apotome

English

Etymology

Borrowing from Ancient Greek ἀποτομή (apotomḗ, “cutting off”). The musical sense originates from the Pythagorean tradition. The mathematical sense is attested in Euclid's Elements (Book X, proposition 73, et seq.).

Pronunciation

• IPA^(key): /əˈpɒtəmi/

• Audio (US):

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• Rhymes: -ɒtəmi

Noun

apotome (plural apotomes)

1. (mathematics, geometry) The difference between two quantities or lengths commensurable only in power, as between 1 and the square root of 2, or between the diagonal and side of a square.
   • 2012, Jade Roskam, “Book X of The Elements: Ordering Irrationals”, in Bharath Sriraman, editor, Crossroads in the History of Mathematics and Mathematics Education, Information Age Publishing, page 210:

     Yet it is not until Book X that the properties of such a line (with greater length is an apotome and lesser length a first apotome) are explained and not until Book XIII that this type of line is applied, which will be discussed in more detail later. […] The likewise is true of apotomes (X. 97).

   • 2014, Jacques Sesiano (translator), Liber Mahameleth, Part Two: Translation, Glossary, [12th c, Anonymous (possibly John of Seville), Liber Mahameleth], Springer, page 767,

     If some number and^[plus] the root of the root of a number are multiplied by the corresponding apotome, the result will be an apotome.

2. (music) The remaining part of a whole tone after a minor second has been deducted from it; an augmented unison. Most commonly used to refer to the Pythagorean chromatic semitone, which has a ratio of 2187/2048.
   • 1813, Music, article in John Mason Good, Olinthus Gregory, Newton Bosworth, Pantologia: A New Cyclopaedia, Volume 8: MID—OZO, unnumbered page,

     This semitone was termed by the Pythagoreans apotome, and the diatonic semitone was termed limma. They contended, that the apotome, or distance from B flat to B natural, was larger than the limma, or distance from A to B flat.

   • 1820, Proclus, translated by Thomas Taylor, The Commentaries of Proclus on the Timæus of Plato, Volume 2, Facsimile edition, published 1967, pages 66–67:

     For the ratio of the excess of the apotome, above that which is truly a semitone, and which cannot be obtained in numbers, is thus called. This then is demonstrated. To what has been said however, it must be added, that we have called the ratio of d b a semitone, not that a sesquioctave is divided into two equal ratios; for no superparticular ratio is capable of being so divided; but because the followers of Aristoxenus assume a semitone after two sesquioctaves, the ratio of a semitone is assumed, as we have said, according to their position, in order to discover what the ratio is of the comma and apotome to the ratio of the leimma.

   • 1984, Mark Lindley, Lutes, Viols, Temperaments, Cambridge University Press, page 9:

     The traditional term, from ancient Greek theory, for the diatonic pythagorean semitone is 'limma'; and for the larger, chromatic semitone, ‘apotome’.
     The oldest extant fretting formula, that of the ninth-century theorist Al-Kindi for the 'ud (the Arabic lute), is pythagorean. It calls for five frets, to make the following succession of semitones down from nut: limma, apotome, limma, apotome, limma.

3. (zoology) A distinct division of an insect which is divided from the other divisions by a pinch point.
   • 1964, Proceedings of the Zoological Society of London, Zoological Society of London, page 287:

     The bands pass inside the margins of the apotome as they approach its constriction and form two pigmented areas broadly based on the frontal sutures and becoming dispersed towards the mid-line.

   • 2008, John L. Capinera, Encyclopedia of Entomology, →ISBN, page 3766:

     In generalized Diplura, Archaeognatha and some Thysanura, five parts or apotomes are distinguished, which from the anterior to the posterior part are: the presternum, the basisternum, the furcasternum, the spinasternum and the poststernum.

Derived terms

• Pythagorean apotome

Translations

difference between two quantities commensurable only in power

• Greek:

  Ancient: ἀποτομή f (apotomḗ)

• Portuguese: apótomo m

See also

• binomial
• limma

Part or all of this entry has been imported from the 1913 edition of Webster’s Dictionary, which is now free of copyright and hence in the public domain. The imported definitions may be significantly out of date, and any more recent senses may be completely missing.
(See the entry for “apotome”, in Webster’s Revised Unabridged Dictionary, Springfield, Mass.: G. & C. Merriam, 1913, →OCLC.)