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To find the fractional part, proceed thus:
Write the several remainders in a horizontal line from right to left, beginning at the left hand with the last; then write the several factors in the same manner to the right of these, but separated by a curve, (. Multiply the first remainder by the first divisor, and to the product add the second remainder, (this can be done mentally in all cases to which this method of division applies), the sum of which is to be placed under the second remainder: multiply this sum by the next divisor, and add the product to the third remainder, putting the sum under the third remainder: multiply this sum by the next divisor, and so on till the last sum falls under the last remainder. This will be the entire remainder which would result from dividing the dividend by the entire divisor *.
* The proof of the truth of this rule may be given as follows; and the example worked will show the nature of the notation employed.
Let the several remainders (reckoned backwards) be r1, 72, 73,
and the divisors which
gave them be d1, d2, dз, . . . . Then the preceding fractions being all to be divided by the successive divisors (they forming parts of the numbers successively divided) we have
Reducing these to a common denominator, we have
r1 d2 dз d4 d5 ... + r2 dz d4 d5. +r3d4 d5 +r4d5. +15
where the continuing dots express that the multiplication and addition are to be carried to the extent of embracing all the terms. We may suppose them to be five, as in the work written down, since the process is the same however many there may be, and the steps are continuous. Then this reduction may be gradually effected thus:
2. Divide 7014596 by 72.
3. Divide 5130652 by 132. 4. Divide 83016572 by 240. IV. Common Division may be performed more concisely, by omitting the several products, and setting down only the remainders; namely, multiply the divisor by the quotient figures as before, and, without setting down the product, subtract each figure of it from the dividend, as it is produced; always remembering to carry as many to the next figure as were borrowed before. This is not, however, to be recommended till considerable practice has conferred on the pupil the power of carrying on two processes at once; namely, multiplication and subtraction.
REDUCTION is the changing of numbers from one name or denomination to another, without altering their value. This is chiefly concerned in reducing money, weights, and measures.
When the numbers are to be reduced from a higher name to a lower, it is called Reduction descending; but when contrariwise, from a lower name to a higher, it is Reduction ascending.
Before we proceed to the rules and questions of Reduction, it will be proper to set down the usual tables of money, weights, and measures, which are as follow.
The full weight and value of the English gold and silver coin, both old and new, are subjoined.
The usual value of gold is nearly 4l. an ounce, or 2d. a grain: and that of silver is nearly 58. an ounce. Also the value of any quantity of gold, was to the value of the same weight of standard silver, as 1524 to 1, in the old coin; but in the new coin they are 147 to 1.
Pure gold, free from mixture with other metals, usually called fine gold, is of so pure a nature, that it will endure the fire without wasting, though it be kept continually melted. But silver, not having the purity of gold, will not endure the fire like it: yet fine silver will waste but a very little by being in the fire any moderate time; whereas copper, tin, lead, &c. will not only waste, but may be calcined, or burnt to a powder.
Both gold and silver, in their purity, are so soft and flexible, (like new lead, &c.) that they are not so useful, either in coin or otherwise (except to beat into leaf gold or silver), as when they are alloyed, or mixed and hardened with copper or brass. And though most nations differ, more or less, in the quantity of such alloy, as well as in the same place at different times, yet in England the standard for gold and silver coin has been for a long time as follows: viz. That 22 parts of fine gold, and 2 parts of copper, being melted together, shall be esteemed the true standard for gold coin: And that 11 ounces and 2 pennyweights of fine silver, and 18 pennyweights of copper, being melted together, be esteemed the true standard for silver coin, called Sterling silver.
In the old coin the pound of sterling gold was coined into 42 guineas, of 21 shillings each, of which the pound of sterling silver was divided into 62. The new coin is also of the same quality or degree of fineness with that of the old sterling gold and silver above described, but divided into pieces of other names or values; viz. the pound of the silver into 66 shillings, of course each shilling is the 66th part of a pound; and 20 pounds of the gold into 934 pieces called sovereigns, or the pound weight into 468 sovereigns, each equal to 20 of the new shillings. So that the weight of the sovereign is 4628ths of a pound, which is equal to 585 pennyweights, or equal to 5 dwt. 3 gr. very nearly, as stated in the preceding tables. And multiples and parts of the sovereign and shilling in their several proportions.
WEIGHTS AND MEASURES,
Agreeably to the Act of Uniformity, which took effect 1st January, 1826.
The term MEASURE is the most comprehensive of the two, and it is distinguished into six kinds, viz. :
3. Solidity, or Capacity.
4. Force of Gravity, or what is commonly called Weight.
The several denominations of these Measures have reference to certain standards, which are entirely arbitrary, and consequently vary among different nations. In this kingdom
Length is a Yard.
Surface is a Square Yard, the 1940 of an Acre.
The standard of... Solidity is a Cubic Yard.
Capacity is a Gallon.
The Standards of Angular Measure, and of Time, are the same in all European and most other countries.
An Inch is the smallest lineal measure to which a name is given; but subdivisions are used for many purposes. Among mechanics, the Inch is commonly divided into eighths. By the officers of the revenue, and by scientific persons, it is divided into tenths, hundredths, &c. Formerly it was made to consist of 12 parts, called lines, but these have properly fallen into disuse.
Imperial Measure of CAPACITY for all liquids, and for all dry goods, except such as are comprised in the third Division.
The four last denominations are used for dry goods only. For liquids, several denominations have been heretofore adopted, viz. :-For Beer, the Firkin of 9 gallons, the Kilderkin of 18, the Barrel of 36, the Hogshead of 54, and the Butt of 108 gallons. These will probably continue to be used in practice. For Wine and Spirits, there are the Anker, Runlet, Tierce, Hogshead, Puncheon, Pipe, Butt, and Tun; but these may be considered rather as the names of the casks in which such commodities are imported, than as expressing any definite number of gallons. It is the practice to gauge all such vessels, and to charge them according to their actual content.
Flour is sold, nominally, by measure, but actually by weight, reckoned at 7 lb. Avoirdupois to a gallon.
Imperial measure of CAPACITY for coals, culm, lime, fish, potatoes, fruit, and other goods, commonly sold by heaped measure:
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 of its depth. The outside diameter of Measures used for heaped goods are to be at least double the depth; consequently, not less than the following dimensions :
The Imperial Measures described in the second and third Divisions were established by Act 5 Geo. IV. c. 74. Before that time there were four different measures of capacity used in England. 1. For wine, spirits, cider, oils, milk, &c.; this was one-sixth less than the Imperial Measure. 2. For malt liquor, this was part greater than the Imperial Measure. 3. For corn, and all other dry goods not heaped, this was 33 part less than the Imperial Measure. 4. For coals, which did not differ sensibly from the Imperial measure.
The Imperial Gallon contains exactly 10 lbs. Avoirdupois of pure water; consequently the pint will hold 1 lb., and the bushel 80 lbs.
This weight is used in almost all commercial transactions, and in the common dealings of