Measuring a segment is asking how many times a unit fits in it. The question is simple; the answer took centuries to complete.

Who discovered

Measuring lengths predates writing, but the person who solved the theoretical problem was Eudoxus of Cnidus (c. 408–355 BC). The Pythagoreans had discovered that the diagonal and the side of a square are incommensurable: there is no unit, no matter how small, that fits an integer number of times in both. Eudoxus created a theory of proportions that works even in this case, recorded in book V of The Elements.

The story

The units started with the human body: span, foot, step, fathom. In Egypt, the royal cubit (about 52 cm) had stone patterns used in large buildings. Each region had its own measurements, which complicated trade for centuries.

Scene from an Egyptian tomb: measurers stretch a knotted rope to measure a field of grain.
Scene from an Egyptian tomb: measurers stretch a knotted rope to measure a field of grain.Collection of the Euler’s Eyes blog.

In 1791, during the French Revolution, the meter was proposed as the ten millionth part of the distance from the North Pole to the equator. Since 1983, it is defined by the distance light travels in a vacuum in 1/299 792 458 of a second. Brazil adopted the metric system in 1862.

In everyday life

  • Measuring tape and ruler. Every measurement is a comparison with the unit recorded on the instrument.
  • Inches of screens. A “50 inch” TV has a diagonal of about 127 cm; the inch measures 2,54 cm.
  • Odometer. The car’s dashboard counts how many revolutions the wheel has made and multiplies it by the length of each revolution.
  • Map scales. On a scale 1 : 50 000, each centimeter on paper represents 500 m on the ground.

Sources: Eudoxo (MacTutor) · Elements, book V · BIPM: definition of the meter.

Theory and constructions

The measurement of a segment $\overline{AB}$ is indicated by $m(\overline{AB})$ or, simply, by $AB$.

DefiniçãoMeasurement (length)

The measure of a non-zero segment is a positive real number associated with it such that:

  1. congruent segments have equal measurements and, conversely, segments with equal measurements are congruent: $\overline{AB} \equiv \overline{CD} \iff m(\overline{AB}) = m(\overline{CD})$;
  2. if one segment is larger than another, its measurement is larger: $\overline{AB} > \overline{CD} \iff m(\overline{AB}) > m(\overline{CD})$;
  3. the sum segment corresponds to the sum of the measurements: $\overline{RS} = \overline{AB} + \overline{CD} \iff m(\overline{RS}) = m(\overline{AB}) + m(\overline{CD})$.

The measurement of a segment is called length of the segment.

In general, we measure a segment by comparing it with another segment chosen as unit. The usual unit is the meter (m), with its multiples — decameter (dam), hectometer (hm) and kilometer (km) — and submultiples — decimeter (dm), centimeter (cm) and millimeter (mm).

Change the size of the unit $u$: the measurement of $\overline{AB}$ changes, but there is always a multiple of $u$ greater than $\overline{AB}$.