Thus, between events E0 and E1, A advances by \(\color{Green}{t_1}\) and B by \(\color{Blue}{t'_1 = a t_1}\) by (1). Therefore
$$ \frac{\color{Green}{\text{rate of A}}}{\color{Blue}{\text{rate of B}}} = \frac{\color{Green}{t_1}}{\color{Blue}{a t_1}} = \frac{1}{a} > 1 \qquad \text{(3)} $$
...
Thus, between events E0 and E2, B advances by \(\color{Brown}{t'_2}\) and A by \(\color{Red}{t_2 = a t'_2}\) by (2). Therefore
$$ \frac{\color{Red}{\text{rate of A}}}{\color{Brown}{\text{rate of B}}} = \frac{\color{Red}{a t'_2}}{\color{Brown}{t'_2}} = a < 1 \qquad \text{(4)} $$
Equations (3) and (4) are contradictory: hence the theory requiring them must be false.
(end quote)
Dingle should have written as follows:
(start correction)
Thus, between events E0 and E1, A, which is not present at both
events, advances by \(\color{Green}{t_1}\) and B, which is present at both events,
by \(\color{Blue}{t'_1 = a t_1}\) by (1). Therefore
$$ \frac{\color{Green}{\text{rate of clock not present at both events E0 and E1}}}{\color{Blue}{\text{rate of clock present at both events E0 and E1}}} = \frac{\color{Green}{\text{coordinate time of E1}}}{\color{Blue}{\text{proper time of E1}}} = \frac{\color{Green}{\text{rate of A}}}{\color{Blue}{\text{rate of B}}} = \frac{\color{Green}{t_1}}{\color{Blue}{a t_1}} = \frac{1}{a} > 1 \qquad \text{(3)} $$
...
Thus, between events E0 and E2, B, which is not present at both
events, advances by \(\color{Brown}{t'_2}\) and A, which is present at both events,
by \(\color{Red}{t_2 = a t'_2}\) by (2). Therefore
$$ \frac{\color{Brown}{\text{rate of clock not present at both events E0 and E2}}}{\color{Red}{\text{rate of clock present at both events E0 and E2}}} = \frac{\color{Brown}{\text{coordinate time of E2}}}{\color{Red}{\text{proper time of E2}}} = \frac{\color{Brown}{\text{rate of B}}}{\color{Red}{\text{rate of A}}} = \frac{\color{Brown}{t'_2}}{\color{Red}{a t'_2}} = \frac{1}{a} > 1 \qquad \text{(4)} $$
Equations (3) and (4) are consistent and say that any event's coordinate time is always larger than its proper time:
hence there is no reason to say that the theory requiring them must be false.
(end correction)
Compare with the following simple analogy. This is how Dingle should have found a trivial contradiction in the laws of geometry and perspective.
\( \color{Green}{\text{X}}\color{Black}{\text{ ———————————————— }}\color{Blue} {\text{Y}} \qquad \) X and Y are twin brothers
\( \color{Black}{\text{A}}\color{Black}{\text{ ———————————————— }}\color{Black}{\text{B}} \qquad \) A and B are observers facing one another with some distance between them
\( \color{Red} {\text{P}}\color{Black}{\text{ ———————————————— }}\color{Brown}{\text{Q}} \qquad \) P and Q are twin sisters
So X and P are standing nearby next to A, and Y and Q are standing nearby next to B.
A looks through a gap between her fingers at the twin brothers X and Y, and she notices that X's gap (nearby A) is twice as large as Y's (nearby B). Therefore for A it is true that
$$ \frac{\color{Green}{\text{gap near A}}}{\color{Blue}{\text{gap near B}}} = 2 \qquad \text{(3)} $$
B looks through a gap between his fingers at the twin sisters P and Q, and he notices that P's gap (nearby A) is half as large as Q's (nearby B). Therefore for B it is true that
$$ \frac{\color{Red}{\text{gap near A}}}{\color{Blue}{\text{gap near B}}} = \frac{1}{2} \qquad \text{(4)} $$
Clearly, equations (3) and (4) are contradictory: hence the theory of perspective behind them must be false.
Or is it?
Of course it isn't. The equations are just poorly expressed, and should be formulated as follows:
For A, looking at twin brothers X and Y, it is true that
$$ \frac{\color{Green}{\text{X-gap near A seen by A}}}{\color{Blue}{\text{Y-gap near B seen by A}}} = \frac{\color{Green}{\text{local gap}}}{\color{Blue}{\text{remote gap}}} = 2 \qquad \text{(3)} $$
For B, looking at twin sisters P and Q, it is true that
$$ \frac{\color{Brown}{\text{Q-gap near B seen by B}}}{\color{Red}{\text{P-gap near A seen by B}}} = \frac{\color{Brown}{\text{local gap}}}{\color{Red}{\text{remote gap}}} = 2 \qquad \text{(4)} $$
Clearly, equations (3) and (4) are consistent: hence there is no reason to think that the theory of perspective behind them must be false.
The trick to make this happen: proper understanding, proper labelling, proper expressing. Trivial.
