exactly with the other flag; then motion to fix the flag, which will be the perpendicular required.

THE PROTRACTOR.

Plate 37. The protractor is used to plot angles taken by the theodolite or other instruments, or for measuring angles that are plotted, and transferring them. It is divided in the same manner as the limb of the theodolite.

They are of various kinds and construction-viz. the ivory protractor A, Fig 3 (such as are generally in a case of instruments).

The semicircular brass protractor B, about five or six inches in diameter.

The semicircular brass protractor. This instrument on extensive surveys is not so convenient as the circular, as the divisions extend only to 180°, so that to plot angles between 180° and 360° it has to be reversed.

The wheel protractor, Fig. 5, is a very useful instrument for measuring angles; it is divided into degrees, minutes, and seconds. The bar A is made to move round the centre, having two verniers; attached to which are two protracting points, a, b, with adjustment screws, c, to bring the two points in a straight line passing through the centre; by means of four fine pins it fixes itself firmly to the paper. All protractors should be engraved with two rows of figures, numbered contrary ways; to some protractors there is attached to the bar A a clamp and tangent screws.

To plot an angle taken from the meridian.

In surveys, when the angles are taken from the meridian line or magnetic needle, the bevel edge of the protractor must always be placed against the meridian line, marking the centre; then prick off as many angles as may be convenient to that part of the survey, noting the number of the station and angle they read. Repeat the same in other parts of the map. (See Plate 22.)

To measure an angle contained between two lines.

Place the protractor so that the bevel edge is against one of the lines and the centre at the station point or meeting of the two lines; read the angle contained between 360°, and the degrees intersected on the edge by the other line will be the angle required.

THE VERNIER.

Figs. 4 and 5, Plate 36. The vernier or dividing plate, so called from its inventor, Peter Vernier, a gentleman of Franche Comte.

It has been commonly called nonius, from Pedro Nunez, or Peter Nonius, an eminent Portuguese mathematician, being very different from that of the vernier (the term nonius is not now used).

The vernier is a scale made for the purpose of subdividing another scale into certain equal proportional parts to any degree

of minuteness.

The vernier is divided into equal parts, one more or less than the scale to which the vernier is attached. In the best instruments they vary in their minuteness or value. They are graduated, sometimes numbering right and left of zero; and in others the numbers are continuous on one side zero.

To find the value of a vernier.

Find the value of each division or subdivision in degrees, minutes, seconds, &c., on the limb; divide the quantity thus found by the number of divisions on the vernier, the quotient will be the value required.

Example 1. The divisions on the lower limb are not subdivided, each being equal to 60 minutes or 1 degree; the vernier has 20 divisions.

Then 6020 = 3 minutes, the value to which the vernier reads

Example 2. Each degree is divided into two parts, each part equal to 30 minutes; the vernier is divided into 15 divisions.

Then 30152 minutes value

Example 3. Each degree is subdivided into five parts, or 12 minutes each; reduce the minutes into seconds.

Thus as 12 x 60 720 seconds, the vernier has 24 divisions

then 720

=

Example 4. Each degree is divided into two parts, or 30 minutes; the vernier has 30 divisions.

Then 3030 = 1 minute value

Example 5. Each degree is divided into six parts, or 10 minutes each; reduce the minutes into seconds; the vernier has 60 divisions.

Then 10 x 60 = 600 ÷ 60 = 10 seconds value

Example 6. Each degree is divided into three parts, or 20 minutes; the vernier has 20 divisions.

Then 2020 = 1 minute value

The nature and use of the vernier is further explained. Let A B, Fig. 4, represent seven divisions or degrees on the limb, each divided into three parts; therefore, in seven degrees there are twenty-one divisions. CD is the vernier or index, equal to seven degrees, divided into twenty equal parts; it is evident that, since twenty of these answer to twenty-one on the limb, each one of these will exceed each on the limb by part, by 1 minute; therefore two on the vernier will exceed two on the limb by two minutes, three on the vernier by three minutes, and so on.

In taking an observation, suppose zero on the vernier be found between the 23° and 23° 20', as shown in the figure; then looking for the first coincidence of lines on the limb and vernier, you find it at the sixth division (marked thus *), therefore the angle reads 23 degrees 6 minutes.

The two 10's in the vernier are the only two divisions that coincide at a time with the divisions on the limb, if the instrument is accurately divided.

If the first line or index of the vernier has moved over a space less than half a division of the limb, then the coincidence

will be on the right hand of zero in the vernier; if it has moved more than half, it will be found in the left hand; if it is just half, the coincidence will be at 10'. This shows the reason why the beginning of zero is placed in the centre.

Fig. 5. In the best instruments, the mode of figuring the vernier is usually adopted as shown by the drawing-which is, by taking nineteen divisions on the limb and dividing it into twenty parts for the vernier; consequently, one division on the limb will exceed th part on the vernier, reading all one way to the left.

2

Beginning at zero, it will be found between 20 degrees 40 minutes and 21 degrees; then looking for the coinciding lines on the limb and vernier, it is found to be at the twelfth division on the vernier (marked thus *); therefore the angle reads 20° 52', that is, by taking 40 seconds on the first degree, and adding 12 minutes on the vernier.

When the index or zero on the vernier coincides with any line on the limb, the angle is read at once; as, for instance, the index of the vernier coincides with the line at 20, then it is 20 degrees; if at the next line, it would be 20 degrees 20 minutes, and so on. As only one line on the vernier can coincide with another line on the limb, that line must decide the angle.

In the best theodolites there are two verniers, and sometimes three; the readings of each should be taken, and, if any difference, the mean difference must be registered as the correct angle.

THE PANTAGRAPH.

Fig. 1, Plate 39. The original inventor of this useful instrument is not exactly known; the earliest account is in a small tract, published about the year 1631, by the Jesuit Scheiner, entitled "Pantagraphice sive Ars nova Delineandi." The principles are self-evident to every geometrician; the mechanical construction was first improved by Mr. W. Jones, instrument maker, Holborn, in the year 1750.

Its chief value is in reducing figures, although it may be used in copying plans, &c., of the same scale; it will also enlarge, which is never at any time recommended.

The pantagraph is made of brass, from 12 inches to 4 feet in length, and consists of four flat bars, two of them long and two short. The two longer ones are joined at the end A by a double pivot, which is fixed to one of the bars, and works in two holes placed at the end of the other; under the joint is an ivory wheel, to support this end of the instrument. The two shorter bars are fixed by pivots at E and H, near the middle of the longer bars, and are also joined together in a similar manner at the other end, G; ivory wheels being also fixed under each joint, marked a.

By the construction of this instrument the four bars always form a parallelogram.

There is a sliding box on the longer bar, B, and another on the shorter bar, D. These boxes may be fixed at any part of the bars by means of the milled screws; each of the boxes are furnished with a cylindric tube, to carry either the tracing point, the pencil, or fulcrum.

The fulcrum or support, K, is a lead weight, to which is fixed a bright iron pin, e; on this the whole instrument moves when in use.

The pencil-holder, tracer, and fulcrum, must in all cases be in a right line, as shown in the drawing marked b, so that when they are set to any number, if a fine string be stretched over them, and they do not coincide minutely, there is an error either in the setting or in the graduations.

The long tube c, which carries the pencil, moves easily up or down in either tube; there is a fine piece of silk, f, fixed to the pencil tube, passing through the holes in the three small knobs to the tracer point d, where it may, if necessary, be fastened. By pulling this string the pencil is lifted up occasionally, and thus prevented from making false or improper marks upon the copy.