ON THE SLIDING RULE. THE Sliding Rule consists of a fixed part and a slider, and is of the same dimensions, and has the same lines marked on it as on a common Gunter's Scale or Plane Scale, which may be used, with a pair of compasses, in the same manner as those scales; and as a description of those lines has already been given, it will be unnecessary to repeat it here, it being sufficient to observe, that there are two lines of numbers, a line of log. sines, and a line of log. tangents, on the slider, and that it may be shifted so as to fix any face of it on either side of the fixed part of the scale, according to the nature of the question to be solved. In solving any problem in Arithmetic, Trigonometry, Plane Sailing, &c., let the proposition be so stated that the first and third terms may be alike, and of course the second and fourth terms alike; then bring the first term of the analogy on the fixed part, against the second term on the slider, and against the third term on the fixed part will be found the fourth term on the slider; or, if necessary, the first and third terms may be found on the slider, and the second and fourth on the fixed part. Multiplication and division are performed by this rule, in considering unity as one of the terms of the analogy. * Thus, to perform multiplication; set 1 on the line of numbers of the fixed part, against one of the factors on the line of numbers of the slider; then against the other factor, on the fixed part, will be found the product on the slider. EXAMPLE. To find the product of 4 by 12; draw out the slider till 1 on the fixed part coincides with 4 on the slider; then opposite 12 on the fixed part will be found 48 on the slider. To perform division; set the divisor on the line of numbers of the fixed part against 1 on the slider; then against the dividend on the fixed part will be found the quotient on the slider. EXAMPLE. To divide 48 by 4; set 4 on the fixed part against 1 on the slider; then against 48 on the fixed part will be found 12 on the slider. EXAMPLES IN THE RULE OF THREE. If a ship sail 25 miles in 4 hours, how many miles will she sail in 12 hours at the same rate? Bring 4 on the line of numbers of the fixed part against 25 on the line of numbers of the slider; then against 12 on the fixed part will be found 75 on the slider, which is the answer required. EXAMPLE. If 3 pounds of sugar cost 21 cents, what will 27 pounds cost? Bring 3 on the line of numbers of the fixed part, against 21 on the line of numbers of the slider; then against 27 on the fixed part will be found 189 on the slider. EXAMPLE IN TRIGONOMETRY. In the oblique-angled triangle ABC, let there be given AB=56, AC-64, angle ABC=46° 30′, to find the other angles and the side BC. In this case we have (by Art. 58, Geometry) the following canons: B AC (64): sine angle B (46° 30′):: AB (56): sine angle C; and sine angle B: AC:: sine angle ABC. Therefore, to work the first proportion by the sliding rule, we must bring 64 on the line of numbers of the fixed part against 46° 30′ on the line of sines of the slider; then against 56 on the former will be 39° 24′ on the latter, which will be *If the first and second terms are alike, instead of the first and third, you must bring the first term on the slider against the third on the fixed part, and against the second term on the slider will be found the fourth term on the fixed part; or, if necessary, the first and second terms may be found on the fixed part, and the third and fourth on the slider. the angle C. The sum of the angles B and C, being subtracted from 180°, leaves the angle A=94° 6'. Then, by the second canon, bring the angle B=46° 30′, on the line of sines of the slider against AC=64, on the line of numbers of the fixed part; then against the angle A = 94° 6' (or its supplement, 85° 54′) on the slider will be found the side BC=88 on the fixed part. In a similar manner may the other propositions in Trigonometry be solved. From what has been said, it will be easy to work all the problems in Plane, Middle Latitude, and Mercator's Sailing, as in the three following examples, which the learner may pass over until he can solve the same problems by the scale. If any one wishes to know the use of the Sliding Rule in problems of Spherical Trigonometry, he may consult the treatises written expressly on that subject; but it may be observed, that in such calculations the Sliding Rule is rather an object of curiosity than of real use, as it is much more accurate to make use of logarithms. EXAMPLE I. Given the course sailed 1 point, and the distance 85 miles; required the difference of latitude and departure. By Case I. of Plane Sailing, we have these canons :— Radius (8 points): Distance (85):: Sine Co. Course (7 points): Difference of Latitude; and Radius (8 points): Distance (85) :: Sine Course (1 point): Departure. Hence we must bring the radius, 8 points, on the fixed part of the sine rhumbs against 85 on the line of numbers on the slider; then against 7 points on the sine rhumbs will be found the difference of latitude, 834, on the slider, and against 1 point will be found the departure, 16 miles. If the course is given in degrees, you must use the line marked (Sin.) EXAMPLE II. Given the difference of latitude, 40 miles, and departure, 30 miles; required the course and distance. By Case VI. of Plane Sailing, we have this canon for the course:- Hence we must bring 40 on the line of numbers of the slider against 45° on the line of tangents on the fixed part; then against 30 on the slider will be found the course, 37°, nearly. Again, the canon for the distance gives Sine Course (37°): Departure (30) :: Radius (90°): Distance. Hence we must bring 37° on the line of sines of the fixed part against 30 on the line of numbers on the slider; then against 90° on the line of sines of the fixed part will be found the distance, 50, on the slider. EXAMPLE III Given the middle latitude, 40°, and the departure, 30 miles; required the difference of longitude. By Case VI. of Middle Latitude Sailing, we have this canon: Sine Comp. Middle Latitude (50°): Departure (30) :: Radius (90°): Difference Long. Hence by bringing 50°, on the line of sines of the fixed part, against 30 on the line of numbers on the slider, then against 90° on the fixed part we shall find 39 on the slider, which will be the difference of longitude required. 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, or 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; 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 III. 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 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? 63 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 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. B E 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. And 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 the 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. 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. 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 A 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° 30angle 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 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 = |