 Playfair's Axiom, though it is actually stated in Proclus on Eucl. I. 31). The proof therefore, apparently ingenious as it is, breaks down. Indeed the method is unsound from the beginning, since (as Saccheri pointed out), before even the definition of...  A History of Greek Mathematics - Page 227by Sir Thomas Little Heath - 1921Full view - About this book
 | Euclid - Mathematics, Greek - 1908 - 550 pages
...not seem to have defined parallels in this way. Saccheri points out that, before such a definition can be used, it has to be proved that " the geometrical...equidistant from a straight line is a straight line." To do him justice, Clavius saw this and tried to prove it : he makes out that the locus is a straight... | |
 | Sir Thomas Little Heath - Mathematics - 1921 - 608 pages
...FL. Therefore QV = LE. Since then EL, QV are equal and parallel, so are EQ, LV, and (says Geminus) it follows that AB passes through Q. What follows...now to the things which follow from the principles (TO. ftfTa T«y dp\ds), we gather from Proclus that Geminus carefully discussed such generalities as... | |
 | Sir Thomas Little Heath - Mathematics - 1921 - 614 pages
...ingenious as it is, breaks down. Indeed the method is unsound from the beginning. since (as Saecheri pointed out), before even the definition of parallels...now to the things which follow from the principles (rc¿ p€r& r&s' dp¿ds¿), we gather from Proclus that Geminus carefully discussed such generalities... | |
 | Sir Thomas Little Heath - Mathematics - 1921 - 610 pages
...Saccheri pointed out), before even the definition of parallels by Geminus can be used, it has to be Droved that ' the geometrical locus of points equidistant...now to the things which follow from the principles (TO. /j.fra. ras ap^as), we gather from Proclus that Geminus carefully discussed such generalities... | |
 | John Stillwell - Mathematics - 2004 - 576 pages
...less obviously equivalent to it. For example, (i) The angle sum of a triangle = n (Euclid). (ii) The locus of points equidistant from a straight line is a straight line. (al-Haytham, around 1000 CE). (iii) Similar triangles of different sizes exist [Wallis (1663); see... | |
 | Euclid - 454 pages
...not seem to have defined parallels in this way. Saccheri points out that, before such a definition can be used, it has to be proved that " the geometrical...equidistant from a straight line is a straight line." To do him justice, Clavius saw this and tried to prove it : he makes out that the locus is a straight... | |
 | Edited by Dean Turner and Richard Hazelett - 1979 - 462 pages
...title, but some form of curve, the perpendiculars cannot all be equal. Saccheri objects to this that it has to be proved that “the geometrical locus...equidistant from a straight line is a straight line.” Does not this apply to our construction, which likewise only takes the two end points of the sides... | |
 | Euclid - Euclid's Elements - 1920 - 304 pages
...This definition is however (as Saccheri pointed out) unsatisfactory because, before such a definition can be used, it has to be proved that the geometrical locus of points equidistant from a given straight line is also a straight line. It is clear that with Aristotle the general notion of... | |
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