ePub Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin (Sources and Studies in the History of Mathematics and Physical Sciences) download

by Jens Høyrup

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Jens Høyrup

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978-1441929457

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1441929452

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English

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Springer; Softcover reprint of the original 1st ed. 2002 edition (December 6, 2010)

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History

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Ancient Civilizations

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Jens Høyrup Ancient Civilizations English

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Babylonian mathematical sources are extremely sparse in words, as is. .

Babylonian mathematical sources are extremely sparse in words, as is understandable since they are laboriously inscribed on clay. This extremely condensed, telegraphic style, allows for a considerable scope of interpretation. We fill in the hole in the L. This hole is a square of side 30', so its area is 15'. So when we fill in the hole the total area is 45'+15' 1. Otto Neugebauer's "The Exact Sciences in Antiquity" was monumental when published in the 1950s.

Request PDF On Jan 1, 2002, Jens Høyrup and others published Lengths, Widths, Surfaces. By the 2 nd millennium, the Old Babylonian mathematicians had developed a so- called pure mathematics involving complex algorithms for calculating answers to artificial as well as practical problems (Friberg, 2007;Høyrup, 2002a, 2002bRobson, 2007Robson,, 2008.

Sources and Studies in the History of Mathematics and Physical . It and AO 8862 belong to a larger group of Old Babylonian school tablets from Larsa.

Sources and Studies in the History of Mathematics and Physical Sciences. For instance AO 8863 and AO 8864, both hexagonal prisms carrying Sumerian literary compositions from the scribal curriculum, are dated to 1739 BCE. Høyrup has drawn together and updated the key insights of his intellectual journey of the past 15 years and presented them in one unified, handsomely produced volume

Sources and Studies in the History of Mathematics and Physical Sciences. Lengths, Widths, Surfaces. A Portrait of Old Babylonian Algebra and Its Kin. Authors: Høyrup, Jens. price for USA in USD (gross). ISBN 978-1-4757-3685-4. Høyrup has drawn together and updated the key insights of his intellectual journey of the past 15 years and presented them in one unified, handsomely produced volume. it contains a wealth of information and can be minded by the interested reader for years to come. It is a worthy testament to a career of deep scholarship.

Jens Egede Høyrup, born 1943 in Copenhagen, is a Danish historian of mathematics, specializing in pre-modern and early modern mathematics, ancient Mesopotamian .

Jens Egede Høyrup, born 1943 in Copenhagen, is a Danish historian of mathematics, specializing in pre-modern and early modern mathematics, ancient Mesopotamian mathematics in particular. He is especially known for his interpretation of what has often been referred to as Old Babylonian "algebra" as consisting of concrete, geometric manipulations. Sources and Studies in the History of Mathematics and Physical Sciences.

Springer, Springer New York.

Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin (Sources and Studies in the History of Mathematics and Physical Sciences). Sources and Studies in the History of Mathematics and Physic. Springer, Springer New York.

Lengths, widths, surfaces: a portrait of Old Babylonian algebra and its kin. Jens Høyrup. Download (pdf, 1. 7 Mb) Donate Read.

Lengths, Widths, Surfaces book. Goodreads helps you keep track of books you want to read. Start by marking Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin as Want to Read: Want to Read savin. ant to Read.

Sources and Studies in the History of Mathematics and Physical Sciences K. Andersen Brook Taylor’s Work on Linear . The problems lead to systems of equations, apparently solved by use of metric algebra in a quite sophisticated way. Then follows another example of an igi-igi. Andersen Brook Taylor’s Work on Linear Perspective K. Andersen The Geometry of An Art . Bos Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction J. Cannon/S. Dostrowsky The Evolution of Dynamics: Vibration Theory From 1687 to 1742 B. Chandler/W. bi problem, and two more, badly preserved problems. The text ends with a summary of the topics in the text.

Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin (Sources and Studies in the .

Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin (Sources and Studies in the History of Mathematics and Physical Sciences). FIND Sources and Studies in the History of Mathematics and Physical Sciences on Barnes & Noble.

In this examination of the Babylonian cuneiform "algebra" texts, based on a detailed investigation of the terminology and discursive organization of the texts, Jens Høyrup proposes that the traditional interpretation must be rejected. The texts turn out to speak not of pure numbers, but of the dimensions and areas of rectangles and other measurable geometrical magnitudes, often serving as representatives of other magnitudes (prices, workdays, etc...), much as pure numbers represent concrete magnitudes in modern applied algebra. Moreover, the geometrical procedures are seen to be reasoned to the same extent as the solutions of modern equation algebra, though not built on any explicit deductive structure.

Bajinn

Hoyrup argues very convincingly for a more geometrical reading of the Babylonian sources than the traditional arithmetical reading. Babylonian mathematical sources are extremely sparse in words, as is understandable since they are laboriously inscribed on clay. This extremely condensed, telegraphic style, allows for a considerable scope of interpretation. A typical phrase such as "30 a-na 7 ta-na-sima 210" (p. 5) can be interpreted alternately in purely arithmetical ("multiply 30 by 7; the result is 210") or geometrical terms ("form a rectangle with sides 30 and 7; it's area is 210"). The former option is the traditional one adopted by Neugebauer et al. The primary argument for this interpretation is the fact that the sources often add lengths and areas together, which is geometrically nonsensical. On the other hand the terms used for multiplication seem to point, linguistically speaking, to the geometrical interpretation ("to raise", "to hold", etc.), and indeed certain words for "multiplication" are only used for multiplying two lengths, never areas. Furthermore, certain words that can be read as "protrude", "break", etc., can be understood quite literally in the geometrical reading, whereas they are basically ignored in the arithmetical readings:

"We may say that the received interpretation made sense of the numbers occurring in the text. But it obliterated the distinction made in the texts which after all need not be synonymous unless the arithmetical interpretation is taken for granted; ... and it had to dismiss some phrases as irrelevant ... or to explain them by gratuitous ad-hoc hypotheses." (p. 13)

Consider an example: BM 13901 #1. Hoyrup's translation is as follows (p. 50).

"The surface and my confrontation I have accumulated: 45' it is."

It is to be understood that "the surface" means the area of a square, and the "confrontation" its side. So the problem is x^2+x=45'.

"1, the projection, you posit."

This step gives a concrete geometrical interpretation of the expression x^2+x. We draw a square and suppose its side to be x. Then we make a rectangle of base 1 protrude from one of its sides. This rectangle has the area 1*x, so the whole figure has the area x^2+x, which is the quantity known.

"The moiety of 1 you break, 30' and 30' you make hold."

We break the rectangle in half and attach the half we cut off to an adjacent side of the square. We have now turned our area of 45' into an L-shaped figure.

"15' to 45' you append: 1."

We fill in the hole in the L. This hole is a square of side 30', so its area is 15'. So when we fill in the hole the total area is 45'+15'=1.

"1 is equalside."

The side of the big square is 1.

"30' which you have made hold in the inside of 1 you tear out: 30' is the confrontation."

The side of the big square is x+30' by construction, and we have just seen that it is also 1. Therefore x must be 1-30'=30', and we have solved the problem.

The difference between the arithmetical and geometrical readings is important since only in Hoyrup's reading does it follow that "The procedure is ... algorithm and proof in one. ... [It] performs all steps in such a way that their correctness is obvious." (p. 98)

But the geometrical reading does not mean that the procedure is applicable to geometrical problems only. On the contrary, the procedure is "functionally abstract": there are examples where a segment represents a number, an area, a volume, or a commercial rate (p. 280). Virtually all texts "use the Sumerograms us and sag unerringly for the lengths and widths of the standard representation; 'real' linear dimensions (the length of a wall, the distance bricks are to be carried, the width of a canal), in contrast, may as well be written in syllabic Akkadian ... This suggests strongly that the Old Babylonian authors were explicitly aware of the functionally abstract character of their standard representation." (pp. 280-281) "In this sense, Neugebauer was right in considering the us and sag as equivalents of the symbols of modern algebra." (p. 10)

Ganthisc

In research for my own book to be, this source is the newest, most up to date, and comprehensive I have found covering Babylonian Mathematics. Otto Neugebauer's "The Exact Sciences in Antiquity" was monumental when published in the 1950s. This work of Hoyrup continues the analysis of some of the same and related sources and updates us with what has been learned in the last 50 years. Hoyrup has a broader survey of tablets that lend support to his new interpretations. This provides a source in english for many tablets that were previously only available in other languages. It also can serve as a list of original sources so that one could easily locate tablets for a specialized presentation.
I am excited about this work and hope that its contents can "trickle down" to the level of common understanding. The modern world will see that Babylonians did not lack in their analytic abilities.

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