When studying properties of the circumcircle of an integer triangle, it quickly becomes evident that the radius of such a circle (circumradius) need not itself be an integer. When it is not an integer, the circumradius can still be rational but it can also be irrational, as exemplified in the following examples. It is left to the reader to verify that the triangle with sides 10, 24, 26 has circumradius 13, and that the corresponding values for the triangles with sides 13, 14, 15 and 1, 1, 1 are 65/8 and respectively. It is shown in Theorem 1 below that a necessary condition for the circumradius to be an integer is that the area of the triangle is itself an integer (Heronian triangle) but this condition is not in itself sufficient. A simple counterexample is given by the 13, 14, 15 triangle above which has area 84. However, as a consequence of Theorem 1, we can restrict ourselves to consideringHeronian triangles, and relevant properties of such triangles, proved in [1], are given in Theorems 2 and 3 below. We also need to quote some well-known results involving the sums of two squares (see e.g. [2]) and these are listed in Lemma 3. In all that follows, we will use the convention that if T is a triangle with sides a, b, c and z > 0, then zT will denote the triangle, similar to T, with sides za, zb, zc.

The present work explores the link between navigational processes and the experience of place by considering the case of Evenki reindeer herders and hunters. Our analysis shows how the idiosyncratic wayfinding methods of the Evenki result in a unique experience of place – a case that elucidates the important question of the impact of navigational processes on environmental experience, and that advances the debate between mental map theory and practical mastery theory in anthropology. We defend that their wayfinding methods – involving a particular gait, path networks, and vast hydrological and toponymical knowledge – allow the Evenki to navigate without a need for integrating egocentric and allocentric frames of reference. As a result, the Evenki experience themselves as free individuals moving through an environment that is alive and rife with possibility. This analysis reveals the ways in which wayfinding processes relying predominantly on route knowledge – as opposed to survey knowledge – affect environmental experience. Alternative methods of wayfinding can be seen as a form of resistance to the uniformisation of landscapes, and as a way of embracing the heterogeneity of space.

IR proceeds on a Eurocentric ontological assumption that sovereignty has universal validity today. How can IR be decolonised, when in spite of countless examples of the enactment of ‘sovereignty otherwise’, the discipline remains unconcerned with the fact that the logic of sovereignty remains uni-versal. The question is as much political as it is intellectual, because as a discipline, we have allowed the inertia of our professional rhythms to marginalise pluri-versal sovereignty, or the organisation of sovereignty along different ontological starting points. I argue IR must abandon its disciplinary love affair with uni-versal sovereignty. The tendency to ‘bring in’ new perspectives by inserting them into an already ontologically constituted set of assumptions works to protect IR’s Eurocentricity, which makes disciplinary decolonisation untenable. I propose that as a starting point, IR needs to be more mature about recognising the decolonisations that are happening under our very feet if we are to stand a chance at disciplinary level decolonisation. As an illustrative example, I explore an ongoing collision of settler-colonial and Mi’kmaw sovereignty through the issue of lobster fisheries in Mi’kma’ki, or Nova Scotia as the territory is known to Canadians.

Although underappreciated in his own day, Catholic convert John Henry Newman was remarkably prophetic about the challenges that lay ahead for the Catholic faith. In his 1873 sermon titled, ‘The Infidelity of the Future’, Newman warned of a time when the Church would face not only the cold indifference of agnosticism but also the targeted hostility of those opposed to both God and religion. Yet Newman was not without hope or wisdom for the future Church. This essay examines Newman's insistence upon the need to cultivate an ‘ecclesiastical spirit’ and an ‘intelligent faith’. It specifically explores how Catholic institutions of higher education can respond to Newman's call and assist in bringing about a renewal in the evangelical mission of the Church, providing a much-needed alternative to the wisdom of the world.

This article revisits the role of cognitive individual differences in creating synergies between second language (L2) writing and second language acquisition research that were proposed by Kormos (2012). It takes stock of the advances in research findings, in the past decade, on the role of working memory and language learning aptitude in L2 writing processes and performance. The article offers an overview of how cognitive factors can mediate learning gains when L2 users engage in writing. Using theoretical accounts of cognitive individual differences, the article discusses how the characteristics of writing tasks can interact with individual variation in cognitive functioning. The article concludes by proposing a Task-Mediated Cognitive Model of L2 Writing and Writing to Learn that describes the role of cognitive factors in L2 writing processes and in learning through writing, and an outline of a research agenda for future studies.

Λιβανωτός is a rare word in the Biblia graeca and means ‘frankincense’. It appears once in the canonical Septuagint in 1 Chron 9.29 as part of a list of ingredients which were under the care of the Levites: flour, wine, olive oil, incense and spices. In the Apocrypha, it appears in 3 Macc 5.2 as a drug, together with unmixed wine, for maddening or running elephants wild. Then it is used only in Rev 8.3, 5 in constructions which made lexicographers unanimously define λιβανωτός as a container (censer or brazier). However, when one examines the usage of this noun in Greek writing at large, he or she observes, not without surprise, that λιβανωτός exhibits impressively stable semantics. Virtually everywhere in the history of Greek, the term is a spice (frankincense). Why then should Rev 8.3, 5 be an exception? The study probes into the claim that λιβανωτός means ‘censer’ in the Johannine Apocalypse, shows how well the regular meaning of incense fits in the scene John witnesses, and draws important implications for the understanding of the text and the lexicographical task.

For centuries, Christians have understood some of the texts included in the New Testament as ‘Jewish,’ in the sense of them being written by (converted) Jews for other Jews. From a historical perspective, a new development in the academy suggests that such approaches do not do justice to the nature of these texts. Indeed, even more recent attempts at understanding the New Testament against the background of Judaism are also found wanting. Instead, placing these texts within the broader context of the diverse ways of embodying Jewish ancestral customs in the pre-rabbinic Second Temple period, this interpretive trajectory, involving scholars from a wide array of backgrounds, insists that Paul, Matthew, Mark, Luke, John, Revelation etc., should be understood as expressions of Judaism. This article highlights key issues involved in such re-readings of New Testament texts, including ways in which they may or may not relate to normative-theological positions among Christians and Jews today. First, the study looks at how the question is asked in our contemporary setting. Then, moving down historical layers, issues related to history and categorisation are addressed before we, finally, return to the present to consider possible implications of our findings.

This article investigates the prolific deployment of images in Pascal Dusapin's scores of the mid to late 1990s. Using Edwin Gordon's concept of ‘audiation’ and Siglind Bruhn's concept of ‘musical ekphrasis’, as well as Neal Curtis’ ideas on the agency and liveness of images, interdisciplinary interpretations of three Dusapin works are offered, using the score image as the main analytical starting point. Beginning with his Piano Études and Loop, these analyses will demonstrate how the score images, derived from the study of ‘catastrophe theory’, prefigure the control of various musical parameters relating to the relationship of variables in this mathematical concept and its forms. An analysis of String Quartet No. 4, notable for its use of both a score image (of a machine) and textual quotation (from Samuel Beckett's Murphy), will then demonstrate how the visual can interact with the verbal to construct a plurality of musical metaphors. The article concludes by positing that Dusapin's elusiveness on the role and function of the images within his work could amount to what W. T. J. Mitchell describes as ‘ekphrastic fear’: an anxiety that the composer's success amalgamating image and music might have broken down the ontological boundary between them.

This article provides an overview of compositional practices that invoke representations of the human voice by acoustic means, without the presence of actual human speakers, and, in identifying the limitations and deficiencies of extant practice, proposes a new method or methodological basis for achieving the same phenomenon, to different ends. Clarence Barlow's seminal synthrumentation method forms a central point of discussion; when considered alongside the work of, among others, Mayuzumi, Grisey and Ablinger, it becomes clear that, in existing synthrumental (or instrumental synthesis) practices, the innate inharmonic components of target material for reproduction have largely been discarded or overlooked. After a brief overview of the basic linguistic and perceptual features of speech, a new method designed to evoke whispered speech, with inharmonicity in mind, is demonstrated.

Elliott Carter's programme notes for his String Quartet No. 5 describe it as being about the embodiment of human interaction within the rehearsal process. This article develops this concept, evaluating the musical figures that are foreshadowed by the fragments that Carter suggests are rehearsal outtakes. Certain motives are reiterated and developed through slight variations, thus exemplifying the rehearsal process, and perhaps the editorial process, in detail. Interactions within this model are suggestive of the character types that Carter has delineated in his previous string quartets, notably No. 2. Using the Practice Session model also alludes to the real-life circumstances of the preparation of previous quartets by ensembles, and anecdotes about the Juilliard Quartet's rehearsals for the premiere of the String Quartet No. 3 can enhance a narratological understanding of the No. 5's construction. Finally, String Quartet No. 5 is considered as an example of one of the transitional works that initiate Carter's late style and its consolidation of material; its use of all-interval chords, their subsets and supersets reflects the constructive elements of human interaction that Carter has stressed as a principal thematic element.

This article focuses on the implications of modernity for human culpability and moral responsibility. Although the sacrament of Penance and Reconciliation is most often approached theologically and pastorally, this article is intended as an answer to Pope Benedict XVI's call to explore catechesis through new lenses by adopting a psychological therapeutic approach. As such, this article will examine how the rejection of religious ascription to God for defining and determining the good and re-ascribing it to humanity leads to a rupture and the psychological conditions of anxiety, depression, and melancholia. The article will go on to argue for a Lacanian reading of Thomas Aquinas’ definition of the good and how the Thomistic understanding provides a more comprehensive approach to determining culpability and overcoming the associated fear which leads to anxiety, depression, and melancholia.