By Carolyn Kieran, Lesley Lee, Nadine Bednarz, N. Bednarz, C. Kieran, L. Lee
In Greek geometry, there's an mathematics of magnitudes within which, when it comes to numbers, purely integers are concerned. This idea of degree is proscribed to targeted degree. Operations on magnitudes can't be truly numerically calculated, other than if these magnitudes are precisely measured through a definite unit. the speculation of proportions doesn't have entry to such operations. It can't be noticeable as an "arithmetic" of ratios. whether Euclidean geometry is completed in a hugely theoretical context, its axioms are primarily semantic. this is often opposite to Mahoney's moment attribute. this can't be acknowledged of the speculation of proportions, that is much less semantic. merely man made proofs are thought of rigorous in Greek geometry. mathematics reasoning can also be artificial, going from the identified to the unknown. eventually, research is an method of geometrical difficulties that has a few algebraic features and consists of a mode for fixing difficulties that's diverse from the arithmetical process. three. GEOMETRIC PROOFS OF ALGEBRAIC ideas until eventually the second one half the nineteenth century, Euclid's parts was once thought of a version of a mathematical idea. this can be one the reason for this is that geometry was once utilized by algebraists as a device to illustrate the accuracy of principles differently given as numerical algorithms. it may possibly even be that geometry used to be a technique to symbolize normal reasoning with no regarding particular magnitudes. to move a section deeper into this, listed here are 3 geometric proofs of algebraic ideas, the frrst by means of Al-Khwarizmi, the opposite by way of Cardano.
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Additional info for Approaches to Algebra: Perspectives for Research and Teaching
6Yrup, 1987). 6yrup (1990) claims that: Old Babylonian "algebra" cannot have been arithmetical, that is, conceptualized as dealing with unknown numbers as organized by means of numerical operations. Instead it appears to have been organized on the basis of "naive," non-deductive geometry. (p. 211) This non-deductive geometry, developed extensively in H0yrup (1985, 1986), consists of a "cut-and-paste geometry" in which the complicated arithmetical calculations resulting from the classical interpretation correspond to simple naive geometric transformations.
112 which you made span you tear out inside 1: 112 the square line. 6Yrup's explanation,2 the procedure consists of projecting a rectangle of base equal to 1 on the line-side of the square (Figure 1). The projected rectangle is cut into two rectangles, the base of each one being equal to 112. The rectangle on the right is then transferred to the bottom (Figure 2). Then a small square is added (Figure 3) in order to obtain a complete square from which the sought square-line can be found (Figure 4).
23 Cardano (1968, p. 82). The problem has been translated into a modern form by Witmer. 24 In French algebra books, such a number was called nombre nombrant (numbering number). 25 Cardano (1968, pp. 96-99). This is in chapter XI. 26 It would have been very interesting to study Fibonacci. Lnis Radford pointed out to me that in the Liber AbOOci (1202), the passage from geometry to algebraic reasoning is not 8S straightforward as in Chuquet. 27 The rule of "first terms" refers to rules of manipulation of the first (prime) power of the unknowns.