From C as a centre, with any convenient opening of the compasses, describe the semicircle DE; then from D and E as centres, with a radius greater than DC, describe two arcs intersecting each other in the point F; and draw CF the perpendicular required.

PROBLEM II.

To erect a perpendicular on the extremity of a given right line.

Let AB be the given right line, and A the extreme point from which a perpendicular is to be erected.

Take at pleasure between A and B, but without the line, and on the side of it on which the perpendicular is to be drawn, the point C, nearer to A than to B. Next, from C as a centre, with the opening CA, describe an arc cutting AB in D, and carried so far beyond A, as that EAD may be a semicircle. Through the points D and C draw the straight line DCE, and then from A to E let the required perpendicular be drawn.

PROBLEM III.

To let fall a perpendicular upon a given right line from a point without it.

Let AB be the given line, and C the point without it, from which the perpendicular is to fall.

Take any point D on the other side of AB, and with the opening CD, describe from C the arc EF, cutting AB in the two points E and F. Then from E and F as centres, with any radius greater than the half of EF, describe two arcs intersecting each other in G. Lastly, lay a ruler from C to G, and draw the perpendicular CD.

PROBLEM IV.

To draw a straight line parallel to a given straight line, through a given point without it.

Let AB be the given straight line, and C the given point without it.

In AB take any point D, and draw the straight line

CD. Then from C and D as centres with the opening CD, describe CE, DF; making the arc DF equal to the arc CE. Lastly, through C and F draw the straight line GH, and it will be parallel to AB.

Note on the first four Problems.

Perpendicular lines are more expeditiously drawn in practice by means of a scale of box-wood or of ivory, with a square end: and parallel lines are best made with a parallel ruler.

PROBLEM V.

To bisect a given finite straight line, that is, to divide it into two equal parts.

Let AB be a given finite straight line to be divided into two equal parts.

From A as a centre, with any opening greater than the half of AB, describe an arc; and from B as a centre, with the same radius, describe a similar arc intersecting the former in two points. Then draw a straight line from one of the points of intersection to the other, and that line will bisect AB.

PROBLEM VI.

To find the centre of a given circle or circular arc. Let ABC be the arc of a circle, whereof the centre is required.

E

B

Draw in the arc any two chords making an angle with one another as AB, BC: then bisect these chords by the perpendiculars DE, DF. the perpendiculars intersect one required.

PROBLEM VII.

The point D where another is the centre

To divide a given right line into any proposed number of equal parts.

Let AB be a right line given to be divided into five equal parts.

From the point A draw an indefinite straight line AC making an angle with AB. Then from A lay off any convenient distance, as AD, five times towards C, which distances are represented by D, E, F, G, and M. From M to B draw the straight line MB, and parallel to MB through the points D, E, F, and G,

draw the straight lines DH, EI, FK, and GL, dividing AB into five equal parts.

PROBLEM VIII.

To draw from the extremity of a given straight line, a straight line making an angle with the former, equal to a given rectilineal angle.

Let AB be the given straight line, and CDE the given rectilineal angle; and let A be the extremity from which the line is to be drawn.

From D as a centre with any convenient opening of the compasses, describe the arc FG; and from A as a centre with the same radius describe the arc IK. Make the arc IK equal to the arc FG, and through the points A and I, draw the straight line AL, which will be the line required.

SECTION II.

CONSTRUCTION OF PLANE FIGURES.

PROBLEM I.

To describe an equilateral triangle upon a given finite straight line.

Let AB be the given line upon which it is required

to describe an equilateral triangle.