*Reference: **Einstein’s 1920 Book*

*This paper presents Part 1, Chapter 6 from the book **RELATIVITY: THE SPECIAL AND GENERAL
THEORY by A. EINSTEIN.
The contents are from the original publication of this book by Henry
Holt and Company, New York (1920).*

*The paragraphs of the original
material (in black) are accompanied by brief comments (in color) based on the present
understanding. Feedback on these comments is appreciated.*

*The heading below is linked to
the original materials.*

.

## The Theorem of the Addition of Velocities Employed in Classical Mechanics

Let
us suppose our old friend the railway carriage to be travelling along the rails
with a constant velocity ** v**, and that a man traverses the
length of the carriage in the direction of travel with a velocity

**. How quickly, or, in other words, with what velocity**

*w***does the man advance relative to the embankment during the process? The only possible answer seems to result from the following consideration: If the man were to stand still for a second, he would advance relative to the embankment through a distance**

*W***equal numerically to the velocity of the carriage. As a consequence of his walking, however, he traverses an additional distance**

*v***relative to the carriage, and hence also relative to the embankment, in this second, the distance**

*w***being numerically equal to the velocity with which he is walking. Thus in total he covers the distance**

*w***relative to the embankment in the second considered. We shall see later that this result, which expresses the theorem of the addition of velocities employed in classical mechanics, cannot be maintained; in other words, the law that we have just written down does not hold in reality. For the time being, however, we shall assume its correctness.**

*W = v + w**The
velocity depends on inertia. If v = F(I _{v}), and w = F(I_{w});
then W = F(I_{v} + I_{w})*

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## FINAL COMMENTS

*The unconstrained uniform velocity of a body increases as its inertia decreases. But for any appreciable increase in velocity in the material domain, the change in inertia is infinitesimally small.*

*Therefore, velocities of different inertial systems in the material domain can simply be added because their inertia can be assumed to be constant. *

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