ited number are inserted. It is advisable, when possible, for the teacher to exhibit the actual coins. The same is true of all the weights and measures represented.

210, 211, 216. Show how square and cubic measure stand related to Long

measure.

211. We see no good reason why a pupil should wait until he studies a treatise on surveying before he knows how a township is subdivided.

By means of cut, p. 192, the teacher may suggest, and the pupil solve, a great number of interesting problems. By means of Section Maps of the Western States, the pupil may locate the principal cities, towns, etc.

214. The edge of a cube is one of its dimensions. The edge of any solid is a line formed by the meeting of two adjacent faces of that solid.

245. Although three methods are given, whether the pupil shall study them all at once should be determined by his stage of advancement.

246, 257, are entirely practical, and while intended for pupils in general, are especially useful for those who have no prospect of taking a course in Mathematics, or who cannot even find the time to go as far as Mensuration proper, as treated in the last pages of this work. Much is given here in a convenient form that is not easy to find in any one book.

248. Nearly every pupil in the Western States will be interested in knowing the manner of dividing lands in his own county.

257. Prob. 4. Since when the sides were 15 inches high, the wagon-bed held of 15 inches high, or 43호

43 cubic feet; in order to hold 50 cubic feet, it must be 17 inches. Hence the depth is increased 23 in.

Probs. 13, 17 and 18 may be solved in the same manner.

50

258. This may be illustrated by the change apparent in a watch, in going from San Francisco to New York, New York to Liverpool, Liverpool to Canton (China), Canton to San Francisco; illustrating that a good time-keeper will lose a day in going round the world eastwardly; and, in like manner, going westwardly will gain a day.

259. The word measure is here used in its general sense; for although 25 consists of a number of parts, yet, for the purpose of measuring, it is a unit. For example: How many centals in 2050 lbs. of wheat? Here we divide 2050 by 100, the number of lbs. in one cental, and 100 lbs. is regarded as the measuring unit.

286. P. 285, Ex. Equimultiples of numbers are the products of those numbers by a given number. Thus : 7 × 5 and 8 × 5 are equimultiples of 7 and 8. 291, 292. Note especially the difference between Rate per cent. and Rate. 305. In Higher Mathematics and Applications, Formulas are deemed invaluable. Why not in Arithmetic?

319. Solution of Ex. 1, in Review Problems: To make 25%, the selling price must be of cost, or of 24c. = 30c. But 30c. is 163% less than asking price. Since 163% = }, { − } ( = §) of asking price, must equal selling price, and of asking price

equals of selling price; therefore, § of asking price equals g of selling price. of 30c. 36c., asking price.

=

353. One method of computation well taught is worth more than all the others poorly taught; and all special methods should be omitted with beginners.

403-412. The five cases correspond to the five cases of simple interest.

426. Both Rules and Formulas are omitted in computations of Stocks and Bonds, because they are the simplest applications of percentage.

Prob. 17. ANOTHER SOLUTION: Dividing the given by the required rate % gives the rate. .75 × 500 375; or, $375 is the price.

427. Letters of Credit, that is, written orders on which partial payments are made at sight, are issued to travellers in all or nearly all civilized countries; thus affording great security in the transportation of necessary funds for travelling expenses. The value of such letters, of course, is reckoned according to the principles governing the calculation of exchange.

436. The table on p. 366 will suggest to the ingenious teacher a great variety of interesting problems.

461. Pupils should also be exercised in solving Problems in Partnership and Bankruptcy by Distributive Proportion, pp. 285-288.

484. Prob. 1. Had Simpson paid the stipulated sums at the times agreed upon, the party from whom he bought would have had at the end of 15 mo. the interest of $500 for 15 mo.; of $600 for 9 mo.; and of $700 for 3 mo.; or, at 6%, he would have had $37.50 + $27 + $10.50 = $75 interest. Simpson then should him pay $2700 ($500+ $600 + $700 + $900) at such a time before the end of 15 mo., that at the end of the 15 mo. its interest at 6% would equal $75. In 12 mo. $2700, at 6%, will gain $162; and will gain $75 in of 12 mo. 58 mo. Simpson, therefore, should pay $2700 in 15 mo. —5; mo. = 9; mo., Ans.

=

485. Let the pupil work a number of examples by selecting both the first and the last dates as focal dates.

509. In the applications of Square and Cube Roots, no demonstrations are attempted.

531. Prob. 1: As the first payment is made at the beginning of the first year, and the tenth payment at the beginning of the tenth year, the entire 10 payments are made within 9 yrs. and 1 da. We must, therefore, regard the annuity as having 10 yrs. to run.

567. It is a matter of regret that the Metric System has not come into general use in the United States.

CHAPTER I.

INTEGERS

SECTION I.-DEFINITIONS.

1. Arithmetic is the science of numbers and the art of computation.

2. A Number is a unit, or a collection of units, and answers the question "How many?"

3. A Unit is one, or a single thing; as, one, one dollar, one house, one bushel, one peck, one half. (5, NOTE, and 150.) Numbers are classified as to object into Concrete and Abstract.

4. A Concrete Number is one that is applied to some particular object; as, three books, four dollars, five miles.

5. An Abstract Number is one that is not applied to any object; as, three, four, five.

Numbers are classified as to their unit into Integral, Fractional, and Mixed.

6. An Integer is a whole number; as, one, five, ten. 7. A Fraction is a number expressing one or more of the equal parts of an integer; as, one half, five tenths.

8. A Mixed Number is composed of an integer and a fraction; as, six and one half, nine and five tenths.

Numbers are classified as to their nature, into Like and Unlike.

9. Like Numbers are those which express the same kind of units; as, two cents and four cents, six hats and one hat.

10. Unlike Numbers are those which express different kinds of units; as, two cents and six hats, four cents and one hat.

Numbers are classified as to the number of kinds of units, into Simple and Compound.

11. A Simple Number is a number expressing one kind of unit; as, four pounds.

12. A Compound Number is a number expressing more than one kind of unit; as, four pounds five ounces.

The terms used in computations in Arithmetic are Solution, Problem, Explanation, Principle, Example, Analysis, and Rule.

13. A Solution is a process of computation used to obtain a required result.

14. A Problem is a question for solution.

15. An Explanation is a statement of the reasons for the manner of solving a problem.

16. A Principle is a general truth upon which a process of computation is founded.

17. An Example is a problem used to illustrate a principle, or to explain a method of computation.

18. An Analysis is a statement of the successive steps in a solution.

19. A Rule is a direction for performing any computation.