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semi-tangents, which must be marked with the corresponding figures of the arc 10, 20, &c.

The preceding are some of the principal lines on the plain scale, and to render them convenient for practice, they are transferred from a general figure, such as figure the 1st, to a plain scale or ruler, such as figure 2d, and arranged in the same order.

Besides the lines already mentioned, there are, on several scales, other lines derived from the circle, such as

(E) Miles of Longitude. Divide the line ac into 60 equal parts, through each of these draw lines a 10; b 20, &c. parallel to dc, and every tenth of them will cut the arc ad in the points a, b, c, d, &c.; from A as a centre, the several divisions of the arc ad may be transferred to the line AD, which will give the divisions of the line of longitude. If this line be laid upon the scale close to the line of chords (as in Fig. III.) both inverted, so that 60 on the scale of longitude be against O on the chords, &c. and any degree of latitude be counted on the chords, there will stand opposite to it, on the line of longitude, the miles contained in one degree of longitude in that latitude, a degree at the equator being 60 miles.

These lines are generally drawn to a larger radius than the lines above described, and consequently are applied to a larger scale of chords, as appears by Fig. III.

(F) Equal Parts. The divisions of the line ac to form the miles of longitude may be considered as a scale of equal parts, and on some plain scales are laid down as such, marked Leag. viz. Leagues or equal parts.

Some scales are divided into equal parts, as in Fig. IV.

The outer one is generally a scale of inches. The others are divided similar to those marked Aj B, C, &c. The scale A contains 40 equal parts to an inch, the scale B 35, c 30, &c. But the most useful scale of equal parts is

(G) The Diagonal Scale. Draw eleven lines parallel to, and equidistant from, each other, as in Fig. V.; cut them at right angles by the lines BC; EF; 1,9; 2, 7, &c. then will hg, &c. be divided into ten equal parts, divide the line op into ten equal parts, also the line m n. From the points of division on the line m n, draw diagonals to the points of division on the line op, viz. join m and the first division on op, the first division on m n, and the second on op, &c.

The chief use of such a scale as this, is to lay down any line from a given measure, or to measure any line. In doing which, the units are counted from m towards 0, the tens from m towards n, and the hundreds from m towards H.

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Thus, for example, to take off the number 242, extend the compasses from m to 2, towards h; with one leg fixed in the point 2, extend the other till it reaches 4 in the line mn; move one leg of the compasses along the line 2, 7, and the other along the line 4, till you come to the line marked 2 in BC, and you will have the number required.

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OF THE LOGARITHMICAL LINES, OR GUNTER'S SCALE.

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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.

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

(H) S. Rhumb, or sine 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.

(I) T. Rhumb, or tangent rhumbs, also corresponds to the logarithms of the tangents 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 four, we must begin at 1 and count towards the right hand; but to take off any number of points above four, we must begin at four and count towards the left hand.

(K)* Numbers, or the line of numbers, is numbered from the left hard 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

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* This line being the principal one on the scale, has been contrived various ways. Gunter first applied it to the two feet rulers, or rather to the cross-staff. Wingate drew the logarithms on two separate rulers sliding against cach other, to save the use of compasses. Oughtred applied the logarithms to concentric circles. Mr. William Nicholson proposed another disposition of them on concentric circles; (vide Philosophical Transactions 1787, page 251 ;), his instrument consists of three concentric circles engraved and graduated on a plate of brass : from the centre, two legs proceed as radii, having straight edges; these

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 left hand 1, and the first 2, 3, 4, &c. is exactly equal to the distance between the iniddle 1 and the 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 left hand 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 ten at the right hand 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, the second 2, 3, 4, 5, &c. will stand for 200, 300, 400, 500, &c. and the ten at the right hand for 1000.

If you consider the first 1 as to of an unit, the 2, 3, 4, &c. following will be 1o, po, is, &c. the middle I will stand for an 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.

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

(M) 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 169; the small divisions are here to be estimated according to the number of them to a degree.

(N) Tangent. The line of tangents begins at the left hand, and is numbered 1, 2, 3, &c. to 10, then 20, 30, 45, where

legs or rulers are moveable about the centre either singly or together, so that if one be placed at the first term of any proportion, and the other at the second, and they be then fixed at this angle and moved together till the leg which was fixed at the first term coincides with the third, the other leg will point out the fourth.

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 arcs 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.

(0) 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 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 mites by multiplying by 60. THE CONSTRUCTION OF THE LOGARITHMICAL LINES ON

GUNTER'S SCALE. (P) The line of numbers on which most of the others depend, sometimes called the Line of Lines, is constructed thus: Let a line equal in length to half the line of numbers be divided into 1000 equal parts, then since the logarithm of 1 is 0, the distance of 1 from the beginning of the line is 0, viz. 1 stands at the beginning of the line. And because the logarithm of 2 is •301, when the logarithm of 10 is 1, or, which is the same thing, the logarithm of 2 is 301, when the logarithm of 10 is 1000; therefore the distance between 1 and 2 is 301 equal parts taken from a plain scale of equal parts; the distance 477, the logarithm of 3, must be set off from 1 to 3; and 602 equal parts, the logarithm of 4, must be set off from 1 to 4, &c. Thus are all the primary divisions on the line formed: And the intermediate divisions are formed in a similar manner, by taking the logarithms of the intermediate numbers, as for example, the logarithm of 116, is 41, the logarithm of 1*. is 79, of 1 } is 114, &c. These numbers set off in a successive order from 1, will divide the primary divisions into ten parts, &c.

(Q) The line of sines is constructed by taking the * arithmetical complements of the logarithmical sines from the same scale of equal parts which the line of numbers was constructed from, and setting them off from 90 backwards, or towards the left hand; thus the arithmetical complement of the logarithmical sines of 80°, 70°, 60°, 50°, 40°, 30°, 20°, 10°, (or, which is the same thing, their cosecants rejecting the indices,) are 7, 27, 116, 192, 301, 466, 760, and these taken from the same scale of equal parts with which the line of numbers was constructed, and set off from 90 towards the left hand, give the sines of the abovementioned degrees, respectively. The reason of setting off these numbers thus is obvious, for the sine of 90°is equal to the radius; and the several arithmetical complements are what ihe sines of the arcs want of radius, viz. their distances from radius or 90°.

In the same manner the intermediate degrees, and divisions of the degrees are found.

(R) The line of sine rhumbs is constructed in a similar manner to the line of sines, by taking the arithmetical complements of the logarithmical sines of the degrees and minutes which are contained in the several points, and quarter-points of the compass : and setting them off from the right hand end of the scale, towards the left.

(S) The line of tangents is constructed in the same manner as that of the sines, by setting off the arithmetical complements of the tangents under 45°, backward from 45° towards the left hand. For the tangent of 45° is equal to the radius, and the arithmetical complement of any logarithmical tangent under 45°, is what that tangent wants of radius. It has been observed that the division at 40 serves both for 40 and 50, that at 30 for 30 and 60, &c. The reason of this will appear

if

we consider that the tangent of any arc is to the radius, as the radius to the

tangent

radius co-tangent; or, which is the same thing,

radius co-tangent or logarithmically speaking, the difference between the tangent and radius is equal to the difference between the radius and cotangent. But the tangent of 45° is equal to the radius, therefore the difference between the tangent of an arc (below 45°) and 45°, is equal to the difference between the co-tangent of that arc and 45°; viz. they are both equidistant from 45°.

(T) The line of tangent rhumbs is constructed in a similar manner as the line of tangents, by taking the arithmetical com

* The arithmetical complement of any logarithm is what that logarithm wants of 10; or what a sine, &c. wants of the logarithmical radius. Thus the arithmetical complement of 2.56820, the logarithm of 370, is 7•43180; the arithmetical complement of 9.53405, the logarithmical sine of 20, is ·46595.

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