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The telling of the proof in .NET Get Code 3 of 9 in .NET The telling of the proof

The telling of the proof using barcode generating for vs .net control to generate, create 3 of 9 barcode image in vs .net applications. Beaware of Malicious QR Codes Z B N A M Q O K H D G L E I Figure context of sty bar code 39 for .NET listic options. This coda is also an intermezzo, positioned as it is halfway through the book.

This section is dedicated to the question of the aftermath whatever happened to the ludic in the later reception of Greek mathematics This follows on naturally from the discussion of the personal voice as, indeed, the perception of mathematics as impersonal seems to be the main force moving later mathematics away from its Hellenistic forms. 2.2 the telling of the proof I begin with a work reputed for its tough-minded seriousness: Apollonius Conics.

For my rst example, I will do my best to follow in detail the key piece of argumentation in an advanced part of book vii, i.e. a very advanced piece of mathematics indeed.

Apollonius proves a series of inequalities resulting with lines constructed on conic sections. The text is extant only in Arabic translation, but it is clear that (with the exception of possible interpolated glosses) the translation is very close to the original. I provide a translation of parts of vii.

adapted from Toomer s edition (in a language somewhat closer to Greek geometrical style than Toomer s). I provide numerals for the steps in the argument. The enunciation, which I do not translate, asserts a number of minimum results, among them the one whose proof we shall follow that the sides of a rectangle constructed on the major axis of the ellipse are smaller than the sides of the same type of rectangle constructed on any other diameter.

I skip the construction, as well, which is implicit in g. ..

It is suf cien t for our purposes here to say that the rectangle involved is produced by the two sides of the diameter and the latus rectum of the ellipse. It will take too long to explain here what a latus rectum is, for which the reader should turn to Apollonius Conics i. .

For our purposes this might be taken as a mystery line associated, according to precise rules, with each given diameter of the ellipse.. The telling of mathematics AG is the majo r axis, DE the minor, and the points N, X are found by GN:AN::AX:XG::(AG:latus rectum).. ( ) And as the .NET barcode code39 square on AG to the square on the line equal to: the diameter AG together with the latus rectum of the gure constructed on it, so the square on NG to the square on NX, ( ) and as the rectangle <contained by> NG,AX to the square on NX, ( ) since the rectangle <contained by> NG,AX is equal to the square on NG. ( ) And as the square on AG to the square on ED, so NG to GX, ( ) since it was proven in proposition of Book i that as the square on AG to the square on DE so AG to its latus rectum.

( ) And as NG to GX, so the rectangle <contained by> NG, GX to the square on GX, ( ) while as the square on DE to the square on the line equal to: the line DE together with the latus rectum of the gure constructed on it, so the square on GX to the square on NX, ( ) also because of what was proven in proposition of Book i. ( ) Therefore as the square on AG to the square on the line equal to: the diameter DE together with the latus rectum of the gure constructed on it, so the rectangle <contained by> NG, GX to the square on GX, ( ) and it was shown that as the rectangle <contained by> NG,AX to the square on GX, so the square on AG to the square on the line equal to the line AG together with the latus rectum of the gure constructed on it, ( ) therefore the ratio of AG to: AG together with the latus rectum is greater than the ratio of AG to: DE together with the latus rectum..

From Step i t follows that AG together with its latus rectum is smaller than DE together with its latus rectum, which is already part of the required proof. I therefore stop at this point, as we can already see here how Apollonius obtains his essential results. The basic structure of the argument is the quick obtaining of Step (based on , ), followed by the somewhat more involved obtaining of Step (based on Steps ).

Step is recalled as Step and then and together yield . For the reader, life is not as easy as that. While the Greek particles would have made the navigation of the text almost as transparent as my numerals and logical chart make it, still the major derivations involve considerable mental labor.

Consider rst the easier case, of Step . This is, in modern notation, AG :(AG+latus) ::(NG AX):NX ..

The Arabic has 3 of 9 for .NET here throughout the expression, less natural in Greek, the ratio of . .

. to . .

. is as the ratio of . .

. to . .

. . I translate as if the original Greek had as .

. . to .

. . , so .

. . to .

. . , and return to the Arabic (more natural in Greek with an inequality) in Step .

Of course I may be wrong in this emendation of the Arabic text; nothing hangs on it. Step is likely to be a gloss added by some Arabic reader, since explicit cross-references are natural in scholia..

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