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DESCRIPTION AND USE OF THE SECTOR

THIS instrument consists of two rules or legs, movable 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 mention a few of the uses of the double scales, 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 45°, 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 scales of lines, chords, sines, and tangents under 45°, 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 60°, the sines at 90°, and the tangents at 45°; 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 45°, 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 75°. 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.

D

B

The use of the Sector depends upon the proportionality of the corresponding sides of similar triangles (demonstrated in Art. 53, Geometry). For if, in the triangle ABC, we take ABAC, and AD AE, and draw DE, BC, it is evident that DE and BC will be parallel; therefore, by the above-mentioned proposition, AB: BC :: 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 arc to the radius AD, BC will be the same to the radius AB.

Use of the line of Lines.

The line of lines is useful to divide a given line into any number of equal parts, ot in any proportion, or to find third and fourth proportionals, or mean proportionals, or to increase a given line in any proportion.

EXAMPLE I. 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 two of the equal parts, or of the whole line, &c.

EXAMPLE II. 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 8 to 8; ther 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 III. Having a chart constructed upon a scare of 6 miles to an ir.ch. y e required to open the Sector, so that a corresponding scale inay De taken from inc hing 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 IV. 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 3; then set off any distance from the chart laterally, and the corresponding transverse distance will be the reduced distance required.

EXAMPLE V. One side of any triangle being given, of any length, to measure the other two sides on the same scale.

Suppose the side AB of the triangle ABC measures 50, what

are the measures of the other two sides?

Take AB in your compasses, and apply it transversely to 50 and

63

45

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 arc cannot be drawn upon it with the radius of a common line of chords.

Suppose it was required to set off an arc of 30' from the point C of the small circle ABC, whose centre is D.

And

B

C

Take the radius, DC, in your compasses, and set it off transversely from 60° to 60° on the chords; then take the transverse extent from 30° to 30° on the chords, and place one foot of the compasses in C; the other will reach to E, and CE will be the arc required. by the converse operation, any angle or arc may be measured, viz. with any radius describe an arc about the angular point; set that radius transversely from 60° to 60°; then take the distance of the arc, 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 60°, 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 arc 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 arc 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 60° and 60° on the chords; then will the same extent reach from 45° to 45° on the tangents, and from 90° to 90° on the sines; so that, to whatever radius the lines of chords is set, to the same are all the others set also. In this disposition, therefore, if the transverse distance between 20° and 20° on tne chords be taken with the compass, it will give the chord of 20 degrees; and if the transverse of 20° and 20° be in like manner taken on the sines, it will be the sine of 20 degrees and lastly, if the transverse distance of 20° and 20° 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 70° were required. For the chord you must first set off the chord of 60° (or the radius) upon the arc, 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 45°; making therefore 2 inches the transverse distance to 45° and 45° 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 arc; make the given radius the transverse distance between 0 and 0 on the secants; then will the transverse distance of 20° and 20°, or

4thly. It 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.

The use of this line is to inscribe a regular polygon in a circle. For example, let it be required to inscribe an octagon or polygon of eight equal sides, 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.

B

All proportions in Trigonometry are easily worked by the double lines on the Sector; observing that the sides of triangles are taken upon the line of lines, and the angles are taken upon 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 angle B (46° 30′) :: AB (56) : sine angle 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 line of lines, and open the Sector to make this a transverse distance of 46° 30 angle 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'angle C. Hence the sum of the angles B and C is 85° 54', which taken from 180°, leaves the angle A94° 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 ne transverse distances on the tangents or secants, instead of measuring them on the ines, 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 win large numbers, a series of numbers, called Logarithms, was invented by Lord Napie., 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 hy simple multiplication, and ...e 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 must 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 integral places in the natural number. Thus the index of the logarithm of any number (integral or mixed), betwee: 10 and 100, is 1; from 100 to 1000, it is 2; from 1000 to 10000 is 3, &c.; the method of finding the logariths from this table will be evident from the following examples.

To find the logarithm 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 pretixed 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 149 was required; this number being found in the lefthand column, against it, in the column marked 0 at the top (or bottom), is found 17319 to which prefixing the index 2, we have the logarithm of 149=2.17319.

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 four places of figures.

Thus, if the logarithm of 1495 was required; opposite to 149, and in the column marked 5 at the top (or bottom), is 17464, to which prefix the index 3, and we have the sought logarithm, 3.17464.

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; ross 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 remaning left-hand figures of this product

than the tumber of integral figures in the given number, and you will have the sought logarithm. To facilitate the calculation of these proportional parts, several small tables are placed in the margin, which give the correction corresponding to the difference D, and to the fifth figure of the proposed number. The use of these tables will be seen in

the following examples.

Thus, if the logarithm of 14957 was required; opposite to 149, and under 5, is 17464, the difference between this and the next greater number, 17493, is 29, the difference D; this multiplied by 7 (the last figure of the given number) gives 203; crossing off the right-hand figure leaves 20.3 or 20 to be added to 17464, which makes 17484; to this prefixing the index 4, we have the sought logarithin, 4.17484. This correction, 20 may also be found by inspection in the small table in the margin, marked at the top with D=29, and opposite to the fifth figure of the number, namely 7, at the side; the corresponding number is the correction, 20.

Again, if the logarithm of 1495738 was required; the logarithm corresponding to 149 at the left, and 5 at the top, is, as in the last example, 17464; the difference between this and the next greater is 29; multiplying this by 738 (which is equal to the given number, excluding the four first figures) gives 21402; crossing off the three right-hand figures of this product (because the number 738 consists of three figures), we have the correction 21 to be added to 17464; and the index to be prefixed is 6, because the given number consists of 7 places of figures; therefore the sought logarithin is 6.17485. This correction, 21, may be found as above, by means of the marginal table, marked at the top with D=29, and at the side 7.38 or 7 nearly, to which corresponds 21, as before.

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 integral part of the given number.

Thus, if the logarithm of the mixed decimal 149.5738 was required; find the logarithm of 1495738, without noticing the decimal point; this, in the last example, was found to be 17485; to this we must prefix the index 2, corresponding to the integral Dart 149; the logarithm sought will therefore be 2.17485.

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 ir the index, which must afterwards be neglected in summing them with other indices thus, instead of writing the index 1, it is usually 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 rule.

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 logarithm of 0,0391 is 8.59218; the logarithm of 0.25 is 9.39794; the logarithin 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 vemainder will be the logarithm of the fraction sought.

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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

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