Spanish dollar,. - - 0.84488 U. S. Customs value = 83j cents. NOTE. -In 1834, English sovereigns and Spanish dollars were made legal tender here and at the Madeiras; the former at the rate of 4120 rels, and the lattecr at 870 reis, each, which corresponds very nearly with their intrinsic values in standard Portuguese gold coins. BALEARIC ISLANDS. - MAJORCA, Palma; MRINORCA, Port Mahon: Same as new system in Spain, see SPAtI. BELGIUM. -Brussels, Antwerp, Ostend, Sc.. Standard for gold and silver coins =- fine, each. Relative values, old to silver as 15.8228* to 1. 100 centimes- 1 Franc = 4:/ grammes of fine silver, - - 0.19452 BERBERA (E. AFRICA.): Same as at Mocha, ARABIA. ~ BERMUDA ISLANDS. - Official, as in Great Britain. In. trade, 100 cents =- 1 dollar - 1 old current Spanish peso, or 50 pence sterling in gold, - - - - — 1.01385 NOTE. -At the Bermudas, in British America, and other British foreign possessions generally, official or government accounts are kept, and duties to the government are assessed, in sterling money; but until 1842 this class of accounts were kept at the Bermudas and Jamaica, in pounds, shillings, and pence, at 12 shillings sterling to the pound, when it was ordered that hereafter they be kept in sterling money, and that all existing contracts in those colonies be settled at the rate of I pound sterling per colonial pound. BOURBON ISLAND. - St. Denis: 100 centimes - 1 franc - - - 0.19452 * Although the silver coins of Belgium, both by law and general usage, have the same intrinsic values as those of France of the same denominations, yet this rule does not hold good with regard to the gold coins. The mint standard for 25 francs of France is 8[fp grammes of mint gold 10 fine, while that for 25 francs of Belgium is only 7191 grammes of mint gold l9Q fine; and so in proportion for the other gold coins. 25 francs, French mint, are worth $4.823816, while the 25-franc piece, Belgic mint, is worth only $4.72541; in other words, the Belgic gold coins are less in value two centimes per frano than the French gold coins, 1OfEOlN aM6NE1tY OF ACCOUNT. a 5 Foreign. U. States. CANADA, DOMINION OF, and British America generally: Standard for silver coins (20-cent pieces or Colonial shillings) 7 fine; for gold coins (British sovereigns) = fine. Relative values, gold to silver as 14.341 to 1. 100 cents = 1 dollar colonial, - - - = $1.00 Also, 4 farthings = 1 penny; 12 p. = 1 shilling; 20 s. = 1 pound colonial, - - 4.00 NOTE. —The standard 20-cent piece, or colonial silver shilling, of British America, contains 661 grains of fine silver, and is, therefore, worth only 18.655 Federal cents in Federal gold coins, or 19.2708 Federal cents in Federal silver:coins. But the money of account shilling in British America is equal to 20 Federal cents in gold; thus, the British pound sterling in gold, the British sovereign, is equal to: $4.8665 in gold; and the shillings in that sovereign are equal to 24.3325 Federal cents, each, in gold; therefore, 4.80 Q = -$4.00, the value of the colonial pound in Federal money (gold, measured by, or payable in, British standard gold. See CANADA, page a 51. CANARY ISLANDS.- Tenerife, Palma, Grand Canary, Fuerteventura, &c.: Official as in Spain; in trade, occasionally, 8 realls (antiquas) of 34 maravedes each = 1 piastre, or peso of exchange, - - - - -0.75623 CANDI ISLAND. - Same as in Turkey. CAPE OF GOOD HOPE (S. AFRICA). - Cape Town, sc.: Same as in Great Britain. CAPE VERDE ISLANDS. —ST. VINCENT, Mindello; ST. JAGO, Porto Praya, ic~.: 1000 reis = 1 milreis. Old Spanish dollars are current here at 870 reis. Central and South America. CENTRAL AMERICA. - HONDURAS, Truxillo, JPort San Lorenzo, Omoa, tic.; NICARAGUA, Realejo, Greytown, &c.; SAN SALVADOR, La Union, Sonsonate, sie.; COSTA RICA, Puntas Arenas, Matina, &c.: GUATEMALA, Ystapa, sc.: Standard fbr silver coins (dollars) = T2 Castilian marco of silver Hi fine, or gu marco of fine silver to the dollar;' for gold coins (double escudos of 32 reals) = ${ Castilian marco of gold - -4 - fine. Relative values, gold to silver as 16.5614 to 1. 100 centavos or 8 redls - 1 dollar -- 355.08 grains of fine silver, -- 0.9946 Spanish dollars and U. S. gold coins circulate here, dollar for dollar. 1* 8 a FOREIGN MONEYS OF ACCOUNT. Foreign.. U. States. BALIZE, Balize: Official accounts are kept here in sterling money. SOUTH AMERICA. — PERU, Callao, Islay, Truxillo, Arica, ic.; CIILI, Valparaiso, Concepcion, Coquirnbo, tc.; NEW GRANADA, Cartagena, Santa Martha, Savanillo, Buenaventura, sc.; ECUADOR, Guayaquil, i'c. The prescribed standard for the mintage of these States is now in conformity with that of France, and there is strong probability that Brazil, and the other States in South America having mints, will soon adopt the same standard. Standard for gold and silver coins - T9 fine, each. Relative values, gold to silver as 151 to 1. 100 cents _ 1 sol, or dollar = 221 grammes of fine silver, - -$0.97461 Also, 100 centesimas -1 duro or old Spanish dollar, - 1.01385 BRAZIL. - Rio Janeiro, Maranham, Bahia, Para, Pernambuco, sc.: Standard for silver coins = - Castilian marco of silver 1 fine per milreis; tor gold coins 5 Castilian marco of gold 1, fine per 10 milreis. Relative values, gold to silver as 14.30556 to 1. 1000 reis 1 milreis = 180.8278 grains fine silver, - - -0.50651 A current Spanish dollar passes for 2 milreis of account, and the modern gold coins (10 milreis and 20 milreis) $0.544375 per milreis. BOLIVIA.- Cobija, tic.: Standard for silver coins (dollars) - -- marco of silver 1 fine; for gold coins (doubloons) -- 2 marco of gold ~ fine. But little gold is coined in Bolivia, and that in circulation has, at present, no nominal mint relation to the silver coins. 100 centavos = 1 dollar 283.03478 grains of fine silver, - 0.79292 NOTE. —The gold coins of'Bolivia are often light of weight, and seldom range above -6o7) fine. The silver dollars are usually minted at full weight, but are often but little if any above -1j fine. The fractional silver coins are worth less, relatively, than the integer. VENEZUELA. - La Guayra, Maracaybo, Cumana, Puerto Cabello, 6c.: In Venezuela, as in Peru, &c., Go0EIGN IONEYS OF ACCOUNT. a Foreign. - U. States. the silver 5-franc piece of France is made the measuring unit of value, and is divided into 100 centavos. In the moneys of account, however, the peso macuquins of 80 centavos is sometimes used; and one dollar in United-States gold coin is assumed to be worth 7 cents more than 5. silver francs, which is the case at the metal ratio of gold to silver as 16.0794 to 1, for equal weights. Hence, 1 peso fuerte Americano de premio — %-0 =- $1.3375, measured by the peso macuquins. A-RGENTINE REPUBLIC. —Buenos Ayres, Parana, &c.;. URUGUAY, Montevideo, &c.; PARAGUAY, Assumption, Neembucu, sc.: Foreign gold and'silver coins circulate here measured generally by the Spanish dollar: 10 decimos = 1 redl; 8 r. = 1 peso. GUIANA, Cayenne: Same as in France. Paramaribo: Sanme as in Holland. Georgetown: 100 cents = 1 dollar =50 d. sterling. FALKLAND ISLANDS. Same as in Great Britain. CHINA. - Canton, Shanghai, Amoy, Ningpo, Foochoo, Iiongkong L, MIacao: Standard for gold, and silver ingots- 94 touch, or _1,4 fine, each. Gold is not treated as money, but as merchandise. 10 cash or lc - 1 candarine or fun; 10 c. = 1 mace or tsien; 10 mi. 1 tael or leing=583~ grains of silver ffU4 fine, or 548k grains of fine silver, - - =$1.53591 NOTE. - Mexican dollars, which are about the only coins, except copper cash, that circulate in China, on account of their convenience generally, bear a premiumn well laid on, upon their intrinsic worth. They commonly pass for about 72 candarines each, or 2- of the tad in ingots of standard purity. A tad weight, or leaIng as it is called by the natives, of wan-yin or sysee (see-sze) silver, that is, of silver as fine as silk, or that may be drawn into a thread as fine as silk (pure silver), is valued by the East India Company at 80 pence sterling. In London, it is commonly valued by the price per ounce paid at the:mint for the fine silver in foreign silver coins, which is about 66 pence sterling, or X-_ 5 -A -- 80.2 pence sterling per tael. In 1799, the Congress of the United States fixed the value of the tacd, in ingots of standard purity, for customs' purposes, at $1.48; but this was when silver was the measure of value in the United States, and when the dollar contained 3711 grains of fine metal. The law, it is believed, has not been repealed. CORSICA ISLAND.- Ajaccio: 100 centimes _ 1 franc, - 0.19452 8a FOIREIGN HONEYS OF ACCOUNT. Foreign. U. States CYPRUS ISLAND. - Same as in Turkey. DENMARK. - Copenhagen, Elsinore, Odense, ~c.: Standard for silver coins - s 77 fine. 16 skillinge = 1 mark; 6 m. 1 rigsdaler -- t Cologne mark of mint silver — former speciesdaler = 195.1233 grains fine silver, - $0.546552 U. S. Customs value of speciesdaler -= $1.05. NOTE. -Denmark coins ducats and 5-thaler and 10-thaler gold pieces; but these and the speciesdaler have no proposed mint-relation to each other. Tho ducat is of the ordinary value of that denomination of coins; and the 10-thaler piece is designedly and practically of the same intrinsic value as that of North Germany; viz., equal to ten Bremen thalers in silver at the metal ratio of 15.409 to 1; or equal, at the common par of exchange, to- $7.90005 in gold. EGYPT (N. AFRICA). - Cairo, Alexandria, Suez, sc.: 40 paras = 1 piastre; in current gold coins $0.04996647; in current silver coins $0.05054979. NOTE. - Tn 1836, British sovereigns were made legal tender throughout Egypt at 97 pi., 20 pa. each; Spanish doubloons or onzas at 313 pi., 29 pa.,; Napoleons (20 francs in gold) at 77 pi., 6 pa.; Venetian sequins at 46 pi., 13 pa.; Dutch ducats at 45 pi., 26 pa.; tallaros (German dollars) at 20 pi.; colonatos (Spanish dollars) at 20 pi.,'28 pa.; 5-franc pieces at 19 pi., 10 pa. The measure of value is oka-drachmas of fine silver to the piastre = 2 Abyssinian dirhem or 18 troy grains = $0.050419109. FRANCE.- Standard for gold and silver coins = fine, each. Relative values, gold to silver as 151 to 1. 10 centimes - 1 decim; 10 d. = 1 franc = 4 grammes of fine silver, - - - = 0.19452 In practice, 100 centimes - 1 franc. U. S. Customs value = $0.186. GERMANY (The Zollverein States, or North German Confederation): PRUSSIA, SAXONY, MECKLENBURG, OLDENBURG, the HANSE CITIES, 6C. Zollverein money: Standard for gold and silver coins == fine, each. Relative values, gold to silver as 15.375 to 1. 12 pfennige = 1 silber groschen; 30 s. g. 1 thaler = ~ Zollverein pfund (16 grams) of fine silver, or 257.2058 grains, - - = 0.72045 369 Zollverein thalers = 40 Zollverein krones; 1 Zoll. krone = -- Zollverein pfund (10 grams) of fine gold, or 154.3235 grains - $6.64614; 27 thalers -- 100 francs. U. S. Customs value of thaler -- NOTE, - S'axony reckons 10 pfennige to the groschen. FOREIGN MONEYS OF ACCOUNT. a 9 Foreign. U. States. Breneen (special): 5 schwarcn - 1 groot; 72 g. 1 thaler -= old Frederic -d'or, - -=-$0.79619 U. S. Customs value = 783 cents. Hamburg, Lubec, Altona (special): 12 pfennige = 1 schilling; 16 s. = 1 mark. 1 mare current = A Cologne mark of fine silver, = 0.28869 1 mare banco, at par (London rate of 13m. 12s. to the ~.), =- =0.35393 U, S. Customs value of mark current - 28 cents; of mark banco = 35 cents. Southern States. - BAVARIA, WURTEMBURG, BADEN, SC.: Standard for silver coins -- T fine. 4 pfennige - 1 kreuzer; 60 k. 1 gullen or florin = SX Zoll. pfund of fine silver (9a1 grams) or, 146.975 grains, - - - 0.41169 7 gulden = 4 Zollvcrein thalers; 189 gulden = 400 francs. U. S. Customs value of gulden - 40 cents. GREAT BRITAIN. - Sterling money: Standard for silver coins = hi fine; for gold coins - fine. Relative values, gold to silver as 14.287891 to 1. 4 farthings = 1 penny; 12 p. 1 shilling; 20 s. - 1 pound, - - - 4.86656 U. S. Customs value = $4.84. NOTE. - Since 1816, the Government of Great Britain has estimated the value of gold compared with that of silver as 14 83 to 1, foi' equal weights. In theory,.by the mint regulations, apound sterzing is equal to 1614T6f grains of fine silver, or 113 grains of fine gold. A pound sterling in silver, therefore, measured by the Federal stlaudard of 345% grains of fine silver to the dollar, is equal to t$41 33 -in Federal silver coins; or a standard silver shilling, measured by this measure, is equal to 23 J7 i cents; and at the ordinary par of exchange it is worth $0.226122, very nearly, in Fedcral gold coin. But a pound sterling in British gold (a sovereign) measured by the Federal standtard of 23 11 grains of fine gold to the dollar, is equal to $4-2 48 3-, or $4.866563529, very nearly, In Federal gold coins. At the former relative values of the two currencies, or relative values before the Federal standard was changed, in 1834, viz., 4 shillings and 6 pence sterling to the dollar,. or $4.444 to the pound, the par of exchange is.09498, or 9S per cent, practically, in favor of sterling money. Gold is the measure of value in Great Britain, as well as in the United States; and the silver coins of that government are not legal tender at home in sums cxceeding ~2. GREECE.- Athens, Patras, the Ionian Islands, &c.: Standard for gold and silver coins -- fine, each. Relative values, gold to silver as 15.549 to 1. 100 lepta = 1 dramia = 1 dramia weight of fine silver, or 62% grains; - - =0.17401 10a FOREIGN MONEYS OF ACCOUNT. Foreign. U. States. 5 dramias = 1 taliron. 20 dramias (gold) = 1 othonion = $3.44137. i00 oboli =1 Spanish, German, or Venetian dollar -- $1.01385. NOTE. - Greece is a party to the present movement (1869) to legalize and enforce the use of the metric system of moneys, weights, and measures, throughout Continental Europe. HOLLAND (NETHERLANDS). — Amsterdam, Rotterdam, The Hague, sc.: Standard for silver coins 1 fine; for gold coins-= A fine. Relative values, gold to silver as 15.7333 to 1. 100 centimes - 1 guilder or florinm = 10 grams of mint silver, - - -$0.40806 Gouden Willem (10 guilders) = 6] grams of mint gold $3.98769. U. S. Customs value of guilder - 40 cents. HAWAIIAN OR SANDWICH ISLANDS.- Honolulu, &c.: 100 cents - 1 dollar, - - - - =1.00 India and Malaysia, or cEast Indies. INDOSTAN. — BENGAL, MADRAS, BOMBAY, Presi, dencies of: Standard for gold and silver coins =- fine, each. Relative values, gold to silver as 15 to 1. Calcutta, Madras, Rangoan, &c.: 12 pice = 1 anna; 16 a. - Irupee —=1 tola, or 180 grains of mint silver, = -- old sicca rupee, 0.46217 1 mohur, or gold rupee (E. I. Co.), 15 silver 0 7 rupees in theory = 1 tola of mint gold, or 165 grains of fine gold, = $7.10594. Bombay, Surat, sc.: 100 reas = 1 quarter; 4 q. = 1 rupee, - - -. 0.46217 U. S. Customs value of rupee -441 cents. Madras, tc. (old usage).: 48 jittas = 1 fanam; 36 f. 1 star pagoda = 4 arcot, or Company rupees, - = 1.84869 48 jittas 1 fanam; 36 f -- 1 India pagoda = 8 shillings sterling in gold, - - - = 1.9466 U. S. Customs value of star pagoda =$1.84; of Indian pagoda,- $1.94. Goa: Official, same as in Portugal. Pondicherry: Same as in France. 0OREIGN MONEYS OF ACCOUNT. a 1t Foreign. U. States. CEYLON ISLAND.- Colombo, Trincomalee, tic.: Official, same as in Great Britain;- also, 1 rixdaalder = 18d.; 1 Spanish dollar 50d.; 1 rupee = 22d., sterling. MALAYA. -Malacca, Pahang, Perak, Sc.: Same as at Singapore. PENANG ISLAND.- Same as in Calcutta; also, 100 cents - 1 Spanish dollar. SIAM. -Bangkok, sc.: 4 prangs, or clams, _ 1 sompay; 4 s. _ 1 salung; 4 salungs = 1 tical 1 tical weight of silver 93 touch, or 9%5 fine, -=$0.61659 U. S. Customs value of tical = 61 cents. SINGAPORE ISLAND. - Singapore: 100 cents = 1 dollar (old Spanish), 1.01385 BANCA ISLAND. -Same'as at Batavia (JAVA I.). BORNEO ISLAND.- Banjermassin, Sarawak, ic.: Generally as at Batavia. CELEBES ISLAND.'- Macassar, and the other Dutch settlements: Same as at Batavia. JAVA ISLAND. —Batavia, Samarang, sc.: 100 cents 1 guilder, - 0.40806 Mexican dollars are current here at 21 guilders, each. M-OLUCCA ISLANDS. - Amboyna,i c.: Same as at Batavia (JAVA I.). PHILIPPINE ISLANDS. -LuZON, Manila, Mindano, fc.: Official, as in Spain; also, 34 maravedis 1 retl; 8 r. = 1 peso, or dollar, 1.00465 SUMATRA ISLAND. - Bencoolen, Padang: Official, as at Batavia; also, 8 satellers -1 soocoo; 4 s. 1 dollar, or rial, - 1 pardow of Acheen. Acheen: 16 copangs- 1 mace; 10 mace of fine silver, or 374.306 troy grains 1 pardow, - 1.04845 Mexican and Spanish dollars pass for 1 pardow, each. ITALY. — The recent formation of the several States in Italy into one kingdom, Rome and in its immediate vicinity only excepted, has had the effect to unify the moneys and moneys of account in that portion of Europe also; and now, throughout Italy proper, or the Kingdom of Italy, including the Island of Sicily and that of Sardinia, the official monetary system, with a slight difference in nomenclature, is identically the same as that of France. Standard for gold and silver coins =, fine, each. Relative values, gold to silver as 15i to 1. 12 a FOREIGN MONEYS OF ACCOUNT. Foreign. U. States. 100 centesimi - 1 lira, or franco, = 4 grams of fine silver, (See SICILY, page a 51.) - $0.19452 JAPAN.- Yeddo, Miaco, Osaka, Simoda, Hakodadi, Nagasaki, Yokohama,.Matsmay, Napa, tic.: New System: Standard for silver coins (assay) -= P fine; for gold coins-= 0 fine. Relative values, gold to silver (assuming the cobang to represent 10 silver itzebous) as 15.01877 to 1. 50 sen - 1 itzebou, or itchibou (often called boo) = 5 monme of mint silver, or t monme of fine silver- 119.1888 grains, 0.333855 1 cobang contains 4-} monme of mint gold, or 79.35992 grains of fine gold, - = 3.417744 U. S. Customs value of itzebou = NOTE. -Mexican dollars circulate in Japan, commonly at 2.87 to 2.90 itzcbou, each; but they are intrinsically worth over 3 itzebous, each, by tale. LIBERIA (W. AFRICA). -Monrovia, &c.: Same as in the United States. MADEIRA ISLANDS. — Funchal, sc.: 1000 reis 1 milreis of account, - - - = 1.01385 See Azore Islands, note relative to. U. S. Customs value of milreis -- $1.00. MALTA ISLAND. - Valetta, sc.: Official, as in Great Britain; also, 20 grani = 1 taro; 12 t. 1 scudo- i ducat of Naples, - - 0.41371 U. S. Customs value of scudo = 40 cents. MAURITIUS ISLAND. - Port Louis, tc.: Official, as in Great Britain; also, 100 cents= 1 dollar. Tallaros of Austria and silver Napoleons (5franc pieces of France) are current here at 1 dollar each; Spanish pesos and Mexican dollars, at 52 pence sterling each. MEXICO. —Acapulco, Mazatlan, San Blas, Campeachy, Tampico, Sisal, Vera Cruz, tc.: Standard for silver coins (dollars, average by assay) = 416.15 grains? fine; for gold coins (doubloons, average by assay)- 416.4 grains * fine. Relative values, gold to silver as 16.618192 to 1. - 100 cents = 1 dollar; also, 6 grani = 1 cuarto; 2 c. 1 medio; 2 m. -1 real; 8 r. 1 dollar 374.535 grains of fine silver, - - 1.0491 NOTE. - The Mexican marco = 3549.81 grains; and the standard silver dollar should weigh tL-1-) = 416.4 grains. F0:TIN XON1~Q1Y5 OFQ ACCOUNT. a13 Foreign. U. States. MOROCCO (N. AFRICA). -Morocco, Fez, Tangier, &e.: 24 fluce = 1 blankeel; 10 b. 1= metical; 4 m. = 1 oncia or ducat = 4 meticals weight of fine silver, or 295.3846 grains 1 old standard ducat of Naples, - - -$0.82739 MOZAMBIQUE (E. AFRICA). —Mozambique, Quizimane, Sofala, Delagoa;Bay, tic.: 1000 reis = 1 milreis of account. NOTE.- A variety of foreign coins are current here, and many of them wide of their true values; viz., Spanish and Mexican dollars at 1000 reis, each; silver 5-franc pieces of France and United-States gold coins per dollar at 900 reis, each; Spanish doubloons at 17120 rels, patriot doubloons at 16000 rels; and British sovereigns at 4500 reis, each. NORWAY.- Christiania, Bergen, ic.: Standard for silver' coins fine. 20 skillinge = 1 mark; 6 m. 1 speciesdaler 2 rixdalers Cologne mark of mint silver or 390.2465 grains of fine silver, - - 1.0931 NOTE. — Norway and Sweden are about to coin gold of- 10 and 20 francs. NUBIA (E. AFRICA).- Suakin, tc.: Same as at Alexandria, Egypt. PERSIA. -Bushire, Gombroon, Astrabad, tc.: 100 mamoodis, or 50 abasse, —.1 toman - miscal weight of fine gold, or 49.23077 grains, - 2.12019 5 dinars = 1 kasbequis; 2 k. = 1 dinars-biste; 2 d.-b. — 1 shatree, shafree, or shahis; 2s. = 1 mamoodi; 2 m. -- 1 abasse; 2M a. = 1 papabat; 2 p. = 1 saheb-keran. NOTE. -Foreign gold and silver coins in great variety circulate in Persia, but at fluctuating prices; the toman, however, is commonly valued at two Mexican dollars. PORTUGAL.- Lisbon, Oporto, St. Ubes, &c.: Standard for gold coins (coroas) = marco of gold jj fine; for silver coins (erusados), marco of silver xo fine. Relative values, gold to silver as 14.72524 to 1. 1000 reis = — 1 milreis = — 2 crusados - 398.4237 grains of fine silver, - - - =1.11601 1 milreis in gold = $1.16525. Method of writing and reading monetary quantities: Example. - Rs. 5: 600 6 750 = 5,600 milreis and 750 reis. U. S. Customs value of milreis = $1.12. NOTE. -In commercial transactions, lately, Mexican dollars commonly pass for 1 milreis each, and Iuited-States gold coinf at 970 reis per dollar. 2 14a FOREIGN MONEYS OF ACCOUNT. Foreign. U. States. ROME and Civita Vecchia: Standard for gold and silver coins-= -0 fine, each. Relative values, gold to silver as 2 =xX - 15.43367 to 1. 5 quatrini 1 baiocho; 10 b. = 1 paolo; 10 p. 1 scudo, or crown, = 24- grams of fine silver=.$1.04609 The gold coins are 24, 5, and 10 scudo pieces, and - contain.-} grams of fine gold per scudo. U. S. Customs value of scudo $1.05. RUSSIA. —St. Petersburg, Riga, Cronstadt, Odessa, Bc.: Standard for silver coins =p fine; for gold coins (assay)- -2 9 fine. Relative values, gold to silver as 15.17295 to 1. 10 kopecks = 1 grieven; 10 g. =-1 rublyu (roublet)= - funt of fine silver, or 280.19 grains = 0.78483 In practice, 100 kopecks - 1 rouble. 100 silver roubles =-360 bank or paper roubles. Bank rouble varies from par to 4 per cent premium. U. S. Customs value of silver rouble -- 75 cents. SENEGAMBIA (W. AFRICA). — Gambia, Bathurst, Sierre Leone, sc.: Official, as in Great Britain. St. Louis: Official, as in France. SOCOTRA ISLAND. - Same as at Muscat, Arabia. SPAIN. - Madrid, Malaga, Cadiz, Santandre, Bilboa, Barcelona,- c.: ANTew system, legalized Oct. 19, 1868, and its use made obligatory to the exclusion of all other systems after Dec. 31, 1870. Standard for gold coins T- fine; for silver coins (5-peseta pieces, duros, or pesos) =Tfine; 1-peseta pieces and less I-o fine. Relative values, gold to silver (duros) as 15j to 1. 100 centesimas = — 1 peseta = 41 gramos of fine silver - 1 franc of France, - - - 0.19452 NOTE. —In this system the peseta of account is equal to 1 silver franc of France; and the 5-peseta silver coin is equal to 5 francs in silver of France; but the 1-peseta coin is worth but $0.18047. The gold coins are worth peseta for franc in gold. Prevailing system last preceding the foregoing: Standard for silver coins (escudos of 10 reils vell6n) =- fine; for gold coins (10 escudos and over) =12 fine. Relative values, gold to silver as 15.5555 to 1. 1000 milesimas = 1 escudo = marco of fine silver, or 1791 grains, - - - 0.50232 FOREIGN MONEYS OF ACCOUNT. a15 NOTE. -The silver coins of less denominitions than 4 reals, and the gold coins of less denominations than 10 escudos, have less purity than those mentioned above; the 80-reAls piece, even, is but 7 3-8-9 fine 3.85446, while the 100-redls piece (double do Isalbel) is worth $4.96493..Foreign. U. States. Gibraltar: 16 quartos 1 real; 12 r. - 1 peso duro or old Spanish dollar,..=$1.01385 SWEDEN.- Stockholm, Gothenburg, Carlscrona, Gefte, Sc.: Standard for silver coins = ~ fine; for gold coins (ducats) =-3 fine. Relative values, gold to silver as 15.18517 to 1. 10(0 ore = 1 riksmynt, or riksdollar riksgaild - speciesdollar,.. - 0.27622 1 speciesdollar = x mark of fine silver; or 393.15 grains,- - - - - 1.10124 U. S. Customs value of speciesdollar = $1.06. 1 ducat - 8 riksmynts, nominally, = - marks of fine gold -$2.23 738. Sweden is about to coin gold coins of 10 and 20 francs. SWITZERLAND. - Basel, Bern, Geneva, Lausanne, Lucerne, Neufchatel, Zurich, arc.: Standard for gold and silver coins = 0 fine, each. Relative values, gold to silver as 151 to I. 10 rappen = 1 batzen; 10 b., or 100 centimes, 1 franc= = 4 grams of fine silver, - - 0.19452 TRIPOLI (N. AFRICA).- Tripoli, sc.: 100 paras 1- piastre, or ghersch; value of piastre same as that of Tunis. TUNIS (N. AFRICA). - Tunis, Soosa, Cabes, &c.: Standard for silver coins = — j fine; for gold coins = -- fine. Relative values, gold to silver as 15.8125 to 1. 2 burbine I asper; 52 a., or 16 karob= 1 piastre - 1 drachma of fine silver, or 44 grains, = 0.123247 TURKEY.- Constantinople, Smyrna, Aleppo, Trebizonde, sc.: Standard for silver coins (assay) = qos- fine; for gold coins (assay)- 1 - fine. Relative values, gold to silver as 15.15625 to 1. 40 paras, or 100 aspers, = 1 piastre A= checki of fine silver, or 15.3856 grains, - - = 0.043096 1 piastre in gold- -1 checki of fine gold= $0.0437181.' 100 piastres in gold= 1 medjdie. 1 purse of silver - 500 piastres; 1 purse of gold 30,000 piastres. U. S. Customs value of pitstre - 5 cents. 16a BOREIGN MONEYS OB ACCOUNT. West Indies. CUBA ISLAND. - Havana, MlIatanzas, Santiago, MIanzanillo, Baracoa, Cardenas, Cienfuegos, Nuevitas, Trinidad, Sc: Official as in Spain; also, 12 dineros, or 16 quartos, =1 redl; 8 r., or 100 centesimas - 1 duro, peso, piastre, or dollar. NOTE. -The full-weight Spanish duro colonato is made the unit, or measure of value, - $1.04872; but the Spanish onza, or doubloon, passes for 17 dollars, and the Mexican and S.' American, for 16 dollars. HAYTI ISLAND.- HAYTI, Port au Prince, Aux Cayes, Cape Haytien, Gonaives, tc.>: DOMINICA, San Domingo, Porto Plate, Samana, sc.: 100 centesimas = 1 dollar, or gourda (old Spanish). NOTE. - Spanish doubloons pass for 16 dollars of account; and the Haytian silver gourda is worth about i Spanish dollar. PORTO RICO ISLAND. -San Juan, Guayama, Ponce, ic.: Official, as in Spain; also, 100 centesimas i dollar. JAMAICA ISLAND. -Kingston, Falmouth, Savana la Mar, 6,c.: Official, as in Great Britain; also, 100 cents-= 1 dollar-= 1 old peso, or duro of Spain =50 pence sterling in gold, - -=$1.01385 CARIBBEE ISLANDS. LEEWARD ISLANDS. - ANTIGUA, - St. John, Falmouth; DOMINICA, MONTSERRAT, TORTOLA,'VIRGIN GORDA, ST. CHRISTOPHER, ANGUILLA, BARBUDA, NEVIS, SABA: Same as in Jamaica. ST. EUSTACIUS: Official, as in Holland. GUADELOUPE, ST. MARTIN,* MARIE GALANTE, DESIRADE, LES SAINTES: Official, as in France; also, 100 centimes = 1 dollar; the colonial livre of these islands = ~o standard duro colonato of Spain. ST. THOMAS, — Charlotte Amalie; SANTA CRUZ, ST. JAN: Official, as in Denmark; also, 100 cents = 1 dollar. S'r. BARTHOLOMEW: Official, as in Sweden; also, 1 Spanish dollar 9 shillings currency. * The southern portion of the island of St. Martin is owned and settled by the Dutch; and the moneys of account, weights, and measures of Holland are in general use there. FOREIGN MONEYS OI' AOCOUNT. - ai Foreign. U. States. WIND WARD ISLANDS. — TRINIDAD, Port Spain; BARBADOES, Bridgetown; GRENADA, ST. VINCENT, ST. LUCIA. TOBAGO: Same as at Jamaica Island. MARTINIQUE, St. Pierre, Port Royal: Official, as in France. BAHAMA ISLANDS. - NEW PROVIDENCE, Nassau; TURKS, ABACO, ANDROS, GREAT BAIIAMA, &c.: Same as at Jamaica Island. LITTLE ANTILLE ISLANDS. - CURACOA, BUEN AYRE, ORUBA: Official, as in Holland. MARGARITA, TORTUGA, BLANQUILLA: Same as in Venezuela. ZANGUEBAR (E. AFRICA).- Zanzibar (Island and Town), Quiloa, Mombas, Magadoxo, Brava, Socotra Island, Arc.: Accounts are now kept in Zanzibar and the Sultan of Muscat's dominions on the east coast of Africa generally in dollars of 100 cents; and by the exertions of William E. Hines, Esq., of New York, late resident consul of the United States at Zanzibar, the present standard dollar in gold of the United States is made the measure of value, or, in other words, is officially rated at par. This was accomplished in 1865. Other foreign gold and silver coins circulate at conventional rates, some of them above and others below their intrinsic values. Thus, the Spanish dollar $1; the Austrian rix-dollar, scudo, or crown, sells for $1.01 to $1.03; the English sovereign is rated at 4]. Austrian rix-dollars; the French franc in gold, at 18- cents; the silver 5-franc piece, at 94 cents; and the Indian silver rupee, at 47 cents. (18a) FOREIGN LINEAL AND SURFACE MEASURES, REDUCED TO THE LINEAL AND SURFACE MEASURES OF THE UNITED STATES. Foreign. U; States. ABYSSINIA. — Massuah: 8 robi- 1 derah, or pic, - - - = 0.682 yard. ALGERIA. - 10 decimetres = 1 metre, - _ 1.0936 " 8 robi =1 pie. Pie, Moorish, for linens, - = 0.519 Pic, Turkish, for silks, &c., - 0.692 ARABIA.- Muscat: 8 gheria 1 covid; 8 c. 1 kassaba, - - 12.86* feet. 500 kassaba =- I coss, 1.2185 miles. Aden: 8 robi i yard or pie, - 0.95 yard. Jidda: 8 robi - pie,'- - - 0743 Mocha:8 robi = 1 gez, - 0.694' AUSTRALIA.- Same as in Great Britain. AUSTRIA.- Imperial and Legal: 12 zollen 1 fuss, -. 1.04 feet. 29* zollen = 1 elle,.. 0.85216 yard. 6 fusse 1 klafter; 10 fusse = 1 ruthe, - 10.39924 feet. 2,400 ruthen =- 1 meile, =- - 4.7269 miles. 192 square ruthen = 1 metzen; 3 m.- 1I joch, - - 1.43 acres.. AZORE ISLANDS. - Same as in Portugal. BALEARIC ISLANDS. —S pie, or 4 palmi, = 1 vara. MAJORCA: 2 varas =-1 cana, - _ 1.711' yards. MINORCA: 2 varas = 1 cana, - -- 1.754 " BELGIUM. —10 streep 1 duim; 10 d.= -1 palm; 10 p. = 1 el = 1 metre, - - = 1.0936 " 10 elen X 1 roed; 100 r. = 1 mijl, or kilometre,. -- 0.6214 mile. 100 square roeds = 1 bunder — = 100 ares - 2.471 acres. The old Brabant el 0. 76006 yard. BERMUDA ISLANDS. - Same as in Great Britain. BOURBON ISLAND. - Same as in France. CANARY ISLANDS, - Same as in Spain. CANDIA ISLAND. 8 rob =.1 pie,. - - 0.697 yard. CAPE VERDE ISLANDS. - Same as in Portugal. FoiEtEN LirNELA AND SURFACE AfESURES. a19 Foreign. U. States. CANADA, DOMINION OF.-Same as in the United States. LOWER CANADA (special): 1 arpent, -- 0.8475 acre. CAROLINE ISLANDS. Same as in Spain. Central and South America. CENTRAL AMERICA. - COSTA RICA, GAUTEMALA, HONDURAS, NICARAGUA, SAN SALVADOR: Same as Spain, — standard of Castile. SOUTH AMERICA. — A-RGENTINE REPUBLIC: 36 pulgadas _ 3 pes 1 vara, -- 34.1 inches. 150 varas 1 cuadra; 40 c.=1 legua, -= 3.229 miles. 27,000 square varas= I suertes des estancia. BRAZIL: 24 pollegadas = 2 pes 1 covado -= m-F nletre, - - - - 0.7218 yard. 40 pollegadas 5 palia 1 vara bracio= 1lW metre, = 1.203 250 varas, or 125 braccios 1 estadio; 8 e. -1 milha; 3 m. 1 legoa, - - 4.101 miles. 4,840 square varas geira, - - = 1.447 acres. CIIILI: 36 pulgadas = 1 vara (custos), - 1.000 yard. In all otherrespects as in Spain, — Castilian.standar1d. VENEZUELA, NEW GRENADA, ECUADOR, IPERU, BOLIVIA, PARAGUAY, URUGUAY: Sanme as Spain, - Castilian Staldard.. GUIANA: Cagenne: Same as in France. Paramiaribo: Same as ill Holland. Georgetown: Same as in Great Britain. FPALKLAND ISLANDS: Sanme as in Great Britain. CHINA. -10 tsun, tr fan, = 1 punt; 10 p. 1 chil, coviid, or cobre (mercers'), - -= 1.226 feet. 17t punts, or 10 tac,c 1 thuoc (tradesmen's) 0.7152 yard. Chih (mathematical), - - -=13.122 inches. Chili (engineers' and surveyors'), - 12.059 " Chih, or kong-pu (architect.'), - - = 12.709 " 10 chih - 1 chang or cheung; 10 chang = 1 yan; 18 y._ 1 li, - - - 0.3425 mile. 100 square cLhih = 1 mow; 240 m.l 1 fu, or king, - -.- - -= 0.5564 acre. CYPRUS ISLAND. - 8 robi1 pi, -= 0.96 yard. 20 a FOREIGN LINEAL AND SURFACE MEASURES. Foreign. U. States. DENMARK. -24 tomme, or 2 fod, = 1 aln, - = 0.6862 yard. 10 fod 1 rode; 2,400 r. = 1 mill, -= 4.6785 miles. 280 square rodes 1 skiepper; 2 s. 1 toende, - - -_ 1.362 acres. 6 fod- 1 favn; and 1 fod =-1 Prussian Rhein-fuss. EGYPT. — 3 kirat - 1 rob; 8 r. 1 -pic or endrasi (for silks and woolens), - 27.06 inches, 1 halebi or archin = 27.9 in.; 1 derah 25.264 inches. 2 derah 1 fedan; 3 f. =1 gasab; 420 g. 1 erri; 3 b. — = parasang, - - 8.015 miles. -4Q0 squ0are gasab= 1 fadden al risach,:- 1.465 acres. FRANCE. - 100 centimbtres, or 10 decimbtres, 1 mbtre, =- 39.37 inches. 100 mbtres, or 10 decambtres, = 1 hectometre; 100 hectombtres, or 10 kilombtres, -- 1 myriambtre, -- 6.2137 miles, 100 square metres _ 1 are; 100 a. I hectare, -.- 2.471 acres. 1 old aune de Paris 1.29972 yard; i aune usuelle, or metrique, = 1} metre. FRIENDLY ISLANDS (FEEJEE AND TONGA GnouPs). - Same as in England. GERMANY. - PRUSSIA, and the Zollverein lidngenmaasse for all Germany: 12 linie - 1 zoll; 12 z.- 1 fuss (Rheinfuss), - - - - = 12.3514 inches. 251 zollen (Rhein-zollen) = 1 elle metre, - - - - 26.24j " 10 land-fusse, or 12 Rhein-fusse, 1 ruthe; 2,000 r. = 1 meile, - - 4.67855 miles. 180 square ruthen = 1 morgen, - - 0.6304 acre. Special and local: — SAXONY: 12 zollen 1 fuss; 2 f.-1 elle,= 0.6196 yard. 1 Lubec-Brabant elle, - - -- 0.7498 " BAVARIA: 120 zollen, or 10 fusse,- 1 ruthe, - 9.579 feet. 34~ zollen = I elle =- metre, - -- 32.808i inches. WURTEMBURG: 100 zollen, or 10 fusse ruthe,..- 9.37326 feet. 21. zollen = 1 elle, - - - 0.67175 yard. BADEN: 20 zollen, or 2 fusse, _ 1 elle = metre,.. - 23.622 inches. FOREIOaN LINEAL AND SUrFACE MEASURES. a21 Foreign. U. States. IEssE SDARMSTADT: 10 zollen = 1 fuss* metre, - - - 0.822111 foot. 24 zollen - I elle - metro, - - 0.65611 yard. MECKLENBURG: Same as at Hamburg. OLDENBURG: 12 zollen 1 fuss; 2 f.- 1 elle, = 0.63528 " BREMEN: 24 zollen, or 2 fhsse, = 1 elle, - 0.63276 " 1 Bremen-Brabant elle = 1- Bremen clle. HAMBURG: 12 zollen = 1 fuss; 2 f. = 1 elle, = 0.62681 " 1 Hamburg-Brabant elle, - - - = 0.75615 " LUBEC: 24 zollen, or 2 fusse 1 elle, = 0.6294 " GREAT BRITAIN. - Same as in the United States. GREECE. - 60 onud = 5 pes = 1 passo = 1] metre. 22 onud _ 1 pichi, for silks, - =0.701734 " 231 onud = 1 pichi, for woolens, 6~., -,0.749579" IHOLLND. — 10 streep 1 duim; 10 d. -1 palm; 10 p. 1 el -= 1 metre, -- 39.37 inches. 10 el = 1 roed; 100 roed = 1 mijl = 1 kilometre. 100 square roede _ 1 bunder - 1 hectare. 1 old Brabant el - =0.75931 yard. India and ztralaysia, or East Indies. ANAM: Same as in China. BURMAH, PEGU: 51 pulgaut = 1 taim; 4 t. 1 sadang; 7 s. - 1 sha, or bambou, -= 154 inches. 1,000 dhas 1 dain, - 2.4306 miles. CEYLON ISLAND: 2 covids = 1 guz,, or yard, - 36 inches. IIINDOSTAN. - Calcutta: 16 tussoos = 8 gheira - haut, or covid; 2 h. = 1 ghes, or guz, = 36 " 2,000 ghes = 1 coss, - - 1.1364 miles. 20 square covids = 1 chattack; 16 c. = 1 cottah; 20 cottahs = 1 biggah, - - 0.3306 acre. Bombay: 16 tussoos = 8 gheira = 1 haut or covid; 1I h. = 1 guz; 2 hauts, or 1- guz, - 1 imperial yard, - -- 36 inches. Madras: 16 tussoos - 8 gheira = 1 haut or covid; 2h. 1 guz, - -=36 " Goa: Same as in Portugal. Massulipatam: 2 palms = 1 span;- 3 s. = 1 cubit or covid, - 19 " Surat: 18 tussoos = 1 haut, for matting, - 21 84 tussoos, or 20 wiswiisa, = 1 wilsa, -= 98 22 a FO(IEIGN LfNEAt AMi) IRtFACE SIEAtJIlES. Foreign. U. States. MALABAR COAST. - Mangalore, Cananore, Calicut, -Cochin; Quilon, Trivandrum: 3 gleria= ady; 2j a. = —1 haut; 2 h.- 1 guz or gujah,.385 inches. COROMANDEL COAST, generally, same as at Madras. SIA. -48 nions = 4 keubs- 2 soks = 1 ken, = 37.836 2 kens = I vouah; 20 v. 1 sen; 4 s. = 1 jod; 25j.-= 1 ro-neng, - 2.3.886: miles. MALACCA. Same as at Calcutta. SI.NGAPORE ISLAND. Same as at Calcutta. PIIILIPPINE ISLANDS. —Same as in Spain,standard of Castile.:. JAVA ISLAND.- 1 voot (Rhyn-voot), - - 1.029 feet. 1 el, or covid (old Amsterdam el), - - 0.7522 yard. English measures also, as at Sumatra. SUMATRA ISLAND. — 2 tempohs- 1 jancal; 2 j.-1 etto; 2 e. = 1 hailoh, - - 36 inches. ITALY.- The metric system of weights and measures is now the official standard, and is in general commercial use throughout the kingdom. Special and local:Milan: 10 atomi 1 dito; 10 d. 1 palmo; 10 p. = 1 metro or braccia, - - 39.37 inches. 1,000 metri 1- miglio (kilometre). 100 metri quadrata = 1 tavolo (are); 100 t. - 1 tornatura (hectare). 2- metri- 1 trabucco; 4 t. = 1 decametre. Florence, Leghorn, Pisa, sc.: 2 palmi 1 braccia; 4 b. -1 canna = 2A metres. Carrar: 1 palm for marble = 9.06 U. S. inches. JAPAN. -10 rin = 1 bun; 10 b. = tsun; 10 t. 1 sasi or sjak. 4 tsun-sasi- 5 kcani-sasi -1 hiro; 5 tsunsasi 1 ink or tattamy, - 2.07103 yards. 7 kani-sasi - 1 kan, kian, or ikjc, - 2.31955 " 60 inks - 1 ting or masti; 36 t. 1 ri, -= 2.54172 miles; MADEIRA ISLANDS.- Same as in Portugal, MAURITIUS ISLANI). - Official as in Great Britain; also, 1 aune (old aune de Paris), - - = 46.79 inches. MEXICO. - Same as in Spain, standard of Castile. FOREIGN LINEAL AND SURFACE MEASURES. a 2~ Foreign. U. States. MOROCCO. - 8 pollegadas = 1 palmo da craveira (Portuguese). 18j pollegadas, or 2k palmi da craveira,1 covado or cadee, - 20.21 inches. 8 rob = 1 pic (Turkish in theory), - 26.03 " NORWAY. - Same as in Denmark. NEW ZEALAND ISLANDS. Same as in England. PERSIA.- Monkelser or royal: 12 fingers — i foot; f.- 1 cubit; 2 c. — 1 guz or gueza (-for silks, sc.), - - - - 36.92 " 1 guz shah (royal), for wooles, - -40 1 guz tabree (of Tabreez), - — 40.4 " 6:000 guz (monkelser) =1 parasang, - 3.496 miles. PORTUGAL. - 8 pollegadas_ 1 palmo da craveira - 4 metre. 12 pollegadas-= 1 p6; 2 p. = 1 cova-do i_ metre. 40 pollegadas = 1 vara- 1=' metre, = 43.307 inches. 24f pollegadas = 1 covado avantejado (retail). 13k pollegadas - vara = 1 terra. 60 pollegadas = 1 passo; 1k p. =1 braga. 780 ps _ 1 estado; 8 e. 1 milha; 3 m. = 1 legua, - - - -=.8386 miles. 4,840 square varas= 1 geira, - - 1.4471 acres. ROME.-12 onze = 1 palmo; 1k p. 1 pie; 5 piddi = 1 passo, - - 58.6104 inches. 10 palmi, or 71 piddi, 1 canna architectona, - - - -87.9156 " B4a canne arch. - 1 catena (chain); 1 catena = 10 stajoli..... 8 palmi mercantile = 1 canna mere., -. 78.43932 " 9 palmi d'are = 1 canna d'are. 1k metre, 44.29- " 1,000 passi = 1 miglio, - - - 0.92504 mile. 31 square catene = 1 quartuccio; 2 q. - 1 scorzo; 4 s. = 1 quarta; 4 quarte (7 pezze) = 1 rubbio, - 4.559 acres. RUSSIA.- 6 verschok -1 -foot, - - = 12 inches. 16 verschok = 1 archine or halebi, - 28 " 3 arehines (7 feet) — 1 sachine, - - 7 feet. 500 sachines - 1 verst, - - - 0.6629 mile. 2,400 square sachines = 1 deciatene, - 2.7 acres. SANDWICHI ISLANDS. Same as in the United States. 24 a FOREIGN LINEAL AND SURFACE MEASURES. Foreign. U. States. SOCIETY ISLANDS. -Official, as in France. SPAIN.- Standard of Castile:- 36 pulgadas = 6 sesma = 4 quarta or palmos - 3 tercia or pie = 1 vara, - - 33.385 inches. 2 varas = 1 estado, braza, brazada, or toesa. 2 estados - 1 estadale; 2,000 cstadalcs = 1 legua, - 4.2153 miles. 560 square estadales 1 fanegada, -= 1.592 acres. 15 pie - 1 codo; 5 pies = 1 passo. Gib'altar: As above; also, 12 inches = 1 foot; 3 f.= 1 yard, — 1 yard. Aiitcante: 36 pulgadas 1 vara, - - - 0.8319 Barcelona: 1 vara or matja-c'ina, - - 0.8641 " Santander: 8 octava, or 4 palma, = 1 vara, -- 0.9142 " Valencia: 36 pulgadas 4 palmos - 3 pie 1 vara, - 1.0044 yards. 21 varas = 1 braza-reale. NOTE. -In 1849, Spain.legalized the use of the metric system of weights and measures in all her dominions; and now, since she is to enforce the employment of the money system of France, to the exclusion of all others (see Foreign Moneys of Account, Spain), it may be expected that she will unify her weights and measures by enforcing the employmcnt of the metric system generally. SWEDEN. 10 linier 1 turnm; i0 t. 1 fot; 10 f. = 1 stWng; 10 s. 1 ref, - - 97.406 feet. 20 tumen = 2 fus = 1 aln a= ol0l aunc de Paris, - 0.64937 yard. 6 fus = 1 fmn; 6,000 f. (3,600 stiinger) - i mil, - - - - -.6413 miles. 560 square stiinger (14,000 square alner): 28 kannland -- 16 kapplandl = 2 spalnland = 1 tmnnland, - - 1.219 75 acres. SWITZERLAND. - Official and general fortlie 22 cantons forming the Republic; 10 zollen =1 -fiiss; 2 f. 1 clle __ 6- metIrC, = 23.622 inches. 4 fuisse = 2 ellen = 1 stab _. aune metric(. 6 filsse 1 klafter; 10 fusse ruthe or toise, 9=.8425 fe t.: 1,600 ruthen 1 meile or hour's way 4t kilometres,. 2.9826 niilcs. 400 square ruthen 1 juchart or' eld-:ker 36 arcs, - - 0.8S96 acre. TRIPOLI. -8 rob = 3 palmi = 1 l)ic or dra, -- 26.42 inches. The pie for silks -= 1 Arabic covid, - - = 19.03. " FOREIGN LINEAL AND SURFACE MEASURES. a 25 Foreign. U. States. TUNIS. - 8 robi - 1 pie; pie for woolens, -- 26.5 inches. Pic for silks, 24.83 " Pic for linens, — 18.62 " TURKEY. - Constantinople and Smyrna: 1 halebi or. archin (Russian archine of 28 U. S. inches in theory),-. - 27.9 " 7,500 halebi = 1 agatch, - = 3.3142 miles. 1 pie, draa, or endrasi, for silks andl woolens, = 27.06 inches. 1 indise, or endese, for cottons,?c., - -= 25.688 " Rassorah, Bagdad: 8 robi= 1 guz, - - 31 West Indies. In the Islands of Cuba, Hayti, Porto Rico, and Isle of Pines, the measures of length and of surface are the same as in Spain, Castilian standard, except that in Port au Prince and the French portion of Hayti, generally, the old piede du roy of 12.78918 U. S. inches, and the old aune de Paris of 1.29972 U. S. yards, is used. In Jamaica, St. Kitts, Antigua, Montserrat, Tortola, Anguilla, Dominica, Barbuda, Nevis, Virgin Gorda; in Trinidad,* Grenada, St. Vincent, St. Lucia, Barbadoes, Toba~go, Grenadines; and i'll New Providence, Great Bahama, Turks, Abaco, and the Bahamas generally, the lineal and surface measures are the same as in Great Britain and the United States. In Gaudeloupe, Mlartinique, Desirade, Les Saintes, 2Marie Galante, the lineal and surface measures are the metric; but the pied tm6ricaine (United-States foot) is chiefly used in measuring timber. ~ in Santa Cruz, St. Thomas, St. John', the official measures of length and of surface are the -same as in Denmark, but those of the United States are much used. In St. Bartholomew: Official, as in Sweden; also as in the United States. In St. Eustatius, Curacoa, Buen Ayer, Oruba: Official, as in, holland. In Margarita, Blanquilla, Tortuqa: Official, as in Venezuela. ZANGUEBAR (EAST AFRICA): Same as at Muscat, Arabia. * The Spanish (Castilian) vyra is still used to some extent bjy the merchants in Trinidad. 3 (26a) FOREIGN COMMERCIAL WEIGHTS, REDUCED TO THEIR EQUIVALENT VALUES IN TUIE UNITEID STATES. Foreign.'U. States Avoirdupoih pounds. ABYSSINIA (E. AFRICA).- MIassuah: 10 dirhem = 1 wakea; 12 w. or 10 mocha = 1 rotl, rotolo or litre = 10 troy ounces,* - - = 0.68571 ALGERIA (Barbary, N.. AFRICA). - Algiers, Bona, Oron: The metric weights, under French denominations, are in official and common use here. ARABIA. -3 2loca: 2331 krat= 1 wakea, wakega, or vacia = 700 troy grains. 15 wakea- 1 rotl; 2 r. = 1 maund or maon, - =- 3.10 maunds = 1 frazil; 15 f. — 1 bahar, - = 450.At the bazaar, 141, frazils 1 bahar for coffee, = 435.Jidda: Official, as at Alexandria, Egypt; also, 3 - drachmas musr (of Egypt) = 1 wakea; 15 w. 1 rotl; 5 r. =1 maund; 10 m.- =frazil; 10 f.= 1 bahar= 662 okes of Egypt, - = 182.8571 Ilodeida, Beit-el-fakih; 15.wakea - 1 rotl; 2 r. = 1 maund; 10 m. = 1 frazil; 40 f. = 1 bahar, = 815.2381 16 rotl= 1 tomaun of rice, - - 168.14286:Muscat, Hasek, sc.: 233A krat - 1 wakea; 10 w. =1 rotl; 9 r. - 1 maund; 200 m. = 1 bahar, -1800.Aden: Official, same as in Great Britain. AUSTRALASIA (OCEANICA). - AUSTRALIA, NEW ZEALAND, TASMANIA: Same as in Great Britain. AUSTRIA (legal for the Empire): 32 lothe = 16 unzen = 4 vierdinge = 2 marken = 1 pfund; 20 p. = 1 stein; 5 s. = 1 centner, - - - 123.468 4 centners = 1 karch; 5 k. - 1 last; 23 centners =1 saum; 1 centners - 1 lagel; 2.=1 - 1 saum for steel. Ragqusa, (DALMATIA): 2. pfunde = 1 oka, - — = 3.0867 AZORE ISLANDS (N. Atlantic Ocean): Same as in Portugal. * See Turkey, weights of, and note relative to. POREIGN COM3IEPRCIAL WEIGHTS. a 27 Foreign. U. States. Avoirdupois pounds. BALEARIC ISLANDS (Mediterranean Sea): 25 rotoli (22k Castilian- libras) = 1 aroba; 4 a. = 1 quintal, 9- 1.3063 3 quintals -1 carga; 110 rotoli = 1 oder; 100 libra menor -=87 Castilian libra = 1 cantaro grosso = 88.2627441 av. lbs.:BELGIUM.- Same as in France (metric system)..BERBERA (E., AFRICA): Same as at Mocha, Arabia. BERMUDA ISLANDS (N. Atlantic Ocean): Same as in Great Britain. BOURBON ISLAND (Mascerene group, Indian Ocean): The metric system, French nomenclature, is in use here. CANADA, DOMINION OF. - Same as in the United States. CANARY ISLANDS (Atlantic Ocean, WV. coast N. Africa): Same as in Spain, Castilian standard. CANDIA ISLAND, on CRETE (E. Medit'errancan Sea): 100 rottoli- 44 okes =l caiitaro, - -..- 116.565 -CAPE OF GOOD HOPE (S. AFRICA): Same as in Great Britain.;CAPE VERDE ISLANDS (Atlantic Ocean, near W. African coast): Same as in Portugal. -Central and South America. GUATEMALA, HONDURAS, SAN SALVADOR; NI- CARAGUA, COSTA RICA: 16 onzas = 2 marcos = 1 libra; 25 1. = 1 arroba; 4 a. - 1 quintal, - - - - 101.4514 2j quintals= 1 carga; 8 c. = 1 tonelada, - 2029.0285 BALIZE. - Official, same as in Great Britain. BRAZIL.-16 onqas 2 marcos = 1 arratel, - 1.01187 32 arratels = 1 arroba; 4 a. - 1 quintal, - 129.5193 ARGENTINE REPUBLIC, or LA PLATA: 16 opn(as =2 marco —1 libra; 25 1. -1 arroba; 4 a. = 1 quintal,..= 101.274 NOTE. -The Argentine Republic has established independent standards of weights and measures, which are now in practice, and which vary more or less in each department from those of Castilo. 28 a FOREIGN COMIERCIAL WEIGIHT8, Foreign. U. States. Avoirdupois pounds. PERU, CHILI,. BOLIVIA, ECUADOR, NEW GRANADA, VENEZUELA, URUGUAY, PARAGUAY: Generally as in Central America. Montevideo: 1 pesado of dry hides (fresh or salted) contains 11 arrobas; 1 pesado of wet salt hides contains 21 arrobas. NOTE. - In Peru, Chili, New Granada Bolivia, Venezuela, and Surinam, the use of the metric system of weights anA measures is sanctioned by law; but as yet (1869) is very little employed in either of the States. GUIANA. - Cayenne: Same as in France. Paramaribo, or Surinam: Same as in Holland; also as in France. Georgetocwn: Same as in Great Britain. FALKLAND ISLANDS. - Same as in Great Britain. CHINA. - 10 tsien = 1 tad or leing; 16 t. = 1 catty or kan; 100 c. - 1 pecul or tam, - 1.133.3333 22- chu = 1 leing; 2 catties 1 yin; 15 y.1 kwan; 33 k.- 1 tam; 1- t. (60 yin) =! shik, - - - 160.CORSICA ISLAND (Mediterranean Sea,): Same as in France, of' which it forms a department. CYPRUS ISLAND (Mediterranean Sea): Same as in Turkey, and forming a part of Turkey in Asia. DENMARK.- 32 lod = 16 unze -- 2 marken -- I pund - kilogram; 100 p. -- 1 centner, - - 110.231i 16 pund = 1 lispundl; 20 1.-1 shifpund; 164G s. 1 last. 12 punds= 1 bismerpund; 3 b. 1 waag or vog. EGYPT (N. AFRICA): 4 gran =1 kara; 16 k. 1 drachma (oka-drachma) -- A troy ounce, or 48 grains. 400 drachmas = 4 oka, - - - = 2.74286 144 drachmas = 1 rottolo; 100 r. (36 okes) = 1 cantaro (customs), 98.7429 FRANCE.-1000 milligrammes= 100 centilrammes 10 decigrammes = 1 gramme 15.43235 troy grains. 1000 grammes = 100 decagrammes = 10 hec togrammes - 1 kilogramme, - -- 2.20462 10 kilogrammes = 1 myriagramme; 100 m. 10 quintals = 1 tonneau,, -. 2204.62143 FOREiWdN COMIiMEiCIAL WEIdITS. a29 Foreign. U. States. Avoirdupois pounds. GERMANY. - Zollverein gewighte for all the States of the tariff-alliance: 10 quentchen =- 1 loth; 30 1. = 1 pfund; 100 p. - 1 centner = 50 kilograms, - - = 110.2311 Special and local, or domestic: - 512 pfennige = 128 quentchene - 32 lothe = 16 unzen - 2 marken - 1 pfund. PRUSSIA: 100 pfunde - 1 centner, - - 103.i194 BAVARIA: 100 pfunde -.1 centner = 56 kilograms, - 123.4588 BREMEN: 116 pfunde = 1 centner, - - - 127.488 7 BRUNSWICK: 100 pfunde = 1 centner, - - 103.0656 IIAMBURG: 112 pfunde = 1 centner, - - 119.6044 HESSE DARMSTADT: 100 pfunde = 1 centner50 kilos, - - 1 -=110.2311 LUBEC: 112 pfunde = 1 centner, - - - 119.6813 MECKLENBURG: 14-pfunde = 1 liespfund; 8 1. = 1 centner, - -=.119.5164 OLDENBURG: 100 pfunde; 1 centner; 3 c. 1 pfundschwer, - - 317.7241 10 pfunde = 1 liespfund; 29 1. - I schiffpfund. WURTEMBERG: 100 pfunde= 1 centner,..- - 103.1153 BADEN: 10,000 as= 1000 dekas- 100 centas = 10 zehnling - 1 pfund. 100 p. = 10 stein= 1 centner _ 50 kilograms, - = 110.2311 SAXONY: 10,000 as = 1000 dekas=- 100 hektas - 10 kilas =1 pfund; 100 p.- -10 halbstein (half-stones) = 1 centner = 50 kilos, - = 110.2311 Leipsic (domestic): 100 pfund = 1 centner, -= 103.0734 GREAT BRITAIN. - Same as in the United States. NOTE. —In Great Britain, in addition to the denominations of weights used in the United States (the values of which are the same), the Clove of wool, - - - 7 lbs. Stone of bubThers' meat or flesh - 8 lbs. Stone " liron, flour, -- 14 I Stone of chtese, - - - - 16 " Tod " " - - - = 2 " Stone of glass, 5. " WVeigh" " - - - 182 Seam of glass, - - - -120 " Sack " " 3- - 364 Stone of hemp, - -= 32 last " " - -4368" Fother of lead, -. -=19i cwt. GIREECE.- 72 cocos =1 dramia; 8 d.- 1 ounghia; 8 o. =1 imilitron; 2 i. = 1 litra, -- 1.136 400 dramias -- 31 litras = 1 oka, - -- 3.55 1371 litras =44 okas = 1 cantaro, - - 156.2 HOLLAND. - 10,000 korrel = 1000 wigtje = 100 lood = 10 onz - 1 pfund = 1 kilogram, -- 2.20462 3* Avoi. rdupois Avoirdupois pounds. itAWAIIAN ISLANDS (Sandwich Islands, Polynesia, N. Pacific Ocean): Same as in the United States. - India and Malaysia, or East Indies. ANNAM.- Kesho (Tonquin): 100 catties 1 pecul, = 132.Hue, Sai-gon,'c. (Cochin China): 16 leiing 1 can; 10 c. -1 yen; 5 y. -1 binh; 2 b.=-1 ta; 5 t. 1 quan, - - - - 688.76 BURMARI (Farther India), Prome, Patanago, Ava, sc.: 100 tical or kiat = 8 abucco = 4 agito 3 catties =1 vis or visay, - - 3.39286 CEYLON ISLAND (Indian Ocean): 500 pond = 1 bahar or candy, - - - - 500.IIINDOSTAN.- Bombay: 72 tanks — 30 pice- 1 maund, - - - - - = 28.Also, 80 tipprees_- 40 seers 1 maund, - = 28.20 maunds - 1 candy_ 5 cwt. Calcutta, Bengal (factory weight)-: 5 siccas or rupees — 1 chattac; 16 c.- 1 seer; 40 s.1 maund; 3 m. = 2 cwt., - - -- 224.4 chattacs = 1 pouah; 5 seers 1 pussaree. 1 maund (bazaar weight) = 100 troy lbs., nominally, - - 82.133 11 factory maunds — 10 bazaar maunds. Madras (Carnatic, Coromandel coast): 10 pagodas or varahuns= 1 polam; 8 p. = 1 seer; 5 s. = 1 vis or visay; 8 v. = 1 maund or maon; 20 m. = 1 candy or baruay, - -= 500.Goa: 32 seers= 1 maund; 20 m., 1 bahar, -=495.Pondicherry: 10 varahuns -- 1 poloin; 40 p. = 1 vis; 8 v. = 1 maund; 20 m. - 1 candy, -= 588.Surat: 16 pice = 2 tipprees = 1 seer; 40 s. = 1 maund; 20 mn. = 1 candy, - - - 300.3 candies = 1 bahur. Tatta: 4 pice = 1 anna; 16 a.= 1 seer- 724 tola; 40 s. = 1 maund, - - -= 72.32 Mysore, Seringapatam, and Malabar coast generally: 40 polams - 1 vis or pussaree; 8 v. = 1 maund or maon, - -- 30.20 mau ds = 1 bahur or candy; 20 b. = 1 garce. rOREIGIN COMMERCIAt WEIdIIS. a t Foreign. U. States. Avoirdupois pounds. Tranquebar, and Coromandel coast generally; 30 chittacks= 1 vis; 6] v. 1 maund; 20 m. = 1 candy, - - - - = 500.MALAYA (Malay Peninsula, Strait of Malacca): Same as at Singapore Island; also, 1 kip for tin = 401 Av. lbs., and 20 buncals = 1 catty for gold and silver = 2I lbs., troy. PENANG ISLAND, or PRINCE OF WALES ISLAND (Areca Island, Strait of Malacca): Same as at Singapore. SIAM (Farther India): 2 tical 1 tael; 20 t. = 1 catty; 100 c. 1 pecul, - 135.2536 20 piculs = 1 cajar. SINGAPORE ISLAND (Off S.'extremity of Malay Peninsula): 16 tael = 1 catty; 100 c. 1 pecul; 3 p. = 1 bahar,.- — 405.BANCA ISLAND (Malay Archipelago): Same as at Batavia, JAVA ISLAND. BORNEO ISLAND.- Same as at Batavia, JAVA, CELEBnES ISLAND. —- Same as at Batavia. JAVA ISLAND. - Bataviac,.c.: 16 tael- -1 catty; 100 c. - 1 pecul; 3 p. = 1 bahur = 200 - goelak, -.._.406.888 4.: peculs (300 goclaks) = 1 great bahar. MOLUCCAS )or SrLcE ISLANDS. -Amboyna, the B3anda Islanls, Bats/dan, Booro, Ceram, Gilolo, Oby, Wlaiqeoo: Official, as at Batavia. PIIILIPPINrI ISLANDS (Luzon, M[indano, Palawan, Miindoro, IPanay, Jliarindique, Negros, Bohol, Zuba, Samar, Mlasbata, Leyte, sc.): 100 catties 1 pecul 371 Castilian libra, - 139.4957 i caban of rice (usual), - - 133.1 caban of cocoa, - 83.5 SOOLOO ISLANDS (Sooloo, Basseelan, Tawee-Tawee, Pilas, Pala, Tapul Isles, 6c.): 10 mace = 1 tael; 16 t. - 1 catty; 50 c. 1 lachsa; 21. -__1 pecul, 1333. — SUMATRA ISLAND.- 16 mace = 1 tael; 25 t. = 1 catty; 36 c. 1 maund; i m. = 1 candil or bahur, - - - - -.423.5 4 catties- 5 goelaks. 1 tael = 133 lbs., Av. 14 salup = 7 o0otan =1 nelli, - 29. 82 a FOE1IGN COMMIERCIAL WEIGHTS. Foreign. U. States. Avoirdupois pounds. ITALY. - The metric system of weights, either under the French denominations or as follows, is now the official, and may be considered the general commercial system throughout Italy, the islands of Sardinia and Sicily included. 10,000 grani - 1000 denari = 100 grossi 10 oncie = I libbra = 1 kilogram. 1000 libbre - 100 rubbi = 10 quintali or centinaji = 1 migliajo or 1 tonnellata, - - - 2204.6214 Special and local: Carrara: 1 cubic palmo of marble = 884.74 cubic inches, - -. 90.82 JAPAN (N.. Pacific Ocean). —Niphon I., KiooSioo I., Sikokf I., the dependencies Yeso I., Bonin I., the Loo-Choo group, sc.: 1000 moo = 100 rin -- 10 sen -- I monme 26.784 troy grains. 160 monme - 1 kan, - - -- 0.612206 350 monme (28- kan) 1 catty; 100 c. = 1 pecul, -- - 133.92 NOTE. -In commercial transactions the pecul is usually reckoned at 1331 lbs., the same as in China; but it is equal to 133.92 lbs. by the weights and analyses of the modern Japanese doins. LIBERIA (W. Africa): Same as in the United States. MADEIRA ISLANDS (Atlantic Ocean, off W. coast of Morocco, N. Africa): Same as in Portugal. MALTA ISLAND (Mediterranean Sea, S. of Sicily): Official, same as in Great Britain. MAURITIUS ISLAND (Mascarene group, Indian Ocean): Official, as in Great Britain; also, 100 livres (old poids de marc of France) = 1 quintal, -. - - 107.9184 MEXICO. -8 ochavas 1 onza; 16 o. 1 libra; 25 I. = 1 arroba; 4 a. = 1 quintal, -=101.4232 MOROCCO (Barbary, N. Africa): 10 onza 1 mark; 2 m. -- 1 rotl; 100 r.= 1 cantaro -=118.65664 NOTE. - The commercial rotl of Morocco, both in theory and practice, is equal to the weight of 20 old standard duros (silver dollars) of Spain; the miner's rotl is equal to 100 meticals (1 old Venetian libbra, peso grosso), or 1.054945 avoirdu-' pois pound; and the market rotl _ 160 meticals. FOREnIGN CO~MMERCIAL WEIGHIITS. a 33 Foreign. U. States. Avoirdupois pounds. MOZAMBIQUE (E. Africa): Same as in Portugal. NORWAY.- Same as in Denmark. NUBIA (E. Africa): Same as at Alexandria, Egypt. PERSIA. 6 dirhem -- 2 mascais -- 1 miscal 73.846154 troy grains. 100 miscals - 1 rotl or ratel, - - = 1.054945 1364 rotl = 144 avoirdupois pounds. 6.4 rotl = 1 maund tabree (customs), - 6.8571.1 6 rotl- 1 "- " (bazaar), - -= 6.3237 74 rotl 1 maund copra (customs), - - 7.91209 7 rotl -1 " " (bazaar), - - 7.3846 Also, 1164 miscals = 1 rotl copra (bazaar), and 6 rotl copra = 1 maund copra, bazaar. 2 maunds tabree (of Tabreez) - 1 maund shah (of Sheeraz). 1600 miscals -_ I reh of Teheran = 16 bazaar rotl of Tabreez - 6 okes of Turkey, - 16.88023 NOTE. -The maund tabree is used chiefly for weighing coarse metals, coffee, sigar, drugs, &c., and the malundl copra for weighing rice and provisions. The maund shah is used chiefly in Sheeraz, Bushire, and Gombroon, although the last-mentioned port now belongs to the Muscat dominion. PORTUGAL. -576 grao 24 cscropulo = 8 outava- i onqa; 16 o. 2 marco -1 arratel, = 1.01187 32 artatel 1 arroba; 4 arroba 1 quintal, - = 129.5193 1 34 quintalo = 1 tonelada. ROME and Civita Vecchia: 24 grao = 1 oncia; 12 o. = 1 libbra; 10 1. = 1 (lecirne; 10 d. = 1 centinajo or cantaro, - - 74.7714 10 centinajo = 1 migliajo. RUSSIA. 96 solotnik- 32 olh -- 12 lana 1 funt, - - 0.902612 40 funt = 20 dowinik 13L trowinik - 8 paterik = 4 desaterik 1 pud, - - 36.10448 10 puds = I berkowitz; 3 b. 1 paken; 2 p. 1 last. - SENEGAMBIA (W. Africa). -Bathurst, Sierre Leone: Official, as in Great-Britain. St. Louis: Official, as in France. SOCOTRA ISLAND (Indian Ocean, off E. coast of Africa): Same as at Muscat, Arabia. SPAIN. - Standard of Castile: 128 ochavas = 16 onzas 2 marcos - 1 libra, - 1.0145143 84 a bOREIGN COMME1iCIAt WtEIIITS. Foreign. - -- U. Stateg, Avoirdupois pounds. -5 libras - 1 arroba; 4 a. = 1 quintal; 20 q. - 1 tonelada, - 2029.028 Special and local, but not oficial: VALENCIA. — Alicante,'c.: 12 onze =1 libra menor (minor). 18 onze = 1 libra mayor (major); 24 1. mayor, or 36 1. menor, — 1 arroba = 1 Castilian ar-.roba, - = 27.3919 4 arrobas 1 quintal; 21 q. = 1 carga; 8 c. 1 tonelada, =. 2191.291 ASTURIAS. — Santander, sc.: 25 libras = 1 arroba = 1k arroba of Castile, - - - = 38,04429 ARAGON. - Saragossa, ~c.: 86 libras 1 arroba, - 27.3919 BISCAY. — Bilboa, sc.: 25 libras = 1 arroba1f1w Castilian arroba, - 26.948-: 146 libras 1 quintal macho (for iron). CATALONIA. - Barcelona, &c.: 25 libras = 1 arroba 7= Castilian arroba, - - 22.1925; ANDALUSIA. Illalaga: 7 arrobas (Castilian) = 11 quintal -1 carga of raisins, - = 177.54$ TTOTE.- The employment of the metric system of weights is sanctioned b~y law in Spain.:. SWEDEN. - New standard: 100 korn = 1 ort; 100 o. =1 skalpund; 100 s. - 1 centner, - 92.8583 SWITZERLAND. - New system: 32 lothe, or 8 gros, = 1 unze; 16 u. - 1 pfund or livre: A kilogram; 100 pfunds = 1 centner, - = 110.23'11 TRIPOLI (Barbary, N. Africa.)- 16 karob 1 — drachma; 10 d. = 1 oncia or usano= miscals or 1 troy ounce. 16 oncia - 1 rotl; 100 r. = 1 cantaro, - = 109.7143: 400 drachma - 1 oka,.2.742857 TUNIS (Barbary, N. Africa).- Same as in Tripoli. TURKEY. -- 4 grani - 1 kara, killot, or taim; 16 k. = 1 dirhem or drachmia miscal or metical = 49 troy grains. 100 dirhems 1 cheki; 4 c. =-1 oka, - = 2.813187 250 dirhems 1 cheki for opium = 1661 miscals, - - - 1.75836: Constantinople, Galata: 2 cheki.- 1 rotl or rotolo; 2 r. = 1 oka; 50 o. — 1 kantar or dantaro grosso, - - - - 140.65934.....40 .0AREIGN C0tEIMmCIAL WE1IOIt$. a 35 Foreign. U. States. Avoirdupois pounds. 176 dirhems -- 1 rotl; 100 r. = 44 okes - 1 cantaro sottile, 123.78834 610 dirhems - 1 teffeh of Brusa silk, - - - 4.29039 800 dirhems = 1 teffeh of goat's wool. Smyrna: 180 dirhems = 120 miscals = 1 rotolo; 100 r. = 45 okes = 1 cantaro, - - = 126.60171 44 okes -- 1 cantaro for tin. In, Bassorah the Arabs commonly employ the light wakea of, 160 krat of Mocha - 1 troy ounce, or 480 grains; and 15 wakea = 1 cheki, - - 1*0285143 40 wakea (21 cheki) =1 oka, -2.742857 50 okes (2,000 wakea) = 1 cuttra or cantaro, = 137.142857 NOTE.- The initial for the avoirdupois values of the Turkish weights, in the absence of documentary statistics on the subject, if any exist, was derived from the Abyssinian dirhem and by comparison; and the result, I find, is almost strictly confirmed by assays carefully made at the United-States mint and elsewhere, of the modern gold and silver coins of Ottoman mintage compared with their.preseiit official standards; viz., l1J cheki of fine silver to the piastre and F cheki of fine gold to the piastre. The troy ounce, it is well known, was derived from the Abyssinian dirhem (drachma) or its multiple by 10, the wakea, vakia, or wakega, and consists of 12 of the first-mentioned units, making the dirhem equivalent to 40 troy or United-States grains, while 120 of these dirhems, or 1 rotl or rotolo of Abyssinia, is equal to 65 miscals, or meticals, or TL maund tabree (customs) of Persia; hence, 120 X 40 65 - 731 troy grains, the value of the Persian miscal. But the miscal, or metical of Persia, and that of Turkey are-alike: in theory it is the same specific weight everywhere; and 1 dirhem of Turkey Is equal to I miscal; hence, 7 3 X 2= 4918 troy grains, the value of the Turkish dirhem, and 4 dirhem of Persia are equal to 1 dirhem of Turkey, and 6} miscals are equal to 1 troy ounee. McCulloch (not to go farther back), in his work published in 1839, says the cantaro of Constantinople of 45 okas is equal to 127.2 avoirdupois pounds; or, in other words, that the oka of Constantinople is equal to 2.82667 pounds; and he states the.oka of Smyrna to be equal to 2 lbs. 13 oz. 5 dr., or 2.83203 pounds, but, at the same time, under the last-mentioned head (Smyrna), states the weights and measures to be the same as those of Constantinople. Alexander, in his work published in 1850, places the oka of Constantinople at 2.828571 pounds, and that of Smyrna at 2.812488 pounds, in a measure reversing the values by McCulloch; while Noback, in a work of more recent date, says the oka of Constantinople is equal to 1278.48 grammes 2.8185644 pounds; making it nearly equal to that of Smyrna by Alexander; and that'the oka of Smyrna is a little heavier, being equal to 2.83286 pounds. From these conflicting statements no tenable idea can be gained except this; -viz., that the initial and leading weights of Asia Minor (Anatolia) are probably theoretically and practically the same as those of Turkey in Europe. But this seems to admit of no question, since 1 batman of Persian silk, containing 1 reh, or 1600 miscals of Teheran, is invariably equal, both in Constantinople and Smyrna, to 6 okas of Turkey; wherefore, the oka is equal to 1`60 -= 2662 2661 X 7311 miscals, or 7000 1 27 4 avoirdupois pounds, being slightly heavier 7000 than that of Smyrna by Alexander, and a trifle lighter than that of Constantinople by Noback. ~86 a POREIGN COMMERCIAL WEIGHTS. Foreign.'U. States. FWest Indies. Avoirdupois pounds. GREAT ANTILLE ISLANDS. CUBA: Standard of Castile. 16 onzas — 1 libra; 25 1. -1 arroba; 4 a. — 1 quintal; 20 q. = 1 tonelada, - - = 2029.028 HAYTI: Poids du marc of France, previous to A.D. 1800. 16 onces = 1 livre; 100 1. = 1 quintaux; 10 q. =1 millier, - - - 1079.176 2 milliers or barriques - 1 tonneau. SAN DOMINGO, OR DOMINICA: Same as in Cuba. PORTO RICO: Same as in- Cuba. JAMAICA: 16 ounces = 1 pound; 28 p. - 1 quart ter; 4 q. - 1 cwt.; 20 cwt.- 1 ton, - = 2240. — LUCAYOS, OR- BAHAMA ISLANDS. —Same as in Jamaica. CARIBBEE ISLANDS. LEEWARD GROUP.- DOMINICA, TORTOLA, VIRGIN GORDA, ST. CHRISTOPHER, ANGUILLA, BARBUDA, NEvIS, SABA: Same as in Jamaica. ANTIGUA, MONTSERRAT: 100 pounds - 1 cental or cwt., -.- 100.ST. EUSTACIUS: Official, as in Holland. GUADELOUPE, MARIE GALANTE, DESIRADE, LES SAINTES: Official, as in France; also as in Hayti. ST. MARTIN: Dutch, as in Holland; French, as in France; also as in Hayti. ST. THOMAS, SANTA CRUZ, ST. JAN: Official, as in Denmark. ST. BARTHOLOMEW: Official, as in Sweden. WVINDWARD GROUP. - BARBADOES, GRENADA, ST. VINCENT, TOBAGO: Same as in Jamaica. MARTINIQUE, ST. PIERRE, ST. LUCIA: Same as in Hayti. TRINIDAD: Same as in Cuba; official, as in Great Britain. LITTLE ANTILLES.- CURACOA, BUEN AYRE, ORUBA: Official, as in Holland. MARGARITA, TORTUGA, BLANQUILLA: Same as in Venezuela. ZANGUEBAR (E. AFRICA). - Zanzibar (island and Town). Consul's report: 12 maunds1 frasler, - - 35. FOREIGN LIQUID MEASURES, REDUCED TO THEIR EQUIVALENT VALUES IN THE UNITED STATES, Foreign. U. States. Wine gallons, ABYSSINIA.- By Weight: see Weights. ALGERIA. - Official, as in France; also, 161 litres - 1 khoulle; 6 k. 1 hectolitre, - - = 26.417 ARABIA. - (Generally by weight.) Mocha: 20 wakeas (weight) =1 nusfiah; 8 n. - 1 cuda or gudda - 16 av. lbs. AUSTRALASIA. - Same as in Great Britain. AUSTRIA (legal for the Empire): 4 seidel — 1 maass; 10 m. = 1 viertel; 4 v. = I eimer or orna, - - 14.9543 32 eimers = 1 fuder; 42j maass = 1 eimer for beer. AZORE ISLANDS. - Same as in Portugal. BALEARIC ISLANDS. -MAJORCA: 8 quartas = 1 quartera; 3j quarteras, or 4 quartinellos, - I1 quartin or barril; 4 quartin 1 carga; 4 c. = 1 botta = 27 fluid arrobas, or 1 pipa of Castile, - - - 114.9692 MINORCA: 8 quartas = 1 quartera; 4 quarteras, or 21 gerah, = 1 quartin or barril; 4 barrils = 1 carga; 4 c. = 1 botta = 31i fluid arrobas of Castilg, - - 133.1635 BELGIUM.- Same as in France. BERBERA. - Same as at Mocha, Arabia. BERMUDA ISLANDS. — Official, as in Great Britain; in trade, generally as in the United States. BOURBON ISLAND. - Same as in France. CANADA, DOMINION OF.- Official, as in Great Britain; in trade, as in the United States. CANARY ISLANDS. -Same as in Spain, Castilian Standard. CANDIA ISLAND. — 1 mistata 81 okes weight, or, of olive oil, - 2.9397 CAPE OF GOOD HOPE. -Same as in Great Britain. CAPE VERDE ISLANDS. - _Same as in Portugal. 4 38 a oR0Ii LIQVIDW MrEALUURJ. Foreign. U. States. Central and South America. GUATEMALA, HIONDURAS, SAN SALVADOR, NICARAGUA, COSTA RICA: Same as in Spain, standard of Castile; also- the wine gallon of the United States is used. BALIZE: Official, as in Great Britain. BRAZIL: 24 quartilhos =-12 garrafa = 6 canada c 3 medida = 1 alqueire or pote = 18 arratels weight, - - - - 2.18418 60 potes - 1 pipa = 1080 arratels weight, - — = 131.051 Bahia: 1 canada. 15. arratels weight= 5= canadas of Rio Janelro, - = 1.88082 72 canadas = 1 pipa of spirits, - - - = 135.4193 100 canadas = 1 lipa of molasses. ARGENTINE REPUBLIC': 4 cuartos, 1 frasco; 8 f. = 1 caneca _ 19 litres in theory, - - - 5.01927 3 frascos= 1 cortan; 16 c. (6 canecas) = 1 carga, - -- 30.11541 4 cargas 1 pipa catalana; also, 8 frascos =5 U. S. gallons, nominally. PERU, CHILI,. BOLIVIA, ECUADOR, NEW GRANADA, VENEZUELA, URUGUAY, PARAGUAY: Chiefly as in Spain, standard of Castile. NOTr. -'In the States last mentioned, the U. S. wine gallon is more or less used in trade; and in Chili it is the customs' unit-measure for liquids. Also, in Chili, Peru, New Granada, Bolivia, and Venezuela, the use of the metric system is sanctioned by law, and may be expected to gradually come into use. GUIANA - Cayenne: Same as in France. Paramraribo (Surinam): Same as in Holland; also as in Cayenne. Georgetown (Demerara): Same as in Great Britain. FALKLAND ISLANDS. Same as in Great Britain. CHINA. -By weight only; for denominations, see DRY MEASURES. DENMARK.- 42 pagel - 4 potte = 2 kande 1 stubehen, -.. 1.02089 40 stubchen - 20 viertel = 4 anker - 1 ohm, - 40.83522 1~ ohm = 1 oxehoved; 2 oxehoved = 1 piba;2 p. 1 fuder; 1 jf. - 1 stykfad. 34 stubchen -= 1 toende for beer; 30 stubehen = 1 toende for tar. EGYPT. - By weight exclusively. See WEIGHTS. FO1ZIGQN: LIQUID MEASURES. a 39 Foreign. U. States. Wine gallons. TFRANCE. — 1,000 millilitres = 100 centilitres - 10 decilitres-= 1 litre, * - -- 0.26417 -100 litres -= 10 decalitres - 1 hectolitre, - - 26.417029 100 hectolitres = 10 kilolitres = 1 myrialitre. GERMANY. - Prussian and Zollverein maasse for all the States:of the tariff-alliance: 2 oessel 1 quart; 30 q. = 1 anker; 2 a. - 1 eimer = 3840 cubic Rhine zollen; or, since 381 zollen = 1 metre, = 682j]j4 litres, - - - 18.126789 2 cimers = 1 aam, ahrm, or ohm; 3 eimers 1 oxhoft; 12 eimers = 1 fuder.'3 eimers-= 2 barrile = 1 fass for beer, - - 60.42263 NOTE. - I have been thus particular in treating of the elimer, because the notion seems to be generally entertained that it is equal in theory to 68.7 litres. Special and local, or domestic: BADEN: 1000 maass = 100 stubehen = 10 ohm=. 1 fuder - 15 hectolitres, - - - 396.2555 BAVARIA: 4 quartile 1 mass, or masskanne; 60 M. l elmner, - - - 16.9452 25 eimer= 1 fass; 64 masse 1 eimerfor beer. BREMEN: 4 mengel _ 1 quartier, or vierling; 4 q. = 1 stubchen, -. = 0.85106 24 stubehen = 1 viertil; 5 v. 1 anker; 4 a. -i ahm, - - - - - = 38.29781 6 ankers = 1 oxhoft; 4 o. = 1 fuder; 44 stubchen = 1 ahm for wine. - 6 stubchen - 1 steckannen; 6 s. 1 tonne for train oil _ 216 pfunde weight, or, at 73 av. lbs to-the gallon, -= 30.6073 BRUNSWICK: 10 stubchen = 1 anker; 4 a. — 1 ahrm; 1. ahm = I oxhoft, - - - = 59.2803 HAMBURG: 8 oessel, plank, or stuck = 4 quartier, or potts = 2 kannen = 1 stubchen, - - 0.956404 8 stubehen - 4 viertel 1 eimer, - - = 7.651224 32 stubchen = 1 anker; 40 stubehen = 1 ohm; 60 stubchen = 1 oxhoft; 240 stubchen = 1 fuder. IIESSE DARMSTADT: 16 schoppen - 4 masschen I viertel; 20 v. = 1 ohmn 16 decalitres, - 42.26725 LUBEC: 16 ort = 8 plank, or nossel = 4 quartier = 2 kanne =- 1 stubchen, - - - = 0.9546 8 stubchen -4 viertcel = 1 eimer, - - 7.6512 5 viertels - 1 anker; 6 a. = 1 fass, - - - 57.384 40 a FOREIGN LIQUID MEAISURES. Foreign. U. States. Wine gallons. MECKLENBURG (legal): Same as in Hamburg. OLDENBURG: 240 quartiers, or 156 kannies - 1 oxhoft (legal) = 1 fuder of Lubec. SAXONY (legal): 144 ndssel = 72 kanne = 24 viertel = 1 eimer, - 17.8107 2 eimers - 1 aam; 3- eimers = 1 oxhoft; 5 eimers - 1 fass; 12 eimers - 1 fuder. WURTEMBERG. - Helleich maiss: 4 quartier, or schoppen 1 maas; 10 m. 1 immer; 16 i. = l eimer; 6 e. - 1 fuder, - - =465.9036 GREAT BRITAIN. — Imperial measure: Denominations and relative values same as in the United States, but capacity values- 20=7j4 per cent- greater. See LIQUID MEASURES, U.S. 1 imperial gallon = i -7- or 1.200320344 wine gallons of the United States. GREECE. - 1 kila, or galloni = 2, okas weight. HOLLAND.- 10 vingerhoed = 1 maatje; 10 m. 1 kan; 100 k. 1 vat = —1 hectolitre, - - 26.417 HAWAIIAN ISLANDS. - Same as in the United States. India and Malaysia, or East Indies. ANNAM, BURMAH, Calcutla, and BENGAL generally, CEYLON I., PHILIPPINE IS., Soo-Loo Is. By weight. See WEIGHTS. Bombay, MIkadras: By weight, chiefly; the wine gallon of the United States is sometimes used. Goa: Same as in Portugal. Pondicherry: Official, as in France. MALACCA.- Capacity measures, same as in the United States. PENANG ISLAND. - Same as in Singapore. SIAM. - 20 canan = 1 cohi- 80 catties weight, or 108.203 av. lbs., - - 12.9757 SINGAPORE ISLAND. - Capacity measures, same as those of the United States. BANCA I., BORNEO I., CELEBES I., JAVA I., Mo. LUCCA IS., SUMATRA I.- Official, same as in Holland. ITALY. — The metric measures of capacity are used here, both under the French nomencla... 0OR1EIQN LIQUIID MEASURES. a 41 Foreign. U. States. Wine gallonsf ture and as follows; viz., 10 coppa 1 pinta; 10 p. = 1 mina; 10 m. - 1 soma -1 hectolitre, - -. - - 26.417 JAPAN. — 1 tsjoo 1- cubic kani-sasi - 106.09603 cubic inches, -. -~ 0.459293 10 tsjoo= 1 to; 10 to=1 kok; 10 sasi=goo; 10 goo = 1 tsjoo. LIBERIA.- Same as in the United States. MADEIRA ISLANDS. - Same as in Portugal. MALTA ISLAND.- Official, same as in Great Britain; also, 1 caffiso of oil — 4.724 gallons, and 1 barrile of wine - 9.448 gallons. MAURITIUS ISLAND. -8 pintes = 1 velt, - = 1.969 MEXICO. - Chiefly as in Spain, Castilian standard; but the use of the metric system is legalized, and may be expected soon to be introduced into practice. MOROCCO.MOZAMBIQUE. - Same as in Portugal. NORWAY. -The Danish capacity measures are used here. NUBIA. - By weight, as at Alexandria. PERSIA. - By weight. See WEIGHTS. PORTUGAL. - 24 quartilhos = 6. canadas = 1 alqueire, or pote = 18 arratels weiht, - = 2.18418 2 potes = 1 almude; 26 a. = 1 bota or pipa, - = 113.5775 2 botas - 1 tonelada; 18 almudes — 1 barril. Oporto: 1 alqueire = 274 arratels weight, or VjoQ alqueires of Lisbon, - - 3.30936 ROME, and Civita Vecchia: 64 cartocci = 16 quartucci = 4 foglietti = I boccale, - = 0.48165.32 boccali = 1 barile; 16 b.= 1 botta, - -246.605 32 boccale for wine = 28 boccale for oil. RUSSIA. - 100 tscharka - 10 krushka = 1 vedro, or wedro, = 3.246 74 3 vedros = 1 anker; 6 a. = 1 oxhoft, - = 58.4413 40 vedros - 1 botschka. SPAIN (Castilian standard): 4 copas= I1 cuartillo; 4 c. - 1 azumbra; 8 a. = 1 arroba or cantaro = 35 libras weight of distilled water at maximum d(ensity, or 35,508 av. lbs., = 4..25812 16 arrobas = 1 moyo; 27 arrobas _ 1 pipa; 30 arrobas - 1 bota; 60 arrobas -- 1 tonelada. 1 arroba menor for oil = 274 libras weight, - _ 3.31525 4* 42 a FOREIGN LIQUID MEASURES8 Foreign. U, States. Wine gallons. Special and local: Alicante and Valencia: 16 cuartillos = 4 cuartos -= 1 arroba - = Castilian arrobas, - - 3.19359 40 arrobas 1 pipa; 2 p. = 1 tonelada, - 255.4872 Also 100 cantaros of ~ Castilian arroba each 1 tonelada. Barcelona: 16 cortans (12 arrobas Catalan weight) 1 carga = 7 fluid arrobas of Castile, - - - - 31.9359 Gibraltar: 38 arrobas menor of Castile 1 pipa, = 125.9795 126 U. S. gallons, or 105 imp. gallons in theory - I pipa. lJlalaga: 334 fluid arrobas of Castile = 1 pipa, - = 141.9373 NOTE. -The employment of the metric capacity measures is sanctioned by law in Spain. SWEDEN.- 4 qwarter 2 stop 1 kanna = i cubit fot. 48 kannas = 8 ottingar - 4 fjerding 1 tunna, - - 33.184106 SWVITZERLAND. (Oficial and legal for the 22 Cantons): 1000 emine = 100 maass or potts 10 gelt = 1 saum = 150 litres, = 39.62555 TRIPOLI. - 14 caraffa = 1 mataro for oil 174 okas weight, or 48 av. lbs. 40 caraffa - 1 barril - 50 okas weight; also 24 bozza = 1 barile = 130 rotolos weight (52 okas), or, of the standard of the United States, - - - - 7.10402 TUNIS. - 2 mettars for wine - I mettar for oil 36 rotoli weight; 34 mettars'= 1 millerolle 120 rotoli weight for oil, or 231.65716 av. lbs. TURKEY.-By weight. See WEIIGHTS. Also 1 almud of oil = 8 okas. West Indies. Cuba, Porto Rico: Same as in Spain, Castilian standard; but in Cuba the U. S. gallon is also used: 36 gallons - 1 bocoy = 36 U. S. gallons. Dominica (San Domingo, or Dominican Republic, HAYTI I.): Same as in Spain, standard of Castile. FOREIGN LIQUID MEASURES. a 43 Porei~g. U. Slates. Wind gallons. HRayti, Empire of(HAYTI 1.). - 60 gallons- 1 tier90o1, - 60.Guadeloupe, Martinique; Ilarie Galtane, Les SairntV, Desirade, northern poltion of' St. Mllartin.Official, as in France; but in trade the UrltldStates fluid gallon is chiefly used; fior molasses, 30 gallons = 1 baral; 65 gallons — 1 tiergon; 105 gallons _ 1 baucaut: for rum, 114 gallons = 1 boucaut. Jamaica, lTrinidad, Bahamas, Barbadoes, St. Christopher, Dominica,- Montserrat, Grenada, St. Lucia, Antigua, Tortola, Tobago, Nevis, Virgin Gorda, Grenadines: Official, as in Great Britain; in trade, mostly: as in the United States. St. Thomas, Santa Cruz, St. Jan: Official, as in Denmark. St. Eustatius, Curacoa, Buen Aiyre, Oruba, southern portion of St. MAartin: Official, as in Holland. Mlargarita, Tortuga, Blanquilla: Same as in Venezuela. St. Bartholomew:' Official, as in Sweden. (44a) FOREIGN DRY MEASURES, REDUCED TO THEIR EQUIVALENT VALUES IN THE UNITED STATES. Foreign. U. States. Winchester bushels. ABYSSINIA. — 24 madega =1 ardeb, 0.33333 ALGERIA. - Official, as in France; also 16 tarrie 8 saa or saha = 1 caffiso,, - = 9.ARABIA. - By weight: 40 kellas = 1 tonaun for rice= 56 maunds weight, or 18 a:v. lbs. AUSTRALIA. - Same as in Great Britain. AUS!RIA (legal for the Empire). -4 beche = 1I massel; 4 m. =- 1viertel; 4 v. A1I metze, - 1.7452 AZORE ISLANDS.-16 quartos _ 4 alqueires = 1 fanga; 15 f. 1 roio= mio of LAs' bon, - - - 20.5298 BALEARIC ISLANDS. -— 6 barcella = 1 quartera = 1 fanega of Castile, - = 2.157 BELGIUM. - 100 kop = 10 schepel-= 1 mudde. = 1 hectolitre, - - - 2.83774 BERBERA. -- BERMUDA ISLANDS. —Official, as in Great Britain; the U. S. bushel is also used. BOURBON ISLAND. - Same as in France. CANADA, DOMINIOx OF. — Official, as in Great Britain; in trade, as in the United States, except that in Lower Canada the old French minot = 1.107436 U. S. bushels is used. CANARY ISLANDS. -12 celamins = — 1 fanega - 136 libras weight, - -= 1.77737 164 celamins = 1 fanega heaped. CANDIA ISLAND. —1 carga, - - - 4.3211 CAPE OF GOOD HOPE. - Same as in Great Britain; also, 4 schepels —- 1 muid - _- 3,1564:. CAPE VERDE ISLANDS. - Same as in Portugal, Central and South America. CUATEMALA, HONDURAS, SAN SALVADOR, NICARAGUA, COSTA RICA: Same as in Spain, Castilian standard but the U. S. bushel is also used. FOREIGN DRY MEAMSURES. a 45 Foreign. U. States. Winchester bushels. BALIZE. - Official, as in Great Britain. BRAZIL. -16 quartas = 4 alqueirs = 1 fanga; 15: f, 1 moo moo io of Portugal, - - 20.11086 Bahia: 1 alqueire 671 arratels weight, or 21 alqueires of Portugal, - - - 0.87985 Maranham: 1 alqueire = 100 arratels weight, - 1.30348 -ARGENTINE REPUBLIC, URUGUAY.- 4 cuartillos = 1 fanega= 134 litres, - - 3.80257 CIIILI. -L 12 celamins = 1 fanega, - - 2.5753 PEnu.- 1 fanega, -= 2.31777 BOLIVIA, ECUADOR, NEW GRANADA, VENEZUELA, PARAGUAY: Chiefly as in Spain, Castilian standard. GUrIANA. — Cayenne: Same as in France. Paramaribo: Same as il IHolland, also as in Cayenne. NOTE. -The use of the metric system is sanctioned by law in Chill, Peru, New Granada, Bolivia, Venezuela, and'rench and Dutch Guiana, and, to some extent, is introduced into practice. Georgetown: Same as in Great Britain. FALKLRLAND ISLANP S.- Same as in Great Britain. CHINA.- 10 h6 - 1 shing; 10 s. =1 tau; 10 t. 1 hwiih, sei, or tane = 120 catties weight 160 av. lbs. DENMARK. -832 sextingkar = 16 ottingkar — 2 skieppe = 1 fjerding, stubehen, or scheffel — 36 potte, - 0.98698 4 fjerding =1 toende; 22 t. - 1 last,- — 86.8546 EGYPT. - Cairo: 24 robi - 6 usbek = 1 ardeb _ 144 okas weight, - = 5.088 Alexandria, Rosetta: 1 kislos = 137 okas weight, = 4.84065 1 rebeb = 126 okas weight, - - - - 4.45198 1 ardeb= 230 okas weight, - - - 8.12663 FRANCE.- 100 litres -- 10 decalitres- 1 hectolitre,- -. --- = 2.83774 100 hectolitres = 10 kilolitres = 1 myrialitre, - _ 283.774 GERMANY. - PRUss IA, and Zollverein mass of all the States of the tariff-alliance: 16 metzen (3072 cubic Rhein zollen, or 48 fluid zollverein quarts) = 1 scheffel, - = 1.55776 Special and local: BADEN: 1000 becher= 100 miisslein = 10 sestcr1 malter -- hectolitre, - - 4.25661 10 malters --- 1 zober. 46a ]011EtdN IRY MEASU.IE. Foreign. U. State. Winchester bushels. BAVARIA: 16 drcissiger 4 mUssel - 1 vicrtl, - = 0.525855 12 viertels (17 fluid mrnsskanne, or six old metzen) = 1 scheffel, 6.310263 BREMEN: 16 spint = 4 viertel = 1 scheffel, - 2.10289 10 scheffels = 1 quarter; 4 q. = 1 last, - = 84.11572 BRUNSWICIK: 4 metzen - 1 himt; 40 h. —1 wispel, - - - 35.3514 AIIMBURG: 8 spint 2 himten 1 fass 1 zollverein scheffel, - - - - 1.55776 10 scheffels = 1 wispel; 6 w. 1 last, - — 93.4656 IILSSE D)ARMSTADT: 64 kopfchen -= 32 maasschen = 8 gescheid = 2 kumpf= 1 metze, - 0.45104 8 metzen = 2 simmer 1 malter - 128 litres, - 3.632308 LUBEC: 16 fass = 4 scheffels = 1 tonne, - - 3.9381 3 tonnen - 1 dromt; 8 d. 1 last, - - =94.5139 MECKLENBURG: 16 spint, or mctzen 4 fass, or viertel — 1 scheffel, - - - 1.1036.28 4 scheffels = 1 wispel; 3 w. =1 last, - - =105.9483 OLDENBURG: 16 kannen = 1 scheffcl; 8 s.. -1 tonne, - - -. - = 5.17536 1- tonne 1 molt; 12 m.= 1 last, - - 93.1565 SAXONY: 16 maisschen = 4 metzen =1 viertel, -= 0.73713 4 viertels= 1 scheffel; 12 s. = 1 malter, - — 35.3823. 2 malters= 1 wispel; 6 w. 1 last, - - -424.5876 WURTEMBERG: 32 viertelein 8 ecklein 1 vierlinog,..0157172 32 vierling - 8 simri = 1 scheffel, - = 5.0295 GREAT BRITAIN. - Imperial measure: Denominations and relative values same as in the United States; but capacity values = 55454. greater; 1 bushel = 1.0315157 U. S. bushels. GREECE.- 1 kila, - - = 0.944 HOLLAND. - 1000 maatje - 100 kopen = 10 schepel - 1 mudde or zac = 1 hcctolitre, - = 2.83774 30 mudden -1 last, - - - 85.1322 2 India and Malaysia, or East Indies. ANNA3, -BURMAHI, CEYLON ISLAND: By weight. See WEIGHTS. ILINDOSTAN. - Bombay:- 8 tipprees = 4 seers 1 adoulie. FOREIGN DRY MEASJRIES. a 47 Foreign. U. States. Winchester bushels. 16 adoulies -1 para= 83 maunds weight, or 245 av. lbs., 3.15607 8 paras= 1 candy 70 maunds weight, - =25.248545 Calcutta: 80 chattac — 16 koonke = 4 raik 1 pallie = 5 seers weight, or 94 av. lbs.: 12 pallies = I1 morah = 1 factory maunds weight, or 112 av. lbs. 20 pallies - 1 soallee = 2I factory maunds, or 1864 av. lbs. 16 soallees (40 maunds, or 2986i av. lbs.) 1 kahoon, - - - 38.4 739 7 Madras: 64 ollock = 8 puddy -1 marcal. 5 marcals = I para = 54 maunds-weight, or 135 av. lbs. 80 paras 1 garce 432 maunds, or 10800 av. lbs., - =139.12464 Goa: Same as in Portugal. Pondicherry: Same as in France. MALACCA. - The Winchester bushel is used, also the coyang of Siam. PENANa ISLAND. - Generally as in tle United States. SIAM. -40 sat - 1 sesti; 40 s. = 1 cohi; 65 c. 1 coyang 52 peculs weight, or 7033.1872 av. lbs., - - 90.6009 SINGAPORE ISLAND. - Generally as in the United States. BANCA ISLAND, BORNEO ISLAND, CELEBES ISLAND, JAVA ISLAND, MOLUCCA ISLANDS, SUMATRA ISLAND. -Official, as in Holland. ITALY. - See LIQUID- MEASURES: 1 soma (hectolitre), - - - 2.83774 JAPAN. — See LIQUID MEASURES: 1 kok (6j cubic kani-sasi), - - - = 4.93376 LIBERIA.- Same as in the United States. MADEIRA ISLANDS. - Same as in Portugal. MALTA ISLAND.- Official, as in Great Britain; also, 1 salma rasa (4 salma colma), - - 8.2202 MAURITIUS ISLANhD.- Official, as in- Great Britain. MEXICO. - Chiefly as in Spain, standard of Castile; but the use of the metric system is sanc. tioned by law. 48 a FOREIGN DRY MEASURES. F oreign. U. Slates. Winchester bushels. MOROCCO.MOZAMBIQUE. -Portuguese measures are used here. NORWAY. - Same as in Denmark. PERSIA.- 8 sextarios — 2 chenicas= 1 capicha. 25 capichas = 8 colothuns = I artaba = 21 maunds tabree (customs), or 144 av. lbs., - = 1.8541 22 sextarios = 1 sabbitha, - - 0.20395 15 capichas = 1 legana, - - - 1.11246 POR TUGAL. - 16 quartos =4 alqucircs 1 fanga -:= 120 arratels weight, - - - 1.56418 15 fangas - 1 moio = 1800 arratels weight, - 23.46267 Op:orto: 1 fanga -= 1 fanga of Lisbon, or 150 arratels weight, - - - - 1.95522 ROME. - 88 quartucci 22 scorzi 16 starclli 12 staja = 8 quarterella = 4 quarte = 2 rub-biatilli - 1 rubbio, - - - - = 8.3562 RUSSIA.-32 garnetz = 16 tschetwertka = 4 tschetwerik= 2 payak = 1 osmin, - - = 2.97607 2 osmins = 1 tschetwert; 1 t.- 1 kuhl. SANDWICH ISLANDS. -- tame as in the United States. SPAIN (Standard of Castile). -16 racion = 4 quartillos - 1 celemin, 0?' almuda;. 12 celamins =4 cuartilla = 1 fancrga 41 arrobas weight of distilled water at maximum density, or 123.64393 av. lbs., - - - 1.59277 12 fanegas- 1 cahiz, - - - - 19.113241 Special and local: Alicante: 4 celamins = 1 barcella; 12 b. 1 cahiz =41 cahiz of Castile, -- = 6.950306 Barcelona: 48 picotin 12 cortan =1 quartera 6- Castilian arrobas weight, - - 2.08285 2 quarteras =1 cargal; 4 qua.rte ras 1 saalm. SWEDEN. - 4 quarter = 2 stop =I kanna: - 1ocubic fot. 7 kanna- 4 kappe= 1 fjerdin, - - - 0.519847 4 fjerding- 1 spann; 2 s. =1 tunna, - - - 4.158777'. 36 kappe-= 1 tunne (firm measure), - - 4.678624 SWITZERLAND. (Official and legal for lthe 2 2 Cantons): 100 immi = 10 viertel, geclt, or quartcron - 1 malter=150 litres, - - - =- -4.25661 Also, 16 missli = 4 vierling = 1 viertel. FOlRElIGN DPr MAEr1U4ES. a 49 Foreign. U. States. Winchester bushels. TRIPOLI. = 2 nufs-orbah - 1 orbah; 4 orbahs - 1 temen; 4 t. = 1 ueba = 216 rotls weight, - 3.05279 TUNIS. — 12 zah, or saha-l quiba; 16 q.= 1 caffiso -- 425 okas weight, -- 15.01663 TURKEY. -4 kiloz - 1 fortin = 110 okas weight, - 3.986315 West Indies. Cuba, Porto Rico, San Domingo: Same as in Spain, standard of Castile. Hayti, Empire of. - 16 litrons 1 boisseau; 12 b. =1 setiere, = 4.4299 Gaudeloupe, Martinique, Marie Galante, Desirade, northern portion of St. Martin, Les Saints. - Official, as in France; also, as in Hayti; and the U. S. bushel is often used. Jamaica, Trinidad, Bahamas, Barbadoes, St. Christopher, Dominica, Montserrat, Grenada, St. Lucia, Antigua, Tortola, Tobago, Nevis, Virgin Goirda, Grenadines.- Official, as in Great Britain; in trade, generally as in the United States, but in Trinidad often as in Hayti. St. Thomas, Santa Cruz, St. Jan. -Official, as in Denmark. St. Eustatius, Curacoa, Buen Ayre, Oruba, southern portion of St. Martin.-Official, as in Holland. M*ar~garita, Tortuga, Blanquilla. - Same as in Venezuela. St. Bartholomew. - Official, as in Sweden...... m... K 50 a MEMORANDA AND ADDENDA. MEMORANDA AND ADDENDA, RELATIVE TO FOREIGN MONEYS OF ACCOUNT, COINS, WEIGHTSt MEASURES, QUOTATIONS OF STOCKS, ETC. GREAT BRITAIN. — In Great Britain, sovereigns weighing not less than 122] grains are a legal tender for a pound sterling each; which makes the minimum value of a pound sterling in gold equal to $4.8458226; and this value of the pound sterling, very nearly, is adopted by the United-States Government in assessing duties on British invoices. Goods by weight, passing through a British custom-house, and subject to duties, are subject to an allowance, called draft or fret, for supposed waste over and above the actual tares, as follows; viz., on I cwt. (112 lbs.), 1 lb.; above 1 cwt. and under 2 cwt., 2 lbs.; on 2 cwt. and under 3 cwt., 3 lbs.; on 3 cwt. and under 10 cwt., 4 lbs; on 10 cwt. and under 18 cwt., 7 lbs; and on 18 cwt. and upwards, 9 lbs. These allowances were also made at the UnitedStates custom-houses, until July 14, 1862, when the discontinuance of the practice was ordered by law. These are the chief reasons why goods by weight from the United States fall short of weight at the British custom-houses. Consols, or Consolidated Annuities, represent a considerable portion of the public debt of Great Britain: they bear interest at the rate of three per cent. a year, payable semi-annually, and are transferable. The quotations in London of the prices of United-States Bonds, and of American Stocks generally, arc in cents per dollar, payable in United-States gold coins; and upon the old basis of $44 to the ~. To these quotations, therefore, 9. per cent. must be added to express the real price. The silver dollars coined by the British government fbr circulation in China weigh 415.4 grains, and are id fine; they are, therefore, intrinsically of the same worth as Mexican dollars. FRANciC.-The prices of United-States Bonds, and of American stocks generally, are quoted in Paris in cents per dollar, payrable in French silver coins, and upon the conventional Bourse rate of 5 francs to the dollar; whereby the rate of exchange affects the prices. From the Paris quotations, therefore, 2.8172 per cent. must be deducted to express the true prices, when exchange is at par, or when a dollar in United-States gold is quoted at f. 5.14086. A tolerance of weight in excess of the standard, and In excess only, is allowed in France; while in the United States and in Great Britain, strict conformity to the standards is required; thus it happens that the model standards of weight, sent abroad from France, are commonly found to be slightly in excess of the true standard. MEMORANDA AND ADDENDA. a 51 The tolerance or remedy spoken of is as follows: On Iron weights of 50 kilogrammes, tolerance 20 grammes; 20 kilogrammes,'tolerance 10 grammes; 5 kilogrammes, tolerance 4 grammes; 1 kilogramme. tolerance 1 gramme. On copper or brass weights of 20 kilogrammes, tolerance 13 grammes; 5 kilogrammes, tolerance:x gramme; 2 kilogrammes, tolerance V gramme; 1 kilogramme, tolerance 13 decigrammes. FRANRFORT.-At Frankfort the prices of United-States Bonds, and of American stocks generally, are quoted in cents per dollar, payable either in United States gold coins or in guilders at the conX'ventional rate of 2j guilders to the dollar. To these quotations,: therefore, when payable in guilders, 2.674 per cent. must be added to express the true prices. -SicILY.-Although the French franc is now the official measure.of value in Sicily, yet in commercial transactions the old nomenclature and values are still chiefly used, viz: 3600 picioli = 600 grani = 30 tari = 1 oncia = 1 oncetta of Naples = 57.636 grains of fine gold = $2.48217. UNITED STATEs.-Silver dollar-pieces of the standard of 1834 to 1853, and no others, are still coined by.this government in greater or less numbers every year; but they are intended mainly for for-.~eign circulation, and bear a premium at home. They are worth, measured by the present standard for silver coins of less denominations than one dollar, $1.0742 each; and measured by the prevailing commercial relations of gold to silver as 15t to 1 for equal weights, they are worth $1.04 each in United-States gold coins of the present standard. See page 2, SEC. A. ~ CANADA.-The new silver coins of the dollar denomination, first coined by the British government in 1870 for circulation in the JDominion of Canada, weigh at the rate. of 360 grains to the dollar, and are ]{ fine., They are therefore worth 96~} cents per nominal dollar's worth, measured by the standard silver coins of the United States of less denominations than one dollar.; or they are worth -90'.93275 per nominal dollar's worth, measured by United-States gold coins. Five of the colonial silver shillings of Canada are equal in value to four quarter-dollars or two half-dollars of the new coin. See CANADA, Dominion of, page 5, SEC. A. 52 ( CUSTOM-HvUSE ALLOWANCES ON DUTIABLE GOODS. CUSTOMS TARES, OR TARES AS ALLOWED ~BY THE UNITED-STATES GOVERNMENT ON DUTIABLE GOODS. By an Act of Congress passed July 14, 1862, it is provided that when the original invoice is produced at the time of making entry thereof, and the tare shall be specified therein, it shall be lawful for the collector, if he shall see fit, with the consent of the consignees, to estimate the said tare according to such invoice; and that in all other cases the real tare shall be allowed, and may be ascertained, under such regulations as the Secretary of tile Treasury may from time to time prescribe; and that hereafter there shall be no allowance for draft. And, in accordance with these provisions, the following rates of tare were adopted:Per cent.' Per cent. Almonds, in bags.. 2 Pepper, in single bags. 2 " in bales.. 22 " in double bags. 4 in frails.. 8 Pimento, in bags.. 2 Alum, in casks.. 10 Raisins, in casks.. 12 " coarse or ground, in. " in boxes. 25 sacks, 2 lbs. per sack. " in half-boxes. 27 Barytes... 3 " in quarter-boxes 29 Cheese, in casks or tubs. 10 " in frails.. 4 Cassia, in mats.. 9 Rice, in bags.. 2 Cinnamon, in bales.. 6 Spanish brown, dry, in Chicory, in bags.. 2 casks... 10 Cocoa, in bags.. 2 Spanish brown, in oil, in " in ceroons. 8 casks.... 12 Coffee, Rio, in single bags 1 Sugar, in mats. 2 " " in double bags 2 " in bags... 2 "all others, actual tare. " in barrels.. 10 Copperas, in casks.. 10 " in tierces.. 12 Currants, in casks.. 10 " in hhds... 12A lhemp, Manilla, in bales, 4 " in boxes 14 pounds per bale. Salt, fine, in sacks, 3 lbs. Hemp, Hamburg, Leghorn, per sack. or-Trieste, in bales, 5 lbs. Tea, China or Japan, inper bale. voice tare. Indigo, in ceroons.. 10 Tea, all others, actual tare. Melado... 11 Tobacco, leaf, in bales, 10 Nails, in bags... 2 pounds per bale. " in casks.. 8 Tobacco, leaf, in bales, exOchre, dry, in casks. -8 tra covers, 12 pounds per " in oil, in casks. 12 bale. Paris white, in casks. 10 Whiting, common, in Peruvian bark, in ceroons 10 casks.,. 10 SECTION B..REDUCTIONS, EXCHANGE, INVESTMENTS, MIXED NEGOTIATIONS, &c., &c. PRoPOSITION 1. — When a dollar in gold is worth a dollar and thirty cents in currency, what is the value of a currency dollar? 1- 1.30=$0.769231, or 76 cents 9.231 mills. Ans. PROP. 2.- When gold is nominally at a premium of 353 per cent., at what discount is currency? 100 — f - = -26.335175 per cent. Ans. PROP. 3.- What is the difference between a given principal, P, and a deduction of r per cent. of it? P,Pr=P (1 -r), the present worth. Ans. PROP. 4. —What is the difference between a given principal, P, and a discount of r per cent. of it? Pr P P p --- _ 7 the present worth. Ans. 1 + r 1 — r PROP. 5. -Express the difference per dollar between deduction at r per cent., and discount at the same rate per cent. (1 - r) 1. Ans. PROP. 6.- Express the discount, d, on a given principal, P, for a given time in days, t, at a given rate, r, per cent. per annum. d=- Ptr - w, i being the interest on 1 365 — tr 1 - i dollar for the given time at the given rate, and w the. present 365P P worth for the same time and rate; whence, w 365 365 + tr 1 + i - P- d. See DIscoUtT, page 129. PROP. 7.-The foregoing proposition (Prop. 6); except that the time, T, is in years, or years and decimal parts of a year? d=PTr-+ -(1+ Tr). Ans. PROP. 8.- The foregoing proposition (Prop. 6), except that the time, m, is given in monts, or months and decimal parts of a month? d=Pmr. (12+mr). Ans. PROP. 9.- To convert time in days into calendar months and decimal parts of a month; m = 12d' 365; but the converse of this, viz., d -= 365m - 12 is not liable to be true in practice, * 1 6lb 2 b INVESTMENTS, MIXE1b NE0OTIATIONS, ETCe. since different calendar months are made up of an unlike numberl of days; and, therefore, those immediately under consideration may contain a greater or less number of days. PnoP. 10.- What is the intrinsic equivalent in Federal money of ~712 10s. 9id. sterling? 71+ 12 X d) X 4.86656=$346 7.6166. Ans. PROP. 11. - Reduce $3467.6166 to its intrinsic equivalent in sterling money. 3467.6166 —. 4.86656 = 712.53958 20 10.7916 12 9.4992 ~712 lOs. 9&d. Ans. PROP. 12. Express the equivalent in Federal money, $, of a given quantity in sterling money, ~, at a given true or direct rate of exchange, r, that is to say, at a rate of exchange based upon the intrinsic equivalent of the two denominations of money,. $ =~ X 4.86656 X r. Ans. PROP. 13.- Express the equivalent in Federal money of a given quantity in sterling money, rated by the former intrinsic par of 4s. 6d. sterling to the dollar, the given rate of exchange, f, being upon the same basis. I$ X 40 X.f_~ X 4.86656 Xf Ans. 9 1.09498 EXAMPLE. — What is the equivalent in Federal money of ~435 7s. 6d. sterling, rated by the former intrinsic par of $4A to the ~, the rate of' exchange upon the same basis being 1:10? 435.375 X 40 X 1.1025 435.375 X 4.86656 X 1.1025 9 1.09498 $2133.3375. Ans. PROP. 14.-Reduce F1172.36 (1172 francs, 36 centimes-of France) to its equivalent in Federal money? 1172.36 X 0.19452 — 1172.36 - 5.14086 -$228.05. Ans. PROP. 1 5. -Express the intrinsic or par equivalent of francs, F, or francs and decimal parts of a franc, in Federal money, —$; and vice versa. $ - F X 0.19452 =- F 5.14086. Ans. F _$'0.19452 $ X 5.14086. Ans. PROP. 16.- Express the intrinsic equivalent of marks banco, M, or marks and decilnal parts of a mark banco of Hamburg, London rate, in Federal money; and vice versa. $ =IM X 0.35393 = ~' 2.8254. Ans. Mi.- $ X 2.8254- $ 0.35393. Ans. INVESTME.NTS, MIXED NEGOTIATIONS, ETC. b3 -EXAMPLE. —-What is the equivalent in Federal money of 6472 marks, 12 schillings banco of Hamburg, London rate? 6472.75 x 0.35393=6472.75+2.8254= $2290.90. Ans. PROP. 17.-Which is the most advantageous purchase, other things being equal; viz., bills of exchange on London at 1.101, rate of 4s. 6d. sterling to the dollar; or on Paris at F5.15 to the dollar; or on Hamburg at 351 cents per mark banco? 1.105 + 1.09498 = 1.009151-, London. 5.15 ~ 5.14086 = 1.00178-, Paris. 35.625 - 35.393 1.00655+, Hamburg. Bills on Paris. Ans. NOTE. — London allows, as the intrinsic par, ~1 for F25.21 =19.304 Federal cents Per franc, instead of 19.452, the prevailing commercial par. But at the same time Great Britain values the silver franc of France, measured by the silver in her own silver coinage, at ~1 = 1=.4~~~ = 23.249+ france = 20.93234+ Federal cents per franc. Of Notes for discount, and their avails at bank. PROP. 18. - When the time of the note is expressly written in days; or when the actual number of days in the specified time are employed; that is, when the true time is taken, and taken in days, 365a P(365. — tr) p 6t'Sa anda= P(3665 - r, p being the principal or 365 — tr 365 face of the note, a the avails or sum advanced by the bank, t the time including grace, and r the rate of the interest or discount per annum. See BANK DISCOUNT, I. 127. EXAMPLE. - What must be the principal of a note payable in 90 days, in order that the equitable avails at bank, for the time of the note and 3 days grace, the rate being 6 per cent., shall be $1 000? 3;5 X 1000 3$1015.53. Ans. 365 — 93 X.06 PROP. 19. - When the time of the note is expressly written in months, and a month is taken to be the He part of a calendar year, p 1i2-a,1 anil a P[12-r(m X.1)]; m being 12-(m- 0.1) 12 the time in months, and.1 being EL of a month, or the usual 3 days grace. ExAMPLE. -$1000 are to be obtained from a bank, on a note having three months to maturity, and three days grace; the rate being 6 per cent., for what sum must the note be drawn? 12 X 100 - X 1015.74. Ans. 12 - 3.1 X.06o NOTE. -3 calendar months and 3 days, mean time, are more than 93 dayis, by 1j day. 4 INVESTMENTS, MIXED NEGOTIATIONS, Ed. PROP. 20. — WVhen the true time of the note is taken in months or days, and a year-is assumed to consist of 12 months of 30 days each, or of 360 days only, 360a a P(360 —tr) P —P and a 360 - tr 360 EXAMPLE. -What must be the principal of a note for discount, payable in 90 days after acceptance, that the proceeds at bank shall be $1000, the rate being 6 per cent., the grace 3 days, and the bank assuming that a year consists of 360 days only? 360 X1000 360 X 1000 $1015.74. Ans. 360 - 93 X.06 NOTE. - The Government of the United States, and the Courts, in matters of interest and discount, reckon times-at 366 days to the year; and in Great Britain, )Franlce,.anid all Europe, a year, for like purposes, contains 365 days. PROP. 21. -A note on tile, without interest, dated Jan. 2, 1868, is to be given in exchange.for the following obligations; the time of the note is required. Note, due March 3, 1868, for $370. Bond, " April 16, " 830.50Note, " June 11,!" 1120. Acc't," June 4, " 127.50 From Jan. 2 to March 3 is 61 days X. 370 = 22570 " " 2 to April 16 is 105 " X 831 = 87255 " " 2 to June 11 is 161 " X 1120 = 180320 " " 2 to " 4 is 154 " X 127 = 19558 2448 )309703(127 days after Jan. 2 = May 8, 1868. Ans. But this is simply equivalent to finding the common time of maturity of the obligations to be transferred, which must be the answer. EXAMPLE. - Due March 3, $370 X 0 = 0 " April 16, 831 X 44 = 36564 " June 11, 1120 X 100 = 112000 -" " 4, 127 X 93 - 11811 2448 ) 160375( 66 days after March 3, = May 8, 1868. See EQUATION OF PAYMENTS, page 132. PROP. 22.- A note dated March 1, 1869, and bearing interest at 7 per cent. from date, is to be given in exchange for the following obligations; the principal of the note is required. 1869, Feb. 16, settlement note on interest, value of, this day, $327.36 " March 27, acceptance of this date for 1000.00 May 4, account, averaging due this date, 658.73 INVESTMENTS, MIXED NE6OTIATlONS, EtC. b5 1869, Feb. 16, due $327 " M'ch. 27, " 1000 X 39- 39000 " May 4, " 659 X 77 50743 $1986 )89743( 45.2 days later than Feb. 16, = April 3, 1869, the day on, which the given obligations collectively become due, or average due; which is 32 days later or after March 1, 1869: then, since the adjustment involves discount instead of interest, 365a. 365 X 1986 Ans. P _=$1973.89-. Ans. 365+ tr 365 + 32 X.07 In which P represents the principal, or face of the new note, a the sum of the obligations to be transferred or cancelled, t the time in days of the adjusting interest or discount, and r the rate of the interest or discount. PROP. 23. — The foregoing proposition (Prop. 22), except that the new note is to be dated July 1, 1869, instead of March 1, 1869. From April 3,'69, to July 1,'69, is 59 days, the number of (lays that April 3d is earlier or previous to July lst: then, since the ad —.justment involves interest instead of discount, pa(365 + tr) 369.13 X 1986 $2008.47. Ans. 365 365 PROP. 24.- Suppose we equate the time of the following demands by the common rule, and then make up the account correctly by interest and discount to the given dates, and thence to the equated time, by way of exhibiting general principles, and the bearing of the common rule of average upon those principles. Interest and discount at 7 per cent. 1868, note due March 10 for $500 "'e " June 8 " 750 X 90 — 67500'- " " (Sept. 6 " 1050 X 180 = 189000 2300 ) 256500( 112 days later than March 1-0, = June 30,'68, the day on which the given demands ($2300) are assumed to collectively mature. $500 due March 10, worth March 10 $500. ~750 " June 8, " " 10 (90 days' disc't.) 737.28 1050 " Sept. 6, " " 10 (180" " ) 1014.96 $2252.24; worth, June 30 (112 days' interest), $2300.62. (True time 65(P -w). wr- 112 -365 (WV- P) * Pr.) $500 due March 10, worth June 8 (90 days' interest) $508.63 -750 " June 8, " 4" 8 750.00 1050 " Sept. 6, " " 8 (90 days' discount) 1032.18 1 s$2290.81; 6t flV ESTMENTS, MIXED NEOOTIATIONS, ETa. worth, June 30 (22 days' interest) $2300.48. (True time 365(P — w) uwr + 90.) $500 due March 10, worth Sept. 6 (180 days' interest) $517.26 750 " June 8, " " 6 (90 " " ) 762.95 1050 " Sept. 6, " " 6 1050.00 $2330.21; worth, June 30 (68 days' discount), $2300.21. (True time_ 180 - 365(w- P) - Pr.) $500 due March 10, worth June 30 (112 days' int.) $510.71 750 " June 8, " " 30 ( 22 " " ) 753.17 1050 " Sept. 6, " " 30 ( 68 " disc't.) 1036.48$2300.36. (True time = 112 -365(w - P) Pr= 111.184 days. NOTE. - On the assumption that simple interest is justly due and payable yearly, it is apparent that no demand should enter the account for average at a present worth more than a year due previous to the maturity of the debt latest due, PnoP. 25. -A man sold two horses for $150 apiece, one at a profit of 25 per cent., and- the other at a loss of 20 per cent.; which was the greater, the profit or the loss on the two sales, and what sum of money expresses the difference? 150 — (1 —.20) -- $187.50, cost of the horse sold at a loss, and 187.50 - 150 = $37.50 loss; 150 + (1 +.25) = 120, cost of the horse sold at a profit, and 150 -120 = $30 profit; then 37.5030 = $7.50 loss greater than the profit. Ans. Pnof. 26. -A merchant sold two packages of goods for the same sum of money each; on one of them he cleared 25 per cent., and on the other he lost 25 per cent.: did he gain or lose in the aggregate; and, if either, what per cent.? 1 * (1-.25) =1-, cost relative to the sum received as 1 of the package sold at a loss; 1 (1 +.25) = A, cost relative to the sum received as 1 of the package sold at a profit; then (1 - 1) +(1 -- ) -- - - -=) 6= per cent. loss. Ans. PRoP. 27. - A and B purchased a farm in company for $8000; A paid $5000 in part payment, and B1 paid the remainder; they then sold A of the farm for $4000; and, to close the copartnership, each took i of the remaining i to his own private interest: how mnuch, if any, of the undivided cash received for the land they disposed of attaches to B's private interest?' 1(4000 - j of 8000) = $500, B's share of the profit on the sale, and 3000 + 500 — e of 8000 = $8331. Ans. PROP. 28. - The sum, S, of two numbers, N and n, given, the difference, d, of their respective factors, C and c, to produce like INVESTMZENTS, MIXED NEIG0TIATIONS, ETC. b 7 products given, and the sum of the products, P, given, to find the numbers. -Let D =the difference of the assumed numbers, and let m P. S; then 1st step S(m +id)N iP-C -m -n S-N=-n kP +N=c N- n =D C- c-.c d', and...- d' ~d-: D d D'. 2d step, S+D_ N -P- C 2 jP-N'=c', C'- c'= d"; d"' d D' D" Sd step, S+-.,-, P' n"= C" 2 P N" =c", and C" —c"f S-NV" = n" i -d"' &c. NOTE. -By this manner of proceeding, all the elements in propositions of this Pnature may be approximated to any degree of exactness desired; and the true values will be between those obtained by the first step and those obtained by the second. The greater required number (AT)-will be less than that obtained by the first step, and greater than that obtained by the second. Often but two steps, and seldom more than three, will be required for ordinary practical purposes. -When the differenece between any trial C and its corresponding c becomes equal to' the'-given difference (d), the elements in that step will be the exact ones sought, if no error has been committed in the work. It is not necessary, however, to obtain the first trial N by the method here proposed; for any number whatever may be assumed in its stead: but, when thus obtained, it has the advantage, generally, of, -being near the true number sought, and of being known to be greater than that number. EXAuMPLE.- A certain farm containing 80 acres is worth in the aggregate $62.50 per acre; but one section of it, to the extent of haIf the gross value, is worth $11 per acre more than the other; h.ow many acres are there in the lesser section? P= 80 X 62.50 = $5000, the gross value of the farm; then (62.5 + 5.5) 4043.52 — N 2500.- 36.48 = 68.5307+ = C 62.5 80- _43.52 ="36.48_ n 2500 43.52 -57.444853-_ c.-.. - 7.04- D 11.085846 -_ d -11.085846 ~ 11 7.04 6.985484+ D — 80 + 6.985484 = 43.492742 - N' 2 80 -43.492742 =-36.507258 = rn' 2500 - 86.507258 - 68.479534 - C' 2500, 43.492742 = 57.480855 = c 10.998679 - d" = 11 nearly; the lesser section, therefore, contains 36.507258 acres, nearly. Ans. 8 I IVES~MENTS, MIXED NEGOTIATIONS, Ed. NOTE. - All propositions of this geaneral class having given relations of parts contain the requisite elements for direct solutions, and need not be worked, as they commonly are- by rules denominated Double Position. For example: "The old sea-serpent's head is 10 feet long, his tall is as long as his head and half the length of his body, and his body is as long as his bead and tail; what is the whole length of the monster?" h 10, t=h+4-b, and b=-2h.&-.b; then b - lb = 2h, therefore b -- 4h 40 t =- +. ib, therefore t = 10+ 20 -= 30 h = 10. 80 feet. Ans, PROP. 29.- Divide $1000 into four such parts that the second shall contain $10 more than the first, the third $6 more than the second, and the fourth 2. times as many as the first and second. Let x represent the smallest part; then 1000=x+x+ 10+x+ 16+5x+25, and 1000 - (10 + 16 t- 25) =949 = 8x, and 949' 8 x,= $118.625, 1st; $128.625; 2d; $134.625, 3d; $618.125, 4th. PROP. 30.- A gentleman being asked his own age and the age of his wife, replied, If you subtract 5 years from' my age, and divide the remainder by 8, the quotient will be j of my wife's age; and if you add 2 years to her age, then multiply the sum by 3, and subtract 7 from the product, the remainder will be my age. Required the age of each. Let.x = the wife's age; then 3x + 6 - 7 the husband's age; but 3x+6 7 5 _; therefore, 9x + 18 - 36 = 8 and 8 3 x = 36-18 = 18, the wife's age, and 18 X 3 6 - 7 = 53, the husband's age. Ans. PnoP. 31.- Two relations between two unknown numbers-given, to find the numbers. BY DOUBLE POSITION. Special cases. Let x = the lesser of the correct numbers sought. "N = the greater and n the lesser of the assumed values of x, or trial numbers. " D = the greater and d the lesser of the differences between the trial numbers and those obtained in their stead; then When both differences are in excess, or show that both the assumed values of x are too high, x (N- n)d. D -d When both differences are in deficiency, or show that both the as qumed values of x are too low, x (N+ n)D D+d When one of the differences is in excess and the other in deficiency, or when one of the assumed values of x is too high and the other too low, _ _ND +.nd-,- D-d D+d EXAMPLE. - Assume the wife's age (last preceding proposition) is 25, then (25 + 2) X 3 - 7 = 74, the husband's age, and (74-5) X 8 _ 25.875, instead of 25, and showing a difference 8 of.875 in excess. Again, assume the wife's age is 20, then (20 + 2)3 - 7 - 59, the husband's age, and (59 5)3-20.25, instead of 20, and showing a difference of.25 in excess; then (25 - 20) X.25 = 18, the wife's age, and.875 -.25 (18 + 2) X 3 - 7 - 53, the husband's age. PROOF. (53 — 5)3- 18 8 NOTE. -It is not necessary to assume the value of the lesser required num. ber instead of that of the greater, for the value of either may be assumed as preferred; and when the greater is assumed, the foregoing formulas are not applicable, but a set that are can be easily made. PRoP. 32. - A said to B, Give me one of your apples and I shall then have as many as you will have left. B replied, Give me one of yours, and I shall then have twice as many as you will have left. How many apples had each'? By the first proposition, A + 1 B - 1; therefore B = A + 2: but by the second proposition, B + 1 = 2(A - 1); therefore B 2A 3: then 2A A +2 +- 3, and A 2 +- 3= 5; B= A - 2 7. A?,s. PROP. 33.- The lesser and half the greater of two casks of wine =82 gallons; and the greater and i the lesser _ 129 gallons: how many gallons are in each cask? Let x = the greater and y the lesser; then x +- by = 129, and y +- x - 82: therefore 2y - 164- x, and jy - 129 -x; consequently 2y — y =164 - 129, and y = 21; also 1-29 — y x-=129 — 7-122. Ans. PROP. 34. - Smith's several cows are in number to their average cost per head as 45 to 138, and they collectively cost him $690; how many cows has he? Let n represent the number of cows, and c their average cost; then n: c: 45: 138; but n X< c}= 690, therefore 1690 X 45 n — 15. Ans. Solved, also, by PO. X, page 138 Solved, also, by PROB. X, page 170. 10 b INVESTMENTS, MIXED NEGOTIATIONS, ETC. PRoP. 35.- An apple-vender bought one-half of a certain lot of apples at the rate of 2 for a cent, and the other half at the rate of 3 for a cent,; and concluded that they collectively cost him 2 cents for 5; being willing to dispose of them at cost, he accordingly mixed them together, and sold them out 5 for 2 cents, and lost 5 cents by so doing: how many apples had he? Let x = the whole number of apples or answer; then X 5 = (Z + 5)5 50x-600 t o x ~- (~2x -I-, therefore 48 x+ 600 -50x, 2x= 600, and x_ 300. Ans. NOTE. -The sum of & of a quantity and of i of a like quantity, is more than 2 of the sum of the quantities, by f one of thewo quantities. PROP. 36. - If a body were to start suddenly into motion, and move 80 miles in the first hour, and in each succeeding hour were to move through three-fourths as much space as in the hour last preceding, and were thus to continue in motion forever, what space would it describe? 80 +4= 320 miles. Ans. NOTE. - This is simply a question in Geometrical Progression descending, in which the greater extreme is 80, the ratio A, the less extreme 0, and the sum of the terms is required. The formula, since the ratio is less than unity, becomes S_ (E-e) (l-r) +_ E_ EX1-e See GEOMETRICAL PRoGREssIoN, P. 151. r-e * r-e PnoP. 37. —What sum in ready money, D, may I pay for $10,000 in stocks, P, that are redeemable at par in T, 3 years, and are bearing interest the while at 7 per cent. a year, payable half-yearly, in order that I may realize 6 per cent. simple interest a year on the investment, supposing that the payments of the interest on the stock are to be kept invested at 7 per cent. a year, from their times of maturity till the stock matures? A =P+pT(1 + R(T1)+ R)=P[l+rn (1r(n1) 2 2 the amount of the stock at the time of its maturity; in which p represents an interest payment, r the rate of the interest per interval between the payments, and n the whole number of the interest payments; and D =A. (1+-R'T) =A (1 an), in which Rt represents the rate of the discount per annum, or rate of the interest on the investment per annum, and a the'rate of the discount per interval between the payments; therefore 10000 + 700 x 3 X (1 +._-1 +. _)= 1.18 10000 X [1+.035 X 6 X (1 + )]_ 1.18 $10,409.96. Ans. -IJNVESTMENTS, MIXED NEGOTIATIONS, ETC. b l1 PROP. 38.- The last preceding proposition, except that my ready money is worth 6 per cent. compound interest yearly? D-A -.(1 -+R)3 12283.75.- 1.191016 - $10,313.67. Ans. The W Z:See ANNUITIr S, p. 156; also, p. 125. PorP.. 39 — The preceding proposition (Prop. 37), except that the payments of the interest are to be invested at 6 per cent. a year, from the times they become due, till the stock matures? 10000 x (1+-.03 X 6(1__ -+QL) =$10,114.41. Ans. 1.18 PROP. 40.- The preceding proposition (Prop. 37), except that the interest on the stock is payable quarterly? 10000 + 700 X 3 X (1 + -. + q X.07) _ 1.18 o100o0 x (1 +-.0175 X 12 X (1 +- _7-~) 1.18 $10,427.22. Ans. PROP. 41. - The first proposition in this class (Prop. 37), except that my ready money is worth 6 per cent. interest a year, with the interest payable semi-annually? A A 12283.75 1 + s(Rl' + R')= 1 + T(r —lo) 1.225 $10,027.55. Ans. PROP. 42.- Express the difference, per dollar, between the,amount of a given algebraic principal for a -given algebraic time and rate, and the present worth of the same principal for the same:time and rate, the time being in days. 365 + tr 1365 ( tr 1.365 365 + tr r 365 1+ EXAMPLE. - What is the difference between the amount of $1,550 for 175 days at 8 per cent. a year, and the present worth of the same sum for'the same time and rate? d P(365 + tr) 365P p 365 + tr 365, 365 365 t+ tr 365 365 + tr/ $116,708. Ans. PROP. 43. - A purchased a bill of goods on six months' credit, amounting to $2,000, with the understanding that he should be:allowed 5 per cent. off for ready cash, in whole or part payment: he paid $1,000 ready cash; for what sum ought he to be credited on the bill? 1000 " (1 —.05) - $1052.63. Ans. PROP. 44. - Which is the lower offer, goods at 1.283, on 4 months' credit; or the same goods at 1.30, on 6 months' credit; allowing money to be worth 8 per cent. interest a year? 12 b INVESTMENTS, MIXED NEGOTIATIONS, ETC. I.282 __ 1.2875 X 12 - 1.254+, the present worth. I +~ AA 12.82 1.30 1.30 X 12 = 1.25, the present worth. 1 + q ~i 12.48 The 1.30 terms, slightly. Ans. Conversion of debts not yet due into others of like sums each, and having a common difference of time from maturity to maturity. S = gross sum to be converted. T -time from the present to the maturity of the gross sum. n = numberiof common substitutes. s - common sum of the substitutes. t = common difference of time from maturity to maturity of the substitutes. t'= assigned time from the present to the maturity of one of the substitutes. When the common difference of time is to apply to all the common substitutes, and is to be measured from the present, s S n, and t T - [I(n 1)]. Puor. 45. - An investment of $2,100 having 90 days to maturity is to be substituted by 3 others of like sums each, which are to become due at the close of a common difference of time from the present, and fiom one to another: the common sum and common difference of the times are required. 2100 3 = $700, the common sum of the substitutes; and 90 2 = 45 days, the common difference, or common interval. The three substitutes of $700 each, therefore, are to be made payable, the first at the expiration of 45 days, the second at the expiration of 90 days, and the third at the expiration of 135 days from the present time. Proof. 700 X 45 31500 700 X 90 — 63000 700 X 135 -- 94500 21 )189000= 90; or700oX (45+90+135)-=2100 X90. PROP. 46. Five notes are to be made for like sums each, and are to become due at the close of equal intervals of time from the present, and from one to-another; and these notes are to be given in exchange for the four following obligations; viz., $1600, due in 90 days, $1250.62 due in 80 days, $852.21 due in 57 days, and: $1865. due in 175 days, from the present time. The common de INVESTMIENTS, MIXED NEd0TIAtIONS, ETC. E t3 nomination of the notes, and the common interval of time are required. 1600.00 X 90= 144000 1250.62 X 80= 100050 852.21 X 57- 48576 1865.00 X 175 - 326375 556 7.83 ) 619001 = 111 days, the mean or average time of maturity from the present of the obligations' to be converted; and 5567.83. 5 =$1113.57, the common denomination of the notes; and 111- (5 + 1) = 37 days, the common difference of time, or common interval. The special times to maturity, therefore, of the five notes of $1113.57 each, are 37, 74, 111, 148,.185 days from the present time, When one of the common substitutes is to be treated as cash, or is to be considered as due at the present time, and the common interval is to be measured from the present, -S =S n, and t= T *[j(n- 1)}. PnoP. 47. -Several matters of indebtedness, amounting in the aggregate to $2175.44, and which will collectively mature, or become due by average, at the close of 68 days from the present time, are to be cancelled by the payment of one-fourth of their sum down, and by passing three notes, made for one-fourth of their sum each, and payable at the close of a common difference of time from the present and from one to another. The common sum and common difference are required. 2175.44 - 4 - $543.86, the common sum; and 68 -- (4 - 1) = 68.-1.5 =46 days, the common interval. The three notes, therefore, are to be made for $543.86 each, and are to be made payable, the first at 46, the second at 92, and the third at 138 days from the present time. WVhen the common interval is to apply -between all the common substitutes, and is to be measured from an assigned time for the maturity of one of them, s = S -n, and t —(T t').[(n-1)]. Pnor. 48.- A debt of $4500, due 123 days hence, without interest, is to be substituted by 4 notes, made for one-fourth of the sum each, which are to run an equal interval of time from maturity to maturity, and one of them is to be made payable at the close of an interval of 30 days from the present time. The common sum and common interval are required. 2 1i4 t INVESTMENTS, MIXED NEGOTIATIONS, Et'. 4500 4 _- $1125, the common denomination of the notes; and (123 - 30). 1.5 = 62 days, the common difference of time.'" The times to maturity of the substitutes, therefore, are 30, 92, 154, 216 days later than the present time. PRoP. 49. - The last preceding proposition (Prop. 47), except that the given debt of $4500 has but 75 days to maturity; and one of the notes is not to become due until the lapse of 105 days from the present time. (75, 105) 1.5 = 20 days, the common interval. The times from the present to the maturities of the notes, therefore, are 45, 65, 85, 105 days. Pitor. 50. - It is proposed to relinquish obligations, amounting in the aggregate to $2554.72, and which will collectively become due by equation at the close of 40 days from the present time, and to receive in their place 3 notes, made for one-fourth of the sum' each, and the balance in ready cash; the said notes to run equal intervals of time from maturity to maturity, measured from the present, and one of them to run 85 days; the adjusting interest, or discount, to be at 7 per cent. (85 -40).' — (4 -1) 30 days, the common interval. The three notes of $638.68 each, therefore, are to be made pay-. able at 85, 55, 25 days; and the present worth of the cash payment of $638.68, due 30- 25 = 5 days hence, is $638.07. To invest a given sum of money in parts, at unlike rates of interest, and the parts to gain like interest in equal intervals of time., p', p", &c. = the parts, or partial investments. S -- the sum of the investments. r, r', r", &c. = the given rates, or these in their relations to each other, expressed in any proportion preferred. m = the product of r, r', r", &c., as expressed. N= the sum of m &c. p, p', p &c. _- S m S. Sm &c., inversely. PN, pY', p, -&c Nr"m I PROP. 51. - Twenty thousand dollars ($20,000) are to be placed at interest in three such parts that the interest on them, at their respective rates of 6, 7, and 8 per cent. a year, shall be equal for all like intervals of time. The parts, or special investments, are required. m= 6 X 7 X 8=336; andl 336 6 = 56 336' — 7-48 336 ~ 8 — 42 146, the value of N; I1VESTMENTS, MIXED NEdOTIATIONS, EC. b 15 20000 X 56 20000 X 28-'P = 146 - 73 =$7,671.23 20000 X 24 73. = 6,575.34 20000 X 21 p" - - *4 —' 5,753.43 73 $20,000.00 To invest a given sum of money in parts, at like rates of interest, and for unequal intervals of tine; and the parts to gain like interest at the close of their respective times. t,', t", &c. =the given time~, or these in their relations to each other, expressed in whatever proportion preferred. m = product of t, t', t", &c., as expressed, N = sum of m m m,&c S, and p, p', p", &c., as in the last preceding proposition. &c., n Sm _ Sm &c., inversely. p) p, p", &c., = Nt Nt' Nt"' PROP. 52. —It is proposed to place $25,000 at interest in four separate sums, one of them for 60, one for 80, one for 110, and one for 150 days' time; and that these sums shall be such, that, at like rates of interest, they will gain like interest at the close of their respective times. The special sums are required. m-r GOX 80X 110X 150, or6X 8 X 11X 15, or 3 X 4 X 5.5 X 7.5 495; and 495 * 3 -- 165. 495 -4 = —123.75 495' 5.5 — 90. 495 7.5 — 66. 444.75 = N; therefore 25,000wX 165 25,000 X 33 p —-- ~ — $9,274.87 444.75 88.95 $ = 25,000 X 12375 25,000 X 495 6956.16 44475 1779,, 25,000 X 90 p = 4447 5,059.02 444.75 p 25,000 X 66 - 3,709.95 444.75 $25,000.00 16 b,. NVESTMENtS, MIXE NEGOTIATIONS, ETO. To invest a given sum of money in parts, at unlike rates of. interest, and for unequal intervals of tlime; and the parts to gain like interest at the close of their respective times. m = product of the given times, or of their relations to each other, by any measure whatever, that s8 common to them; or of the given rates, if preferred. S, N, p, p', p", &c., as in the preceding.,, o Sra S'm Sm p,', n", &c.:'- X &c. Ntr Nt'r' Nt"' PorP. 53. - Ten thousand dollars ($10 000) are to be placed a:t interest in three separate sums, one of them at 4 per cent., for 240 days; one at 6 per cent., for 120 days; and one at 8 per cent., fbr 80 days; and these sums are to be such that they will gain like interest, one with another, at the close of their respective times. The s pecal sums are required. 240 X 1IX 80, or 24 X 12 X 8, or 6 X 3 X 2 36 = tX t' X t" m. 4, 6,8, or 2, 3, 4, =r,r',r" 12, 9, 8, the products of tr, t'r', t"r", and 36' 2-=3 36 - 9-4 36' 8 -4.5 11.5 N, therefore 1p; = i0,000 X 30 10,000 X 6 $2,608.70 115 23 10,000 X 40 10,000 X 8 3478.26 115 23 10,000 X 45 10,000 X 9 3 913.04. $10,000.00. 115 23 To invest a given sum of money in parts, at- like rates of interest, and for unequal intervals of time; and the amount (principal and interest) of the pdrts to be equal at the close of their respective times. Pnor. 54. - It is proposed to place $16,000 at interest in four separatecsums, each at 7 per cent. a year: one of them for 80, one for 100, one for 150, and one for 200 days' time; and that these sums shall be such that their amount shall be equal, one with another, at the close of their respective times. The special sums are required. suo 65 + 365 35 &c. Nsuo_365 sum of + -' 365 +E"'C. 365 + it 365 f+- t' 365 + ri' INVESTMENTS, MIXED NEOOTIATIONS, ETC. b 17,365S - 365S 365S P, p', _p"; &c.; P N(365 + rt) N(365 + rt') -N(365 +- rt")' &c.,= = q- x'r' &c.;,_then 365 _ 370.6. 0.98489 365 372. 0.98118 365' 375.5 =:0.97204 365' 379. -_ 0.96306 3.90117, the value of N; and 5,840,000 p = 0.6N= $4039.36 370.6N, 6,840,000=- 4'24. 3-72N,:_ 5,840,000 = 3986.65 375.5N,, 5,840,000 3949.83. $16,000.00 379N Therefore, A = S.- N= $4101.33, the common amount. If the times, t, 1', t", &c., be taken in years instead of days, then N=1 -+ 1rt 1 + &c.; and = N(1 + rt); -- 1 rt 1 q- rt' 1 q rt' p'= S N(1 + r'), &c. And, if the times be taken in months, 12 12 12 rt 12- rt" &c.; and'p _-12S N(12 + rt); p'= 12+rt 12+rt' 12S - N(12 rt'), &c. To invest a given sum of money in parts, at unlike rates of interest, and for unequal intervals of time;. and the amount of the parts-to be equal, one with another, at the close of their respective times. PROP. 55. — The last preceding proposition, except that the rates are to be 6 per cent. for the -80 days' term, 7 per cent. for the 100 days' term, 8 per -cent, for the 150 days' term, and 9 per cent. for the 200 days'term, instead of 7 per cent, for each of the terms. 365 — 369.8 = 0.98702 365 372. — 0.98118 365 377. 0.96817 365 383. = 0.95300 3.88937, the value of N;. and p 5,840,000 369.8N $4060.38 - p' = 5,840,000 372N = 4036.37'.'p" = 5,840,000 377N 3982.83 p"' 5,840,000 383N = 3920.44. $16,000.02. 2* 18 b INVESTMENTS, MIXED NEGOTIATIONS, ET0. To solve Problems in Medial Proportion by the common rule of Simple Proportion; making use of two or more equations when there are three or more given extreme rates, of one or more awsumed, mean rates when there are three or more given extreme rates, and using the first assumed mean rate as an extreme rate in the second equation, the second assumed mean rate as an extreme rate in the third eguation, &c. Let a represent the higher, c thelower, and b the mean rate, employed in each equation; also, let n represent the sum of the proportional terms already found, or quantity taken at one oft the rates, and let x represent the required proportional term, or quantity to be taken at the other rate, in the same equation; then When the quantity taken at the higher rate in the equation is given, (a —b)n' xt b-c; or, The difference of the lower rate and mean rate, b-c is to the difference of the higher rate and mean rate, as the quantity taken at the higher rate, is to the quantity required, and to be taken at the lower rate. Conversely — When the quantity taken at thle lower rate in the equation is given. (b —c)n x t- t'; or, The difference of the higher rate and mean rate, is to the difference of the lower rate and mean rat9, as the quantity taken at the lower rate, is to the quantity required, and to be taken at the higher rate. From the foregoing it will be perceived that the initial proportional term, or first quantity, n, may be taken at any number whatever; but, commonly, it will be best to take it at 1. The assumed mean rates may be taken at any stage between their respective extremes; and, by varying them, an almost endless: number of different proportions may be obtained. Even when there are only three extreme rates in the given proposition, the assumed mean rate can commonly be so taken as to obtain the relation of the terms, one to another, that may be desired. PRor. 56.-In what proportion must two kinds of tea, one rated at 80 cents a pound, and the other at 96 cents a pound, be taken, that the mean rate, or rate of the sum of the quantities taken, shall be 85 cents a pound? Let 1 represent the initial proportional term, or quantity taken of the kind rated at 96 cents a pound; then (96-85) x 1 21, the required corresponding term, or quantity to be 85-80 taken of the kind rated at 80 cents a pound. The two kinds of tea, therefbre, must be taken in the proportion of one pound at 96 cents a pound, to 2k pounds at 80 cents a pound, thai the mean rate, or rate of the mixture, shall be at 85 cents a pound; or, they must be INVESTMENTS, MIXED NEGOTIATIONS, ETC. b 19, taken in the proportion of 1 x 5 = 5 pounds at 96 cents a pound, to 21 x = 11 pounds at 80 cents a pound, &c. Conversely.-How much tea at 96 cents a pound, must be taken with 1 pound at 80 cents a pound, that the mean rate, or rate of the mixture, shall be at 85 cents a pound? (85-80) x 1 9(8 5 -= ~-l- of a pound. Ans. The two kinds of tea, therefore, taken in the proportion of 1 pound at 80 cents a pound, to -, of a pound at 96 cents a pound, or in the proportion of 1 times 1 1 =11 pounds at 80 cents a pound, to -A: x 11 = 5 pounds at 96 9e0lvis a pound, as before, will form a mixture at the rate or plice 8.cents a pound. See ALLIGATION, page 140; also, see page 117. PROP. 57.-In what proportion may oats at 50 cents a busliel, and rye at f00 cents a bushel, be mixed with 1 bushel of corn at 80 cents a bushel,-that the mean-rate, or rate of the mixture, shall be 75 cents a bushel? Let the assumed mean rate be 60 cents a bushel; then 1 bus. of corn, mliaking 3 bus. (80 —60)x1 at 60 cents a? at 50,1 at 80, m 60-=6 0 bus. ofoats, bus., and (75 —60) x 3? at 100, 3 at 60, im75 (7 -60)x3 1- bus. of rye. 100-75 On the contrary, let the assumed mean rate be 70 cts. a bushel; then 1 bus. of corn, (80-70) x 1? at 50, 1 at 80, n 70 = (0 bus. of oats, 70-50 Ans.?at 100, 1 at 70, m 75 (5-70x bus. of 100-75. Let the assumed mean rate be 65; then 1 bus. at 80 cents, at50, at 80, m 65= (80-65) x 1 P?; atZ0, 1 at 80, m ( 65_-5) -= 1 bus. at 50 cents, Ans? at 100, 2 at 65, m 75 (75-65) = x bus. at 100 cents. 100-75 PROOF (last example). 1 at 80=80 1 at 50 - 50 0.8 at 100 - 80 2.8 ) 20( 75. PRoP. 58.-In what proportion may one quality of wine, rated at $1.00 a gallon, and another quality, rated at $1.50 a gallon, be mixed with 1 gallon of water, rated at 0 a gallon, that the mean rate, or average rate of the mixture, shall be $1.25 a gallon? Let the assumed mean rate be 25 cents a gallon; then 20 b INVESTMENTS, MIXED NEGOTIATIONS ETC. 1 gal. at 0-? at $1.00,1 at 0, m 0.25 = (2 —0)x=25 = " "$1.00 (1.25e-'25x x~ I? at $1.50, 1k at.25, m $1.25='1.2-50 —25 --- " " 50A. Let the assumed mean rate be 50 cents a gallon; then 1 gallon at 0? at 100, 1 at 0, m 50 = 00 1 1 " $1.00ns (125 —50) x 2J? at 150, 2 at 50, m 125 = ) x 2 6 $.50 Let the assumed mean rate be 70 cents a gallon; then 1 gallon at 0? 100, 1 at 0, m 70 = (70- ) 2x $10 100 —70 =2 " "$1.0 0 Ans.? 150, 83 at 70, m 125 ( 150 —125 " " $150 Let the assumed mean rate be 90 cents a gallon; then 1 gallon at 0]? at 100, 1 at 0, m 90 = ( " $10 A 100-90 =n9 (1255-90) x 10 14? at 150, 10 at 90, m 125 = 150 -25 14 $1.50 PROP. 59.-In what proportions may four grades of sugar, rated at 8, 10, 14, and 17 cents a pound, respectively, be taken, that the mean rate shall be 12 cents a pound? Let the assumed mean rates be 9 and 11; then 1 at 10? at 8, lat 10, m9-9 1 at 8 9-8? at 14,2 at 9, Il l1 = ) 1W at 14 A 1 (12 —11)x 3j?at 17, 3 ati 11 2 - i 2 =- ] at 17 17-12 J Let the assumed mean rates be 9- and 11; thenl 1 at 10 = 3 at 10? at 8,-1 at 10, min 91 = 9 - W at 8 = 1 at 8? at 14, 1k at 91, m 11 (= 9)1- at 14 aa t 14 Ari.? at 17 at, ul = (at 17 12- 11)t 17 17-12 a INVEST1MENTS), MIXED NEGOTIATIONS, ETC. b 21 Let the assumed mean rates be 9 and 101; then 1 at 8 e 7 at 8 (9-8) x 1? at 10,1 at 8, m 9 1 -9 = 1 at 10 = 7 at 10 (101-9))x 2 Ans.? at 14, 2 at 9, m 10 - 0 = ( at 14 = 6 at 14? at17, 21 at 0g,= 1 _ —2 -- 2 at 17 = 6 at 17 17-12 NoTE.-All the assumed mean rates, it will be perceived, must be taken short of the given, or final, mean rate. To find the Proportional Coefficients of the dividend, or the numerical relation of each part to the whole, in Partitive Proportion, when the Proportional terms, or terms of the ratio of the parts, one to another, are given. R-ULE I.-Divide each of the given proportional terms by the sum of those terms, or by that sum divided by any common measure, or the greatest common measure, of both its parts; the quotients thus obtained, or these reduced to lower or their lowest terms, will be the coefficients demanded; for they will be to one another as their respective proportional terms, which are to one another as the parts of the dividend required; and, collectively taken, they will be equal to a unit, or the whole. RULE II.-Reduce the given proportional terms to integers, by reducing them to a common denominator, and take the sum of the integers, or numerators of the fractions thus obtained, for the denominator of each integer; the fractions thus formed, or these divided by any common measure of their respective parts, will be the proportional coefficients of the dividend, or coefficients required; for the denominator of any proportional coefficient in partitive proportion, is to its numerator, as the dividend to be divided into parts, is to the part corresponding to that numerator. It is, therefore, evident that when the proportional terms are given in integers, the integers are to be taken as the numerators, and their sum as the denominator of each integer, to express the numerical relation of each part to the whole, in partitive proportion. PROP. 60. —It is required to divide a certain, quantity into four parts which shall be to one another as i, i, ~, i; that is, the first part to the second as i to ~, the second to the third as + to i, and the third to the fourth as X to *; the proportional coefficients of the quantity, or numerical relations of the proposed parts to the whole quantity, are demanded. By Rule I. — ~+~+X+ — I = i —-; and — t =~, + A= -- Set, -,-AS= —, and i. — Id q-, the coefficients required; or, I, 7Qo 911, 114,- aI,, + = 7, u% 19, 9A, quantities of the same 6114 95, 57 7, 7 19 relative value, one to another, as the preceding; and, therefore, the proportional ooefficients required, if preferred. 22 b INVESTMENTS, MIXED NEIGOTIATIONS, ETC. By Rule II.-', 1, r, A, reduced to firactions having a common denominator, become ~36-, ~.-9a,* -7-, g;, 4 aiad by removing the denominators, we obtain the integers 120, 90, 72, 60, quantities bearing the same relation,'one to another, as the quantities from which -they were derived; and, therefore, since the sum of these integers is equal'to 342, we have 4,.c, a W-, 314' - = - -, *+, i +,,~-s, ~, the numerical relations of the proposed parts to the whole, or the proportional coefficients of the dividend required.. Thus, allowing the proportional coefficients to be f.-l,&-, -4-, it and the dividend to be $120; the required parts, in the denomina120 x 20 120 x 5 tion of the dividend, will be - $42- - _ = $431, 19 =$25, 67 = 1$21-. 120 x 4 1.20 x 10 Proof.-42+31 +25+21+~L = 120. See FELLOWSHIP, page 138. PnoP. 61.-A; B, and C entered into copartnership in a nmanufacturing business with a gross capital of $31,000, of which A furnished $12,500, B $10,500, and C $8,000; their collective gain in the business is $32,800, which is to be divided between the partners in proportion to their respective investments, or share in the gross capital; what is each partner's share of the gain? The given proportional terms, 12500, 10500, 8000, taken in lower corresponding terms.= 125, 105, 80= 25, 21, 16, the sum of "which last set = 62; then it, ]j, t =, -i, -3a, are the proportional coefficients of the dividend for the parts required; therefore, - 32,800 x 25~62 = $13,225.81, A's share, 32,800 x 21~. 62 = $ 11,109.68, B's share, Ans. 32,800 x 16~62 = $ 8,464.51, C's share. PRoP. 62.-Two travelers start from two points 80 miles apart, fnd- travel- uniformly toward each other, one at the rate of 3A miles an hour, and the other at the rate of 4J miles an hour; how many miles will each travel till they meet, supposing the one making 3A miles an hour starts 3 hours earlier than the other? 80-3 x 3-69.5, and 3A,4~ = i - 1, 1-4z, coef.= 4,then 81 69.5 x 14~+-31+3 x 31 = 41.8871 A 69.5 x 17~31 =38.1129 PROP. 63.-Two travelers, each under like circumstances making 4 miles an hour, start at the same time from two points 30 miles apart, and travel toward each other, but one, having a down grade, makes 4 mile more per hour in consequence, while the other, having an up grade, makes A mile less per hour in consequence; how'many miles will each have traveled when they meet? 4+~, 4-A = 4i, 3A, and 41+3~ = 8; then 30 x 4A-8 = 16 } An 30x3+ 8 =13-. IN'VESTIENTS, MIXED NtEGOTIATIONS, ETC. b 23 When the required parts are given in relation as greater or less, one than another, by a given part of one of them, to find the proportional terms, &c. RULE.-Let 1 represent the first or initial proportional term in all cases, and let 1 plus the given part of that term greater, or 1 minus the given part of that term less, as the case may be, represent the next proportional term, and so on for all the proportional terms involved in the proposition; then proceed for the proportional coefficients of the dividend, &c., by Rule I or II, as in the foregoing. Paor. 64. —How shall I divide a certain number of apples between two boys, so that one of them shall receive as many as the other and 4 of as many more? that is, so that the difference of their shares shall be equal to i of the smaller share? 7..9 7..9 1,1,+= -; coef. - —. Give one of them -& of the number, 1,14=; ~7+9 and the other -l of it. Ans. Suppose the number to be divided is 20. Then give one of them 20 x 7+16 = 81 apples, and the.other 20 x 9+16 = 114 apples. Proof.-_-g-: 84:: -: 114, 84+I of 84 = 114, and 84+114 =20. PROP. 65.-A certain quantity of wheat is to be divided into two parts, such that the difference of the parts shall be equal to i of the greater part; the proportional relations of the parts to the whole are required. 1_, 1 — = -5; coef.=, 4. Ans. PROP. 66.-Divide $1,572 between two persons, so that one shall have -9 as much as, the other. I-, 1- - ~-~ =coef., then 11 20 $1,572 x 11 +20= $864.60, and $1,572 x 9~+20 = $707.40. PRoP. 67.-It is required to divide $5,000 into three parts, such that the second shall be greater than the first by i of the first, and the third less than the first by i of the first; what are the parts in the denomination of the dividend? 12..14..9 1,1+, 1 — = 4, i,4; coef. — -: 1st = $.,714-; 2d = $2,000; 3d = $1,2854. Ans. PROP. 68.-D's interest in the ship IIunter is to be transferred to three parties, A, B, and C, in such proportion, one to another, that B's share shall be less than A's by 4 of A's, and C's greater than B's by - of B's. The proportional coefficients of the dividend for the respective shares are required. 24 b INV:ESTMRINTS, MIXED NEGOtIATiONS, ETC. Let 1 represent the proportional term of A's share; then A = 1, B = 1- i, C = (1 —)+ —; reduced = A, -; coef. = 20..15..18 53 Let I represent the proportional term of 13's share; then 15..18..20 B=1, C=1+J,A-=1+-I=-, ~,; coef.= 1820, as before. 53 Let 1 represent the proportional term of C's share; then, since by the stipulations C (1) is greater than B by i of B, 6 B = 5 and B --; also, since by the stipulations B (i) is less than A by i of A, A —A = 9, and A g = 9; then18..15..20 C = 1, B = t, A =.,, a; coef. = 53, as before. 53 When thte required parts are given in relation as greater or less, one than another, by a given independent quantity of their kind, to find their algebraic relation, one to another, and theirvalues in the denomination of the dividend. RULE.-Let x represent the first or initial algebraic part in all cases, and let x plus the given number greater than that part, or x minus the given number less than that part, as the case may be, represent the next algebraic part, and so on for all the parts involved in the proposition; then find the value of x in the denomination of the dividend for the first part, and to that part add, or fitrom it subtract, as the case may demand, the given difference between that part and the next for the second part, and so on for all the parts required. PROP. 69.-It is required to divide $5000 into three parts, such that the second shall be greater than the first by $600, and the third less than the first by $400. The algebraic relations of the parts, and the parts in the denomination of the dividend, are demanded. A, x+600,x-400, the algebraic relations of the parts, in their order, that, collectively taken, are equal to the dividend; then x+x+600+x —400 = 3x+200 = $5000, and 3x = 5000-200 = $4800, therefore - = 4800-.-3 = 1600, 1st part. 1600+600 = 2200, 2d part. 1600-400 = 1200, 3d part. PROP. 70.-How shall 50 acres of land be parcelled between A, B, and C, so that B's share shall contain 6 acres more than As, and C's 14 acres less than B's? The sum of, x+6, x+ 6-14 = 3x-2 = 50, therefore 3x = 52, and x = 17~ acres, A's share, ) 17i+6 = 23k acres, B's share, - Ans. 23J -14 = 9k acres, C's share. INVESTMENTSj MIXED NEGOTIATIONS, ETC. b 2~ When of the three parts that make up the whole, one of them and the difference of the other two are given, to find the remaining parts, or coefficients of the dividend. Let 1 represent the sum of the parts, A, B, and C the parts, and. d the given difference; then 1-A = B+ C, and (B+ C+d)+2 - B; B-d-= C. PROP. 71.-A, B and C own a ship; A owns -27- of it, and B owns 1 of it more than 0. What shares of it do B and C own respectively? 1 —- = Wt, and (a1~+-#4) 2 g = = B's share, ancd;k —4-e -= = C's share. Ans. When the quantities of two or more articles of unlike rates of value per nit of measure are separately given, ant the relations of the rates as greater or less, one than another, by a stipulated independent rate of the same kind are given, and the sum of the products of the quantities by their rates is given, to find the rates and the values of the quantities. Let A, B, C, &c., represent tle given quantities: Let b represent the given rate that B is greater or less-thlan A;1 Let c represent the given rate that G is greater or less than A or B, &c.; Let D represent the given dividend or sum of the products of the quantities by their rates; Let r represent the rate per unit of measure of lot A, lot B, lot C, &C. When two quantities only are involved in the proposition, and A is the lesser, D-Bb D+Ab r of lot A - A+ andrcl of lot B -A+ therefore (D —Bb)A ( -+B- = value of lot A, or A's share of the dividend; and A+B (D+Ab+B - value of lot B, or B's share of the dividend. Also, If we let x represent the rate of lot A or the lower of the two D-bB rates, x(A+B)+bB = D, x(A+ B) = D-b.B, and x = D —bB+(A~+B)b D+Ab A+B A+B' PROP. 72. —Two lots of cheese, one of three hundred pounds (A's lot), and the other of 400 pounds (B's lot), were sold together by consent of parties for $126, and it was agreed between the parties: that in dividing thle imoney B's cheese,should be reckoned wortlh 26 b INVESTMENTS, MIXED NEGOTIATIO-S, ETC. 2 cents a pound more than A's. How much of the money received for the cheese belonged to A, and how much to B? 126 - 400x.02 118 826300+400 =-700 = 161 cents a pound, the rate of A's cheese, and.164 x 300 = $50, A's share of the money. 126 + 300 x.02 182 7(00 x2= 70 —-0 = 18~ cents a pound, the rate of B's cheese, and.185 x 400 = $751, B's share of the money. PROP. 73.-A lumber dealer sells 8000 feet of boards consigned by A, and 4500 feet consigned by B, together for $496, and wishes to enter the sales to the credit of the two accounts, calling B's lot wvorthi $2i per M less than A's.' At what prices per M must he enter the two lots? A0+cB) —Ab = 496-8 x 2- = $475, and x = 475-t12.5 = $38 per M, B's lot. Ans. 38+21 or (475+12.5 x'2D)+12.5 = $40.625 per M, A's lot..TVen nwre than two quantities are involved in the proposition, and x is taken to represent the rate of one of them. x(A ~B+ U(,&e.) ~ Bb f Cc ~ Dd, &ec. = D. Pror. 74.-Three lots of wool, A's of 380 pounds, B's of 450 pounds, worth 41 cents a pound more than A's, and C's of 620 pounds, worth 2 cents a potind less than B's, were sold together itr $920.25. What was A's, what B's, and what C's share of the -honey, predicating the division upon the given conditions? D = x(A+B+ aC)+Bb+ a(b-c); x = [D-Bb- C(b-c)]. (A+B)+ C); then 920.25-37.75 884.50 380+450+620 = 1450 = 61 cents, the rate per pound of A's wool; therefore.61 x,380 $231.80, A's share, (.61+.045) x 450 = 294.75, B's share, Ans. (.61+.045-.02) x 620 = 393.70, C's share. The individual relations of a plurality of agents to a proposed end given, to find their combined relation to the same end, when time is an element in the calculations. GENERAL RULE.- -Reduce the given relations to simple fractions (if they are not already so reduced) without changing their values, and invert them; then take the denominator of their sum for the numerator of their combined relation, and take the numerator of their sum for the denominator of their combined relation. PROP. 75. —A can do a given piece of work in 5 days, and B can do it in 7. In what time can both together do it? Since A can do it in 5 days he can do I of it in 1 day, and since B can do it in 7 days he can do I of it in 1 day; both together. INVESTMENTS, MIXED NEGOTIATIONS, ETC. b 27 therefore, can do i++ -- ~ of it in 1 day, and together they can do the whole in jt = 21. days. Ans. PROP. 76.-A reservoir has two receiving pipes and one discharging pipe; by one of the pipes it can be filled in 108 hours, by' the other in 168 honrs, and by the discharging pipe it can be emptied, when full, in 84 hours. In what time can it be filled when the three pipes are acting together, supposing the velocity of the water through the pipes to be uniform? rTAS iT -l5 = r —sQ%4-, and 14-20-% = 302.4 hours. Ans. PnoP. 77.-A can do a given piece of work in 12 days, B can do 3 times as much in 27 days, and C can do 4 times as much in 35 days. In what time can they do the proposed work, working at it together? A2+-A-+- A-5' =N2W6, and 1.%oa S8.239+ days. Ans. Or, since A can do -al ofit in 1 day, B i of it in 1 day, and C of it in 1 1 2911 day, they can collectively do i-t'-+-+ - =9 of it in 1 day, and 945 they can do the whole in 8.239 + days, as before. PROP. 78.-If A can accomplish a given task in 24 days, B in 3~ days, and C in 5 days; in what time can they together accomplish it? 2;, 3~, 5 i-,'4, -, inverted =-, -As, 1, and + —.-+j-~ =, the relation that one day's labor of all three bears to the whole task, and ]2] inverted --- fit — 1 s days, the time required by all to accomplish it. Ans. PROP. 79.-What number diminished by the difference between $ and * of itself will equal 128? Let x represent the number; then 4x, l1a 7x 8 a 128 x 15 -5 3 -.- X-l- = 128, and x - 8 = 240. Ans. PROP. 80.-There are two numbers whose difference is 117A, and one of them is I less than the other, what are the numbers? Let ~ represent the greater, then x —7 will represent the less, and -- =117f; therefore m-411~, and 57 41 293aa Ans. PROP. 81.-A merchant lost ] of the money he invested in trade, and then gained $1250, when he had $3680, How much did he lose? f28 b INVESTMENTS, MIXED NEGOTIATIONS,, ETC. Let x represent the sum invested; then 3 x 5x 2430x 8 x — +1250 = $3680, = 3680-1250 = 2430, and x = = $3888, therefore 3888 x 3+8 = $1458 loss. Ans. PROP. 82.-A merchant owning 4 of a ship sold i of his share for $12000. How much at that rate is the whole ship worth? Since I of q (~s) of the ship is worth $12000, it of it must be worth It = 2k times more than $12000- $12000 x 35 + 15 = $28000. Ans. PROP. 83.-If A can do a certain job of work in 16k days, what part of it at that rate can he do in 14k days? Since he can do - of it in 1 day, he can do 14- of it in 143 days = t of it in 14a4 days. Ans. PROP. 84.-A certain quantity is to be divided into three parts such that the 1st shall be 16+, the 2d equal to ~ of the whole, and the 3d as much as the other two. What are the parts? Let x represent the sum of the parts; then, by the stipulations, 2x ~4~2*4x+165 2-+161 I 10 65, therefore 5x- 4xz+165, and x= 165, -~-+ 16~z — ~x = 1then 1st part - 16) 2d part = 165 x 2. 5 =66 Ans. 3d part = 16t+-66 = 8. ) PROP. 85.-A certain estate is to be divided into three parts as follows, viz: the 1st is to be equal to ~- of it, thIe 2d equal to, of the remainder, and the difference between tile 1st ancd 2d is to be $450. What are the parts? Let x represent the whole estate; then 4 \ 3x 16X 16z 8x 3x 4.)0 x2 45 - J= -,and 3 — = 450 - therefore 7,' 35 7 245' 7 -- $15750, then Ist part- 15750 x 3 —7- 1750,) 2d part = (15705-6750)x 4 +5- 7200, An?.. 3d part = (750750(6750- L7200) - 1800., Pnor. 86. —What is the least number of pounds of tea that will fill canisters holding either 2, 3, 4 or 5 pounds? The number required must be equal to thle least common multiple of the given capacities of the vessels; then 1... 2..5 60., Ans. INNESTMAENTSMt- X1i) N FGoT iATt() NoS., hice. b PROrP S7.-'What. is the least number of gallons of liquor that will fill bottles conitaiiinig either A, 7, i or -IA- of a gallon? 2 2..4..83..8 2 1. 2..3..4 1....8..2= 24. Ans. PROP. 88.-How many bottles of either size, viz: A, -, a or -~, gallons capacity will be required to hold 24 gallons? 24 x =36,24 x = 42, 24 x =32,24 x L=39. Ans. PRno. 89.-Froim a cask containing 45 gallons of 96 per cent. alcohol, how much must be drawn out and replaced with water, to make 45 gallons of 90 per cent. alcohol? (96-90) x 45~-+96 = 2}{ gallons. Ans. Proof:-(45-2+ ) x 96+45 = 90. See page 117. PROP. 90.-Three persons start together from the same point in the circumference of a circle and travel around it in the same direction; one makes I of a revolution in a day, another -la, and the third ft; in how many days will they be together again at the point of starting? RuLE.- -Reduce the given relations to a common denominator, and divide that denominator by the greatest common measure of the numerators; or reduce the given relations to their lowest common numerator and divide the least common multiple of the denominators by that numerator; or reduce the given relations to their lowest terms, and divide the least common multiple of the denominators by the greatest common measure of the numerators. 3375..1800..1440 EXAMPLE'-a''.-'' ~ t = 4 and 4 7, 4500 45 3375..1800..1440 4500 75 40 32 = 45 -100 days. Ans. Or...7- 4 — 4., and 4 32..60..75 4~-16-~ 39 —t'~o.1~ 15 8..15..75 8.. 1.. 5 - 2400.24 = 100 days. Ans. Or]..a. -- =...._,and 5 4..5..25 44..1. 5-= 100 = 100 days. 13..2..8= 1 Ans. PROP. 91. —Two travelers start from two points which are 76, miles apart, and travel toward each other till they meet, when one has traveled 11a miles more than the other. How many miles has each traveled? Let x represent the shorter distance; then;r+x+11a = 761, 2x = 761-111 = 641, and x = 32 — An x+111 = 44} A n 30.& INVESTME'rs, MIXED NEGOTIATIONS, ETC. PROP. 92.-There are two numbers whose sum is 76-1, and whose difference is 11*. What are the numbers? Let x represent the greater number;, then x+x-11 = 76, 2x=88, x=44 -- A. 44 —11 = 32X i See PROBLEMS, page 169. PROP. 93.-A man traveled in a straight line 9 hours, consecutively, from home at the rate of 8 miles ah hour, and toward home (on his return) at the rate of 3 miles an hour. How far from home did he proceed? GENERAL RUIE.-Divide the product of the different speeds per hour by their sum,-and multiply the quotient by the whole time consumed. Or, let a represent half the number of miles traveled, or the number of miles traveled outward; then = time consumed in traveling outward, and - = time consumed in traveling back; therefore xa x ~+_ = whole time consumed = 9 hours, and 11x = 216, x = 216+11-= 19-, miles. Ans. PROP. 94. —A starts on a Journey and travels at the rate of 4 miles an hour. With what speed must B travel, who starts to overtake A 3 hours later, that he may overtake him in 5 hours? s(t+t') vt+vt' 4 x 3 -=12 f. 5' 61 miles per hour. Ans. PROP. 95.-A horse started suddenly at a speed of 16 miles an hour and travelled at a uniformly diminishing speed for 3 hours, when his speed had become at the rate of 7 miles an hour. How many miles did he travel? RULE.-Multiply the square root of the product of the given extremes of speed into the given time, for the answer; then n = t /(v') = 3 x 4/(16 x 7) = 31.749 miles. Ans. PROP. 96.-What are the eight special weights with which, upon a pair of grocer's scales, any number of ounces from 1 to 3280, inclusive, may be weighed? 1, 3, 9, 27, 81, 243, 729, 2187 oz. Ans. PROP. 97.-What sum in ready money may I pay for a mortgage of $3000, drawing interest at 7 per cent., and payable in sums of $1000 and interest at the close of cach year from the present time, till paid, in order that I mnay realize 10 per cent. interest on the investment? INVESTMENTS, MIXED NEGOTIATIONS, ETC. b 31 [(p + Pr) x (1 + )(n-) +p (P-p)r x (1 + t)(n-) &c.] ( — a)" - - present worth, allowing compound discount, and conmpound interest on the payments, then Amount (principal and interest) three years hence of 1st payment = (1000+3000 x.07) x (1.07)2 $1385.33 2d paynient = (1000+2000 x.07) x 1.07 = 1219.80 3d payment = (1000+ 1000 x.07) x 1 1070.00 (1.10)3 = 1.831) 3675.13 = $2761.18. Ans. (p+Pr) x [1 +r(sn-l)] +[ p+(P-p)r] x [1~+-(n-2)] &c.+(l+rn) = present worth, allowing simple discount, and simple interest on the payments; then Amount (principal and interest) three years hence of 1st payment = (1000+3000 x.07) x 1.14 $1379.40 2d payment = (1000+2000 x.07) x 1.7 1219.80 3d payment = (1000+ 1000 x.07) x 1 _ 1070.00 1+3a = 1.30) 3669.20 = $2822.46. Ans. Pnop. 98. - Ganson & Co. are paying $12,000 a year in monthly payments in advance for the use of the premises they occupy; and they wish to know how much less the rent would be per annum, were it fixed at the same price by the year, but payable quarterly, at the expiration of each quarter-year; allowing money to be worth 7 per cent. interest a year. P - Pr(n + 1) 12p ~ pr(n + 1) - $12,455.00, the amount, 2n 2 by the present manner of paying, at the close of the year; and P + Prn _ 12p{ -pr(12-n) _ $12,315.00, the amount, 2(n+1) 2by the proposed manner of paying, at the close of the year, in both of which cases n represents the number of payments to be made per annum; then 12455 - 12315 = $140.00. Ans. Pnor. 99.-A and B took a job in company of digging a ditch of a certain width and depth and 100 rods in length for $100. They agreed between themselves to dig separately from the opposite ends of the ditch toward each other, and that each sliould do one-half of the work andc receive one-half of the pay, and that, because of the difference in the soil, A's allowance per rod for digging should be to that of B's as 75 cents to $1.25. How many rods must each dig? - $50-0.75-= 66} 50 —.1.25= 40 106k 32 b INVESTMENTS, MIXED NEGOTIATIONS, ETC. 106i: 100:: 66]: 62& rods, to be dug by A Ans. 1061: 100:: 40: 37t4 rods, to be dug by B A 62k: 66:: 75:i 80 cents per rod, A's price for digging. 37j: 40 1.25:: $1t per rod, B's price for digging. CONFIRMATION. 6.:5 x 0.80 = $50 37.5x1* = 50 100 rods $100.75:.80:; 1.25: 1.33k. PROP. 100.-A and B own a stock of goods worth $10,000, and one-third of A's interest in the stock is equal to two-fifths of B's. What is the interest of each? k..* = -_, then 10,000 x 6+11 = $5454A-l, A's share An. 10,000 x 5~+11 = $4545-Pr, B's share f Or, let x = A's share, then 10,000 - will represent B's-share, (10,000-x)2 x 20,000-2x x and by the conditions' 5 =; therefore 3 and 60,000-6x = 5x, then 11x = 60,000, x = $5454-&, and 10,000 _-5454-l _ $4545-,A-.