probably, before that of the Decimal Notation; and accident, which, we have seen, determined this Notation, (§ vr.) 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 to be supposed that rude and barbarous nations would be likely 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 measures, no question was made of their propriety, until sometime during the last century. Near the commencement of the French revolution, the National convention resolved 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, oné 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.

FRENCH MEASURES OF LENGTH.

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.

In measuring SURFACE, a SQUARE DECA-METRE, equal to 100 square metres, was made the unit. This unit is called the ARE, from the Latin area, meaning surface. The names of the denominations, all of course, graduated by the deci mal scale, are as follows, beginning with the least, and proceeding to the greatest. Centiare, (square metre); Deciare; ARE, (unit of surface); Decare; Hectare; Kilare; Myriare. The square deci-metre, centi-metre, and milli-metre are used for smaller measures.

In measuring CAPACITY, a CUBIC DECI-METRE, equal to .001 of a cubic metre, was made the unit. This unit is called the LITRE, from litron, the name of an old French measure, of one-fifth less capacity. The denominations increase decimally, as before, and are Centi-litre; Deci-litre; LITRE, (unit of capacity); Deca-litre; Hecto-litre; Kilo-litre, (cubic metre); Myria-litre. In measuring SOLIDS, a CUBIC METRE was made the unit. This unit is called the STERE, and will be seen to be equal to the Kilo-litre, in the above measures of Capacity. The denominations, increasing decimally, are Deci-stere; STERE, (cubic metre and unit of solidity); Deca stere. We have given the length of the metre, and from that may be found, if necessary, all the other dimensions. In measuring TIME, the day was divided into 10 hours; each hour, into 100 minutes; and each minute into 100 seconds. Thus a French hour became 2h. 24m. English, or common hours; and the French minute, Im. 26.4 sec. English. In measuring THE CIRCLE, 100" made I'; 100' made 10; 100° made I quadrant, (quarter of a circle); and 4 quadrants, or 400° made the whole circle. By this division, one degree became 54 minutes as they are commonly measured. WEIGHTS were connected with measures of extension, by assuming as the unit of weight, the weight of a CUBIC CENTI-METRE of pure WATER at a medium temperature. This unit was called a GRAMME, and the names of the other denominations were formed, as above, by prefixes. The French MONEY was made very sim ple. 100 centiemes, (hundredths,) make I franc. The centieme was made to weigh 2 grammes, and the silver franc, 5 grammes. Even the thermometer was decimally divided, and the Paris CENTIGRADE THERMOMÈTER has 0 at the freezing, and 100 at the boiling point. Experience proved, however, that these divisions, though they rendered calculations extremely easy, and, in fact, reduced all compound quantities to simple, were not as convenient in practice, as many of those, of the systems exploded. On the decimal system, no even quar ter, third, or sixth of a foot could be obtained, as on the duodecimal, and as these fractions are of frequent occurrence in practice, much inconvenience was experi enced. It was found, too, that "for the operations of weighing, and measuring capacities, the continual division by 2 renders it practicable to make up a given quantity with the smallest number of standard weights and measures, and is far preferable, in this respect, to any decimal scale." (Report Brit. House of C. June 24, 1819-] These circumstances rendered the people dissatisfied. The govern ment found it necessary, therefore, to decree, on the 28th March, 1812, that the old names of the toise, the ell or aune, the foot, the inch, and the bushel, should be allowed; but that, in order to preserve the metrical system, they should represent new measures, related to that system as follows:

The toise 2 metres.

The footmetre.

2

The inch metre.
The aune 1 metres.

The bushel of a hecto-litre.

These measures were likewise subdivided duodecimally, instead of decimally. on the ground that "the decimal system, though favorable for calculations, is not equally so for the daily operations of the people, and is not easily comprehended.". (Decree, Mar. 28, 1812.)

Thus we see, that the French weights and measures, at the present day, are. for the most part, only decimal in theory, while, in practice, they are duodecimal. Similar in this respect, in our own country, are the measures of land. Surveyors, by the use of Gunter's chain, are enabled, as has been shown, § XXXIII, to make their calculations decimally. But, at the same time, the measures in common use, so far from being on the decimal scale, or from increasing uniformly, are

among the most inconvenient we have. This shows the great difficulty of making innovations upon systems in common use, even where convenience seems to demand it.

It should be recollected, that the French decimal system was introduced at the recommendation of philosophers and men of learning, who, having little occasion to engage in those occupations, in which it becomes necessary to make frequent use of weights and measures, over-looked entirely one most important part of their subject, viz. the practical; and kept in view only that which most concerned them, viz. the theoretical. The result of this experiment has been to show, that it is out of the question ever to think of reducing compound numbers to correspond entirely with the decimal Notation. Much might doubtless be done towards rendering them more uniform and simple than they are at present, were it not for the difficulty of the attempt, as well as the inconvenience, and the confusion that must result from any material change in a system so long established, and so interwoven with all the transactions of commerce, as well as the processes of the arts. There has probably never been, in the history of the world, an occa sion, on which such a change could have been effected, if we except the French revolution. The people of that country were, at that time, so carried away by their rage for innovation, that they were ready to root up and destroy every thing ancient and venerable; and that, too, for the very reason which induces other nations to cherish and esteem their institutions, viz. their antiquity and long familliarity. They conceived every thing old, to savor of the monarchy, and therefore, they waged against it a most furious warfare. Had this zeal been turned into another channel, it might have been productive of much good. Had the favorable moment been seized for the introduction of the duodecimal scale of Notation, (see § VI.) more would have been accomplished towards promoting the cause of science, than it will probably ever again be within the scope of a legis. lative act to effect.

The subject of weights and measures having been submitted in 1819, by the Prince Regent of Great Britain, to a committee of distinguished men, they reported decidedly against the decimal scale, but proposed some slight alterations in the existing system. Accordingly, in 1825, by act of Parliament, the following changes took place.

The bushel which had been before used in England, was called the Winchester bushel, and was a cylinder 8 inches deep, and 18 inches in diameter, containing 2,150.4 cubic inches. The gallon, by dry measure, therefore, contained one eighth as much, or 268.8 cubic inches. The beer gallon contained 282, and the wine gallon 231 cubic inches.

These measures were abolished, and the bushel ordered to contain 2,217.6 cubic inches, and the gallon, one-eighth as much=277.2 cubic inches. This latter measure is used both for dry and liquid measure.

The capacity of these measures is determined by weighing them, at first empty, and afterwards filled with pure distilled water, at the temperature of 62° of Fahrenheit's thermometer. The bushel ought to hold 80, and the gallon 10 lbs. Avoirdupois.

The weights, however, were first determined from measures of capacity, 19 cubic inches of distilled water, at the temperature of 50°, being declared to weigh 10 oz. or 4,800 grs. Troy, and 7,000 grs. Troy, to be equal to 1 lb. Avoirdupois. From this, also, we have the relation of the weights Avoirdupois and Troy. 7,000 grs. reduced, make 1 lb. 2 oz. 11 dwt. 16 gr., which expresses the weight of 1 lb. Avoirdupois, in Troy weight. 1 lb. Troy, in Avoirdupois weight, is 13 oz. 2 dr. Thus the lb. Avcirdupois is greater than the lb. Troy, and the oz. Troy, greater than the oz. Avoirdupois.

114

It may still be enquired, how is the measure of length determined, on which these measures of capacity and weight depend? It was found that, according to the scale adopted, the pendulum vibrating seconds on a level with the sea, and in a vacuum, at London, was 39.1372 inches, and the French metre, being the ten millionth part of the quadrant of a meridian on the earth, was 39.3694 inches. Hence, there are two natural standards with which it may always be compared, and by which it may be rectified.

1000000

In 1,827, the legislature of the state of New-York passed an act, regulating weights and measures. The yard was made the standard of length, and was ordered to be 188 of the pendulum vibrating seconds at Columbia college, in the city of New-York, this pendulum being 39.10107 inches. The pound was made the unit of weight, and its magnitude determined by the weight of a cubic foot of water, which, at 40° temperature, weighs, on this scale, 624 lbs., or 1000 oz. The gallon holds 10lb. and the bushel 80 lb. Hence, in the bushel, are 2,211.84 cubic inches, and in the gallon 276.48 cu. in.

In the state of Connecticut, the bushel contains 2,198 cubic inches, and the galJon, dry measure, 2744. The ale and wine gallons are of the same dimensions with the old English measures of the same kind mentioned above. No scientific standard has been adopted, nor has the mode of testing capacities by weight been established by law. Most of the states of the uinon have been as inattentive to this subject as Connecticut. They have all, indeed, legislated on the subject, but their enactments have been made without the advice or assistance of scientific men. It is to be hoped that other states will soon follow the example of New-York.

PROMISCUOUS EXAMPLES.
MENTAL EXERCISES.

§ XL. 1. of 35 are how many times 4? of 40 are how many times 7? 2.

of 60 are how many times 5?

many times 8?

3. of 24 are how many times 10? many times 9?

4. of 40 are how many 7ths of 21? many ninths of 72?

5. 1 of 24 are how many fourths of 16?

how many fifths of 50?

of 45 are how

of 42 are how

of 16 are how

of 40 are

of 63 are

6. of 48 are how many elevenths of 44? how many eighths of 72?

7. 24 is of how many times 6? 36 is of how many

many times 5? 10. 77 is

& of how

of how many fourths of 32? 64 is many sevenths of 49? 11. 42 is of how many fifths of 30? 49 is of how many thirds of 36?

12. 39 is of how many twelfths of 48? 60 is how many eighths of 72?

of how many times 6? of 15 is

13. of 24 is of how many times 3?

of

14. g of 32 is

of how many times 5?

of 44 is

of how many times 8? 15. 1 of 45 is 20 of how many times 11? 16. 7 of 64 is 1 is of of how many of 66 is of of how many times 9? T of 80 of how many times 9?

of how many times 4? 1 of 15 is

17. 1

is of 18.

of

of how many times 7? g of 40 times 8 ?

of 15 is of of how many times 12? is of of how many times 5? 19. 2 of 40 is

of 24

of how many 4ths of 28? 1

of 72 is

of how many 3ds of 15? 22. of 35 is

54 is

of of how many 5ths of 45? § of

of of how many 8ths of 56?

23 of 32 is

of of how many 6ths of 54? 12

of 32 is of 4 of how many 9ths of 54? 24. 14 of 34 is of 18 of how many 8ths of 3 of 40? of 30 is of of how many 3ds of of 44? 25. of 45 is

of 39 is

of of how many 4ths of 3 of 10? of of how many 7ths of of 28?

The following are to be written. 26. A man collected debts, of the following amounts; 12£. 9s. 1d. 3 qrs.; 19£. 2s. 11d. 1qr.; 47£. 10s. 9d. 2 qrs. ; and 14£. 13s. 5d. How much did he receive in all ? Ans. 90£. 16s. 3 d.

27. I pay for cloth £14; 19; 6; for flannel, £11; 4; 9; for grain, £25; 10; for sugar, £4; 0; 6; for coffee, £3; 6; 8; and for molasses 19s. 6d. What cost the whole? Ans. £60; 0; 11.

28. A farmer brought to market butter, which brought him 54s. cheese, which brought 59s.; a load of wood, 49s. 9d. 2 qrs.; eggs, 398. 8d.; and apples, 478. 9d. In part payment he received 4 lb. of tea, at 4s. 9d. pr. lb.; 12 lb. sugar, at 8d pr. lb.; 3 shovels at 6s. apiece; 5 hoes, at 3s. apiece; 8 yds. of cloth, at 1£. 3s. 6d. pr. yd., and the rest in money. How much money did he receive? 2s. 2 d.

29. What cost 11 T. of hay at 2£. 1s, 10d. pr. T.?

0; 2.

30. 5 men shared equally a prize of £1,444; 1; 6. each man's share? Ans. £288; 16; 3.

Ans.

Ans. £23;

What was

31. In 9 bales of cloth, each containing 12 pieces, and each piece, 27 yds. 1 qr. 2 nls. how many yds? Ans. 2,956 yds. 2 qrs. 32. A man, having £1,000, lost 41d. What had he left? Ans. £999; 19; 7.