This essay describes the author\u27s recent encounter with two well-known passages in Plutarch th... more This essay describes the author\u27s recent encounter with two well-known passages in Plutarch that touch on a crucial episode in the history of the Greek mathematics of the fourth century BCE involving various approaches to the problem of the duplication of the cube. One theme will be the way key sources for understanding the history of our subject sometimes come from texts that have much wider cultural contexts and resonances. Sensitivity to the history, to the mathematics, and to the language is necessary to tease out the meaning of such texts. However, in the past, historians of mathematics often interpreted these sources using the mathematics of their own times. Their sometimes anachronistic accounts have often been presented in the mainstream histories of mathematics to which mathematicians who do not read Greek must turn to learn about that history. With the original sources, the tidy and inevitable picture of the development of mathematics disappears and we are left with a m...
ABSTRACT This appendix will discuss several computer algebra systems that can be used in conjunct... more ABSTRACT This appendix will discuss several computer algebra systems that can be used in conjunction with the text. We will describe AXIOM, Maple, Mathematica and REDUCE in some detail, and then mention some other systems. These are all amazingly powerful programs, and our brief discussion will not do justice to their true capability.
A canonically-embedded curve of genus $g$ is a pure 1-dimensional, non-degenerate subscheme $C$ o... more A canonically-embedded curve of genus $g$ is a pure 1-dimensional, non-degenerate subscheme $C$ of ${\bf P}^{g-1}$ over an algebraically closed field $k$, for which ${\cal O}_C(1) \cong \omega_C$, (the dualizing sheaf)$ and $h^0(C, {\cal O}_C) = 1$, $h^0(C, \omega_C) = g$. The singularities of $C$ (if any) are Gorenstein, and $C$ is connected of degree $2g-2$ and arithmetic genus $g$. In a recent paper, Schreyer has proved that Petri's normalization of the homogeneous ideal $I(C)$ of a smooth canonically-embedded curve can be also carried out for singular curves, provided that the curve has a simple $(g-2)$-secant (a linear ${\bf P}^{g-3}$ intersecting $C$ transversely at exactly $g-2$ (smooth) points). We use the Petri normalization to study the Hilbert scheme of curves of degree $2g-2$ and arithmetic genus $g$ in ${\bf P}^{g-1}$ in the low-genus cases $g = 5,6$. The main results are that the Hilbert points of all curves for which Petri's approach applies lie on one irred...
The recent extensive work on different approaches to the Schottky problem has produced marked pro... more The recent extensive work on different approaches to the Schottky problem has produced marked progress on several fronts. At the same time, it has become apparent that there exist very close connections between the various characterizations of Jacobian varieties described in Mumford's classic lectures {\it Curves and Their Jacobians\/}. Until now, the approach via double translation manifolds has seemed to be quite different from other approaches to the Schottky problem. The purpose of this paper is to bring this last approach ``into the fold'' as it were, and to show precisely how it relates to characterizations of Jacobians based on trisecants and flexes of the Kummer variety, and the K.P. equation.
We study the set W(𝓛) of Weierstrass points of all positive tensor powers of an invertible sheaf ... more We study the set W(𝓛) of Weierstrass points of all positive tensor powers of an invertible sheaf 𝓛 on an irreducible rational curve X with g ≧ 2 ordinary cusps. Using an idea from B. Olsen's study of the analogous question on smooth curves, and an explicit formula for the "theta function" of a cuspidal rational curve, we show that W(𝓛) is never dense on X (in contrast to the case of smooth curves of genus g ≧ 2).
John B. Little is the translator. This is a Latin to English translation of Geometria Practica by... more John B. Little is the translator. This is a Latin to English translation of Geometria Practica by Chrisopher Clavius, S.J. (1538-1612), the preeminent Jesuit mathematician and mathematical astronomer of his time. The first edition of Geometria Practica appeared in 1604. This translation is of the second edition from 1606, produced by the printshop of Johann Albin in Mainz. In preparing this translation we have made use of the electronic version of the 1606 edition of the Geometria Practica maintained by the Bayerische StaatsBibliothek. In particular, all of the figures have been copied from the scanned images here. The typesetting was done with the LaTeX system. In an attempt to duplicate the organization of the original book as much as possible, the marginal references and labels as in Clavius\u27s original have been included. References in the form Book X, Prop. Y are references to Clavius\u27s own edition of Euclid\u27s Elements. This was very influential and a standard text in J...
We consider the Geometria Practica of Christopher Clavius, S.J., a suprisingly eclectic and compr... more We consider the Geometria Practica of Christopher Clavius, S.J., a suprisingly eclectic and comprehensive textbook of practical geometry, whose first edition appeared in 1604. Our focus is on four particular sections from Books IV and VI where Clavius has either used his sources in an interesting way or where he has been uncharacteristically reticent about them. These include the treatments of Heron’s Formula, Archimedes’ Measurement of the Circle, four methods for constructing two mean proportionals between two lines, and finally an algorithm for computing nth roots of numbers.
This essay describes the author\u27s recent encounter with two well-known passages in Plutarch th... more This essay describes the author\u27s recent encounter with two well-known passages in Plutarch that touch on a crucial episode in the history of the Greek mathematics of the fourth century BCE involving various approaches to the problem of the duplication of the cube. One theme will be the way key sources for understanding the history of our subject sometimes come from texts that have much wider cultural contexts and resonances. Sensitivity to the history, to the mathematics, and to the language is necessary to tease out the meaning of such texts. However, in the past, historians of mathematics often interpreted these sources using the mathematics of their own times. Their sometimes anachronistic accounts have often been presented in the mainstream histories of mathematics to which mathematicians who do not read Greek must turn to learn about that history. With the original sources, the tidy and inevitable picture of the development of mathematics disappears and we are left with a m...
ABSTRACT This appendix will discuss several computer algebra systems that can be used in conjunct... more ABSTRACT This appendix will discuss several computer algebra systems that can be used in conjunction with the text. We will describe AXIOM, Maple, Mathematica and REDUCE in some detail, and then mention some other systems. These are all amazingly powerful programs, and our brief discussion will not do justice to their true capability.
A canonically-embedded curve of genus $g$ is a pure 1-dimensional, non-degenerate subscheme $C$ o... more A canonically-embedded curve of genus $g$ is a pure 1-dimensional, non-degenerate subscheme $C$ of ${\bf P}^{g-1}$ over an algebraically closed field $k$, for which ${\cal O}_C(1) \cong \omega_C$, (the dualizing sheaf)$ and $h^0(C, {\cal O}_C) = 1$, $h^0(C, \omega_C) = g$. The singularities of $C$ (if any) are Gorenstein, and $C$ is connected of degree $2g-2$ and arithmetic genus $g$. In a recent paper, Schreyer has proved that Petri's normalization of the homogeneous ideal $I(C)$ of a smooth canonically-embedded curve can be also carried out for singular curves, provided that the curve has a simple $(g-2)$-secant (a linear ${\bf P}^{g-3}$ intersecting $C$ transversely at exactly $g-2$ (smooth) points). We use the Petri normalization to study the Hilbert scheme of curves of degree $2g-2$ and arithmetic genus $g$ in ${\bf P}^{g-1}$ in the low-genus cases $g = 5,6$. The main results are that the Hilbert points of all curves for which Petri's approach applies lie on one irred...
The recent extensive work on different approaches to the Schottky problem has produced marked pro... more The recent extensive work on different approaches to the Schottky problem has produced marked progress on several fronts. At the same time, it has become apparent that there exist very close connections between the various characterizations of Jacobian varieties described in Mumford's classic lectures {\it Curves and Their Jacobians\/}. Until now, the approach via double translation manifolds has seemed to be quite different from other approaches to the Schottky problem. The purpose of this paper is to bring this last approach ``into the fold'' as it were, and to show precisely how it relates to characterizations of Jacobians based on trisecants and flexes of the Kummer variety, and the K.P. equation.
We study the set W(𝓛) of Weierstrass points of all positive tensor powers of an invertible sheaf ... more We study the set W(𝓛) of Weierstrass points of all positive tensor powers of an invertible sheaf 𝓛 on an irreducible rational curve X with g ≧ 2 ordinary cusps. Using an idea from B. Olsen's study of the analogous question on smooth curves, and an explicit formula for the "theta function" of a cuspidal rational curve, we show that W(𝓛) is never dense on X (in contrast to the case of smooth curves of genus g ≧ 2).
John B. Little is the translator. This is a Latin to English translation of Geometria Practica by... more John B. Little is the translator. This is a Latin to English translation of Geometria Practica by Chrisopher Clavius, S.J. (1538-1612), the preeminent Jesuit mathematician and mathematical astronomer of his time. The first edition of Geometria Practica appeared in 1604. This translation is of the second edition from 1606, produced by the printshop of Johann Albin in Mainz. In preparing this translation we have made use of the electronic version of the 1606 edition of the Geometria Practica maintained by the Bayerische StaatsBibliothek. In particular, all of the figures have been copied from the scanned images here. The typesetting was done with the LaTeX system. In an attempt to duplicate the organization of the original book as much as possible, the marginal references and labels as in Clavius\u27s original have been included. References in the form Book X, Prop. Y are references to Clavius\u27s own edition of Euclid\u27s Elements. This was very influential and a standard text in J...
We consider the Geometria Practica of Christopher Clavius, S.J., a suprisingly eclectic and compr... more We consider the Geometria Practica of Christopher Clavius, S.J., a suprisingly eclectic and comprehensive textbook of practical geometry, whose first edition appeared in 1604. Our focus is on four particular sections from Books IV and VI where Clavius has either used his sources in an interesting way or where he has been uncharacteristically reticent about them. These include the treatments of Heron’s Formula, Archimedes’ Measurement of the Circle, four methods for constructing two mean proportionals between two lines, and finally an algorithm for computing nth roots of numbers.
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