The eighteenth meeting was held January 26-27, 2007, at Xavier University.  This was the first of two meetings planned to celebrate the Euler tricentennial.  We read selections (from v.1, Sections I.xxi - I.xxiii on logarithms; II.v on series; IV.viii on square roots of binomials; IV.x - IV.xv on cubics and quartics; from v.2, Sections IV - VII on solutions to Pell's equation) from Leonhard Euler's Vollständige Anleitung zur Algebra (St. Petersburg Akademie der Wissenschaften, 1770), available in a 1984 Springer reprint of John Hewlett's 1828 English translation as Elements of Algebra.