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 XIX. THEOREM.

IN every Triangle, the greatest Angle is opposite to the greatest Side.

Let the Angle A of the Triangle BAC, be greater than the Angle ABC. I say that the Side BC, opposite to the Angle A, is greater then the Side AC, opposite to the Angle B.

Demonstration. If the Side BC, was equal to the Side AC; in this case the Angles A and B would be equal (by the 5^th.) which is contrary to the Hypo∣thesis: If the Side BC was less than AC, then the Angle B would be grea∣ter than A, which is also contrary to the Hypothesis. Wherefore I conclude that the Side BC is greater than AC.

USE.

* 1.1 WE prove by these Propositions, not only that from the same Point to a Line given, there can be but one Perpendicular drawn, but also that that Perpendicular is the shortest Line of all those Lines which might be drawn to the said

Line. As for instance, if the Line RV be Perpendicular to ST; it shall be shorter then RS: because the Angle RVS being Right, the Angle RSV shall be Acute (by the Cor. of the 17^th) and the Line RV shall be shorter than RS (by the Preceding.) For this Reason Geometri∣cians always make use of the Perpendicu∣lars in their measuring; and Reducing irregular Figures into those whose Angles are Right.

I further add, that seeing there can only be drawn Three Perpendiculars to one and the same Point, it cannot be ima∣gined that there are more than Three Spe∣cies of quantity; viz. A Line, a Surface, and a Solid.

We also prove by these Propositions, that a Boul which is exactly round, being put on a Plain, cannot stand but on one deter∣minate Point. As for Example, Let the Line AB Represent the Plain, and from the Center of the Earth C, let the Line CA be drawn Perpendicular to the Line AB. I say that a Boul being placed on the Point B, ought not to stand on that Point, because no heavy Body will stand still while it may Descend. Now the Boul B going towards A, is always De∣scending, and cometh nearer and nearer

the Center of the Earth C: Because in the Triangle CAB, the Perpendicular CA is shorter than BC.

We also prove in like manner, that a liquid Body ought to Descend from B to A, and that the surface ought to be ••ound.

Notes

• * 1.1

  Use 19.