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to the radius CD, and we shall have the sines of 10, 20, 30, &c. and if from A we describe the arcs 10, 10: 20, 20: 30, 30, &c. from every division of the arc AD; we shall have a line of chords. The same way we may have the sine, tangents, &c. to every one single degree on the quadrant, by subdividing every of the 9 former divisions into 10 equal parts. By this method the sines, tangents, &c. may be drawn to any radius; and if after, they be transferred to lines on a rule, we shall have the scales of sines, tangents, &c. ready for use.

Concerning Scales of equal Parts.

If an inch be divided into any assigned number of equal parts, and if these parts be continued on in a right line, and if the last of them be subdivided into 10 equal parts, and thence if the first divisions be numbered with 1, 2, 3, 4, &c. as far as the ruler upon which they are transferred will admit, the scale is completed.

These numbers, 1, 2, 3, 4, &c. usually stand for 10, 20, 30, 40, &c. and every one of the subdivisions is called 1; but if the numbers 1, 2, 3, 4, &c. be called 100, 200, 300, 400, &c. then every one of the subdivisions will be 10, and the units must be guessed at.

On one side of most surveying scales, there are lines or scales, marked at the end with 50, 45, 40, 35, 30, 25, 15, 10, and sometimes with other numbers; these are scales of so many parts to an inch (whether of feet, yards, perches, or miles) as the respective number at

they are counted to be so many perches to an inch, and sometimes so many feet to an inch.

On the contrary side there are two scales, one of 10, and the other of 20; or one of 100, and the other of 200; or one of 1000, and the other of 2000 parts to an inch, diagonally divided; (a view of the scale will make all easy :) the first of these surveyors call a scale of 10, and the other a scale of 20 perches to an inch; and are thus counted: every large division is 10, every one of the subdivisions is 1, and every one downwards is one tenth of a perch; or sometimes thus, every large division is called 100, every subdivision 10, and every one downwards 1: or again, frequently by navigators, every large division is called 1000, every subdivision 100, every one downwards, 10, and the tenth part of the distance between the lines 1.

Hence it is easy to measure the length of any line knowing the scale by which it was laid down ; and on the contrary, to set off any given distance from any scale.

10

OF LOGARITHMS.

IF to a series of numbers in geometrical progression, whose common ratio is 10, and first term 1, we annex another series of numbers in arithmetical progression, whose first term is 0, and common difference 1; these latter numbers will be the logarithms of the former.

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If several geometrical means be taken, and the like number of arithmetical ones, to the corresponding numbers, the latter will be the logarithms of the former.

The nature therefore of logarithms is such, that addition of them answers to the multiplication of their corresponding numbers; and subtraction to division ; that is, when two numbers proposed are to be multiplied into each other, if we take the logarithms answering to those numbers, and add them together, the sum will be the logarithm answering to the number,

Again, when one number is proposed to be divided by another; if from the logarithm of the dividend, we subtract the logarithm of the divisor, the remainder shall be the logarithm of the quotient.

Most tables of logarithms contain the logarithms of all numbers from 1 to 40000, the column marked at the top N, is that in which you must find your number; in the same line with which in the adjacent column, is the logarithm of that number.

EXAMPLE.

Required, the logarithm of 365.

Answer, 2,56229.

And though most tables of logarithms run but to 10000, yet by them the log. of any number not exceeding 10,000,000 may be found, and on the contrary, the number to any such logarithm, thus,

1. Find the log. of the first four figures of the given Rumber.

2. Take that log. from the log. of the number next following, and note their difference.

3. Multiply that difference by the remaining figures of the given number; and from the product cut off as many figures as remain in the given number, or as the given number is more than four (counting from the right to the left) as in decimals.

4. The whole number in the product, added to the first log. is the log. required; but the first figure, which is called the index, or characteristick must be changed; and always be one less than the number of figures in the logarithm.

EXAMPLE I.

is

is

Required, the logarithm of the number 3567194

The log. of 3567, which are the first four figures,

3.55230

The log. of the following or next number, viz. 3568,

3.55242

Their difference,

Mult. by the remaining fig. viz.

'Cut off 3 figures, because 894 is three figures,

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12

.894

10.728

3.55230

3.55240

But because the given number consists of 7 figures, the index must be one less, which is 6; so the above index, 3, must be changed to 6, and we have 655240

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