APPENDIX XII.RULES FOR THE APPLICATION OFARITHMETIC TO GEOMETRY.

The student should make himself familiar with the most common terms of geometry, after which the following rules will present no difficulty. In them all, it must be understood, that when we talk of multiplying one line by another, we mean the repetition of one line as often as there are units of a given kind, as feet or inches, in another. In any other sense, it is absurd to talk of multiplying a quantity by another quantity. All quantities of the same kind should be represented in numbers of the same unit; thus, all the lines should be either feet and decimals of a foot, or inches and decimals of an inch, &c. And in whatever unit a length is represented, a surface is expressed in the corresponding square units, and a solid in the corresponding cubic units. This being understood, the rules apply to all sorts of units.

To find the area of a rectangle.Multiply together the units intwo sides which meet, or multiply together two sides which meet; the product is the number of square units in the area. Thus, if 6 feet and 5 feet be the sides, the area is 6 × 5, or 30 square feet. Similarly, the area of a square of 6 feet long is 6 × 6, or 36 square feet (234).

To find the area of a parallelogram.Multiply one side by the perpendicular distance between it and the opposite side; the product is the area required in square units.

To find the area of a trapezium.[77]Multiply either of the two sides which are not parallel by the perpendicular let fall upon it from the middle point of the other.

To find the area of a triangle.Multiply any side by the perpendicular let fall upon it from the opposite vertex, and take half the product. Or, halve the sum of the three sides, subtract the three sides severally from this half sum, multiply the four results together, and find the square root of the product. The result is the number of square units in the area; and twice this, divided by either side, is the perpendicular distance of that side from its opposite vertex.

To find the radius of the internal circle which touches the three sides of a triangle.Divide the area, found in the last paragraph, by half the sum of the sides.

Given the two sides of a right-angled triangle, to find the hypothenuse.Add the squares of the sides, and extract the square root of the sum.

Given the hypothenuse and one of the sides, to find the other side.Multiply the sum of the given lines by their difference, and extract the square root of the product.

To find the circumference of a circle from its radius, very nearly.Multiply twice the radius, or the diameter, by 3·1415927, taking as many decimal places as may be thought necessary. For a rough computation, multiply by 22 and divide by 7. For a very exact computation, in which decimals shall be avoided, multiply by 355 and divide by 113. See (131), last example.

To find the arc of a circular sector, very nearly, knowing theradius and the angle.Turn the angle into seconds,[78]multiply by the radius, and divide the product by 206265. The result will be the number of units in the arc.

To find the area of a circle from its radius, very nearly.Multiply the square of the radius by 3·1415927.

To find the area of a sector, very nearly, knowing the radius and the angle.Turn the angle into seconds, multiply by the square of the radius, and divide by 206265 × 2, or 412530.

To find the solid content of a rectangular parallelopiped.Multiply together three sides which meet: the result is the number of cubic units required. If the figure be not rectangular, multiply the area of one of its planes by the perpendicular distance between it and its opposite plane.

To find the solid content of a pyramid.Multiply the area of the base by the perpendicular let fall from the vertex upon the base, and divide by 3.

To find the solid content of a prism.Multiply the area of the base by the perpendicular distance between the opposite bases.

To find the surface of a sphere.Multiply 4 times the square of the radius by 3·1415927.

To find the solid content of a sphere.Multiply the cube of the radius by 3·1415927 ×⁴/₃, or 4·18879.

To find the surface of a right cone.Take half the product of the circumference of the base and slanting side.To find the solid content, take one-third of the product of the base and the altitude.

To find the surface of a right cylinder.Multiply the circumference of the base by the altitude.To find the solid content, multiply the area of the base by the altitude.

The weight of a body may be found, when its solid content is known, if the weight of one cubic inch or foot of the body be known. But it isusual to form tables, not of the weights of a cubic unit of different bodies, but of the proportion which these weights bear to some one amongst them. The one chosen is usually distilled water, and the proportion just mentioned is called thespecific gravity. Thus, the specific gravity of gold is 19·362, or a cubic foot of gold is 19·362 times as heavy as a cubic foot of distilled water. Suppose now the weight of a sphere of gold is required, whose radius is 4 inches. The content of this sphere is 4 × 4 × 4 × 4·1888, or 268·0832 cubic inches; and since, by (217), each cubic inch of water weighs 252·458 grains, each cubic inch of gold weighs 252·458 × 19·362, or 4888·091 grains; so that 268·0832 cubic inches of gold weigh 268·0832 × 4888·091 grains, or 227½ pounds troy nearly. Tables of specific gravities may be found in most works of chemistry and practical mechanics.

The cubic foot of water is 908·8488 troy ounces, 75·7374 troy pounds, 997·1369691 averdupois ounces, and 62·3210606 averdupois pounds. For all rough purposes it will do to consider the cubic foot of water as being 1000 common ounces, which reduces tables of specific gravities to common terms in an obvious way. Thus, when we read of a substance which has the specific gravity 4·1172, we may take it that a cubic foot of the substance weighs 4117 ounces. For greater correctness, diminish this result by 3 parts out of a thousand.

THE END.

WALTON AND MABERLY’S

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NATURAL PHILOSOPHY, ASTRONOMY, Etc.Lardner’s Museum of Science and Art.Complete in 12 Single Volumes, 18s., ornamental boards; or 6 Double Ones, £1 1s., cl. lettered.⁂Also, handsomely half-bound morocco, 6 volumes, £1 11s. 6d.Contents:—The Planets; are they inhabited Worlds?Weather Prognostics. Popular Fallacies in Questionsof Physical Science. Latitudes and Longitudes. LunarInfluences. Meteoric Stones and Shooting Stars. RailwayAccidents. Light. Common Things.—Air. Locomotionin the United States. Cometary Influences. CommonThings.—Water. The Potter’s Art. Common Things.—Fire.Locomotion and Transport, their Influence and Progress.The Moon. Common Things.—The Earth. The ElectricTelegraph. Terrestrial Heat. The Sun. Earthquakes andVolcanoes. Barometer, Safety Lamp, and Whitworth’sMicrometric Apparatus. Steam. The Steam Engine. TheEye. The Atmosphere. Time. Common Things.—Pumps.Common Things.—Spectacles—The Kaleidoscope. Clocksand Watches. Microscopic Drawing and Engraving. TheLocomotive. Thermometer. New Planets.—Leverrier andAdams’s Planet. Magnitude and Minuteness. CommonThings.—The Almanack. Optical Images. How to Observethe Heavens. Common Things.—The Looking Glass. StellarUniverse. The Tides. Colour. Common Things.—Man.Magnifying Glasses. Instinct and Intelligence. The SolarMicroscope. The Camera Lucida. The Magic Lantern. TheCamera Obscura. The Microscope. The White Ants; theirManners and Habits. The Surface of the Earth, or FirstNotions of Geography. Science and Poetry. The Bee. SteamNavigation. Electro-Motive Power. Thunder, Lightning,and the Aurora Borealis. The Printing Press. The Crustof the Earth. Comets. The Stereoscope. The Pre-AdamiteEarth. Eclipses. Sound.Lardner’s Animal Physics, orthe Body and its Functionsfamiliarly Explained. 520 Illustrations. 1 vol., small 8vo. 12s. 6d. cloth.Lardner’s Animal Physiology for Schools(chiefly taken from the “Animal Physics”). 190 Illustrations. 12mo. 3s. 6d. cloth.Lardner’s Hand-Book of Mechanics. 357 Illustrations. 1 vol., small 8vo., 5s.Lardner’s Hand-Book of Hydrostatics, Pneumatics, and Heat.292 Illustrations. 1 vol., small 8vo., 5s.Lardner’s Hand-Book of Optics.290 Illustrations. 1 vol., small 8vo., 5s.Lardner’s Hand-Book of Electricity, Magnetism, and Acoustics.395 Illustrations. 1 vol., small 8vo. 5s.Lardner’s Hand-Book of Astronomy and Meteorology, forming a companion work to the “Hand-Book of Natural Philosophy.” 37 Plates, and upwards of 200 Illustrations on Wood. 2 vols., each 5s., cloth lettered.Lardner’s Natural Philosophy for Schools.328 Illustrations. 1 vol., large 12mo., 3s. 6d. cloth.Lardner’s Chemistry for Schools.170 Illustrations. 1 vol., large 12mo. 3s. 6d. cloth.

NATURAL PHILOSOPHY, ASTRONOMY, Etc.

Lardner’s Museum of Science and Art.Complete in 12 Single Volumes, 18s., ornamental boards; or 6 Double Ones, £1 1s., cl. lettered.

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Lardner’s Animal Physics, orthe Body and its Functionsfamiliarly Explained. 520 Illustrations. 1 vol., small 8vo. 12s. 6d. cloth.

Lardner’s Animal Physiology for Schools(chiefly taken from the “Animal Physics”). 190 Illustrations. 12mo. 3s. 6d. cloth.

Lardner’s Hand-Book of Mechanics. 357 Illustrations. 1 vol., small 8vo., 5s.

Lardner’s Hand-Book of Hydrostatics, Pneumatics, and Heat.292 Illustrations. 1 vol., small 8vo., 5s.

Lardner’s Hand-Book of Optics.290 Illustrations. 1 vol., small 8vo., 5s.

Lardner’s Hand-Book of Electricity, Magnetism, and Acoustics.395 Illustrations. 1 vol., small 8vo. 5s.

Lardner’s Hand-Book of Astronomy and Meteorology, forming a companion work to the “Hand-Book of Natural Philosophy.” 37 Plates, and upwards of 200 Illustrations on Wood. 2 vols., each 5s., cloth lettered.

Lardner’s Natural Philosophy for Schools.328 Illustrations. 1 vol., large 12mo., 3s. 6d. cloth.

Lardner’s Chemistry for Schools.170 Illustrations. 1 vol., large 12mo. 3s. 6d. cloth.

Pictorial Illustrations of Science and Art.Large Printed Sheets,each containing from 50 to 100 Engraved Figures.Part I. 1s. 6d.1. Mechanic Powers.2. Machinery.3. Watch and Clock Work.Part II. 1s. 6d.4. Elements of Machinery.5. Motion and Force.6. Steam Engine.Part III. 1s. 6d.7. Hydrostatics.8. Hydraulics.9. Pneumatics.

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Lardner’s Popular Geology. (From “The Museum of Science and Art.”) 201 Illustrations. 2s. 6d.Lardner’s Common Things Explained. Containing:Air—Earth—Fire—Water—Time—The Almanack—Clocks and Watches—Spectacles—Colour—Kaleidoscope— Pumps—Man—The Eye—The Printing Press—The Potter’s Art—Locomotion and Transport—The Surface of the Earth, or First Notions of Geography. (From “The Museum of Science and Art.”) With 233 Illustrations. Complete, 5s., cloth lettered.⁂Sold also in Two Series, 2s. 6d.each.Lardner’s Popular Physics. Containing:Magnitude and Minuteness—Atmosphere—Thunder and Lightning—Terrestrial Heat—Meteoric Stones—Popular Fallacies— Weather Prognostics—Thermometer—Barometer—Safety Lamp—Whitworth’s Micrometric Apparatus—Electro-Motive Power—Sound—Magic Lantern—Camera Obscura—Camera Lucida—Looking Glass—Stereoscope—Science and Poetry. (From “The Museum of Science and Art.”) With 85 Illustrations. 2s. 6d. cloth lettered.Lardner’s Popular Astronomy. Containing:How to Observe the Heavens—Latitudes and Longitudes —TheEarth—The Sun—The Moon—The Planets: are they Inhabited?—The New Planets—Leverrier and Adams’s Planet—The Tides—Lunar Influences—and the Stellar Universe—Light—Comets—Cometary Influences—Eclipses—Terrestrial Rotation—Lunar Rotation—Astronomical Instruments. (From “The Museum of Science and Art.”) 182 Illustrations. Complete, 4s. 6d. cloth lettered.⁂Sold also in Two Series, 2s. 6d.and2s.each.Lardner on the Microscope.(From “The Museum of Science and Art.”) 1 vol. 147 Engravings. 2s.Lardner on the Bee and White Ants; their Manners and Habits; with Illustrations of Animal Instinct and Intelligence. (From “The Museum of Science and Art.”) 1 vol. 135 Illustrations. 2s., cloth lettered.Lardner on Steam and its Uses;including the Steam Engine and Locomotive, and Steam Navigation. (From “The Museum of Science and Art.”) 1 vol., with 89 Illustrations. 2s.Lardner on the Electric Telegraph, Popularised.With 100 Illustrations. (From “The Museum of Science and Art.”) 12mo., 250 pages. 2s., cloth lettered.⁂The following Works from “Lardner’s Museum of Science and Art,” may also be had arranged as described, handsomely half bound morocco, cloth sides.Common Things. Two series in one vol.7s. 6d.Popular Astronomy. Two series in one vol.7s. 0d.Electric Telegraph, with Steam and its Uses. In one vol.7s. 0d.Microscope and Popular Physics. In one vol.7s. 0d.Popular Geology, and Bee and White Ants. In one vol.7s. 6d.Lardner on the Steam Engine, Steam Navigation, Roads, and Railways.Explained and Illustrated. Eighth Edition. With numerous Illustrations. 1 vol. large 12mo. 8s. 6d.A Guide to the Stars for every Night in the Year. In Eight Planispheres. With an Introduction. 8vo. 5s., cloth.Minasi’s Mechanical Diagrams. For the Use of Lecturers and Schools. 15 Sheets of Diagrams, coloured, 15s., illustrating the following subjects: 1 and 2. Composition of Forces.—3. Equilibrium.—4 and 5. Levers.—6. Steelyard, Brady Balance, and Danish Balance.—7. Wheel and Axle.—8. Inclined Plane.—9, 10, 11. Pulleys.—12. Hunter’s Screw.—13 and 14. Toothed Wheels.—15. Combination of the Mechanical Powers.

Lardner’s Popular Geology. (From “The Museum of Science and Art.”) 201 Illustrations. 2s. 6d.

Lardner’s Common Things Explained. Containing:

Air—Earth—Fire—Water—Time—The Almanack—Clocks and Watches—Spectacles—Colour—Kaleidoscope— Pumps—Man—The Eye—The Printing Press—The Potter’s Art—Locomotion and Transport—The Surface of the Earth, or First Notions of Geography. (From “The Museum of Science and Art.”) With 233 Illustrations. Complete, 5s., cloth lettered.

⁂Sold also in Two Series, 2s. 6d.each.

Lardner’s Popular Physics. Containing:

Magnitude and Minuteness—Atmosphere—Thunder and Lightning—Terrestrial Heat—Meteoric Stones—Popular Fallacies— Weather Prognostics—Thermometer—Barometer—Safety Lamp—Whitworth’s Micrometric Apparatus—Electro-Motive Power—Sound—Magic Lantern—Camera Obscura—Camera Lucida—Looking Glass—Stereoscope—Science and Poetry. (From “The Museum of Science and Art.”) With 85 Illustrations. 2s. 6d. cloth lettered.

Lardner’s Popular Astronomy. Containing:

How to Observe the Heavens—Latitudes and Longitudes —TheEarth—The Sun—The Moon—The Planets: are they Inhabited?—The New Planets—Leverrier and Adams’s Planet—The Tides—Lunar Influences—and the Stellar Universe—Light—Comets—Cometary Influences—Eclipses—Terrestrial Rotation—Lunar Rotation—Astronomical Instruments. (From “The Museum of Science and Art.”) 182 Illustrations. Complete, 4s. 6d. cloth lettered.

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Lardner on the Microscope.(From “The Museum of Science and Art.”) 1 vol. 147 Engravings. 2s.

Lardner on the Bee and White Ants; their Manners and Habits; with Illustrations of Animal Instinct and Intelligence. (From “The Museum of Science and Art.”) 1 vol. 135 Illustrations. 2s., cloth lettered.

Lardner on Steam and its Uses;including the Steam Engine and Locomotive, and Steam Navigation. (From “The Museum of Science and Art.”) 1 vol., with 89 Illustrations. 2s.

Lardner on the Electric Telegraph, Popularised.With 100 Illustrations. (From “The Museum of Science and Art.”) 12mo., 250 pages. 2s., cloth lettered.

⁂The following Works from “Lardner’s Museum of Science and Art,” may also be had arranged as described, handsomely half bound morocco, cloth sides.

Lardner on the Steam Engine, Steam Navigation, Roads, and Railways.Explained and Illustrated. Eighth Edition. With numerous Illustrations. 1 vol. large 12mo. 8s. 6d.

A Guide to the Stars for every Night in the Year. In Eight Planispheres. With an Introduction. 8vo. 5s., cloth.

Minasi’s Mechanical Diagrams. For the Use of Lecturers and Schools. 15 Sheets of Diagrams, coloured, 15s., illustrating the following subjects: 1 and 2. Composition of Forces.—3. Equilibrium.—4 and 5. Levers.—6. Steelyard, Brady Balance, and Danish Balance.—7. Wheel and Axle.—8. Inclined Plane.—9, 10, 11. Pulleys.—12. Hunter’s Screw.—13 and 14. Toothed Wheels.—15. Combination of the Mechanical Powers.

LOGIC.De Morgan’s Formal Logic; or,The Calculus of Inference,Necessary and Probable. 8vo. 6s. 6d.Neil’s Art of Reasoning:a Popular Exposition of the Principles of Logic, Inductive and Deductive; with an Introductory Outline of the History of Logic, and an Appendix on recent Logical Developments, with Notes. Crown 8vo. 4s. 6d., cloth.

LOGIC.

De Morgan’s Formal Logic; or,The Calculus of Inference,Necessary and Probable. 8vo. 6s. 6d.

Neil’s Art of Reasoning:a Popular Exposition of the Principles of Logic, Inductive and Deductive; with an Introductory Outline of the History of Logic, and an Appendix on recent Logical Developments, with Notes. Crown 8vo. 4s. 6d., cloth.

ENGLISH COMPOSITION.Neil’s Elements of Rhetoric; a Manual of the Laws of Taste, including the Theory and Practice of Composition. Crown 8vo. 4s. 6d., cl.

ENGLISH COMPOSITION.

Neil’s Elements of Rhetoric; a Manual of the Laws of Taste, including the Theory and Practice of Composition. Crown 8vo. 4s. 6d., cl.

DRAWING.Lineal Drawing Copies for the earliest Instruction.Comprising upwards of 200 subjects on 24 sheets, mounted on 12 pieces of thick pasteboard, in a Portfolio. By the Author of “Drawing for Young Children.” 5s. 6d.Easy Drawing Copies for Elementary Instruction.Simple Outlines without Perspective. 67 subjects, in a Portfolio. By the Author of “Drawing for Young Children.” 6s. 6d.Sold also in Two Sets.Set I.Twenty-six Subjects mounted on thick pasteboard, in a Portfolio. 3s. 6d.Set II.Forty-one Subjects mounted on thick pasteboard, in a Portfolio. 3s. 6d.The copies are sufficiently large and bold to be drawn from by forty or fifty children at the same time.

DRAWING.

Lineal Drawing Copies for the earliest Instruction.Comprising upwards of 200 subjects on 24 sheets, mounted on 12 pieces of thick pasteboard, in a Portfolio. By the Author of “Drawing for Young Children.” 5s. 6d.

Easy Drawing Copies for Elementary Instruction.Simple Outlines without Perspective. 67 subjects, in a Portfolio. By the Author of “Drawing for Young Children.” 6s. 6d.

Sold also in Two Sets.

Set I.Twenty-six Subjects mounted on thick pasteboard, in a Portfolio. 3s. 6d.

Set II.Forty-one Subjects mounted on thick pasteboard, in a Portfolio. 3s. 6d.

The copies are sufficiently large and bold to be drawn from by forty or fifty children at the same time.

SINGING.A Musical Gift from an Old Friend, containing Twenty-four New Songs for the Young. ByW. E Hickson, author of the Moral Songs of “The Singing Master.” 8vo. 2s. 6d.The Singing Master.Containing First Lessons in Singing, and the Notation of Music; Rudiments of the Science of Harmony; The First Class Tune Book; The Second Class Tune Book; and the Hymn Tune Book. Sixth Edition. 8vo. 6s., cloth lettered.Sold also in Five Parts, any of which may be had separately.I.—First Lessons in Singing and the Notation of Music.Containing Nineteen Lessons in the Notation and Art of Reading Music, as adapted for the Instruction of Children, and especially for Class Teaching, with Sixteen Vocal Exercises, arranged as simple two-part harmonies. 8vo. 1s., sewed.II.—Rudiments of the Science of HarmonyorThorough Bass.Containing a general view of the principles of Musical Composition, the Nature of Chords and Discords, mode of applying them, and an Explanation of Musical Terms connected with this branch of Science. 8vo. 1s., sewed.III.—The First Class Tune Book. A Selection of Thirty Single and Pleasing Airs, arranged with suitable words for young children. 8vo. 1s., sewed.IV.—The Second Class Tune Book.A Selection of Vocal Music adapted for youth of different ages, and arranged (with suitable words) as two or three-part harmonies. 8vo, 1s. 6d.V.—The Hymn Tune Book.A Selection of Seventy popular Hymn and Psalm Tunes, arranged with a view of facilitating the progress of Children learning to sing in parts. 8vo. 1s. 6d.⁂ The Vocal Exercises, Moral Songs, and Hymns, with the Music, may also be had, printed on Cards, price Twopence each Card, or Twenty-five for Three Shillings.

SINGING.

A Musical Gift from an Old Friend, containing Twenty-four New Songs for the Young. ByW. E Hickson, author of the Moral Songs of “The Singing Master.” 8vo. 2s. 6d.

The Singing Master.Containing First Lessons in Singing, and the Notation of Music; Rudiments of the Science of Harmony; The First Class Tune Book; The Second Class Tune Book; and the Hymn Tune Book. Sixth Edition. 8vo. 6s., cloth lettered.Sold also in Five Parts, any of which may be had separately.

I.—First Lessons in Singing and the Notation of Music.Containing Nineteen Lessons in the Notation and Art of Reading Music, as adapted for the Instruction of Children, and especially for Class Teaching, with Sixteen Vocal Exercises, arranged as simple two-part harmonies. 8vo. 1s., sewed.

II.—Rudiments of the Science of HarmonyorThorough Bass.Containing a general view of the principles of Musical Composition, the Nature of Chords and Discords, mode of applying them, and an Explanation of Musical Terms connected with this branch of Science. 8vo. 1s., sewed.

III.—The First Class Tune Book. A Selection of Thirty Single and Pleasing Airs, arranged with suitable words for young children. 8vo. 1s., sewed.

IV.—The Second Class Tune Book.A Selection of Vocal Music adapted for youth of different ages, and arranged (with suitable words) as two or three-part harmonies. 8vo, 1s. 6d.

V.—The Hymn Tune Book.A Selection of Seventy popular Hymn and Psalm Tunes, arranged with a view of facilitating the progress of Children learning to sing in parts. 8vo. 1s. 6d.

⁂ The Vocal Exercises, Moral Songs, and Hymns, with the Music, may also be had, printed on Cards, price Twopence each Card, or Twenty-five for Three Shillings.

CHEMISTRY.Gregory’s Hand-Book of Chemistry.For the use of Students. ByWilliam Gregory, M.D., late Professor of Chemistry in the University of Edinburgh. Fourth Edition, revised and enlarged. Illustrated by Engravings on Wood. Complete in One Volume. Large 12mo. 18s. cloth.⁂The Work may also be had in two Volumes, as under.Inorganic Chemistry.Fourth Edition, revised and enlarged. 6s. 6d. cloth.Organic Chemistry.Fourth Edition, very carefully revised, and greatly enlarged. 12s., cloth.(Sold separately.)Chemistry for Schools.ByDr. Lardner.190 Illustrations. Large 12mo. 3s. 6d. cloth.Liebig’s Familiar Letters on Chemistry,in its Relations to Physiology, Dietetics, Agriculture, Commerce, and Political Economy.Fourth Edition, revised and enlarged, with additional Letters. Edited byDr. Blyth. Small 8vo. 7s. 6d. cloth.Liebig’s Letters on Modern Agriculture.Small 8vo. 6s.Liebig’s Principles of Agricultural Chemistry;with Special Reference to the late Researches made in England.Small 8vo. 3s. 6d., cloth.Liebig’s Chemistry in its Applications to Agriculture and Physiology.Fourth Edition, revised. 8vo. 6s. 6d., cloth.Liebig’s Animal Chemistry; or,Chemistry in its Application to Physiology and Pathology.Third Edition. Part I. (the first half of the work). 8vo. 6s. 6d., cloth.Liebig’s Hand-Book of Organic Analysis;containing a detailed Account of the various Methods used in determining the Elementary Composition of Organic Substances. Illustrated by 85 Woodcuts. 12mo. 5s., cloth.Bunsen’s Gasometry; comprising the Leading Physical and Chemical Properties of Gases, together with the Methods of Gas Analysis. Fifty-nine Illustrations. 8vo. 8s. 6d., cloth.Wöhler’s Hand-Book of Inorganic Analysis;One Hundred and Twenty-two Examples, illustrating the most important processes for determining the Elementary composition of Mineral substances. Edited byDr. A. W. Hofmann, Professor in the Royal College of Chemistry. Large 12mo.Parnell on Dyeing and Calico Printing. (Reprinted from Parnell’s “Applied Chemistry in Manufactures, Arts, and Domestic Economy, 1844.”) With Illustrations. 8vo. 7s., cloth.

CHEMISTRY.

Gregory’s Hand-Book of Chemistry.For the use of Students. ByWilliam Gregory, M.D., late Professor of Chemistry in the University of Edinburgh. Fourth Edition, revised and enlarged. Illustrated by Engravings on Wood. Complete in One Volume. Large 12mo. 18s. cloth.

⁂The Work may also be had in two Volumes, as under.

Inorganic Chemistry.Fourth Edition, revised and enlarged. 6s. 6d. cloth.

Organic Chemistry.Fourth Edition, very carefully revised, and greatly enlarged. 12s., cloth.(Sold separately.)

Chemistry for Schools.ByDr. Lardner.190 Illustrations. Large 12mo. 3s. 6d. cloth.

Liebig’s Familiar Letters on Chemistry,in its Relations to Physiology, Dietetics, Agriculture, Commerce, and Political Economy.Fourth Edition, revised and enlarged, with additional Letters. Edited byDr. Blyth. Small 8vo. 7s. 6d. cloth.

Liebig’s Letters on Modern Agriculture.Small 8vo. 6s.

Liebig’s Principles of Agricultural Chemistry;with Special Reference to the late Researches made in England.Small 8vo. 3s. 6d., cloth.

Liebig’s Chemistry in its Applications to Agriculture and Physiology.Fourth Edition, revised. 8vo. 6s. 6d., cloth.

Liebig’s Animal Chemistry; or,Chemistry in its Application to Physiology and Pathology.Third Edition. Part I. (the first half of the work). 8vo. 6s. 6d., cloth.

Liebig’s Hand-Book of Organic Analysis;containing a detailed Account of the various Methods used in determining the Elementary Composition of Organic Substances. Illustrated by 85 Woodcuts. 12mo. 5s., cloth.

Bunsen’s Gasometry; comprising the Leading Physical and Chemical Properties of Gases, together with the Methods of Gas Analysis. Fifty-nine Illustrations. 8vo. 8s. 6d., cloth.

Wöhler’s Hand-Book of Inorganic Analysis;One Hundred and Twenty-two Examples, illustrating the most important processes for determining the Elementary composition of Mineral substances. Edited byDr. A. W. Hofmann, Professor in the Royal College of Chemistry. Large 12mo.

Parnell on Dyeing and Calico Printing. (Reprinted from Parnell’s “Applied Chemistry in Manufactures, Arts, and Domestic Economy, 1844.”) With Illustrations. 8vo. 7s., cloth.

GENERAL LITERATURE.De Morgans Book of Almanacs.With an Index of Reference by which the Almanac may be found for every Year, whether in Old Style or New, from any Epoch, Ancient or Modern, up toa.d.2000. With means of finding the Day of New or Full Moon, fromb.c.2000 toa.d.2000. 5s., cloth lettered.Guesses at Truth. By Two Brothers. New Edition. With an Index. Complete 1 vol. Small 8vo. Handsomely bound in cloth with red edges. 10s. 6d.Rudall’s Memoir of the Rev. James Crabb; late of Southampton. With Portrait. Large 12mo., 6s., cloth.Herschell (R. H.) The Jews;a brief Sketch of their Present State and Future Expectations. Fcap. 8vo. 1s. 6d., cloth.

GENERAL LITERATURE.

De Morgans Book of Almanacs.With an Index of Reference by which the Almanac may be found for every Year, whether in Old Style or New, from any Epoch, Ancient or Modern, up toa.d.2000. With means of finding the Day of New or Full Moon, fromb.c.2000 toa.d.2000. 5s., cloth lettered.

Guesses at Truth. By Two Brothers. New Edition. With an Index. Complete 1 vol. Small 8vo. Handsomely bound in cloth with red edges. 10s. 6d.

Rudall’s Memoir of the Rev. James Crabb; late of Southampton. With Portrait. Large 12mo., 6s., cloth.

Herschell (R. H.) The Jews;a brief Sketch of their Present State and Future Expectations. Fcap. 8vo. 1s. 6d., cloth.

Footnotes:[1]Some separate copies of these Appendixes are printed, for those who may desire to add them to the former editions.[2]It has been supposed thatelevenandtwelveare derived from the Saxon forone leftandtwo left(meaning, after ten is removed); but there seems better reason to think thatlevenis a word meaning ten, and connected withdecem.[3]The references are to the preceding articles.[4]Any little computations which occur in the rest of this section may be made on the fingers, or with counters.[5]This should be (23)a×a, but the sign × is unnecessary here. It is used with numbers, as in 2 × 7, to prevent confounding this, which is 14, with 27.[6]In this and all other processes, the student is strongly recommended to look at and follow thefirst Appendix.[7]Those numbers which have been altered are put in italics.[8]As it is usual to learn the product of numbers up to 12 times 12, I have extended the table thus far. In my opinion, all pupils who shew a tolerable capacity should slowly commit the products to memory as far as 20 times 20, in the course of their progress through this work.[9]To speak always in the same way, instead of saying that 6 does not contain 13, I say that it contains it 0 times and 6 over, which is merely saying that 6 is 6 more than nothing.[10]If you have any doubt as to this expression, recollect that it means “contains more than two eighteens, but not so much as three.”[11]Among the even figures we include 0.[12]Including both ciphers and others.[13]For shortness, I abbreviate the wordsgreatest common measureinto their initial letters, g. c. m.[14]Numbers which contain an exact number of units, such as 5, 7, 100, &c., are calledwhole numbersorintegers, when we wish to distinguish them from fractions.[15]A factor of a number is a number which divides it without remainder: thus, 4, 6, 8, are factors of 24, and 6 × 4, 8 × 3, 2 × 2 × 2 × 3, are several ways of decomposing 24 into factors.[16]The method of solving this and the following question may be shewn thus: If the number of days in which each could reap the field is given, the part which each could do in a day by himself can be found, and thence the part which all could do together; this being known, the number of days which it would take all to do the whole can be found.[17]A formula is a name given to any algebraical expression which is commonly used.[18]Or remove ciphers from the divisor; or make up the number of ciphers partly by removing from the divisor and annexing to the dividend, if there be not a sufficient number in the divisor.[19]These are not quite correct, but sufficiently so for every practical purpose.[20]The 1′ here means that the 1 is in the multiplier.[21]This is written 7 instead of 6, because the figure which is abandoned in the dividend is 9 (151).[22]Meaning, of course, a really fractional number, such as ⅞ or ¹⁵/₁₁, not one which, though fractional in form, is whole in reality, such as ¹⁰/₅ or ²⁷/₃.[23]By square number I mean, a number which has a square root. Thus, 25 is a square number, but 26 is not.[24]The term ‘root’ is frequently used as an abbreviation of square root.[25]Or, more simply, add the second figure of the root to the first divisor.[26]This is a very incorrect name, since the term ‘arithmetical’ applies equally to every notion in this book. It is necessary, however, that the pupil should use words in the sense in which they will be used in his succeeding studies.[27]The same remark may be made here as was made in the note on the term ‘arithmetical proportion,’ page 101. The word ‘geometrical’ is, generally speaking, dropped, except when we wish to distinguish between this kind of proportion and that which has been called arithmetical.[28]A theorem is a general mathematical fact: thus, that every number is divisible by four when its last two figures are divisible by four, is a theorem; that in every proportion the product of the extremes is equal to the product of the means, is another.[29]Ifbxbe substituted forain any expression which is homogeneous with respect toaandb, the pupil may easily see thatbmust occur in every term as often as there are units in the degree of the expression: thus,aa+abbecomesbxbx+bxborbb(xx+x);aaa+bbbbecomesbxbxbx+bbborbbb(xxx+ 1); and so on.[30]The difference between this problem and the last is left to the ingenuity of the pupil.[31]It is not true, that if we choose any quantity as a unit,anyother quantity of the same kind can be exactly represented either by a certain number of units, or of parts of a unit. To understand how this is proved, the pupil would require more knowledge than he can be supposed to have; but we can shew him that, for any thing he knows to the contrary, there may be quantities which are neither units nor parts of the unit. Take a mathematical line of one foot in length, divide it into ten parts, each of those parts into ten parts, and so on continually. If a point A be taken at hazard in the line, it does not appear self-evident that if the decimal division be continued ever so far, one of the points of division must at last fall exactly on A: neither would the same appear necessarily true if the division were made into sevenths, or elevenths, or in any other way. There may then possibly be a part of a foot which is no exact numerical fraction whatever of the foot; and this, in a higher branch of mathematics, is found to be the case times without number. What is meant in the words on which this note is written, is, that any part of a foot can be represented as nearly as we please by a numerical fraction of it; and this is sufficient for practical purposes.[32]Since this was first written, the accident has happened. Thestandard yardwas so injured as to be rendered useless by the fire at the Houses of Parliament.[33]The minute and second are often marked thus, 1′, 1″: but this notation is now almost entirely appropriated to the minute and second ofangularmeasure.[34]The measures in italics are those which it is most necessary that the student should learn by heart.[35]The lengths of the pendulums which will vibrate in one second are slightly different in different latitudes. Greenwich is chosen as the station of the Royal Observatory. We may add, that much doubt is now entertained as to the system of standards derived from nature being capable of that extreme accuracy which was once attributed to it.[36]The inch is said to have been originally obtained by putting together three grains of barley.[37]‘Capacity’ is a term which cannot be better explained than by its use. When one measure holds more than another, it is said to be more capacious, or to have a greater capacity.[38]This measure, and those which follow, are used for dry goods only.[39]Since the publication of the third edition, theheapedmeasure, which was part of the new system, has been abolished. The following paragraph from the third edition will serve for reference to it:“The other imperial measure is applied to goods which it is customary to sell byheaped measure, and is as follows:2 gallons1 peck4 pecks1 bushel3 bushels1 sack12 sacks1 chaldron.The gallon and bushel in this measure hold the same when only just filled, as in the last. The bushel, however, heaped up as directed by the act of parliament, is a little more than one-fourth greater than before.”[40]Pure water, cleared from foreign substances by distillation, at a temperature of 62° Fahr.[41]It is more common to divide the ounce into four quarters than into sixteen drams.[42]The English pound is generally called apound sterling, which distinguishes it from the weight called a pound, and also from foreign coins.[43]The coin called a guinea is now no longer in use, but the name is still given, from custom, to 21 shillings. The pound, which was not a coin, but a note promising to pay 20 shillings to the bearer, is also disused for the present, and the sovereign supplies its place; but the name pound is still given to 20 shillings.[44]Farthings are never written but as parts of a penny. Thus, three farthings being ¾ of a penny, is written ¾, or ¾. One halfpenny may be written either as 2/4 or ½; the latter is most common.[45]When a decimal follows a whole number, the decimal is always of the same unit as the whole number. Thus, 5ᔆ·5 is fivesecondsand five-tenths of asecond. Thus, 0ᔆ·5 means five-tenths of a second; 0ʰ·3, three-tenths of an hour.[46]Before reading this article and the next, articles (29) and (42) should be read again carefully.[47]Any fraction of a unit, whose numerator is unity, is generally called analiquot partof that unit. Thus, 2s.and 10s.are both aliquot parts of a pound, being £⅒ and £½.[48]A parallelepiped, or more properly, arectangularparallelepiped, is a figure of the form of a brick; its sides, however, may be of any length; thus, the figure of a plank has the same name. A cube is a parallelepiped with equal sides, such as is a die.[49]This generally comes in the same member of the sentence. In some cases the ingenuity of the student must be employed in detecting it. The reasoning of (238) is the best guide. The following may be very often applied. If it be evident that the answer must be less than the given quantity of its kind, multiply that given quantity by the less of the other two; if greater, by the greater. Thus, in the first question, 156 yards must cost more than 22; multiply, therefore, by 156.[50]It is usual to place points, in the manner here shewn, between the quantities. Those who have read Section VIII. will see that the Rule of Three is no more than the process for finding the fourth term of a proportion from the other three.[51]Commission is what is allowed by one merchant to another for buying or selling goods for him, and is usually a per-centage on the whole sum employed. Brokerage is an allowance similar to commission, under a different name, principally used in the buying and selling of stock in the funds.Insurance is a per-centage paid to those who engage to make good to the payers any loss they may sustain by accidents from fire, or storms, according to the agreement, up to a certain amount which is named, and is a per-centage upon this amount. Tare, tret, and cloff, are allowances made in selling goods by wholesale, for the weight of the boxes or barrels which contain them, waste, &c.; and are usually either the price of a certain number of pounds of the goods for each box or barrel, or a certain allowance on each cwt.[52]Here the 4s.from the dividend is taken in.[53]Here the 3d.from the dividend is taken in.[54]Sufficient tables for all common purposes are contained in the article on Interest in the Penny Cyclopædia; and ample ones in the Treatise on Annuities and Reversions, in the Library of Useful Knowledge.[55]This rule is obsolete in business. When a bill, for instance, of £100 having a year to run, isdiscounted(as people now say) at 5 per cent, this means that 5 per cent of £100, or £5, is struck off.[56]This question does not at first appear to fall under the rule. A little thought will serve to shew that what probably will be the first idea of the proper method of solution is erroneous.[57]The teacher will find further remarks on this subject in theCompanion to the Almanacfor 1844, and in theSupplement to the Penny Cyclopædia, articleComputation.[58]And at discretion one hundredth more for a large fraction of three inches.[59]The student should remember all the multiples of 4 up to 4 × 25, or 100.[60]The treatises on book-keeping have described this difference in as peculiar a manner. They call these accounts thefictitious accounts. Now they represent the merchant himself; their credits are gain to the business, their debits losses or liabilities. If the terms real and fictitious are to be used at all, they are therealaccounts, end all the others are asfictitiousas the clerks whom we have supposed to keep them.[61]This theorem shews that what iscalledreducing a fraction to its lowest terms (namely, dividing numerator and denominator by their greatest common measure), is correctly so called.[62]For that which measures a measure is itself a measure; so that if a measure ofacould have a measure in common withb,aitself would have a common measure withb.[63]A prime number is one which is prime to all numbers except its own multiples, or has no divisors except 1 and itself.[64]Expand (a-1)ᵇ by the binomial theorem; shew thatwhen b is a prime numberevery coefficient which is not unity is divisible byb; and the proposition follows.[65]The principle of this mode of demonstration of Horner’s method was stated in Young’s Algebra (1823), being the earliest elementary work in which that method was given.[66]Various exceptions may arise when an equation has two nearly equal roots. But I do not here introduce algebraical difficulties; and a student might give himself a hundred examples, taken at hazard, without much chance of lighting upon one which gives any difficulty.[67]This form might be also applied to the integer portions; but it is hardly needed in such instances as usually occur. See the articleInvolution and Evolutionin theSupplementto thePenny Cyclopædia.[68]After the second step, the trial will rarely fail to give the true figure.[69]The solution ofx³ + 0x² + 0x-2 = 0.[70]Taken from a paper on the subject, by Mr. Peter Gray, in theMechanics’ Magazine.[71]Taken from a paper on the subject, by Mr. Peter Gray, in theMechanics’ Magazine.[72]Taken from a paper on the subject, by Mr. Peter Gray, in theMechanics’ Magazine.[73]Taken from the late Mr. Peter Nicholson’s Essay on Involution and Evolution.[74]Taken from the late Mr. Peter Nicholson’s Essay on Involution and Evolution.[75]Taken from the late Mr. Peter Nicholson’s Essay on Involution and Evolution.[76]Taken from the late Mr. Peter Nicholson’s Essay on Involution and Evolution.[77]A four-sided figure, which has two sides parallel, and two sides not parallel.[78]The right angle is divided into 90 equal parts calleddegrees, each degree into 60 equal parts calledminutes, and each minute into 60 equal parts calledseconds. Thus, 2° 15′ 40″ means 2 degrees, 15 minutes, and 40 seconds.

Footnotes:

[1]Some separate copies of these Appendixes are printed, for those who may desire to add them to the former editions.

[2]It has been supposed thatelevenandtwelveare derived from the Saxon forone leftandtwo left(meaning, after ten is removed); but there seems better reason to think thatlevenis a word meaning ten, and connected withdecem.

[3]The references are to the preceding articles.

[4]Any little computations which occur in the rest of this section may be made on the fingers, or with counters.

[5]This should be (23)a×a, but the sign × is unnecessary here. It is used with numbers, as in 2 × 7, to prevent confounding this, which is 14, with 27.

[6]In this and all other processes, the student is strongly recommended to look at and follow thefirst Appendix.

[7]Those numbers which have been altered are put in italics.

[8]As it is usual to learn the product of numbers up to 12 times 12, I have extended the table thus far. In my opinion, all pupils who shew a tolerable capacity should slowly commit the products to memory as far as 20 times 20, in the course of their progress through this work.

[9]To speak always in the same way, instead of saying that 6 does not contain 13, I say that it contains it 0 times and 6 over, which is merely saying that 6 is 6 more than nothing.

[10]If you have any doubt as to this expression, recollect that it means “contains more than two eighteens, but not so much as three.”

[11]Among the even figures we include 0.

[12]Including both ciphers and others.

[13]For shortness, I abbreviate the wordsgreatest common measureinto their initial letters, g. c. m.

[14]Numbers which contain an exact number of units, such as 5, 7, 100, &c., are calledwhole numbersorintegers, when we wish to distinguish them from fractions.

[15]A factor of a number is a number which divides it without remainder: thus, 4, 6, 8, are factors of 24, and 6 × 4, 8 × 3, 2 × 2 × 2 × 3, are several ways of decomposing 24 into factors.

[16]The method of solving this and the following question may be shewn thus: If the number of days in which each could reap the field is given, the part which each could do in a day by himself can be found, and thence the part which all could do together; this being known, the number of days which it would take all to do the whole can be found.

[17]A formula is a name given to any algebraical expression which is commonly used.

[18]Or remove ciphers from the divisor; or make up the number of ciphers partly by removing from the divisor and annexing to the dividend, if there be not a sufficient number in the divisor.

[19]These are not quite correct, but sufficiently so for every practical purpose.

[20]The 1′ here means that the 1 is in the multiplier.

[21]This is written 7 instead of 6, because the figure which is abandoned in the dividend is 9 (151).

[22]Meaning, of course, a really fractional number, such as ⅞ or ¹⁵/₁₁, not one which, though fractional in form, is whole in reality, such as ¹⁰/₅ or ²⁷/₃.

[23]By square number I mean, a number which has a square root. Thus, 25 is a square number, but 26 is not.

[24]The term ‘root’ is frequently used as an abbreviation of square root.

[25]Or, more simply, add the second figure of the root to the first divisor.

[26]This is a very incorrect name, since the term ‘arithmetical’ applies equally to every notion in this book. It is necessary, however, that the pupil should use words in the sense in which they will be used in his succeeding studies.

[27]The same remark may be made here as was made in the note on the term ‘arithmetical proportion,’ page 101. The word ‘geometrical’ is, generally speaking, dropped, except when we wish to distinguish between this kind of proportion and that which has been called arithmetical.

[28]A theorem is a general mathematical fact: thus, that every number is divisible by four when its last two figures are divisible by four, is a theorem; that in every proportion the product of the extremes is equal to the product of the means, is another.

[29]Ifbxbe substituted forain any expression which is homogeneous with respect toaandb, the pupil may easily see thatbmust occur in every term as often as there are units in the degree of the expression: thus,aa+abbecomesbxbx+bxborbb(xx+x);aaa+bbbbecomesbxbxbx+bbborbbb(xxx+ 1); and so on.

[30]The difference between this problem and the last is left to the ingenuity of the pupil.

[31]It is not true, that if we choose any quantity as a unit,anyother quantity of the same kind can be exactly represented either by a certain number of units, or of parts of a unit. To understand how this is proved, the pupil would require more knowledge than he can be supposed to have; but we can shew him that, for any thing he knows to the contrary, there may be quantities which are neither units nor parts of the unit. Take a mathematical line of one foot in length, divide it into ten parts, each of those parts into ten parts, and so on continually. If a point A be taken at hazard in the line, it does not appear self-evident that if the decimal division be continued ever so far, one of the points of division must at last fall exactly on A: neither would the same appear necessarily true if the division were made into sevenths, or elevenths, or in any other way. There may then possibly be a part of a foot which is no exact numerical fraction whatever of the foot; and this, in a higher branch of mathematics, is found to be the case times without number. What is meant in the words on which this note is written, is, that any part of a foot can be represented as nearly as we please by a numerical fraction of it; and this is sufficient for practical purposes.

[32]Since this was first written, the accident has happened. Thestandard yardwas so injured as to be rendered useless by the fire at the Houses of Parliament.

[33]The minute and second are often marked thus, 1′, 1″: but this notation is now almost entirely appropriated to the minute and second ofangularmeasure.

[34]The measures in italics are those which it is most necessary that the student should learn by heart.

[35]The lengths of the pendulums which will vibrate in one second are slightly different in different latitudes. Greenwich is chosen as the station of the Royal Observatory. We may add, that much doubt is now entertained as to the system of standards derived from nature being capable of that extreme accuracy which was once attributed to it.

[36]The inch is said to have been originally obtained by putting together three grains of barley.

[37]‘Capacity’ is a term which cannot be better explained than by its use. When one measure holds more than another, it is said to be more capacious, or to have a greater capacity.

[38]This measure, and those which follow, are used for dry goods only.

[39]Since the publication of the third edition, theheapedmeasure, which was part of the new system, has been abolished. The following paragraph from the third edition will serve for reference to it:“The other imperial measure is applied to goods which it is customary to sell byheaped measure, and is as follows:2 gallons1 peck4 pecks1 bushel3 bushels1 sack12 sacks1 chaldron.The gallon and bushel in this measure hold the same when only just filled, as in the last. The bushel, however, heaped up as directed by the act of parliament, is a little more than one-fourth greater than before.”

[39]Since the publication of the third edition, theheapedmeasure, which was part of the new system, has been abolished. The following paragraph from the third edition will serve for reference to it:

“The other imperial measure is applied to goods which it is customary to sell byheaped measure, and is as follows:

The gallon and bushel in this measure hold the same when only just filled, as in the last. The bushel, however, heaped up as directed by the act of parliament, is a little more than one-fourth greater than before.”

[40]Pure water, cleared from foreign substances by distillation, at a temperature of 62° Fahr.

[41]It is more common to divide the ounce into four quarters than into sixteen drams.

[42]The English pound is generally called apound sterling, which distinguishes it from the weight called a pound, and also from foreign coins.

[43]The coin called a guinea is now no longer in use, but the name is still given, from custom, to 21 shillings. The pound, which was not a coin, but a note promising to pay 20 shillings to the bearer, is also disused for the present, and the sovereign supplies its place; but the name pound is still given to 20 shillings.

[44]Farthings are never written but as parts of a penny. Thus, three farthings being ¾ of a penny, is written ¾, or ¾. One halfpenny may be written either as 2/4 or ½; the latter is most common.

[45]When a decimal follows a whole number, the decimal is always of the same unit as the whole number. Thus, 5ᔆ·5 is fivesecondsand five-tenths of asecond. Thus, 0ᔆ·5 means five-tenths of a second; 0ʰ·3, three-tenths of an hour.

[46]Before reading this article and the next, articles (29) and (42) should be read again carefully.

[47]Any fraction of a unit, whose numerator is unity, is generally called analiquot partof that unit. Thus, 2s.and 10s.are both aliquot parts of a pound, being £⅒ and £½.

[48]A parallelepiped, or more properly, arectangularparallelepiped, is a figure of the form of a brick; its sides, however, may be of any length; thus, the figure of a plank has the same name. A cube is a parallelepiped with equal sides, such as is a die.

[49]This generally comes in the same member of the sentence. In some cases the ingenuity of the student must be employed in detecting it. The reasoning of (238) is the best guide. The following may be very often applied. If it be evident that the answer must be less than the given quantity of its kind, multiply that given quantity by the less of the other two; if greater, by the greater. Thus, in the first question, 156 yards must cost more than 22; multiply, therefore, by 156.

[50]It is usual to place points, in the manner here shewn, between the quantities. Those who have read Section VIII. will see that the Rule of Three is no more than the process for finding the fourth term of a proportion from the other three.

[51]Commission is what is allowed by one merchant to another for buying or selling goods for him, and is usually a per-centage on the whole sum employed. Brokerage is an allowance similar to commission, under a different name, principally used in the buying and selling of stock in the funds.Insurance is a per-centage paid to those who engage to make good to the payers any loss they may sustain by accidents from fire, or storms, according to the agreement, up to a certain amount which is named, and is a per-centage upon this amount. Tare, tret, and cloff, are allowances made in selling goods by wholesale, for the weight of the boxes or barrels which contain them, waste, &c.; and are usually either the price of a certain number of pounds of the goods for each box or barrel, or a certain allowance on each cwt.

Insurance is a per-centage paid to those who engage to make good to the payers any loss they may sustain by accidents from fire, or storms, according to the agreement, up to a certain amount which is named, and is a per-centage upon this amount. Tare, tret, and cloff, are allowances made in selling goods by wholesale, for the weight of the boxes or barrels which contain them, waste, &c.; and are usually either the price of a certain number of pounds of the goods for each box or barrel, or a certain allowance on each cwt.

[52]Here the 4s.from the dividend is taken in.

[53]Here the 3d.from the dividend is taken in.

[54]Sufficient tables for all common purposes are contained in the article on Interest in the Penny Cyclopædia; and ample ones in the Treatise on Annuities and Reversions, in the Library of Useful Knowledge.

[55]This rule is obsolete in business. When a bill, for instance, of £100 having a year to run, isdiscounted(as people now say) at 5 per cent, this means that 5 per cent of £100, or £5, is struck off.

[56]This question does not at first appear to fall under the rule. A little thought will serve to shew that what probably will be the first idea of the proper method of solution is erroneous.

[57]The teacher will find further remarks on this subject in theCompanion to the Almanacfor 1844, and in theSupplement to the Penny Cyclopædia, articleComputation.

[58]And at discretion one hundredth more for a large fraction of three inches.

[59]The student should remember all the multiples of 4 up to 4 × 25, or 100.

[60]The treatises on book-keeping have described this difference in as peculiar a manner. They call these accounts thefictitious accounts. Now they represent the merchant himself; their credits are gain to the business, their debits losses or liabilities. If the terms real and fictitious are to be used at all, they are therealaccounts, end all the others are asfictitiousas the clerks whom we have supposed to keep them.

[61]This theorem shews that what iscalledreducing a fraction to its lowest terms (namely, dividing numerator and denominator by their greatest common measure), is correctly so called.

[62]For that which measures a measure is itself a measure; so that if a measure ofacould have a measure in common withb,aitself would have a common measure withb.

[63]A prime number is one which is prime to all numbers except its own multiples, or has no divisors except 1 and itself.

[64]Expand (a-1)ᵇ by the binomial theorem; shew thatwhen b is a prime numberevery coefficient which is not unity is divisible byb; and the proposition follows.

[65]The principle of this mode of demonstration of Horner’s method was stated in Young’s Algebra (1823), being the earliest elementary work in which that method was given.

[66]Various exceptions may arise when an equation has two nearly equal roots. But I do not here introduce algebraical difficulties; and a student might give himself a hundred examples, taken at hazard, without much chance of lighting upon one which gives any difficulty.

[67]This form might be also applied to the integer portions; but it is hardly needed in such instances as usually occur. See the articleInvolution and Evolutionin theSupplementto thePenny Cyclopædia.

[68]After the second step, the trial will rarely fail to give the true figure.

[69]The solution ofx³ + 0x² + 0x-2 = 0.

[70]Taken from a paper on the subject, by Mr. Peter Gray, in theMechanics’ Magazine.

[71]Taken from a paper on the subject, by Mr. Peter Gray, in theMechanics’ Magazine.

[72]Taken from a paper on the subject, by Mr. Peter Gray, in theMechanics’ Magazine.

[73]Taken from the late Mr. Peter Nicholson’s Essay on Involution and Evolution.

[74]Taken from the late Mr. Peter Nicholson’s Essay on Involution and Evolution.

[75]Taken from the late Mr. Peter Nicholson’s Essay on Involution and Evolution.

[76]Taken from the late Mr. Peter Nicholson’s Essay on Involution and Evolution.

[77]A four-sided figure, which has two sides parallel, and two sides not parallel.

[78]The right angle is divided into 90 equal parts calleddegrees, each degree into 60 equal parts calledminutes, and each minute into 60 equal parts calledseconds. Thus, 2° 15′ 40″ means 2 degrees, 15 minutes, and 40 seconds.

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