Problem 20.

Two Right Lines being given, to find a third Proportional.

Let the two Lines given be A and B, and let it be required to find a third Line in Proportion thereto.

Construction. First, at the Point C, on the End of the Line CE, make any Angle at Pleasure, as DCE. Then, taking the Line A in the Compasses, set it from C to a, on the Line CD. Next, take the Line B, and set it from C to b, on the Line CD; and also on the Line CE, from C to e. Lastly, draw a e, and make b d parallel to it; fo will C d, on the Line CÉ, be the third Proportional required.

For, as Ca is to Cb, so is Ce to Cd.

The Construction and Use of the Sector is founded

on this and the following Problem.

Problem 21.

Three Right Lines being given, to find a fourth Proportional.

Let ABC be the three Lines given, and let it be required to find a fourth Line in Proportion thereto.

Construction. First, at the Point D, at the End of the Line DF, , make an Angle of any Quantity at Pleasure, as EDF. Then, taking the Line A in the Compasses, fet it from D to b, on the Line DE. Next, take the Line B, and fet it from D to con the Line D F. Then take the Line C, and set it from D to d on the Line DE. Lastly, join bc, and draw de parallel to it, so will the Line D'e be the fourth Proportional required.

For, as Db is to Dc, so is D d to De.

These two Problems do the Work of the Rule of Three, without the Use of Arithmetic.

Problem 22.

To find a Mean Proportional between two Right Lines given.

Let A and B be the two given Lines, and let it be required to find a Mean Proportional between them.

Construction. First, draw the Line DE at Pleasure, upon which set with the Compasses the Line A from D to b; and also the Line B from b to E. Next, divide the whole Line D E into two equal Parts in the Point C. Then, upon C, with the Distance CD or CE, describe the Semicircle DFE. Lastly, from the Point & draw the Line & F perpendicular to the Line DE, and it will be the mean Proportional required.

For, as Db is to b F, so is b F to E.

By the Help of this Problem we can find the Square Root of any given Number; and readily reduce a long Square to a perfect One.

Problem 23.

To divide a Right Line given into extreme and mean Proportion; that is, to cut a Line so that the Product of the whole Line and one of the Parts shall be equal to the Square of the other Part.

Let A B be the Right Line given to be so divided.

Conftruction. First, on the Point A, erect the Perpendicular AD; and produce it downwards also toward C. Next, make AC equal to half A B. Then, upon the Point C, with the Distance CB, describe the Arch BD; and upon the Point A, with the Distance A D, describe the Arch DE, which will cut the Line AB in the Point E in extreme and mean Proportion, as required. For the Area, Rectangle or Product ABed made of the whole Line A B, and the Part BE, will be equal to the Area or Square AEFD made on the other Part A E.

For, as BE: EA::EA:А В.

By this Problem we find (Geometrically) the Length

of the Sides of fome of the regular or Platonic Solids.

Problem 24.

To describe a spiral Line about a given Line.

Let A B be the given Line about which the spiral Line is to be described.

Construction. First, divide Half the Line Abor & B into as many Parts as there are to be Revolutions of the Spiral, as ince A and df B, which in this Case we will suppose to be Three. Next, divide bc into two equal Parts in a; then, upon the Point a describe the upper Semicircles cb, e d, Af. Lastly, upon the Point b describe the under Semicircles cd, ef, AB, which will complete the Spiral required.

This Problem may be useful to Architects in drawing the Capitals of some Orders in Building; particularly the Ionic.