## Principles of Geometry, Volume 4Henry Frederick Baker (1866-1956) was a renowned British mathematician specialising in algebraic geometry. He was elected a Fellow of the Royal Society in 1898 and appointed the Lowndean Professor of Astronomy and Geometry in the University of Cambridge in 1914. First published between 1922 and 1925, the six-volume Principles of Geometry was a synthesis of Baker's lecture series on geometry and was the first British work on geometry to use axiomatic methods without the use of co-ordinates. The first four volumes describe the projective geometry of space of between two and five dimensions, with the last two volumes reflecting Baker's later research interests in the birational theory of surfaces. The work as a whole provides a detailed insight into the geometry which was developing at the time of publication. This, the fourth volume, describes the principal configurations of space of four and five dimensions. |

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### Contents

INTRODUCTORY RELATIONS OF | 1 |

The generalised Miquel theorem 810 | 8 |

Wallaces theorem for four circles and Moebiuss figure of | 18 |

Theorems for circumscribed circles for any number of lines in | 29 |

Spheres as determined from sections of a quadric in fourfold space 3638 | 36 |

Planes lying entirely on the quadric fourfold in the space of five | 44 |

Number of lines common to two congruences in threefold space 50 | 50 |

The section of Veroneses surface by a fourold space | 54 |

The bipolar character of the figure obtained 76 | 76 |

Eight sections divided into four pairs | 77 |

Relations with metrical geometry | 78 |

The same Hart section arises in different ways | 80 |

Four circles arising from the intersections of three circles | 82 |

A particular form of the equation of the quadric 8586 | 85 |

THE PLANE QUARTIC CURVE WITH | 89 |

Inversion of the curve into itself 96 | 96 |

Lies correspondence between lines and spheres in space of three dimensions 55 | 55 |

Algebraic formulation of the correspondence 56 | 56 |

Comparison with deduction of sphere from fourfold space | 57 |

The eight spheres touching four planes | 58 |

Exx 15 Proofs of the theorems 6064 | 60 |

Note Another proof of Wallaces theorem and extensions | 64 |

HARTS THEOREM FOR CIRCLES IN A PLANE OR FOR SECTIONS OF A QUADRIC The sections of a quadric which touch three given sections 65 | 65 |

A pair of variable sections of a quadric touching one another and two fixed sections | 66 |

Circles cutting three given circles at equal angles | 67 |

Solution of generalised Malfattis problem 68 | 68 |

Equations of sections touching three sections of a quadric 69 | 69 |

Exx 1015 Various algebraic results 70 | 70 |

The Hart circles of three circles when the four have a common orthogonal circle 71 | 71 |

The Hart circles of three circles in general 7275 | 72 |

The various cases | 75 |

A PARTICULAR FIGURE IN SPACE | 104 |

A FIGURE OF FIFTEEN LINES | 113 |

The six systems of planes 123 | 123 |

The equations of the singular solids 129 | 129 |

The dual of the figure which has been considered The cubic locus S 151158 | 151 |

A QUARTIC SURFACE IN SPACE | 161 |

The selfpolar pentad for the quartic surface 168170 | 168 |

Confocal Cyclides Three through arbitrary point with given focal | 180 |

RELATIONS IN SPACE OF FIVE | 203 |

Relation of the transformation to the theory of Rummers surface 211 | 211 |

Kummers surface in the geometry of space of five dimensions 218221 | 218 |

The common singular points and planes of the six congruences 225227 | 225 |

The rationality of the Quadratic Complex and of the Quadratic | 234 |

The singular points and planes of a Quadratic Congruence | 240 |

### Common terms and phrases

Absolute conic arbitrary plane arbitrary point arising centre circles common point conical sheet conies consider contains the line contains the point coordinates corresponding cubic curve cubic surface Cyclide denote double points drawn Dupin Cyclide equation fifteen five dimensions five points fivefold space four dimensions four lines four planes four points fourfold space given lines harmonic conjugates Kummer surface lies line joining line meets line of curvature line-cone linear complex locus meet the line original threefold space pairs passing plane sections planes meet point of contact polar plane pole prove quadric cone quadric point-cone quadric surface quartic curve regard to ft respectively shew shewn singular solids six planes six points sixteen lines sixteen points space of four space of three sphere symbols tangent fourfold tangent plane tangent prime tangent solid tetrads theorem three dimensions three lines three points touch triad vertex