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 directed line segments are equipollent when they have the same length and direction.
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:
- Equipollences continue to hold when one substitutes for the lines in them, other lines which are respectively equipollent to them, however they may be situated in space. From this it can be understood how any number and any kind of lines may be summed, and that in whatever order these lines are taken, the same equipollent-sum will be obtained...
- In equipollences, just as in equations, a line may be transferred from one side to the other, provided that the sign is changed...
Thus oppositely directed segments are negatives of each other:
- The equipollence where n stands for a positive number, indicates that AB is both parallel to and has the same direction as CD, and that their lengths have the relation expressed by AB = n.CD.
- Giusto Bellavitis (1835) "Saggio di applicazioni di un nuovo metodo di Geometria Analitica (Calcolo delle equipollenze)", Annali delle Scienze del Regno Lombardo-Veneto, Padova 5: 244–59.
- Giusto Bellavitis (1854) Sposizione del Metodo della Equipollenze, link from Google Books.
- Giusto Bellavitis (1858) Calcolo dei Quaternioni di W.R. Hamilton e sua Relazione col Metodo delle Equipollenze, link from HathiTrust.
- Charles-Ange Laisant (1887) Theorie et Applications des Equipollence, Gauthier-Villars, link from University of Michigan Historical Math Collection.
- Lena L. Severance (1930) The Theory of Equipollences; Method of Analytical Geometry of Sig. Bellavitis, link from HathiTrust.