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. 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: 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. 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.

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