PROPOSITION XXXI. THEOREM.
THe Angle which is in a Semi-circle is Right, that which is comprehended in a greater Segment, is Acute; and that in a lesser Segment, is Obtuse.
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.
- Rights/Permissions
-
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.
- Link to this Item
-
http://name.umdl.umich.edu/A38722.0001.001
- 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 June 4, 2024.
Pages
Page 170
If the Angle BAC be in a Semi-circle: I demonstrate that it is Right. Draw the Line DA.
Demonstration. The Angle ADB ex∣teriour in respect of the Triangle DAC is equal (by the 32d. of the 1st.) to the Two Interiours DAC, DCA; and those being equal (by the 5th. of the 1st.) seeing the Sides DA, DC, are equal; it shall be double to the Angle DAC. In like manner the Angle ADC is double to the Angle DAB: therefore the Two Angles ADB, ADC, which are equal to Two Right, are double to the Angle BAC, and by consequence the Angle BAC is a Right Angle.
Secondly, the Angle AEC which is in the Segment AEC, is obtuse; for in the Quadrilateral ABCE, the Opposite Angles E and B, are equal to Two Right (by the 22d.) the Angle B is Acute; therefore the Angle E shall be Obtuse.
Thirdly, the Angle B which is in the Segment ABC, greater than a Semi-circle, is Acute; seeing that in the Triangle ABC, the Angle BAC is a Right Angle.
Page 171
USE.
* 1.1 THe Workmen have drawn from this Proposition the way of trying if their Squares be exact; for having drawn a Semi-circle BAD, they apply the Point A of their Square BAD, on the Circum∣ference of the Circle, and one of its Sides AB, on the Point B of the Diameter; the other Side AD must touch the other Point D, which is the other end of the Diameter.
Ptolomy makes use of this Proposition to make the Table of Subtendants or Chords, of which he hath occasion in Trigonometry.
* 1.2 We have also a practical way to erect a perpendicular on the end of a Line, which is founded on this Proposition. For Ex∣ample, to erect a Perpendicular from the Point A of the Line AB, I put the Foot of the Compass on the Point C, taken at discretion; and extending the other to A, I describe a Circle which may cut the Line AB in the Point B. I draw the Line BCD. It is evident that the Line DA shall be perpendicular to the Line AB; seeing the Angle BAD is in the Semi-circle.
Notes
-
* 1.1
Use 31.
-
* 1.2
Use 31.