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.

IF on the same Base you draw a lesser Triangle in a greater, the Sides of the lesser shall be shorter than the greater, but contain a greater Angle.

Let the less Triangle ADB, be drawn within the greater ACB on the same Base AB. I say in the first place that the Sides AC, BC, are greater than the Sides AD, BD. Continue the Side AD unto E.

Demonstration. In the Triangle ACE the Sides, AC, CE, taken to∣gether, are greater than the Side AE, (by the 20^th.) Add thereto the Side EB, the Sides AC, CEB, shall be greater than the Sides AE, EB. Likewise in the Triangle DBE, the Two Sides BE, ED, taken together, are greater than BD; and adding thereto AD, the Sides AE, EB, shall be greater than AD, BD.

Moreover, I say that the Angle ADB, is greater than the Angle ACB: For the Angle ADB is exteriour in respect of the Triangle DEB; it is therefore greater than the Interiour DEB (by the 16^th.) Likewise the Angle DEB being Exteriour in respect of the Tri∣angle ACE, is greater than the Angle ACE: Therefore the Angle ADB, is greater than the Angle ACB.

USE

* 1.1 WE Demonstrate in Opticks by this Proposition, that if from the Point C, one should see the Base AB, it would seem less than if one should see the same from the Point D; according to this Principle, that quantities seen under a greater Angle, appear greater, for which reason Vitruvius would that the Tops of very high Pillars should be made but little tapering, because that their Tops being at a good distance from the Eyes, will of themselves appear very much diminished.

Notes

• * 1.1

  Prop. XXI