tween the vertex and the points of intersection shall have to each other a given ratio.

6. Given a straight line, to divide it into two parts such that the square of one of the parts shall have a given ratio to the rectangle contained by the whole line and the other part.

7. Given the first and fourth of four proportional lines and the difference between the second and third, to find the second and third.

8. Given a rectilineal figure, to describe a square which shall have a given ratio to it.

9. Given a trapezium in which the sides about one angle are equal, and the sides about the opposite angle also equal, to inscribe in it a square.

10. Given two similar rectilineal figures, to find a third similar rectilineal figure which shall be a mean proportional between them.

11. Given a segment of a circle, to divide it into two parts so that the chords of these parts shall have a given ratio to each other.

12. To trisect a given circle.

SECT. II.-THEOREMS.

13. If a straight line be drawn from the vertex of an isosceles triangle at right angles to one of the equal sides and produced until it meets the base produced, either of the equal sides is a mean proportional between the base and one-half of the base produced.

14. If the three sides of a triangle be bisected and straight lines drawn from the points of bisection to the opposite angles, they shall intersect each other in the same point.

15. If two or more lines meet three parallel lines, they are cut proportionally.

16. If a straight line touch two circles that also touch each other, the mean proportional between the diameters of the circles is that part of the line lying between the points of contact.

17. If parallelograms be equiangular, the ratio between them is the same as the ratio between the rectangles contained by the sides about equal angles in each.

THE END.

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