CONSTRUCTION OF THE PLANE SCALE.

1st. WITH the radius you intend for your scale, describe a semicircle, ADB (Plate II. fig. 1), and from the centre, C, draw CD perpendicular to AB, which will divide the semicircle into two quadrants, AD, BD; continue CD towards S, draw BT perpendicular to CB, and join BD and AD.

2dly. Divide the quadrant BD into 9 equal parts; then will each of these be 10 degrees; subdivide each of these parts into single degrees, and, if your radius will adinit of it, into minutes or some aliquot parts of a degree greater than minutes.

3dly. Set one foot of the compasses in B, and transfer each of the divisions of the quadrant BD to the right line BD, then will BD be a line of chords.

4thly. From the points 10, 20, 30, &c., in the quadrant BD, draw right lines parallel to CD, to cut the radius CB, and they will divide that line into a line of sines which must be numbered from C towards B.

5thly. If the same line of sines be numbered from B towards C, it will become a line of versed sines, which may be continued to 180°, if the same divisions be transferred on the same line on the other side of the centre C.

6thly. From the centre C, through the several divisions of the quadrant BD, draw right lines till they cut the tangent BT; so will the line BT become a line of tangents. 7thly. Setting one foot of the compasses in C, extend the other to the several divisions, 10, 20, 30, &c., in the tangent line, BT, and transfer these extents severally to the right line, CS; then will that line be a line of secants.

8thly. Right lines drawn from A to the several divisions, 10, 20, 30, &c., in the quadrant BD, will divide the radius CD into a line of semi-tangents.

9thly. Divide the quadrant AD into eight equal parts, and from A, as a centre, transfer these divisions severally into the line AD; then will AD be a line of rhumbs, each division answering to 11° 15' upon the line of chords. The use of this line is for protracting and measuring angles, according to the common division of the mariner's compass. If the radius AC be divided into 100 or 1000, &c., equal parts, and the lengths of the several sines, tangents, and secants, corresponding to the several arcs of the quadrant, be measured thereby, and these numbers be set down in a table,* each in its proper column, you will by these means have a collection of numbers by which the several cases in trigonometry may be solved. Right lines, graduated as above, being placed severally upon a ruler, form the instrument called the Plane Scale (see Plate II. fig. 2), by which the lines and angles of all triangles may be measured. All right lines, as the sides of plane triangles, &c., when they are considered simply as such, without having any relation to a circle, are measured by scales of equal parts, one of which is subdivided equally into 10, and this serves as a common division to all the rest. In most scales, an inch is taken for a common measure, and what an inch is divided into is generally set at the end of the scale. By any common scale of equal parts, divided in this manner, any number less than 100 may be readily taken; but if the number should consist of three places of figures, the value of the third figure cannot be exactly ascertained, and in this case it is better to use a diagonal scale, by which any number consisting of three places of figures, may be exactly found. The figure of this scale is given in Plate II. fig. 3; its construction is as follows:

Having prepared a ruler of convenient breadth for your scale, draw near the edges thereof two right lines, af, cg, parallel to each other; divide one of these lines, as af, into equal parts, according to the size of your scale; and, through each of these divisions draw right lines perpendicular to af, to meet cg; then divide the breadth into 10 equal parts, and through each of these divisions draw right lines parallel to af and cg; divide the lines ab, cd, into 10 equal parts, and from the point a to the first division

* In Table XXIV. are given the sine and cosine to every minute of the quadrant, to five places of decimals.

+ The length of one of these equal parts at the end of the scale to which this description refers is ab

in the line cd, draw a diagonal line; then, parallel to that line, draw diagonal lines through all the other divisions, and the scale is complete. Then, if any number, consisting of three places of figures, as 256, be required from the larger scale, gd, you must place one foot of the compasses on the figure 2 on the line gd, then the extent from 2 to the point d will represent 200. The second figure being 5, count five of the smaller divisions from d towards c, and the extent from 2 to that point will be 250. Move both points of the compasses downwards till they are on the sixth parallel line below gd, and open them a little till the one point rests on the vertical line drawn through 2, and the other on the diagonal line drawn through 5; the extent then in the compasses will represent 256. In the same way the quantities 25.6, 2.56, 0.256, &c., are measured.

Besides the lines already mentioned, there is another on the Plane Scale, marked ML, which is joined to a line of chords, and shows how many miles of easting or westing correspond to a degree of longitude in every latitude.* These several lines are generally put on one side of a ruler two feet long; and on the other side is laid down a scale of the logarithms of the sines, tangents, and numbers, which is commonly called Gunter's Scale; and, as it is of general use, it requires a particular description.

As it would confuse the adjoined figure to describe on it the line of longitudes, it is neglected, but the construction is as follows; divide the line CB into 60 equal parts (if it can be done), and through each point draw lines parallel to CD, to intersect the are BD; about B, as a centre, transfer the several points of intersection to the line of chords, BD, and then number it from D towards B, from 0 to 60, and it will be the line of longitudes, corresponding to the degrees on the line of chords.

GUNTER'S SCALE.

ON Gunter's Scale are eight lines, viz.

1st. Sine rhumbs, marked (SR), corresponding to the logarithms of the natural 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, 1 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 1 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 I 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 1 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.

* The description and use of logarithms are given in page 28, et seq. The log. sines, tangents, &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 smaller 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 or 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 be 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; that 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 1 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 left 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, and 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 logarithms.