The fectoral lines are like fo many fimilar triangles, namely, that their correfponding fides are proportional, thus: let AC, AE, reprefent in plate 1. fig. 1. a pair of fectoral lines, forming the angle CAE, divide each leg into any number of equal parts (fay 10) draw lines to any of the corresponding numbers, and each will be a fimilar triangle to CAE, and if the lines AC, AE, fhould reprefent the line of chords, fines, or tangents, and CE the radius, and D on the chord, fine, or tangent, any propofed number, then the tranfverfe meafure BD will be the chord,, fine, or tangent of that number.

In defcribing the use of the sector, the term lateral distance is the distance on one leg, only taken from the centre to any part of a fectoral line; and the tranfverfe diftance is that taken between any two correfponding divifions on a scale of the fame name. All are measured on the lines of each scale that are nearest each other.

The Line of Lines, or Proportional Scale.

The line of lines is ufed to divide a given line into any number of equal parts: fuppofe for example 8 deg. take the length of the line given in the compaffes, and make it a tranfverfe diftance from 8 to 8, then will the tranfverfe distance from 1 to I be one of the equal parts, or of the whole; from 2 to 2 will be the 2d, &c.; but if the line to be divided be too long for the legs of the fector, make any divifion fo that it may be applied to the sector, multiplying each tranfverfe distance by the fame number you divided by.

To find a fourth proportional to any 3 given lines or numbers, as fuppofe 6, 2, and 4, take the lateral distance of 2 in your compaffes, and make it the tranfverfe diftance at 6, then the tranfverfe diftance of 4 will give the lateral diftance of and. Or if a fhip failed 64 miles in 8 hours, how many miles did the fail in 5 hours at the fame rate of failing? Make the lateral diftance of 64 the tranfverfe distance at 8 and 8, then the tranfverfe diftance of 5 and 5 will give the lateral diftance of 40, the fourth proportional. Having a chart constructed upon a fcale of 5 miles to an inch, the fector is adjusted to a correfponding feale, by making the tranfverfe diftance from 5 to 5 equal to one inch. And to reduce a chart of 6 inches to a degree, to one of 4 inches to a degree, make the transverse distance of 6, 6, equal to the lateral diflance of 4, then any distance from the chart fet off laterally the corref ponding tranfverfe diftance will be the diftance required. And if you have a chart of 3 inches to a mile, to enlarge to 5 inches to a mile, make the tranfverfe diftance of 3, 3, equal to the lateral diftance of 5, and proceed as before. A third proportional is found to two numbers; thus having 6 and 4 given to find a third proportional, make the tranfverfe diftance at 4 and 4, the lateral diftance

of

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

Ufe of the Line of Chords.

The line or scale of chords is ufed for protracting any angle; you open the sector to any radius within compafs of the inftrument, and the tranfverfe distance of 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 inftance, to one inch, make the transverse distance between 60 and 60 equal to 1 inch, then you may take off tranfverfly any degree under 60, but for any degree above 60, lay off the radius firft on the circumference, and the excefs above 60 taken tranfverfely, are to be laid off on the circumference from the radius juft before laid down. The measure of any angle is found by taking the diftance of the legs on the circumference, and applying it tranfverfely on the line of chords.

Of the Lines of Sines, Tangents, and Secants.

The tranfverfe diftance on the line of fines fhews the degrees, &c. required; and the tranfverfe diftance on the line of tangents to 45, do the fame. 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 juft taken may juft reach to 45,45 on the line of upper tangents marked t, or on the beginning of the fcale of fecants, then the fector is adjufted to take any tangent above 45 degrees, or any fecant to 75 degrees.

The Line of Poligons.

Open the sector that 6,6 be equal to the radius, then the tranfverse distance of any of the numbers on the fcale will divide the circle into as many fided poligons.

LOGARITHM S.

OGARITHMS are a series of numbers, invented by Lord Napier, Baron of Marchinfton, in Scotland, by which the work of multiplication may be performed by addition, and the operation of divifion may be done by fubtraction; fo that great time and trouble are faved thereby in the performance of all arithmetical operations; for if the logarithm of any two numbers be added together, the fum will be the logarithm of the product; and if from the logarithm of the dividend you fubtract the logarithm of the divifor, 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 fquare 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 feries now made ufe of is the following: 5 &c. index.

о

2

I 10

100

3

4

1000 10000 100000, &c. logarithms. By which you perceive the index of any logarithm always one lefs 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 fide column, and against it is 0.84510. This number having but one figure, the index thereto is o.

I would find the logarithm of 79?

Look in the table for the number of 79 in the fide column, and againft it is 1.89763; to which I is the index, because the number contains two figures.

I would find the logarithm of 763?

Against 763, in the firft fide 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 fide column as before; and, cafting 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. Suppofe 76345?

Look out the logarithm of the three first figures 763 in the fide 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 88278; to which prefix the index 4, as it contains five places of figures; and that makes the logarithm of 76345 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 fay, 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 763458 to be 5.88278.

To

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 fame as if they were whole numbers as before, to which prefix always the index of the integer, which in this number is 2; fo that the log. of 763.458 is 2.88278, nearly the fame as above, only differing in its index.

To find the Number anfwering to any Logarithm to 4 Places of

Figures.

Seek under the column o, 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 neareft logarithm next lefs 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 fide column.

What is the number to the logarithm 3.77342?—I look in column o, and find under it, against the number 593, the logarithm 7705; and, guiding my eye along that line, I find the given logarithm 77342 under the column, with 5 at the top; fo that the number is 5935.

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

To find the Number anfwering 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; alfo take the difference betwixt the next greater logarithm, and next lefs to the given one; then fay, as the difference of the next greater and next lefs is to 10, fo is the former difference to the correction fought ;-as, fuppofe 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 fay, 11:10:: 5:5 nearly the correction; which I add to the number 39470, makes the number fought to be 39475, anfwering 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 laft 5 is nearly the half of 11, as 5, the number fought, is of 10, the difference of the two numbers belonging to the greater and Jefs logarithms, which will often fave time and trouble.

MULTI

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

When both, or either, 0.265 Log. 9.42325 0.031 8.49136

0.73239

=2.61548

of the fractions are less than unity, as if Here the index of a fraction is 9, when the firft decimal figure, as 2, ftands in the first decimal place; but if it should .008215 =7.91461 stand in the fecond decimal place, as the 3 in .031, the index will be 8; if it ftood in the third decimal place, as .co31, the index would be 7. Thus the number of cyphers prefixed to any decimal, and the index of that decimal, always together make 9; fo that if you take the number of cyphers 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 reafon, if 9 reprefent the index of a fraction, 10 will reprefent, in this cafe, the index of unity. Indeed the index of unity may be affumed either o, 10, 100, &c. as you pleafe; but generaily, for moft ufes, is not wanted to be more than To, as in the fines, tangents, fecants, &c. As 7 or 8 places of decimals are generally fufficient for all purposes, take these two more examples:

Here the remainder to 9 is 2 in the index; therefore prefix two cy phers 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 lefs whole or mixed Number, RULE. From the logarithm of the dividend fubtract the logarithm of the divifor, and the remainder is the logarithm of the quotient.