Consideration of the objections raised against the geometrical repre­sentation of the square roots of negative quantities

1833 ◽  
Vol 2 ◽  
pp. 371-372
Keyword(s):  
A Priori ◽  
The Other ◽  
Square Root ◽  
Square Roots ◽  
Negative Direction ◽  
The Third ◽  
Oblique Direction ◽  
Positive Quantity ◽  
Algebraic Operations ◽  
The One

It has always appeared a paradox in mathematics, that by em­ploying what are called imaginary or impossible quantities, and sub­jecting them to the same algebraic operations as those which are performed on quantities that are real and possible, the results ob­tained should always prove perfectly correct. The author inferring from this fact, that the operations of algebra are of a more compre­hensive nature than its definitions and fundamental principles, was led to inquire what extension might be given to these definitions and principles, so as to render them strictly applicable to quantities of every description, whether real or imaginary. This deficiency, he conceives, may be supplied by having recourse to certain geometrical considerations. By taking into account the directions as well as the lengths of lines drawn in a given plane, from a given point, the ad­dition of such lines may admit of being performed in the same man­ner as the composition of motions in dynamics; and four such lines may be regarded as proportional, both in length and direction, when they are proportionals in length, and, when also the fourth is inclined to the third at the same angle that the second is to the first. From this principle he deduces, that if a line drawn in any given di­rection be assumed as a positive quantity, and consequently its op­posite a negative quantity, a line drawn at right angles to the posi­tive or negative direction will be represented by the square root of a negative quantity ; and a line drawn in an oblique direction will be represented by the sum of two quantities, the one either positive or negative, and the other the square root of a negative quantity. On this subject, the author published a treatise in April 1828; since which period several objections have been made to this hypothesis. The purpose of the present paper is to answer these objections. The first of these is, that impossible roots should be considered merely as the indications of some impossible condition, which the pro­position that has given rise to them involves; and that they have in fact no real or absolute existence. To this it is replied by the author, that although such a statement may be true in some cases, it is by no means necessarily so in all; and that these quantities re­semble in this respect fractional and negative roots, which, whenever they are excluded by the nature of the question, are indeed signs of impossibility, but yet in other cases are admitted to be real and significant quantities. We have therefore no stronger reasons, à priori , for denying the real existence of what are called impossible roots, because they are in some cases the signs of impossibility, than we should have for refusing that character to fractional or negative roots on similar grounds.

scholarly journals XIX. —Consideration of the objections raised against the geometrical representation of the square roots of negative quantities

1829 ◽  
Vol 119 ◽  
pp. 241-254 ◽  
Keyword(s):  
The Other ◽  
Square Root ◽  
Square Roots ◽  
The Third ◽  
Oblique Direction ◽  
Real Nature ◽  
Positive Quantity ◽  
Algebraic Operations ◽  
The One ◽  
Straight Lines

Some years ago my attention was drawn to those algebraic quantities, which are commonly called impossible roots or imaginary quantities: it appeared extraordinary, that mathematicians should be able by means of these quan­tities to pursue their investigations, both in pure and mixed mathematics, and to arrive at results which agree with the results obtained by other independent processes; and yet that the real nature of these quantities should be entirely unknown, and even their real existence denied. One thing was evident re­specting them; that they were quantities capable of undergoing algebraic operations analogous to the operations performed on what are called possible quantities, and of producing correct results: thus it was manifest, that the operations of algebra were more comprehensive than the definitions and funda­mental principles; that is, that they extended to a class of quantities, viz. those commonly called impossible roots, to which the definitions and funda­mental principles were inapplicable. It seemed probable, therefore, that there was a deficiency in the definitions and fundamental principles of algebra ; and that other definitions and fundamental principles might be discovered of a more comprehensive nature, which would extend to every class of quantities to which the operations of algebra were applicable; that is, both to possible and impossible quantities, as they are called. I was induced therefore to examine into the nature of algebraic operations, with a view, if possible, of arriving at these general definitions and fundamental principles: and I found, that, by considering algebra merely as applied to geometry, such principles and definitions might be obtained. The fundamental principles and definitions which I arrived at were these: that all straight lines drawn in a given plane from a given point, in any direction whatever, are capable of being algebra­ically represented, both in length and direction; that the addition of such lines (when estimated both in length and direction) must be performed in the same manner as composition of motion in dynamics; and that four such lines are proportionals, -both in length and direction, when they are proportionals in length, and the fourth is inclined to the third at the same angle that the second is to the first. From these principles I deduced, that, if a line drawn in any given direction be assumed as a positive quantity, and consequently its oppo­site, a negative quantity, a line drawn at right angles to the positive or nega­tive direction will be the square root of a negative quantity, and a line drawn in an oblique direction will be the sum of two quantities, the one either posi­tive or negative, and the other, the square root of a negative quantity.

Distinctions, Differentiations, Ontology, and Non-humans in Theories of Religion

2010 ◽  
Vol 22 (4) ◽  
pp. 354-374 ◽  
Author(s):  
Michael Stausberg
Keyword(s):  
Network Theory ◽  
Final Section ◽  
A Priori ◽  
Actor Network Theory ◽  
Functional Differentiation ◽  
The Other ◽  
Process Stage ◽  
The Third ◽  
Theories Of Religion ◽  
The One

AbstractThis essay has four main parts. (1) Reviewing previous theories of religion, it suggests that it may be helpful not to conflate, a priori, the notions of (the) religious on the one hand and religion\s on the other, and that it may be useful to explore concepts such as (the) sacred and transcendence as independent yet related to the business of theorizing religion. (2) Distinguishing social/cultural from biological/genetic evolution, it outlines the occurrence of three processes/stages of the evolution of religious affairs and religions(s), here called attributive, structural, and functional differentiation respectively. While the first two processes/stages occurred in the remote and ancient past respectively, the third process/stage is typical of modernities and has by now globalized. (3) The article argues that recent criticisms of the validity of the category of religion are informed by a reverse sui generis approach characterized by a tacit claim that religion is an anomaly, by virtue of its supposedly being inherently different from similar concepts. The article suggests that John Searle’s philosophy may throw light on the mode of existence (ontology) of religion as an example of social and institutional reality, as an intentionality- and observer-relative yet real and empowering structure. (4) In the final section, the article engages some lines of thinking of Bruno Latour’s interpretation of Actor-Network-Theory, in particular the category of non-humans and the importance of things (objects) for social reality, including religion.

Introduction

2017 ◽  
Author(s):  
Patrick Colm Hogan
Keyword(s):  
Queer Theory ◽  
Social Constructionism ◽  
The Other ◽  
Sexual Identities ◽  
The Third ◽  
Sexuality And Gender ◽  
And Gender ◽  
Sexual Part ◽  
The One ◽  
And Behavior

The introduction first sets out some preliminary definitions of sex, sexuality, and gender. It then turns from the sexual part of Sexual Identities to the identity part. A great deal of confusion results from failing to distinguish between identity in the sense of a category with which one identifies (categorial identity) and identity in the sense of a set of patterns that characterize one’s cognition, emotion, and behavior (practical identity). The second section gives a brief summary of this difference. The third and fourth sections sketch the relation of the book to social constructionism and queer theory, on the one hand, and evolutionary-cognitive approaches to sex, sexuality, and gender, on the other. The fifth section outlines the value of literature in not only illustrating, but advancing a research program in sex, sexuality, and gender identity. Finally, the introduction provides an overview of the chapters in this volume.

Introducing Morphology

2021 ◽  
Author(s):  
Rochelle Lieber
Keyword(s):  
Mental Lexicon ◽  
The Other ◽  
Morphological Data ◽  
Theoretical Frameworks ◽  
The Third ◽  
Hands On ◽  
Morphological Theory ◽  
Basic Concepts ◽  
The One

A lively introduction to morphology, this textbook is intended for undergraduates with relatively little background in linguistics. It shows students how to find and analyze morphological data and presents them with basic concepts and terminology concerning the mental lexicon, inflection, derivation, morphological typology, productivity, and the interfaces between morphology and syntax on the one hand and phonology on the other. By the end of the text students are ready to understand morphological theory and how to support or refute theoretical proposals. Providing data from a wide variety of languages, the text includes hands-on activities designed to encourage students to gather and analyse their own data. The third edition has been thoroughly updated with new examples and exercises. Chapter 2 now includes an updated detailed introduction to using linguistic corpora, and there is a new final chapter covering several current theoretical frameworks.

Space, Time and Mind-Dependence

2011 ◽  
Vol 16 (2) ◽  
pp. 175-190 ◽  
Author(s):  
Sorin Baiasu
Keyword(s):  
Transcendental Idealism ◽  
Space Time ◽  
The Other ◽  
Special Issue ◽  
Critical Philosophy ◽  
The Third ◽  
The One

AbstractThe interpretation of Kant's Critical philosophy as a version of traditional idealism has a long history. In spite of Kant's and his commentators’ various attempts to distinguish between traditional and transcendental idealism, his philosophy continues to be construed as committed (whether explicitly or implicitly and whether consistently or inconsistently) to various features usually associated with the traditional idealist project. As a result, most often, the accusation is that his Critical philosophy makes too strong metaphysical and epistemological claims.In his The Revolutionary Kant, Graham Bird engages in a systematic and thorough evaluation of the traditionalist interpretation, as part of perhaps the most comprehensive and compelling defence of a revolutionary reading of Kant's thought. In the third part of this special issue, the exchanges between, on the one hand, Graham Bird and, on the other, Gary Banham, Gordon Brittan, Manfred Kuehn, Adrian Moore and Kenneth Westphal focus on specific aspects of Bird's interpretation of Kant's first Critique. More exactly, the emphasis is on specific aspects of Bird's interpretation of the Introduction, Analytic of Principles and Transcendental Dialectic of Kant's first Critique.The second part of the special issue is devoted to discussions of particular topics in Bird's construal of the remaining significant parts of the first Critique, namely, of the Transcendental Aesthetic and the Analytic of Concepts. Written by Sorin Baiasu and Michelle Grier, these articles examine specific issues in these two remaining parts of the Critique, from the perspective of the debate between the traditionalist and revolutionary interpretation. The special issue begins with an Introduction by the guest co-editors. This provides a summary of the exchanges between Bird and his critics, with a particular focus on the debates stemming from the differences between traditional and revolutionary interpretations of Kant.

‘How, or Why, do we Come to Think of a World of Things in Themselves?’

2011 ◽  
Vol 16 (2) ◽  
pp. 221-233 ◽  
Author(s):  
Manfred Kuehn
Keyword(s):  
Transcendental Idealism ◽  
The Other ◽  
Special Issue ◽  
Critical Philosophy ◽  
The Third ◽  
The One

AbstractThe interpretation of Kant's Critical philosophy as a version of traditional idealism has a long history. In spite of Kant's and his commentators’ various attempts to distinguish between traditional and transcendental idealism, his philosophy continues to be construed as committed (whether explicitly or implicitly and whether consistently or inconsistently) to various features usually associated with the traditional idealist project. As a result, most often, the accusation is that his Critical philosophy makes too strong metaphysical and epistemological claims.In his The Revolutionary Kant, Graham Bird engages in a systematic and thorough evaluation of the traditionalist interpretation, as part of perhaps the most comprehensive and compelling defence of a revolutionary reading of Kant's thought. In the third part of this special issue, the exchanges between, on the one hand, Graham Bird and, on the other, Gary Banham, Gordon Brittan, Manfred Kuehn, Adrian Moore and Kenneth Westphal focus on specific aspects of Bird's interpretation of Kant's first Critique. More exactly, the emphasis is on specific aspects of Bird's interpretation of the Introduction, Analytic of Principles and Transcendental Dialectic of Kant's first Critique.The second part of the special issue is devoted to discussions of particular topics in Bird's construal of the remaining significant parts of the first Critique, namely, of the Transcendental Aesthetic and the Analytic of Concepts. Written by Sorin Baiasu and Michelle Grier, these articles examine specific issues in these two remaining parts of the Critique, from the perspective of the debate between the traditionalist and revolutionary interpretation. The special issue begins with an Introduction by the guest co-editors. This provides a summary of the exchanges between Bird and his critics, with a particular focus on the debates stemming from the differences between traditional and revolutionary interpretations of Kant.

scholarly journals Note sur la théorie néo-keynésienne du producteur

2009 ◽  
Vol 57 (2) ◽  
pp. 137-147
Author(s):  
Marie Allard ◽  
Camille Bronsard ◽  
Gilles McDougall
Keyword(s):  
Classical Theory ◽  
A Priori ◽  
Comparative Statics ◽  
The Other ◽  
Structural Form ◽  
Other Hand ◽  
The One

ABSTRACT While the meaningful theorems of neo-classical theory of the producer are well known, the neo-keynesian counterparts are not. Therefore, this paper will present those new meaningful theorems and their relations with neo-classical theory. On the one hand, this paper is of interest to the theoretician who would want to use the properties of comparative statics of the producer with quantitative rationing. On the other hand, since a neo-keynesian structural form is presented, the econometrician will be interested in imposing the meaningful theorems of this theory as a priori restrictions.

The Impact of Harmful Habits on Academic Performance and Sports Activities among Young People

2021 ◽  
Vol 9 (3) ◽  
pp. 267-273
Author(s):  
Victoria Yermilova ◽  
◽  
Natalia Stroiteleva ◽  
Zhanna Egorova ◽  
Ekaterina Vanina
Keyword(s):  
Gender Differences ◽  
Academic Performance ◽  
Alcohol Consumption ◽  
Young People ◽  
The Other ◽  
Control Group ◽  
The Third ◽  
The One ◽  
The Impact ◽  
Sports Activities

Smoking and alcohol consumption is a growing trend among young people worldwide. The purpose of this study was to provide students with a comparative analysis of adherence to harmful habits (smoking and alcohol) on the one hand and the frequency of sports and academic performance on the other, taking into account gender differences. The research was conducted in 2019-2020 in 5 cities of Russia; the sample included 1500 people aged 18.4 ± 1.1 years, divided into three equal groups. The control (first) group had students who are not engaged in sports, and the second group comprised students practicing sports but not professionally. The third group was made up of student-athletes. All participants were surveyed to determine the frequency of adherence to harmful habits. In the control group, boys smoked 50% more often than girls (p ≤ 0.05), while in the third group, smoking among boys was registered 70 times less often (p ≤ 0.001). Alcohol consumption in controls was 0.5 times more likely among boys (p ≤ 0.05). Harmful habits affect young people's free time and reduce their academic performance and ability to practice sports.

Marginality in the Mainstream, Lesbian Love in the Third Person

2016 ◽  
Author(s):  
Emily Van Buskirk
Keyword(s):  
French Literature ◽  
Gender And Sexuality ◽  
The Other ◽  
The Third ◽  
The World ◽  
Third Person ◽  
Philosophical Texts ◽  
And Gender ◽  
The One ◽  
The Third Person

This chapter undertakes a treatment of the rhetoric of personal pronouns in Ginzburg's writings on love and sexuality, drawing on Michael Lucey's study of the first person in twentieth-century French literature about love. It brings together questions of genre and narrative, on the one hand, and gender and sexuality, on the other. The chapter is divided into two sections, treating writings from two different periods on two kinds of love Ginzburg thought typical of intellectuals: in “First Love,” it discusses the unrequited and tragic love depicted in Ginzburg's teenage diaries (1920–23); in “Second Love,” it analyzes the love that is realized but in the end equally tragic, depicted in drafts related to Home and the World (1930s). The chapter examines the models the author sought in literary, psychological, and philosophical texts (Weininger, Kraft-Ebbing, Blok, Shklovsky, Oleinikov, Hemingway, and Proust).

IV.—On a Question in Absorption Spectroscopy

1909 ◽  
Vol 29 ◽  
pp. 68-74
Author(s):  
Robert A. Houstoun ◽  
Alexander S. Russell
Keyword(s):  
Absorption Spectrum ◽  
Absorption Spectroscopy ◽  
Copper Sulphate ◽  
Potassium Dichromate ◽  
The Other ◽  
Ammoniacal Solution ◽  
The Third ◽  
Physical Action ◽  
Apparent Shift ◽  
The One

In the third volume of his Spectroscopie, p. 91, Kayser has raised the question whether on mixing two coloured solutions which do not act on one another chemically the absorption spectrum of each of the components remains unchanged. Melde thought he had discovered such an effect; he stated that when a solution of carmine in ammonia which has two sharp bands in the green, was added to a solution of potassium dichromate which absorbs the violet end of the spectrum, or to an ammoniacal solution of copper sulphate which absorbs the red end, that the carmine bands were in each case displaced towards the end absorption in question. He ascribed this to a physical action between the molecules. It was, however, pointed out by Schuster that a shift of this nature would be seen if, instead of mixing, the one solution was merely placed behind the other. Bostwick and Krüss repeated Melde's work, and came to the conclusion that there was a real shift in addition to the apparent shift pointed out by Schuster. Since then additional evidence has been adduced by Formánek and has been quoted by Kayser in the section cited above. The object of the research recorded in this paper was to investigate those cases with the most accurate means possible, and, if a shift was established, to decide if it was physical.

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