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 I. THEOREM.

A Strait Line cannot have one of its parts in a Plane, and the other with∣out it.

If the Line AB be in the Plane AD, it being continued, shall not go without, but all its parts shall be in the same Plane. For if it could be that BC were a part of AB continued. Draw in the Plane CD, the Line BD, perpendi∣cular to AB: draw also in the same Plane, BE perpendicular to BD.

Demonstration. The Angles ABD, BDE, are both Right Angles; thence (by the 14th. of the first,) AB, BE, do make but one Line; and consequently BC, is not a part of the Line AB con∣tinued; otherwise two strait Lines CB, EB, would have the same part AB: that is AB would be part of both: which we have rejected as false in the Thirteenth Maxim of the first Book.

USE.

WE establish on this Proposition a principle in Gnomonicks, to

prove that the shadow of the stile falleth not without the Plane of a great Circle, in which the Sun is. Seeing that the end or top of the stile is taken for the Center of the Heavens; and consequently for the Center of all the great Circles: the shadow being always in a streight Line, with the Ray drawn from the Sun to the Opaque Body; this Ray being in any great Circle, the shadow must also be therein.

Notes

• Plate VII. Prop. I.