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GUNTER'S SCALE.

ON Gunter's Scale are eight lines, viz.

1st. Sine rhumbs, marked (SR), corresponding to the logarithms of the natura1 sines of every point of the mariner's compass, numbered from the left hand towards the right, with 1, 2, 3, 4, 5, 6, 7, to 8, where is a brass pin. This line is also divided, where it can be done, into halves and quarters.

2dly. Tangent rhumbs, marked (TR), correspond to the logarithms of the tangents of every point of the compass, and are numbered 1, 2, 3, to 4, at the right hand, where there is a pin, and thence towards the left hand with 5, 6, 7; it is also divided, where it can be done, into halves and quarters.

3dly. The line of numbers, marked (Num.), corresponds to the logarithms of numbers, and is marked thus: near the left hand it begins at 1, and towards the right hand are 2, 3, 4, 5, 6, 7, 8, 9; and 1 in the middle, at which is a brass pin; then 2, 3, 4, 5, 6, 7, 8, 9, and 10, at the end, where there is another pin. The values of these numbers and their intermediate divisions depend on the estimated values of the extreme numbers 1 and 10; and as this line is of great importance, a particular description of it will be given. The first 1 may be counted for 1, 10, 100, or 1000, &c., and then the next 2 will be 2, 20, 200, or 2000, &c., respectively. Again, the first 1 may be reckoned 1 tenth, I hundredth, or 1 thousandth part, &c.; then the next will be 2 tenth, or 2 hundredth, or 2 thousandth parts, &c.; so that if the first 1 be esteemed 1, the middle 1 will be 10; 2 to its right, 20; 3, 30; 4, 40; and 10 at the end, 100. Again, if the first is 10, the next 2 is 20, 3 is 30, and so on, making the middle 1, 100; the next 2 is 200, 3 is 300, 4 is 400, and 10 at the end is 1000. In like manner, if the first 1 be esteemed 1 tenth part, the next 2 will be 2 tenth parts, and the middle 1 will be 1; the next 2, 2; and 10 at the end will be 10. Again, if the first 1 be counted 1 hundredth part; the next, 2 hundredth parts; the middle 1 will be 10 hundredth parts, or 1 tenth part: and the next 2, 2 tenth parts; and 10 at the end will be but one whole number or integer.

As the figures are increased or diminished in their value, so in like manner must all the intermediate strokes or subdivisions be increased or diminished; that is, if the first I at the left hand be counted 1, then 2 (next following it) will be 2, and each subdivision between them will be 1 tenth part; and so all the way to the middle 1, which will be 10; the next 2, 20; and the longer strokes between 1 and 2 are to be counted from 1 thus, 11, 12 (where is a brass pin); then 13, 14, 15, sometimes a longer stroke than the rest; then 16, 17, 18, 19, 20, at the figure 2; and in the same manner the short strokes between the figures 2 and 3, 3 and 4, 4 and 5, &c., are to be reckoned as units. Again, if 1 at the left hand be 10, the figures between it and the middle 1 will be common tens, and the subdivisions between each figure will be units; from the middle 1 to 10 at the end, each figure will be so many hundreds; and between these figures each longer division will be 10. From this description it will be easy to find the divisions representing any given number, thus: Suppose the point representing the number 12 were required; take the division at the figure 1 in the middle, for the first figure of 12; then for the second figure count two tenths, or longer strokes to the right hand, and this will be the point representing 12, where the brass pin is.

Again, suppose the number 22 were required; the first figure 2 is to be found on the scale, and for the second figure 2, count 2 tenths onwards, and that is the point representing 22.

Again, suppose 1728 were required; for the first figure 1, I take the middle 1, for the second figure 7, count onwards as before, and that will be 1700. And, as the remaining figures are 28, or nearly 30, I note the point which is nearly of the distance between the marks 7 and 8, and this will be the point representing 1728.

&c.

If the point representing 435 was required, from the 4 in the second interval count towards 5 on the right, three of the larger divisions and one of the smaller (this sinaller division being midway between the marks 3 and 4), and that will be the division expressing 435. In a similar manner other numbers may be found.

All fractions found in this line must be decimals; and if they are not, they must be reduced into decimals, which is easily done by extending the compasses from the denominator to the numerator; that extent laid the same way, from 1 in the middle o right hand, will reach to the decimal required.

EXAMPLE. Required the decimal fraction equal to . Extend from 4 to 3; that extent will reach from 1 on the middle to .75 towards the left hand. The like may bo observed of any other vulgar fraction.

Multiplication is performed on this line by extending from 1 to the multiplier; that extent will reach from the multiplicand to the product.

Suppose, for example, it were required to find the product of 16 multiplied by 4; extend from 1 to 4; that extent will reach from 16 to 64, the product required.

Division being the reverse of multiplication, therefore extend from the divisor to unity; that extent will reach from the dividend to the quotient.

Suppose 64 to be divided by 4; extend from 4 to 1; that extent will reach from 64 to 16, the quotient.

Questions in the Rule of Three are solved by this line as follows: Extend from the first term to the second; that extent will reach from the third term

to the fourth.

And it ought to be particularly noted, that if you extend to the left, from the first number to the second, you must also extend to the left, from the third number to the fourth; and the contrary.

EXAMPLE. If the diameter of a circle be 7 inches, and the circumference 22, what is the circumference of another circle, the diameter of which is 14 inches? Extend from 7 to 22; that extent will reach from 14 to 44, the same way.

The superficial content of any parallelogram is found by extending from 1 to the breadth hat extent will reach from the length to the superficial content.

EXAMPLE. Suppose a plank or board to be 15 inches broad and 27 feet long, the content of which is required. Extend from 1 to 1 foot 3 inches (or 1 25); that extent will reach from 27 feet to 33.75 feet, the superficial content. Or extend from 12 inches to 15, &c.

The solid content of any bale, box, chest, &c., is found by extending from 1 to the breadth; that extent will reach from the depth to a fourth number, and the extent from I to that fourth number will reach from the length to the solid content.

EXAMPLE I. What is the content of a square pillar, whose length is 21 feet 9 inches, and breadth 1 foot 3 inches? The extent from 1 to 1.25 will reach from 1.25 to 1.56, the content of one foot in length; again, the extent from 1 to 1.56, will reach from the length 21.75 to 33.9, or 34, the solid content in feet.

EXAMPLE II. Suppose a square piece of timber, 1.25 feet broad, .56 deep, and 36 long, be given to find the content. Extend from 1 to 1.25; that extent will reach from .56 to 7; then extend from 1 to .7; that extent will reach from 36 to 25.2, the solid content. In like manner may the contents of bales, &c., be found, which, being divided by 40, will give the number of tons.

4thly. The line of sines, marked (Sin.), corresponding to the log. sines of the degrees of the quadrant, begins at the left hand, and is numbered to the right, 1, 2, 3, 4, 5, &c.. to 10; then 20, 30, 40, &c., ending at 90 degrees, where is a brass centre-pin, as there is at the right end of all the lines.

5thly. The line of versed sines, marked (V. S.), corresponding to the log. versed sines of the degrees of the quadrant, begins at the right hand against 90° on the sines, and from thence is numbered towards the left hand, 10, 20, 30, 40, &c., ending at the left hand at about 169°; each of the subdivisions, from 10 to 30, is in general two degrees; from thence to 90 is single degrees; from thence to the end, each degree is divided into 15 minutes.

6thly. The line of tangents, marked (Tang.), corresponding to the log. tangents of the degrees of the quadrant, begins at the left hand, and is numbered towards the right, 1, 2, 3, &c. to 10, and so on, 20, 30, 40, and 45, where is a brass pin under 90° on the sines; from thence it is numbered backwards, 50, 60, 70, 80, &c. to 89, ending at the eft hand where it began at 1 degree. The subdivisions are nearly similar to those of the sines. When you have any extent in your compasses, to be set off from any number less than 45° on the line of tangents, towards the right, and it is found to reach

*Or you may extend from the first to the third; for that extent will reach from the second to the fourth. This method must be adopted when using the lines of sines, tangents, &c., if the first and third terms are of the same name, and different from the second and fourth.

beyond the mark of 45°, you must see how far it extends beyond that mark, and set it off from 45° towards the left, and see what degree it falls upon, which will be the number sought, which must exceed 45°; if, on the contrary, you are to set off such a distance to the right from a number greater than 45°, you must proceed as before, only remembering, that the answer must be less than 45°, and you must always consider the degrees above 45°, as if they were marked on the continuation of the line to the right hand of 45°.

7thly. The line of the meridional parts, marked (Mer.), begins at the right hand, an is numbered, 10, 20, 30, &c., to the left hand, where it ends at 87 degrees. This line. with the line of equal parts, marked (E. P.), under it, are used together, and only in Mercator's Sailing. The upper line contains the degrees of the meridian, or latitude in a Mercator's chart, corresponding to the degrees of longitude on the lower line.

The use of this Scale in solving the usual problems of Trigonometry, Plane Sailing Middle Latitude Sailing, and Mercator's Sailing, will be given in the course of this work; but it will be unnecessary to enter into an explanation of its use in calculating the common problems of Nautical Astronomy as it is much more accurate to perform those calculations by logarithmns.

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 terins 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. If3 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 sid 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 A94° 5. 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 BC88 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 chumbs will be found the difference of latitude, 834, on the slider, and against 1 point will be found the departure, 164 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 :—

Difference of Latitude (40): Radius (45°) :: Departure (30): Tangent 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 he 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.

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