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 ...

LEMMA.

A Line which is Perpendicular to one Parallel, is also Perpendicu∣lar to the other.

Let the Lines AB, CD, be Parallel to each other, and let EF be Perpendicular to CD. I say that it is Perpendicular to AB. Cut off two equal Lines CF, FD; at the Points C and D erect two Perpen∣diculars to the Line CD, which shall also be equal to FE, by the Definition of Parallels; and draw the Lines EC, ED.

Demonstration. The Triangles CEF, FED, have the Side FE common; the Sides FD, FC, are equal; the Angles at F are Right, and by consequence equal. Therefore (by the 4^th) the Bases EC, ED, the Angles FED, FEC; FDE, FCE, are equal; and those two last being taken away from the Right Angles ACE, BDF, leaveth the equal Angles EDB, ECA: Now the Triangles CAE, DBE, shall (by the 4^th.) have the Angles DEB, CEA, equal; which Angles being added to the equal Angles CEF, FED, maketh equal Angles

FEB, FEA. Therefore EF is Per∣pendicular to AB.

Notes

• Lem. 26.