Cosmographia, or, A view of the terrestrial and cœlestial globes in a brief explanation of the principles of plain and solid geometry applied to surveying and gauging of cask : the doctrine of primum mobile : with an account of the Juilan & Gregorian calendars, and the computation of the places of the sun, moon, and fixed stars ... : to which is added an introduction unto geography / by John Newton ...

CHAP. VII.

Of the Measuring of a Circle.

THe squaring of a Circle, or the finding of a Square exactly equal to a Circle given, is that which many have endeavoured, but none as yet have attained: Yet Archimedes that Famous Mathe∣matician hath sufficiently proved, That the Area of a Circle is equal to a Rectangle made of the Ro∣dius and half the Circumference: Or thus, The Area of a Circle is equal to a Rectangle made of the Diameter and the fourth part of the Circum∣ference. For Example, let the Diameter of a Circle be 14 and the Circumference 44; if you multiply half the Circumference 22 by 7 half the Diameter, the Product is 154; or if you multiply 11 the fourth part of the Circumference, by 14 the whole Diameter, the Product will still be 154. And hence the Superficies of any Circle may be found though not exactly, yet near enough for any use.

2. But Ludolphus Van Culen finds the Circum∣ference of a Circle whose Diameter is 1.00 to be 3.14159 the half whereof 1.57095 being mul∣tiplied by half the Diameter 50, &c. the Product is 7.85395 which is the Area of that Circle, and from these given Numbers, the Area, Circumfe∣rence and Diameter of any other Circle may be found by the Proportions in the Propositions fol∣lowing.

Proposition I.

The Diameter of a Circle being given to find the Circumference.

As 1. to 3.14159: so is the Diameter to the Circumference. Example. In Fig. 13. Let the Diameter IB be 13. 25. I say as 1. to 3. 14159. so IB. 13.25 to 41.626 the Circumference of that Circle.

Proposition II.

The Diameter of a Circle being given to find the Superficial Content.

As 1. to 78539; so is the Square of the Dia∣meter given, to the Superficial Content required. Example, Let the Diameter given be as before IB 13.25 the Square thereof is 175.5625 therefore.

As 1. to 78539: so 175.5625 to 137.88 the Superficial Content of that Circle.

Proposition III.

The Circumference of a Circle being given, to find the Diameter.

This is but the Converse of the first Propositi∣on: Therefore as 3.14159 is to 1: so is the Circumference to the Diameter; and making the Circumference an Unite, it is. 3. 14159. 1∷ 1. 318308, and so an Unite may be brought into the first place. Example, Let the given Cir∣cumference

be 41. 626. I say,

As 1. to 318308: so 41.626 to 13. 25. the Diameter required.

Proposition IV.

The Circumference of a Circle being given to find the Superficial Content.

As the Square of the Circumference of a Cir∣cle given is to the Superficial Content of that Circle: so is the Square of the Circumference of another Circle given to the Superficial Con∣tent required. Example, As the Square of 3.14159 is to 7853938: so is 1. the Square of another Circle to 079578 the Superficial Content required, and so an Unite for the most easie work∣ing may be brought into the first place: Thus the given Circumference being 41. 626. I say,

As 1. to 0.79578: so is the Square of 41.626 to 137.88 the Superficial Content required.

Proposition V.

The Superficial Content of a Circle being given, to find the Diameter.

This is the Converse of the second Propositi∣on, therefore as 78539 is to 1. so is the Superfici∣al Content given, to the Square of the Diameter required. And to bring an Unite in the first place: I say.

As 7853978. 1∷1. 1. 27324, and there∣fore if the Superficial Content given be 137.88, to find the Diameter: I say,

As 1. to 1.27324: so 137.88 to 175.5625 whose Square Root is 13.25, the Diameter sought.

Proposition VI.

The Superficial Content of a Circle being given, to find the Circumference.

This is the Converse of the Fourth Propositi∣on, and therefore as 079578 is to 1 : so is the Su∣perficial Content given, to the Square of the Cir∣cumference required, and to bring an Unite in the first place: I say,

As 079578. 1 :: 1. 12.5664, and therefore if the Superficial Content given be 137.88, to find that Circumference: I say,

As 1. to 12.5664: so is the 137.88 to 1732.7 whose Square Root is 626 the Circumference.

Proposition VII.

The Diameter of a Circle being given to find the Side of the Square, which may be inscribed within the same Circle.

The Chord or Subtense of the Fourth Part of a Circle, whose Diameter is an Unite, is 7071067, and therefore, as 1. to 7071067: so is the Dia∣meter of another Circle, to the Side required. Example, let the Diameter given be 13.25 to find the side of a Square which may be inscribed in that Circle: I say,

As 1. to 7071067: so is 13.25 to 9.3691 the side required.

Proposition VIII.

The Circumference of a Circle being given, to find the Side of the Square which may be inscribed in the same Circle.

As the Circumference of a Circle whose Dia∣meter is an Unite, is to the side inscribed in that Circle; so is the Circumference of any other Circle, to the side of the Square that may be in∣scribed therein. Therefore an Unite being made the Circumference of a Circle.

As 3.14159 to 7071067: so 1. to 225078.

And therefore the Circumference of a Circle being as before 41.626, to find the side of the Square that may be inscribed: I say,

As 1. to 225078. so is 41.626 to 9.3691 the side inquired.

Proposition IX.

The Axis of a Sphere or Globe being given, to find the Superficial Content.

As the Square of the Diameter of a Circle, which is Unity, is to 3.14159 the Superficial Content, so is the Square of any other Axis given, to the Superficial Content required. Ex∣ample, Let 13.25 be the Diameter given, to find the Content of such a Globe: I say,

As 1. to 3.14159: so is the Square of 13.25 to 551.54 the Superficial Content required.

Proposition X.

To find the Area of an Ellipsis.

As the Square of the Diameter of a Circle, is to the Superficial Content of that Circle; so is the Rectangle made of the Conjugate Diame∣ters in an Ellipsis, to the Area of that Ellipsis; And the Diameter of a Circle being one, the Area is 7853975, therefore in Fig. 26. the Diameters AC8 and BD5 being given, the Area of the Ellipsis ABCD may thus be found.

As 1. to 7853975: so is the Rectangle AC in BD40 to 3.1415900, the Area of the Ellipsis required.