PROPOSITION IV. THEOREM.
IF a Line be perpendicular to two other Lines which cut one another, it shall be also perpendicular to the Plane of those Lines.
If the Line AB be perpendicular to
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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To the extent possible under law, the Text Creation Partnership has waived all copyright and related or neighboring rights to this keyboarded and encoded edition of the work described above, according to the terms of the CC0 1.0 Public Domain Dedication (http://creativecommons.org/publicdomain/zero/1.0/). This waiver does not extend to any page images or other supplementary files associated with this work, which may be protected by copyright or other license restrictions. Please go to http://www.textcreationpartnership.org/ for more information.
- Subject terms
- Geometry -- Early works to 1800.
- Mathematical analysis.
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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 313
the Lines CD, EF, which cut one ano∣ther in the point B; in such manner that the Angles ABC, ABD, ABE, ABF, be right, which a flat figure cannot re∣present; it shall be perpendicular to the Plane CD, EF; that is to say, that it shall be Perpendicular to all the Lines that are drawn in that Plane through the point B: as to the Line GBH. Let equal Lines be cut BC, BD, BE, BF; and let be drawn the Lines EC, DF, AC, AD, AE, AF, AG, and AH.
Demonstration. The four Triangles ABC, ABD, ABE, ABF, have their Angles Right in the Point B; and the Sides BC, BD, BE, BF, equal with the side AB common to them all. Therefore their Bases AC, AD, AE, AF, are equal (by the 4th. of the 1st.)
2. The Triangles EBC, DBF, shall be equal in every respect, having the Sides BC, BD, BE, BF, equal; and the Angles CBE, DBF, opposite at the vertex being equal; so then the Angles BCE, BDF, BEC, BFD, shall be equal (by the 4th. of the first,) and their Bases EC, DF, equal.
3. The Triangles GBC, DBH, having their opposite Angles CBG, DBH, equal, as also the Angles BDH,
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BCG, and the sides BC, BD, they shall then have (by the 26th. of the 1st.) their Sides BG, BH, CG, DH, equal.
4. The Triangles ACE, AFD, having their sides AC, AD, AE, AF, equal, and the Bases EC, DF, equal, they shall have (by the 8th. of the 1st.) the Angles ADF, ACE, equal.
5. The Triangles ACG, ADH, have the Sides AC, AD, CG, DH, equal; with the Angles ADH, AGC: Thence they shall have their Bases AG, AH, equal.
Lastly, the Triangles ABH, ABG, have all their sides equal; thence (by the 27th. of the 1st.) the Angles ABG, ABH, shall be equal, and the Line AB perpendicular to GH. So then the Line AB shall be perpendicular to any Line which may be drawn through the point B, in the Plane of the Lines CD, EF, which I call perpendicular to the Plane.
USE.
THis Proposition cometh often in use in the first Book of Theodosius; for ex∣ample, to Demonstrate that the Axis of the World
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is perpendicular to the Plane of the Equi∣noctial. In like manner in Gnomonicks, we Demonstrate by this Proposition, that the Equinoctial is perpendicular to the Me∣ridian in Horizontal Dials, it is not less useful in other Treatises; as in that of Astrolabes, or in the sections of Stone.