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 XXXI. PROBLEM.

TO Draw a Line Parallel to another, through a Point given.

It is proposed to draw a Line through the Point C, which shall be Parallel to the Line AB. Draw the Line CE, and make the Angle ECD, equal to the Angle CEA: I say that the Line CD is Parallel to AB.

Demonstration. The Alternate Angles DCE, CEA, are equal: Therefore the Lines CD, AB, are Parallel.

One might easily demonstrate the Eleventh Maxim, that is to say, that if a Line falling on Two other Lines, maketh the Interiour Angles less than Two Right, those Two Lines shall Intersect.

* 1.1 Let the Line AC falling on the Lines AB, CD, make the Interiour Angles ACD, CAB, less than two Right: I say the Lines AB, CD, shall Intersect. Let the Angles ACD, CAE, be equal to Two Right, the Lines AE, CD, are Parallels

(by the 29th.) Take at discretion the Line AB, and through the Point B draw the Line EF Parallel to CA. Then take EB as many times as shall be necessary to fall below the Line CD, as in this Figure I have taken it but twice, wherefore EB, BF, are equal. Draw from the Point F a Parallel FG, equal to AE, and joyn GB. I say that the Line ABG, is but one streight Line; and that so the Line AB concurring with FG, if CD be continued, since it cannot cut the Paral∣lel FG, it shall cut the Line BG between B and G.

Demonstration. The Triangles AEB, BFG, have the Sides AE, FG; BE, BF, equal, as also the Alternate Angles AEB, BFG, (by the 29th.) There∣fore they are equal in every respect (by the 4th.) and the opposite Angles ABE, FBG, are equal; and consequently (ac∣cording to the 15th.) AB, BG, maketh one streight Line.

USE

THe use of Parallel Lines is very ne∣cessary: First in perspective, since that the Appearances or Images of the Parallel Lines on the Draft are Parallels

between themselves. In Navigation, like Rumbs in our Charts, are Parallel to each other. Likewise the hour Lines on Polar Dials, the Compass of Proportion or Sector is grounded also on Parallel Lines.

Notes

• * 1.1

  Ax. 11.