EXAMPLE 1. What is the cube root of 378.4?

NOTE. If the given number is a decimal without integers, count off the figures in groups of two or three each, according as the square or cube root is required, commencing at the decimal point and separating the groups by # commas; then remove the decimal point to the place of the comma which follows the first significant figure. When the root of the number thus altered has been found, restore the decimal point to its proper place by removing it to the left as many places as the number of groups it previously passed over to the right. Thus, observe whether the decimal point in the given number was transferred to the place of the first, second, or third comma, and accordingly replace it one, two, or three figures further to the left.

EXAMPLE 2. Extract the square root of the fraction ⚫000,003,567.

00,00,03,56,7.

Num. 3.567.........2)0·552303 Log.

Root 1.889-... 0.276151.
Ans. 001,889-.

EXERCISE 1. Extract the square roots of the numbers 144; 2; 47.32; 1.909; 17;690,000; and 384.59.

Answers: 12; 1·414+; 6.879; 1.382 -; 4,206 -; and 19.61 +.

2. Extract the cube roots of the following: 6,859; 36; 3;497,000; 563.1; and 2.5838.

Answers: 19; 3·302 —; 151·8; 8.258 —; and 1·372 + . 3. Find the square roots of '079,63; 935,62; ·007,26; 000,537; and 000,016,9. (See Note).

Answers: 2822-; 9673-; ·0852 +; 023,17+; and ⚫004,111-.

4. Find the cube roots of 802,88; 005,831; 000,076; and 000,000,532,7.

Answers: 9294 +;·1800 —; ·042,36 —; and ·008,106 + .

CHAPTER III.

ON THE APPLICATIONS OF LOGARITHMS.

Logarithms are applicable to all operations involving multiplication, division, involution or evolution; and, in combinations of those operations, they save much labour.

Their grand use is in Trigonometrical calculations—a use which will be fully explained in the Second Part of this volume. But they may also be applied to all questions under the Arithmetical rules of Reduction, Simple and Compound Proportion,* Interest, Exchange, Annuities, Geometrical Progressions, &c., and to questions in Mensuration. Of all these applications, however, it is for the operations of Compound Interest and Annuities, and for finding the areas of triangles when the three sides are given, that logarithms are particularly advantageous and are often employed.

Examples of some of the applications mentioned, and Exercises in them, will be found in the following pages. In these it will be taken for granted that the scholar is already acquainted with the rules of Arithmetic and of Mensuration, leaving nothing more to be exhibited here than the mode of applying the logarithms, which will be a mere exemplification of the rules given in the preceding Chapter.

There are other purposes, both in science and in the higher branches of Mathematics, to which logarithms are applied. But these belong to a more advanced stage of the student's progress, and are invariably explained in treatises on the particular subjects of their application.

EXAMPLE 1. Reduce 764 old corn gallons to imperial gallons.

* Including, of course, the numerous applications of Proportion to which other names are applied, such as Profit and Loss, Discount, &c.

†This would be 277-274 correctly.

EXAMPLE 2. If the rent of 38 acres 3 roods of land is £75, 10s., what should be the rent of 884 acres, at the same rate?

EXAMPLE 3. If the rent of a farm of 38 acres 3 roods of land, in England, is £75, 10s., what should be the rent of a farm of equal quality in France, containing 1,677 ares, supposing it let on the same terms; an imperial acre containing 4,840 square yards, and an are of French measure containing 119.6 square yards, imperial?

As 38.75 x 4,840 1,677 × 119.6 :: 75.5.

EXAMPLE 4. What is the interest of £74, 12s., for 96

days, at 4 per cent. per annum?

The decimal point is removed one place to the right and restored one place

at the conclusion.

EXAMPLE 5. The first term of a geometrical progression is 5.45, and the common ratio 1.874; what is the tenth term?

EXAMPLE 6. What is the compound interest of £854 for six years, at 4 per cent. per annum?

EXAMPLE 14. The diameter of a circle being 25.84,

what is its area?

EXAMPLE 15. Taking the Sun and Earth as spheres,the former of 883,000 miles in diameter, and the latter

* For the uses and values of II and its parts, see Part II, Ch. III, Pr. vi, &c.

of 7,912, how many times the bulk of the Earth is the Sun?

EXERCISE 1. An imperial gallon contains 277.274 cubic inches. Reduce 379 cubic feet to gallons.

Ans. 2362-gals.

2. If the construction of a railway, 48 miles, cost £736,800, what will it cost to continue the same railway an additional distance of 152 miles, at the same rate?

Ans. About £239,300.

3. If 939 men consume 702 quarters of corn in 14 months, how many men can be maintained 5 months on 1,464 quarters?

Ans. 5483 +.

4. An estate of 386 imperial acres was purchased for £19,740. A small piece of land lies contiguous, said to contain 3 Scotch acres, and of equal quality: the owner offers to let me have this on the same terms as the estate. What will be the price, the Scotch acre being equivalent to 1.261 imperial acre?

Ans. £225, 14s. + .

5. What is the interest of £1,578 for 139 days at 41 per cent. per annum? Ans. £25: 109+.

8. What is the amount of £320, 10s., for four years, at 5 per cent. per annum, compound interest?

Ans. £389, 12s., nearly.

9. What is the compound interest of £15, 10s., for nine years, at 3 per cent. per annum ?

Ans. £5: 12: 6.