![]() ![]() I would guess that a clearer reference to CN4 would have been appropriate here, if Heron had known of it. ![]() if the figures are themselves congruent). if the parts of the figures are congruent one by one) or “according to the configuration” (i.e. 117), equality among figures is defined as two figures being in conformity with one another (Heron uses the word ἁρμόττειν rather than Euclid’s ἐφαρμόζειν) either “according to the part” (i.e. ![]() The same happens in Heron’s Definitiones, where the notions of whole and parts are dealt with at length ( Def. For a recent edition of this work, see Acerbi and Vitrac ( 2014). Nowhere in the work is mention made of the principle stating that the whole is greater than the part (or even of the Euclidean expression regarding the lesser and the greater). Proclus, In Euclidis 237–238, for a similar reference). In the case of segments, Heron does not even mention that they have to be equal and may have simply had in mind that they are homeomeric (cf. the verb ἐφαρμόζειν) in this work is in the Preface to the First Book, in which Heron tersely says that right angles and segments are congruent to one another. In particular, Heron’s Metrica would have been a perfect place to employ both CN4 and CN5 for a theory of measure. ![]() The passage reports a demonstration by Nicomedes, and we cannot say whether the reference to superposition was already in Pappus’ source. Pappus himself makes an inference from congruence to equality in Collectiones Δ 39 but I am not aware of any further passage in the corpus and the latter seems to be too isolated to make a case for the existence of an explicit axiom to this effect. These theorems are to be found in Theon’s commentary on Ptolemy’s Almagest and again in Pappus’ Collectiones Ε 1–19 (see also Hultsch 1875–1878, vol. Zenodorus’ theorems on isoperimetric figures (second century BCE) would have easily allowed inferences through CN4 and CN5, but they are nowhere mentioned. Neither CN4 and CN5 are mentioned by Proclus in relation to Geminus’ many considerations on axiomatics. Again, however, this statement does not suggest the reference to any principle entailing equality from congruence. Α 9, on the other hand, Archimedes says that two figures coincide (ἐφαρμόζειν) when they are “equal and similar” (ἴσα καὶ ὁμοῖα). Archimedes is rather interested, throughout, only in congruence, and equality of size is never at issue. No consideration of measure, however, is involved here, and even though in the last sentence of the demonstration Archimedes does employ the words ἐφαρμόζειν and ἴσος, the equality of the two halves is not even mentioned in the statement of the proposition. It first appears as an interpolation in Gerardo da Cremona’s Latin translation (from the twelfth century) and passed from there over into several early modern editions of the Elements, such as Clavius’ ( 1574).Īrchimedes’ only use of something like CN4 is in Conoids and Spheroids 18, where he shows that the two halves of a spheroid cut by a plane passing through its center are “equal”, by showing that they are congruent. We are informed by Proclus ( In Euclidis 197) that Pappus had proposed to add a similar common notion to the principles of Euclid, but we have no trace of it in any Greek manuscript of the Elements. 154b: “For adding equals to unequals, in time or anything else whatever, always makes the difference equal in the amount by which the unequals originally differed” (transl. If we suppose that Euclid’s common notions were developed in the generation of Theaetetus and Eudoxus, it may perhaps be considered a stroke of Platonic irony to imagine that they were notions advanced by an aged Socrates while a receptive stripling well versed in mathematics nodded in approval. It is remarkable, however, that they appear in a dialogue on scientific knowledge conducted by a young mathematician. The correspondence with Euclid’s common notions is admittedly quite vague, and the Platonic principles have a much more metaphysical aspect. Is it not so? … and secondly, that anything to which nothing is added and from which nothing is subtracted, is neither increased nor diminished, but is always equal” (transl. 155a: “And as we consider them, I shall say, I think, first, that nothing can ever become more or less in size or number, so long as it remains equal to itself. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |