Equipollence (geometry)
In Euclidean geometry, equipollence is a binary relation between directed line segments. A line segment "AB" from point "A" to point "B" has the opposite direction to line segment "BA". Two parallel line segments are equipollent when they have the same length and direction. Parallelogram property. A definitive feature of Euclidean space is the parallelogram property of vectors: If two segments are equipollent, then they form two sides of a parallelogram: History. 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: formula_2 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. Extension. Geometric equipollence is  also used on the sphere: On a great circle of a sphere, two directed circular arcs are equipollent when they agree in direction and arc length. An equivalence class of such arcs is associated with a quaternion versor