In this example, the second divisor is not contained in the corresponding dividend, so that o is the third figure of the root. The first line of the new divisor will then be formed by adding two cyphers to the 86700 already found.

The cube root of a fraction may be found by taking the cube roots of the numerator and denominator; mixed numbers being first expressed as improper fractions.

But if the denominator be not a perfect cube, it will be best to reduce the fraction to a decimal, and then extract the root.

N.B.-The first line of each divisor after the first in the above process, may be obtained more shortly by adding to the preceding divisor its second line and twice its third line, and then annexing two cyphers,

Thus in Ex. (2) 3484+720+128=4332; in Ex. (3) 559+210+98=867; and 8715981+45900+162=8762043.

(7) 17183498535125.

(4) 304957115891.

(6) 104600290750613.

(8) 10, 3, and 8 to four places of decimals.

(9) Find a number whose cube equals the sum of the cubes of 14'1, 18°8, 23°5.

(10) Determine the edge of a cube whose volume is 16 yds. 18 ft. 1088 in.

APPENDIX.

NOTES ON CHAPTER I.

Art. 2.-THE use of 10 as the base of the common system of notation had no doubt its origin in the number of figures on the two hands. Any other integral number would have served the same purpose.

Though generally called the Arabic notation, our system was not invented by the Arabs, but was obtained by them from India, on which account it is also called Hindoo.

Art. 6.-The following test of correctness may be sometimes usefully applied to multiplication: Divide the sum of the digits in the multiplicand and in the multiplier by 9, and multiply the remainders together. The remainder when this is divided by 9, should be the same as the remainder when the sum of the digits of the product is divided by 9. This is often known as the test by casting out the nines.

For instance, in Ex. (3), the sum of the digits in 2549 is 20, which when divided by 9 gives a remainder 2; the sum of the digits in 58 is 13, which gives a remainder 4; and 2×4=8. Also the sum of the digits in 147842 is 26, which gives the same remainder, 8.

Similarly in Ex. (6), the remainders from the multiplicand and multiplier are 6 and 4; and the remainder from 6×4 is 6, which is the same as the remainder from the sum of the digits in the product, 33.

This however is not to be considered as a complete test; for although it will be satisfied whenever the multiplication has been performed correctly, there are several errors, such as the interchange of digits, the wrong position of a line, &c., which it will not detect.

NOTES ON THE TABLES OF MONEY, &c.

The letters £. s. d. q. stand for Libræ, Solidi, Denarii, Quad

rantes.

The fineness of gold is expressed by the number of carats of fine (i, e. pure) gold that there are in a lb. Troy. Thus standard gold is said to be 22 carats fine. The Mint price of standard gold is £3 178. 10d. per oz., and of standard silver 58. 6d. per oz.

TROY weight takes its name from Troy-novant, the old monkish name for London.

AVOIRDUPOIS from avoirs, the old Norman word for "goods and chattels," and pois "weight."

The original unit of all our weights was a grain of wheat, 32 of which taken from the middle of the ear and well dried, were in Edward I.'s time equal in weight to a silver penny. penny-weight was reduced by Henry III. to 24 grains.

8 lbs. Troy of wheat determined the size of a gallon.

The

The original unit of our measures of length was the grain of barley, 3 of which from the middle of the ear and well dried, were taken as an inch.

The present standards of measure and weight are the Imperial Standard Yard and the Imperial Standard Pound Troy, which were made and legalised in 1836; the earlier standards (made respectively in 1760 and 1758) having been injured by the fire at the Houses of Parliament in 1834. The standard yard is kept at the Greenwich Observatory, and consists of a bar of gun metal 38 inches long and 1 inch square, near the ends of which are two small cylindrical holes half through the metal. The distance between the centres of the two holes, when the metal is at the temperature of 62° Fahrenheit, is the standard yard. If the standard yard be lost or injured, it is enacted that it shall be restored by determining the length of a pendulum which in the latitude of London at the level of the sea would in a vacuum vibrate seconds. Such a pendulum would measure 39°1393 inches, and 36 of these would be the yard.

If the standard pound be lost or injured, it is to be restored by means of the weight of a cubic inch of distilled water, which at a temperature of 62° Fahrenheit, would in a vacuum weigh

252 724 grains; and 5760 such grains would be the pound Troy.

The imperial gallon is to contain 10 lbs. Avoirdupois weight of distilled water, the barometer being at 30 inches, and the temperature 62°.

TIME.-A solar day is the interval between two successive passages of the sun across the meridian of a place. Its length varies very slightly at different periods of the year, and its average length, called the mean solar day, is the day of which an hour is the twenty-fourth part.

A year is the time in which the earth performs one revolution in its orbit round the sun, and consists of 365°242218 mean solar days. Its exact length was only gradually ascertained. Before the time of Julius Cæsar it had been estimated at 365 days; and correcting this on the supposition that 365 days was the true length, he introduced a calendar in which 365 days was the length of the ordinary year, but every fourth year (bissextile or leap-year) was to consist of 366 days. This correction, however, was too large by (365°25 - 365*242218=) *007781 of a day, and consequently an error existed which would in four centuries amount to 3.1124 days.

In the new calendar introduced by Pope Gregory XIII. in 1582, a correction for this error is made by considering three leap years in every four centuries as common years; and the error is thus reduced to 1124 of a day in 400 years, i. e. one day in about 3600 years. The leap-years appointed to be so passed over are the centuries whose numbers are not divisible by 4, e.g. 1700, 1800, 1900; 2100, &c.

On establishing this new calendar in 1582, Pope Gregory decreed-in order that the vernal equinox might fall on March 21st (on which day it fell in A.D. 325, the year of the Council of Nicaea)—that the accumulated error since 325 should be rectified by the omission of 10 days in October of that year; and when the Gregorian calendar was introduced into England in 1752, a similar correction was made, the eleven days between September 2nd and September 14th in that year being omitted. The Julian calendar, which is still used in Russia, is known as the Old Style, and the Gregorian as the New Style: the difference in their reckoning is now 12 days.

It may be mentioned that before A.D. 1752, the legal year in