The Theory of the Imaginary in Geometry: Together with the Trigonometry of the Imaginary

Note moyenne 0
( 0 avis fournis par GoodReads )
 
9781515094166: The Theory of the Imaginary in Geometry: Together with the Trigonometry of the Imaginary
Présentation de l'éditeur :

THE word theory in the title is to be understood in a very non-technical sense. Indeed, apart from the idea of the invariant elements of an elliptic involution on a straight line, no theory is found at all. The purpose of the book is rather to furnish a certain graphical representation of imaginaries under a number of conventions more or less well known. Three concepts run through the work: first, an incompletely defined idea of the nature of an imaginary; second, the analogy with the geometry of reals; third, the use of coordinate methods, assuming the algebra of imaginaries.

Given a real point O and a real constant k, an imaginary point P is defined by the equation OP2 = -k - 2. The two imaginary points P and P' are the double points of an involution having O for center, and ik for parameter. The algebra of imaginaries is now assumed, and a geometry of imaginary distances on a straight line is built upon it. The reader is repeatedly reminded that in themselves there is no difference between real and imaginary points; that differences exist solely in their relations to other points. In the extension to two dimensions both x and ix are plotted on a horizontal line, while x and xy are plotted on a vertical line. Imaginary lines are dotted, and points having one or both coordinates imaginary are enclosed by parentheses, but otherwise the same figures are used for proofs, either by the methods of elementary geometry, or by coordinate methods. In the algebra of segments it is shown that an imaginary distance O'D' can be expressed in the form iOD, wherein OD is a real segment, or at most by OD times some number. Now follows a long development of the extension of cross ratios, etc., to imaginaries. In fact every word of this is found implicitly in any treatment of the invariance of cross ratios under linear fractional transformation. In Chapter II the conic with a real branch is introduced, beginning with involutions of conjugate points on lines having imaginary points on the conic. If the coefficients in the equation of a circle are real, the usual graph of x2 + y2 = a2 for real x and real y is followed by replacing y by iy, then proceeding as before. The former locus is called the (1, 1) branch, and the latter the (1, i) branch of the circle. Similarly, it has a (i, 1) branch, and another, (i, i) , but the latter has no graph. This idea is applied in all detail to ellipses, hyperbolas, and parabolas; in the case of the central conies it is also followed by replacing rectangular coordinates by a pair of conjugate diameters. The ordinary theorems of poles and polars, and the theorems of Pascal, Brianchon, Desargues, Carnot are shown to apply. Indeed, after having established the applicability of cross ratios in the earlier chapters, all these proofs can be applied in the same manner as to reals, without changing a word…. —An excerpt from Bulletin of the American Mathematical Society, Vol. 27 [1921]

Les informations fournies dans la section « A propos du livre » peuvent faire référence à une autre édition de ce titre.

Meilleurs résultats de recherche sur AbeBooks

1.

Hatton, J. L. S.
ISBN 10 : 1515094162 ISBN 13 : 9781515094166
Neuf(s) Paperback Quantité : > 20
impression à la demande
Vendeur
BargainBookStores
(Grand Rapids, MI, Etats-Unis)
Evaluation vendeur
[?]

Description du livre Paperback. État : New. This item is printed on demand. Item doesn't include CD/DVD. N° de réf. du libraire 8924354

Plus d'informations sur ce vendeur | Poser une question au libraire

Acheter neuf
EUR 7,45
Autre devise

Ajouter au panier

Frais de port : EUR 3,69
Vers Etats-Unis
Destinations, frais et délais

2.

Hatton, J. L. S.
ISBN 10 : 1515094162 ISBN 13 : 9781515094166
Neuf(s) Quantité : > 20
impression à la demande
Vendeur
PBShop
(Wood Dale, IL, Etats-Unis)
Evaluation vendeur
[?]

Description du livre 2015. PAP. État : New. New Book. Shipped from US within 10 to 14 business days. THIS BOOK IS PRINTED ON DEMAND. Established seller since 2000. N° de réf. du libraire IQ-9781515094166

Plus d'informations sur ce vendeur | Poser une question au libraire

Acheter neuf
EUR 7,79
Autre devise

Ajouter au panier

Frais de port : EUR 3,69
Vers Etats-Unis
Destinations, frais et délais

3.

J L S Hatton
Edité par Createspace Independent Publishing Platform, United States (2015)
ISBN 10 : 1515094162 ISBN 13 : 9781515094166
Neuf(s) Paperback Quantité : 10
impression à la demande
Vendeur
The Book Depository US
(London, Royaume-Uni)
Evaluation vendeur
[?]

Description du livre Createspace Independent Publishing Platform, United States, 2015. Paperback. État : New. 229 x 152 mm. Language: English . Brand New Book ***** Print on Demand *****. THE word theory in the title is to be understood in a very non-technical sense. Indeed, apart from the idea of the invariant elements of an elliptic involution on a straight line, no theory is found at all. The purpose of the book is rather to furnish a certain graphical representation of imaginaries under a number of conventions more or less well known. Three concepts run through the work: first, an incompletely defined idea of the nature of an imaginary; second, the analogy with the geometry of reals; third, the use of coordinate methods, assuming the algebra of imaginaries. Given a real point O and a real constant k, an imaginary point P is defined by the equation OP2 = -k - 2. The two imaginary points P and P are the double points of an involution having O for center, and ik for parameter. The algebra of imaginaries is now assumed, and a geometry of imaginary distances on a straight line is built upon it. The reader is repeatedly reminded that in themselves there is no difference between real and imaginary points; that differences exist solely in their relations to other points. In the extension to two dimensions both x and ix are plotted on a horizontal line, while x and xy are plotted on a vertical line. Imaginary lines are dotted, and points having one or both coordinates imaginary are enclosed by parentheses, but otherwise the same figures are used for proofs, either by the methods of elementary geometry, or by coordinate methods. In the algebra of segments it is shown that an imaginary distance O D can be expressed in the form iOD, wherein OD is a real segment, or at most by OD times some number. Now follows a long development of the extension of cross ratios, etc., to imaginaries. In fact every word of this is found implicitly in any treatment of the invariance of cross ratios under linear fractional transformation. In Chapter II the conic with a real branch is introduced, beginning with involutions of conjugate points on lines having imaginary points on the conic. If the coefficients in the equation of a circle are real, the usual graph of x2 + y2 = a2 for real x and real y is followed by replacing y by iy, then proceeding as before. The former locus is called the (1, 1) branch, and the latter the (1, i) branch of the circle. Similarly, it has a (i, 1) branch, and another, (i, i) , but the latter has no graph. This idea is applied in all detail to ellipses, hyperbolas, and parabolas; in the case of the central conies it is also followed by replacing rectangular coordinates by a pair of conjugate diameters. The ordinary theorems of poles and polars, and the theorems of Pascal, Brianchon, Desargues, Carnot are shown to apply. Indeed, after having established the applicability of cross ratios in the earlier chapters, all these proofs can be applied in the same manner as to reals, without changing a word. -An excerpt from Bulletin of the American Mathematical Society, Vol. 27 [1921]. N° de réf. du libraire APC9781515094166

Plus d'informations sur ce vendeur | Poser une question au libraire

Acheter neuf
EUR 11,54
Autre devise

Ajouter au panier

Frais de port : Gratuit
De Royaume-Uni vers Etats-Unis
Destinations, frais et délais

4.

J L S Hatton
Edité par Createspace Independent Publishing Platform, United States (2015)
ISBN 10 : 1515094162 ISBN 13 : 9781515094166
Neuf(s) Paperback Quantité : 10
impression à la demande
Vendeur
The Book Depository
(London, Royaume-Uni)
Evaluation vendeur
[?]

Description du livre Createspace Independent Publishing Platform, United States, 2015. Paperback. État : New. 229 x 152 mm. Language: English . Brand New Book ***** Print on Demand *****.THE word theory in the title is to be understood in a very non-technical sense. Indeed, apart from the idea of the invariant elements of an elliptic involution on a straight line, no theory is found at all. The purpose of the book is rather to furnish a certain graphical representation of imaginaries under a number of conventions more or less well known. Three concepts run through the work: first, an incompletely defined idea of the nature of an imaginary; second, the analogy with the geometry of reals; third, the use of coordinate methods, assuming the algebra of imaginaries. Given a real point O and a real constant k, an imaginary point P is defined by the equation OP2 = -k - 2. The two imaginary points P and P are the double points of an involution having O for center, and ik for parameter. The algebra of imaginaries is now assumed, and a geometry of imaginary distances on a straight line is built upon it. The reader is repeatedly reminded that in themselves there is no difference between real and imaginary points; that differences exist solely in their relations to other points. In the extension to two dimensions both x and ix are plotted on a horizontal line, while x and xy are plotted on a vertical line. Imaginary lines are dotted, and points having one or both coordinates imaginary are enclosed by parentheses, but otherwise the same figures are used for proofs, either by the methods of elementary geometry, or by coordinate methods. In the algebra of segments it is shown that an imaginary distance O D can be expressed in the form iOD, wherein OD is a real segment, or at most by OD times some number. Now follows a long development of the extension of cross ratios, etc., to imaginaries. In fact every word of this is found implicitly in any treatment of the invariance of cross ratios under linear fractional transformation. In Chapter II the conic with a real branch is introduced, beginning with involutions of conjugate points on lines having imaginary points on the conic. If the coefficients in the equation of a circle are real, the usual graph of x2 + y2 = a2 for real x and real y is followed by replacing y by iy, then proceeding as before. The former locus is called the (1, 1) branch, and the latter the (1, i) branch of the circle. Similarly, it has a (i, 1) branch, and another, (i, i) , but the latter has no graph. This idea is applied in all detail to ellipses, hyperbolas, and parabolas; in the case of the central conies it is also followed by replacing rectangular coordinates by a pair of conjugate diameters. The ordinary theorems of poles and polars, and the theorems of Pascal, Brianchon, Desargues, Carnot are shown to apply. Indeed, after having established the applicability of cross ratios in the earlier chapters, all these proofs can be applied in the same manner as to reals, without changing a word. -An excerpt from Bulletin of the American Mathematical Society, Vol. 27 [1921]. N° de réf. du libraire APC9781515094166

Plus d'informations sur ce vendeur | Poser une question au libraire

Acheter neuf
EUR 11,62
Autre devise

Ajouter au panier

Frais de port : Gratuit
De Royaume-Uni vers Etats-Unis
Destinations, frais et délais

5.

Hatton, J. L. S.
Edité par CreateSpace Independent Publishing Platform
ISBN 10 : 1515094162 ISBN 13 : 9781515094166
Neuf(s) PAPERBACK Quantité : > 20
Vendeur
Russell Books
(Victoria, BC, Canada)
Evaluation vendeur
[?]

Description du livre CreateSpace Independent Publishing Platform. PAPERBACK. État : New. 1515094162 Special order direct from the distributor. N° de réf. du libraire ING9781515094166

Plus d'informations sur ce vendeur | Poser une question au libraire

Acheter neuf
EUR 10,71
Autre devise

Ajouter au panier

Frais de port : EUR 6,47
De Canada vers Etats-Unis
Destinations, frais et délais

6.

Hatton, J. L. S.
ISBN 10 : 1515094162 ISBN 13 : 9781515094166
Neuf(s) Quantité : > 20
impression à la demande
Vendeur
Books2Anywhere
(Fairford, GLOS, Royaume-Uni)
Evaluation vendeur
[?]

Description du livre 2015. PAP. État : New. New Book. Delivered from our UK warehouse in 3 to 5 business days. THIS BOOK IS PRINTED ON DEMAND. Established seller since 2000. N° de réf. du libraire IQ-9781515094166

Plus d'informations sur ce vendeur | Poser une question au libraire

Acheter neuf
EUR 7,50
Autre devise

Ajouter au panier

Frais de port : EUR 10,39
De Royaume-Uni vers Etats-Unis
Destinations, frais et délais