PROBLEM XII.

To find a third proportional to two given lines A B and B C..

UPON the extremity of A B draw B C perpendicular; also draw the hypothenuse A C, and bisect it in D, with the perpendicular DE: then upon E, with the distance A, describe the arc A C F, and produce A B to F.-B F is a third proportional to the lines A B and B C ; or, A B: BC:: BC: B F.

PROBLEM XIII.

To find a fourth proportional to three given lines a b, b c, and a d.

MAKE A B equal to the first, and A D equal to the third, and from B, the extremity of the first, draw B C equal to the second, at any convenient angle to A B; also, through the point C, draw A C produced to E.-Draw D E parallel to B C, meeting A C in E, and DE is the fourth proportional; or, A B: BC::AD: D E, and AC:CB::AE: ED.

PROBLEM XIV.

The side of a polygon being given, to describe the polygon to any number of sides whatever.

B

UPON one extreme of the given side A B, describe a semicircle of any radius, and divide it into the same number of equal parts, as the sides of the required polygon, for instance five. Then draw lines from the centre through the points of division, but omitting the two last; and with the distance of the side A B, from A or B intersect each successively from the next.-Join these intersections, which will complete the polygon.

PROBLEM XV.

On a given diagonal to describe a square,

BISECT the given diagonal A B, by the perpendicular D E, and upon C, the point of bisection, with the distance A or B, describe the circle A E B D.-Join A E, E B, B D, and D A, and the square is complete.

PROBLEM XVI.

To inscribe a square in a given triangle.

DRAW the perpendicular C D, and make B E perpendicular and equal to A B; also join E D, and draw F G parallel, and G H and F I perpendicular to A B.-H G F I will be the inscribed square.

UPON one extreme of the given side A B, describe a semicircle of any radius, and divide it into the same number of equal parts, as the sides of the required polygon, for instance five. Then draw lines from the centre through the points of division, but omitting the two last ; and with the distance of the side A B, from A or B intersect each successively from the next.-Join these intersections, which will complete the polygon.

PROBLEM XV.

On a given diagonal to describe a square,

BISECT the given diagonal A B, by the perpendicular D E, and upon C, the point of bisection, with the distance A or B, describe the circle A E B D.-Join A E, E B, B D, and D A, and the square is complete.

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PROBLEM XVI.

To inscribe a square in a given triangle.

DRAW the perpendicular C D, and make BE perpendicular and equal to A B; also join E D, and draw F G parallel, and G H and F I perpendicular to A B.-H G F I will be the inscribed square.