experience, often longer than is held judicious by after times. We only desire to avail ourselves of this feeling until the book is produced which is to supplant Euclid; we regret the manner in which it has allowed the retention of the faults of Euclid; and we trust the fight against it will rage until it ends in an amended form of Euclid.

APPENDIX III.

The enunciations of Tables I-IV, stated in full: with references to the various writers who have assumed or proved them.

[N.B.-Contranominals are bracketed. A''assumed ';

'P''proved.']

TABLE I. [See p. 32.]

'Different

Lines have not two common points; i. e. they have all points, but one, separate.' (Nothing is asserted of the excepted point.) This Through two given points only one Line can be drawn.' 2 (2). c8 PX. Lines, which have a common point and a separate point,' (or 'Different Lines, which have a common point,') are intersectional; i. e. have one point common and all others separate.'

3. ct PC.

m

'Lines, which have a common point and

make equals with a transversal, are coincidental.'

4. XP Tm'

'Intersectional Lines make unequal s

with any transversal.' (N.B. This is equivalent to XP Ta T, T, Ta", Te", and T': the full statements of which are as follows:

Té

XP Td

Intersectional Lines make unequal

alt s with any transversal.'

XP Te

'Intersectional Lines make unequal

'Intersectional Lines make with any

ext and int opp Zs with any transversal.'
XP T.
transversal two int 4s two right angles.

XP T

to one of two
other.'

'Any transversal, which is perpendicular intersectional Lines, is not so to the

XP Tď:

'Intersectional Lines make unequal

That is

alt s with any transversal, that which is further from
their point of intersection being > the other.'
(since two intersectional Lines and a transversal constitute
a triangle) 'An ext ≤ of a ▲ > the int alt .'

XP T.
PS.

5. stm

'Two int s of a ▲ < 2 Ls.'

'Different Lines, which make equal s

with a transversal, are separational.' (N.B. This is equivalent

to 'stɑ, or ste, or st¿, or str, P S.')

6. ce P C.

'Lines, which have a common point and of which one contains 2 points on the same side of the other and equidistant from it, are coincidental.'

7. X PF.

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

8. se PS.

'Different Lines, of which one contains 2 points on the same side of the other and equidistant from it, are separational.'

9. SS real.

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

10. XP V.

'Intersectional Lines diverge without limit; i. e. a point can be found on each, whose distance from the other shall exceed any assigned length.'

13. sD' real.

'If there be given a Line and a point

not on it; a Line can be drawn, through the given point, and having a direction different from that of the given Line.'

14. ce P D.

'Lines, which have a common point and of which one contains 2 points on the same side of the other and equidistant from it, have the same direction.'

15 (1). D' PS-1•

'Lines, which have different direc

tions, have all points, but one, separate.' (Nothing is asserted

of the excepted point.)

15(2). c D'PX.

and different directions, are intersectional.'

P

'Lines, which have a common point

"A Line, which has a point

16. Cc common with one of two coincidental Lines, has a point common with the other also.'

17. ccSPX T ;=PXX.

'A Line,

which has a point common with one of two separational Lines, and also a point common with the other, intersects the one.' This 'it intersects both.'