it is taken, represents the triangle and one of the rules, as being used to draw a series of parallel lines. Either rule is one foot long, and has, parallel to each of its edges, two scales one placed close to the edge and the other immediately within this, the outer being termed the artificial, and the inner, the natural scale. The divisions upon the outer scale are three times the length of those upon the inner scale, so as to bear the same proportion to each other that the longest side of the triangle bears to the shortest. Each inner, or natural scale, is, in fact, a simply divided scale of equal parts (see p. 9), having the primary divisions numbered from the left hand to the right throughout the whole extent of the rule. The first primary division on the left hand is subdivided into ten equal parts, and the number of these subdivisions in an inch is marked underneath the scale, and gives it its name. On one of the pair of Marquois's scales now before us, we have, on one face, scales of 30 and 60, on the obverse scales of 25 and 50, and on the other we have on one face scales of 35 and 45, and on the obverse scales of 20 and 40. In the arti ficial scales the zero point is placed in the middle of the edge of the rule, and the primary divisions are numbered both ways from this point to the two ends of the rule, and are, every one, subdivided into ten equal parts, each of which is, consequently, three times the length of a subdivision of the corresponding natural scale.

The triangle has a short line drawn perpendicular to the hypothenuse near the middle of it, to serve as an index or pointer; and the longest of the other two sides has a sloped edge.

To draw a Line parallel to a given Line, at a given Distance from it.-1. Having applied the given distance to the one of the natural scales which is found to measure it most conveniently, place the triangle with its sloped edge coincident with the given line, or rather at such small distance from it, that the pen or pencil passes directly over it when drawn along this edge. 2. Set the rule closely against the hypothenuse, making the zero point of the corresponding artificial scale coincide with the index upon the triangle. 3. Move the triangle along the rule, to the left or right, according as the required line is to be above or below the given line, until the index coincides with the division or subdivision corresponding to the number of divisions or subdivisions of the natural scale, which measures the given distance; and the line drawn

along the sloped edge in its new position will be the line. required*.

Note. The natural scale may be used advantageously in setting off the distances in a drawing, and the corresponding artificial scale in drawing parallels at required distances.

To draw a Line perpendicular to a given Line from a given Point in it.-1. Make the shortest side of the triangle coincide with the given line, and apply the rule closely against the hypothenuse. 2. Slide the triangle along the rule until a line drawn along the sloped edge passes through the given point; and the line so drawn will be the line required.

The advantages of Marquois's scales are: 1st, that the sight is greatly assisted by the divisions on the artificial scale being so much larger than those of the natural scale to which the drawing is constructed; 2nd, that any error in the setting of the index produces an error of but one-third the amount in the drawing.

If the triangle be accurately constructed, these scales may e advantageously used for dividing lines with accuracy and despatch; our figure, as well as the sliding rule (fig. 4), was drawn by the aid of Marquois's scales alone.

We proceed to explain a method which we have found to answer well for dividing lines with accuracy and despatch, and which is altogether independent of any error in the construction of the triangle.

Let ab represent the line to be divided into any number, n, of equal parts; select one of the natural scales, of which n divisions or some multiple p n of n divisions, are nearly

equal to a b, but less than it; then setting the sloped edge of the triangle perpendicular to a b, so that a line drawn along it passes through a, place the rule closely against the hypothenuse, making a division of the artificial scale, corresponding *If A B C represent the triangle in its new position, and the dotted lines represent its original position, we have, by similar triangles, A B C, A' A D,

D

A

A

to the selected natural scale, coincide with the index upon the triangle; and, moving the triangle along to the rule to the right, until the index coincides with the p nth division from that to which it was set, draw a line along the sloped edge intersecting a b in c. Upon ab as hypothenuse describe the right-angled triangle a bd, having the side a d equal to a c, and placing the sloped edge of the triangle perpendicular to a d, so that a line drawn along it passes through a or d; slide the triangle along the rule to the right or left, and drawing a line as the index comes into contact with every pth division of the artificial scale, the line will be divided as required.

THE VERNIER. (Plate II. Figs. 1 and 2.)

The property of this ingenious little subsidiary instrument will be readily comprehended from what has been already said of the construction and use of a vernier scale (p. 11). It is so constructed as to slide evenly along the graduated limb of an instrument, and enables us to measure distances, or read off observations, with remarkable nicety. In the vernier scale before described, the divisions on the lower or subsidiary scale were longer than those on the upper or primary scale; but in the vernier now to be described the divisions are usually shorter than those upon the limb to which it is attached, the length of the graduated scale of the vernier being exactly equal to the length of a certain number (n - 1) of the divisions upon the limb, and the number (n) of divisions upon the vernier being one more than the number upon the same length of the limb.

Let, then, L represent the length of a division upon the limb, and v, the length of a division upon the vernier: so that (n - 1) L = n V ;

1

L; n

or the defect of a division upon the vernier from a division upon the limb is equal to the nth part of a division upon the limb, n being the number of divisions upon the vernier *

* If n divisions of the vernier were equal to (n + 1) divisions of the limb, or (n+1) L= nv,

or the excess of a division upon the vernier above a division upon the limb would be equal to the nth part of a division upon the limb. With this arrangement, however, we should have the inconvenience of reading the vernier backwards.

In fig. 1, six divisions of the vernier are equal to five divisions of the limb, and, consequently, the above defect, or L- v, is equal to a sixth part of a division upon the limb, or to 20', since a division of the limb is equal to 2o.

In fig. 2, ten divisions of the vernier are equal to nine divisions of the limb, and, consequently, L v is equal to a tenth part of a division upon the limb, or to the hundredth part of an inch, a division of the limb being equal to the tenth part of an inch.

In reading off we must first look to the lozenge, as point ing out the exact place upon the limb at which the required measurement is indicated. If, then, the stroke upon the vernier at the lozenge exactly coincides with a stroke upon the limb, the reading at this stroke gives the measurement required; but, if the stroke at the lozenge be a distance. beyond a stroke upon the limb, then will this distance be equal to once, or twice, or thrice, &c., the difference of a division upon the limb and upon the vernier, according as the stroke at the end of the first, or second, or third, &c., division upon the vernier coincides with a stroke upon the limb.

In fig. I the stroke upon the vernier at the lozenge falls beyond the stroke indicating 22° upon the limb, and the stroke at the end of the second division upon the vernier coincides with a stroke upon the limb; the reading therefore is 22° 40'.

In fig. 2, the stroke upon the vernier at the lozenge falls beyond the stroke indicating one inch and three-tenths upon the limb, and the stroke at the end of the sixth division upon the vernier coincides with a stroke upon the limb: the reading, therefore, is 1.36 inches, or one inch three tenths and six hundredths.

The limbs of the best sextants are now divided at every 10 minutes, and 59 of these parts are made equal to 60 divisions of their verniers. In this case

so that these instruments can be read off by the aid of their verniers to an accuracy of 10 seconds.

on the limbs spaces equal to 9° 50′*.

The verniers occupy

* That is, according to the graduation of the instrument; but, as the angles observed by a sextant are double the angles moved over by the inaex, the limb of the instrument is graduated, as though it were double the size; so that the verniers really cccupy an arc of 4 55' only.

The limbs of small theodolites are generally divided at every 30 minutes, and 29 of these parts are made equal to 30 divisions of their verniers, which therefore enables us to read

In the mountain barometer the scale being divided into ths of an inch, 9 of these parts are made equal to 10 divisions of the vernier, which therefore enables us to read off to an accuracy of 5ths of an inch.

T000

In the above explanations we have only considered the case of an exact coincidence between some one of the strokes upon the vernier and a stroke upon the limb. Suppose now that in fig. 1 the stroke at the end of the second division, instead of coinciding with a stroke upon the limb, fell a little beyond it, while the stroke at the end of the third division fell a little short of a stroke upon the limb; then the measurement indicated would be something between 22° 40′ and 23°, which the observer, should there be no other mechanism attached to the vernier, must estimate by guess, according to the best of his judgment. By the aid, however, of a piece of mechanism, which is called a micrometer, and which we proceed to describe, the excess of the angle indicated above 22° 40′ might be exactly computed.

The instrument having been nearly set by the hand alone, the vernier is fixed in this position by turning a screw, called the clamping screw, which is shown on the top of the vernier in fig. 2, but is not seen in fig. 1, being at the back of the instrument. The instrument is then set more accurately by the screw at the side of the vernier, shown in both figures, which gives a slow motion to the vernier plate, and to the limb or index bar attached to the vernier. This screw is called a tangent or slow motion screw, and the micrometer consists of a graduated cylindrical head, BB, attached to this screw, and an index, I, attached to the vernier. Suppose, now, the tangent screw to be of that fineness that, whilst it is turned once round, by means of the milled head н, so that the graduated head BB makes one complete revolution, the vernier is advanced on the limb of the instrument, a distance

B

B