1 7 0 D O C U M E N T 2 6 6 O C T O B E R 1 9 2 1

Let there be a system of points K in which the axioms I–V are

satisfied.[3]

At the

origin A, instead of uniform time t, a function is introduced and a system

of points is constructed according to Def.

3.[4]

In synchronism and the

lengths from A being equal are defined just as in K. If in axiom III is supposed

to be

valid,[5]

it is evidently sufficient that for a signal between two arbitrary points

in , the time be independent of . It suffices if one imagines the

points from speeding radially away from A, because a system winding in

upon itself cannot lead to the goal. The azimuth hence transforms identically.

The transformation formulas from K to then read for the radius (= distance from

A) and the time (velocity of light set = 1)

(1) In K: t, r

(2) In :

One can now require that for 2 points in , , the sig-

nal time be independent of and then obtain the following condition for

f(t). If , the inverse function of , then must have the follow-

ing characteristic: 3 arbitrary finite quantities are given; hence from

, 3 quantities: are determined by

argument function

If one now varies entirely

arbitrarily for constant , the thus emerging should satisfy the following

relation:

(3)

This is satisfied not just by a straight line but also by such that, depending

on the choice of f(t), is left over. This is the only possibility. If one plugs

f t =

K K

K

B C K

2 1

–

K K

K

1

2

-- - f t r + f t r – + =

1

2

-- - f t r + f t r – – = K ,

K

1 1

and

2 2 1 2

2 1

–

t = f t =

1 2 3

1 2 3

t 1

2 3

1

2

3

1

+

2

+

3

+

t =

t

1

+

t

2

+

t

3

+

1 2 3

1 2 2 3

2

1 3

– +

2 3 1

–

------------------------------------------------ - const

1 2 3

= =

1 – 1. +

t e =

logt =