Page:CarmichealPostulates.djvu/23

which the relative velocity of two systems may approach but can never reach. This may be formulated in the following theorem:

Theorem X. The velocity of light is a maximum which the velocity of a material system may approach but can never reach (MVLR).

It should be pointed out that this theorem may also be proved directly from theorem IX, as one can readily show. This fact will be useful in the next section.

§12. Logical Equivalents of the Postulates. — We shall now show that theorem IX. is in a certain sense a logical equivalent of R. From IX. it follows, as we have seen in theorem X., that the velocity of a material body is less than the velocity of light. But the source of light is always a material body; and therefore no light source can have a velocity as great as that of light. Now, the following is a natural hypothesis:

Postulate B. The velocity of the light source cannot add^[1] to the velocity of light a greater velocity than that of the source itself. Likewise, the velocity of a system of reference cannot add to the velocity of light a greater velocity than that of the source itself.

Now, from theorem IX. it follows that if any velocity less than that of light is compounded with that of light the resultant is the velocity of light. Hence, if we assume theorem IX. and postulate B we can conclude as a consequence postulates $R'$ and $R''$. Hence we have the following result:

Theorem XI. Postulates (MVLB) and theorem IX. are a logical equivalent of postulates (MVLR).

That is, in our system of postulates R may be replaced by theorem IX. and postulate B, and the resulting total body of postulates and theorems will be unaltered.

Now theorem IX. was proved by means of formulae (B) alone; and formulae (B) are a direct consequence of formulae (A) above. Hence postulate R may be proved solely from postulate B and formulae (A) if these are assumed to be true. Further, the third and fourth equations in (A) are equivalent to postulate L. We shall show that a special case of the first two formulae (A) is postulate V. Putting x' = 0 we have x/t = v. That is, to an observer on S the system $S'$ appears to move with the velocity v. Now the first two equations in ($A_{1}$) may be obtained algebraically by solving the first two in (A) . Then in the second equation of ($A_{1}$) put x = 0; thus we have x't' = -v. That is, to an observer on $S'$ the system S appears to move with the velocity -v. These two results together constitute our postulate V. Combining the several conclusions thus reached we have the following theorem:

1. Addition is defined by saying that the sum of two velocities is the result of compounding them.