Of the line of longitudes. The line of longitudes is a line. divided into sixty unequal parts, and so applied to the line of chords as to show, by inspection, the number of equatorial miles contained in a degree on any parallel of latitude. The graduated line of chords is necessary, in order to show the latitudes; the line of longitude shows the quantity of a degree on each parallel in sixtieth parts of an equatorial degree, that is, miles.

The lines of tangents, semitangents and secants serve to find the centres and poles of projected circles in the stereographical projection of the sphere.

The line of sines is principally used for the orthographic projection of the sphere.

The lines of latitudes and hours are used conjointly, and serve very readily to mark the hour lines in the construction of dials: they are generally on the most complete sorts of scales and sectors; for the uses of which see treatises on dialling.

OF THE PROTRACTOR.

This is an instrument used to protract or lay down an angle containing any number of degrees, or to find how many degrees are contained in any given angle. There are two kinds put into cases of mathematical drawing instruments; one in the form of a semicircle, the other in the form of a parallelogram. The circle is undoubtedly the only natural measure of angles; when a straight line is therefore used the divisions thereon are derived from a circle or its properties, and the straight line is made use of for some relative convenience: it is thus the parallelogram is often used as a protractor instead of the semicircle, because it is in some cases more convenient, and that other scales, &c. may be placed upon it.

The semicircular protractor is divided into 180 equal parts or degrees, which are numbered at every tenth degree each way, for the conveniency of reckoning either from the right towards the left, or from the left towards the right; or the more easily to lay down an angle from either end of the line, beginning at each end with 10, 20, &c. and proceeding to 180 degrees. The edge is the diameter of the semicircle, and the mark in the middle points out the centre, in a protractor in the form of a parallelogram: the divisions are, as in the semicircular one, numbered both ways; the blank side represents the diameter of a circle. The side of the protractor to be applied to the paper is made flat, and that whereon the degrees are marked is chamfered or sloped away to the edge, that an angle

may be more easily measured, and the divisions set off with greater exactness.

Application of the protractor to use. 1. A number of degrees being given, to protract or lay down an angle whose measure shall be equal thereto.

Thus, to lay down an angle of 60 degrees from the point of a line, apply the diameter of the protractor to the line, so that the centre thereof may coincide exactly with the extremity; then, with a protracting pin make a fine dot against 60 upon the limb of the protractor; now remove the protractor, and draw a line from the extremity through that point, and the angle contains the given number of degrees.

2. To find the number of degrees contained in a given angle. Place the centre of the protractor upon the angular point, and the fiducial edge or diameter exactly upon the line; then the degree upon the limb that is cut by the line will be the measure of the given angle, which, in the present instance is found to be 60 degrees.

3. From a given point in a line to erect a perpendicular to that line.

Apply the protractor to the line, so that the centre may coincide with the given point, and the division marked 90 may be cut by the line; then a line drawn against the diameter of the protractor will be the perpendicular required.

OF PARALLEL RULES.

Parallel lines occur so continually in every species of mathematical drawing, that it is no wonder so many instruments have been contrived to delineate them with more expedition than could be effected by the general geometrical methods. For this purpose rules of various constructions have been made, and particularly recommended by their inventors; their use, however, is so apparent as to need no explanation.

GUNTER'S SCALE.

The scale generally used is a ruler two feet in length, having drawn upon it equal parts, chords, sines, tangents, secants, &c. These are contained on one side of the scale, and the other side contains the logarithms of these numbers. Mr. Edmund Gunter was the first who applied the logarithms of numbers and of sines and tangents to straight lines drawn on a scale or ruler, with which proportions in common numbers and trigonometry may be solved by the application of a pair of compasses only. The method is founded on this property,

That the logarithms of the terms of equal ratios are equidifferent. This was called Gunter's Proportion and Gunter's Line; hence the scale is generally called the Gunter.

Of the Logarithmical Lines on Gunter's Scale.

The logarithmical lines on Gunter's Scale are the eight following:

SRhumb, or fine rhumbs, is a line containing the logarithms of the natural sines of every point and quarter point of the compass, numbered from a brass pin on the right-hand towards the left with 8, 7, 6, 5, 4, 3, 2, 1..

TRhumb, or tangent rhumbs, also corresponds to the logarithm of the tangent of every point and quarter point of the compass. This line is numbered from near the middle of the scale with 1, 2, 3, 4, towards the right-hand, and back again with the numbers 5, 6, 7, from the right-hand towards the left. To take off any number of points below 4, we must begin at 1 and count towards the right-hand; but to take off any number of points above 4, we must begin at 4 and count towards the left-hand.

Numbers, on the line of numbers, is numbered from the lefthand of the scale towards the right, with 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, which stands exactly in the middle of the scale; the numbers then go on 2, 3, 4, 5, 6, 7, 8, 9, 10, which stands at the right-hand end of the scale. These two equal parts of the scale are divided equally, the distance between the first or lefthand 1 and the first 2, 3, 4, &c. is exactly equal to the distance between the middle 1 and numbers 2, 3, 4, &c. which follow it. The subdivisions of these scales are likewise similar, viz. they are each one-tenth of the primary divisions, and are distinguished by lines of about half the length of the primary divisions.

These subdivisions are again divided into ten parts, where room will permit; and where that is not the case the units must be estimated or guessed at by the eye, which is easily done by a little practice.

The primary divisions on the second part of the scale are estimated according to the value set upon the unit on the lefthand of the scale: if you call it one, then the first 1, 2, 3, &c. stand for 1, 2, 3, &c.; the middle 1 is 10, and the 2, 3, 4, &c. following stand for 20, 30, 40, &c.; and the 10 at the righthand is 100. If the first 1 stand for 10, the first 2, 3, 4, &c. must be counted 20, 30, 40, &c.; the middle 1 will be 100, and the second 2, 3, 4, 5, &c. will stand for 200, 300, 400, 500, &c.; and the 10 at the right for 1000.

If you consider the first

as of a unit, the 2, 3, 4, &c.

following will be a,,, &c.; the middle 1 will stand for a unit, and the 2, 3, 4, &c. following will stand for 2, 3, 4, &c.; also, the division at the right-hand end of the scale will stand for 10. The intermediate small divisions must be estimated according to the value set upon the primary ones.

Sine. The line of sines is numbered from the left-hand of the scale towards the right, 1, 2, 3, 4, 5, &c. to 10; then 20, 30, 40, &c. to 90, where it terminates just opposite 10 on the line of numbers.

Versed sine. This line is placed immediately under the line of sines, and numbered in a contrary direction, viz. from the right-hand towards the left, 10, 20, 30, 40, 50, to about 160; the small divisions are here to be estimated according to the number of them to a degree. By comparing the line of versed sines with the line of sines, it will appear that the versed sines do not belong to the arches with which they are marked, but are the half versed sines of their supplements. Thus, what is marked the versed sine of 90 is only half the versed sine of 90, the versed sine of 120° is half the versed sine of 60°, and the versed sine marked 100° is half the versed sine of 80°, &c.

The versed sines are numbered in this manner to render them more commodious in the solution of trigonometrical and astronomical problems.,

Tangents. The line of tangents begins at the left-hand, and is numbered 1, 2, 3, &c. to 10, then 20, 30, 45, where there is a little brass pin just under 90 in the line of sines, because the sine of 90° is equal to the tangent of 45°. It is numbered from 45° towards the left-hand 50, 60, 70, 80, &c. The tangents of arches above 45° are therefore counted backward on the line, and are found at the same points of the line as the tangents of their complements.

Thus the division at 40 represents both 40 and 50, the division at 30 serves for 30 and 60, &c.

Meridional Parts. This line stands immediately above a line of equal parts, marked Equal Pt., with which it must always be compared when used. The line of equal parts is marked from the right-hand to the left with 0, 10, 20, 30, &c.; each of these large divisions represents 10 degrees of the equator, or 600 miles. The first of these divisions is sometimes divided into 40 equal parts, each representing 15 minutes or miles,

The extent from the brass pin on the scale of meridional parts to any division on that scale, applied to the line of equal parts, will give (in degrees) the meridional parts answering to

the latitude of that division. Or the extent from any division to another on the line of meridional parts, applied to the line of equal parts, will give the meridional difference of latitude between the two places denoted by the divisions. These degrees are reduced to leagues by multiplying by 20, or to miles by multiplying by 60.

The use of the Logarithmical Lines on Gunter's Scale.

By these lines and a pair of compasses all the problems of trigonometry, &c. may be solved.

These problems are all solved by proportion. Now, in natural numbers the quotient of the first term by the second is equal to the quotient of the third by the fourth: therefore, logarithmically speaking, the difference between the first and second term is equal to the difference between the third and fourth; consequently, on the lines on the scale the distance between the first and second term will be equal to the distance between the third and fourth. And for a similar reason, because four proportional quantities are alternately proportional, the distance between the first and third terms will be equal to the distance between the second and fourth. Hence the following

General Rule.

The extent of the compasses from the first term to the second will reach, in this same direction, from the third to the fourth term. Or, the extent of the compasses from the first term to the third will reach, in the same direction, from the second to the fourth.

By the same direction in the foregoing rule is meant, that if the second term lie on the right-hand of the first the fourth will lie on the right-hand of the third, and the contrary. This is true, except the two first or two last terms of the proportion are on the line of tangents, and neither of them under 45°; in this case, the extent on the tangents is to be made in a contrary direction for had the tangents above 45° been laid down in their proper direction, they would have extended beyond the length of the scale towards the right-hand; they are therefore, as it were, folded back upon the tangents below 45°, and consequently lie in a direction contrary to their proper and natural order.

If the two last terms of a proportion be on the line of tangents, and one of them greater and the other less than 45°, the extent from the first term to the second will reach from the third beyond the scale. To remedy this inconvenience, apply the extent between the two first terms from 45° backward upon the line of tangents, and keep the left-hand point of the