La teoría de las cuasi-proporciones de Pietro Mengoli

Pietro Mengoli presenta en 1659 la teoría de las cuasi-proporciones, fundamento de su método de cálculo de cuadraturas. Esta se basa en la teoría de las proporciones del Libro V de Elementos y en la obra en algebra simbólica de Viète. Inicia trabajando con números naturales generalizados por medio de variables y sumatorias de estos, ordenándolos en seis tablas triangulares. Con estos resultados, define una relación —cuasi-razón—, una relación de relaciones —cuasi-proporción—, y construye su teoría definiendo los conceptos de cantidad indeterminada determinable y la razón indeterminada determinable, conceptos donde subyace la idea del límite como proceso de aproximación Pietro Mengoli presented in 1659 the quasi-proportions theory, basis of his own method for calculating quadratures. This theory is based on the Euclidean theory of proportions, shown in the fifth book of Elements and on symbolic algebra work of Viète. He started working with generalized natural numbers using variables and sums of these, ordering these in six triangular tables. With these results, he defined a relation —quasi-ration— and a relation of relations —quasi-proportion— and constructs his theory defining the concepts of determined indeterminate quantity and determined indeterminate ratio, concepts where the idea of limit underlies as a process of approximation.

On the Syllogistic of G. Boole

Introduction. This article focuses on the investigation of Boole’s theory of categorical syllogism, exposed in his book “The Mathematical analysis of Logic”. That part of Boolean legacy has been neglected in the prevailed investigations on the history of logic; the latter provides the novelty of the work presented.Methodology and sources. The formal reconstruction of the methods of algebraic presentation of categorical syllogism, as it is exposed in the original work of Boole, is conducted. The character of Boolean methods is investigated in the interconnections with the principles of symbolic algebra on the one hand, and with the principles of signification, taken from R. Whately, on the other hand. The approaches to signification, grounding the syllogistic theories of Boole and Brentano, are analyzed in comparison, wherefrom we explain the reasons why the results of those theories are different so much.Results and discussion. It is demonstrated here that Boole has borrowed the principles of signification from the Whately’s book “The Elements of Logic”. The interpreting the content of the terms as classes, being combined with methods of symbolic algebra, has determined the core features of Boolean syllogism theory and its unexpected results. In contrast to Whately, Boole conduct the approach to ultimate ends, overcoming the restrictions imposed by Aristotelean doctrine. In particular, he neglects the distinction of subject and predicate among the terms of proposition, the order of premises, and provide the possibility to draw conclusions with negative terms. At the same time Boole missed that the forms of inference, parallel to Bramantip and Fresison, are legitimate forms in his system. In spite of the apparent affinities between the Boolean and Brentanian theories of judgment, the syllogistics of Boole appeared to be more flexible. The drawing of particular conclusion from universal premises is allowable in Boolean theory, but not in Brentanian one; besides, in his theory is allowable the drawing of conclusion from two negative premises, which is prohibited in Aristotelian syllogistic.Conclusion. Boole consistently interpreted signification of terms as classes; being combine with methods symbolic algebra it led to very flexible syllogism theory with rich results.

Numerical Implementation of Drop Spin and Tilt Method For Five-Axis Tool Positioning For Tensor Product Surfaces

This paper presents an extension of multi point machining technique, called the Drop Spin and Tilt (DST) method, that spins the tool on two axes, allowing for the generation of multiple contact points at varying distances around the first point of contact. The multiple DST second points of contact were used to manually generate a toolpath with uniform spacing between the two points of contact. The original DST method used a symbolic algebra package to position the tool on a bi-quadratic surface; our extension is a numer- ical solution that allows positioning a toroidal tool on a tensor product Bezier surface. Further, we investigate the spread of possible second points of contact as the tool is spun around these two axes, demonstrating the feasability of using the method to control the machining strip width.

The mapping of raster images on plane isometric grid

Drawing images on curvilinear shapes with the least distortion takes place in many design tasks. In most ways, build a grid, each elementary cell of which is painted a given color. In this problem it is necessary to solve two main problems: the first - to carry out the formation of a given curvilinear grid with elementary cells in the form of squares, which are called isometric (or isothermal); the second is to paint each cell of the curved area with the corresponding pixel color of the original raster The aim of the study is to reveal the way of displaying raster images on flat curvilinear areas represented by isometric grids, and with the help of a computer model in the Maple symbolic algebra to analyze the influence of isometric grid parameters on the position and size of displayed raster images. The mapping of images onto curvilinear forms with minimal distortion takes place in many design tasks. A method of conformal mapping of arbitrary raster images onto plane curvilinear region is proposed, which are represented by isometric (also called isothermal) grids. The essence of the proposed method is as follows. Any raster image, for example, digital photography in jpg format, is characterized by the dimensions N×M - the number of pixels in width and height. In addition, each pixel has a color and brightness, which are arranged in rows and columns. To apply a raster image to a curvilinear region, it is also necessary to divide the curvilinear domain into N×M, the number of elementary squares, each of which is assigned the corresponding color from the raster. The influence and arguments of the various isometric grids constructed on the sizes and positions of an arbitrary raster image are investigated in the article. It is shown how the isometric grid, depending on and localizes the raster image - it can be located both within the limits of the isometric grid coordinate lines and beyond it, can also be oriented in different directions with respect to the and coordinate lines. It is shown the possibility of scaling a raster image that can be performed relative to the relative dimensions of an isometric grid. Since there is a correspondence between the pixel matrix of the original raster image and the - cells of the isometric grid, the rotation of the image will affect its position in the isometric grid. For example, rotating the original bitmap image at an angle 90 degrees will change its location on a plane isometric grids – from along the coordinate lines to along the coordinate lines. Note that, the curvilinear cells of the constructed isometric grids differ somewhat from the shape of the squares because the values and of the corresponding arguments and of their coordinate lines were taken somewhat too large. Otherwise, cells would degenerate into points and the corresponding grid image would not be so clear.

A Synergy between History of Mathematics and Mathematics Education: A Possible Path from Geometry to Symbolic Algebra

This paper proposes an experimental path aimed at guiding upper secondary school students to overcome that discontinuity, often perceived by them, between learning geometry and learning algebra. This path contributes to making students aware of how the algebraic language, formalized in the most powerful form by Descartes, grafts itself onto the geometric language. This is realized by introducing a problem included in a text written by Abū Kāmil before the year 870. This awareness acquired by the students, when accompanied by some semiotic considerations, allows the translation of the problem from “spoken” algebra to “symbolic” algebra, and it represents the background for a possible use of the same problem within the framework of analytic geometry. This proposition manifests a didactic and popular efficacy that supports and favors the recognition of the object it is talking about in different contexts, helping to create a unitary vision of mathematics.

Classification of All Non-Isomorphic Regular and Cuspidal Arm Anatomies in an Orthogonal Metamorphic Manipulator

This paper proposes a classification of all non-isomorphic anatomies of an orthogonal metamorphic manipulator according to the topology of workspace considering cusps and nodes. Using symbolic algebra, a general kinematics polynomial equation is formulated, and the closed-form parametric solution of the inverse kinematics is obtained for the coming anatomies. The metamorphic design space was disjointed into eight distinct subspaces with the same number of cusps and nodes plotting the bifurcating and strict surfaces in a cartesian coordinate system { θ π 1 , θ π 2 , d 4 } . In addition, several non-singular, smooth and continuous trajectories are simulated to show the importance of this classification.