Euclid obviously puts the whole difficulty into an assumption ; which, though the most direct course, is not that which is best calcu lated to give the highest degree of evidence to geometrical truths. For it is a more obvious proposition that two lines which intersect one another cannot both be parallel to a third line, and this being granted, Euclid's axiom readily follows. If it should be objected that this is merely Euclid's axiom in another form, it is replied that the form is a more easy one, and therefore preferable : just as it would be wiser to assume " Every A is is and every n is a," than the identical but more complicated proposition " Every A is n, and everything which is not A is not n." It is known then that the difficulty is entirely removed if we grant that "two lines which intersect are not both parallel to any third line," or, which is the same thing, that "through a given point not more than one line can be drawn parallel to a given line." The theory of Euclid being thus impro6ed so far as it is capable of being done by a mere difference of statement, it remains to ask, 1. Whether assump tion can be dispensed with altogether and a direct proof of Euclid's axiom, or something equivalent. to it, given/ 2. If the preceding question be answered in the negative, can any more simple assumption be made the foundation of the theory? The attempts to answer one or the other question in the affirmative have been very numerous, and have (without any exception but one in which new axioms of another sort lire introduced) tacitly contained the defect which their authors were desirous of avoiding. The author of ',Geometry without Axioms' (General Perronet Thompson) has collected and commented on thirty instances, of which we here make a brief abstract, adding one or two more.
1. The axiom of Euclid in question. 2. Ptolemy ; his proof assumes the symmetricality of parallel lines on one side and the other of any line which cuts both. 3. Proclus assumes that intersectors diverge infinitely from the point of intersection, and that parallels do not. 4. Clavius assumes that a line which is everywhere equidistant from a straight line, is itself straight. 5 and 6. Two demonstrations of Dr. Thomas Oliver (1604) assume Clavius's axiom. 7. Wolf, Boscovich, T. Simpson (in the first edition of his Elements), and Bonnycastle, define parallel lines as those which always preserve the same distance, which is Clavius's axiom in the disguise of a definition. 8. D'Alembert would define parallels as lines one of which has two points equidistant from the other, but acknowledged that he could not complete the proof of the axiom of Clavius. 9. T. Simpson (Ele ments, 2nd edition) proposed to assume that two lines, one of which has two points unequally distant from the other, must meet. 10. Robert Simpson proposed to assume that a straight line cannot first approach to and then recede from another, without cutting it. 11. Varignon, Bezout, &c., would define parallels as lines which make the same angle with a third line if a third line mean some one third line, the difficulty remains just as before; if any third line, the difficulty is tacitly removed by an assumption. 12. Ludlam, Playfair, he.,
recommend the axiom which we have also recommended, namely, that two intersecting straight lines cannot be both parallel to a third. 13. Leslie proposes to attain the same axiom in a sort of experimental manner, by making a line revolve about a point. 14. Playfair (in his Notes) proposes to assume that a straight line which turns completely round, and thus regains its first position, must turn through four right angles, whether it constantly revolves about one point, or whether the pivot of revolution changes. 15 and 10. Fmnceschini (1787) proposes to assume that the projections of a straight line on a line making an acute angle with it, increase without limit with the projected line. 17. Sonic have proposed to define paralicla as "lines having the same assuming it to be ob viously contained in the conception of direction, that two similar directions make the same angle with any other direction. 18. Mr. Exley (1818) proposes to assume that if four equal straight lines, each at right angles to the preceding, do not meet and enclose a space, a fifth such line would do so. 19. Dr. Creswell proposes to assume that through any point within an angle less than two right angles, a straight line may be supposed to be drawn cutting the two straight lines which contain the angle. 20. Professor Thompson makes it an axiom that "if a triangle be moved along a plane, so that its base may always be on the same straight line, its vertex describes a straight line equal to that described by either extremity of the base. 21. M. Legendre (in the earlier editions of his Elements) makes a direct appeal to the senses. 22. In the seventh edition he assumes (as in instance 15) that a magnitude increases without limit, where perpetual increase is all that is demonstrable. 23. In the twelfth edition he fairly brings the disputed proposition to rest upon the axiom, that if two angles of a triangle diminish without limit, the third (whatever the base may be) approaches without limit to two right angles, a proposition not admis sible when, as in M. Legendre's final construction, the base at the same time increases without limit. 24. In a note to the same edition, he demands as an axiom that no straight line can be entirely included between two straight lines which make an angle less than two right angles, which may easily be shown to be nothing more or less than Euclid's axiom. 25. He attemps a proof of the last, which fails. 26.
Legendre's analytical proof, which we shall presently examine. 27. M. Lacroix would confine the assumption of Euclid to the case in which one of the internal angles is a right angle and the other less. 23. M. Bertrand extends the use of the term equality ; we shall afterwards examine his proof. 29. Mr. Ivory assumes a right to construct a series of triangles in a manner which cannot be certainly done unless an assumption as difficult as in (20) be made : and 30, Professor Young makes a modification of the preceding, which does not remove the difficulty.