# Non-hypothetical problem of using rulers to measure lengths

Consider two factories on earth's Equator producing rulers of the same length. One factory produces rulers made of wood and the other one produces rulers made of aluminium. Both types of rulers are exported all over the world and used to measure lengths.

Here's the problem:

If I take both types of rulers and make a trip to the north pole, they will no longer have the same length. If I take the wooden ruler as reference, I conclude that the one made of aluminium shrank, but if I take the latter as reference, I conclude that the wooden ruler expanded. Even worse, a third reference leads to the conclusion that the other rulers both changed in size. It seems impossible to tell which conclusion is right. However, tables of thermal expansion coefficients give a definite answer - how is that possible?

The solution (?)

Assuming that temperatures we perceive as different correpond to different ratios of the lengths of the two types of rulers, we can define temperature as that ratio, measure the length of one ruler in dependence of temperature (defining the length of one ruler at some specific temperature as unit of length) and use the resulting graph together with both types of rulers to measure lengths everywhere on the world.

Does this procedure sound legit? Do you know another (better) solution?

• Famous rule in engineering design: always have an odd number of redundancies, so that the final "vote" will never be a tie. In your case, it would appear that measuring & calibrating both rulers (vs temperature) before leaving the factory would be in order. Jan 4 at 13:36
• @CarlWitthoft I like that rule :D But wouldn't I have two "votes" if I calibrated both rulers since I could use both rulers to measure a length? Jan 4 at 13:43
• Or did I missunderstand what you mean by votes/redundancies? Jan 4 at 13:50
• OT, but for a useful application of your problem, check out Thermocouples. Jan 4 at 14:04
• @simonatrcl Interesting, thank you for the comment :) Jan 4 at 14:07