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The carnival of calculation in .NET Encoder Code 3/9 in .NET The carnival of calculation




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The carnival of calculation generate, create none none for none projects datamatrix Pappus Coll none for none ection. This, in turn, is extant only in part (the whole of book i, as well as the beginning of book ii of the Collection, were lost at an early stage from the codex upon which our knowledge of Pappus rests, Vat. Gr.

). It seems that the commentary is extant from proposition onwards, where the original treatise by Apollonius contained twenty-six propositions followed by a sequence of calculations. Pappus must have thought highly of this treatise.

Among the extant books of the Collection, the only one to have the formal structure of a commentary on a single treatise is book ii. Of course, Pappus is also the author of a commentary on Euclid s Elements x, extant only in Arabic, and it is likely that he produced even more commentaries. But it is still striking that he considered this work by Apollonius to be worthy of commentary.

In fact, the need for commentary may have been precisely a consequence of Apollonius non-utilitarian numerical practices. Since we do have such a detailed commentary, we may reconstruct those practices in detail. Thus, for instance, we may quote the commentary to proposition , where Pappus in turn quotes the enunciation by Apollonius (Pappus ii.

. ):. Let there be a multitude of numbers, that on which are the A s, of which <numbers> each is smaller than hundred while being measured by ten, and let there be another multitude of numbers, that on which are the B s, of which each is smaller than thousand while being measured by hundreds, and it is required to state the solid number <produced> by the A s, B s, without multiplying them.. Pappus comm none none entary then goes on to offer a concrete example with speci c numerical values. All of this is continuous with what we have seen so far: mainstream Greek demonstrative practice based on the diagram alone; pedagogic explications of the same practice then introduce simple numerical relations. But there is more to the non-utilitarian character of numerals in mainstream Greek mathematics in this case.

I rst explain the basic theoretical tools of Apollonius treatise. The rst is the notion of the base where, e.g.

, the base of is , of is etc. The second is the notion of double myriad, triple myriad, etc.: a double myriad is a myriad-myriads, a triple myriad is a myriad-myriad-myriads, etc.

Pappus commentary rst puts forward the case where there are four A s, whose bases are , , , (the A s are , , , ), and four B s, whose bases are , , , (the B s are , , , ). He then multiplies all eight bases to obtain , . Then he moves back to the language of Apollonius himself in his proof.

Apollonius constructs Z as the number of A s and twice the number of. Non-utilitarian calculation B s, which h none for none e then has shown to be relevant to the solution as follows: the solid number <produced> of all the <numbers> on which are the A s, B s, is as many myriads, of the same name [ = i.e of the same power of ten] as the number Z, as there units in E. Pappus in his commentary unpacks this to mean that, since Z in the case produced by Pappus himself is , the number resulting is a triple myriad (we can see that a myriad is the equivalent of ten taken four times), and the value as a whole, Pappus nally shows, is triple myriads.

The few explicit quotations Pappus makes of Apollonius suf ce to determine the original nature of the Apollonian proposition: it was an abstract, quasi-geometrical proof, accompanied by a diagram showing line segments on which lay the letters A and B (several times each) as well as E and Z (once each). The argument was all couched in general terms, no speci c values being mentioned going as far as treating even the number of A s and B s (tacitly determined by the diagram) as if it were indeterminate. That is: the diagram would have had (I suspect) exactly four lines next to each of which was written the letter A; and the text would still speak of as many as there are A s.

All of this would be standard Greek demonstrative practice. Notice also how opaque the resulting text would be. No doubt Apollonius himself did not provide concrete numerical examples, or otherwise Pappus commentary would have been truly otiose: for indeed Pappus hardly does anything besides providing those numerical values.

And note, even in my brief example above, how important those numerical values are for clarifying the text! Once again, we see the value of simple numbers as expository aids, the numbers in this case being very simple indeed no more than those going from to . We may also begin to perceive the purpose of the treatise. It has to do with the calculation of the multiplication of numbers, each of which is an integer multiple (the integer no greater than ) of a power of ten: numbers of the form k n , where k .

This is of very little practical value: in practice, multiplications become truly complex where the multiplicands are each a sum of different powers of ten. Not to mention the fact that multiplications, after all, are not a major practical obstacle for calculation: the real dif culty usually begins with division. But Apollonius study has one immediate application.

The major type of numerals used in the Greek literary context are known as alphabetic numerals, where the twenty-four characters of the Greek alphabet (to which are added three ad-hoc symbols) are associated with the twenty-seven values one obtains with k n , where k as well as n , that is the values , , , . . .

, , , , , . . .

, , , , , . . .

, . Apollonius .
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