into two equal parts, at the point a5 place the compasses on a, as a centre, and extending them to d, describe the arc c d e; and the points of intersection of this arc, with the circumference of the given circle, will give the true points of contact of the required tangents ae-af,

It should be observed that the cord ab, which is the measure between the points of contact of the two tangents, is less than ca, the diameter of the circle; and the nearer the point, whence the two tangents are drawn, is placed with reference to the ' n l circle, the greater must necessarily be the differ ence between the measure of the cord giving the distance between the points of contact and the diameter of the circle, as shown in the tangents gk-sl, drawn from the point g, compared with former example.

29. To draw within a circle an equilateral triangle, hexagon, dodecagon, etc.-This operation consists in a simple division of the circumference of the given circle into Three, Six, and Twelve equal parts, etc. First, therefore, for the Equilateral Triangle, draw a diameter (a a); then from a, as a centre, describe an arc (b b c), passing through the centre of the circle; and the points where this arc cuts the circumference of the given circle, at b and c, will give its required division into three equal parts, and abc, the equilateral triangle required. To trace a Hexagonâ€”the radius, or half-diameter, will give the true measure of the divisions of the circumference into six parts. For a Dodecagon, divide one or more, if necessary, of the ascertained sides of a hexagon, as figured, etc.

30. To draw either, or all of these figures, as well as such-like that follow, outside of the circle, the process is so similar that it will be only necessary to figure the Equilateral Triangle and Hexagon thus produced, and leave the student to exercise his ingenuity and practise his hand upon such others as he may have occasion to draw; in doing 16

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