In Euclidean geometry, equipollence is a homogeneous relation between directed line segments. Two segments are said to be equipollent when they have the same length and direction. Two equipollent segments are parallel but not necessarily colinear nor overlapping. For example, a segment AB, from point A to point B, has the opposite direction to segment BA; thus AB and BA are not equipollent.

A property of Euclidean spaces is the parallelogram property of vectors: If two segments are equipollent, then they form two sides of a (possibly degenerate) parallelogram:

If a given vector holds between a and b, c and d, then the vector which holds between a and c is the same as that which holds between b and d.

— Bertrand Russell, The Principles of Mathematics, page 432

The concept of equipollent line segments was advanced by Giusto Bellavitis in 1835. Subsequently, the term vector was adopted for a class of equipollent line segments. Bellavitis's use of the idea of a relation to compare different but similar objects has become a common mathematical technique, particularly in the use of equivalence relations. Bellavitis used a special notation for the equipollence of segments AB and CD:

The following passages, translated by Michael J. Crowe, show the anticipation that Bellavitis had of vector concepts:

Thus oppositely directed segments are negatives of each other: ${\displaystyle AB+BA\bumpeq 0.}$

The segment from A to B is a bound vector, while the class of segments equipollent to it is a free vector, in the parlance of Euclidean vectors.