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The telling of mathematics in .NET Drawer barcode code39 in .NET The telling of mathematics




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The telling of mathematics generate, create barcode code39 none for .net projects iPad This second thought-experi 3 of 9 for .NET ment is very well known in the history of mathematics, as Archimedes here seems to be suggesting some kind of integration. The interesting aspect for us is the breathtaking ease with which this suggestion is broached.

Archimedes does not really imply that there is some kind of major thought-experiment involved and, for the na ve reader, this appears like a standard demonstrative move where a self evident truth is asserted. The effect in this case is quite different from that of Step a mentioned above. There, the reader was invited to share the delight in discovering, retrospectively, a beautiful correlation.

Here, a further effect is involved, intended for some very discerning readers only. For here Archimedes has laid a kind of trap. Is the argument at all valid That is, are we allowed to transform a result for arbitrary lines into a result concerning the objects through which the lines are drawn Are we allowed to conceive of a gure as if it were many lines glued together Indeed Archimedes suggests at the introduction to the Method that the results proved through the special method introduced in the treatise are in some sense not quite valid.

However he does not explain in the introduction what the special method is, and where its invalidity lies (I shall return to discuss this in greater detail below). Steps may well be understood to constitute the Method s Achilles heel. But if so, how carefully Archimedes avoided exposing it! The na ve reader would not notice any dif culty; the very sophisticated one will realize the dif culty but will also be at a loss to state Archimedes true position.

As against the puzzle of Apollonius vii. a tough but doable problem, a challenging exercise for the sophisticated geometrical reader this moment in the Method is a trap for the na ve and an insoluble riddle for the expert. We have seen three types of experience elicited by the telling of the proof: the challenging puzzle, meant for the erudite (Apollonius vii.

); the gasp of delight at a clever combination, retrospectively grasped (Archimedes Method , Step a); the trap or the riddle (the same, Steps ). These are but examples, of course. How representative are they I will not try to study this question in a quantitative way.

But the following should be noted. First, the bulk of the transmission of Greek mathematics in the West concentrated on Euclid s Elements, i.e.

the Greek work least marked for its use of the challenging puzzle (while some of its books, such as book x, are indeed dif cult, this is not because they set the readers explicit puzzles, but rather because of the complex, non-intuitive character of their results). This is obvious for a tradition made by compilers with little geometry and produced for the bene t of readers with little geometry. There is no fun in a puzzle you cannot solve.

As for the more advanced works of Greek geometry, many of them survive in Arabic only (such as. The telling of the proof Apollonius Conics, books VS .NET Code 39 Full ASCII v vii from which our example was taken or Diocles On Burning Mirrors, where we have seen an analysis with the synthesis left as an exercise). The major body of advanced Greek geometry transmitted in the West is that of the works of Archimedes, and it is striking to see how often the text is interpolated by readers who wished, precisely, to solve Archimedean puzzles.

(Because their solutions are often clumsy, we have reason to believe those are indeed interpolations rather than Archimedes own words.) The detection of such interpolations of course always conjectural was the main editorial task faced by Heiberg in his edition of Archimedes. And nally, it should be noted that the sense of frustration with the puzzling was not limited to medieval geometrical mediocrities.

Heath, for instance, commenting on Archimedes Method, sounds a familiar complaint, somewhat different from that of medieval readers but essentially related to it: . Nothing is more characteri stic of the classical works of the great geometers of Greece, or more tantalizing, than the absence of any indication of the steps by which they worked their way to the discovery of their great theorems. As they have come down to us, these theorems are nished masterpieces which leave no traces of any rough-hewn stage, no hint of the method by which they were evolved..

Heath s notion that Arch Code 3/9 for .NET imedes Method was, in his words, a sort of lifting of the veil is of course nonsense. If the veil was lifted, how come we have spent the last century arguing over what s underneath No, the Method is another, more subtle veil.

But this is not my point here. What I want to stress is that the modern frustration with the Greek synthetic presentation of results is precisely what I have described so far in this section: a frustration with a certain puzzling appearance, where the author attempts no pedagogic intervention explaining the signi cance of the ow of the text, instead demanding that the reader work out for himself the line of thought. The challenging puzzle, meant for the erudite, is standard in the more advanced parts of Greek geometry.

As for the trap or the riddle, this is perhaps rare, as indeed only some very special combinations will allow for it. It seems, however, to have been at the very least an Archimedean stylistic principle. We are reminded of course of the false claims sent out by Archimedes as challenges for the mathematical community, so as to trap his competitors (see my discussion of Spiral Lines, p.

above). But more can be. I discuss this editorial p ractice in detail in Netz forthcoming. Heath : . Heath repeats a topos: the persistent early modern myth of a hidden, ancient ars inveniendi (see e.

g. Bos : ). The myth must have been false but it is nevertheless revealing: Greek mathematics was perceived to hide a secret.

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