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13. A man bought a keg of brandy for 3£. Os. 6d., but in carry. ing it home there leaked out as much as would come to l£. 3s. 8d. What was the value of the remainder ? Here we borrowed a shilling. Το £.
d. make up for this, we increased the 3
3; 0; 6; shillings in the lower number by 1, in.
8; stead of diminishing the upper number, which is a cypher.
10; Ans. Note. The intelligent pupil will perceive, that, as there are no shillings in the minuend, the borrowed shilling comes from the pounds. If we had taken a pound from the 3 pounds, and after borrowing ls., put down the remaining 19 in place of shillings, there would have been no occasion to increase the lower number. For 19—3=16 and 20—4=16. This however, in compound num. bers, would be perplexing.
It will be recollected that this method of allowing, after having borrowed, was noticed in Subtraction of Simple Numbers- (Sxxii.)
14. A man commenced business with 1,850£. Os. 10 d. and at the end of the year, found that he was worth 2,570£. 9s. 6fd. How much had he gained ? Ans. 720£. 8s. 8d.
15. I borrowed £317 ; 6, and afterwards paid £178; 18; 54. How much was then due ? Ans. £138; 7; 61.
16. A man purchased cloth to the amount of £27; 11. In turn he gave flour to the amount of £19; 17; 6, and the rest in money. How much money did he give ? Ans. £7; 13; 6.
From the above we derive the following rule.
PLACE THE SAME DENOMINATIONS UNDER EACH OTHER. TAKE EACH DENOMINATION IN THE SUBTRAHEND, BEGINNING WITH THE LOWEST, FROM THE SAME IN THE MINUEND, AND TO COMPENSATE FOR BORROWING FROM ONE DENOMINATION TO ANOTHER, INCREASE THE NEXT HIGHER DENOMINATION OF THE SUBTRAHEND BY I.
EXAMPLES FOR PRACTICE. 18. A merchant had 200 bls. 16.gal. of brandy, of which he sold 82 bls. 15 gal. 1 pt. How much had he left ? Ans. 118 bls. 3 qts. 1 pt.
19. A man borrowed £60 ; 10,: and paid, at one time £17; 11 ; 6 ; at another, £9; 8; at another, £7; 9; 6; and at another, 19s. 6}d. How much then remained unpaid? Ans. £25; 1; 53.
20. I pay a debt of £105; 10, as follows; viz. I give an order on another person for £15; 14; 9, and two notes, one for £30; 0; 6, and another for £39 ; 11. The rest I pay down. How much do I pay down?
21. The war between England and America commenced April 19, 1775, and continued until Jan. 20, 1783. How long did it con. tinue ? Set it down thus. Yrs.
22. From 2 cwt. 2 qrs. 27 lb. 3 oz. 8 dr. take 3 qrs. 0 lb. 9 oz. 7 dr.
23. From 7T. 9 cwt. 3 qis. 16 lb. 8 oz. 3 dr., take IT. 11 cwt. 3 qrs. 17 lb. 9 oz. 6 dr.
24. From 15 yrs. 3 mo. 3 w. 2 d. 5 h. 3 m. 3 sec., take 13 yrs. 9 d. 27 sec.
25. From 289 acres, 3 roods 7 rds., take 196 acres 3 roods, 30 rde.
♡ XXXVIII. 1. A man divided 4 bushels and 2 pecks of grain, between 2 poor persons.
How much had each ?
2. A man divided 6 acres, 2 roods, and 8 rods of land into 2 equal fields. How much land was there in each field ?
3. Five boys agreed to share 2 qts. and 1 pt. of nuts equally. How much ought each to have?
4. What is the 4th part of 5d. ? What is the 6th part of 1d. 2 qrs. ?
5. What is the 8th part of 1 lb. Avoidupois ? 6. What is the 7th part of 1 lb. 2 oz. Troy? 7. What is the 8th part of 93 ? of 116. 43 ? 8. What is the 4th part of 6d.? of 9d. ? of 1s. 1d. ?
2d.? 9. What is the 5th part of 6 gals. 1 qt. ? of 12 gals.
io. What is the 9th part of 2 square yds. ? of 8 square yds.?
11. What is the 11th part 1£ 2s. ? 'of 3£ 6s. ? of 12£ 2s.?
The following are to be written.
12. A man paid 4 labourers an equal sum each. To the whole he gave £10; 8. What was that apiece ?
In dividing 10£ by 4, 2£ are left. £. s. £. s. 4 will not go in 2, but if 2£ be redu. 4)10 8(2 12 ced to shillings, 4 will divide the 8 number of shillings. Let this reduc. tion be made, by multiplying by 20, and let the 8s in the given sum be 20 added in, at the same time. The whole is 488., which divided by 48 4=128.
48 Hence, to divide compound numbers,
I. DIVIDE EACH DENOMINATION OF THE DIVIDEND, RATELY BY THE DIVISOR, AND THE SEVERAL QUOTIENTS WILL
BELONG TO THE DENOMINATIONS OF THEIR RESPECTIVE DIVIDENDS.
II. IF A REMAINDER BE LEFT, IN DIVIDING ANY DENOMINATION, REDUCE IT TO THE NEXT LOWER DENOMINATION, ADD IN ALL OF THAT DENOMINATION, IN THE GIVEN DIVIDEND, AND THEN DIVIDE AS BEFORE.
EXAMPLES FOR PRACTICE. 13. If 11 tons of hay cost £23; 0; 2, what is that per ton ?
A. £2; 1; 10. 14. If 48 lbs. of cheese cost £l ; 16, what is that per lb. ?
A. 9d. 15. If 13 persons pay equally towards a bill of 5£. 8s. 101d. how much must each pay A. 8s. 4 d. 16. If a nobleman's salary be £150,000 a year, what is that a day?
A. 410£ ; 198. ; 2d. 17. If 1 cwt. of raisins cost £3; 10, what is that a lb. A. 74d. 18. If 12 quarts of wine cost £4; 15; 6, what is that a qt.
A. 7s. 114d. 19. Divide £115 ; 10, by 90. A. £l; 5; 8. 20. Divide £136; 16; 6, by 108. A. £l; 5; 4. 21. Divide 6 T. 11 cwt. 3 qrs. 19 lb. by 4.
A. 1 T. 12 cwt. 3 qrs. 25 lb. 12 oz. 22. Divide 26 lb. 1 oz. 5 dwt. by 24. A. 1 lb. 1 oz. 1 dwt. 1 qr. 23. Divide 666 £ 15s. 9d. 1 qr. by 125. 24. Divide 32 yrs. 5 mo. 2 w. 5 d. 17 h. 27 sec. by 306. 25. Divide 441 11.3 02. 16 dwt, 17 gr. by 509. 26. Divide 81 T. 16 cwt. 3 qrs. 25 lb. 15 oz. 13 dr. by 572.
OBSERVATIONS ON COMPOUND NUMBERS, FOR ADVANCED
PUPILS. 8 XXXIX. Man could not have lived long in the earth, without perceiving the necessity of different measures. In estimating dimensions of length, for example, he would find those minute divisions of space, which, for the measurement of small objects were not only convenient but even necessary, very inadequate to his purpose when applied to the height of a tree, or the breadth of a river ; and much more so, when employed to express the elevation of a mountain, or the distance traversed on a long journey. Hence, the origin of different denominations. For the purpose of easily comparing the different denominations with one another, and, in some cases, of substituting one for another without altering the value, it seemed best to make each higher denomination such as to contain an exact number of the next lower.
Space was, doubtless, first measured by man; and for this purpose were employed the dimensions of various members of the human hody ; as the breadth of the hand, its extent when spread, called
the span, the breadth of the nail, and of the thumb, the length of the foot, and of the arm, and also the length of a step or pace. These things seem to have formed the basis of measures of length, in all nations. From the thumb is derived the inch; from the foot, as being, in length, about twelve times the thumb's breadth, the meas. ure still used of the same name; from the arm, as being about three times the length of the foot, the yard; and from the hand, the span and the pace, the measures, respectively bearing those names at the present time. It is said that the yard now in use in England, “was adjusted from the arm of Henry I. in 1,101, and that the old French pied du roi, (king's foot) had a similar origin." From these sprung higher denominations ; as the mile, being mille passuum, that is a thousand paces, &c. From lineal measurements, the transition was easy to those of surfaces and solids.
In the ruder ages, men weighed with the natural balance of the arms and hands; a balance, indeed, quite as rude as the age, in which it was employed. When greater accuracy seemed to become necessary, the artificial balance was constructed, on the hint, thus afforded by nature. From what circumstance weights derive their actual, or their comparative sizes, we do not know.
As the process of coining implies weight, money, properly speaking, was not probably employed, until long after the invention of the balance. Its denominations have usually been entirely arbitrary.
The division of time was naturally suggested by the succession of days and nights, by the revolutions of the moon, and the returns of . the seasons. The subdivisions into 60s, seem to have had their origin in Ptolemy's sexagesimal Notation. (see g vi.) It has been supposed by some, on the other hand, however, that, since the moon was observed to make 12 complete revolutions in a year, man naturally made the subdivisions of the day, and likewise of the night the same in number. 12 subdivisions of the day, and 12 of the night made 24 hours. Then, as the month contained 30 days and 30 nights, making 60 parts in the whole, the subdivisions of the hour were made by 60s..[See American Almanac for 1830.] This, however seems improbable, and the circumstance that the year contains considerably more than 12 revolutions of the moon, and likewise, that one revolution, (which measured the original month, does not contain 30 days, shows that the supposition has little foundation in fact. The division of time into periods of 7 days, or weeks, has been found to have been very extensively employed, by rude nations.
Any one who will compare the operations on Federal Money, with those on any other set of Compound quantities, which we have exhibited, will be ready to inquire, or will perhaps rather be able to answer the inquiry, why it is that the former are so much more simple and easy, than the latter. It is evidently because the law of increase, in Federal Money, is the same as that of simple numbers. The radix of each is 10. The pupil will now perceive how much reason we have to regret, that weights, measures, &c. should have been originally made to increase by ratios so irregular and so incona venient. But it must be recollected that they had their origin,
probably, before that of the Decimal Notation; and accident, which, we have seen, determined this Notarion, (vi.) determined like. wise the divisions of Compound Numbers. The ease with which we conduct processes on Federal money, results only from the fact that its ratio is the radix of our scheme of Notation. Then, before this Notation was established, there was nothing to give the decimal division of Compound Numbers any advantage over many others. And, if there had been, it is not t be supposed that rude and barba. rous nations would be l kely to form systems of division on scientif. ic principles. Present convenience or caprice would be their only guide. Yet notwithstanding all the inconvenience of our present divisions of weights and me sures, no question was made of their propriety, until sometime during the last century. Near the com. mencement of the French revolution, the National convention re. solved on a reform. They determined, to destroy, at a blow, the old systems, and to establish others, increasing decimally. This was accordingly done, and, of course, the numerical operations upon Compound Numbers, in France, immediately became as easy, as those upon Simple Numbers. The following is a brief account of THE FRENCH MONEY, WEIGHTS AND MEASURES.
The National Convention of France, on the first of August, 1793, resolved to introduce a new and uniform system of weights, measures, and generally of all compound quantities. The French Academy of Science, were requested by the Assembly, to draw up such a system; and, accordingly, in the year 1795, they submitted that which we are about to explain. This was immediately adopted, and its use enforced for 16 years, when the alterations took place which we have noticed below.
Commencing with measures of length, the Academy took great pains to obtain some fixed and universal standard. It seemed necessary to determine on some dimension in nature, which should not be liable to change with time, but which, existing always the same, should afford the means of rectifying any error which might, by any accident, or by variations insensibly creeping in, disturb the uni. formity of the system. It was, at length, determined that this should be the ten-millionth part of the distance from the equator of the earth to the pole, or, in other words, one ten-millionth part of the quadrant, (quarter of a circle,) of the terrestrial meridian.
By very accurate measurement, the length of this was discovered to be something more than 3.078 old French feet, or 3.281 English feet, nearly. This was taken as the unit of length, and called the METRE. The larger measures were made by multiplying this unit by 10, 100, &c., and the sub-divisions, by dividing it by 10, 100, &c.; in order that the whole might proceed on the decimal scale, and thus, correspond with the decimal Notation of numbers.
FRENCİL MEASURES OF LENGTH.
1 METRE, (the unit of length.)
1 deca-metre. 10 deca-metres
1 hecto-metre. 10 hecto-metres
1 kilo-metre. 10 kilo-metres
1 myria-metre. The word metre means measure. The names of the other denominations are formed from this, by PREFIXES. The prefixes for the sub-divisions of the unit, are taken from the Latin ; those for the larger measures, from the Greek,