The use of this line is for protracting and measuring of 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 arches 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 triangular canon of numbers, by which the several cases in Trigonometry may be solved, the right lines, graduated as above, being placed severally upon a ruler, form the instrument called the Plane Scale; 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 to determine their largeness and number of parts; what an inch is divided into is generally set at the end of the scale, as in the scales A, B, and C; the numbers 10, 20, 30, 45, shew that so many parts, of the scales A, B, C, are contained in an inch. By any scale of equal parts, divided as above, 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 can only be guessed at; wherefore, in these scales, it is better to use such a scale as D, called a diagonal scale, by which any number of three figures may be exactly found.

Having prepared a ruler of convenient breadth for your scale, (which may be an inch, more or less), first, near the edges thereof, draw two right lines, af, eg, parallel to each other; then divide one of these lines, as af, into equal parts, according to the largeness you intend your scale; and through each of these divisions draw perpendicular right lines as far as the line c d; next divide the breadth into 10 equal parts, and through each of these divisions draw right lines parallel to the former a f and eg; again divide the length a, c, d, g, each into 10 equal parts, and from the point c to the first division in the line d g, draw a right line; then parallel to that line, draw right lines through all the other divisions, and the scale is done.

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, easting or westing, make 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 are 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.

WHILE the reader is perusing the following, it is proper he

should have a GUNTER'S SCALE before him.

Gunter's Scale hath set upon it these eight lines following:

Ist. Sine rhumbs, marked (SR) is a line which contains the logarithms of the natural sine of every point and quarter point of the Mariner's Compass, figured from the left hand towards the right, with 1, 2, 3, 4, 5, 6, 7, to 8, where is a brass pin, and where it can be done, into halves and quarters.

2d. Tangent rhumbs, marked (TR) also corresponds to the logarithm of the tangent of every point of the compass, and is figured 1, 2, 3, 4, where there is a pin, and from thence towards the left hand with 5, 6, 7.

3d. The line of numbers marked (Num.) contains the logarithms of the numbers, and is figured thus; near the left hand it begins at 1, and towards the right hand is 2, 3, 4, 5, 6, 7, 8, 9 ; and then 1, at which is a brass centre pin, going still on 2, 3, 4, 5, 6, 7, 8, 9, and 10 at the end, where there is another brass pin; (as this line is generally much used, it requires a larger description.) The first one may be counted for 1, or 10, or 100, or 1000, and then the next 2 is accordingly 2, or 20, or 200, or 2000, &c. Again, the first 1 may be reckoned 1 tenth, or 1 hundredth, or 1 thousandth part, &c. then the next is 2 tenth, or 2 hundredth, or 2 thousand parts, &c. so that if the first one be esteemed 1, the middle is then 10, and 2 to its right is 20, 3 is 30, 4 is 40, and 10 at the end is 100; again, if the first I is 10, the next 2 is 20, 3 is 30, so on, making the middle 1 now 100, the next 2 is 200, 3 is 300, 4 is 400, and 10 at the end is now 1000. In like manner, if the first 1 be esteemed 1 tenth part, the next 2 is 2 tenth parts, and the middle 1 is 1, and the next 2 is 2, and 10 at the end is now 10. Again, if the first 1 be counted 1 hundredth part, the next is 2 hundredth parts, the middle one is now 10 hundredth parts, or 1 tenth part, and the next 2 is 2 tenth parts, and 10 at the end is now 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 (on the right hand of it) is 2, and each subdivision between them now is 1 tenth part, and so all the way to the middle 1, which now is 10, the next 2 is 20: now the longer strokes between 1 and 2 are 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 all the shorter strokes between them longer, are now each to be counted for 1 tenth part from the middle one to the next 2, now 20, from whence the longer strokes between the figures are units, thus 21, 22, 23, &c. to 3, which now is 30, and the shorter strokes each between them, now is the tenth part of an integer; from 3, each short stroke or division, is 1 tenth part of a unit. Again, if 1 at the left hand be 10, the figures between it.and the middle 1 are common tens; and the subdivisions between each figure are units; from the middle 1 to 10 at the end, each figure is so many hundredths; and between these figures each longer division is 10; from the middle 1 to 2, each less division is 2 units; and, from 2 to the end, each shorter division is 5 units. From this description it will be easy to find the divisions representing any given number, thus: Suppose the point representing the number 12 was required: Take the division at the figure 1, in the middle, for the first figure, of 12; then, for the second figure, count 2 tenths, or longer strokes to the right hand, and this last is the point representing 12, where is the brass pin.

Again, Suppose the number 22 were required, the first figure being 2, I take the division to the figure 2, and for the 2d 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 is 1700, then for the third 2 count 2 tenths from the last, and it represents 1720; lastly, for the 4th figure 8, estimate 8 parts out of 10 of the next smaller division, or a little less than 10, this point, last found, represents 1728.

Required the point, representing the number 435: from the 4 in the 2d interval count towards 5 on the right, three of the larger divisions, and one of the smaller, and that will be the division expressing 435, and the like of other numbers, which by a little practice is readily done.

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 thenumerator; that extent laid upon 1 in the middle will reach to the decimal required. Example. Required the decimal fraction equal to 2, 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 was 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.