caa a22 4500 606223967 CHVBK 20210128101141.0 cr unu---uuuuu 210128e20151101xx s 000 0 eng 10.1007/s00407-015-0156-x doi (NATIONALLICENCE)springer-10.1007/s00407-015-0156-x Revisiting Al-Samaw'al's table of binomial coefficients: Greek inspiration, diagrammatic reasoning and mathematical induction [Elektronische Daten] [Sanaa Bajri, John Hannah, Clemency Montelle] In a famous passage from his al-Bāhir, al-Samaw'al proves the identity which we would now write as $$(ab)^n=a^n b^n$$ ( a b ) n = a n b n for the cases $$n=3,4$$ n = 3 , 4 . He also calculates the equivalent of the expansion of the binomial $$(a+b)^n$$ ( a + b ) n for the same values of n and describes the construction of what we now call the Pascal Triangle, showing the table up to its 12th row. We give a literal translation of the whole passage, along with paraphrases in more modern or symbolic form. We discuss the influence of the Euclidean tradition on al-Samaw'al's presentation, and the role that diagrams might have played in helping al-Samaw'al's readers follow his arguments, including his supposed use of an early form of mathematical induction. Springer-Verlag Berlin Heidelberg, 2015 Bajri Sanaa Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand aut Hannah John Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand aut Montelle Clemency Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand aut Archive for History of Exact Sciences Springer Berlin Heidelberg 69/6(2015-11-01), 537-576 0003-9519 69:6<537 2015 69 407 https://doi.org/10.1007/s00407-015-0156-x text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00407-015-0156-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bajri Sanaa Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand aut NATIONALLICENCE 700 1- Hannah John Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand aut NATIONALLICENCE 700 1- Montelle Clemency Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand aut NATIONALLICENCE 773 0- Archive for History of Exact Sciences Springer Berlin Heidelberg 69/6(2015-11-01), 537-576 0003-9519 69:6<537 2015 69 407