of 6, then the lateral distance of 4 will give the transverse distance of 2,66 nearly.

Use of the Line of Chords.

The line or scale of chords is used for protracting any angle; you open the sector to any radius within compass of the instrument, and the transverse distance to any degree required is to be laid down on the circumference of the circle; but if you want it to any particular radius, as, for instance, to one inch, make the transverse distance between 60 and 60 equal to 1 inch, then you may take off transversely any degree under 60, but for any degree above 60, lay off the radius first on the circumference, and the excess above 60 taken transversely, are to be laid off on the circumference from the radius just before laid down. The measure of any angle is found by taking the distance of the legs on the circumference, and applying it transversely on the line of chords.

Of the Lines of Sines, Tangents, and Secants.

The transverse distance on the line of sines shows the degrees, &c. required; and the transverse distance on the line of tangents to 45, do the same. But to lay off a tangent above 45 degrees, you must take the radius of the tangent 45, and open the sector that the radius just taken may just reach to 45,45 on the line of upper tangents marked t, or on the beginning of the scale of secants, then the sector is adjusted to take any tangent above 45 degrees, or any secant to 75 degrees. The Line of Polygons.

Open the sector that 6,6 be equal to the radius, then the transverse distance of any of the numbers on the scale will divide the circle into as many sided polygons.

LOGARITHM S.

LOGARITHMS are a series of numbers, invented by Lord Napier, Baron of Marchinston, in Scotland, by which the work of multiplication may be performed by addition, and the operation of division may be done by subtraction; so that great time and trouble are saved thereby in the performance of all arithmetical operations; for if the logarithm of any two numbers be added together, the sum will be the logarithm of the product; and if from the logarithm of the dividend you subtraet the logarithm of the divisor, the remainder will be the logarithm of the quotient. Again, if the logarithm of any number be divided by 2, the quotient will be the logarithm of the square root of that number; or, if the logarithm of any number be divided by 3, the quotient will be the logarithm of the cube root of that number.

The most convenient series now made use of is the following:

&c. index. 100000, &c. logarithms.

By which you perceive the index of any logarithm always one less than the number of figures the integer contains.

To find the Logarithm of any Number containing less than 5 Figures. EXAMPLES.

I would find the logarithm of 7.

Look in the table for the number of 7 in the side column, and against it is 0.84510. This number having but one figure, the index thereto is 0.

I would find the logarithm of 79.

Look in the table for the number of 79 in the side column, and against it is 1.89763; to which 1 is the index, because the number contains two figures.

I would find the logarithm of 763.

Against 763, in the first side column, is 2.88252; to which prefix the index 2, as the number contains 3 places of figures, 2.88252.

To find the Logarithm of 7634.

Find the logarithm of the three first figures in the side column as before; and, casting your eye on the numbers on the top line of the table, look for the remaining figure 4, bring your eye to bear down that column, and right against 763 is the logarithm 88275, to which prefix the index 3, as it contains four places of figures, thus: 3.88275 is the logarithm of 7634.

To find the Logarithm of any whole Number to 5 Places of Figures. Suppose 76315.

Look out the logarithm of the first three figures 763 in the side column, and the next figure 4 in the top column as before, and against the angle of meeting is 88275, as before. Take the difference between this logarithm and the next greater; that is, the difference between 275 and 281, which is 6; then say, by the rule of three, if 10 gives 6, what will 5 give? that is its half or 3; which added to the logarithm 88275, makes $8278; to, which prefix the index 4, as it contains five places of figures; and that inakes the logarithm of 76815 to be 4.88278.

Again, to find the Logarithm of any Number to 6 Places of figures,

as 763458.

Find the logarithm of the 4 first places of figures as before 88275, as above; then say, if 100 gives 6 difference, what will 58 give? Answer 3; which, added to 88275, makes 88278; to which prefix its index 5, makes the logarithm of 763-158 to be 5,88278.

To find the Logarithm of any mixed Number, as 763.458. Where the integer is 763, or has only three places of figures, the rule is: Find the logarithm to all the figures, the same as if they were whole numbers as before, to which prefix always the index of the integer, which in this number is 2; so that the log. of 763.458 is 2.88278, nearly the same as above, only differing in its index.

To find the Number answering to any Logarithm to 4 Places of Figures.

Seek under the column 0, at the top of the table, the next less logarithm; note the number against it, and carry your eye along that line until you find the nearest logarithm next less than the given one, and you will have the fourth figure at the top of the table, which affix to the three given ones in the first side column.... What is the number to the logarithm 3.77342?-I look in column 0, and find under it, against the number 593, the logarithm 77305; and guiding my eye along that line, I find the given logarithm 77342 under the column, with 5 at the top; so that the number is 5935.

The number, if taken out by this precept, will be either the number required, or the next less.

To find the Number answering any Logarithm to 5 Places of
Figures nearly.

Find the next less logarithm to the given one, and take the difference betwixt it and the given one; also take the difference betwixt the next greater logarithm, and next less to the given one; then say, as the difference of the next greater and next less is to 10, so is the former difference to the correction sought;-as, suppose you would find the number to the logarithm 4.59632.

4.59632

4.59627 The nearest next log. I can find is 59627 its num. 39470 · The next greater ditto is 59638=

39480

10

Then say, 11: 10:5: 5 nearly the correction; which I add to the number 39470, makes the number sought to be 39475, answering to the logarithm 4.59632.

NOTE.-Aliquot or even parts may be taken of the difference. between the less and greater logarithms, where it can be done, thus: In this last 5 is nearly the half of 11, as 5, the number.sought, is of 10, the difference of the two numbers belonging to the greater and less logarithms, which will often save time and trouble.

MULTIPLICATION BY LOGARITHMS.

CASE I.

To find the Product of two whole or mixed Numbers.

Multiply 76

by 54

Product 4104

Log.1.88081 Multiply 76.4 Log.=1.88309 1.73239 by 5.4

3.61320 Product 412.56

CASE II.

0.73239

=2.61548

When both, or either, of the fractions are less than unity, as if 0.265 Log. 9.42325 0.031 8.49136

Here the index of a fraction is 9, when the first decimal figure, as 2, stands in the first decimal place; but if it should .008215 =7.91461 stand in the second decimal place, as the 3 in .031, the index will be 8; if it stood in the third decimal place, as .0031, the index would be 7. Thus the number of ciphers prefixed to any decimal, and the index of that decimal, always together make 9; so that if you take the number of ciphers prefixed to the decimal from 9 remains its proper index. In the addition reject 10 in the sum of the indices; and the proper product, or value of the product, will be obtained: By reason, if 9 represent the index of a fraction, 10 will represent, in this case, the index of unity. Indeed the index of unity may be assumed either 0, 10, 100, &c. as you please; but generally, for most uses, is not wanted to be more than 10, as in the sines, tangents, secants, &e. As 7 or 8 places of decimals are generally sufficient for all purposes, take these two more examples: Multiply 3.72 by 0.00064

Product .0023808

Log. 0.57054 Multiply 59.4 Log 1.77379 6.80618 by .000031

7.37672 Product .0018414

Here the remainder to 9 is 2 in the index; therefore prefix two ciphers to the number of the log. 23808 for the product required.

DIVISION BY LOGARITHMS.
CASE I.

5.49136

7.26515

To divide a whole or mixed Number by a less whole or mixed Number, RULE. From the logarithm of the dividend subtract the logarithm of the divisor, and the remainder is the logarithm of the quotient.

When both, or either, fractions are less than unity.

As divide .008215 by .031. .008215 Its log. is

.031 Its log. is

NOTE-If I had assumed the 7.91461 index of unity 100, then the index 8.49136 of the first number would have been 97 or 97.91461,

.0001916 Its quotient 6.28243

NOTE. Whatever index you make represent unity, omit it in the sum of the indices, and borrow it in the subtraction of indices, the sum or remainder will be the true index required.

TO EXTRACT THE ROOTS IN LOGARITHMS.

As the multiplying the logarithm of any number by the index of its power produces the logarithm of that power; so the division of any logarithm by its proposed index, the quotient will be the logarithm of the root required.

What is the square root of 324? | What is the cube root of 10678? 324 Its logarithm is 2)2.51054 10678 Its log. is

18 Log, of the root is

1.25527 22 log. of the root is

3)4.02726

1.34242

To find any proposed root of any decimal fraction, you must first prepare the index for the division of the proposed power, thus:For the square you must add 10 to the index before you divide it; for the cube you must add 20 to its index before you divide it; and so on for the root of any power proposed.