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SCHOLIUM.

It is evident from the above deduction, that the double of the perpendicular, let fall from the vertex of a triangle on its base, may very readily be found by the compass alone: And a reference to the Scholium in the twenty-seventh page, of this Appendix, will shew that this expedient might be of great utility in Practical Geometry, for the mensuration of the surfaces of triangles and other rectilineal figures.

PROP. III.

(II.)

Through a given point within a circle, which is not the center, to draw a chord which shall be bisected in that point. ..1

PROP. XIV.

(III.)

If two isosceles triangles be of equal altitudes, and the side of the one be equal to the side of the other, their bases shall be equal.

(IV.)

Any two chords of a circle which cut a diameter in the same point and at equal angles are equal to one another.

(v.)

Through a given point, within a given circle, to draw two equal chords, making with one another an angle equal to a given rectilineal angle.

PROP. XVI.

(VI.)

If the diameters of two circles are in the same straight line, and have a common extremity, the two circles shall touch one another.

(VII.)

To draw a tangent to a circle, which shall bẹ parallel to a given finite straight line.

COR. Hence, to draw a tangent to a circle, which shall make, with a given straight line, an angle equal to a given rectilineal angle.

(VIII.)

To describe a circle which shall have a given radius, and its center in a given straight line, and shall also touch another straight line, inclined at a given angle to the former.

(IX.)

To describe a circle, the circumference of which shall pass through a given point, and touch a given straight line in another given point.

(x.)

To describe a circle, the circumference of which shall pass through a given point, and touch a given circle in another given point; the two points not lying in a tangent to the circle.

(XI.)

To describe a circle, which shall touch a given straight line in a given point, and also touch a. given circle.

(XII.)

Hence, if two circles touch each other externally, to describe another circle, which shall touch the one of them in a given point, and also touch the other.

(XIII.)

To describe two circles, each having a given radius, which shall touch the same given straight line, both on the same side of it, and shall also touch each other..

(XIV.)

To describe two equal circles, each having it's diameter equal to a given straight line, each touching a given circle, and each also passing through a given point without that circle: The given straight line being greater than the shortest distance, between the given point and the circumference of the given circle.

PROP. XVII.

(xv.)

To find a point in the diameter, produced, of a given circle, from which, if a tangent be drawn to the circle, it shall be equal to a given straight line.

(XVI.)

Through a given point, either within, or without a given circle, to draw a straight line, so that

the part of it within the circle shall be equal to a given finite straight line, which is less than the diameter.

(XVII.)

To draw a tangent to a given circle, such that it's segment, contained between the point of contact, and an indefinite straight line, given in position, shall be equal to a given finite straight line.

PROP. XVIII.

(XVIII.)

If a straight line touch the interior of two concentric circles, and be terminated both ways by the circumference of the outer circle, it shall be bisected in the point of contact.

(XIX.)

If a polygon be described about a circle, the straight lines joining the several points of contact will contain a polygon of the same number of angles as the former; and any two adjacent angles of the circumscribed figure shall be, together, the double of that angle, of the inscribed figure, which lies between them.

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