Euclid's Elements

I S S N 2 3 4 7-1 9 2 1 V o l u m e 1 3 N u m b e r 4 J o u r n a l o f A d v a n c e i n M a t h e m a t i c s 7308 | P a g e A u g u s t , 2 0 1 6 h t t p s : / / c i r w o r l d. c o m / ABSTRACT This paper is devoted to exposition of a provable classical solution for the ancient Greek " s classical geometric problem of angle trisection [3]. (Pierre Laurent Wantzel, 1837), presented an algebraic proof based on ideas from Galois field showing that, the angle trisection solution correspond to an implicit solution of the cubic equation; , which he stated as geometrically irreducible [23]. The primary objective of this novel work is to show the possibility to solve the trisection of an arbitrary angle using the traditional Greek " s tools of geometry, and refute the presented proof of angle trisection impossibility statement. The exposed proof of the solution is theorem , which is based on the classical rules of Euclidean geometry, contrary to the Archimedes proposition of using a marked straightedge construction [4], [11].

We argue against the claim that the employment of diagrams in Euclidean geometry gives rise to gaps in the proofs. First, we argue that it is a mistake to evaluate its merits through the lenses of Hilbert’s formal reconstruction. Second, we elucidate the abilities employed in diagram-based inferences in the Elements and show that diagrams are mathematically reputable tools. Finally, we complement our analysis with a review of recent experimental results purporting to show that, not only is the Euclidean diagram-based practice strictly regimented, it is rooted in cognitive abilities that are universally shared

► JOHN CORCORAN AND JUSTIN LEGAULT, Reformalizing Euclid’s first axiom: Euclitivity.
Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA
E-mail: [email protected]
  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 also are non-logical principles governing magnitude types both geometrical (e.g., lengths, areas) and non-geometrical (e.g., durations, weights). No logical axioms or rules are stated or even mentioned. 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 (1) was reported in [2].

(1) Given two things x, y, if for something z, x and y equal z, then x equals y.

Translation (1) is closer to Euclid’s meaning than (2), which resembles but doesn’t imply transitivity, as reported in [3].

(2) Given any things x, y, z, if x equals z and y equals z, then x equals y.

We recognize its Euclidean origin as well as its resemblance to and distinctness from transitivity by calling is Euclitivity, specifically for reasons given below, simple Euclitivity. The idea of (2) is used widely in the literature ([1], p. 121).
  Translations (1) and (2) both overlook Euclid’s reflexive construction ‘equal one another’.  Translation (3), which we call conjunctive Euclitivity, avoids that objection.

(3) Given any things x, y, z, if x equals z and y equals z, then x equals y and y equals x.

It is easy to see that translations (1), (2), and (3) are logically equivalent. Thus by [3] none of them imply reflexivity, symmetry, or transitivity.
  We treat several other translations and formalizations.
  [1] ALFRED TARSKI, Introduction to Logic, Dover, 1995. 
  [2] JOHN CORCORAN AND DANIEL NOVOTNÝ, Formalizing Euclid’s first axiom, this BULLETIN, vol. 20 (2014), pp. 404–5. 
[3] JOHN CORCORAN AND JOSÉ MIGUEL SAGÜILLO, Euclid’s weak first axiom, this BULLETIN, vol. 20 (2014), p. 405. 
HOW CAN THIS BE IMPROVED?