*Project Gutenberg*. Public domain.

December, 1916

A. *Einstein*

## PART I

**THE SPECIAL THEORY OF RELATIVITY**

## PHYSICAL MEANING OF GEOMETRICAL PROPOSITIONS

In your schooldays most of you who read this book made acquaintance with the noble building of Euclid’s geometry, and you remember — perhaps with more respect than love — the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers. By reason of our past experience, you would certainly regard everyone with disdain who should pronounce even the most out-of-the-way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if some one were to ask you: “What, then, do you mean by the assertion that these propositions are true?” Let us proceed to give this question a little consideration.

Geometry sets out form certain conceptions such as
“plane,” “point,” and “straight
line,” with which we are able to associate more
or less definite ideas, and from certain simple propositions
(axioms) which, in virtue of these ideas, we are inclined
to accept as “true.” Then, on the
basis of a logical process, the justification of which
we feel ourselves compelled to admit, all remaining
propositions are shown to follow from those axioms,
*i.e*. they are proven. A proposition is then
correct ("true”) when it has been derived in
the recognised manner from the axioms. The question
of “truth” of the individual geometrical
propositions is thus reduced to one of the “truth”
of the axioms. Now it has long been known that
the last question is not only unanswerable by the
methods of geometry, but that it is in itself entirely
without meaning. We cannot ask whether it is
true that only one straight line goes through two
points. We can only say that Euclidean geometry
deals with things called “straight lines,”
to each of which is ascribed the property of being
uniquely determined by two points situated on it.
The concept “true” does not tally with
the assertions of pure geometry, because by the word
“true” we are eventually in the habit of
designating always the correspondence with a “real”
object; geometry, however, is not concerned with the
relation of the ideas involved in it to objects of
experience, but only with the logical connection of
these ideas among themselves.