Linguistic Elements
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Verbal: Verbal expressions related with geometric magnitudes. |
Verbal: Verbal expressions as “infinitely small” and “infinity of infinitely small elements” |
Verbal: Physics expressions as: velocity, flow, velocity of flow, magnitude of moment |
Verbal: Expressions as “infinitesimal”, “differential” and continuum. |
Verbal: Expressions like: Limits of a magnitude, variable and constant quantity, infinitesimal, successions, series, function, continuity and differential. |
Verbal: Expressions like continuity of geometric lines, rational and irrational numbers, set, etc. |
Geometric: Characteristic shapes from synthetic geometry. |
Geometric: Characteristic shapes from synthetic geometry, but with a different meaning of generation. |
Geometric: Characteristic shapes from Analytic and Euclidean Geometry. |
Geometric: Combination of elements of Euclidean geometry with elements of analytic geometry, for example: the characteristic triangle. |
Geometric: Use of graphic representations characteristic of analytic geometry. |
Geometrics: Great use of Analytic Geometry as representation of analytic expressions. |
Symbolic: Use of Greek alphabet letters to represent geometric shapes. |
Symbolic: Lack of adequate symbology. |
Symbolic: Predominant use of algebraic language. |
Symbolic: Notations of differential (dx and dy) and integral calculus. |
Symbolic: Algebraic language to represent constant and variable limits. |
Symbolic: Algebraic language to represent constant and variable limits |
Concepts and/or definitions
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Commensurable and incommensurable magnitudes.
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Indivisibles |
Fluents, Fluxions. Moments of flowing quantities. |
Characteristic triangle. |
Limit of a magnitude (D’ Alembert). |
Continuity of a line. |
Potential infinity. |
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Nascent and Evanescent quantities. |
Differential. |
Limit of a variable quantity (Cauchy). |
Completeness of real numbers. |
Proportions between magnitudes. |
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First and Ultimate ratios. |
Continuum. |
Actual infinity. Sequence limit |
Weierstrass’ definition of limits. |
Problems and/or tasks
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To determine relations between geometric commensurable and incommensurable magnitudes in general.
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Difficulties of quadrature and cubature. Problems of application in daily life or of other disciplines. |
To determine the velocity of a motion, knowing its distance and its inverse problem. |
Constitution of the continuum. Determining the tangent to a curve and problems of quadrature and cubature |
Typical of mathematics to substantiate infinitesimal calculus and mathematic analysis. |
Completeness of real numbers. Continuity of the actual geometrical line. |
Procedures
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Method of exhaustion. Double reduction to absurdity. |
Use of indivisibles through the properties of Euclidean geometry. |
Calculus algorithm of fluxions:
Inversed algebraic algorithm of fluxions calculus.
Method of convergent series.
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Operatory (addition, subtraction, etc.) with infinitesimal quantities and infinite processes. Use of differential for the resolution of problems. |
Algebra of limits. Operations are made with infinitesimals of different orders. |
Demonstration of limits of functions and algebra of limits based on the Weierstrass’ definition of limits. |
Properties or propositions
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Archimedes’ axiom.
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Every geometric magnitude is generated by the addition of an infinity of elements belonging to an immediately inferior magnitude. |
Preconceptions of limits in a geometric context. |
Introduction of the characteristic triangle. |
D’Alembert: the limit of a magnitude cannot be reached in the approximation process. Cauchy: The limit of a variable as a process of approximation to a fixed quantity, can be reached. |
Completeness of the set of real numbers. |
Proposition I from book X of Euclid’s elements (foundation underpinning exhaustion method) |
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Preconception of limit in an arithmetic context. |
Algebra of differentials. Acceptation of actual infinity. Principle of continuity of Leibniz. Generation of the finite by infinite processes. |
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Continuity Axiom of the geometric line. Bijective correspondence between the points of a geometric line and real numbers. |
Arguments
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Based on the axiomatic-deductive method of synthetic geometry. |
The surfaces and solids are composed by elements that generate “indivisibles”, which allows to determine the relation between unknown magnitudes comparing them with known magnitudes. |
Conception of infinitesimal calculus as a mathematical model of physics. Operatory justification of the algorithmic processes. The geometric intuition as a basis of mathematic propositions. |
Based on the Principle of Geometric and Arithmetic Continuity. The current state of mathematical objects and things in general is the result of infinite processes. |
D’Alembert: the basis of differential and integral calculus is the concept of limits. Cauchy: Concepts of infinitesimals and limits can be consistently integrated. |
The set of real numbers as an organized and complete body. The geometric line as a representation of the set of real numbers. Assumption of different types of infinities, particularly actual infinity. |