Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

SCHOLIUM.

IT may not be amiss to note these few things here, concer∣ning the Inscription of Regular Figures in a Circle.

I. Having described a Circle on any Semidiameter AC,(a) 1.1 (Fig. 33. N. 1.) that Semidiameter being placed in the Circumference, will precisely cut off one sixth part of it, and so become the Side of a Re∣gular Hexagon: and so the Triangle ABC will be an Equilateral one, and consequently the An∣gle ACB and the Arch AB 60 Degrees, by Cons. 1. Definit. 13.

II. Hence a Right Line AD, omitting one point of the di∣vision B, and drawn(b) 1.2 to the next D, gives you the Side of a Regular Triangle inscrib'd in the Circle, and subtends twice 60, i. e. 120 Degrees.

[illustration]

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III. If the Diameters of the Circle AD and DE (N. 2.) cut one another at Right Angles in the Center C, the Right Lines AB, BD, &c. will be the Sides of an inscribed Square ABDE: For(c) 1.3 the Sides AB, BD, &c. are the equal Chords of Quadrants, or Quadrantal Arches, and the Angles ABD, BDE, &c. will be all Right ones, as being Angles in a Semicircle (per Schol. 6. Definit. 10.) composed each of two half Right ones, by Consect. 2. Definit. 13.

IV. Euclid very ingeniously shews us how to Inscribe a Re∣gular Pentagon also, Lib. 4. Prop. 10 & 11. and also a Quin∣decagon (or Polygon of 15 Sides) Prop. 16. But though the first is too far fetch'd to be shewn here, yet (supposing that) the second will easily and briefly follow:

In a given Circle from the same point A (N. 3.) inscribe a Regular Pentagon AEFGHA, and also a Regular Triangle ABC; then will BF be the Side of the Quindecagon, or 15 Sided Figure. For the two Arches AE and EF make together 144 Degrees, and AB 120: (a) Therefore the difference BF will be 24, which is the 15th. part of the Circumference.

V. The Invention of Renaldinus would be very happy, if it could be rightly Demonstrated; (as he supposes it to be in his Book of the Circle) which gives an Universal Rule of dividing the Periphery of the Circle into any number of equal Parts re∣quired, in his 2d Book De Resol. & Comp. Mathem. p. 367. which in short is this: Upon the Diameter of a given Circle AB Fig. 34.) make an Equilateral Triangle ABD, and having divided the Diameter AB into as many equal Parts, as you design there shall be Sides of the Polygon to be Inscribed, and omitting two, e. g. from B to A, draw thro' the beginning of the third from D, a Right Line, to the opposite Concave Circumference, and thence another Right Line to the end of the Diameter B, which the two parts you omitted shall touch thus, e. g. for the Triangle, having divided AB into three equal parts, if omitting the two B2, thro' this beginning of the 3d you draw the Right Line DIII, and thence the Right Line III.B, which will be the Side of the Triangle; and so IV.B will be the Side of the Square, VB the Side of the Pentagon, &c.

N. B. The Demonstration of these (Renaldinus adds, p. 368.) we have several ways prosecuted in our Treatise of the Circle: Some of the most noted Antient Geometricians, have spent a great deal of pains in the Investigation and Effection of this Problem, and several of the Moderns have lost both time and pains therein: Whence, we hope, without the imputation of Vain Glory, we may have somewhat obliged Posterity in this point.

Notes

• (a) 1.1

  Euclid. Prop. 15. lib. 4.
• (b) 1.2

  Eucl. 2d Coroll. of the same.
• (c) 1.3

  Eucl. 6. lib. 4.