Definition:Reciprocal Relation

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In the words of Euclid:

Two figures are reciprocally related when there are in each of the two figures antecedent and consequent ratios.

(The Elements: Book $\text{VI}$: Definition $2$)

Historical Note

Many commentators on Euclid's The Elements either omit this or sideline it, as in its original Greek form (a translation of which is given in this definition) it appears to be meaningless.

As noted by Thomas L. Heath in his Euclid: The Thirteen Books of The Elements:

No intelligible meaning can be attached to "antecedent and consequent ratios" here; the sense would require rather "an antecedent and a consequent of (two equal) ratios in each figure."

Candalla and Peyrard read ratios as terms of ratios. Camerer makes a similar interpretation.

Robert Simson believes that this definition may have originated with Heron. He suggests that this definition be deleted, and perhaps substitute this definition instead:

Two magnitudes are said to be reciprocally proportional to two others when one of the first is to one of the other magnitudes as the remaining one of the last two is to the remaining one of the first.

For this definition, all the magnitudes must be of the same kind for it to make sense.

This definition is never actually used throughout the rest of The Elements, and may in fact be removed (as Heath effectively does, relegating it to a footnote).

Also see