shall be equal to a given line, which is not greater than the diameter. 20. To draw a straight line which shall touch two given circles. 21. In the diameter of a circle produced, to determine a point from which a tangent drawn to the circumference shall be equal to the diameter. 22. Two circles being given in magnitude and position, to find a point in the circumference of one of them, to which if a tangent be drawn cutting the circumference of the other, the part intercepted between the two circumferences may be equal to a given line. 23. Upon a given straight line to describe a segment of a circle, which shall be similar to a given segment of another circle. 24. To divide a given circular arch into two parts, so that the sum of their chords may be equal to a given straight line, greater than the chord of the whole arch, but not greater than the double of the chord of half the arch. 25. The perimeter, the vertical angle, and the altitude of a triangle being given, to construct it. 26. Two circles being given in position and magnitude, to draw a straight line cutting them so that the chords in each circle may be equal to a given line, not greater than the diameter of the smaller circle. 27. Having given the radii of two circles which cut each other, and the distance between their centres, to draw a line of given length through their point of intersection, so as to terminate in their circumferences. 28. To describe a rectangle which shall be equal to a given square, and have its adjacent sides together equal to a given line. 29. Having given the distance of the centres of two equal circles which cut each other, to inscribe a square in the space included between the two circumferences. - 30. In a given circle to inscribe a rectangle equal to a given rectilineal figure. 31. To inscribe a circle in a given quadrant. 32. The vertical angle, the base, and the sum of the three sides of a triangle being given, to construct it. 33. The vertical angle, the base, and the excess of the greater of the two remaining sides, of a tri. angle, above the less, being given, to construct the triangle. 34. To produce a given line, so that the rectangle contained by the whole line thus produced, and the part of it produced, shall be equal to a given square. 35. From the obtuse angle of an obtuse-angled triangle, to draw a straight line to the base, the square of which shall be equal to the rectangle contained by the segments, into which it divides the base. 36. A flag-staff having a given height stands on the top of a tower whose height is also given ; at what point in the horizontal line, drawn from the foot of the tower, will the flag-staff appear under the greatest angle? : 37. Given the vertical angle, the difference of the two sides containing it, and the difference of the segments of the base made by a perpendicular from the vertex; to construct the triangle. THEOREMS. 1. If two circles cut each other, the straight line joining their two points of intersection is bisected, N at right angles, by the straight line joining their centres. 2. If two isosceles triangles be of equal altitudes, and the side of one be equal to the side of the other, their bases shall be equal. 3. Any two chords of a circle, which cut a diameter in the same point and at equal angles, are equal to one another. 4. If a straight line be drawn to touch a circle and parallel to a chord, the point of contact will be the middle point of the arc cut off by that chord. 5. If from a point without a circle, two straight lines be drawn to the concave part of the circumference, making equal angles, with a line joining the same point and the centre, the parts of the lines which are intercepted within the circle are equal. 6. If a circle be described on the radius of another circle, any straight line drawn from the point where they meet to the outer circumference, is bisected by the interior one. 7. The two straight lines in a circle, which join the extremities of two parallel chords, are equal to each other. 8. The straight lines joining the extremities of the chords of two equal arches of the same circle, towards the same parts, are parallel to each other. 9. If two circles cut each other, and from either point of intersection diameters be drawn, the extremities of these diameters and the other point of intersection shall be in the same straight line. 10. If a straight line touch the interior of two concentric circles and be placed in the exterior one, it will be bisected in the point of contact. 11. If any number of equal straight lines be placed in a circle, the locus of their points of bisection will be a circle. 12. If from any two points in the circumference of the greater of two given concentric circles, two straight lines be drawn so as to touch the less circle, they shall be equal to one another. 13. If a quadrilateral rectilineal figure be described about a circle, the angles subtended, at the centre of the circle, by any two opposite sides of the figure, are, together, equal to two right angles. 14. If two circles touch each other externally or internally, two straight lines drawn through the point of contact, will intercept arcs the chords of which are parallel. 15. If two equal circles cut each other, and from either point of intersection a circle be described cutting them, the points where this circle cuts them, and the other point of intersection of the equal circles, are in the same straight line. 16. If a straight line touch the interior of two concentric circles, and be terminated both ways by the circumference of the outer circle, its squares shall be equal to the difference of the squares of the diameters of the circles. 17. The perpendiculars let fall from the three angles of any triangle upon the opposite sides, intersect each other in the same point. 18. If two opposite angles of a quadrilateral figure be together equal to two right angles, a circle may be described about it. 19. If from the extremities of the diameter of a circle tangents be drawn, and produced to intersect a tangent to any point of the circumference; the straight lines joining the points of intersection and the centre of the circle form a right angle. 20. If a semicircle be described on the side of a quadrant, and from any point in the quadrantal are à radius be drawn; the part of this radius intercepted between the quadrant and semicircle, is equal to the perpendicular let fall from the same point on their common tangent. 21. If the chord of a quadrant be made the diameter of a semicircle, and from its extremities two straight lines be drawn to any point in the circumference of the semicircle ; the segment of the greater line intercepted between the two circumferences shall be equal to the less of the two lines. 22. If two circles cut each other so that the circumference of one passes through the centre of the other, and from either point of intersection a straight line be drawn cutting both circumferences; the part intersected between the two circumferences will be equal to the chord drawn from the other point of intersection to the point where it meets the inner circumference. 23. If through any point in the common chord of two circles, which intersect one another, there be drawn any two other chords, one in each circle, their four extremities shall all lie in the circumference of a circle. 24. If from the extremities of any diameter of a given circle, perpendiculars be drawn to any chord of the circle, they shall meet the chord, or the chord produced, in two points which are equidistant from the centre. 25. If two circles cut each other, and from any point in the straight line produced, which joins their intersections, two tangents be drawn, one to each circle, they shall be equal to one another. 26. If from any two points in the circumference of a circle there be drawn two straight lines to a point in a tangent to that circle, they will make the greatest angle when drawn to the point of contact. 27. If any chord in a circle be bisected by |