Louis A. Girifalco
- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199228966
- eISBN:
- 9780191711183
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199228966.003.0004
- Subject:
- Physics, History of Physics
Descartes believed he was meant to create the ultimate natural philosophy. He relied primarily on reason and philosophic principles, so most of the physics he developed was wrong. But he was a great ...
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Descartes believed he was meant to create the ultimate natural philosophy. He relied primarily on reason and philosophic principles, so most of the physics he developed was wrong. But he was a great mathematician and created analytic geometry, which was a major step in developing modern mathematics. Mathematics had always primarily meant geometry, which was regarded as being the only absolute truth. Galileo, and even Newton, presented their results in geometric form. Descartes showed that there was a close connection between geometry and algebra. This ultimately led to modern powerful analytic tools. His contributions to philosophy were important because they stressed the need for rigorous logic and for making as few assumptions as possible.Less
Descartes believed he was meant to create the ultimate natural philosophy. He relied primarily on reason and philosophic principles, so most of the physics he developed was wrong. But he was a great mathematician and created analytic geometry, which was a major step in developing modern mathematics. Mathematics had always primarily meant geometry, which was regarded as being the only absolute truth. Galileo, and even Newton, presented their results in geometric form. Descartes showed that there was a close connection between geometry and algebra. This ultimately led to modern powerful analytic tools. His contributions to philosophy were important because they stressed the need for rigorous logic and for making as few assumptions as possible.
Victor J. Katz and Karen Hunger Parshall
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691149059
- eISBN:
- 9781400850525
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691149059.003.0010
- Subject:
- Mathematics, History of Mathematics
This chapter follows up on the mathematical advances made during the sixteenth century, especially in the work of François Viète as he aspired to transform his algebra to realize his aim to “solve ...
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This chapter follows up on the mathematical advances made during the sixteenth century, especially in the work of François Viète as he aspired to transform his algebra to realize his aim to “solve every problem.” Though Viète's algebra was not up to the task, two of his followers—Thomas Harriot and Pierre de Fermat—helped to transform that algebra into the problem-solving tool he had envisioned, and René Descartes would later recognize the significance of this work and begin circulating these ideas, thus jumpstarting the transformation of algebra, which this chapter explores through a number of noted intellectuals during the period.Less
This chapter follows up on the mathematical advances made during the sixteenth century, especially in the work of François Viète as he aspired to transform his algebra to realize his aim to “solve every problem.” Though Viète's algebra was not up to the task, two of his followers—Thomas Harriot and Pierre de Fermat—helped to transform that algebra into the problem-solving tool he had envisioned, and René Descartes would later recognize the significance of this work and begin circulating these ideas, thus jumpstarting the transformation of algebra, which this chapter explores through a number of noted intellectuals during the period.
John P. Burgess and Gideon Rosen
- Published in print:
- 1999
- Published Online:
- November 2003
- ISBN:
- 9780198250128
- eISBN:
- 9780191597138
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198250126.003.0003
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
Develops in some detail a strategy of nominalistic interpretation that assumes that points of spacetime are legitimate, concrete, physical entities. What makes the strategy possible is the fact that ...
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Develops in some detail a strategy of nominalistic interpretation that assumes that points of spacetime are legitimate, concrete, physical entities. What makes the strategy possible is the fact that analytic geometry in the style of Descartes can be interpreted in synthetic geometry in the style of Euclid, using triples of points to represent real numbers (namely, as ratios of pairs of line segments connecting each of a pair of points with a third). This strategy can handle classical theories based on Euclidean space and special‐relativistic theories based on Minkowski space. The extension of the strategy to general relativity and quantum mechanics remains to be worked out, as does the treatment of the higher branch of geometry known as descriptive set theory.Less
Develops in some detail a strategy of nominalistic interpretation that assumes that points of spacetime are legitimate, concrete, physical entities. What makes the strategy possible is the fact that analytic geometry in the style of Descartes can be interpreted in synthetic geometry in the style of Euclid, using triples of points to represent real numbers (namely, as ratios of pairs of line segments connecting each of a pair of points with a third). This strategy can handle classical theories based on Euclidean space and special‐relativistic theories based on Minkowski space. The extension of the strategy to general relativity and quantum mechanics remains to be worked out, as does the treatment of the higher branch of geometry known as descriptive set theory.
Victor J. Katz and Karen Hunger Parshall
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691149059
- eISBN:
- 9781400850525
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691149059.001.0001
- Subject:
- Mathematics, History of Mathematics
What is algebra? For some, it is an abstract language of x's and y's. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and ...
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What is algebra? For some, it is an abstract language of x's and y's. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. This book considers how these two seemingly different types of algebra evolved and how they relate. The book explores the history of algebra, from its roots in the ancient civilizations of Egypt, Mesopotamia, Greece, China, and India, through its development in the medieval Islamic world and medieval and early modern Europe, to its modern form in the early twentieth century. Defining algebra originally as a collection of techniques for determining unknowns, the book traces the development of these techniques from geometric beginnings in ancient Egypt and Mesopotamia and classical Greece. It shows how similar problems were tackled in Alexandrian Greece, in China, and in India, then looks at how medieval Islamic scholars shifted to an algorithmic stage, which was further developed by medieval and early modern European mathematicians. With the introduction of a flexible and operative symbolism in the sixteenth and seventeenth centuries, algebra entered into a dynamic period characterized by the analytic geometry that could evaluate curves represented by equations in two variables, thereby solving problems in the physics of motion. This new symbolism freed mathematicians to study equations of degrees higher than two and three, ultimately leading to the present abstract era. The book follows algebra's remarkable growth through different epochs around the globe.Less
What is algebra? For some, it is an abstract language of x's and y's. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. This book considers how these two seemingly different types of algebra evolved and how they relate. The book explores the history of algebra, from its roots in the ancient civilizations of Egypt, Mesopotamia, Greece, China, and India, through its development in the medieval Islamic world and medieval and early modern Europe, to its modern form in the early twentieth century. Defining algebra originally as a collection of techniques for determining unknowns, the book traces the development of these techniques from geometric beginnings in ancient Egypt and Mesopotamia and classical Greece. It shows how similar problems were tackled in Alexandrian Greece, in China, and in India, then looks at how medieval Islamic scholars shifted to an algorithmic stage, which was further developed by medieval and early modern European mathematicians. With the introduction of a flexible and operative symbolism in the sixteenth and seventeenth centuries, algebra entered into a dynamic period characterized by the analytic geometry that could evaluate curves represented by equations in two variables, thereby solving problems in the physics of motion. This new symbolism freed mathematicians to study equations of degrees higher than two and three, ultimately leading to the present abstract era. The book follows algebra's remarkable growth through different epochs around the globe.
Emily R. Grosholz
- Published in print:
- 1991
- Published Online:
- October 2011
- ISBN:
- 9780198242505
- eISBN:
- 9780191680502
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198242505.003.0003
- Subject:
- Philosophy, History of Philosophy, Logic/Philosophy of Mathematics
This chapter investigates Descartes's treatment of curves and his notion of genre. The exposition shows in detail how his method acts as a powerful ...
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This chapter investigates Descartes's treatment of curves and his notion of genre. The exposition shows in detail how his method acts as a powerful problem-solving device which nonetheless weakens its own results and diverts Descartes from important mathematical questions which some of his contemporaries were already exploring. It argues that even if he discovered new higher curves, his discoveries were surprisingly restricted, and that he did not fully exploit the possibilities opened up by his own new analytic geometry for investigating them. The first section argues that Descartes's commitment to his homogenous starting points and line segments, and to the reductive claims of his method, prevents him from being able to focus on his mathematical curves as such.Less
This chapter investigates Descartes's treatment of curves and his notion of genre. The exposition shows in detail how his method acts as a powerful problem-solving device which nonetheless weakens its own results and diverts Descartes from important mathematical questions which some of his contemporaries were already exploring. It argues that even if he discovered new higher curves, his discoveries were surprisingly restricted, and that he did not fully exploit the possibilities opened up by his own new analytic geometry for investigating them. The first section argues that Descartes's commitment to his homogenous starting points and line segments, and to the reductive claims of his method, prevents him from being able to focus on his mathematical curves as such.
Tony Robbin
- Published in print:
- 2006
- Published Online:
- October 2013
- ISBN:
- 9780300110395
- eISBN:
- 9780300129625
- Item type:
- chapter
- Publisher:
- Yale University Press
- DOI:
- 10.12987/yale/9780300110395.003.0005
- Subject:
- Society and Culture, Technology and Society
This chapter serves as a brief lesson on projective geometry. It starts by discussing two basic theorems of projective geometry, which are concerned with perspectives from a point and perspectives ...
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This chapter serves as a brief lesson on projective geometry. It starts by discussing two basic theorems of projective geometry, which are concerned with perspectives from a point and perspectives from a line. The discussion then looks at projectivities and perspectivities, which result from several perspectivities and relate a range, respectively. This is followed by the study of analytic projective geometry and the connection between projective and projection. The chapter concludes with a discussion on the complex projective line, the projective three-space, and Felix Klein's thoughts on the presence of non-Euclidean geometries in projective geometry, specifically the topology of the projective plane.Less
This chapter serves as a brief lesson on projective geometry. It starts by discussing two basic theorems of projective geometry, which are concerned with perspectives from a point and perspectives from a line. The discussion then looks at projectivities and perspectivities, which result from several perspectivities and relate a range, respectively. This is followed by the study of analytic projective geometry and the connection between projective and projection. The chapter concludes with a discussion on the complex projective line, the projective three-space, and Felix Klein's thoughts on the presence of non-Euclidean geometries in projective geometry, specifically the topology of the projective plane.
Ehud Hrushovski and François Loeser
- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691161686
- eISBN:
- 9781400881222
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161686.001.0001
- Subject:
- Mathematics, Geometry / Topology
Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model ...
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Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed and model-theoretic prerequisites are reviewed in the first sections.Less
Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed and model-theoretic prerequisites are reviewed in the first sections.
- Published in print:
- 2009
- Published Online:
- March 2013
- ISBN:
- 9780226168067
- eISBN:
- 9780226168081
- Item type:
- chapter
- Publisher:
- University of Chicago Press
- DOI:
- 10.7208/chicago/9780226168081.003.0005
- Subject:
- History, History of Science, Technology, and Medicine
This chapter presents a set of selections from Marquise Du Châtelet's major work of natural philosophy, the Foundations of Physics, published in Paris in 1740, when she was just thirty-four. “Natural ...
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This chapter presents a set of selections from Marquise Du Châtelet's major work of natural philosophy, the Foundations of Physics, published in Paris in 1740, when she was just thirty-four. “Natural philosophy” is a phrase coined in seventeenth-century England. Like Du Châtelet, the “natural philosopher” sought to describe not only the mechanics of the natural world but also the first causes of phenomena such as motion and gravity, and the role of God. The French used three words to describe those who excelled in this kind of knowledge: philosophe [philosopher], géomètre [mathematician], and physicien [physicist]. Du Châtelet's letter to her bookseller and publisher, Laurent François Prault, included in this collection, gives some idea of the scope of her reading as she worked to become proficient in each of these categories. The Foundations demonstrates Du Châtelet's mastery of mathematics, in particular René Descartes' analytic geometry. This chapter also includes Du Châtelet's letters to Pierre-Louis Moreau de Maupertuis and to Johann Bernoulli II.Less
This chapter presents a set of selections from Marquise Du Châtelet's major work of natural philosophy, the Foundations of Physics, published in Paris in 1740, when she was just thirty-four. “Natural philosophy” is a phrase coined in seventeenth-century England. Like Du Châtelet, the “natural philosopher” sought to describe not only the mechanics of the natural world but also the first causes of phenomena such as motion and gravity, and the role of God. The French used three words to describe those who excelled in this kind of knowledge: philosophe [philosopher], géomètre [mathematician], and physicien [physicist]. Du Châtelet's letter to her bookseller and publisher, Laurent François Prault, included in this collection, gives some idea of the scope of her reading as she worked to become proficient in each of these categories. The Foundations demonstrates Du Châtelet's mastery of mathematics, in particular René Descartes' analytic geometry. This chapter also includes Du Châtelet's letters to Pierre-Louis Moreau de Maupertuis and to Johann Bernoulli II.
Princess Elisabeth of Bohemia and Rene Descartes
- Published in print:
- 2007
- Published Online:
- February 2013
- ISBN:
- 9780226204413
- eISBN:
- 9780226204444
- Item type:
- book
- Publisher:
- University of Chicago Press
- DOI:
- 10.7208/chicago/9780226204444.001.0001
- Subject:
- Literature, Women's Literature
Between the years 1643 and 1649, Princess Elisabeth of Bohemia (1618–80) and René Descartes (1596–1650) exchanged fifty-eight letters—thirty-two from Descartes and twenty-six from Elisabeth. Their ...
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Between the years 1643 and 1649, Princess Elisabeth of Bohemia (1618–80) and René Descartes (1596–1650) exchanged fifty-eight letters—thirty-two from Descartes and twenty-six from Elisabeth. Their correspondence contains the only known extant philosophical writings by Elisabeth, revealing her mastery of metaphysics, analytic geometry, and moral philosophy, as well as her keen interest in natural philosophy. The letters are essential reading for anyone interested in Descartes's philosophy, in particular his account of the human being as a union of mind and body, as well as his ethics. They also provide an insight into the character of their authors, and the way ideas develop through intellectual collaboration. Philosophers have long been familiar with Descartes's side of the correspondence. Elisabeth's letters add context and depth both to Descartes's ideas and the legacy of the princess. This annotated edition also includes Elisabeth's correspondence with the Quakers William Penn and Robert Barclay.Less
Between the years 1643 and 1649, Princess Elisabeth of Bohemia (1618–80) and René Descartes (1596–1650) exchanged fifty-eight letters—thirty-two from Descartes and twenty-six from Elisabeth. Their correspondence contains the only known extant philosophical writings by Elisabeth, revealing her mastery of metaphysics, analytic geometry, and moral philosophy, as well as her keen interest in natural philosophy. The letters are essential reading for anyone interested in Descartes's philosophy, in particular his account of the human being as a union of mind and body, as well as his ethics. They also provide an insight into the character of their authors, and the way ideas develop through intellectual collaboration. Philosophers have long been familiar with Descartes's side of the correspondence. Elisabeth's letters add context and depth both to Descartes's ideas and the legacy of the princess. This annotated edition also includes Elisabeth's correspondence with the Quakers William Penn and Robert Barclay.
Vera S. Candiani
- Published in print:
- 2014
- Published Online:
- September 2014
- ISBN:
- 9780804788052
- eISBN:
- 9780804791076
- Item type:
- chapter
- Publisher:
- Stanford University Press
- DOI:
- 10.11126/stanford/9780804788052.003.0006
- Subject:
- History, Latin American History
This chapter examines the first push for changes in the Desagüe during the eighteenth century, when oidores and military engineers drew the project into both the efforts of the Bourbons to rein in ...
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This chapter examines the first push for changes in the Desagüe during the eighteenth century, when oidores and military engineers drew the project into both the efforts of the Bourbons to rein in the administration of the colonies and the colonists’ resistance to their reforms. Oidores restricted local producers’ usage of land and water around the Desagüe, fixing what and where each were to be. European royal military engineers proposed methods and techniques drawn from their technical and scientific training to improve and accelerate the completion of the open trench that clashed with the mechanisms by which urban elites reduced for themselves the costs of their own flood protection, the water-sweeping method, the rotational nature of repartimiento labor, and the delegation of maintenance duties to indigenous communities. As a result, none of their proposals prospered. But their enlightened culture did: elites appropriated the features that suited them best.Less
This chapter examines the first push for changes in the Desagüe during the eighteenth century, when oidores and military engineers drew the project into both the efforts of the Bourbons to rein in the administration of the colonies and the colonists’ resistance to their reforms. Oidores restricted local producers’ usage of land and water around the Desagüe, fixing what and where each were to be. European royal military engineers proposed methods and techniques drawn from their technical and scientific training to improve and accelerate the completion of the open trench that clashed with the mechanisms by which urban elites reduced for themselves the costs of their own flood protection, the water-sweeping method, the rotational nature of repartimiento labor, and the delegation of maintenance duties to indigenous communities. As a result, none of their proposals prospered. But their enlightened culture did: elites appropriated the features that suited them best.
Marcel Danesi
- Published in print:
- 2020
- Published Online:
- January 2020
- ISBN:
- 9780198852247
- eISBN:
- 9780191886959
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198852247.003.0003
- Subject:
- Mathematics, History of Mathematics, Educational Mathematics
Numerals are symbols that represent numbers. The most commonly used numerals, which are easy to read after one has learned to use them, are the decimal ones. The principle used to construct them is ...
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Numerals are symbols that represent numbers. The most commonly used numerals, which are easy to read after one has learned to use them, are the decimal ones. The principle used to construct them is an efficient one—the position of each digit in the numeral indicates its value as a power of ten. But for such numerals to work this system requires a symbol as a place-holder for a position that has “nothing” in it. That symbol is 0, which makes it possible to differentiate between numbers such as “eleven” (= 11) “one hundred and one” (= 101), and “one thousand and one” (= 1001) without the need for additional numerals. The 0 tells us, simply, that the position is “empty.” This chapter looks at the origin of this extraordinary symbol, which over time became a number like any other, but with peculiar properties. It led to concepts such as negative numbers and the number line, which became crucial to the evolution of mathematics itself.Less
Numerals are symbols that represent numbers. The most commonly used numerals, which are easy to read after one has learned to use them, are the decimal ones. The principle used to construct them is an efficient one—the position of each digit in the numeral indicates its value as a power of ten. But for such numerals to work this system requires a symbol as a place-holder for a position that has “nothing” in it. That symbol is 0, which makes it possible to differentiate between numbers such as “eleven” (= 11) “one hundred and one” (= 101), and “one thousand and one” (= 1001) without the need for additional numerals. The 0 tells us, simply, that the position is “empty.” This chapter looks at the origin of this extraordinary symbol, which over time became a number like any other, but with peculiar properties. It led to concepts such as negative numbers and the number line, which became crucial to the evolution of mathematics itself.
Gerhard Oertel
- Published in print:
- 1996
- Published Online:
- November 2020
- ISBN:
- 9780195095036
- eISBN:
- 9780197560792
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195095036.003.0011
- Subject:
- Earth Sciences and Geography, Geology and the Lithosphere
The simplest relationship between stress and strain is Hooke’s law, describing the linear elastic response of solids to stress. Elastic strain (almost in all cases ...
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The simplest relationship between stress and strain is Hooke’s law, describing the linear elastic response of solids to stress. Elastic strain (almost in all cases small) is proportional to the applied stress, with one proportionality factor expressing the relationship between normal, and another that between tangential stress and strain. An ideally elastic strain is completely reversed upon removal of the stress that has caused it. Most materials obey Hooke’s law somewhat imperfectly, and that only up to a critical yield stress beyond which they begin to flow and to acquire, in addition to the elastic strain, a permanent strain that does not revert upon stress release. Hooke’s law in this form is applied to materials that are elastically isotropic, or can be assumed to be approximately so. Crystals, however, never are elastically isotropic, nor are crystalline materials consisting of constituent grains with a distribution of crystallographic orientations that departs from being uniform. The response of a crystal to a stress (at a level below the yield stress) consists of a strain determined by a matter tensor of the fourth rank, the compliance tensor s i j k l : . . . ɛij = s i j k l σkl, (7.1) . . . the 81 components of which are constants. Any tensor that describes the linear relationship between two tensors of the second rank is necessarily of the fourth rank, and like other tensors of the fourth rank, the compliance tensor can be referred to a new set of reference coordinates by means of a rotation matrix aij: s i j k l = aimajnakoalp smnop. (7.2) . . . The components of the compliance tensor are highly redundant, first because both the stress and the strain tensors are symmetric, and second because the tensor itself is symmetric. The number of independent components for crystals of the lowest, triclinic (both classes) symmetry is 21, and with increasing crystal symmetry the redundancies become more numerous; only three independent compliances are needed to describe the elastic properties of a cubic crystal.
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The simplest relationship between stress and strain is Hooke’s law, describing the linear elastic response of solids to stress. Elastic strain (almost in all cases small) is proportional to the applied stress, with one proportionality factor expressing the relationship between normal, and another that between tangential stress and strain. An ideally elastic strain is completely reversed upon removal of the stress that has caused it. Most materials obey Hooke’s law somewhat imperfectly, and that only up to a critical yield stress beyond which they begin to flow and to acquire, in addition to the elastic strain, a permanent strain that does not revert upon stress release. Hooke’s law in this form is applied to materials that are elastically isotropic, or can be assumed to be approximately so. Crystals, however, never are elastically isotropic, nor are crystalline materials consisting of constituent grains with a distribution of crystallographic orientations that departs from being uniform. The response of a crystal to a stress (at a level below the yield stress) consists of a strain determined by a matter tensor of the fourth rank, the compliance tensor s i j k l : . . . ɛij = s i j k l σkl, (7.1) . . . the 81 components of which are constants. Any tensor that describes the linear relationship between two tensors of the second rank is necessarily of the fourth rank, and like other tensors of the fourth rank, the compliance tensor can be referred to a new set of reference coordinates by means of a rotation matrix aij: s i j k l = aimajnakoalp smnop. (7.2) . . . The components of the compliance tensor are highly redundant, first because both the stress and the strain tensors are symmetric, and second because the tensor itself is symmetric. The number of independent components for crystals of the lowest, triclinic (both classes) symmetry is 21, and with increasing crystal symmetry the redundancies become more numerous; only three independent compliances are needed to describe the elastic properties of a cubic crystal.
