The thirteen books of Euclid's Elements Volume 3 - Brossura

Euclid

Brossura

ISBN 10: 1232124486 ISBN 13: 9781232124481

Casa editrice: RareBooksClub.com, 2012

Riassunto

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: ... to one of two parallel planes, it is perpendicular to the other also. For suppose that MN, PQ are two parallel planes, and that AB is perpendicular to MN Through AB draw any plane, and let it intersect the planes MN, PQ in AC, BD respectively. Therefore AC, BD are parallel. xi. 16 But A C is perpendicular to AB; therefore AB is also perpendicular to BD. That is, AB is perpendicular to any line in PQ passing through B; therefore AB is perpendicular to PQ. It follows as a corollary that Through a given point one platie, and only one, can be drawn parallel to a given plane. In the above figure let A be the given point and PQ the given plane. Draw AB perpendicular to PQ. Through A draw a plane MN at right angles to AB (see note on xi. 5 above). Then MN is parallel to PQ. xi, 14 If there could pass through A a second plane parallel to PQ, AB would also be perpendicular to it. That is, AB would be perpendicular to two different planes through A: which is impossible (see the same note). Also it is readily proved that, If two planes are parallel to a third plane, they are parallel to one another. Proposition 1 7. If two straight lines be cut by parallel planes, they will be cut in the same ratios. For let the two straight lines AB, CD be cut by the parallel planes GH, KL, MN at the points A, E, B and C, F,D; I say that, as the straight line AE is to EB, so is CF to FD. For let AC, BD, AD be joined, let AD meet the plane KL at the point 0, and let EO, OF be joined. Now, since the two parallel planes KL, MN are cut by the plane EBDO, their common sections EO, BD are parallel. xi, 16 For the same reason, since the two parallel planes GH, KL are cut by the plane AOFC, their common sections A C, OF are parallel. id-And, since the straight line EO has been drawn paralle...

Su questa edizione

• Editore: RareBooksClub.com
• Data di pubblicazione: 2012
• ISBN 10: 1232124486
• ISBN 13: 9781232124481
• Rilegatura: Copertina flessibile
• Numero di pagine: 170