Draw the indefinite line AB; and with the chord of 60° in your compasses describe the arc DE. Set off 90° from D to C; and from C to G set off the excess above 90°, which is 38° 35′. Draw the line AG; and GAD will be the required angle.

PROBLEM XXV.

To find the number of degrees contained in any given angle BAC.

With the chord of 60, and A as a centre, describe the arc mn. Take the distance mn in your compasses, and apply it to the line of chords; and it will shew the number of degrees required.

NOTE. Angles may be more expeditiously laid down or measured by means of a semi-circle of brass, called "a Protrector," the arc of which is divided into 180 degrees.

PROBLEM XXVI.

Upon a given line AB, to make a regular polygon of any proposed number of sides.

quo

Divide 360° by the number of sides, subtract the quotient from 180°, and divide the difference by two. Make the angles ABC and BAC each equal to the tient last found; and the point of intersection C, will be the centre of the circumscribing circle. With the radius AC or BC describe the circle; and apply the chord AB to the circumference the proposed number of times, and it will form the polygon required.

PROBLEM XXVII.

In any given circle to inscribe a regular polygon of any proposed number of sides: or to divide the cir cumference into any number of equal parts.

Divide 360° by the number of sides, and make the angle ACB, at the centre, equal to the number of degrees contained in the quotient; and the arc AB will be one of the equal parts of the circumference; hence the polygon may be formed.

NOTE. The sum of all the interior angles of any polygon, whether regular or irregular, is equal to twice as many right angles, wanting four, as the figure has sides.

PROBLEM XXVIII.

To find a mean proportional between two given lines. Let the given lines be AB = 32, and BC = 18,

D

A

Make AC = 50, with the radius Ao the semicircle ADC.

o B

the sum of the given lines; and 25, and o as a centre, describe From the point B erect the perpendicular BD, and it will measure 24, the mean proportional sought.

NOTE. A mean proportional between any two numbers may also be found by multiplying them together, and extracting the square root of their product.

PROBLEM XXIX.

To raise a perpendicular from any point D, in a given line AB, by a scale of equal parts.

Make Dm = 3; and from the points D and m, with the distances 4 and 5, describe arcs intersecting each other in n. From D, through the point n, draw the line DC, and it will be the perpendicular required.

NOTE. This Problem may be performed by any other numbers in the same proportion; but 3, 4, and 5 are the least whole numbers that will form a right-angled triangle.

PROBLEM XXX.

Given the span or chord line, and height or versed sine of the arch of a bridge or cellar, to find the radius of the circle that will strike the arch.

B

Draw the unlimited line CE; and take the versed sine from a scale of equal parts, and set it from C to D. Through the point D draw the line AB perpendicularly to CE, and make AD and BD each equal to half the given chord; also draw the chords AC, BC. Bisect either of these chords perpendicularly, with a line meeting CE in o, which will be the centre of the circle; hence the arch ACB may be described. Or, divide the square of half the chord by the versed sine; to the quotient add the versed sine, and the sum will be the diameter of the circle.

NOTE. This Problem is extremely useful to joiners in striking circular arcs, forming centres for bridges, cellars, &c. and also to masons and bricklayers, in describing circular pediments and other ornamental arches.

D

GEOMETRICAL THEOREMS,

THE

DEMONSTRATIONS

OF WHICH

May be seen in the Elements of Euclid, Simpson, and Emerson

THEOREM I.

IF two straight lines AB, CD cut each other in the point E, the angle AEC will be equal to the angle DEB, and CEB to AED. (Euclid I. 15. Simp. I. 8. Em. I. 2.)

The greatest side of every triangle is opposite to the greatest angle. (Euc. I. 18. Simp. I. 13. Em. II. 4.)

THEOREM III.

Let the right line EF fall upon the parallel right lines AB, CD; the alternate angles AGH, GHD are equal to each other; and the exterior angle EGB is equal to the interior and opposite, upon the same side