184. With the above explanation (Arts. 180 and 182) on the subject of pointing, the rule for extracting the square root of a decimal, or of a number consisting partly of a whole number and partly of a decimal, will be the same as that before given (Art. 181) for finding the square root of a whole number. As the decimal notation is only an extension or continuance of the ordinary integral notation, and quite in agreement with it, the reason before given for the process, will in fact apply also here.

185. To extract the square root of a vulgar fraction, if the numerator and denominator of the fraction be perfect squares, we may find the square root of each separately, and the answer will thus be obtained as a vulgar fraction; if not, we can first reduce the fraction to a decimal, or to a whole number and decimal, and then find the root of the resulting number. The answer will thus be obtained either as a decimal, or as a whole number and decimal, according to the case. Also a mixed number may be reduced to an improper fraction, and its root extracted in the same

way.

Ex. 4. Extract the square root of 4 to four places of decimals.

Ex. 5. Extract the square root of 0006 to four places of decimals.

Ex. 6. Extract the square root of '0365 to five places of decimals. 0365000000 (*19104

This may be done by first reducing to a decimal, and then by extracting the square root of the decimal, thus = '714285...

(1) 289; 576; 1444; 4096. (2) 6561; 21025; 173056.

(3) 98596; 37249; 11664.

(4) 998001; 978121; 824464.

(5) 29506624; 14356521; 5345344.

(6) 236144689; 282429536481; 282475249.

(7) 295066240000; 4160580062500.

2. Find the square roots of

(1) 167-9616; 28 8369; 57648 01. (2) 3486784401;39-15380329
(3) 042849; 00139876; 00203401. (4) 5774409; 5·774409.
(5) 120888-68379025; 240398′012416.

to four places of decimals in each case where the root does

not terminate.

4. Extract the square root of ⚫0019140625 and reduce the result to the corresponding equivalent fraction in its lowest terms.

5. Find the side of a square field equal in area to a rectangular field 700 yards wide and 2800 yards long.

6. A square field contains 1 ac., 22 po., 7 yds.; find the length of its side.

7. A rectangular field measures 225 yards in length, and 120 yards in breadth; what will be the length of a diagonal path across it?

8. Find the length of the side of a square enclosure, the paving of which cost £27. 1s. 6d. at 8d. per sq. yard.

9. The hypothenuse of a right-angled triangle is 51 yards, and the perpendicular is 24 yards, find the base.

10. A ladder, whose length is 91 feet, reaches from the extremity of a path 35 feet wide, to a point in a building on the other side, which is within 9 inches of the top of it; find the height of the building.

11. Extract the square root of '0050722884, and find within an inch the length of a side of a square field the area of which is 2 acres.

12. Two persons start from a certain point at the same time, the one goes due east at the rate of 12 miles an hour, and the other due north at the rate of 9 miles an hour; how far are they distant from each other at the end of six hours?

13. A ladder 36 feet long will reach to a window 28 feet from the ground, on one side of a street; and if the foot of the ladder be retained in the same position, will reach to a window 26 feet high on the other side. Find the breadth of the street.

14. A society collected among themselves for certain purposes a fund of £45. 18s. 9d.: each person paid as many pence as there were members in the whole society. Find the number of members.

15. The area of a circular lake is 295066-24 square yards, how many yards are contained in the side of a square of equal superficies?

CUBE ROOT.

186. The CUBE of a given number is the product which arises from multiplying that number by itself, and then multiplying the result again by the same number. Thus 6 × 6 × 6 or 216 is the cube of 6.

The cube of a number is frequently denoted by placing the figure 3 above the number, a little to the right. Thus 63 denotes the cube of 6, so that 636 × 6 × 6 or 216.

187. The CUBE ROOT of a given number is a number, which, when multiplied into itself, and the result again multiplied by it, will produce the given number. Thus 6 is the cube root of 216; for 6×6×6 is=216. The cube root of a number is sometimes denoted by placing the sign before the number, or placing the fraction above the number, a little to the right. Thus /216 or (216)* denotes the cube root of 216; so that 3/216 or (216)*= 6.

3

188. The number of figures in the integral part of the cube root of any whole number may readily be known from the following considerations:

Hence it follows that the cube root of any number between 1 and 1000 must lie between 1 and 10, that is, will have one figure for its integral part; of any number between 1000 and 1000000, must lie between 10 and 100, that is, will have two figures in its integral part; of any number between 1000000 and 1000000000, must lie between 100 and 1000, that is, must have three figures in its integral part; and so on. Wherefore, if a point be placed over the units' place of the number, and thence over every third figure to the left of that place, the points will shew the number of figures in the integral part of the root. Thus the cube root of 677 consists, so far as it is integral, of one figure; that of 198999 of two figures; that of 134198999 of three figures; and so on,

189. The following Rule may be laid down for extracting the Cube Root of a whole number.

RULE. "Place a point or dot over the units' place of the given number, and thence over every third figure to the left of that place, thus dividing the whole number into several periods. The number of points will shew the number of figures in the required root. (Art. 188.)

Find the greatest number whose cube is contained in the first period at the left; this is the first figure in the root, which place in the form of a quotient to the right of the given number.

Subtract its cube from the first period, and to the remainder bring down the second period.

Divide the number thus formed, omitting the two last figures, by 3 times the square of the part of the root already obtained, and annex the result to the root.

Now calculate the value of 3 times the square of the first figure in the root (which of course has the value of so many tens) +3 times the product of the two figures in the root+the square of the last figure in the root. Multiply the value thus found by the second figure in the root, and subtract the result from the number formed, as above mentioned, by