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Mathematical Treasure: Clavius' Extended Euclid

Frank J. Swetz (Pennsylvania State University)

This is the title page of Christopher Clavius' Euclidis Posteriores Libri IX from 1654; the original edition appeared in 1607. In translation this page reads approximately as

Beyond Euclid’s Nine Books.

Included is Book XVI, On the Comparison of Regular Solids of all Kinds, all clearly demonstrated and illustrated with the most accurate commentary; now published in quarto and enriched with much material not contained in the first edition.

By the author Christopher Clavius

Bamburg, Society of Jesus

Frankfurt: Published by the heirs of Jonah Rocia in the year 1654

The publisher’s seal is particularly interesting showing the biblical scene of a whale vomiting Jonah onto the shore. The whale appears to be an imaginary creature conceived from the mind of the artist.

The Posteriores appears to be an extension of Clavius’ Euclidis Elementorum of 1574, in which he considered fifteen books of Euclid. Here we find a sixteenth book added by François de Foix-Candale (1512-1594). De Foix was a French mathematician and alchemist and a noted geometer who established a Chair of Geometry at the University of Bordeaux in 1591. The three books comprising an addendum to Euclid’s Elements, Books XIV-XVI, all consider the regular polygons and their properties.

Page 275 of Posteriores discusses two propositions of Book X:

Proposition 35: To find two straight lines in square which make the sum of the squares on them medial and the rectangle contained by the medial and moreover incommensurable with the sum [of] the squares on them.

Proposition 36: If two rational straight lines commensurable in square only are added together, then the whole is irrational; let it be called binominal.


Professor Michael Twomey of Ithaca College, Ithaca, New York, assisted with the Latin translation.

The images above are provided courtesy of Archives and Special Collections, Dickinson College, Carlisle, Pennsylvania.

Frank J. Swetz (Pennsylvania State University), "Mathematical Treasure: Clavius' Extended Euclid," Convergence (August 2015)