DESCRIPTION AND USE OF THE SECTOR.

THIS instrument consists of two rules or legs, moveable round an axis or joint, as a centre, having several scales drawn on the faces, some single, others double; the single scales are like those upon a common Gunter's Scale; the double scales are those which proceed from the centre, each being laid twice on the same face of the instrument, viz. once on each leg. From these scales, dimensions or distances are to be taken, when the legs of the instrument are set in an angular position.

The single scales being used exactly like those on the common Gunter's Scale, it is unnecessary to notice them particularly; we shall therefore only enumerate a few of the uses of the double scale, the number of which is seven, viz. the scale of Lines, marked Lin. or L. the scale of chords, marked Cho. or C. the scale of Sines, marked Sin. or S. the scale of Tangents to 450, and another scale of tangents from 45° to about 76°, both of which are marked Tan. or T. the scale of Secants, marked Sec. or S. and the scale of Polygons, marked Pol.

The scale of lines, chords, sines, and tangents, under 450, are all of the same radius, beginning at the centre of the instrument, and terminating near the other extremity of each leg, viz. the lines at the division 10, the chords at 600, the sines at 90°, and the tangents at 450; the remainder of the tangents, or those above 45°, are on other scales, beginning at a quarter of the length of the former, counted from the centre, where they are marked with 450, and extend to about 76 degrees. The secants also begin at the same distance from the centre, where they are marked with 0, and are from thence continued to 750. The scales of polygons are set near the inner edge of the legs, and where these scales begin, they are marked with 4, and from thence are numbered backward or towards the centre, to 12.

In describing the use of the sector, the terms lateral distance and transverse distance often occur. By the former is meant the distance taken with the compasses on one of the scales only, beginning at the centre of the sector; and by the latter, the distance taken between any two corresponding divisions of the scales of the same name, the legs of the sector being in an angular position.

B

D

The use of the sector depends upon the porportionability of the corresponding sides of similar triangles, (demonstrated in art. 53, Geometry.) For if in the triangle ABC we take AB AC and AD= AE, and draw DE, BC, it is evident that DE and BC will be parallel; therefore by the

above-mentioned proposition AB:ABC:: AD: DE; so that whatever part AD is of AB, the same part DE will be of BC; hence, if DE be the chord, sine, or tangent of any arch to the radius AD, BC will be the same to the radius AB.

Use of the line of Lines.

E

The line of lines is useful to divide a given line into any number of equal parts, or in any proportion, or to find 3d and 4th proportionals, or mean proportionals, or to increase a given line in any proportion.

EXAMPLE 1. To divide a given line into any number of equal parts, as suppose 9: make the length of the given line a transverse distance to 9 and 9, the number of parts proposed; then will the transverse distance of 1 and 1 be one of the parts, or the ninth part of the whole; and the transverse distance of 2 and 2 will be 2 of the equal parts or of the whole line, &c.

EXAMPLE 2. If a ship sails 52 miles in 8 hours, how much would she sail in 3 hours at the same rate?

Take 52 in your compasses as a transverse distance, and set it off from & to 8, then the transverse distance 3 and 3 being measured laterally, will be. found equal to 19 and a half, which is the number of miles required.

EXAMPLE 3. Having a chart constructed upon a scale of 6 miles to an inch. it is required to open the sector, so that a corresponding scale may be taken from the line of lines?

Make the transverse distance 6 and 6, equal to 1 inch, and this position of the sector will produce the given scale.

EXAMPLE 4. It is required to reduce a scale of 6 inches to a degree, to another of 3 inches to a degree?

Make the transverse distance 6 and 6, equal to the lateral distance 3 and S: then set off any distance from the chart laterally, and the corresponding transverse distance will be the reduced distance required.

EXAMPLE 5. One side of any triangle

being given, of any length, to measure the other two sides on the same scale.

B

Suppose the side AB of the triangle ABC measures 50, what are the measures of the other two sides?

50

A

63

45

с

Take AB in your compasses, and apply it transversely to 50 and 50; to this opening of the sector apply the distance AC in your compasses to the same number on both sides of the rule transversely; and where the two points fall will be the measure on the line of lines of the distance required; the distance AC will fall against 63, 63, and BC against 45, 45, on the line of lines.

Use of the line of Chords on the Sector.

The line of chords upon the sector is very useful for protracting any angle, when the paper is so small that an arch cannot be drawn upon it with the radius of a common line of chords.

Suppose it was required to set off an arch of 30°, from the point C of the small circle ABC.

B

E

D

C

Take the radius DC in your compasses, and set it off transversely from 600 to 60° on the chords, then take the transverse extent from 30° to 300 on the chords; and place one foot of the compasses in C, the other will reach to E, and CE will be the arch required. And by A the converse operation any angle or arch may be measured, viz. with any radius describe an arch about the angular point; set that radius transversely from 60° to 60°; then take the distance of the arch, intercepted between the two legs, and apply it transversely to the chords, which will show the degrees of the given angle.

NOTE. When the angle to be protracted exceeds 60°, you must lay off 600, and then the remaining part; or if it be above 120°, lay off 60° twice, and then the remaining part. And in a similar manner any arch above 60° may be measured.

Uses of the lines of Sines, Tangents, and Secants.

By the several lines disposed on the sector, we have scales of several radii, so that,

1st. Having a length or radius given, not exceeding the length of the sector when opened, we can find the chord, sine, &c. of an arch to that radius; thus, suppose the chord, sine, or tangent of 20 degrees to a radius of 2 inches be required. Make 2 inches the transverse opening to 600 and 60° on the chords; then will the same extent reach from 450 to 450 on the tangents, and

from 90° to 90° on the sines; so that to whatever radius the line of chords is set, to the same are all the others set also. In this disposition, therefore, if the transverse distance between 200 and 200 on the chords be taken with the compass, it will give the chord of 20 degrees; and if the transverse of 20° and 200 be in like manner taken on the sines, it will be the sine of 20 degrees; and lastly, if the transverse distance of 200 and 200 be taken on the tangents, it will be the tangent of 20 degrees to the same radius of two inches.

2dly. If the chord or tangent of 700 were required. For the chord you must first set off the chord of 60° (or the radius) upon the arch, and then set off the chord of 10°. To find the tangent of 70 degrees, to the same radius, the scale of upper tangents must be used, the under one only reaching to 450; making therefore 2 inches the transverse distance to 450 and 450 at the beginning of that scale, the extent between 70° and 70° on the same will be the tangent of 70 degrees to 2 inches radius.

3dly. To find the secant of any arch; make the given radius the transverse distance between 0 and 0 on the secants; then will the transverse distance of 200 and 200, or 700 and 700, give the secant of 200 or 70° respectively.

4thly. If the radius and any line representing a sine, tangent, or secant, be given, the degrees corresponding to that line may be found by setting the sector to the given radius, according as a sine, tangent, or secant is concerned; then taking the given line between the compasses, and applying the two feet transversely to the proper scale, and sliding the feet along till they both rest on like divisions on both legs; then the divisions will show the degrees and parts corresponding to the given line.

Use of the line of Polygons.

For exam

The use of this line is to inscribe a regular polygon in a circle. ple, let it be required to inscribe an octagon in a circle. Open the sector till the transverse distance 6 and 6 be equal to the radius of the circle; then will the transverse distance of 8 and 8 be the side of the inscribed polygon.

Use of the sector in Trigonometry.

A.

All proportions in trigonometry are easily worked by the double lines on the sector; observing that the sides of triangles are taken off the line of lines, and the angles are taken off the sines, tangents, or secants, according to the nature of the proportion. Thus, if in the triangle ABC we have given AB=56, AC=64, and the angle ABC=46° 30' to find the rest. In this case we have (by art. 58, Geometry) the following proportions, as AC (64): sine < B (46° 30') :: :: AB (56): sine < C, and as sine B: AC :: sine A: BC. Therefore to work these proportions by the. sector, take the lateral distance 64-AC from the B

C

line of lines, and open the sector to make this a transverse distance of 460 30 B on the sines; then take the lateral distance 56=AB on the lines, and apply it transversely on the sines, which will give 39° 24'=< C. Hence the sum of the angles B and C is 85° 54', which taken from 180°, leave the angle A=94° 6′. Then to work this second proportion, the sector being set at the same opening as before, take the transverse distance of 94° 6' the angle A on the sines, or, which is the same thing, the transverse distance of its supplement 85° 54'; then this applied laterally to the lines, gives the sought side BC=88. In the same manner we might solve any problem in trigonometry, where the tangents and secants occur, by only measuring the transverse distances on the tangents or secants, instead of measuring them on the sines, as in the preceding example. All the problems that occur in Nautical Astronomy may be solved by the sector, but as the calculation by logarithms is much more accurate, it will be useless to enter into a further detail on this subject.

LOGARITHMS.

IN order to abbreviate the tedious operations of multiplication and division with large numbers, a series of numbers, called logarithms, were invented by Lord Napier, Baron of Marchinston in Scotland, and published in Edinburgh in 1614; by means of which the operation of multiplication may be performed by addition, and division by subtraction; numbers may be involved to any power by simple multiplication, and the root of any power extracted by simple division.

In Table XXVI. are given the logarithms of all numbers from 1 to 9999; to each one ought to be prefixed an index, with a period or dot to separate it from the other part, as in decimal fractions; the numbers from 1 to 100 are published in that table with their indices; but from 100 to 9999 the index is left out for the sake of brevity, but it may be supplied by this general rule, viz. the index of the logarithm of any integer, or mixed number, is always one less than the number of integer places in the natural number. Thus the index of the logarithm of any number (integer or mixed) between 10 and 100 is 1, from 100 to 1000 it is 2, from 1000 to 10000 is 3, &c. the method of finding the logarithms from this table will be evident from the following examples. To find the logarithms of any number less than 100.

RULE. Enter the first page of the table, and opposite the given number will be found the logarithm with its index prefixed.

Thus, opposite 71 is 1.85126, which is its logarithm.

To find the logarithm of any number between 100 and 1000.

RULE. Find the given number in the left hand column of the table of logarithms, and immediately under 0 in the next column, is a number, to which must be prefixed the number 2 as an index (because the number consists of three places of figures) and you will have the sought logarithm.

Thus, if the logarithm of 649 was required; this number being found in the left hand column, against it in the column marked 0 at the top (or bottom) is found 81224, to which prefixing the index 2, we have the logarithm of 649-2.81224.

To find the logarithm of any number between 1000 and 10000. RULE. Find the three left hand figures of the given number, in the left hand column of the table of logarithms, opposite to which, in the column that is marked at the top (or bottom) with the fourth figure, is to be found the sought logarithm; to which must be prefixed the index 3, because the number contains 4 places of figures.

Thus, if the logarithm of 6495 was required; opposite to 649, and in the column marked 5 at the top (or bottom) is 81258, to which prefix the index 3 and we have the sought logarithm 3.81258.

To find the logarithm of any number above 10000.

RULE. Find the three first figures of the given number, in the left hand column of the table, and the fourth figure at the top or bottom, and take out the corresponding number as in the preceding rule; take also the difference between this logarithm and the next greater, and multiply it by the given number exclusive of the four first figures, cross off at the right hand of the product as many figures as you had figures of the given number to multiply by; then add the remaining left hand figures of this product to the logarithm taken from the table, and to the sum prefix an index equal to one less than the number of integer figures in the given number, and you will have the sought logarithm.

Thus, if the logarithm of 64957 was required: opposite to 649 and under 5 is 81258, the difference between this and the next greater number 81265 is 7, this multiplied by 7 (the last figure of the given number) gives 49, crossing off the right hand figure leaves 4.9 or 5 to be added to 81258, which makes 81263, to this prefixing the index 4, we have the sought logarithm 4.81263.

Again, if the logarithm of 6495738 was required; the logarithm corresponding to 649 at the left. and 5 at the top, is as in the last example 81258,

the difference between this and the next greater is 7, which multiplied by 738 (which is equal to the given number excluding the four first figures) gives 5166, crossing off the three right hand figures of this product (because the number 738 consists of three figures) we have the correction 5 to be added to 81258; and the index to be prefixed is 6 because the given number consists of 7 places of figures, therefore the sought logarithm is 6.81263.

To find the logarithm of any mixed decimal number.

RULE. Find the logarithm of the number as if it was an integer by the last rule, to which prefix the index of the integer part of the given number.

Thus, if the logarithm of the mixed decimal 649.5738 was required;find the logarithm of 6495738 without noticing the decimal point; this, in the last example, was found to be 81263, to this we must prefix the index 2, corresponding to the integer part 649; the logarithm sought will therefore be 2.81263.

To find the logarithm of any decimal fraction less than unity.

The index of the logarithm of any number less than unity is negative, but to avoid the mixture of positive and negative quantities, it is common to borrow 10 or 100 in the index, which must afterwards be neglected in summing them with other indices; thus instead of writing the index - 1, it is generally written +9 or +99; but in general it is sufficient to borrow 10 in the index, and it is what we shall do in the rest of this work. In this way we may find the logarithm of any decimal fraction by the following rules.

RULE. Find the logarithm of a fraction as if it was a whole number;-see how many ciphers precede the first figure of the decimal fraction, subtract that number from 9 and the remainder will be the index of the given fraction. Thus the log. of 0,0391 is 8.59218; the log. of 0,25 is 9.39794; the log. of 0,0000025 is 4.39794, &c.

To find the logarithm of a vulgar fraction.

RULE. Subtract the logarithm of the denominator from the logarithm of the numerator (borrowing 10 in the index when the denominator is the greatest) the remainder will be the logarithm of the fraction sought.

To find the number corresponding to any logarithm.

RULE. In the column marked 0 at the top (and bottom) of the table, seek for the next less logarithm, neglecting the index; note the number against it, and carry your eye along that line until you find the nearest less logarithm to the given one, and you will have the fourth figure of the given number at the top, which is to be placed to the right of the three other figures; if you wish for greater accuracy, you must take the difference between this tabular logarithm and the next greater, also the difference between that least tabular logarithm and the given one; to the latter difference annex 2 or more ciphers at the right hand, and divide it by the former difference, and place the quotient to the right hand of the four figures already found; and you will have the number sought expressed in a mixed decimal, the integer part of which will consist of a number of figures (at the left hand) equal to the index of the logarithm increased by unity.†

Thus, if the number corresponding to the logarithm 1.52634 was required; I look for 52634 in the column marked 0 at the top or bottom, and find it standing opposite to 336; now the index being 1, the sought number must consist of two integer places, therefore it is 33,6.

*This quotient must consist of as many places of figures as there were ciphers annexed, conformatie to the rules of the division of decimals. Thus, if the divisor was 40, and the number to which two ciphers were annexed was 2, making 2,00, the quotient must not be estimated as 5, but as 05, and then two figures must be placed to the right of the four figures before found.

If the index corresponds to a fraction less than unity, you must place as many ciphers to the left of that number as are equal to the index subtracted from 9, the decimal point being placed to the left of these ciphers; in this manner you will obtain the sought number.