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* 1.1 WE often make use of the two Pre∣ceding Propositions, in Catop∣tricks, to prove that of all the Lines which can be drawn from the Point A, by re∣flection, to the Point B, those are the shortest which make the Angles of Incidence equal to the Angles of Reflection. For
Example, if the Angles BED, AEF, are equal; the Lines AE, EB, are shorter then AF, FB. Draw from the Point B, the Perpendicular BD, and let the Lines BD, DC, be equal: Then draw EC, FC. Now in the first place, in the Triangles BED, CED, the Side DE being common, and the Sides BD, DC, being equal, and the Angles CDE, BDE, being also equal, the Bases BE, CE, shall be equal; as also the Angles BED, DEC, (by the 4th) I might prove by the same manner of Argument, that BF, CF, are equal.
Demonstration. The Angles BED, DEC, are equal, it is also supposed that the Angles BED, AEF, are equal: Thence the opposite Angles DEC, AEF, shall be equal; and (by the Cor. of the 15th Prop.) AEC is a streight Line. By consequence AFC is a Tri∣angle, in which the Sides AF, FC, are greater than AEC; that is to say AE, EB: For the Lines AE, FC, are equal to AF, EB: Therefore the Lines AF, FB, are greater then AE, BE; and since natural Causes do act by the most shortest Lines; therefore all reflections are made after this sort; that the
Angles of Reflection, and of Incidence, are equal.
* 1.2 Moreover, because we can easily prove that all the Angles which are made upon a Plain, by the meeting of never so many Lines in the same Point, are equal to Four Right Angles; since that in the first Figure of this Proposition the Angles AEC, AED, are equal to two Right, as also BED, BEC; we make this General Rule to determine the number of Polygons, which may be joyned together to Pave a a Floor, so we say that four squares, six Triangles, three Hexigones, may Pave the same; and that it is for this reason the Bees make their cellules Hexagonal.
Use 15.
Prop. XV.