Greek Geometers Research Articles

Fighting on Land and Sea in Greek Geometric Art. Gudrun Ahlberg

Previous articleNext article No AccessBook ReviewsFighting on Land and Sea in Greek Geometric Art. Gudrun Ahlberg J. K. AndersonJ. K. Anderson Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Classical Philology Volume 70, Number 1Jan., 1975 Article DOIhttps://doi.org/10.1086/366144 Views: 2Total views on this site PDF download Crossref reports no articles citing this article.

Prothesis and Ekphora in Greek Geometric Art. By Gudrun Ahlberg.

Previous articleNext article No AccessBook ReviewsProthesis and Ekphora in Greek Geometric Art. By Gudrun Ahlberg.J. L. BensonJ. L. Benson Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by American Journal of Archaeology Volume 77, Number 1January 1973 The journal of the Archaeological Institute of America Views: 4Total views on this site Article DOIhttps://doi.org/10.2307/503249 Views: 4Total views on this site Copyright © 1973 by the Archaeological Institute of America. All rights reserved.PDF download Crossref reports no articles citing this article.

Greek Geometric Pottery, A Survey of Ten Local Styles and Their Chronology

List of plates Acknowledgements Abbreviations Preface to the second edition (2008) 1. Introduction 2. Attic Geometric 3. Corinthian Geometric 4. Argive Geometric 5. Protogeometric survivals in Thessaly, Skyros, Euboea and the Cyclades 6. Thessalian Geometric 7. Cycladic and Euboean Geometric 8. Boeotian Geometric 9. Laconian Geometric 10. West Greek Geometric 11. Cretan Geometric 12. East Greek Geometric 13. Absolute Chronology 14. Historical Conclusions Bibliography, exclusive of site publications Glossary of linear motifs Site Index Index of collections Supplement (2008) Text Bibliography Site Index General Index

The Beginning of Narrative Art in the Greek Geometric Period

The very familiarity of Greek art is apt to make us take its development very much for granted. It has a beginning in the isolated silhouettes and matchstick processions of Geometric art; progresses through an ever more faithful, that is to say illusionistic, rendering of the objects of the natural world to classic formalism; goes on to the remarkable technical proficiency of the third century B.C.; and finally declines in late Hellenistic sterilities before being absorbed and transformed by Imperial Rome and the Early Christians. It seems to be a sort of organic process: birth, flowering, and death follow in such a logical and apparently inevitable fashion that it is difficult not to be seduced by the metaphor.

Fighting on Land and Sea in Greek Geometric Art. By Gudrun Ahlberg.

<i>Fighting on Land and Sea in Greek Geometric Art</i>. By Gudrun Ahlberg.

The Geography of Pappus of Alexandria: A Translation of the Armenian Fragments

pAPPUS OF ALEXANDRIA, the noted Greek geometer, probably flourished in the time of Diocletian-at the turn of the third century A.D. His most famous work is the Mathematical Collection (MaOCJXTUC1)K' 2vVoycXyi), the greater part of which is still extant. He also wrote a commentary on Euclid's Elements, part of which has come down to us in an Arabic translation, as well as commentaries on Ptolemy's Almagest and Harmonics, both of which have been lost. Most intriguing of all his works, however, is his lost Geography (Xopoypceot O1OKVE11KVW), known only from the Byzantine dictionary of Suidas' until an Armenian adaptation of it was published in 1683.2 The significance of this Armenian Geography (Asxarhac 'oyc') for students of Pappus of Alexandria has been obscured for decades by lengthy debates over its date and authorship and by the fact that despite the Armenian redactor's specific statement that his work is based on the Geography of Pappus, specialists have spent considerable time attempting to prove-or disprove-that the redactor worked directly from Ptolemy. 3 In a recent study I have attempted to demonstrate that on the basis of internal evidence the Armenian Geography dates from between c. 615 and c. 636 A.D. and that it was written byAnanias of Sirak (c. 590-c. 670), the onlyArmenian author of the seventh century known to possess the necessary education and interests to undertake such a work-an idea which must now be revised.4 I also attempted to show that despite his

J. N. COLDSTREAM, Greek Geometric Pottery. A Survey of Ten Local Styles and their Chronology (Methuen's Handbooks of Archaeology). London, Methuen, 1068. XXXIX, 465 p. 64 pl. Pr. £. 12/12/0

J. N. COLDSTREAM, Greek Geometric Pottery. A Survey of Ten Local Styles and their Chronology (Methuen's Handbooks of Archaeology). London, Methuen, 1068. XXXIX, 465 p. 64 pl. Pr. £. 12/12/0

Greek Geometric - J. N. Coldstream: Greek Geometric Pottery: a Survey of Ten Local Styles and their Chronology. Pp. xxxix+465; 64 plates. London: Methuen, 1968. Cloth, £12. 12s. net.

Greek Geometric - J. N. Coldstream: Greek Geometric Pottery: a Survey of Ten Local Styles and their Chronology. Pp. xxxix+465; 64 plates. London: Methuen, 1968. Cloth, £12. 12s. net.

Greek Geometric Pottery. By J. N. Coldstream

Greek Geometric Pottery. By J. N. C<scp>oldstream</scp>

A Note on the Absolute Chronology of the Eighth and Seventh Centuries B.C.

For absolute dates down to the early sixth century B.C. students of Hellenic archaeology look to countries further east where reliable chronologies have been established. So far very few useful contexts have been discovered, and all or almost all of them are open at one end or the other. What has not been made clear enough is that occasionally two such open-ended contexts can be bracketed to give a more precise fix.One pair is the end of Period V at Samaria and grave 102 at Pithecusae (Ischia). The deposit of Period V at Samaria contained a few fragments of Attic Geometric pottery, and most Palestinian experts are confident that by comparison with other sites in their region it was closed little if any later than 750. Grave 102 at Pithecusae, which was used only for the burial of one child, contained three Corinthian pots and a scarab with the cartouche of the Egyptian king Bocchoris, who came to the throne in or very near to 718; this scarab had had some wear and anyhow ought not to be appreciably later in date than the pottery, since it is unlikely that only old pots were chosen to be buried with a child. So from their contexts the interval between the Attic sherds from Samaria and the Corinthian pots of Pithecusae should not be less than thirty years or more probably, if one allows five years for the wear of the Bocchoris scarab, thirty-five years. Now the development of the relevant phases of Attic and Corinthian pottery is known in fair detail, thanks most recently to J. N. Coldstream's Greek Geometric Pottery; and if one accepts his classification, which seems to me convincing, the latest of the Samaria sherds belong to the end of Attic Middle Geometric II and the latest of the Corinthian pots of Pithecusae is of the middle of the Early Protocorinthian phase, that is distinctly earlier than the end of Attic Late Geometric IIb. To judge by the stylistic character of Attic Late Geometric (which succeeds Middle Geometric II) and the sequence of its workshops, the two finds should be separated by not more than sixty years and, if thirty-five of these years are fixed absolutely as 750–715, then—again absolutely—there is a play of only twenty-five years in the archaeological chronology of this period.

J. N. Coldstream: Greek Geometric Pottery. London: Methuen, 1968. 504 pp., 64 pls. £12 12s.

J. N. Coldstream: Greek Geometric Pottery. London: Methuen, 1968. 504 pp., 64 pls. £12 12s.

Greek Geometric Pottery: A Survey of Ten Local Styles and Their Chronology

Greek Geometric Pottery: A Survey of Ten Local Styles and Their Chronology

Greek Geometrical Analysis

THIS article offers an interpretation of the passage in Pappus describing geometrical analysis (Collection, vii., Preface, pp. 634-6 in Hultsch; full text also in Thomas, Greek Mathematical Works, II, pp. 596-9, Loeb). The principal difficulty is that Pappus appears to give two different accounts of the direction of the analysis. (i) as an upward movement to prior assumptions from which an initial assumption follows (ev IL'V y&'p -T 0oV0F.XUxcC iT4. xCXX0UtV (FVOAsLv). (ii) as a downward movement of deduction from an initial assumption (8vTv. npo6fX-%toc). Two interpretations have been offered. The first accepts (ii) as the proper formulation of what the Greeks called geometrical analysis, and explains (i) as merely an alternative way of describing (ii). The second accepts (i) as the proper formulation, and explains (II) as merely an alternative way of describing (i). The first is the commonly accepted interpretation. Lucid and detailed accouats of the method on this interpretation are given by Heath (The Thirteen Books of Euclid's Elements, I, pp. I37-142), Robinson (Mind, N.S. XLV, I936, pp. 464-473), and Cherniss (Review of Metaphysics, IV, i9gi, pp. 4I4-4I9). The method is one of assuming to be true a geometrical proposition which it is required to prove, or assuming a geometrical problem to be solved, and, by analysis, deducing con-sequences until one reaches either a proposition known independently to be true, or a construction which it is possible to satisfy, or a proposition known to be false, or a construction which it is impossible to satisfy. In the first case, the last step in the analysis becomes the first in the synthesis, which repeats the steps of the analysis in reverse order until the original assumption is reached and so proved. In the second case one may conclude, without resort to the synthesis, that the original assumption is false, or the solution impossible. A requirement of the method is that the implications at each step are reciprocal. The second part (ii) of Pappus' account seems, quite clearly, to be describing this method. So also do the definitions of analysis and synthesis interpolated in Euclid XIII (see Heath, op. cit. p. I 38). Moreover, as Robinson emphasizes, there are excellent examples of its use in Archimedes, in Pappus, in the alternative proofs of XIII, i-g, interpolated in Euclid, and elsewhere (loc. cit. pp. 469-472; see also Heath, op. cit., I, pp. I4I-142). That this, and no other method is what the Greek geometers called analysis is, says Robinson (Plato's Earlier Dialectic,

The Logic of the Indirect Proof in Geometry: Analysis, Criticism and Recommendations

There are few topics in plane geometry that have had as strange a career as that of the indirect proof. In Euclid's Elements it does not appear separated from the other method of proof. There is no comment indicating that it is a unique device. There are no reasons quoted for each of the steps taken, as is done so meticulously in the other proof. Moreover, this type of mathematical reasoning invented by Eudoxus in 375 b.c. had become so much a part of the methodology of the Greek geometers that Euclid uses it no less than 14 times in 35 proofs in Book I, and 16 times in 31 theorems in Book III of his Elements.

Greek Geometric Art, its Symbolism and its Origin.By A. Roes

Greek Geometric Art, its Symbolism and its Origin.<i>By A. Roes</i>

Greek Geometric Art, Its Symbolism and Its Origin. By A. Roes.

<i>Greek Geometric Art, Its Symbolism and Its Origin</i>. By A. Roes.

Greek Geometric Art; its symbolism and its origin. By Anna Roes. Pp. 128; 104 figs. Haarlem: H. D. Tjeenk Willink, and London: H. Milford, 1933. 9s.

Greek Geometric Art; its symbolism and its origin. By Anna Roes. Pp. 128; 104 figs. Haarlem: H. D. Tjeenk Willink, and London: H. Milford, 1933. <i>9s.</i>

The Relation between Geometry and Einstein's Theory of Gravitation

THE term “geometry” has been used ever since the time of Euclid to denote two completely distinct subjects; but the formal similarity of their propositions has been so close as to obscure until recently the entire dissimilarity of their status in scientific knowledge. The Greek geometers seem to have been inspired originally by the need for a satisfactory method of surveying; at the same time, their logical turn of mind led them to present their results in the now| familiar form of a deductive science. The characteristics of such a science are that a certain number of primitive propositions P1 now called postulates, are stated at the beginning, and that from these, by a process of pure logic, further propositions q1 are one by one developed. But this development is quite a separate process from that of deciding whether the primitive propositions are true or not, and if this is not done it is impossible to assert that the deduced propositions are true.

Note on the Mathematical Theory of Miller's Trisector, and its Relation to other Solutions of the Problem of Trisection

The problem of the trisection of an angle, analytically considered, depends on the solution of a cubic equation, say 4x3−3x−cos a = 0, which is in general irreducible, and therefore not soluble by means of quadratic radicals. It follows that the trisection of an angle cannot be effected by means of the ruler and compass alone. This, in fact if not in theory, appears to have been known to the early Greek geometers, and they proposed the use of various curves, the continuous mechanical construction of which must, of course, be postulated, for the solution of the problem. The Quadratrix and the Spiral of Archimedes, both transcendental curves, may be mentioned. It was early discovered that the conic sections could be used for the purpose, and in modern times various solutions have been suggested involving their use.

Scientific Serials

The Mathematical Gazette, No. 1, April 1893, 8 pp. (London: Macmillan.)—It is now maiter of ancient history that a correspondence in the columns of NATURE resulted in the formation, in January 1871, of the Association for the Improvement of Geometrical Teaching. The original objects of the association were threefold: to collect and distribute information as to the prevailing methods of instruction in geometry practised in this and other countries, to use its influence to induce examining bodies to frame their questions in geometry without reference to any particular text-book, and to stamp with its approval some text-book already submitted, or to bring out a new one under its own auspices. Ten years later, viz. in January 1881, the association widened its basis, though after some discussion it retained its name. The objects it had in view were now sought to be carried out by the reading of papers and raising discussions, and by the appointment of committees to report on existing defects in the usual methods, order, range, &c, in teaching special subjects—all branches of elementary mathematics and mathematical physics being included in the widened basis. Now that the association has passed its majority, it is thought that it owes its continued existence to a “widespread desire on the part of teachers of mathematics to become acquainted with the methods of other teachers.” The editor of the Gazette, Mr. E. M. Langley, to whose long-continued and enthusiastic advocacy of its aims the Association for the Improvement of Geometrical Teaching owes so much as late secretary, hopes, through the agency of its columns, to extract from experienced teachers MSS. which have long been lurking in desk or pigeonhole for want of a suitable organ for making them known. This new venture, which has been started in consequence of a resolution passed at the association's annual meeting in January last, is proposed to be “a terminal journal for students and teachers.” The editor has to feel his way: words of encouragement have come from the far East and West, as well as from many teachers in this country. The number before us opens with a short paper, by the editor, on the eccentric circle of Boscovich. We borrow from Dr. C. Taylor's classical book on Conies the following verdict on Bosovich's work:—It is “a clear and compact treatise, which for simplicity, depth, and suggestiveness will not readily be surpassed.” Dr. j. S. Mackay abstracts the first book of Gino Loria's treatise on “the exact sciences in ancient Greece,” viz. that on the Greek geometers before Euclid. In addition to the works cited by Dr. Mackay, we may call attention to three notes on the history of mathematics by the Danish mathematician, H. G. Zeuthen (which have recently been published in the Bulletin de l'Acadénie Royale des Sciences de Danemark (1893). Prof. A. Lodge gives some useful approximations and reductions. Then follow some elegant solutions of examination questions, and a select number of questions for solution. A commendation of the new French journal, l'Intermédiare des Mathematiciens closes this No. 1. The size of the page, the clear type, and the excellent paper, should secure for the Gazette far more than a mere succés d'estime. The figures are lithographed on a separate sheet. We note one little slip—Adam's property for Adams' (p. S.)