118. Formalizing Euclid's first axiom. Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný) ► JOHN CORCORAN AND DANIEL NOVOTNÝ, Formalizing Euclid's first axiom. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: corcoran@buffalo.edu Euclid's Elements divides its ten premises into two groups of five. The first five (postulates)-applying in geometry but nowhere else-are specifically geometrical. The first: "to draw a line from any point to any point"; the last: the parallel postulate. The second five (axioms) apply in geometry and elsewhere. They are non-logical principles governing magnitude types both geometrical (e.g., lengths, areas) and non-geometrical (e.g., durations, weights). Euclid called axioms koinai ennoia: koinai ("shared", "communal", etc.), ennoia ("designs", "thoughts", etc.). The first axiom is: Ta toi autoi isa kai allelois estin isa. Things that equal the same thing equal one another. One first-order translation in variable-enhanced English (cf. [2], p. 121) is: (1) Given two things x, y, if for something z, x and y equal z, then x equals y. Translation (1) overlooks Euclid's plural construction not limited to two. Second-order translations avoid that objection. (2) For any set S, if for something z, everything x in S equals z, then anything x in S equals anything y in S. Translations (1) and (2) are "too broad": they cover all magnitude types but by amalgamating them into a hodgepodge universe containing all magnitude types-a universe violating category restrictions and not itself a magnitude type. Translation (3) is a second-order axiom schema (cf. [1]) having one instance for each magnitude type. 'MAG' is placeholder for magnitude words such as length, area, etc. (3) For any set S, if for some MAG z, every MAG x in S equals z, then any MAG x in S equals any MAG y in S. We treat several other translations and formalizations. [1] JOHN CORCORAN, Schemata, Bulletin of Symbolic Logic, vol. 12 (2006), pp. 219–40. [2] ALFRED TARSKI, Introduction to Logic, Dover, New York, 1995.