of 14 to 17 nearly; but the troy ounce is greater than the avoirdupois in the proportion of 192 to 175, or of 79 to 72 nearly. (Encyclopædia Britannica, 7th ed. vol. xxi. p. 850.) Imperial measure of capacity for coals, culm, lime, fish, potatoes, fruit, and other goods, commonly sold by heaped

The goods are to be heaped up in the form of a cone, to a height above the rim of the measure, of at least three fourths of its depth. The outside diameters of measures, used for heaped goods, are to be at least double the depth, consequently not less than the following dimensions:

Gallon............ 93 inches
Half gallon...... 73

The imperial gallon contains exactly 10 lbs. avoirdupois of pure or distilled water; consequently, the pint will hold 1 lb. and the bushel 80 lbs.

A roll of parchment, or vellum, is five A load of straw, 36 trusses, of 36 lbs.

dozen, or 60 skins.

A dicker of hides is 10 skins.

A last of hides is 20 dickers.

A dicker of gloves is 10 dozen pairs.

A firkin of butter is 56 lbs.

A firkin of soap is 64 lbs.

A tierce of rice is about 5 cwt.

each. A chaldron of coals, in London, is 36 bushels. Coals are sold by the ton.

A chaldron of coals, in Newcastle, is
53 cwt.

A cart of coals, in Scotland, is 12 cwt.
A deal of coals, in Scotland, is 23 cwt.

A hogshead of tobacco is from 9 to 10 A grain of gold is worth about 2d.

cwt.

A barrel of gunpowder is 1 cwt.

An ounce of fine silver is worth from 5s. to 5s. 6d.

Measures of Time.

Time is a measure of duration, ascertained by the motions of the heavenly bodies, and is either apparent, mean, or sidereal. (See the Table at the end of the Work.)

The Julian year is equal to 365 days 6 hours.

Solar year

Civil year

Lunar month..

Civil, or natural day.

365 days, 5 hours, 48 min. 48 sec.
365 days, or 12 calendar months.

4 weeks, or 28 days.

24 hours.

60 minutes.

60 seconds.

Subsect. II.

RULE FOR CALCULATING INTEREST AT

5 PER CENT.

Multiply the pounds by the days, and divide the product by 365. The quotient gives the interest at 5 per cent. in shillings.

If the wages be guineas instead of pounds, for each guinea add one penny to each month, or one farthing to each week.

First term, or Lady-day, falls on the 25th March.

Note.In Scotland, the first quarter term is on Candlemas, or the 2d of February; the second term, Whitsunday, is on the 15th of May; the third term, or Lammas, on the 1st of August; and the fourth term, Martinmas, on the 11th of November.

SECTION V.

PROPORTION AND INTEREST.

Subsect. I. OF PROPORTION, OR THE RULE OF THREE.

PROPORTION is that rule by which, from a comparison of circumstances, arising from certain conditions or stipulations, certain conclusions are drawn, and consequences deduced and ascertained.

In simple proportion, three numbers are always given to find a fourth; of which the first two are always conditional, and the third implies a demand, and in consequence moves the question. In all direct processes the answer, or fourth proportional, bears the same ratio to the third as the second bears to the first; wherefore, the greater the second term is in respect to the first, the greater will the fourth term be in respect to the third; and the less the second term is in respect to the first, the less will the fourth term be in respect to the third. Hence in all direct proportions, the product of the extremes will always be equal to the product of the means. On the other hand, if the terms are in reciprocal proportion, the fourth proportional must always bear the same ratio to the second as the third does to the first; consequently, the

greater the third term is in respect of the first, the less must the fourth be in respect to the second; and the less the third is, compared with the first, the greater will the fourth be, compared with the second. Hence again, in reciprocal proportion, the product of the first and second terms will always be equal to the product of the third and fourth.

To state the question.

Rule I. Write down that number or term which is of the kind, whether money, weight, measure, time, &c. with the answer, for the middle term.

Rule II. On the right of the middle term already written down, place the term upon which the demand lies.

Rule III. On the left of the middle term, place that term of the two conditional ones which is of the same kind with the term on the right. Then will the terms be placed in a

proportional order.

Or, write down that term on which the demand lies for the middle term, and the term homogeneal with it, on the left, and the term homogeneal with the answer on the right.

To find a fourth proportional.

Rule. If, upon comparing the first and third, more be found to require more, or less to require less, then will the terms be in direct proportion, and the product of the two last divided by the first, will quote the answer; but if less require more, or more require less, then will the product of the two first, divided by the last, quote the answer.

Note. Similar terms, that is, the first and third, or second and fourth, must always be of the same denomination, and the preparation may be made as in Reduction, Vulgar or Decimal Fractions.

Ex. 1. Bought 1751 yards of cloth, and paid 1057. 12s. Od. what did it cost me per piece of 25 yards?

ގ

Another question naturally suggests itself to the reader, which is this: if there are 25 yards in one piece, how many pieces are there in 175 yards? And if he operate by saying as 25: 1 :: 175.5 to the answer, he will find that there are 7 pieces, and a remainder of half a yard.

Illustration. Because the answer is to be money, the price is put for the middle term, on the right of which stands the term which implies a demand, 25 yards, and on the left 1751 yards, being the term which is of the same kind with 25. Now, as 25 yards, when compared with 1751, reckon less, and in consequence must bring less, the two last terms are multiplied together, and their product divided by the first. Previous to any multiplication or division, because in the first term there is a fraction, the first and third terms are reduced into parts expressed by the denominator; and since there are shillings in the second, as well as pounds, the shillings are reduced to a decimal.

The same answer may be effected decimally, thus: 175.5 105.6 :: 25: 15.0. 10 nearly. 105.6 x 25

For

175.5

15.042 15. 0. 10 nearly.

Then £15.042 × 7·02 = £105. 12s. Od. for the whole.

In questions of this character, the product of the extremes is equal to the product of the mean terms; but in the present question we must consider that the 25 yards constitute but 7.02 parts of the whole quantity; therefore the proof is effected by multiplying the 4th term, 157. Os. 10d., by 7.02, and the product will be 1057. 12s. nearly.