*Glen Van Brummelen*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175997
- eISBN:
- 9781400844807
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175997.003.0008
- Subject:
- Mathematics, History of Mathematics

This chapter deals with stereographic projection, which is superior to other projections of the sphere because of its angle-preserving and circle-preserving properties; the first property gave ...
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This chapter deals with stereographic projection, which is superior to other projections of the sphere because of its angle-preserving and circle-preserving properties; the first property gave instrument makers a huge advantage and the second provides clear astronomical advantages. The earliest text on stereographic projection is Ptolemy's Planisphere, in which he explains how to use stereographic projection to solve problems involving rising times, suggesting that the astrolabe may have existed already. After providing an overview of the astrolabe, an instrument for solving astronomical problems, the chapter considers how stereographic projection is used in solving triangles. It then describes the Cesàro method, named after Giuseppe Cesàro, that uses stereographic projection to project an arbitrary triangle ABC onto a plane. It also examines B. M. Brown's complaint against Cesàro's approach to spherical trigonometry.Less

This chapter deals with stereographic projection, which is superior to other projections of the sphere because of its angle-preserving and circle-preserving properties; the first property gave instrument makers a huge advantage and the second provides clear astronomical advantages. The earliest text on stereographic projection is Ptolemy's *Planisphere*, in which he explains how to use stereographic projection to solve problems involving rising times, suggesting that the astrolabe may have existed already. After providing an overview of the astrolabe, an instrument for solving astronomical problems, the chapter considers how stereographic projection is used in solving triangles. It then describes the Cesàro method, named after Giuseppe Cesàro, that uses stereographic projection to project an arbitrary triangle ABC onto a plane. It also examines B. M. Brown's complaint against Cesàro's approach to spherical trigonometry.

*Mark Ladd*

- Published in print:
- 2014
- Published Online:
- April 2014
- ISBN:
- 9780199670888
- eISBN:
- 9780191781124
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199670888.003.0002
- Subject:
- Physics, Crystallography

The early history of crystallography is discussed, including particularly the work of Steno, Hauy, de l’Isle, Miller and Weiss. Reference axes are introduced together with the mathematical ...
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The early history of crystallography is discussed, including particularly the work of Steno, Hauy, de l’Isle, Miller and Weiss. Reference axes are introduced together with the mathematical description of planes, leading to the designation of crystal faces, or possible faces, and zones. The projection of crystal faces and, therefore, also crystal symmetry on to a plane diagram is elaborated by the techniques of goniometric measurements on and stereographic projection of a crystal. The geometry of molecules is discussed in terms of VSEPR, experimental and theoretical procedures. Molecular geometry and its precision are examined.Less

The early history of crystallography is discussed, including particularly the work of Steno, Hauy, de l’Isle, Miller and Weiss. Reference axes are introduced together with the mathematical description of planes, leading to the designation of crystal faces, or possible faces, and zones. The projection of crystal faces and, therefore, also crystal symmetry on to a plane diagram is elaborated by the techniques of goniometric measurements on and stereographic projection of a crystal. The geometry of molecules is discussed in terms of VSEPR, experimental and theoretical procedures. Molecular geometry and its precision are examined.

*Glen Van Brummelen*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175997
- eISBN:
- 9781400844807
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175997.001.0001
- Subject:
- Mathematics, History of Mathematics

This book traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the ...
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This book traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the Earth. Once at the heart of astronomy and ocean-going navigation for two millennia, the discipline was also a mainstay of mathematics education for centuries and taught widely until the 1950s. The book explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. The book conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. It is illustrated throughout with stunning historical images and informative drawings and diagrams. It also features easy-to-use appendices as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.Less

This book traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the Earth. Once at the heart of astronomy and ocean-going navigation for two millennia, the discipline was also a mainstay of mathematics education for centuries and taught widely until the 1950s. The book explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. The book conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. It is illustrated throughout with stunning historical images and informative drawings and diagrams. It also features easy-to-use appendices as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

*A. R. P. Rau*

- Published in print:
- 2014
- Published Online:
- December 2014
- ISBN:
- 9780198709916
- eISBN:
- 9780191780189
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198709916.003.0006
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Maps originated as cartographic representations of the local neighbourhood or the entire global surface. They have a wider context for representations more generally and, therefore, occur throughout ...
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Maps originated as cartographic representations of the local neighbourhood or the entire global surface. They have a wider context for representations more generally and, therefore, occur throughout mathematics and physics. Representations, transformations, and maps can often be used interchangeably. Various illustrations of this in physics are discussed, including iterative maps for fractals and stereographic projections for quantum systems such as qubits. In particular, in the reduction from a higher-dimensional object to a lower-dimensional map lie quantum concepts, such as non-locality.Less

Maps originated as cartographic representations of the local neighbourhood or the entire global surface. They have a wider context for representations more generally and, therefore, occur throughout mathematics and physics. Representations, transformations, and maps can often be used interchangeably. Various illustrations of this in physics are discussed, including iterative maps for fractals and stereographic projections for quantum systems such as qubits. In particular, in the reduction from a higher-dimensional object to a lower-dimensional map lie quantum concepts, such as non-locality.