The elements of Euclid, explained and demonstrated in a new and most easie method with the uses of each proposition in all the parts of the mathematicks / by Claude Francois Milliet D'Chales, a Jesuit ; done out of French, corrected and augmented, and illustrated with nine copper plates, and the effigies of Euclid, by Reeve Williams ...

PROPOSITION XXI. THEOREM.

THe Angles which are in the same Segment of a Circle, or that have the same Arch for Base, are equal.

If the Angles BAC, BDC, are in the same Segment of a Circle, greater than a Semicircle; they shall be equal. Draw the Lines BI, CI.

Demonstration. The Angles A and D are each of them half of the Angle BIC, by the preceding Proposition; therefore they are equal. They have also the same Arch BC for Base.

Secondly, let the Angles A and D be in a Segment BAC, less than a semi-circle; they shall notwithstanding be equal.

Demonstration. The Angles of the Triangle ABE are equal to all the Angles of the Triangle DEC: The Angles ECD, ABE, are equal (by the preceding case;) since they are in the same Segment ABCD, greater than a Semi-circle: the Angles in E are like∣wise equal (by the 15^th. of the 1^st.) therefore the Angles A and D shall be equal, which Angles have also the same Arch BFC, for Base.

USE.

IT is proved in Opticks, that the Line BC shall appear equal, being seen from A and D; since it always ap∣peareth under equal Angles.

We make use of this Proposition to de∣scribe a great Circle, without having its Center; For Example, when we would give a Spherical figure, to Brass Cauldrons to the end we may work thereon; and to pollish Prospective or Telescope Glasses. For having made in Iron an Angle BAC equal to that which the Segment ABC contains; and having put in the Points B, and C, two small pins of Iron, if the Triangle BAC be made to move after such a manner, that the Side AB may always touch the Pin

B; and the Side AC, the Pin C: the Point A shall be always in the Circum∣ference of the Circle ABCD. This way of describing a Circle may also serve to make large Astrolabes.

Notes

• Prop. XXI.