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(2)

Having a common point and a separate point;

(3)

Having„ no common point.

Four kinds of 'properties'; ....................................................................................................................................................................................................................................................

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(1)

common or separate points;

(2)

equality, or otherwise, of angles made with transversals;

(3)

equidistance, or otherwise, of points on the one from the others;

(4)

direction.

Conventions as to language ....................................................................................................................................................................................................................................................

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Propositions divisible into two classes:— ....................................................................................................................................................................................................................................................

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(1)

Deducible from undisputed Axioms;

(2)

Deducible from„ disputable Axioms„

Three classes of Pairs of Lines:— ....................................................................................................................................................................................................................................................

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(1)

Coincidental;

(2)

Intersectional;

(3)

Separational.

Subjects and predicates of Propositions concerning these three classes:—

Coincidental ....................................................................................................................................................................................................................................................

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Intersectional ....................................................................................................................................................................................................................................................

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Separational ....................................................................................................................................................................................................................................................

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Table I. Containing twenty Propositions, of which some are undisputed Axioms, and the rest real and valid Theorems, deducible from undisputed Axioms ....................................................................................................................................................................................................................................................

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Subjects and predicates of other propositions concerning Separational Lines ....................................................................................................................................................................................................................................................

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Table II. Containing eighteen Propositions, of which no one is an undisputed Axiom, but all are real and valid Theorems, which, though not deducible from undisputed Axioms, are such that, if any one be admitted as an Axiom, the rest can be proved ....................................................................................................................................................................................................................................................

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