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 XVIII. THEOREM.

TO describe a Poligon like to another on a Line given.

There is proposed the Line AB, on which one would describe a Poligon like unto the Poligon CFDE. Having di∣vided the Poligon CFDE into Tri∣angles, make on the Line AB a Triangle ABH like unto the Triangle CFE; that is to say, make the Angle ABH equal to the Angle CFE, and BAH equal to FCE. So then the Triangles ABH, CFE, shall be equiangled (by the 32d. of the first.) make also on BH, a Triangle equiangled to FDE.

Demonstration. Seeing the Triangles which are parts of the Poligons, are equiangular, the two Poligons are equiangular. Moreover, seeing the Tri∣angles ABH, CFE, are equiangular, there is the same Ratio of AB to BH, as of CF to FE, (by the 4th.) In like manner, the Triangles HBG, EFD, being equiangular, there shall be the

same Ratio of BH to BG, as of FE to FD; and by equality there shall be the same Ratio of AB to BG, as of CF to FD. And so of the rest of the sides. Thence (by the first Definition,) the Poligons are like to each other.

USE.

IT is on this Proposition we establish the greatest part of the practical ways to take the plane of a place, of an Edifice, of a Field, of a Forrest, or of a whole Country; for making use of the equal parts of a Line for Feet or for Chains; we de∣scribe a figure like unto the Prototype, but lesser, in which we may see the Proportion of all its Lines. And because it is easier on paper than on the ground; we may compre∣hend in this Proposition all Geodes•…•…, all Chorography, all Geographical Charts, and ways of reducing of the greater into a lesser; wherefore this Proposition extends almost to all Arts, in which it is necessary to take a design or Model.