PREFACE. 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 fractions as soon as possible, thoroughly mastered; common fractions postponed as long as possible.” The Metric System in a few years will be in common use, and will supersede other systems, as dollars and cents have superseded pounds, shillings, and pence.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. THOMAS HILL. INDEX. The black nurnbers refer to pages; the other numbers to sections. common measures ADDITION, 44; tests, 56; com- | Chemical symbols and problems, 462. areas, 204–206; 188. Clapboards, 320. Commission, 349. Common measure: greatest, 233, and common fractions, 426-429. Common multiple: least, 239, 240, Condensation of sulphuric acid and water, 266. logarithms, 413. Cylinder, 445. DECIMAL fractions, 25; reading, common, 276, 277; circulating, 279–283; shortening decimals, 145. note. Division, 149–168; by reciprocals, 162; contracted, 168; of two ens, 174. V kinds, 150; compound, 308; by Height of objects in horizon, 458, 459. Horizon, distance of, 458, 459. Hydraulic press, 453. of, 68. EARTH, circumference of, 97. INSURANCE, 350. Interest, 352; compound, 367, 368; rithms, 442. Involution and Evolution, 379– note; of air, ibid; of iron, 224; KNOT, 287, 302. LEAP year, 301. Lever, 343. ing prime, 220–226; multiply- 401 et seq.; calculation of, 402; characteristic and mantissa, 403– mon, 244-284; terms of, 250; tient, 415-419; of reciprocal, 416, 417. 293–300; comparison of metric and common, 322; miscellane- of angle, 302; temperature, 304. 220; specific gravity, 101. circles, 201-206, 188; cubes and rectangular parallelopipeds, 210; | Principal, 353. compound, 339–340, also 267. ical, or knot, 287, 302, notes. Ratio, 323. 128; contracted, 143; by com- Reduction, 291, 292; time and Roots, 379-397; by logarithms, 413. Rule of Three, 336, 338; of false, 358. SCREW, 344. Shingles, 321. Sinking fund, 317 (Exs. 17, 18). 216. 365; Vermont, New Hampshire, iron, 461; in water, 461. Specific gravity explained, 212, 214, 215; table of some common substances, 348; problems in, 199. ume of, 211. Stock, 370, 371; investments in, 254, 259. pound, 306. Taxes, 351. Thermometers, 304, in |