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PREFACE.

THE

HE object of a text-book on Arithmetic should be to teach the pupil to cipher, to learn by doing. The shortest and surest road to a knowledge of Arithmetic is by solving problems, not by memorizing rules or by demonstrating propositions. The pupil should be trained to obtain results rapidly and correctly. He should be taught, in questions involving decimal fractions, to limit the answers to the number of decimals required by the nature of the examples, and to avoid all superfluous work. He should not be expected to discover the reason of a process until he fully understands the process; then he should be allowed to state the reason in his own language.

This Arithmetic is not intended for beginners; but it is presumed that pupils will have a thorough knowledge of our "Lessons in Number," and be at least twelve years of age, before entering upon the study of this book.

Decimal fractions are introduced at the beginning of the book. Experience proves that when thus taught they present no difficulty. The difficulty of decimal fractions arises solely from comparing them with common fractions, and is avoided by teaching decimals first. The pupil learns the notation on both sides of the decimal point as easily as on one side; provided the notation on both sides is presented at the same time. Much time is saved by strict adherence to the motto, "Decimal possible, thoroughly mastered; common fractions postponed as long as possible."

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The Metric System in a few years will be in common use, and will supersede other systems, as dollars and cents have superseded pounds, shillings, and pence. Taught immediately after decimal fractions, the system is easily learned. A great number of examples is given to show the simplicity of the system in its application to questions of common occurrence, and to furnish additional practice in operations with decimal fractions. The abbreviations used are such as have been adopted throughout Germany.

Many of the problems are original, but some have been obtained from French, English, and German sources. Though the problems are very numerous, it has been found, by actual trial, that a class of pupils fourteen to fifteen years old can accomplish the whole work of this Arithmetic, with one recitation a day, in a school year. The examples are intended to convey, incidentally, a great deal of accurate and valuable information; so that, by means of the index, the book becomes a book of reference for many physical and mathematical constants.

The introduction of logarithms will be welcomed by all who know the ease of learning the practical use of a four-place table, and the increased power given by it over mathematical questions. Teachers who have never taught or learned logarithms are assured that they will find no difficulty in the subject as here presented.

The method of "Supposition," called in old Arithmetics “Position," has been restored to its rightful place, and is fully explained in the chapter on Approximations. This method is applicable to a large variety of problems, and is made very simple by logarithms.

We gladly acknowledge our obligations to many friends who have improved this work by their advice; and we also give assurance that any suggestions for its further improvement will be thankfully received.

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INDEX.

The black numbers refer to pages; the other numbers to sections.

ADDITION, 44; tests, 56; com- | Chemical symbols and problems,
pound, 305.

462.
Circle

Air, composition of, 216, 220.
Alligation, 105, 265–267.

Amount, 353.

Annuity, 317; in reversion, 317 Cologarithms, 416.
Commission, 349.

(Ex. 16).

Antilogarithms, 421.
Approximations, 424 et seq.; to
decimals, 145; general, 425; to
common fractions, 426-429.
Average, 265; of payments, 267–
271.

BELL metal, composition of, 215.
Board measure, 209, 317, 318.
Bowl, measure of, 449.

Brass, one variety, 215.
Brokerage, 349.

linear ratios, 201, 202;
areas, 204-206; 188.
Clapboards, 320.

ens, 174.

Catenary, 334.

Centrifugal force, 457.

Common measure: greatest, 233,
284; common measures and
metric, 285 et seq.
Common multiple: least, 239, 240,
243; of fractions, 284.
Condensation of sulphuric acid
and water, 266.

Cone, 448; frustum, 450.
Cube, 192; cube root, 389; by
logarithms, 413.
Cylinder, 445.

CALENDAR, Julian and Gregorian,
301.

DECIMAL fractions, 25; reading,
best way, 27; changing to
common, 276, 277; circulating,
279-283; shortening decimals,
145.

Cancelling factors, 155.

Carpeting rooms, 207, 189.

Casting out nines, 170-173; elev- Discount, 356, 358; true, 361, 362,

note.

Division, 149–168; by reciprocals,
162; contracted, 168; of two

V

kinds, 150; compound, 308; by | HEIGHT of objects in horizon, 458,
logarithms, 415-419.

459.

Double position, 425.
Drafts, 372-374.

Duties, 351; ad valorem, 231.

EARTH, circumference of, 97.
Ellipse, 451.

Equation: solution of, 66; of pay-

ments, 267.

Exchange, 372, 374; foreign, Invoice, 231.

263.
Expansion

coefficient of, 340,
note; of air, ibid; of iron, 224;
of glass and steel, 320.
Exponent, 148; negative, 169;
logarithms, 411.

FACTORS: cancelling, 155; detect-
ing prime, 220-226; multiply-
ing by, 132; with negative ex-
ponents, 228.

Falling bodies, 454.
Fractions: decimal, 25, 27; com-
mon, 244-284; terms of, 250;
improper, 252, 258; multiplica-
tion of, 262, 264; division of,
266; common denominator, 268;
addition, 270; subtraction, 271;
simplification, 272-274; chang-
ing common and decimal, 276,
277; in compound numbers,
309-312.

GRAVITY, accelerating force of,
454.

Horizon, distance of, 458, 459.
Hydraulic press, 453.
Hydrostatic pressure, 453.

Gun metal, composition of, 215.
Gunpowder composition of, 215,
220; specific gravity, 101.

INSURANCE, 350.

Interest, 352; compound, 367, 368;
annual, 369; computed by loga-
rithms, 442.

Involution and Evolution, 379-
397; by logarithms, 412-419.

KNOT, 287, 302.

LEAP year, 301.
Lever, 343.

Light: intensity of, 342; velocity
of, 68.

Logarithms, 399 et seq.; common,
401 et seq.; calculation of, 402;
characteristic and mantissa, 403–
410; exponents, 411; of quo-
tient, 415-419; of reciprocal,
416, 417.
Longitude, 313-316; reduction to
time and the reverse, 315.

MEASURES: metric, length, 184;
surface, 189; volume, 194, 197;
weight, 199, 200; common, 287,
293-300; comparison of metric
and common, 322; miscellane-
ous, 160; measure of time, 301;
of angle, 302; temperature, 304.
Mensuration of squares and rec-
tangles, 187, 293; triangles, 443;
circles, 201-206, 188; cubes and

rectangular parallelopipeds, 210; | Principal, 353.

of prisms, 446; cones, 448; pyr-
amids, 446; frustums, 450; bowls
and boilers, 449; cylinder, 445;
sphere, 205, 211, 189.
Miles, 287; nautical, or geograph- | Pyramid, 446; frustum, 450.
ical, or knot, 287, 302, notes.
Money: U. S., 182; foreign,
303.
Multiplication, 126; of decimals,
128; contracted, 143; by com-
plements, 137; by reciprocals,
164; by factors, 132; com-
pound, 307.

NOTATION, 2, 16.

Notes of hand and bank discount,
358.

Numeration, 2, 16.

Poll tax, 351, Ex.

Position, 425.

Progression, arithmetical, 430–
434; geometrical, 435–441. .
Proportion, 323-338; test of, 335;
compound, 339–340, also 267.

RATE per cent, 355.
Ratio, 323.
Reciprocal, 161.

Reduction, 291, 292; time and
longitude, 315, 316.
Representative numbers, 345-347.
Roots, 379-397; by logarithms,

413.

Rule of Three, 336, 338; of false,
or double position, 425.

SCREW,
344.
Shingles, 321.

ONCOMETRICS, 194, 196, 325–329. Similarity, geometrical, 398.
Sinking fund, 317 (Exs. 17, 18).
PARENTHESIS, when needed, 111; Solder, composition of common,
how to use, 67, 68.

216.

Partial payments, 363; U.S. rule,
365; Vermont, New Hampshire,
and Connecticut, 252.
Partnership, 342.

Sound: velocity of, 460, 461; in
iron, 461; in water, 461.
Specific gravity explained, 212,
214, 215; table of some common
substances, 348; problems in,
199.

Pendulum, 452.

Percentage, 343 et seq.

Planets, approximate distances of, Sphere: surface of, 205, 189; vol-

69.

ume of, 211.

Stock, 370, 371; investments in,
254, 259.

TAXES, 351.
Thermometers, 304.

Pound, weight, 298, note; Eng- Subtraction, 71; tests, 72; com-

lish money, 303.

pound, 306.

Powers, 148, 378-397.
Present worth, 360.
Prism, 446.

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