The Thirteen Books of Euclid's Elements, Tr. from the Text of Heiberg Volume 3

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General Books, Mar 6, 2012 - 166 pages
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1908 Excerpt: ... right angles to the plane of reference. For let the plane DE be drawn through AB, let CE be the common section of Z / the plane DE and the plane of reference, let a point F be taken at random on CE, and from F let FG be drawn in the plane DE at right angles to CE. 1. n Now, since AB is at right angles to the plane of reference, AB is also at right angles to all the straight lines which meet it and are in the plane of reference; xi. Def. 3 so that it is also at right angles to CE; therefore the angle ABF is right. But the angle GFB is also right; therefore AB is parallel to FG. 1. 28 But AB is at right angles to the plane of reference; therefore FG is also at right angles to the plane of reference. xi. 8 Now a plane is at right angles to a plane, when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane. xi. Def. 4 And FG, drawn in one of the planes DE at right angles to CE, the common section of the planes, was proved to be at right angles to the plane of reference; therefore the plane DE is at right angles to the plane of reference. Similarly also it can be proved that all the planes through AB are at right angles to the plane of reference. Therefore etc. Q. E. D. Starting as Euclid does from the definition of perpendicular planes as planes such that all straight lines drawn in one of the planes at right angles to the common section are at right angles to the other plane, it is necessary for him to show that, if F be any point in CE, and FG be drawn in the plane DE at right angles to CE, FG will be perpendicular to the plane to which AB is perpendicular. It is perhaps more scientific to make the definition, as Legendre makes it, a particular case of the definition of...

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