At the root of geometry one has primitive notions such as length. Since line segments may have different lengths, an early step in building a foundation for geometry consists of inventing a systematic way to compare distinct lengths. Based on experiment, presumably, the Greeks originally embraced the false conclusion that they could always multiply two given lengths up to one common length. By then observing the multipliers they had used they thought they could set any two lengths in a fixed whole number ratio. Even though this conclusion turns out false, we can understand easily how the Greeks came to believe it.
All practical measurement has only a limited tolerence.
This renders a sufficiently small discrepancy between two lengths undetectable. Though we cannot always multiply two distinct lengths up to a single common length, we can indeed do this up the limits of measurement. This explains why, in this case, experiment tends to confirm something that mathematics refutes. According to legend, the discovery of irrational numbers first occurred among the Pythagoreans. Indeed from the famous theorem of Pythagoras we can compute that the length of the hypoteneuse of an isoceles right triangle with unit sides equals the square root of 2.