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 VII. THEOREM.

IF in a Circle several Lines be drawn from a Point which is not the Center, unto the Circumference; first, that which passeth through the Center shall be great∣est; secondly, the least shall be the Re∣mainder of the same Line; thirdly, the nearest Line to the greatest is greater than any Line which is farther from it; fourthly, there cannot be drawn more than two equal Lines.

Let there be drawn several Lines from the Point A, which is not the Center of the Circle, to the Circumfe∣rence; and let the Line AC pass through the Center B: I demonstrate that it shall be the greatest or longest, that it is greater than AF. Draw FB.

Demonstration. The sides AB, BF, of the Triangle ABF, are greater than the side AF, (by the 20^th. of the 1^st.) Now BF, BC, are equal (by the defini∣tion of a Circle) therefore AB, BC,

that is to say, AC, is greater than AF. Moreover, in the second place, I say, that AD is the shortest; for Example, shorter than AE. Draw EB.

Demonstration. The Sides EA, AB, of the Triangle ABE, are greater than BE, which is equal to BD; thence EA, AB, are greater than BD; and taking away AB, which is common, AE shall be greater than AD. More∣over, AF, which is nearer to AC than AE, is greater than AE.

Demonstration. The Triangles FBA, EBA, have the Sides BF, BE, equal, and BA common to both: the Angle ABF is greater than the Angle ABE; thence (by the 24^th. of the first) AF is greater than AE.

In fine, I say that there cannot be drawn but two equal Lines from the Point A, to the Circumference, let the Angles ABE, ABG, be equal, and let be drawn the Line AE, AG.

Demonstration. The Triangles ABG, ABE, have their Bases AE, AG, equal; and all the Lines which may be drawn on either side of these, shall be either nearer to AC than the Lines AE, AG, or farther: and so they shall be either shorter or longer than

AE, AG. Therefore, there cannot be drawn more than two equal Lines.

USE.

THeodosius doth very well make use of this Proposition, to prove, that if from a Point in the Superficies of a Sphere, which is not the Pole of a Circle, there be drawn several Arks of great Cir∣cles unto the Circumference of a Circle: that which passeth through the Pole of that Circle to which the great Circles are drawn, is greatest. For Example, if from the Pole of the World, which is not the Pole of the Horizon (for the Zenith is the Pole thereof) there be drawn several Arks of great Circles unto the Circumfe∣rence of the Horizon: the greatest Ark of all shall be that part of the Meridian which passeth through the Zenith. This Proposition is also made use of, to prove, that the Sun being in Apoge, is farthest distant from the Earth.