polygon, will give the radius of its circumscribing circle.

Or, the three angles, and one side being given of a triangle, the radius, or remaining sides, may be found by the rules of trigonometry.

ITS USE IN DRAWING PARALLEL LINES AT ANY GIVEN DISTANCE, WITHOUT THE ASSISTANCE OF A PAIR

OF COMPASSES.

If the side of your polygon was taken from the scale C. Move the slider from 0 to 1, 2, 3, &c. on the scale D, and you will draw a parallel line of the width of 1, 2, 3, &c. divisions on the scale C. This scale is 20 fathoms to an inch.

In a similar manner the scales G, L, and P, are to be used.

If the side of your polygon was taken from the scale C, calling the divisions two each. Move the slider from O, on the scale A, to 1, 2, 3, &c. on the same scale, and you will draw a parallel line of half the width of 1, 2, 3, &c. divisions on the scale C. This scale is 40 fathoms to an inch.

Similar instructions may be applied to the scales G, L, and P.

',

If the side of your polygon was taken from the scale C, calling the divisions three each. When the middle index stands at O, the divisions marked C on the slider will make straight lines with 5 and 6 on the scale D. By moving the slider to the right or left till the other divisions thereon make straight lines successively with 5, 6, &c. you will draw a parallel line of one-third of the width of 1, 2, 3, &c. divisions on the scale C. This scale is 60 fathoms to an inch.

Similar directions must be observed in using the scales G, L, and P.

If the side of your polygon was taken from the scale A. Move the slider from O, to 1, 2, 5, &c, on

the scale B, and you will draw a parallel line of the width of 1, 2, 3, &c. divisions on the scale A. This scale is 10 fathoms to an inch.

Similar directions must be observed in using the scales E, I, and N.

If the side of your polygon was taken from the scale D. Move the slider from O, to 2, 4, 6, &c. on the scale B, and you will draw a parallel line of the width of 1, 2, 3, &c. divisions on the scale D. This scale is five fathoms to an inch.

In making use of H, M, and Q, similar directions must be observed.

Note. There is no absolute necessity for always moving the slider from O, and it may be used either side uppermost. Any one of the three indices may likewise be made use of, &c.

To render the scale still more perfect, Mr. Keith has made the following additions.

1. The divisions on the scale A, and the fourth scale from A, have been subdivided for the purpose of constructing sections, &c. Ten of the divisions on the scale A make one inch; if, therefore, you call these divisions six each, the small divisions on the same scale, nearest to the left hand, will be one each; you call the large divisions on the scale A, five, four, or three each, then the sets of smaller divisions in a successive order from the left hand division above-mentioned, towards the right, will be one each.

if

2. Five of the divisions on the fourth scale from the edge make an inch: if, therefore, you call these divisions 3, 5, 7, 9, or 11 each, then each of the sets of small divisions, from the left hand towards the right, will be subdivided into parts of one cach. Thus you have a scale divided into 2, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, and 60 equal parts to an inch, exclusive of the scales on the three, other edges of the ruler.

3. The line P on the slider stands for polygons; R, radius; the figures in this line being the radi

of the several polygons under which they stand, the side being 180 toises in each. If the side of your polygon be different from 180 toises, say, as 180 toises are to the radius under the number of sides your polygon contains, so is the side of your polygon to its radius. Er. What is the radius of an octagon, the exterior sides being 120 toises? as 180 : 235 18: 120: 156 78 Ans. The remaining lines are to be read thus-The English foot is to the French foot as 107 is to 114, or as 1 is to 1.065, a toise six French feet, a fathom six English feet. The English foot is to the Rhynland foot as 9715 is to 10000; the Rhynland foot is to the French foot as 1033 is to 1068: the Rhynland rod is 12 Rhynland feet.

It may not be amiss to remark, that either the scale or the slider will erect a perpendicular instantaneously, for the long divisions across the scale make right angles with the edges.

OF THE GUNNER'S CALLIPERS, OR COMPASSES.

This article is generally included in the magazine case of instruments for the military officer, or engineer; is a very useful mathematical instrument in the artillery service, and, as its description was omitted by our late author, I have taken the opportunity of inserting some account of it here.

The principal uses of this instrument are to take the diameters of common shot, the bore or caliber of a piece of ordnance, estimate the weight of shot, quantity of powder, &c. for guns of given dimensions, and other particulars in practical gunnery.

Fig. 6 and 7, are the representations of the two faces of the callipers marked A, B, C, D. They consist of two thin flat brass rulers moving on a joint, curved internally to admit the convex figure of a ball, whose diameter is to be taken by the points at the end A; these points are of steel,.to prevent much

wear. The rulers are made from six to 12 inches in length from the centre, according to the number of lines and tables to be engraved upon them. The usual length for pocket cases is six inches, the scales upon which I shall now describe.

On Fig. 6, ruler A, is contained, 1. A scale of inches, divided into tenths, and continued to 12 inches on the ruler B.

2. A table shewing the quantity of powder necessary for charging the chambers of brass mortars and howitzers.

3. On the ruler B, is à line marked Inches, be ginning from the steel point, for giving the diame、 ters of the calibers of guns in inches.

4. A line marked Guns, contiguous to the preceding, shewing the nominal pounders, or weight of shot for the respective bores of the guns in inches.

5. On the semi-circular head of the rule is a semicircle divided into degrees, figured in contrary direc tions, to measure angles by, and give the elevation of cannon, &c.

6. Next to the preceding, is a circular scale marked Shot Diameters, which, with the chamfered edge marked Inder, shews the convex diameter in inches of a shot, or other object placed between the points. The quadrant part of the joint prevents this being represented in the figure.

On the other face of the callipers, C and D, fig. 7, are engraved.

7. On the ruler C, a table marked Brass Guns, shewing the quantity of powder necessary for the proof and service charges of brass guns.

8. The line of lines, marked Lin.

9. Two circular scales on the head of the ruler, marked Shot, shewing the weights of iron shot, as taken by the points of the callipers.

On the ruler D, besides another corresponding line of lines.

10. A table shewing the quantity of powder ne

cessary for proof and service charges of iron guns, from to 42 pounders.

11. To these are sometimes added various figures of a circle, cube, &c. with numbers.

12. A table of the weights and specific gravities of a cubic foot of various metals, ivory, wood, waters, &c.

EXPLANATION OF THE LINES AND TABLES.

1. The line of inches, graduated contiguous to the exterior edges of the sides A and B, fig. 6, when opened to a straight line, makes a measure of 12 inches and tenths for the purpose of a common rule.

2. The table shewing the quantity of powder necessary for mortars and howitzers. This table is adapted for both sea and land. Thus, by inspection merely is shewn that a 13-inch brass mortar at sea requires 30 pounds of powder; by land only 10 pounds. Mortars and howitzers under 10 inches have only the land quantities inserted.

3. The line of inches for concave diameters. It commences from the steel point at 2 inches, and continues on to 10 inches, and is subdivided to halves and quarters. When used, the legs are placed across each other, and the steel points brought to a contact with the internal concave surface of the gun, at the dian.etrical, or greatest possible distance; the callipers then being taken out, and inspected where in the scale one external edge of the rule is upon the other scale, that division will give in inches and quarters the caliber of the gun required.

4. The line shewing the weight of shot for the bores. Adjoining to the preceding line is the one marked Guns, proceeding from 1 to 42 pounds.

It is therefore evident, that at the same time the caliber in inches is given, the weight of the shot is also given by inspection on the scale, by the side of the ruler crossing both; and, from the weight of