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dimensions the Mathematician will find the following formula pretty accurate, g being the reading in degrees &c. Diameter

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It is to be observed, that while this instrument is theoretically perfect as a gauge for great dimensions, yet, that the most ready way to arrive at the diameter of a cask or tree, (the outsides of which are frequently too rough and irregular to give accurate results to any Callipers), is, to girth it with a piece of string, and then to divide that measure by folding it into three; one-third of its length being the diameter of the object measured: this, though not geometrically correct, is really as exact as the thing can be done at all; for if the whole line be a trifle too long, that excess goes out in the bending of the string, and it may be in all cases relied upon as sufficiently exact for military measurements.

Furthermore, if the dimension to be gauged do not fall in within any of the diameter graduations of the disc (which will almost invariably happen in the case of foreign ordnance), then the diameter of the shot may be found by this simple rule. Supposing for example that the diameter required be found to fall somewhere between 3 and 4, as shewn by the arrow on the instrument, then the instrument must be opened at 3, and the ball be tangented by the legs. Then take the Square distance in inches from

and, with it, measure the shortest the central point of the circle of the hinge to the circumference of the ball. This distance multiplied by 3 will be the diameter of the ball. It will be of no consequence of what diameter the ball may be that the sector so expanded may enclose; the rule will be equally applicable, whether the

1 N.B. The truth of this rule depends upon the property of similar plane figures, in which the ratio of homologous parts is constant.

diameter be 4, 5, 6, 7, 8, or 9 inches, and equally also whether the index 1, 2, 4, or any other be used, only remembering that the diameter of the ball in inches must be greater than that employed in the index.

To calibre a diameter internally, the sector must be opened so that the extreme angles of the sector may make contact with the interior of the circle to be calibred. Up to 6 inches of internal measure, the dimension will be taken off by the scale of equal parts which the Square presents to go of an inch. Should the dimension exceed 6 inches, the Square is to be used by drawing it gradually out of its groove up to 12 inches. For a bore of from 12 to 18 inches, the rule must be fully expanded to a straight line, which will then be 12 inches long, and the Square slid out, as before, until the internal length be realized.

III. THE PROTRACTOR.

The instrument for its next simple use may be employed as a Protractor, an implement for protracting angles upon paper or on the ground. When used to protract an angle, it can set off the work to one minute of a degree. The disc of the joint shews a graduation of 180 degrees, to which there is a double numeration, in order that the instrument may be used both ways; by means also of the Vernier Scale it reads also to minutes of a degree: this also reads both ways, by means of a double numeration reading from 0 to 60. This minuteness of reading has not been introduced to enable the draughtsman to work with his pencil or pen with any such pretended precision, but because the Vernier is wanted in the observing instrument. This Protractor will make good work to a quarter of a degree or 15 minutes; and that, not because it is

graduated to minutes, but because it has a good long arm, viz. 6 inches. It is the length of the arm of the Protractor, not the fineness and minuteness of its graduation, that enables a man to make good work.

To constitute the ring-joint Sector a Protractor, in the centre of the ring, the columnar ring pivot has been furnished with a quadrant, one of the radii of which lies in the prolongation of one of the legs of the sector on its inner edge; and at the extreme end of which is the angle of the quadrant.

When it is desired to lay down an angle, the draughtsman expands the legs of the sector to shew the required angle. He then lays one leg of the sector along the line drawn on his plan; and next adjusts the angle of the small quadrant over his point or position on the paper. When the instrument is so placed that the angle of the quadrant is thus over the point, and one leg of the instrument is aligned with the line, as above drawn, the draughtsman must draw a line along the other leg of the sector; the angle will then be protracted.

For the purpose of measuring or laying off on the plan the measured distances obtained by pacing or otherwise, the Sector is furnished within its legs with a set of graduations. These give 4 inches, divided into 1760 parts, i.e. a scale of 4 inches to a mile. This will be quite sufficient for practical purposes: as either directly, or by multiples or sub-multiples of it, scales of 1, 2, 4, 8, 12, 16, 20 or 24 inches to a mile can be made by the draughtsman as he works in the field. The best way of using this will be, to lay the sector with its centre over your position, carefully aligning one edge of the sector with the line of direction on which the dimension is to be marked off. The numbering of the yards is made to read from the centre outwards, on both legs of the instrument. Each numbered interval indicates 100 yards, and therefore each small division

represents 50 yards. The whole length of the leg of the sector represents a mile and a half or 2640 yards. If the distance to be marked off be greater than this, the opening of the legs to a straight line will give you any length by prolonging it, when drawn by shifting the ruler sideways against the Metal Ruler, so held that it may be slid along in a right line.

On the outer edge of the Sector, on the side with the Spirit-Level, there is sometimes a scale of 3 inches to a mile in yards; the numbers are made to read both ways.

When the instrument is opened at an angle of 90° it will evidently form a SQUARE, by which all the work done by a carpenter's square can be effected.

THE VERNIER SCALE TO THE PROTRACTOR.

On the short leg of the sector there will be observed an arc of the circle of 59° graduated into 60 equal parts. That is the Vernier Scale, and is for the purpose of dividing a degree of the circle into 60 parts. These divisions of the degree are called minutes. It will be observed, that there is a double set of readings on the Vernier Scale: one set reading from right to left, the other from left to right. It is to be noted, that when the reading of degrees on the limbs of the annular disc is to the right, the Vernier's readings to the right will be used for reading off the minutes of the degree: when the reading of the degrees is from the right to the left, the other numeration on the Vernier, viz. that from right to left, must be in use. If the half-arrow shew 20 for example, then 20° will be the required angle. Should, however, the half-arrow not exactly coincide with any graduation, but lie between some two graduations,

then the observer must run his eye to the right, where he will see, on the leg of the sector, the above-named 59° divided into a set of graduations numbered in the order, 0, 10, 20, 30, 40, 50, 60, and will observe which of those graduations, thus numbered, in inverse order, or which of those lying between these numbers, coincides with some graduation on the annular disc. Such coinciding graduation will give the number of minutes of a degree which are to be added to the number of degrees indicated by the half-arrow at the end of the sector's leg. Thus, if the half-arrow stand between 190 and 20o on the annular disc, and should the graduation 30 on the outer set of graduations cut in with some one of those on the inner set of graduations, then the reading off would be 19o. 30'. If the coincidence should happen under the 38th of the outer graduations, then will the reading-off be 19o.38'.

Observe. Should the reading be on the left-hand quadrant, the numbers on the Vernier reading from 0 to 60 must be used; if on the right-hand quadrant, those from 60 to 0.

IV. THE REFLECTING SECTOR OR QUINTANT.

It will be perceived, that, from the joint of the sector, the middle of it has been so much turned out, as to leave the joint itself a mere annulus or ring, in which, as a Columnar Pivot, is another ring.

Into this latter circular recess, a Metallic Mirror, semi-circular, has been fitted for the purpose of Measuring Angles, by means of reflection, as in the instrument called the Sextant.

The mirror, which stands at right angles to the plane of the sector when it is in use, lies down flat in this circular cavity for the convenience of travel; in order that the instrument, of

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