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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 ...

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Title
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 ...
Author
Dechales, Claude-François Milliet, 1621-1678.
Publication
London :: Printed for Philip Lea ...,
1685.
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Subject terms
Geometry -- Early works to 1800.
Mathematical analysis.
Cite this Item
"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 ..." In the digital collection Early English Books Online 2. https://name.umdl.umich.edu/A38722.0001.001. University of Michigan Library Digital Collections. Accessed May 29, 2024.

Pages

PROPOSITION IX. THEOREM.

IF a Line be equally Divided, and un∣equally; the Squares of the unequal parts shall be double to the Squares of half the Line, and of the part between.

Let the Line AB be Divided equally, in the Point C; and unequally, in the Point D. The Squares of the unequal parts AD, DB, shall be double of the

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Squares of AC, which is the half of AB, and the Square between both CD. Draw AB, and the Perpendi∣cular CE, equal to AC: Draw also the Lines AE, BE, and the Perpendi∣cular DF; as also FG Parallel to CD. Then draw the Line AF.

Demonstration. The Lines AC, CE, are equal, and the Angle C is Right; thence (by the 6th of the 1st.) the Angles CAE, CEA, are equal to a half Right Angle. In like manner, the Angles CEB, CBE, CFE, DFB, are half Right; the Lines GF, GE, DF, DB, are equal; and the total Angle AEF is Right. The Square of AE (by the 47th of 1st.) is equal to the Squares of AC, CE, which are equal: Thence it is double to the Square of AC. After the same manner the Square of EF is double to the Square of GF, or CD: Now the Square of AF is equal to the Squares of AE, EF; seeing that the Angle AEF is Right: There∣fore the Square of AF is double to the Squares of AC, CD. The same Square AF is equal to the Squares of AD, DF, or DB; seeing that the Angle D is Right: There∣fore the Squares of AD, DB, are

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double to the Squares of AC, CD.

ARITHMETICALLY.

LEt AB be 10; AC, 5; CD, 3; DB, 2: The Squares of AD, 8, and DB, 2; that is to say 64, and 4, which together are 68, are double to the Square AC, 5, which is 25; and of the Square CD, 3, which is 9; for 25 and 9 are 34, which doubled make 68.

USE.

I Have not found this Proposition, nor the insuing, but only in Algebra.

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