In Part I of his paper Cooper gives indisputable evidence regarding Plato's use of the man-made image as a step to the apprehension of a Form under discussion, whether that image be in fact a diagram or a model, or simply a verbal picture, such as his imaginative account of Justice within a community, which we find used to provide us with in Republic 443 c 4 ff. However, Cooper goes on to assure us that the divided-line figure offers us (...) only three types of object: ‘We have three kinds of objects which differ from one another in clearness and esteem, firstly, the Forms, secondly, the objects of ordinary sense-perception, and thirdly, images, shadows and reflections.’ Now the admitted fact that, as he notes, Republic 10 gives the same three orders of reality, does not entirely absolve Cooper from all the implications of Plato's decision to divide his line into four parts rather than three; for it is made quite clear in 509 b 6–10 that the Form of the Good, is the source of being as well as of knowledge, so the Line must also classify both. (shrink)

In Part I of his paper Cooper gives indisputable evidence regarding Plato's use of the man-made image as a step to the apprehension of a Form under discussion, whether that image be in fact a diagram or a model, or simply a verbal picture, such as his imaginative account of Justice within a community, which we find used to provide us with in Republic 443 c 4 ff. However, Cooper goes on to assure us that the divided-line figure offers us (...) only three types of object: ‘We have three kinds of objects which differ from one another in clearness and esteem, firstly, the Forms, secondly, the objects of ordinary sense-perception, and thirdly, images, shadows and reflections.’ Now the admitted fact that, as he notes, Republic 10 gives the same three orders of reality, does not entirely absolve Cooper from all the implications of Plato's decision to divide his line into four parts rather than three; for it is made quite clear in 509 b 6–10 that the Form of the Good, is the source of being as well as of knowledge, so the Line must also classify both. (shrink)

This is a companion paper to my preceding one on Harriot's experimentations in the field of the sign-rule of multiplication in algebra. Cardano had earlier attacked the conventional rule in a chapter of his De Aliza regula liber, published in 1570 as an appendix to the second edition of his Ars magna. He returned to the subject in a brief tract, published nearly a century later in his collected works as Sermo de plus et minus. Only Cardano's valid contention that (...) the standard ‘proof’ of the accustomed sign-rule was faulty made impact, in the Euclidean background to which the ‘proof’ belonged. His cause was to preserve the purity of the Euclidean tradition. It overshadows the true aim in the De Aliza chapter, but the Sermo reveals it plainly as far from retrograde: he wished to retain all the new techniques of algebra that brought in not only ‘minus’ quantities but also their square roots, and to escape the ‘impossible’ status of these last. This the rebel sign-rule did for him by making every root of a ‘minus’ just another ‘minus’. (shrink)

Torporley's ‘Congestor analyticus’, completed in 1627 in the library of the Earl of Northumberland at Petworth, was seen by Rigaud in the 1830s among the mathematical manuscript collection of the Earl of Macclesfield. Torporley's additional copy of the introductory part, preserved at Sion College, has been used for the present report. Torporley's prime objective was the presentation of some of Harriot's work. His first example concerns a classical problem in number theory. The complete solution, by an inductive process based on (...) tables of differences, is given in a few folios of the British Library collection of Harriot manuscripts. Torporley's transcription and commentary contrast forcibly today with the immediate illuminating impact of Harriot's original. (shrink)

The ratification of the United Nations Convention on the Rights of the Child (UNCRC) demands that those participating nations, adopt the aims of the convention as state responsibilities toward their child citizens. The central premise of the convention is clear: that it is the right of all children to develop to their full potential. The authors propose six basic interdependent developmental requirements if the child is to reach ‘full potential’. Without prioritising any one need, but instead concentrating on the facilitation (...) of complete care of the child, the paper aims to examine how the UNCRC may finally be realised. This is achieved by highlighting several key situations and associated factors, which contribute to deficient care. Whilst the factors leading to the creation of such situations are often culturally specific, the consequences for the child, and the implications for care providers (parents, aid agencies, and policy makers), are often very similar within and between continents. (shrink)

According to the “fitting-attitudes” (FA) account of value, for a thing to be valuable is for it to be the fitting object of a pro-attitude. Value here is analyzed in terms of reasons for and against favoring, admiring, desiring, preferring, loving, etc. a thing. Whichever particular FA analysis you prefer, the basic idea is just that a thing’s value depends on extant reasons to be favorably (or disfavorably) disposed toward it. Of course, proponents of FA analyses deny that just any (...) such reasons suffice to ground a thing’s value. The reasons must be of the right sort. If I threaten to stab you in the face unless you become favorably disposed toward Rob Schneider movies, you now have a reason to do just that. But it seems clear that while my threat does make Rob Schneider movies to you worth liking, it does not make them valuable or good. The difficulty then is to distinguish the right kind of reasons from the wrong kind. Several writers have recently tried to offer principled ways of resolving the so-called “wrong kind of reasons” (WKR) problem, though I will not closely examine them here. Instead I wish to focus on one particular FAaccount of value whom some have suggested is immune to WKR-style problems, specifically the one Michael Zimmerman offers in The Nature of Intrinsic Value. I argue that even Zimmerman’s account incurs WKR difficulties, and that it can actually help illustrate a certain deep problem with FA accounts in general. (shrink)

The following essay is adapted from one with the same title read to the British Society for the History of Science on 20 October 1958—the anniversary, by a striking coincidence, of the birth of W. H. Young . To his memory I dedicated the talk, and now rededicate its publication, not only because I am his daughter and of all that means, but because he invented a method, the method of monotone sequences, which shows the powerfulness of inequalities as a (...) mathematical tool supremely. (shrink)

I. By Ictus we mean in this paper the sounds emphasized in the pattern of an utterance in the given language under discussion. So in languages like Chinese which depend on variation of tone we mean that the high notes in the intonation tune of a sentence or the rhythmic scheme of a verse carry an ictus; while in a language based, like English, on speech stress, we mean that the syllables uttered most loudly and clearly bear the ictus. Again, (...) some languages, such as Italian, emphasize a syllable by a combination of stress and raised pitch, so in these languages we say that the syllables so emphasized receive the ictus. Lastly, in a language like ancient Greek or Vedic Sanskrit where quantity is decisive in utterance, the syllables which receive emphasis are the long ones, and thus it is these which have an ictus. (shrink)

I. By Ictus we mean in this paper the sounds emphasized in the pattern of an utterance in the given language under discussion. So in languages like Chinese which depend on variation of tone we mean that the high notes in the intonation tune of a sentence or the rhythmic scheme of a verse carry an ictus; while in a language based, like English, on speech stress, we mean that the syllables uttered most loudly and clearly bear the ictus. Again, (...) some languages, such as Italian, emphasize a syllable by a combination of stress and raised pitch, so in these languages we say that the syllables so emphasized receive the ictus. Lastly, in a language like ancient Greek or Vedic Sanskrit where quantity is decisive in utterance, the syllables which receive emphasis are the long ones, and thus it is these which have an ictus. (shrink)

The manuscripts of Harriot discussed in this paper are essentially rough notes marginal to his systematic treatment of algebra, of which a small part was published posthumously. The central theme is the sign-rule for multiplication; but the incidentals open up an aspect of symbolism in mathematics entirely new for the time. A more restricted aspect of the same theme was touched on by Commandino in his Euclid, quoted by Harriot as rightly blaming ‘those that thinke that minus per minus shal (...) produce plus’; Commandino's authority was Cardano, in a similarly marginal departure from his great Ars magna, the de Aliza, which Harriot also cites in the same reference. The ostensible motivation, to preserve intact the Euclidean tradition in geometry, plays a very minor role in Harriot's notes. The basic aim, to elude the fatal link with imaginaries, they do touch on. More dimly underlying the arguments are the crucial logical facets of the problem, which Harriot faced with a greater conceptual clarity, formulating in words, however, only his own conclusion: the choice of sign-rule is the user's—a fact not fully recognised by mathematicians generally until the 19th century. (shrink)