Fibonacci’s De Practica Geometrie

Hughes, Barnabas

doi:10.1007/978-0-387-72931-2

Cited by 32 publications

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“…Contrary to methods often called evolution methods [20,21] which evolve one digit at the time, Fibonacci's method is a (front-end) recursion. In Hughes' translation of De Practica Geometrie [10] (p. 36), he demonstrates Fibonacci's square root computation of √ 864 in seven steps all based on the binomial expansion (1).…”

Section: Integer Part Of the Square Rootmentioning

confidence: 99%

“…The recursive thinking is further evident in the example of √ 9,876,543 in De Practica Geometrie where Fibonacci consider the root of a seven digit number but first finds the root of the first five digits. The method used by Fibonacci is not the Hindu method described by Datta and Singh [9] (p. 169-175) as claimed in [10] (p. 35) or the computational method of al-Nasawī (c. 1011-c. 1075) where √ 57,342 is demonstrated [11]. In computing the integer part of the square root, a further misconception is that this is the Indian-Arabic algorithm [12].…”

Section: Introductionmentioning

confidence: 99%

“…Hughes [10] (p. 36) illustrates Fibonacci's computation with an evolution of Tables 1 to 6 in Figure 1 With the notation used in this paper, the corresponding computation is given in Figure 2 where the numbered lines correspond to the seven steps and to the tables in Figure 1. The subscripts where the digit is inserted in Figure 3 correspond to table in Figure 1 and line number in Figure 2.…”

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confidence: 99%

“…Fibonacci's computation of the remainder is just an rearrangement of the binomial expression for N − (α 1 10 + α 0 ) 2 which is shown below. Hughes [10] translates to English De Practica Geometrie in 2008 based on the transcript of Boncompagni from 1862 [24]. The following example from De Practica Geometrie of computing the integer part of √ 864 is based on [10] (p. 39), via the use of Google translate from Latin of [24] (p. 19), and OpenAI.…”

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confidence: 99%

“…Hughes [10] translates to English De Practica Geometrie in 2008 based on the transcript of Boncompagni from 1862 [24]. The following example from De Practica Geometrie of computing the integer part of √ 864 is based on [10] (p. 39), via the use of Google translate from Latin of [24] (p. 19), and OpenAI. The blue inserted text in the quoted material below, [L:x] refers to line x in the table for the square root computation in Figure 2.…”

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On the Square Root Computation in Liber Abaci and De Practica Geometrie by Fibonacci

Steihaug

2024

Mathematics

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We study the square root computation by Leonardo Fibonacci (or Leonardo of Pisa) in his MSS Liber Abaci from c1202 and c1228 and De Practica Geometrie from c1220. In this MSS, Fibonacci systematically describes finding the integer part of the square root of an integer in numerous examples with three to seven decimal digits. The results of these examples are summarized in a table in the paper. Liber Abaci also describes in detail finding an approximation to the fractional part of the square root. However, in other examples in Liber Abaci and De Practica Geometrie, only the approximate values of the fractional part of the square roots are stated. This paper further explores these approximate values using techniques like reverse engineering. Contrary to many claims that Fibonacci also used other methods or approximations, we show that all examples can be explained using one digit-by-digit method to compute the integer part of the square root and one approximation scheme for the fractional part. Further, it is shown that the approximation scheme is tied to the method to compute the integer part of the square root.

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Section: Integer Part Of the Square Rootmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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