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In Book 7, Prop. 1. Euclid uses repeated subtraction to prove that two numbers are relatively prime. As explained here the Greek word for repeated subtraction is "antenaresis". There isn't much information about Antenaresis online. I found this video (and the text mentioned in the video).

I have two questions. (1) Do you have more information on antenaresis and (2) What does Heath do in his commentary to the mentioned proposition?

Heath's commentary can be found on page 234 of this link:

This is the screen shot:

enter image description here

I understand what Euclid does but I don't understand Heath's notation.

For instance, how does Heath's notation work for a=85 and b=54?

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    $\begingroup$ Isn't $b$ the divisor of $a$ and $p$ the quotient in the first line? Then $c$ is the remainder, which becomes the divisor of $b$ in the 3rd line? And the process repeats. (Instead of repeated subtraction, the equivalent division is performed. Since multiplication = repeated addition, division = repeated subtraction in modern terms.) $\endgroup$
    – Michael E2
    Nov 18, 2023 at 14:33
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    $\begingroup$ The WIkipedia page Euclidean algorithm not only describes the repeated subtraction form but also points out that it can be more efficient to do it using division as @MichaelE2 and I both read your screenshot. $\endgroup$
    – mdewey
    Nov 18, 2023 at 14:37
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    $\begingroup$ ἀνθυφαιρουμένου is not the same as ἀντεναρεσις (sp?). They both might have the root αἱρέω, to take (away). The first is a participle of the verb ἀνθυφαιρέω. It is formed from anti-/anthi- (against or back/in return— we could use the Latin re- or counter- perhaps); plus hyp-/hyph- (below — Latin sub-); plus -haireo (Latin -tract). So "countersubtracting" or "resubtracting", so to speak, meaning to take away in turn. The second term replaces hyp- by en- (in, on, among). It does not appear in my dictionary. There are several words with the compound prefix anten- (in turn, instead, counter-).... $\endgroup$
    – Michael E2
    Nov 18, 2023 at 18:28
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    $\begingroup$ @MichaelE2 The more common transliteration is anthyphairesis, see Fowler, p. 816, Hewitt, etc. However, Joyce's translation of Elements does transliterate it as antenaresis instead (in the commentaries). Neither is used in Heath's translation. $\endgroup$
    – Conifold
    Nov 19, 2023 at 9:01
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    $\begingroup$ @Conifold I've read Books VII-IX in Greek. Here's what I think I know: Antenaresis is a mistake (by Joyce and others). It is not from a Greek word, does not appear in any source, and is not an alternate form of anthyphairesis. It was not used by scholars such as Heath, Knorr, or Fowler, who used the correct term. It does not appear in English before ca. 2000. There is a word, ἀνταναίρεσις (antanæresis or antaneresis), with transposed e/a (Latin/English form) and a different meaning. (My knowledge is not comprehensive, so I'd welcome any correction.) $\endgroup$
    – Michael E2
    Nov 19, 2023 at 15:19

1 Answer 1

2
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54 ) 85 ( 1
     54
    ---
     31 ) 54 ( 1
          31
         ---
          23 ) 31 ( 1
               23
              ---
                8 ) 23 ( 2
                    16
                   ---
                     7 ) 8 ( 1
                         7
                        --
                         1
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  • $\begingroup$ Thanks. This was helpful. $\endgroup$
    – zeynel
    Nov 19, 2023 at 6:21
  • $\begingroup$ From Wikipedia: "Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers)". It is interesting that Euclid does not mention "division", he only talks about repeated substruction. But now, including Heath, this algorithm is assoiated with division. $\endgroup$
    – zeynel
    Nov 19, 2023 at 10:31
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    $\begingroup$ @zeynel The terms Euclid, "measures" or "measuring," are translated from two related words, katametreo ("measures out") and metreo ("measures"). They mean in modern terms, "divides," sometimes with remainder and sometimes evenly. The statement of prop. VII.1 uses katametreo and means divides evenly: "if the number which is left never measures the one before it." The proof uses repeated division with remainder (metreo): "And let $CD$ measuring $BF$ leave $FA$ less than itself, and let $AF$ measuring $DG$ leave $GC$ less than itself...." It's Euclid who uses division in the proof. $\endgroup$
    – Michael E2
    Nov 19, 2023 at 20:08
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    $\begingroup$ @zeynel Here's a screen shot from Weston's Arithmetick (1736) p. 93 showing long division in Heath's format: i.stack.imgur.com/pp4Xt.png -- I guess it was the standard format that Heath was taught. $\endgroup$
    – Michael E2
    Nov 22, 2023 at 13:58
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    $\begingroup$ Even recently, I have seen division arranged this way by people from Europe or Asia. $\endgroup$ Nov 22, 2023 at 14:35

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