PROPOSITION IV. THEOREM.
EQuiangular Triangles have their Sides Proportional.
If the Triangles ABC, DCE, are equiangular; that is to say, that the
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 ...
About this Item
- 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.
- Link to this Item
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http://name.umdl.umich.edu/A38722.0001.001
- Cite this Item
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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 ..." In the digital collection Early English Books Online 2. https://name.umdl.umich.edu/A38722.0001.001. University of Michigan Library Digital Collections. Accessed June 4, 2024.
Pages
Page 268
Angles ABC, DCE; BAC, CDE, be equal: There will be the same Rea∣son of BA to BC, as of CD to CE. In like manner the reason of BA to AC, shall be the same with that of CD to DE. Joyn the Triangles after such a manner, that their Bases BC, CE, be on the same Line; and continue the sides ED, BA: seeing the Angles ACB, DEC, are equal; the Lines AC, EF, are parallel; and so CD, BF, (by the 29th. of the 1st.) and AF, DC, shall be a parallelogram.
Demonstration. In the Triangle BFE, AC, is parallel to the Base FE, thence (by the 2d.) there shall be the same reason of BA to AF, or CD, as of BC to CE; (and by exchange) there shall be the same reason of AB to BC, as of DC to CE. In like manner in the same Triangle, CD being parallel to the Base BF; there shall be the same Reason of FD, or AC to DE, as of BC to GE (by the 2d.) and by ex∣change, there shall be the same reason of AC to BC, as of DE to CE.
Page 269
USE.
THis Proposition is of a great extent, and may pass for a universal Princi∣ple in all sorts of Measuring. For in the first place the ordinary practice in measu∣ring inaccessible Lines, by making a little Triangle like unto that which is made or imagined to be made on the ground, is founded on this Proposition, as also the greatest part of those Instruments, on which are made Triangles like unto those that we would measure, as the Geometrical Square, Sinical Quadrant, Jacobs Staff, and others. Moreover we could not take the plane of a place, but by this Proposi∣tion: wherefore to explain its uses, we should be forced to bring in the first Book of practical Geometry.