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* '''Scholium.'''  The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, § 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic.  And of course there is the strange but true story of how the Turin machines of the 1840's prefigured the Turing machines of the 1940's [Men, 225-297].  At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].

* '''Scholium.'''  The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, § 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic.  And of course there is the strange but true story of how the Turin machines of the 1840's prefigured the Turing machines of the 1940's [Men, 225-297].  At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].

Given this language, the particular Example that I take up here can be described as the family of 4^th gear curves through E⁴''X'' = 〈''A'', d''A'', d²''A'', d³''A'', d⁴''A''〉.  These are the trajectories generated subject to the dynamic law d⁴''A'' = 1, where it is understood in such a statement that all higher order differences are equal to 0.  Since d⁴''A'' and all higher d^''k''''A'' are fixed, the temporal or transitional conditions (initial, mediate, terminal - transient or stable states) vary only with respect to their projections as points of E³''X'' = 〈''A'', d''A'', d²''A'', d³''A''〉.  Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of E³''X''.  It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure 16.

Given this language, the particular Example that I take up here can be described as the family of 4^th gear curves through E⁴''X'' = 〈''A'', d''A'', d²''A'', d³''A'', d⁴''A''〉.  These are the trajectories generated subject to the dynamic law d⁴''A'' = 1, where it is understood in such a statement that all higher order differences are equal to 0.  Since d⁴''A'' and all higher d^''k''''A'' are fixed, the temporal or transitional conditions (initial, mediate, terminal - transient or stable states) vary only with respect to their projections as points of E³''X'' = 〈''A'', d''A'', d²''A'', d³''A''〉.  Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of E³''X''.  It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure 16.

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