...tread." Euclid's definition of parallel lines is straight lines which, being in the same plane and produced indefinitely in both directions, do not meet one another in either direction. Euclid's fifth postulate differs from all the others, and as Staekel remarks, "It requires a certain...

...") in his proofs, so that he might well have omitted the definitions, as we often do. 23. PARALLELS. Parallel straight lines are straight lines which,...directions, do not meet one another in either direction. This definition of parallels, simplified in its language, is the one commonly used to-day. Other definitions...

...") in his proofs, so that he might well have omitted the definitions, as we often do. 23. PARALLELS. Parallel straight lines are straight lines which, being in the same plane and being produced indefin,tely in both directions, do not meet one another in either direction. This definition of parallels,...

...symmetrical relationship between two straight lines, in accordance with Euclid's definition of parallels as " straight lines which, being in the same plane and...directions, do not meet one another in either direction." If we take along with this definition Euclid's fifth postulate, or Playfair's equivalent, it defines...

...foundation of his Elements of Geometry. It will be remembered that he defines Parallel Lines as follows : Parallel straight lines are straight lines which,...directions, do not meet one another in either direction.* Then in I. 27 he proves that If a straight line falling on two straight lines make the alternate angles...

...Lines. The word "parallel"6 means "alongside one another." Euclid defined parallel straight lines as "straight lines which, being in the same plane and...directions, do not meet one another in either direction." Rather less satisfactory is the definition of Poseidonius (c. 100 BC) as those lines "which, in one...

...neither equilateral nor rightangled. And let quadrilaterals other than these be called trapezia. 23. Parallel straight lines are straight lines which,...directions, do not meet one another in either direction. The Postulates and the Common Notions (or Axioms) appear next in Book I. In reading these you might...

...to require such substitutions as the following: Euclidean Def. 23: Parallel straight lines are those which, being in the same plane and being produced indefinitely in both directions, do not meet in either direction. Aristotelian Def. 23: Parallel straight lines are those which, being in the same...

...circumference of the circle, and such a straight line also bisects the circle. The final Definition is: 23. Parallel straight lines are straight lines which, being in the same plane and being produced indeiinitely in both directions, do not meet one another in either direction. It is possible to discuss...

...is a 'practical corollary'. Euclid's definition of parallel lines, however, is not so obliging: they are "straight lines which, being in the same plane...directions, do not meet one another in either direction", which offers neither a 'rule' JUDSON WEBB for constructing parallels, nor suggests any 'synthesis'...