It is well known that when resolution occurs in the stichic iambics and trochaics of tragedy word-end is not found between the two shorts so produced: w or, more accurately, that the first short of resolution must not be the last syllable of a polysyllabic word. Moreover, the syllables in resolution most often form part of the same word as the following short or anceps, e.g.: Ion 1143.

The elements which every schoolboy learns on beginning Latin Verse Composition include a number of rules which seem arbitrarily designed to make the game harder. In hexameters, he is told, he must have a masculine caesura either in the third foot or in the second and fourth, and end normally with a disyllabic or a trisyllable; in pentameters he must end with a disyllabic; and in neither line may a single monosyllable stand at the end. Rarely, in my experience, is (...) any reason given him by way of redress, and he will search for one in vain in most of the school text-books, in introductions like Postgate's to Tibullus and Propertius, and in histories of Latin literature like Wight Duff's and Mackail's. This reticence may be due to the dissensions of experts on this subject and on the subject of Latin accentuation in general, but the theory that predominates in England, among those who hold a theory at all, explains so many of the phenomena that it deserves to be more widely and precisely known. The most detailed exposition of it is by E. H. Sturtevant, who summed up his analyses in a pair of articles in the Transactions of the American Philological Association in 1923–4. While his careful work is invaluable as marshalling the statistics and evidence, it errs on the side of excessive minuteness, and leaves room not merely for some additions, but for a different kind of treatment concerned less with bare statistics and more with poetic principles and historical development. Since Latin Verse Composition plays such an important part in English higher classical education, it seems desirable that a less technical and more accessible account should be available for English readers. Such an account I attempt to give here, keeping separate as far as possible the exposition and the consideration of the criticisms and rival theories that have been advanced. (shrink)

Paul Maas extended the law still further: ‘The following rule applies to several metres which contain the rhythm : no word can end after a long anceps, except at the caesura in the middle of the line.’ He lists the types of metre to which the rule applies as the stichic iambic trimeters and trochaic tetrameters of the early iambographers and the Attic tragedians, the dactylo-epitrites of Bacchylides, the trochaic trimeters and dimeters of Alcman's Partheneion, the end of the iambic (...) tetrameters of Sophocles' Ichneutae, and ‘certain rare metres, whose conformity to this rule may be due to accident’. (shrink)

It has been pointed out to me that in my article on ‘The Augustan Rules for Dactylic Verse’ I misrepresented an observation of Maas as reported by Wilamowitz in his Griechische Verskunst, p. 53. Wilamowitz' words are: ‘Wenn Tibull und Ovid den Pentameter so bauen, dass die vorletzte Silbe betont wird, tun sie das nach dem Vorgange gleichzeitiger griechischer Epigrammatiker.’ This means, of course, that the Greek writers mentioned ended with a paroxytone word, not necessarily with a disyllable, as I (...) assumed it to mean, and I apologize for the oversight. (shrink)

A fundamental problem in the construction of local electromagnetic interactions in the framework of relativistic wave equations of Klein-Gordon or Dirac type is discussed, and shown to be resolved in a relativistic quantum theory of events described by functions in a Hilbert space on the manifold of space-time. The relation, abstracted from the structure of the electromagnetic current, between sequences of events, parametrized by an evolution parameter τ (“historical time”), and the commonly accepted notion of particles is reviewed. As an (...) illustration of these ideas, a perturbative calculation is made for photon emission from a charged two-body system in which the electromagnetic field is quantized in the usual way. The result is in essential agreement with calculations in which the charged particles are treated in the framework of nonrelativistic quantum mechanics, and provides them with a relativistic interpretation. In particular, we obtain a relativistically invariant form of the Bohr radiation condition. (shrink)

The fragmentary columns of the Fourth Book of Philodemus' περ μονσικς were the first-fruits of Herculaneum, published in 1793, with venturesome reconstructions and learned notes, by the Academici of Naples. Fragments of the other books occur in four volumes of the Collectio Altera of 1862–5. A. Teubner Text by J. Kemke, a pupil of Bücheler, appeared in 1884, upon which Gomperz made a number of improvements in a pamphlet published in the following year. Otherwise, save for a few pages in (...) H. Abert's Die Lehre vom Ethos in der griechischen Musik published in 1899, the work has attracted very little attention; which is scarcely surprising in view of the state of the text and the peculiar awkwardness of Philodemus' Greek. (shrink)

One of the notable arguments in the history of philosophy is Descartes' thesis: cogito, ergo sum. However, the question as to the logical nature of that argument has not yet been entirely clarified.

Purpose: One of my goals in the paper is to investigate why realists reject radical constructivism (RC) as well as social constructivism (SC) out of hand. I shall do this by means of commenting on Peter Slezak’s critical paper, Radical Constructivism: Epistemology, Education and Dynamite. My other goal is to explore why realists condemn the use of RC and SC in science and mathematics education for no stated reason, again by means of commenting on Slezak’s paper. Method: I restrict my (...) comments to Slezak’s paper and leave it to the reader to judge which, if any, of the reasons that I advance for these two states of affairs are not specific to Slezak’s paper. Other readers might not agree with my interpretations of Slezak’s paper, including Slezak himself, but I offer them after having worked with von Glasersfeld in interdisciplinary research in mathematics education for over 25 years. Findings: My findings are that Slezak: (1) rejects RC and SC on the basis of unjustified criticisms, (2) does not explore basic tenets of RC nor of SC beyond the unjustified criticisms, (3) rejects how SC and RC have been used in science and mathematics education, based at least in part on the unjustified criticisms, (3) dislikes how SC has been used in science and mathematics education, a dislike that fuels his rejection of any constructivism, and (4) doesn’t explore how RC has been used in scientific investigations in mathematics education. On the basis of these findings, I conclude that how epistemological models of knowing might be used in science or mathematics education would be better left to the educators who use them in interdisciplinary work. (shrink)

Open peer commentary on the article “Negotiating Between Learner and Mathematics: A Conceptual Framework to Analyze Teacher Sensitivity Toward Constructivism in a Mathematics Classroom” by Philip Borg, Dave Hewitt & Ian Jones. Upshot: My commentary has two general goals. First, I investigate how basic principles of radical constructivism might be used in constructing models of mathematics teaching. Toward that end, I found that I was not in complete intersubjective agreement with Borg et al.’s use of some basic terms. Second, I (...) explore what mathematics teaching might look like in what is construed as constructivist mathematics teaching. Toward that end, I comment on Borg et al.’s use of “negotiation” and explain how a constructivist teacher can establish experiential models of students’ mathematics. (shrink)

---- t. 6. O nachalakh : opyt khristīanskoĭ metafiziki. [no.] 1. Bog i tvarnyi mīr. [no.] 2-3. Bog i mīr, v Bogochelovi︠e︡ki︠e︡ vi︠e︡chnyĭ ad pobi︠e︡zhdai︠u︡shchīĭ.

As described by the ancient metricians, catalexis is a matter of arithmetic rather than rhythm. They develop the idea in their usual way, mechanically and mathematically, adding and subtracting elements, so as to produce ‘brachycatalexis’ and ‘hypercatalexis’. These are now mere metrical-glossary terms, but in catalexismodern metricians have seen a genuine relationship between cola and a rhythmic effect more or less comprehensible even to us. Wilamowitz, T.D. Goodell, and A.M. Dale explore the concept to some extent, but current hand books (...) and general treatises give it only perfunctory and superficial treatment, while casual appearances of the term offer the reader glimpses of a theoretical substructure which hasnot been explicitly and coherently explained. (shrink)