Number Of Equal Parts Research Articles

The last chapter of the Disquisitiones of Gauss

This exposition reviews what exactly Gauss asserted and what did he prove in the last chapter of {\sl Disquisitiones Arithmeticae} about dividing the circle into a given number of equal parts. In other words, what did Gauss claim and actually prove concerning the roots of unity and the construction of a regular polygon with a given number of sides. Some history of Gauss's solution is briefly recalled, and in particular many relevant classical references are provided which we believe deserve to be better known.

ФОРМУВАННЯ ІНТЕГРАЦІЙНОЇ КОМПЕТЕНТНОСТІ СТУДЕНТІВ ПРИ ВИВЧЕННІ КУРСУ «НАРОДНІ РЕМЕСЛА»

The article considers some aspects of the problem of integration of students' knowledge and the implementation of interdisciplinary links in the modern educational process. It is established that these problems at different times were the subject of consideration of many scientists in whose works interdisciplinary, interdisciplinary connections were studied from different points of view: pedagogical, psychological, philosophical. In particular, the works of teachers, psychologists studied the integration of science, technology and education, a detailed analysis of the structure of the educational process, considered the problems of interdisciplinary links on the cognitive activity of students, classified types and types of interdisciplinary links by various criteria. Also, a significant amount of scientific work is devoted to identifying ways to eliminate duplication of educational material, methods of conducting classes using interdisciplinary links, the problems of integration of knowledge in the training of future teachers. This publication offers elements of the formation of integration competence in students when they study the course "Folk Crafts". The content and structure of the discipline "Folk Crafts", which is a discipline of professional training of future teachers of labor education and technology, describes the interdisciplinary links in the study of this discipline. There are specific examples of problems in solving which students must use previously acquired knowledge in other disciplines. In particular, when studying geometric wood carving, carving of complex ornaments there is a need to build high-quality drawings, which are associated with geometric constructions on the plane: dividing the circle into a number of equal parts, drawing parallel, perpendicular lines, dividing the segment in half, tangent to angle bisectors and so on. In performing such tasks, students use knowledge of algebra, descriptive geometry, computer graphics and other disciplines. It is proved that such an approach contributes to the formation of subject and integration competencies as the ability of students to holistically perceive the acquired knowledge, to use their professional potential in future pedagogical activities.

Trisection of Any Angle and Consequentially the Division of Any Angle Into Any Number of Equal Parts

Historically, a contrived trisected line was used to trisect any other line, using the principle of projection. This is in essence about relationship and its accomplishment is about working backwards. Loosely speaking, any angle comprises two connecting lines. Attempts at trisecting any angle, which is dividing it into three equal parts, failed. In this paper any angle is defined as a unique pair of arc and chord of sector of a circle irrespective of arc radius. Two theorems viz. Equal arcs have equal central angles and equal chords have equal central angles are combined to establish a unique relationship between a pair of arc-chord and its composite of three identical pairs of arc-chord, thereby revealing a CYCLIC TRAPEZIUM, where the base defines the angle, and each equal edge defines each of the equal trisected parts of this angle. For a range of angles between 0o and 360o, this relationship is expressed as Lorna Graph, which becomes the practical tool for trisection of any angle, using the working backwards approach. This approach is extended to division of any angle into any number of equal parts.

Digital image hiding technology based on chaotic system with variable parameters

To ensure the security and robustness of hiding image information under the conditions of visibility,a digital image hiding method based on a chaotic system with variable parameters was proposed. Firstly,with the pseudo-random sequences generated by the chaos system,a hiding image was encrypted. Secondly,the one-dimensional vector which transformed from the template image was divided into a number of equal parts according to the size of the hiding image,and each part corresponding to each pixel of the hiding image. Finally,with the new chaotic sequences and digital image blending technique,the pixel of hiding image was blended into a position of the part. This digital image hiding technique improves the imperceptibility and security of hiding image. Simulation results show that the proposed technique has good imperceptibility and high robustness.

Another Way to Divide a Line Segment into n Equal Parts

In summer 1995, two high school students, David Goldenheim and Dan Litchfield, discovered a way to divide a line segment into any number of equal parts. Their method differed from the standard method of Euclid. Together with their teacher Charles Dietrich, they wrote an article on their method, which appeared in the January 1997 issue of the Mathematics Teacher (Litchfield, Goldenheim, and Dietrich 1997). The discovery received considerable publicity in the popular media and was written up in the Wall Street Journal and the New York Times. The authors gave talks at several professional conferences and were invited to meet the secretary of education.

Dividing Any Angle into Any Number of Equal Parts

Two problems, easily solved in three dimensions, that are often approached in two.

Current Jam

This article focuses on the latest technology, called the insulated gate bipolar transistor that has increased switching frequencies almost twentyfold from the last few years. It turns out that the newer variable-frequency drives, while working full-time shifts as speed controllers of ac induction motors, may be moonlighting with a little part-time electrical discharge machining (EDM). The new drives are suspected of setting up bearing currents. These currents, acting in a fashion identical to the mechanism of EDM—the manufacturing technique by which metal is removed from a workpiece through the use of high-energy sparks—have been eroding the material from the races of motor bearings. In some cases, stray currents are destroying bearings a few months after startup. ABB recently patented a motor winding designed to eliminate circulating bearing currents. The design divides the stator winding into an even number of equal parts per phase. The groups are then distributed uniformly between ac supply connections at both ends of the stator.

Rational Points on the Number Line

It Is now fashionable for eighth-grade mathematics texts to include a chapter on intuitive geometry before the chapter in which rational numbers are assigned points on a number line, but the geometry is seldom brought to bear upon the latter problem. The reason is probably that the usual construction is difficult for eighth graders. Recall that it involves dividing a segment into some integral number of equal parts.

Shear Center of a Leading-Edge Wing Beam

Simple formulas based on Simpson's Rule are given for the coordinates of the shear center and the shear force per unit length of arc in terms of the lengths and thicknesses of the nose and web sheets and the coordinates of a set of points which divide the nose contour into a number of equal parts.

892. To divide an angle into any number of equal parts

892. To divide an angle into any number of equal parts

Lenses for Light Distribution

The principle on which lenses for securing a required distribution of light from a given source have been designed is illustrated by a two-dimensional example. The principle employed is to divide the incident and emergent energy into a number of equal parts, and compute the lens system so that the rays which separate off these portions of incident light from one another are refracted as rays which separate the corresponding portions of the emergent light. The surfaces obtained are in general of varying curvature, and the lenses must, therefore, be moulded. It is shown how the effect of the finite size of the light source may be determined.

3. Abstract of Professor Kelland's Exposition of the Views.

The fundamental hypothesis of the author was stated to be this:— That the eye is capable of appreciating the exact subdivision of spaces, just as the ear is capable of appreciating the exact subdivisions of intervals of time; so that tho division of space into an exact number of equal parts will affect the eye agreeably in the same way that the division of the time of vibration in music, into an exact number of equal parts, agreeably affects the ear. But the question now arises, What spaces does the eye most readily divide? It was stated that the author supposes those spaces to be angles, not lines; believing that the eye is more affected by direction than by distance. The basis of his theory, accordingly, is, that bodies are agreeable to the eye, so far as symmetry is concerned, whenever the principal angles are exact submultiples of some common fundamental angle.

On the theory of the elliptic transcendents

Fagnani discovered that the two arcs of the periphery of a given ellipse may be determined in many ways, so that their difference shall be equal to an assignable straight line; and proved that any arc of a lemniscate, like that of a circle, may be multiplified any number of times, or may be subdivided into any number of equal parts, by finite algebraic equations. What he had accomplished with respect to the arcs of the lemniscates, which are expressed by a particular elliptic integral, Euler extended to all transcendents of the same class. Landen showed that the arcs of the hyperbola may be reduced, by a proper transformation, to those of an ellipse. Lagrange furnished us with a general method for changing an elliptic function into another having a different modulus; a process which greatly facilitates the numerical calculation of this class of integrals. Legendre distributed the elliptic functions into distinct classes, and reduced them to a regular theory, developing many of their properties which were before unknown, and introducing many important additions and improvements in the theory. Mr. Abel of Christiana happity conceived the idea of expressing the amplitude of an elliptic function in terms of the function itself, which led to the discovery of many new and useful properties. Mr. Jacobi proved, by a different method, that an elliptic function may be transformed in innumerable ways into another similar function, to which it bears constantly the same proportion. But his demonstrations require long and complicated calculations; and the train of deductions he pursues does not lead naturally to the truths which are proved, nor does it present in a connected view all the conclusions which the theory embraces. The author of the present paper gives a comprehensive view of the theory in its full extent, and deduces all the connected truths from the same principle. He finds that the sines or cosines of the amplitudes, used in the transformations, are analogous to the sines or cosines of two circular arcs, one of which is a multiple of the other; so that the former quantities are changed into the latter when the modulus is supposed to vanish in the algebraic expression. Hence he is enabled to transfer to the elliptic transcendents the same methods of investigation that succeed in the circle: a procedure which renders the demonstrations considerably shorter, and which removes most of the difficulties, in consequence of the close analogy that subsists between the two cases.

XX. On the theory of the elliptic transcendents

The branch of the integral calculus which treats of elliptic transcendents originated in the researches of Fagnani, an Italian geometer of eminence. He discovered that two arcs of the periphery of a given ellipse may be determined in many ways, so that their difference shall be equal to an assignable straight line; and he proved that any arc of the lemniscata, like that of a circle, may be multiplied any number of times, or may be subdivided into any number of equal parts, by finite algebraic equations. These are particular results; and it was the discoveries of Euler that enabled geometers to advance to the investigation of the general properties of the elliptic functions. An integral in finite terms deduced by that geometer from an equation between the differentials of two similar transcendent quantities not separately integrable, led immediately to an algebraic equation between the amplitudes of three elliptic functions, of which one is the sum, or the difference, of the other two. This sort of integrals, therefore, could now be added or subtracted in a manner analogous to circular arcs, or logarithms; the amplitude of the sum, or of the difference, being expressed algebraically by means of the amplitudes of the quantities added or subtracted. What Fagnani had accomplished with respect to the arcs of the lemniscata, which are expressed by a particular elliptic integral, Euler extended to all transcendents of the same class. To multiply a function of this kind, or to subdivide it into equal parts, was reduced to solving an algebraic equation. In general, all the properties of the elliptic transcendents, in which the modulus remains unchanged, are deducible from the discoveries of Euler. Landen enlarged our knowledge of this kind of functions, and made a useful addition to analysis, by showing that the arcs of the hyperbola may be reduced, by a proper transformation, to those of the ellipse. Every part of analysis is indebted to Lagrange, who enriched this particular branch with a general method for changing an elliptic function into another having a different modulus, a process which greatly facilitates the numerical calculation of this class of integrals. An elliptic function lies between an arc of the circle on one hand, and a logarithm on the other, approaching indefinitely to the first when the modulus is diminished to zero, and to the second when the modulus is augmented to unit, its other limit. By repeatedly applying the transformation of Lagrange, we may compute either a scale of decreasing moduli reducing the integral to a circular arc, or a scale of increasing moduli bringing it continually nearer to a logarithm. The approximation is very elegant and simple, and attains the end proposed with great rapidity. The discoveries that have been mentioned occurred in the general cultivation of analysis; but Legendre has bestowed much of his attention and study upon this particular branch of the integral calculus. He distributed the elliptic functions in distinct classes, and reduced them to a regular theory. In a Mémoire sur les Transcendantes Elliptiques, published in 1793, and in his Exercices de Calcul Intégral, which appeared in 1817 he has developed many of their properties entirely new; investigated the easiest methods of approximating to their values; computed numerical tables to facilitate their application; and exemplified their use in some interesting problems of geometry and mechanics. In a publication so late as 1825, the author, returning to the same subject, has rendered his theory still more perfect, and made many additions to it which further researches had suggested. In particular we find a new method of making an elliptic function approach as near as we please to a circular arc, or to a logarithm, by a scale of reduction very different from that of which Lagrange is the author, the only one before known. This step in advance would unavoidably have conducted to a more extensive theory of this kind of integrals, which, nearly about the same time, was being discovered by the researches of other geometers.

LXXIX. Theorems for calculating the Temperaments of such regular douzeaves as are commensurable, or defined by a certain number of equal parts, into which the Octave is divided

LXXIX. <i>Theorems for calculating the Temperaments of such regular douzeaves as are commensurable, or defined by a certain number of equal parts, into which the Octave is divided</i>