` I've found a misplaced reference showing that ``generatics`` did not start with me in content, only in name. The book, ``Learn from the Masters``, edited by Dwetz, Fauvel, Bekken, Johannsson, Katz (Mathematical Association of America, 1991), says on p. 286, ``It was not until 1894 that J. Tannery introduced the arithmetic of rationals as pairs [vectors] of integers.`` (Jules Tannery (1818-1910), French, is cited ONLINE.)In 1957, I received a grant from the National Science Foundation to organize the first NSF Workshop in Puerto Rico, planned for high math school teachers (some from States). I taught ``Foundations of Mathematics``. I was sent papers (later lost) from previous Workshops. One set described Tannery's work and Hamilton's formulation of complex numbers as pairs or vectors of reals. The formulator filled in by deriving integers from pairs or vectors of natural numbers. The latter shows how ``the law of signs`` derives from CLOSURE on DEFINED DIFFERENCES (DDs) of naturals  (a - b), s.t. subtrahend is not greater than minuend, hence, a natural number. Critical is multiplication law for DD. From standard multiplication algorithm, find that, for DDs, (a - b) * (c - d) = (a*c + (-b)*(-d)) + (a*(-d) + (-b)*c). Applying, 10 = 5*2 = (9 - 4)*(2 - 0) = 18 + (-4)*(2) + 0 = 10, hence, (-4)*(2) must act as a subtrahend -8, leading to ``negative times positive equals negative`` rule. Applying product rule to 30 = 6*5 = (9 - 3) - (7 - 2) = (63 + (-3)*(-2)) - (18 + 21) = (63 - 39) + (-3)*(-2) = 24 + x = 30, hence, x = 6 = (-3)*(-2), leading to ``negative times negative equals positive`` rule. This is forced by CLOSURE on DDs. However, in the ``Generating arithmetic`` article which I initiated, some one put in that CLOSURE is a concept from category theory, very advanceed math. Yet, the above book, on p. 260, says, ``For Galois (1830), Jordan (1870), and even in Klein's ``Lectures on the Icosohedron`` (1884), groups were defined by the one axiom of closure. The other axioms were implicit in the context of their discussions  finite groups of transformations.`` So CLOSURE goes back at least to 1830.Jonhays0`