18. c│SPs.

'A Line, which has a point common with one of two separational Lines, has a point separate from the other.'

Lines, the second (not assumed to be straight,) be equidistant from the first; a third Line, which has a point common with, and diverges without limit from the first, is intersectional with the second.'

1 (1). Angles, which have the sides of the one separational from those of the other, each from each, are equal.'

equals with a transversal, are intersectional.' (N.B. This includes the following as a particular case.)

stPX.

Euclid's Axiom.

'Different Lines, which make with

a transversal 2 int Ls < 2 Ls, are intersectional.'

3. sTm real.

'If there be given a Line and a point not on it; a Line can be drawn, through the given point, and such that it and the given Line make equals with any transversal.' 4. stm P Tm 'Different Lines, which make equal

Zs with a certain transversal, do so with any transversal.'

5. stm PTm'

'Different Lines, which make unequal

/s with a certain transversal, do so with any transversal.'

6. SPE.

7. sf P X.

'Separational Lines are equidistant.'

Simpson's Axiom.

'Different Lines, of which one contains

2 points unequally distant from the other, are intersectional.'

8. sE real. 'If there be given a Line and a point not on it; a Line can be drawn, through the given point, and such that it and the given Line shall be equidistant from each other.' (N.B. This includes the following as a particular case.)

Clavius' Axiom.

'If there be given a Line and a point not on it; a Line can be drawn, through the given point, equidistant from the given Line.

9. se P Tm

'Different Lines, of which one has 2 points on the same side of the other and equidistant from it, make equal As with any transversal.'

ន

10. stm PF.

'Different Lines, which make unequal Zs with a transversal, are such that all points on each, which lie on the same side of the other, are unequally distant from it.'

11. stm P E.

'Different Lines, which make equal

Zs with a transversal, are equidistant.'

12. sf P Tm

'Different Lines, of which one has 2

points unequally distant from the other, make unequal Zs with any transversal.'

13. se PE.

'Different Lines, of which one has 2

points on the same side of the other and equidistant from it, are equidistant from each other.'

14. sf P F.

'Different Lines, of which one has 2

points unequally distant from the other, are such that all points on each, which lie on the same side of the other, are unequally distant from it.'

15 (1). SSPX SSPX T.

separational from a third Line, are not intersectional.'

Lines, which are

'Lines,

15 (2). SSPC; SS unique. which have a common point and are separational from a third Line, are coincidental.' This 'If there be given a Line and

a point not on it; only one Line can be drawn, through the given point, separational from the given Line.'

15 (3). 8|SS PS│.

'Different Lines, which

are separational from a third Line, are separational from each other.'

Playfair's Axiom.

16 (1). X| | PSSY.

'Intersectional Lines

cannot both be separational from a third Line.'

16 (2). X|S| P||x.

'A Line, which is in

tersectional with one of two separational Lines, is intersectional with the other also.'

The Ahmedabad Axiom.

17. 'A Line cannot recede from and then approach another; nor can one approach and then recede from another on the same side of it.'

18 (1).

'An ext 4 of a A = 2 int opp 4s.' 18 (2). The 3 4s of a ▲ 2 Ls.'