To inscribe a hexagon in a circle.

Bisect the arcs A C, BC in E, and F, and join A D, DB, BF, &c., which will form the hexagon. Or carry the radius six times round the circumference, and the hexagon will be obtained. Fig. 11.

To inscribe a dodecagon in a circle.

Bisect the arc AD of the hexagon in G, and AG being carried twelve times round the circumference, will form the dodecagon, Fig. 11.

To inscribe a pentagon, hexagon, or decagon, in a circle.

Draw the diameter A B, and make the radius D C perpendicular to A B. Bisect D B in E. From E as a centre, and with E C as radius, describe an arc cutting A D in F. Join C F, which will be the side of the 'pentagon, CD that of the hexagon, and DF that of the decagon. Fig. 12.

To find the angles at the centre, and circumference of a regular polygon.

Divide 360 by the number of the sides of the given polygon, and the quotient will be the angle at the centre; and this angle being subtracted from 180°, the difference will be the angle, at the circumference, required.

Tuble, showing the angles at the centre, and circumference.

To inscribe any regular polygon in a circle.

From the centre c draw the radii CA, CB, making an angle equal to that at the centre of the proposed polygon, as contained in the preceding table. Then the distance A B will be one side of the polygon, which, being carried round the circumference the proper number of times, will complete the polygon required. Fig. 13.

To circumscribe a circle about a triangle.

Bisect any two of the given sides, A B, B C by the perpendiculars EF, D f. From the intersection F as a centre, and with the distance of any of the angles, as a radius, describe the circle required. Fig. 14.

To circumscribe a circle about a square.

Draw the two diagonals A C, B D intersecting each other in o. From o as a centre, and with o A, or O B, as a radius, describe the required circle. Fig. 15.

To circumscribe a square about a circle.

Draw the two diameters A B, C D perpendicular to each other, through the points A, C, B, D, draw the tangents EF, EG, G H, F H, and EGHF will be the square required. Fig. 16.

·To reduce a map, or plan, from one scale to another.

Divide the given figure A C by cross lines, forming as many squares as may be thought necessary. Draw a line E F, on which set off as many parts from the scale M, as A B contains parts of the scale N. Draw EH, and F G perpendicular to E F, and each equal to the proportional parts contained in A D, or B C. Join H G, and divide the figure EG into the same number of squares as the original A C. Describe in every square what is contained in the corresponding square of the given figure; and EFGH will be the reduced plan required. The same operation will serve either to reduce, or enlarge any map, plan, drawing, or painting. Fig. 17.

MENSURATION OF PLANES, AND SOLIDS.

Mensuration is of three kinds, viz., lineal, superficial, and solid.
Lineal measure has reference to length only.

Superficial measure (or the surface) includes length, and breadth.
Solid measure (or the content) comprehends length, breadth, and

thickness.

MENSURATION OF PLANES.

The area of any plane figure is the superficial measure contained within its extremes, or bounds. This area is estimated by the number of small squares that may be contained in it, the side of these measuring squares being an inch, a foot, or any other fixed quantity, and hence the area is said to be so many square inches, square feet, &c. Vide Table, Square measure. Page 281.

To find the area of a parallelogram, whether a square, rectangle, &c. Multiply the length by the breadth, or perpendicular height, for the area required.

Example.-Required the area of a rectangle, whose length is 9 feet, and breadth 4 feet.

9x4=36 feet. The required area, or surface.

B

H

2

H

B