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Book from Project Gutenberg: The Foundations of Geometry

Book from Project Gutenberg: The Foundations of Geometry

An Ordered Incidence Geometry, that is a geometry with certain axioms of incidence and order is proposed as a minimal setting for the fundamental convexity theorems, such as the hyperplane separation theorem and the theorems of Radon and Helly. These theorems are usually stated, proved, understood and/or applied in the context of a linear vector space, but they require only incidence and order, (and for separation, completeness), and none of the linear structure of a vector space.

"A Simplified Proof of the Distributive Law of Multiplication Given in Hilbert's "The Foundations of Geometry"" is an article from The American Mathematical Monthly, Volume 23 . View more articles from The American Mathematical Monthly . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2973409

We explore the possibility of replacing point set topology by higher category theory and topos theory as the foundation for quantum general relativity. We discuss the BC model and problems of its interpretation, and connect with the construction of causal sites.

We explore the possibility of replacing point set topology by higher category theory and topos theory as the foundation for quantum general relativity. We discuss the BC model and problems of its interpretation, and connect with the construction of causal sites.

Book from Project Gutenberg: The Foundations of Mathematics A Contribution to the Philosophy of Geometry

viii, 285 p. 19 cm

Assuming a general timelike congruence of worldlines as a reference frame, we derive a covariant general formalism of inertial forces in General Relativity. Inspired by the works of Abramowicz et. al. (see e.g. Abramowicz and Lasota, Class. Quantum Grav. 14 (1997) A23), we also study conformal rescalings of spacetime and investigate how these affect the inertial force formalism. While many ways of describing spatial curvature of a trajectory has been discussed in papers prior to this, one particular prescription (which differs from the standard projected curvature when the reference is shearing) appears novel. For the particular case of a hypersurface-forming congruence, using a suitable rescaling of spacetime, we show that a geodesic photon is always following a line that is spatially straight with respect to the new curvature measure. This fact is intimately connected to Fermat's principle, and allows for a certain generalization of the optical geometry as will be further pursued in a companion paper (Jonsson and Westman, Class. Quantum Grav. 23 (2006) 61). For the particular case when the shear-tensor vanishes, we present the inertial force equation in three-dimensional form (using the bold face vector notation), and note how similar it is to its Newtonian counterpart. From the spatial curvature measures that we introduce, we derive corresponding covariant differentiations of a vector defined along a spacetime trajectory. This allows us to connect the formalism of this paper to that of Jantzen et. al. (see e.g. Bini et. al., Int. J. Mod. Phys. D 6 (1997) 143).

"THE FOUNDATIONS OF GEOMETRY" is an article from The Monist, Volume 13 . View more articles from The Monist . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-27899411

"THE FOUNDATIONS OF GEOMETRY" is an article from The Monist, Volume 13 . View more articles from The Monist . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-27899411

"With special reference to "An essay on the foundation of geometry, by B. A. W. Russell

"With special reference to "An essay on the foundation of geometry, by B. A. W. Russell

In this paper, the third in the series, we define the generalized orthocenter $H$ corresponding to a point $P$, with respect to triangle $ABC$, as the unique point for which the lines $HA, HB, HC$ are parallel, respectively, to $QD, QE, QF$, where $DEF$ is the cevian triangle of $P$ and $Q=K \circ \iota(P)$ is the $isotomcomplement$ of $P$, both with respect to $ABC$. We prove a generalized Feuerbach Theorem, and characterize the center $Z$ of the cevian conic $\mathcal{C}_P$, defined in Part II, as the center of the affine map $\Phi_P = T_P \circ K^{-1} \circ T_{P'} \circ K^{-1}$, where $T_P$ is the unique affine map for which $T_P(ABC)=DEF$; $T_{P'}$ is defined similarly for the isotomic conjugate $P'=\iota(P)$ of $P$; and $K$ is the complement map. The affine map $\Phi_P$ fixes $Z$ and takes the nine-point conic $\mathcal{N}_H$ for the quadrangle $ABCH$ (with respect to the line at infinity) to the inconic $\mathcal{I}$, defined to be the unique conic which is tangent to the sides of $ABC$ at the points $D, E, F$. The point $Z$ is therefore the point where the nine-point conic $\mathcal{N}_H$ and the inconic $\mathcal{I}$ touch. This theorem generalizes the usual Feuerbach theorem and holds in all cases where the point $P$ is not on a median, whether the conics involved are ellipses, parabolas, or hyperbolas, and also holds when $Z$ is an infinite point. We also determine the locus of points $P$ for which the generalized orthocenter $H$ coincides with a vertex of $ABC$; this locus turns out to be the union of three conics minus six points. All our proofs are synthetic, and combine affine and projective arguments.

In this paper, the third in the series, we define the generalized orthocenter $H$ corresponding to a point $P$, with respect to triangle $ABC$, as the unique point for which the lines $HA, HB, HC$ are parallel, respectively, to $QD, QE, QF$, where $DEF$ is the cevian triangle of $P$ and $Q=K \circ \iota(P)$ is the $isotomcomplement$ of $P$, both with respect to $ABC$. We prove a generalized Feuerbach Theorem, and characterize the center $Z$ of the cevian conic $\mathcal{C}_P$, defined in Part II, as the center of the affine map $\Phi_P = T_P \circ K^{-1} \circ T_{P'} \circ K^{-1}$, where $T_P$ is the unique affine map for which $T_P(ABC)=DEF$; $T_{P'}$ is defined similarly for the isotomic conjugate $P'=\iota(P)$ of $P$; and $K$ is the complement map. The affine map $\Phi_P$ fixes $Z$ and takes the nine-point conic $\mathcal{N}_H$ for the quadrangle $ABCH$ (with respect to the line at infinity) to the inconic $\mathcal{I}$, defined to be the unique conic which is tangent to the sides of $ABC$ at the points $D, E, F$. The point $Z$ is therefore the point where the nine-point conic $\mathcal{N}_H$ and the inconic $\mathcal{I}$ touch. This theorem generalizes the usual Feuerbach theorem and holds in all cases where the point $P$ is not on a median, whether the conics involved are ellipses, parabolas, or hyperbolas, and also holds when $Z$ is an infinite point. We also determine the locus of points $P$ for which the generalized orthocenter $H$ coincides with a vertex of $ABC$; this locus turns out to be the union of three conics minus six points. All our proofs are synthetic, and combine affine and projective arguments.

In this paper, the third in the series, we define the generalized orthocenter $H$ corresponding to a point $P$, with respect to triangle $ABC$, as the unique point for which the lines $HA, HB, HC$ are parallel, respectively, to $QD, QE, QF$, where $DEF$ is the cevian triangle of $P$ and $Q=K \circ \iota(P)$ is the $isotomcomplement$ of $P$, both with respect to $ABC$. We prove a generalized Feuerbach Theorem, and characterize the center $Z$ of the cevian conic $\mathcal{C}_P$, defined in Part II, as the center of the affine map $\Phi_P = T_P \circ K^{-1} \circ T_{P'} \circ K^{-1}$, where $T_P$ is the unique affine map for which $T_P(ABC)=DEF$; $T_{P'}$ is defined similarly for the isotomic conjugate $P'=\iota(P)$ of $P$; and $K$ is the complement map. The affine map $\Phi_P$ fixes $Z$ and takes the nine-point conic $\mathcal{N}_H$ for the quadrangle $ABCH$ (with respect to the line at infinity) to the inconic $\mathcal{I}$, defined to be the unique conic which is tangent to the sides of $ABC$ at the points $D, E, F$. The point $Z$ is therefore the point where the nine-point conic $\mathcal{N}_H$ and the inconic $\mathcal{I}$ touch. This theorem generalizes the usual Feuerbach theorem and holds in all cases where the point $P$ is not on a median, whether the conics involved are ellipses, parabolas, or hyperbolas, and also holds when $Z$ is an infinite point. We also determine the locus of points $P$ for which the generalized orthocenter $H$ coincides with a vertex of $ABC$; this locus turns out to be the union of three conics minus six points. All our proofs are synthetic, and combine affine and projective arguments.

In this paper, the third in the series, we define the generalized orthocenter $H$ corresponding to a point $P$, with respect to triangle $ABC$, as the unique point for which the lines $HA, HB, HC$ are parallel, respectively, to $QD, QE, QF$, where $DEF$ is the cevian triangle of $P$ and $Q=K \circ \iota(P)$ is the $isotomcomplement$ of $P$, both with respect to $ABC$. We prove a generalized Feuerbach Theorem, and characterize the center $Z$ of the cevian conic $\mathcal{C}_P$, defined in Part II, as the center of the affine map $\Phi_P = T_P \circ K^{-1} \circ T_{P'} \circ K^{-1}$, where $T_P$ is the unique affine map for which $T_P(ABC)=DEF$; $T_{P'}$ is defined similarly for the isotomic conjugate $P'=\iota(P)$ of $P$; and $K$ is the complement map. The affine map $\Phi_P$ fixes $Z$ and takes the nine-point conic $\mathcal{N}_H$ for the quadrangle $ABCH$ (with respect to the line at infinity) to the inconic $\mathcal{I}$, defined to be the unique conic which is tangent to the sides of $ABC$ at the points $D, E, F$. The point $Z$ is therefore the point where the nine-point conic $\mathcal{N}_H$ and the inconic $\mathcal{I}$ touch. This theorem generalizes the usual Feuerbach theorem and holds in all cases where the point $P$ is not on a median, whether the conics involved are ellipses, parabolas, or hyperbolas, and also holds when $Z$ is an infinite point. We also determine the locus of points $P$ for which the generalized orthocenter $H$ coincides with a vertex of $ABC$; this locus turns out to be the union of three conics minus six points. All our proofs are synthetic, and combine affine and projective arguments.

"On the Lie-Riemann-Helmholtz-Hilbert Problem of the Foundations of Geometry" is an article from American Journal of Mathematics, Volume 41 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2370289

"On the Lie-Riemann-Helmholtz-Hilbert Problem of the Foundations of Geometry" is an article from American Journal of Mathematics, Volume 41 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2370289

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

"ON THE FOUNDATIONS OF GEOMETRY" is an article from The Monist, Volume 9 . View more articles from The Monist . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-27899007

"THE FOUNDATIONS OF GEOMETRY. (CONCLUDED)" is an article from The Monist, Volume 13 . View more articles from The Monist . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-27899433

"THE FOUNDATIONS OF GEOMETRY. (CONCLUDED)" is an article from The Monist, Volume 13 . View more articles from The Monist . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-27899433

"THE FOUNDATIONS OF GEOMETRY. (CONCLUDED)" is an article from The Monist, Volume 13 . View more articles from The Monist . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-27899433

"THE FOUNDATIONS OF GEOMETRY. (CONCLUDED)" is an article from The Monist, Volume 13 . View more articles from The Monist . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-27899433

"THE FOUNDATIONS OF GEOMETRY. (CONCLUDED)" is an article from The Monist, Volume 13 . View more articles from The Monist . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-27899433

xvi, 201, [1] p. 23 cm

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

"The material contained in the following translation was given in substance by Professor Hilbert as a course of lectures on euclidean geometry at the University of Göttingen during the winter semester of 1898–1899." (Original source from Project Gutenberg)

Speaker: Olaf Neumann Date: April 2003

Book digitized by Google and uploaded to the Internet Archive by user tpb.

Book digitized by Google and uploaded to the Internet Archive by user tpb.

The original formed the first part of "Festschrift zur feier der enthüllung des Gauss-Weber-denkmals in Göttingen, Grundlagen der geometrie," with the additions made by the author in the French translation, Paris, 1901, incorporated

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

In this paper we give a completely synthetic proof of the TCC-perspector theorem, that the isogonal conjugate $\gamma(H)$ of the generalized orthocenter $H$ (defined in Part III of this series of papers), with respect to a triangle $ABC$ and a point $P$, is the perspector of the tangential triangle of $ABC$ and the circumcevian triangle (both with respect to the circumcircle) of the isogonal conjugate $\gamma(Q)$, where $Q$ is the complement of the isotomic conjugate $P'$ of the point $P$.

We investigate a geometric structure on ${\rm Sym}^+(p)$, the set of $p \times p$ symmetric positive-definite matrices, based on eigen-decomposition. Eigenstructure determines both a stratification of ${\rm Sym}^+(p)$, defined by eigenvalue multiplicities, and fibers of the "eigen-composition" map $F:M(p):=SO(p)\times{\rm Diag}^+(p)\to{\rm Sym}^+(p)$. The fiber structure leads to the notions of {\em scaling-rotation distance} between $X,Y\in {\rm Sym}^+(p)$, the distance in $M(p)$ between fibers $F^{-1}(X)$ and $F^{-1}(Y)$, and {\em minimal smooth scaling-rotation (MSSR) curves} [Jung et al. (2015)], images in ${\rm Sym}^+(p)$ of minimal-length geodesics connecting two fibers. In this paper we study the geometry of the triple $(M(p),F,{\rm Sym}^+(p))$, where $M(p)$ is given a suitable product Riemannian metric, and use this basic geometry to begin a systematic characterization and analysis of MSSR curves. This task raises basic questions: For which $X,Y$ is there a {\em unique} MSSR curve from $X$ to $Y$? More generally, what is the set ${\cal M}(X,Y)$ of MSSR curves from $X$ to $Y$? This set is influenced by two ways that non-uniqueness can potentially occur, "Type I" and "Type II". We translate the question of whether Type II nonuniqueness can occur into a question about the geometry of Grassmannians $G_m({\bf R}^p)$, with $m$ even, that we answer for $p\leq 4$ and $p\geq 11$. Our method of proof also yields an interesting half-angle formula concerning principal angles beween subspaces of ${\bf R}^p$ whose dimensions may or may not be equal. The general-$p$ results concerning MSSR curves and scaling-rotation distance that we establish here underpin the $p=3$ results in [Groisser et al. (2017)], where they enable us to find new explicit formulas for MSSR curves and scaling-rotation distance, and to identify ${\cal M}(X,Y)$, in all nontrivial cases.

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

xvi, 201, [1] p. 23 cm

This paper continues the investigation of Part I, by studying the conic $\mathcal{C}_P$ on the five points $ABCPQ$, where $ABC$ is a given ordinary triangle and $Q$ is the isotomcomplement of $P$, defined as the complement of the isotomic conjugate $P'$ of $P$ with respect to triangle $ABC$. We show that $\mathcal{C}_P$ also lies on the points $P'$ and $Q'$, where $Q'$ is the isotomcomplement of $P'$. The conic $\mathcal{C}_P$ lies on six other points which are the images of the vertices of $ABC$ under the affine mapping $\lambda=T_{P'} \circ T_P^{-1}$ and its inverse, where $T_P$ and $T_{P'}$ are the unique affine maps taking $ABC$ to the cevian triangles of $P$ and $P'$, respectively. In the paper we characterize the center $Z$ of $\mathcal{C}_P$ as the unique fixed point of $\lambda$ in the extended plane, when $\mathcal{C}_P$ is a parabola or an ellipse, and the unique ordinary fixed point of $\lambda$, when $\mathcal{C}_P$ is a hyperbola. We also show that $Z=GV \cdot T_P(GV)$, where $G$ is the centroid of $ABC$ and $V=PQ \cdot P'Q'$. When $P$ is the Gergonne point of $ABC$, this gives a new characterization of the Feuerbach point $Z$. All of our arguments are purely synthetic.

This paper continues the investigation of Part I, by studying the conic $\mathcal{C}_P$ on the five points $ABCPQ$, where $ABC$ is a given ordinary triangle and $Q$ is the isotomcomplement of $P$, defined as the complement of the isotomic conjugate $P'$ of $P$ with respect to triangle $ABC$. We show that $\mathcal{C}_P$ also lies on the points $P'$ and $Q'$, where $Q'$ is the isotomcomplement of $P'$. The conic $\mathcal{C}_P$ lies on six other points which are the images of the vertices of $ABC$ under the affine mapping $\lambda=T_{P'} \circ T_P^{-1}$ and its inverse, where $T_P$ and $T_{P'}$ are the unique affine maps taking $ABC$ to the cevian triangles of $P$ and $P'$, respectively. In the paper we characterize the center $Z$ of $\mathcal{C}_P$ as the unique fixed point of $\lambda$ in the extended plane, when $\mathcal{C}_P$ is a parabola or an ellipse, and the unique ordinary fixed point of $\lambda$, when $\mathcal{C}_P$ is a hyperbola. We also show that $Z=GV \cdot T_P(GV)$, where $G$ is the centroid of $ABC$ and $V=PQ \cdot P'Q'$. When $P$ is the Gergonne point of $ABC$, this gives a new characterization of the Feuerbach point $Z$. All of our arguments are purely synthetic.

This paper continues the investigation of Part I, by studying the conic $\mathcal{C}_P$ on the five points $ABCPQ$, where $ABC$ is a given ordinary triangle and $Q$ is the isotomcomplement of $P$, defined as the complement of the isotomic conjugate $P'$ of $P$ with respect to triangle $ABC$. We show that $\mathcal{C}_P$ also lies on the points $P'$ and $Q'$, where $Q'$ is the isotomcomplement of $P'$. The conic $\mathcal{C}_P$ lies on six other points which are the images of the vertices of $ABC$ under the affine mapping $\lambda=T_{P'} \circ T_P^{-1}$ and its inverse, where $T_P$ and $T_{P'}$ are the unique affine maps taking $ABC$ to the cevian triangles of $P$ and $P'$, respectively. In the paper we characterize the center $Z$ of $\mathcal{C}_P$ as the unique fixed point of $\lambda$ in the extended plane, when $\mathcal{C}_P$ is a parabola or an ellipse, and the unique ordinary fixed point of $\lambda$, when $\mathcal{C}_P$ is a hyperbola. We also show that $Z=GV \cdot T_P(GV)$, where $G$ is the centroid of $ABC$ and $V=PQ \cdot P'Q'$. When $P$ is the Gergonne point of $ABC$, this gives a new characterization of the Feuerbach point $Z$. All of our arguments are purely synthetic.

Book digitized by Google from the library of the University of Michigan and uploaded to the Internet Archive by user tpb.

"The Relation of the Principles of Logic to the Foundations of Geometry" is an article from Transactions of the American Mathematical Society, Volume 6 . View more articles from Transactions of the American Mathematical Society . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1986227