LOGARITHMS, SINES, AND TANGENTS.

The radius or logarithmic sine of 90 is 10. Of the natural sine is 1. The tangent of 90, whether natural or logarithmic, is infinite.

To find the logarithm of any sine, tangent, &c.:

Look for the degrees in the first column, and for the minutes on the top line, opposite to which will be the logarithm required. And in the same manner will the measure of an angle be found in degrees and minutes by referring to the table.* If the logarithm should not agree with the degree and minutes in the table, seek in the table for that which is next less and next greater, and take the difference, thus:

As this difference is to 3 minutes,

So is the difference between the given logarithm, and the next less

Is to a fourth number of minutes,

Add this fourth number to the minutes in the less logarithm, and it will be the degree and minute required.

Example.

What degree and minute of the sine answers to 9.467996:

As 822: 3 :: 411 : 1 which minute add to 17° 4' the

less logarithm, will be 17° 5', the sine required.

OF THE SQUARE ROOT.

When a number is multiplied by itself, as 6 x 6 or 9 × 9, &c., it produces the square or second power of that number, and the number itself is called the root of that square.

The same numbers are also the co-sines and co-tangents, reading from the bottom of the table.

A root consisting of a single figure is found by inspection of the following table:

To extract or find the square root of any number which consists of more figures than one, observe the following rule:

Divide the given number into periods of two figures each, by setting a point over the place of units, another over the place of hundredths, and so on over every second figure, both to the left hand in integers and to the right hand in decimals.

Find the greatest square in the first period on the left hand, and set its root on the right hand of the given number, after the manner of a quotient figure in division.

Subtract the square thus formed from the said period, and to the remainder annex the two figures of the next following period, for a dividend.

Double the root above mentioned for a divisor, and find how often it is contained in the said dividend, exclusive of its right hand figure, and set that quotient figure both in the quotient and divisor.

Multiply the whole augmented divisor by this last quotient figure, and subtract the product from the said dividend, bringing down to the next period of the given number, for a new dividend.

Repeat the same process over again-viz. find another new divisor, by doubling all the figures now found in the root; from which, and the last dividend, find the next figure of the root as before; and so on through all the periods to the last.

Note.-The best way of doubling the root, to form the new divisors, is by adding the last figure always to the last divisor, as appears in the following examples. Also, after the figures belonging to the given number are all exhausted, the operation may be continued into decimals at pleasure, by adding any number of periods of cyphers, two in each period.

The rules usually given in books of arithmetic for the cube and higher roots are very tedious in practice; on which account it is advisable to work either by means of approximating rules, or by means of logarithms.

We shall merely point out here Dr. Hutton's approximating rule for the cube root, which may sometimes be serviceable when logarithmic tables are not at hand.

Rule. By trials take the nearest rational cube to the given number, whether it be greater or less, and call it the assumed cube.

Then say, by the rule of three, as the sum of the given

number and double the assumed cube is to the sum of the assumed cube and double the given number, so is the root of the assumed cube to the root required, nearly;

Or as the first sum is to the difference of the given and assumed cube, so is the assumed root to the difference of the roots, nearly.

Again, by using in like manner the cube of the root last found as a new assumed cube, another root will be obtained still nearer; and so on as far as we please, using always the cube of the last found root for the assumed cube.

Example.

To find the cube root of 210358.8.

Here we soon find the root lies between 20 and 30, and then between 27 and 28, taking therefore 27 its cube is 19683, which is the assumed cube; then

PURCHASING LEASES, ANNUITIES, OR ESTATES.

Rule. For finding the value of leases, estates, annuities, held for a term of years.

Seek in the first column, Table 25, the number of years the lease is to continue. In the horizontal line with this number

will be found the number of years' purchase, or value of one pound per annum, at the several rates of 3, 4, 5, 6, 7, 8, 9, and 10 per cent.; if this value be multiplied by the annual rental it will give the real value of the lease.

Example 1.

A is desirous to purchase of B the lease of an estate, producing a net rental of 301. per annum, held for an unexpired term of 20 years, to pay 6 per cent.

By reference to the Table 25, in a line with the number of years 20, and in the column under 6 per cent., will be found 11.47, the number of years' purchase or value of 17., which being multiplied by 301., will give 344.10, or 3441. 2s. the value.

Example 2.

What is the value of freehold to pay 5 per cent. ?

By reference to Table 25, on the lower line (perp.), which is for freeholds, in the column under 5 per cent., will be found 20.00, the number of years' purchase or value of 17., which, if multiplied by 30%. (the rental), will give 600.00, or 6007. the value.

Rule. For finding the value of leases, estates, or annuities, held for a single life.

Seek in the first column, Table 25, the age of the life; on the horizontal line with this number will be found the number of years' purchase, or value of 17. per annum, at the several rates of 1, 2, 3, 4, 5, 6, 7, and 8 per cent.

Example.

What is the value of a lease or annuity producing 231. per annum, held during the life of a person 30 years of age, to pay 7 per cent.

By reference to Table 26, in the first column, will be found age 30; and in a line with this number, in the column under the 7 per cent., will be found 10.54, the number of years' purchase