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.
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:
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.
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:
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.
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
Properties of the equivalence classes of equipollent segments can be abstracted to define affine space:
If A is a set of points and V is a vector space, then (A, V) is an affine space provided that for any two points a,b in A there is a vector in V, and for any a in A and v in V there is b in A such that and for any three points in A there is the vector equation
Evidently this development depends on previous introduction to abstract vector spaces, in contrast to the introduction of vectors via equivalence classes of directed segments. [3]
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Giusto Bellavitis was an Italian mathematician, senator, and municipal councilor. According to Charles Laisant,
