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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. 1 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

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so that these instruments can be read off by the aid of their verniers to an accuracy of 10 seconds. The verniers occupy on the limbs spaces equal to 9° 50′*.

* 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 index, the limb of the instrument is graduated, as though it were double the size; so that the verniers really occupy 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

30' 30"

off to an accuracy of or 1'.

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 5 ths of an inch.

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

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B

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

equal to the difference of a division of the limb, and of the vernier: then, in fig. 1, one revolution of the screw advances the vernier a distance equal to 20'; and, if the cylindrical head BB be divided into 60 equal parts, a revolution of the screw through one of these parts would advance the vernier a distance equal to 20".

Suppose, then, that in the illustration above given the screw has to be turned back, so that 14 of these graduations pass the index 1, in order to bring the stroke at the end of the second division upon the vernier into coincidence with a stroke upon the limb; then the corresponding space moved through by the vernier would be equal to 20" x 14, or 4′ 40′′, and the reading of the instrument would be 22° 44′ 40′′.

Similarly, by means of a micrometer divided into ten equal parts, a distance to the thousandth part of an inch may be read off by the vernier in fig. 2. If the micrometer in this case were divided into one hundred equal parts, a distance might be read off to the ten-thousandth part of an inch; or the same effect may be produced by dividing the micrometer into ten equal parts, and making the screw of such fineness that ten complete revolutions move the vernier through a distance equal to the difference of a division of the limb and of the vernier, or the hundredth part of an inch.

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The above engraving represents this instrument, which consists of a beam, AA, of any length required, generally made of well-seasoned mahogany. Upon its face is inlaid throughout its whole length a slip of holly, or boxwood, aa, upon which are engraved the divisions or scale, either feet and decimals, or inches and decimals, or whatever particular scale may be required. Those made for the use of the persons engaged on the Ordnance survey of Ireland were divided to a scale of chains, 80 of which occupied a length equal to six inches, which, therefore, represented one mile, six inches to

the mile being the scale to which that important survey is plotted *. Two brass boxes, в and c, are adapted to the beam; of which the latter may be moved, by sliding, to any part of its length, and fixed in position by tightening the clamp screw E. Connected with the brass boxes are the two points of the instrument, G and H, which may be made to have any extent of opening by sliding the box c along the beam, the other box, B, being firmly fixed at one extremity. The object to be attained, in the use of this instrument, is the nice adjustment of the points G, H, to any definite distance apart. This is accomplished by two verniert or reading plates, b, c, each fixed at the side of an opening in the brass boxes to which they are attached, and affording the means of minutely subdividing the principal divisions, a a, on the beam, which appear through those openings. D is a clamp screw for a similar purpose to the screw E, namely, to fix the box B, and prevent motion in the point it carries after adjustment to position. F is a slow motion screw, by which the point & may be moved any very minute quantity for perfecting the setting of the instrument, after it has been otherwise set as nearly as possible by the hand alone.

The method of setting the instrument for use may be understood from the above description of its parts, and also by the following explanation of the method of examining and correcting the adjustment of the vernier, b, which, like all other mechanical adjustments, will occasionally get deranged. This verification must be performed by means of a detached scale. Thus, suppose, for example, that our beam compass is divided to feet, inches, and tenths, and subdivided by the vernier to hundredths, &c. First set the zero division of the vernier to the zero of the principal divisions on the beam, by means of the slow motion screw F. This must be done very nicely. Then slide the box c, with its point G, till the zero on the vernier c exactly coincides with any principal division on the beam, as twelve inches, or six inches, &c. To enable us to do this with extreme accuracy some superior kinds of beam compasses have the box c also furnished with a tangent or slow motion screw, by which the setting of the points or divisions may be performed with the utmost precision. Lastly, apply the points to a similar detached scale, and, if the adjustment be perfect, the interval of the points GH will * The survey of the metropolis is plotted to a scale of 60 inches to the mile. + For a description of the vernier, see preceding article.

D

measure on it the distance to which they were set on the beam. If they do not, by ever so small a quantity, the adjustment should be corrected by turning the screw F till the points do exactly measure that quantity on the detached scale; then, by loosening the little screws which hold the vernier b in its place, the position of the vernier may be gradually changed, till its zero coincides with the zero on the beam; and, then tightening the screws again, the adjustment will be complete.

PLOTTING SCALES.

Plotting scales, also called feather-edged scales, are straight rulers, usually about ten or twelve inches long. Each ruler has scales of equal parts, decimally divided, placed upon its edges, which are made sloping, so that the extremities of the strokes marking the divisions lie close to the paper. The primary divisions represent chains, and the subdivisions, consequently, ten links each, as there are 100 links on the surveying chain. Plotting scales may be procured in sets, each with a different number of chains to the inch.

The advantages of this arrangement are, that the distances required can be transferred with great expedition from the scale to the paper by the aid of the pricking-point alone, and the marks denoting the divisions are in no danger of becoming defaced, as upon the plain scale, by the frequent application of the compasses.

One of the best plotting scales consists of two feather-edged rulers, one sliding along the other in a dovetailed groove, so that the two are always at right angles to each other. We shall describe this instrument more particularly when we come to speak of plotting, after describing the instruments used in surveying.

at A.

THE PANTAGRAPH.

The pantagraph consists of four rulers, A B, AC, DF, and EF, made of stout brass. The two longer rulers, AB and a c, are connected together by, and have a motion round a center The two shorter rulers are connected in like manner with each other at F, and with the longer rulers at D and E, and, being equal in length to the portions AD and AE of the longer rulers, form with them an accurate parallelogram, ADFE, in every position of the instrument. Several ivory castors support the machine parallel to the paper, and allow it to move freely over it in all directions The arms, AB and

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