Applications of Quadratic Equations

1945 ◽  
Vol 38 (3) ◽  
pp. 120-125
Author(s):  
William A. Cordrey

The advent of quadratic equations antedates the dawn of the Christian era by about two millennia. The study of conics, however, did not get under way until the fourth century prior to the birth of Christ. The first writer on this subject, Menaechmus, used the parabola and hyperbola in duplicating the cube. Several years later Euclid wrote a treatise on conic sections. This work was continued by Apollonius, whose investigations added much to the existing knowdedge of the subject.

1938 ◽  
Vol 18 (1) ◽  
pp. 68-74 ◽  
Author(s):  
Philip Corder ◽  
I. A. Richmond

The Roman Ermine Street, having crossed the Humber on the way to York from Lincoln, leaves Brough Haven on its west side, and the little town of Petuaria to the east. For the first half-mile northwards from the Haven its course is not certainly known: then, followed by the modern road, it runs northwards through South Cave towards Market Weighton. In the area thus traversed by the Roman road burials of the Roman age have already been noted in sufficient quantity to suggest an extensive cemetery. The interment which is the subject of the present note was found on 10th October 1936, when men laying pipes at right angles to the modern road, in the carriage-drive of Mr. J. G. Southam, having cut through some 4 ft. of blown sand, came upon a mass of mixed Roman pottery, dating from the late first to the fourth century A.D. Bones of pig, dog, sheep, and ox were also represented. Presently, at a depth of about 5 ft., something attracted closer attention. A layer of thin limestone slabs was found, covering two human skeletons, one lying a few feet from the west margin of the modern road, the other parallel with the road and some 8 ft. from its edge. The objects described below were found with the second skeleton, and the first to be discovered was submitted by Mr. Southam to Mr. T. Sheppard, F.S.A.Scot., Director of the Hull Museums, who visited the site with his staff. All that can be recorded of the circumstances of the discovery is contained in the observations then made, under difficult conditions. ‘Slabs of hard limestone’, it was reported, ‘taken from a local quarry of millepore oolite and forming the original Roman road, were distinctly visible beneath the present roadway—one of the few points where the precise site of the old road has been located. On the side of this… a burial-place has been constructed. What it was like originally it is difficult to say, beyond that a layer of thin … slabs of limestone occurred over the skeletons. This had probably been kept in place or supported by some structure of wood, as several large iron nails, some bent at right angles, were among the bones.’ If this were all that could be said about the burials, they would hardly merit a place in these pages. The chief interest of the record would be its apparent identification of the exact course of the Roman road at a point where this had hitherto been uncertain. Three objects associated with the second skeleton are, however, of exceptional interest.


Author(s):  
Anthony Grafton

This chapter examines the centrality of early modern ecclesiastical history, written by Catholics as well as Protestants, in the refinement of research techniques and practices anticipatory of modern scholarship. To Christians of all varieties, getting the Church's early history right mattered. Eusebius's fourth-century history of the Church opened a royal road into the subject, but he made mistakes, and it was important to be able to ferret them out. Saint Augustine was recognized as a sure-footed guide to the truth about the Church's original and bedrock beliefs, but some of the Saint's writings were spurious, and it was important to be able to separate the wheat from the chaff. To distinguish true belief from false, teams of religious scholars gathered documents; the documents in turn were subjected to skeptical scrutiny and philological critique; and sources were compared and cited. The practices of humanistic scholarship, it turns out, came from within the Catholic Church itself as it examined its own past.


1994 ◽  
Vol 13 (1) ◽  
pp. 17-37 ◽  
Author(s):  
Jeremy McInerney

Jacoby's influential opinion that the Atthidographers were part of the political discourse of the fourth century has been the subject of revision in recent years. His critics have argued that the genre of Atthidography is primarily antiquarian and that to look for partisan political attitudes in the Atthides is a mistake. An examination of the work of Kleidemos, however, reveals a coherent presentation of the Athenian past designed to vindicate the democratic constitution and to demonstrate the close connection between the democracy and Athens' naval power. This emerges most clearly in Kleidemos's treatment of three important democratic heroes: Theseus, Kleisthenes, and Themistokles. By the fourth century, Theseus had already emerged as the most popular Athenian hero. His accomplishments were modeled in part on the deeds of Herakles and were recorded in vase painting and relief sculpture, and on the walls of the Stoa Poikile. Kleidemos presented a distinctive account of Theseus, emphasizing his role in founding the Athenian navy in preparation for the expedition to Krete. Kleidemos portrayed him as a leader capable of defending Athens and making peace with Athens' enemies, first the Kretans and later the Amazons. This is a king in the tradition of Euripides' Theseus in the Suppliants, the ruler of a free and democratic city. The connection between democratic leadership, Athenian might, and the naval power of Athens is also underscored in Kleidemos's handling of Kleisthenes. Again, the information provided by Kleidemos is distinctive, inasmuch as he reports that it was Kleisthenes who was responsible for the system of naukrariai, which he likens to the symmories of the fourth century. Unlike the version of the Ath. Pol., which imagines the Kleisthenic demes replacing the Solonian naukrariai, Kleidemos saw the demes and naukrariai as complementary divisions, the former organizing the state's resources for the upkeep of the navy, and the latter establishing the political basis for the democracy. Themistokles is also given unique treatment. Kleidemos records the anecdote according to which Themistokles was responsible for the Battle of Salamis because he found sufficient money to man the ships when the generals had run out of funds and had ordered the abandonment of the city. He used the disappearance of the gorgoneion of the statue of Athena as an excuse to ransack the baggage of the Athenians and collect enough wealth to pay the fleet. The story is as tendentious as the account in the Ath. Pol., which gives the credit to the Areopagos. Both versions demonstrate how Athens' past had become a battleground in the political debates of the mid-fourth century. Unlike the epitaphios logos with its emphasis on the eternal and unchanging glory of Athens, the "Atthis" of Kleidemos attempted to prove that the greatness of Athens rested historically on three foundations: the heroes of the democracy, the democratic constitution, and the navy.


1940 ◽  
Vol 9 (27) ◽  
pp. 129-136
Author(s):  
T. B. L. Webster

‘If a man were a good painter, he could deceive children and fools by painting a carpenter and showing it from a long way off, because it would seem really to be a carpenter.’ Plato here (Republic, 598c) is undoubtedly describing realistic painting, perhaps not so photographically realistic as the paintings in Pompeii or the painting that we know to-day, but painting which aimed at producing a likeness and rendering the appearance of the original. Such pictures can be seen on the vases of the fourth century and of the late fifth century b.c., for instance the two women on a red-figured perfume vase in the Manchester School of Art, which was painted about the time of the dramatic date of the Republic (Pl. I). But if we go back rather over a hundred years to the black-figure vase reproduced by Mr. Austin in Greece and Rome, vol. vii, we are in a completely different world of flat figures in conventional poses, and if we go further back still to the Geometric vase (Fig. I) which forms the subject of Mr. Austin's article, we are yet further from the world of ‘likeness’. These early Greek painters cannot have wanted to produce likenesses; but what was their aim ?


1954 ◽  
Vol 4 (3-4) ◽  
pp. 151-157
Author(s):  
H. C. Baldry

This article is a survey of familiar ground—those passages of the Poetics of Aristotle which throw light on the treatment of legend by the tragic poets. Although sweeping generalizations are often made on the use of the traditional stories in drama, our evidence on the subject is slight and inconclusive. We have little knowledge of the form in which most of the legends were known to the Attic playwrights, for the few we find in the Iliad and Odyssey appear there in very different versions from those they take on in the plays, and the fragmentary remains of epic and lyric poetry between Homer and the fifth century B.C. present us with a wide field for speculation, but few certain facts; while vase paintings and other works of art supplement only here and there the scanty information gained from literature.The comments of ancient writers on this aspect of tragedy are surprisingly few, and carry us little farther. The Poetics stands out as the one source from which we can draw any substantial account of the matter. Even Aristotle, of course, is not directly concerned with the history of drama, and deals with it only incidentally in isolated passages; and in considering these it must constantly be borne in mind that he is discussing tragedy as he knew it in the late fourth century, for the benefit of fourth-century readers. But even so, his statements are the main foundation on which our view of the dramatists' use of legend must be built.


1996 ◽  
Vol 116 ◽  
pp. 33-46 ◽  
Author(s):  
C. Carey

Forensic oratory must of necessity deal with the subject of law, and rhetoric which aspires to be of use in the courts must offer the potential litigant or logographer guidance on the way to deal with questions of law. Accordingly, Aristotle devotes some space to this issue in the Rhetoric. Although the morality of Aristotle's advice has been debated, little attention has been paid to the more basic question of the soundness of his advice. The aim of this paper is to examine Aristotle's presentation of the rhetoric of law in the Rhetoric in comparison with actual practice in surviving forensic speeches. The fourth century Rhetorica ad Alexandrum, commonly ascribed to Anaximenes of Lampsakos, also offers advice on the manipulation of argument from law, and the general similarity of that advice to Aristotle's suggests either direct influence or a common source. Anaximenes' discussion of the use of law in forensic oratory is both more brief and less systematic, and will be given more cursory treatment.


Urban History ◽  
2005 ◽  
Vol 32 (1) ◽  
pp. 132-169
Author(s):  
BÄRBEL BRODT ◽  
PAUL ELLIOTT ◽  
BILL LUCKIN

Few – if any - would deny that cartography is one of the most essential disciplines within the multi-layered scope of urban history. Elizabeth Baigent pays tribute to the possibilities and problems posed by maps in her ‘Fact or fiction? Town maps as aids and snares to the historian’, Archives: The J. of the British Records Association, 29, 110 (2004), 24–37. By looking at a map of Gloucester, compiled in 1455, and two late medieval Bristol maps (one by Robert Ricart, the other by William Smith), she outlines their usefulness as well as the problems that the modern urban historian faces. Although medieval maps can clearly help to identify ‘lost’ streets, and to elucidate the town's social geography, it is essential to consider the purpose for which any individual map was drawn, the context in which it was published (and re-published) and not least the skills of the cartographer concerned. Cartography may be an essential tool for the urban historian, but there are many other tools and topics, and this year's medieval urban periodical literature again reflects the wide scope of the subject. This is especially true for the German language periodicals which tend to relate to traditionally powerful concepts rather than to recent departures. This trend largely reflects the nature of those periodicals concerned for they are almost entirely devoted to strictly local, or at most regional concerns. They are naturally home to brief essays on mainly local matters, particularly the commemoration of anniversaries of urban charters (e.g., Paul Wietzorek, ‘Zum Titelbild: 100 Jahre Stadtwappen Zons – 1904–2004’, Der Niederrhein. Die Zeitschrift des Vereins Niederrhein, 71, 1 (2004), 2–5; Paul Wietzorek, ‘Zum Titelbild: 750 Jahre Stadtrechte Grieth 1254–2004’, ibid., 71, 2 (2004), 54–5; Paul Wietzorek, ‘Zum Titelbild: 650 Jahre Stadt Dahlen (Rheindalen) 1354–2004’, ibid., 71, 3 (2004), 114–15), overviews of town histories (e.g. Eberhard Lebender, ‘Die Weizackerstadt Pyritz. Ein Gang durch die Geschichte – von der Bronzezeit bis zur Zerstörung 1945’, Pommern. Zeitschrift für Kultur und Geschichte, 42, 2 (2004), 8–17) and recent archaeological excavations (e.g., Sven Spiong, ‘Archäologische Ausgrabung an der Paderborner Stadtmauer’, Die Warte, 65, 123 (2004), 23–6; Sven Spiong, ‘Den Stiftsherren auf der Spur: Archäologische Ausgrabung nördlich der Busdorfkirche in Paderborn’, ibid., 65, 124 (2004), 9–10). Anna Helena Schubert's ‘Archäologische Untersuchungen im Bereich der “Untersten Stadtmühle” in Olpe’, Heimatstimmen aus dem Kreis Olpe, 75, 3 (2004), 195–202, is another example of local archaeological case studies. Olpe received its urban charter in 1311; in the German context such an urban charter necessarily involved fortification. Schubert is concerned whether the ‘lower mill’ which was situated outside the first urban wall was erected at the same time or at a later date than this wall, yet has to admit that despite extensive archaeological excavation this question has to remain – at least for the time – unanswered. English articles on local excavations are too numerous to be dealt with adequately in this short review. Two examples may suffice: Robert Cowie's ‘The evidence from royal sites in Middle Anglo-Saxon London’, Medieval Archaeology, 48 (2004), 201–8, looks at the evidence for palaces c. 650 – c. 850 that emerged from recent archaeological investigations in the Cripplegate area of the City and at the Treasury in Whitehall. Mary Alexander, Natasha Dodnell and Christopher Evans have published ‘A Roman cemetery in Jesus Lane, Cambridge’, Proceedings of the Cambridge Antiquarian Society, 93 (2004), 67–94. 32 corpses were unearthed (three of them decapitated), and modest grave goods were found. This cemetery seems to have served a suburban settlement within the lower Roman town. Pottery assemblage indicates industrial activity. The excavation added significantly to our knowledge of the layout and scale of Roman Cambridge. Cambridge clearly remained a significant centre during the fourth century and sustained an economic and commercial role.


2010 ◽  
Vol 6 (1) ◽  
pp. 229-263
Author(s):  
David M. Gwynn

The so-called ‘Arian Controversy’ that divided the Christian Church in the 4th c. has been the subject of considerable scholarly debate in recent decades. The literary sources from which the majority of our knowledge of the controversy derives are highly polemical and distorted, written almost exclusively from the perspective of those whose positions would come to be accepted as ‘orthodox’, and this in turn has directly influenced scholarly interpretations of the material evidence from this crucial period in the history of the Church. In this paper I wish to reconsider that material evidence and ask how an archaeological approach independent of the biases of our literary sources might broaden our understanding of the controversy and its impact upon the 4th c. Roman empire.


1954 ◽  
Vol 7 (4) ◽  
pp. 353-375 ◽  
Author(s):  
John Mcintyre

The subject of this paper abounds in historical problems of an extremely intricate nature—some of which arise through the theologically close association of the doctrine of the Holy Spirit with the doctrine of the Holy Trinity; but others of which are rooted in a number of more particular controversies in which the Greek Fathers were involved concerning the Holy Spirit. Among the latter are to be found such questions as the precise identification of the persons against whom many of the Greek writings were directed, for example, who the ‘Tropici’ were, with whom Athanasius deals in the Letters to Serapion, whether ‘Pneumatomachi’ was a generic term used to describe a variety of different heretics, or a proper name referring to an identifiable group existing in one particular place, whether Macedonius was a Macedonian, and so on. Included in the intricate historical problems raised in our period, there is that of accounting for the revival towards the middle of the fourth century of an interest in the doctrine of the Holy Spirit— whether it was due to the developing influence of asceticism, or simply a reaction to a latter-day Arianism working itself out belatedly in heresy concerning the Holy Spirit, or more subtly, whether it was due to a necessity felt by the Church to give substance to the Person and Work of the Holy Spirit, now that the doctrine of the Trinity had replaced the doctrine of the Logos as her central and dominant doctrine.


1965 ◽  
Vol 85 ◽  
pp. 62-71 ◽  
Author(s):  
R. A. Higgins ◽  
R. P. Winnington-Ingram

The primary purpose of this article is to publish two terracotta representations of lute-players in the British Museum. The subject is rare, but not quite so rare as might be supposed from the scarcity of literature about it. It has, therefore, seemed worth while to add a Ust of the examples known to us—a list which does not claim to be exhaustive—and to discuss briefly some of the problems which they raise. We do this in the hope that it may stimulate further investigation of a neglected theme.Between lutes and lyres there is a difference of principle which could hardly be more fundamental. The strings of the lyre are relatively numerous, but, in default of a fingerboard, fret-board, or neck, against which they could be firmly pressed (or ‘stopped’), the possibilities of obtaining more than one note from each string, in so far as they existed, must have been limited as to the number and quality of notes obtainable. The lute has few strings, but they are stretched over a solid neck, or a prolongation of the sound-box, against which they can be pressed so as to shorten the string-length and produce notes of higher pitch than those of the open strings; each string can thus provide a number of notes of approximately equal quality. Lutes and lyres were both common in Asia and in Egypt. In Greek lands the lyre predominated, and no examples of the lute are found in art before the fourth century B.C. The examples known to us are mostly terracottas.


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