Part 1 — APPENDIX III.

How should Parallels be defined?

We know that, if a Pair of Lines has either of the following properties

(1) they are equally inclined to a certain transversal,

(2) one of them contains 2 Points on the same side of, and equidistant from, the other,

it has all the following properties

(3) they are equally inclined to all transversals,

(4) any 2 Points, on either of them, are on the same side of, and equidistant from, the other,

(5) they do not meet, however far produced.

Any one of these properties may be used as a Definition. Let us take them one by one.

No. (1). This has the advantage that we need not begin by proving that such Lines exist, but may assume it as axiomatic. It has been used as a Definition by Varignon, Bezout, Cooley, &c.: at least, they say "which make equal angles with a transversal," leaving it uncertain whether they mean "a certain transversal" or "any transversal": if the latter, it is of course No. (3) they are proposing to use. From No. (1) we can prove No. (5) without using any disputable Axiom (see Euc. I. 27, 28), but not No. (3) or No. (4).

No. (2). This has the same advantage as No. (1). It has been used as a Definition by D'Alembert. From it, also, we can prove No. (5) without using any disputable Axiom; and it, also, fails to prove No. (3) or No. (4).

No. (3). This cannot be used, as a Definition, till we have proved that such Lines exist—which has not yet been done without employing some disputable Axiom. If Varignon, &c. mean this to be their Definition, they are assuming the existence of such Lines,—a very un-axiomatic Axiom. But, when once their existence has been granted, or proved, No. (4) can be deduced.

No. (4). This cannot be used, as a Definition, till we have proved that such Lines exist—which has not yet been done without employing some disputable axiom. When once their existence has been granted, or proved, No. (3) can be deduced. No. (4) has been used as a Definition by Wolf, Boscovich, T. Simpson, and Bonnycastle.

No. (5). This has the advantage that it is easy to prove (as in Euc. I. 27, 28) that such Lines exist. It has been used as a Definition by Euclid and a host of other geometers. It has, however, the enormous disadvantage that, whereas Nos. (3), (4), give us a unique Pair of Lines (e.g. given a Line and a Point not on it, we can prove that only one Line can be drawn, through the Point, such that the Pair shall have property No. (3)), No. (5) does not: on the contrary, given a Line and a Point not on it, a whole 'pencil' of Lines may be drawn, through the Point, and not meeting the given Line: all we need to do is to take care, after drawing one such Line, that the others shall make with it angles which are Infinitesimals of the second order with regard to a right angle.

We see, then, that the word "Parallels" has been already used with four, and possibly with five, different meanings: so that any fresh writer, who uses the word, is liable to be misunderstood unless he first defines it. The derivation of the word would seem to suggest No. (4) as its Definition; but Euclid's adoption of No. (5) has led to that being the popular meaning attached to the word.

It is easy, however, to avoid all this ambiguity by the use of new terms. Rejecting Nos. (1) and (2), as useless for the purpose of definition, we may call a Pair of Lines, which possesses

No. (3), "equiangulated";

No. (4), "equidistantial";

No. (5), "separational";

and thus avoid the dangerous word "parallel" altogether.

My own belief is that No. (3) is, on the whole, the property best adapted for practical use as a Definition.