product divided by 3, the quotient will be 36; what is the number? Ans. 33.

47. A watch has an hour, a minute and a second hand, turning upon the same centre-staff. At 12 o'clock the three hands are together. How long will it be (1) before the second-hand will be equally distant from the other two? (2) before the minutehand will be equidistant from the other two? (3) before the hour-hand will be equidistant from the other two?

Hence the proportion:-As the distance round the clock is to the distance the hour-hand has moved, so is the number of seconds while the hour-hand is going round the clock to the number of seconds required; i. e.,

390

As 1427: 1 :: 43200 sec.: 30 sec., Ans.

REMARK.—In a similar manner we may determine when the second-hand shall have the position s' represented in Fig. 1; also the questions (2) and (3), and all similar examples respecting the positions of the hands of a watch may be analyzed.

48. At what time between 12 and 1 o'clock will the hour and minute hands of a watch make equal acute angles with the line extending from the centre-staff to 12? Ans. 55 m. past 12.

10.

9+

8

11i

FIG. 2.

H

12

6

ST

h'h 1

2

4

3

ANALYSIS.-At half-past 12 the minute-hand is at 6 and the hourhand is at h', half way from 12 to 1. Now, if the hour-hand would stand still at h' while the minute-hand moved forward to ', half-way from 11 to 12, 27 minutes from the point 6, the hands would have the required positions; but, while the minute-hand is advancing, the hour-hand goes from h' to h; ..the minute-hand must stop at i, as

much short of 'as h is in advance of h'; i. e., the hour and minute hands together move over the space represented by 271⁄2 minutes on the dial; but the hour and minute hands together move over 13 spaces while the minute-hand alone moves over 12 spaces; hence the proportion:-13 : 12 : : 271⁄2 m. : 25+ m., the number of minutes beyond half-past 12 when the hands will have the required positions.

49. At what time between 5 and 6 o'clock do the hour and minute hands make equal acute angles with the line from 12 to 6?

50. At what time between 2 and 3 o'clock do the hour and minute hands point in opposite directions?

51. At a certain time between 8 and 9 o'clock the minutehand was between 9 and 10. Within an hour afterwards the hour and minute hands had changed places. What was the 1st mentioned time?

52. An intelligent farmer, having a good pair of oxen fully shod, offered to sell them if any one would pay 1 cent for the first shoe, 2 cents for the second, and so on in geometrical progression for all the shoes. Having found a ready purchaser, it is required to find the price of the oxen. Ans. $655.35.

53. The ignorant purchaser of the oxen mentioned in the above example, being astonished at the price, soon negotiated the sale of his horse, a noble animal, on the same terms. What did he receive for his horse? Ans. 15 cents.

54. Mr. Ignoramus, being chagrined by the bargains mentioned in Ex. 52 and 53, and resolving to retrieve his fortune by the purchase of a splendid mansion, offered the owner 1 cent for the 1st door, 2 cents for the 2d, 4 for the 3d, and so on. The house having 65 doors, what did it cost?

Ans. $368934881474191032.31.

55. The purchaser of the above-named house, being again astonished above measure, called in legal advice. The lawyer being a shrewd, practical man, counselled his client to say to the inexorable creditor:-"It is but reasonable, sir, that your legal demands be satisfied; please be seated till I can count the money, and you shall have your pay." Now, suppose the debtor can count $1 per second, 10 hours a day and 300 days in a year, how long must the creditor wait?

Ans. 34160637173 yr. 6 m. 13 d.

56. What would be the simple interest on the above-named sum for the time required to count it?

57. What the compound interest, allowing it to double once in 12 years?

SUPPLEMENT.

§ 46. MISCELLANY.

390. ARITHMETICAL operations by the Roman Notation, are very cumbersome, and this may be both cause and consequence of there being few, if any, Roman mathematicians of eminence.

391. The directions usually given for the manner of performing certain operations are merely for convenience; thus, in Subtraction we are directed to write the subtrahend under the minuend; but one versed in figures will subtract as readily when the subtrahend is over the minuend or elsewhere, and it is frequently more convenient than to follow the rule.

392. In Multiplication, the result will be the same, whichever figure of the multiplier is used first; still, there is usually no gain in departing from the common course, but a decided loss in the lack of system. The following example illustrates these points:

Product, 1 6 0 6 4 4 0 6 16 0 6 4 406

16064406

393. In Long Division, there may be a real gain in writing the divisor at the right instead of at the left of the dividend; for the work will be more compact, and the divisor and quotient will have the usual relative position of factors in multiplication; thus,

RULE.

Annex 00 to the multiplicand (or move the decimal point two places towards the right) and divide the result by 4.

Why? Because 25 is of 100, and .. we wish to obtain of 100 times the multiplicand; thus,

Multiply 796 by 25. Also 7.96 by 25.

4)79600

19900, Ans.

4)79 6.

19 9., Ans.

395. To multiply by 331,

RULE.

· Annex 00 (or move, etc.) and divide by 3. Why?

Multiply 7824 by 33. Also 7.824 by 331.

3) 782400

260 800, Ans.

3) 78 2.4

2 6 0.8, Ans.