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The Detection of Absolutes (3)Even if infinity and parallelity were found to be absolutely linked, it would not solve everything. Questions like those listed on the right remain to be answered in strictly projective terms. 
Some unanswered questions


In neither the Rotational nor the Translational case can one, by strictly projective methods, exactly divide an interval into N subintervals, where N is any odd integer equal to or greater than 3. Translational and Rotational stepmeasures may be made
Other examples follow below of a kind essentially similar to the foregoing: A real rope (such as might tether a real goat) is assumed to be, for all practical purposes, of constant length whether coiled or uncoiled (some might say that this is, “selfevident.” I say it is empirical, and hypothetical.) A haberdasher's cloth tapemeasure is taken to preserve calibration whether pulled taut, or wrapped. (Again some say this is, “selfevident”, and again I say it is empirical and hypothetical.) And this is the question (item 5, top right) Do strictlyprojectivelymade curves (given that such things altogether exist) preserve length as belts, ropes and tapemeasures appear to do? 
Re item 5, above, a pen is expected to deliver ink at a fairly constant rate of somany litres per unit distance of travel on the written surface, whether or not its path is curved. Can projective pens do this? Are there projective pens in sober reality, as opposed to fantasy? Can they write "joined up"? It is hard to see how they could  they would need to dispense dimensionless ink, indefinitely. (This question relates to the production of locii.) Do projectivelymade surfaces preserve measure like tapemeasures and ropes seem to do? As to that, just how are surfaces projectively made? Note that, if this can be managed at all, it cannot be managed quantitatively, only "incidentally"—that is, via the axioms of elementary incidence, and these alone. Can there be a projectivelybuilt “printer's cylinder”? How can such curves or surfaces be made  purely projectively  to roll on each other without slipping? In other words, can there be a projective evolute — “Evolute: a curve that is the locus of the centers of curvature of another curve (its involute).” There are geometric involutes, but, unlike the above, they are not metrically defined. Are projectivelymade curves continuous enough to completely support infinitesimals and differentials of the Newtonian/Leibnitzian kind? 

Home Site Map Tutorial Material Elements(1) Elements(2) Proscribed Ops Absolutes(1) Absolutes(2) Projection Perspective