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.

A

B

D.

E

Conftruction. 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 Compaffes, fet it from C to a, on the Line CD. Next, take the Line B, and fet it from C to b, on the Line CD; and alfo on the Line CE, from C to e. Laftly, draw a e, and make b d parallel to it; fo will C d, on the Line CE, be the third Proportional required.

For, as Ca is to Cb, fo is Ce to C d.

The Construction and Ufe 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.

Conftruction. Firft, at the Point D, at the End of the Line D F, make an Angle of any Quantity at Pleasure, as EDF. Then, taking the Line A in the Compaffes, fet it from D to b, on the Line D E. Next, take the Line B, and fet it from D to c on the Line D F. Then take the Line C, and fet it from D to d on the Line D E. Laftly, join bc, and draw de parallel to it, fo will the Line D be the fourth Proportional required.

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

e

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

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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.

Conftruction. Firft, draw the Line D E at Pleasure, upon which fet with the Compaffes the Line A from D to b; and alfo 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 C D or C E, defcribe the Semicircle D F E. Laftly, from the Point b draw the Line F perpendicular to the Line DE, and it will be the mean Proportional required.

For, as Db is to b F, fo 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 fo 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 fo divided.

Conftruction. First, on the Point A, erect the Perpendicular AD; and produce it downwards alfo toward C. Next, make AC equal to half A B. Then, upon the Point C, with the Distance C B, describe the Arch BD; and upon the Point A, with the Distance A D, defcribe the Arch DE, which will cut the Line A B 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 B E, will be equal to the Area or Square A EFD made on the other Part A E.

For, as BE EA:: EA: A B.

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.

Conftruction. Firft, divide Half the Line Ab or B into as many Parts as there are to be Revolutions of the Spiral, as in ce A and df B, which in this Cafe we will fuppofe to be Three. Next, divide be into two equal Parts in a; then, upon the Point a describe the upper Semicircles cb, e d, Af. Laftly, upon the Point b defcribe the under Semicircles c d, ef, AB, which will complete the Spiral required.

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