It is to be borne in mind that, while the grain is one definitely fixed weight, being specified as 70th part of the Imperial Standard Pound Avoirdupois, the troy ounce and pound are different weights from the avoirdupois ounce and pound. The troy ounce contains 480 grains, and the avoirdupois ounce contains 522 grains, and the troy pound is 5760 of the avoirdupois pound. 7000 STANDARD OF LENGTH. By the Act of Parliament already referred to, 18th and 19th of Victoria, Chapter 72, July 30, 1855, it is enacted that the distance between the centres of the two gold plugs in a certain bronze bar, which had been prepared with extreme scientific care before the passing of the Act, and is accurately described in the Act, and stated to be deposited in the Office of the Exchequer, shall, when the temperature of the bronze bar is sixty-two degrees by Fahrenheit's thermometer, be deemed to be the IMPERIAL Standard Yard, * Troy weight was introduced into Europe from Cairo, in Egypt, about the time of the crusades, and was first adopted in Troyes, a city in France, where great fairs were held, and whence it has its name. This weight was formerly used for weighing articles of every kind; it is now employed in weighing gold, silver, jewels, and liquors; and grains counted in any numbers large or small, without the use of penny-weights or ounces are employed in philosophical experiments, but the French grams and milligrams are rapidly coming into use in their stead. Troy weight is also employed by apothecaries in mixing their medicines, though they buy and sell them by avoirdupois weight. When troy weight is thus used it is called APOTHECARIES' WEIGHT; but in this case the ounce (3) is divided into 8 drams (3), the dram into 3 scruples (9), and the scruple into 20 grains. † 3 barleycorns make an inch. The barleycorn, however, is never employed now as a measure. Instead, also, of being divided into lines, the inch is now generally divided into eighths, sixteenths, and thirty-seconds, or into tenths and hundredths. The perch is also sometimes called a pole, or rod. Each of the names given to this measure is expressive of the instrument by which it was formerly measured a rod, a pole, or perche, a French word of the same import. In some counties of England the perch is 6 yards, in some 7 yards, and in others 8 yards. In Cunningham measure it is 6 yards; in Forest measure, 8 yards; and in Woodland, or Burleigh measure, 6 yards. None of these, however, is now legal. The yard is said to have been taken from the length of the arm of Henry I. of England. § From this table, by an easy reduction, it will appear that a mile contains 320 perches, 1760 yards, or 5280 feet. Also 4 perches or 66 feet = 1 chain = 100 links; and 80 chains = 1 mile. A fathom is 2 yards, or 6 feet; a pace, 5 feet; a hand (used in measuring horses), 4 inches; a span, 9 inches, and a nautical or geographical mile, by Admiralty Regulation, is the length of one minute of longitude at the equator, and is ascertained to be about 6086 feet, or approximately 1000 fathoms. In nautical language 1th of a nautical mile is sometimes called a cable, and in connexion with this, th of a cable is called a fathom, being nearly equal to a true fathom. TABLE OF CLOTH MEASURE. 4 nails.........1 quarter | 4 quarters......= 1 yard.* A Flemish ell is 3 quarters of a yard; an English ell, 5 quarters, or a yard and a quarter; and a French ell, 6 quarters, or a yard and a half. Till the year 1826, the perch in Ireland contained 7 yards instead of 51, so that 11 Irish miles were equivalent to 14 British ones, and the Irish mile contained 2240 yards, or 6720 feet. *Cloth measure is a species of long measure, and the yard is the same in both. Hence, a quarter of a yard is 9 inches, and a nail 2 inches. † A square is a figure which has four equal sides, each perpendicular to the adjacent ones. A square inch is a square, each of whose sides is an inch in length; a square yard, a square, each of whose sides is a yard in length, &c. The table of square measure is formed from the table of long measure by multiplying the number there belonging to each lineal dimension by itself, as in the following example :-A square foot is=12 × 12=144 square inches, &c. In measuring land, surveyors use the chain, which, as is stated above, is 4 perches in length, and is divided into 100 equal parts, called links. They also compute by square chains and square links, but exhibit the result in acres, roods, and perches. Four lineal perches in Imperial or English statute measure being equivalent to 792 inches, it follows, from dividing by 100, that the length of a link is 7,92 inches. It may be observed, also, that 640 acres are a square mile and that a hide of land, mentioned by old writers, is 100 acres. 100 The relations among the chief land measures which have been, or are still, in use in England, Scotland, and Ireland, may be stated as follows: One Imperial acre = One Imperial acre = One Imperial acre = One Imperial acre = 121 of the acre in Irish Plantation Measure. 196 1089 of the Scotch acre. 1369 TABLE OF CUBIC OR SOLID MEASURE, OR MEASURE OF VOLUME. * 1728 cubic inches .......................... 1 cubic foot 27 cubic feet...............= 1 cubic yard. TABLE OF LIQUID AND DRY MEASURE.t For the use of readers who may be already acquainted with proportion, and reduction of fractions, and also with some of the principles of mensuration, it may here be explained that the magnitudes of the acres in the different Land Measures are proportional to the squares of the lineal perches in those Land Measures. Thus, on reference to the Table of Long Measure on page 58, and to one of the foot notes annexed to that Table, it will be seen that an Imperial perch lineal is 11 half-yards, and a lineal perch in Irish Plantation Measure is 14 half-yards; and then, by taking the squares of those two numbers, we have :one Imperial acre: one Irish acre :: 121; 196, and hence, and the numerical relations in the other cases may be brought out in like manner, with the aid of processes in reduction of fractions when necessary. In order to prevent a common mistake, it may be proper to remark that the difference in the comparative magnitudes of the acres is much greater than that of the lineal perches. Thus, while 121 Irish are equivalent to 196 English square perches, 121 Irish lineal perches are equivalent to only 154 English ones. The chain in Scotland, before 1826, was fixed at 74 feet; and hence, since the Imperial chain is 66 feet, we may find by comparison of square chains, which will give the same result as a comparison of square perches, that one Imperial acre: one Scotch acre:: 66 x 66: 74×74; and hence that one Imperial acre is 1089 of a Scotch acre. 1369 It may be observed that now the only legal measure for land is the English statute measure, called also Imperial measure. A cube is a figure contained by six equal squares. Dice afford a familiar instance of this figure. A cubic inch is a cube whose faces are each a square inch; a cubic foot, a cube whose faces are each a square foot, &c. It may be remarked that 1728 is equal to 12 x 12 x 12, and 27 to 3×3 x 3. ↑ By this measure, which is evidently a species of solid measure or measure of volume, liquids, and also grain and other dry goods are sold. The peck, bushel, and quarter are used only for dry goods. In reference to such goods, also, the wey or load, containing 5 quarters, and the last, 10 quarters, are sometimes spoken of but the sooner that the number of various units used in what is called "Liquid and Dry Measure," or generally in Measure of Volume, is greatly diminished, the better. The use of heaped measure was done away by Act of Parliament in 1835. The year is divided into 12 portions, called calendar months, the names of which are January, February, March, April, May, June, July, August, Sep TABLE OF ANGULAR MEASURE, OR OF ANGULAR DIVISION OF THE CIRCLE. An angle is the opening between two straight lines which meet. The point in which they meet is called the vertex of the angle; and the lines radiating out from that point may be spoken of as radial lines. If a straight line is kept constantly in a plane, and with one end centred at a fixed point, and is made to revolve with a motion like that of the hand of a clock, it moves through an angular space called a round, in turning one revolution. One fourth of a round is called a right-angle, or a quarter-round. Right-angles, degrees, minutes, and seconds are often considered as marked out by spaces on the circumference of a circle having any radius, and described round the vertex of the angle as centre.* Thus, if the circumference is divided into 360 equal parts, each of thes shows a degree of angular space round the centre, and is often spoken of as a degree on that circumference. If a degree space on the circumference is divided into 60 equal parts, each of these parts shows a minute of angular space round the centre. If a minute space on the circumference is divided into 60 equal parts, each of these parts shows a second of angular space round the centre.t tember, October, November, December. Of these, April, June, September, and November have 30 days each; and the rest, except February, have 31 days each. In leap years, February has 29 days; in common years, 28 days; so that a leap year contains 366 days, and any other 365. The precise length of the year is found to be 365 days, 5 hours, 48 minutes, 48 seconds; it is, therefore, 365 days, 6 hours, nearly. Leap years occur at intervals of 4 years, and may be known by dividing by 4 the number expressed by the last two figures in the number of the year, according to the Christian era: if there be no remainder, it is a leap year; otherwise, the remainder shows how many years it is after leap year. To this there is one exception, as the exact centuries are not leap years, except when the number of centuries is divisible by 4, without remainder. Thus, the year 1840 was a leap year, because 40 is divisible by 4; but 1839 was the third year after leap year, because 3 remain when 39 is divided by 4. Also, the year 2000 will be a leap year, but 1900 not, as 20 is divisible by 4, but 19 not. Hence, in 400 years there are 97 leap years. Learners may easily remember the number of days contained in each month by recollecting that the months are long and short alternately, with the exception of August, which, as well as July, is long, while the months after it follow the rule. * A circle is a plane figure bounded by one curved line, which line is such that all straight lines drawn to it from a certain point within the figure are equal. That point within the figure is called the centre of the circle. The bounding line is properly called the circumference of the circle. The name "circle," however, has come often to be applied to the circumference, so that it now sometimes means the bounding line, and sometimes the plane figure or space enclosed by that line. A radius of a circle is any straight line drawn from the centre to the circumference. The plural of the word radius is radii. A diameter of a circle is any straight line passing through the centre and terminated both ways by the circumference. An arc of a circle is any part of the circumfe rence. † Degrees, minutes, and seconds are marked thus: °,',". Hence, the expression, 41° 24' 54", is read, 41 degrees, 24 minutes, 54 seconds. The reason of these marks being employed will appear evident from the consideration that minutes and seconds are only abbreviated expressions for first minutes, or minutes of the first order, and second minutes, or minutes of the second order; minutes, in each instance, signifying small parts. It may also be remarked that seconds, both in time and in the circle, were formerly divided each into 60 thirds, but that they are now divided into tenths and hundredths. The angle between two radii of a circle, which comprise between them an arc of the circle equal in length to the radius, is a very important unit of angular measurement, and is called a radian. The radian is not commensurable with the round, quarter-round, degree, minute, or second. Its relation to these other units may, however, be given almost quite exactly by stating that: = One radian is 206265 seconds, approximately, this being true to the nearest second; or that : One radian is = 3438 minutes, approximately, this being true to the nearest minute:-or otherwise, by stating, in a decimal fractional expression, that: One radian is = {57.2958 degrees) approximately; or that or One half-round is = { 10000 3.1416 radians or 314010 radians j approximately. TABLES OF FRENCH WEIGHTS AND MEASURES ON THE METRIC SYSTEM will be found farther on in this treatise in the chapter entitled METRIC SYSTEM OF WEIGHTS AND MEASURES. It is a very important system, not in France alone, but throughout the world. By an Act of Parliament entitled the "Metric Weights and Measures Act, 1864," its use is legalized in Great Britain and Ireland. The subject can be better treated after the teaching of Decimal Fractions than at the present stage, and it is therefore deferred, REDUCTION. REDUCTION is the process of changing the numerical expression of a quantity, from its numerical expression by units of any one denomination, or set of denominations, to its numerical expression by units of any other denomination, or set of denominations. RULE I. To reduce a quantity in one denomination to a lower denomination. Multiply the number which expresses the quantity in the higher denomination by the number of units of the lower denomination which make one unit of the higher denomination. |