20/Oct/1997
Not all the propositions proved in the Elements are of equal importance. Some propositions are "local," in the sense that they are never used again after the book in which they appear though they could be useful in further arguments, while there are other propositions which, though not very impressive to us, are repeatedly used also by other authors.
It seems that Greek mathematicians had a set of theorems and techniques, repeatedly used for their investigations. This set may have been considerably different from the set of propositions that seem important to us. Let us call this set the "tool box" of Greek mathematicians.
This article reports a tentative research of restoring this "tool box." For this purpose, every proposition of Book 7 of Pappus' Collection is divided into clauses (according to the punctuation marks as printed by Jones), and the propositions used in each clause have been identified and listed (though of course, a perfect identification and classification is not possible). An "index of all the propositions used in Book 7 of the Collection" is thus obtained and its substantial part is printed here. The whole index is available via internet. [i.e., available here in Part 3.]To make this index, first of all it is necessary to identify the propositions used, and to give them names.
In principle, I tried to find propositions in the Elements which could justify Pappus' argument, not because I believe that Pappus really had in mind some of Euclid's proposition whenever he made a mathematical argument (what an absurd assumption this would be!), but only because it is too inconvenient to restore the "tool box" from scratch.
For propositions and theorems not found in the Elements,
one of the following solutions was taken.
The third interpretation sometimes generated a series of specific sequence of propositions used repeatedly. In this case, this sequence was recognized as a "tool," and a name was given to it. Proposition 6-4-x (equivalent to 6-4 + 6-16) is one of the tools found and named in this way.
The list of the propositions applied (Part 1 of the index below) and the list of the places of their application (Part 2) have been thus created together, and both have been revised several times. The index contained also the third and the last part, where the propositions are listed in the order of Pappus' text. This part is available via internet. [So it is available here].
Part 1 is a list of the propositions used in Pappus' Book 7 established in the way just explained above.
The order of propositions in Part 1 is:
The second part is the list of all the places in Book 7 where each of the propositions listed in Part 1 is used. The order of the propositions is the same as in Part 1. The propositions used implicitly and too often, such as Elements 1-3, or 5-11 are omitted from the index.
There are also several axioms and assumptions Pappus used, which are only partly included in Euclidean postulates and axioms, but they are not included neither in Part 1 nor in Part 2, because it is too hard to identify them, without exception and with perfect consistency. In the third part (available via internet) [available here] however, I noted them whenever I was aware of them.
Here I report some of the results, or by-products obtained from the index, so that the readers will have an idea of its potential usefulness.
It is evident for us that if a=b and c=d, then the proportion a:c::b:d holds. This seems to have been slightly less obvious for Pappus, who argues in the following way (Book 7, 69)(reference to the paragraph number):
The numbers at the beginning of each sentence is the sentence number in my index. (6) is the sixth sentence in this propostion. This argument can be presented as follows:(6) ... the rectangle contained by AE plus GB and ED equals the rectangle contained by AE, EG.
(7) But it has been proved that the rectangle contained by by AE plus GB and BD also equals the rectangle contained by AB, BG.
(8) Hence inverting [enallax], as is the rectangle contained by AE plus GB and BD to the rectangle contained by AE plus GB and DE, so is the rectangle contained by AB, BG to the rectangle contained by AE, EG. (Jones, 69, p.142; Greek alphabets are romanized)
(6) r(AE+GB, ED) = r(AE,EG)
(7) r(AE+GB, BD) = r(AB, BG)
Inverting,
(8) r(AE+GB, BD):r(AE+GB, DE) ::r(AB, BG):r(AE, EG)
Though (8) can be directly derived from (6) and (7), Pappus inserts here the word "inverting" (enallax). This is an well-known word to invoke Elements 5-16.
The proportion to be inverted must have been:
(8') r(AE+GB, BD):r(AB,BG):: r(AE+GB, DE):r(AE, EG)
For Pappus, this (and not (8)) is the relation directly deduced from (6) and (7), so his argument can be resumed as follows:
Since a=b and c=d, a:b::c:d; and enallax, a:c=b:d.
The theorem
a=a', b=b' —> a:a'::b:b'is an trivial case of Elements 5-7 (this is 5-7-cor-1 in this index and is identical to Mueller's VD; see [Mueller 1981, Appendix 3]). It was more fundamental than
a=a', b=b' —> a:b::a':b'
which can be proved by 5-7 and 5-11.
Similar arguments with "enallax." appears in 80 (p.153, 22-25) and 83 (p.155, 29-35) in Book 7. Pappus' expression is quite consistent, following the formula:
A is equal to B. But it was shown that C is equal to D. Therefore, "enallax," as is C is to A as D is to B.
Now, we have to examine other passages where Pappus wrote differently. Paragraph 78 (p.151, 26-31) contains similar argument, but the phrase where "enallax" is expected, is corrupt, and has been emended by Commandino. Commandino did not insert the word "enallax," followed by subsequent editors. But it is quite possible, that the text had "enallax" here. Paragraph 87 (p.159, l.10) on the other hand, clearly lacks the word "enallax." The text however, is considerably disturbed around here and it is possible that some emendation had wiped away the word. The worst example for us is paragraph 98, where the word "enallax" is omitted, and the text is by no means corrupt. But this is a lemma for book 2 of the Determinate Section (all other propositions are for book 1 of the same work), and we can still assume that Pappus was quite consistent while writing the lemmas for book 1.
Now, the existence or omission of "enallax" has no great significance, for the result is anyway obvious. This small example, however, illustrates the difference between the implicit premises for Greek mathematicians and those for us.
(5-14) a:b::c:d, a>=<c —> b>=< d
But this is never used in other books of the Elements, and the arguments that would require 5-14 are covered by:
(5-14-a) a:b::c:d, a>=<b —> c>=< d
And in the cases where 5-14 is directly applicable, the proportion a:b::c:d is transformed into a:c::b:d by "enallax," then 5-14-a is used. Book 12 of the Elements contains several arguments of this type, whose genuineness have been doubted by [Gardies 1991], and defended by [saito 1994].
Which of these did Pappus use? He uses almost always 5-14-a, not 5-14. With regard to this problem, paragraph 123 of Book 7 is very curious.
(13) Therefore AZ is greater than DE.
(14) And as is AZ to GE, so is DE to ZB.
(15) Hence GE too is greater than ZB.
Here, 5-14 seems to be used. The manuscripts, however, read unanimously:
(14', Codices ABS) And as is AZ to DE, so is GE to ZB
Here, the logical foundation of (14) or (14') is rec(AZB)=rec(GED)(p.199, l.18-20), so that either (14) or (14a) is correct from mathematical point of view. The difference lies in the theorem invoked (implicitly) in the deduction of (15). (14) requires 5-14, while (14') requires 5-14-a.
Since Pappus almost always uses 5-14-a in other instances in his Book 7, it is not necessary to adopt (14) against all the extant manuscripts. The manuscripts are prefereble to Hultsch's emendation (followed by Jones).
The following proposition (paragraph 124) has similar argument (p.201, l.12). Although the situation is slightly more complicated because Codex A is corrupt here, the editorial principle should be the same. The text should be read as to require proposition 5-14-a.
These two examples are not very important in themselves, but sufficiently illustrate the potential value of this kind of indexes.
For the index, it was of course necessary to
add several propositions not found in the Elements,
used by Pappus. The most striking one is what I named
as 6-4-x, and its converse 6-4-x-conv:
let there be a triangle ABC and a point D between AC,
if angle ACB = angle ABD, then r(AC, CD) =