web pages barcode The telling of mathematics in .NET Receive 39 barcode in .NET The telling of mathematics

The telling of mathematics using barcode integrating for none control to generate, create none image in none to make barcode reader in parabola KBM, with none none axis AZ, and let half the parameter of the squares on the ordinates be line BH . . .

Here is Diocles decisive choice. The main point is the following. A Greek mathematical proposition is typically introduced by a general enunciation, where the theorem to be proved, or problem to be solved, is asserted in its most general form ( in every triangle .

. . ).

Then comes a speci c construction in the diagram, where once again the goal of the proof is asserted explicitly, though this time in the particular terms ( I say that the line AB is . . .

). Only then comes the proof proper. This is the typical structure of a Greek mathematical proposition (typically, but not universally applied see Netz a).

Its narrative consequence is obvious: the reader has a clear sense of the goal of the demonstrative discourse, so that he is able to position the steps of the argument in relation to this goal. The author may then be more explicit in which case the parsing of the argument, in terms of the contribution it makes towards the goal, becomes obvious or less explicit, where such parsing becomes more challenging and potentially interesting from a narrative point of view. Diocles, in a move rare in Greek mathematics, in this treatise avoids both general enunciation as well as particular setting out.

The propositions simply move from construction to proof. Only at the end of each proof are consequences drawn, not quite in general geometrical terms but in optical and then practical terms having to do with the construction of the burning mirror. Thus, following a rather complex proof through which, I repeat, the reader has no idea what the proof is about! the text reaches the following passage which I quote extensively (slightly expanding Toomer s translation in pp.

; see g. ):. Microsoft Official Website So the angle T, rem ainder, is equal to the angle PQ, remainder. So when line SQ meets line AQ it is re ected to point D, forming equal angles, PQ and T, between itself and the tangent AQ. [End of proof, though the reader has no reason to suspect this.

] Hence it has been shown that if one draws from any point on KBM a line tangent to the section, and draws the line connecting the point of tangency with point D, e.g. line QD, and draws line SQ parallel to AZ, then in that case line SQ is re ected to point D, i.

e. the line passing through point Q is re ected at equal angles from the tangent to the section. And all parallel lines from.

For whatever its wo none for none rth, the diagram the marker of closure in early mathematical manuscripts, as it is invariably situated at the end of the text to which it refers is here positioned not following this end of the proof, but later on, following the discussion of the optical and practical consequences (Toomer : ; we have here a positioning of the diagram in the form of a blank space, while the diagrams themselves are missing from this manuscript, a state of affairs not uncommon with medieval manuscripts. At some stage of the transmission, the scribe responsible for putting in diagrams happened to be on holiday)..

The telling of the treatise L N O Q T R F O K S Figure all points on KBM h none for none ave the same property, so, since they make equal angles with the tangent, they go to point D. [Here ends the general geometrical exposition of the import of the proof a tantalizing exposition as I shall return to explain below.] Hence, if AZ is kept stationary, and KBM revolved (about it) until it returns to its original position, and a concave surface of brass is constructed on the surface described by KBM, and placed facing the sun, so that the sun s rays meet the concave surface, they will be re ected to point D, since they are parallel to each other.

. The text now moves on to offer various variations on the construction of the paraboloid, with resulting consequences for the re ecting rays. I concentrate just on the simplest case discussed right now. The most crucial feature of the construction is that point D about which angles T, PQ are equal is independent of the choice of point Q.

This is what makes all parallel lines such as QS re ect towards the same, single point D. However, since the de nition of point D is never described in general terms, the reader is to pick up this vital information for himself. Otherwise it could well be that each point Q de nes its own point D with no singularity emerging and no burning.

Diocles made the choice to delay his general exposition and. Diocles effectively none for none assumes as did Eratosthenes in his calculation of the size of the earth an in nitely distant sun. The astronomical discussions in the introduction contribute to make this assumption more plausible. (There, Diocles describes his assumption as treating every point on the surface of the earth as if it were its center.

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