Download A History of Algebra: From al-Khwārizmī to Emmy Noether by Prof. Dr. Bartel Leenert van der Waerden (auth.) PDF

By Prof. Dr. Bartel Leenert van der Waerden (auth.)

ISBN-10: 3642516017

ISBN-13: 9783642516016

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Extra resources for A History of Algebra: From al-Khwārizmī to Emmy Noether

Sample text

Chapter 14 is devoted to calculations with square roots and cube roots. Leonardo begins by presenting some theorems from Euclid's Book II in numerical form, omitting the proofs, "because they are all in Euclid". For From Leonardo da Pisa to Luca Pacioli 37 square roots he has the well-known approximation For the cube root Leonardo presents a first approximation (4) 3,t:3-;-: +r~a+( Va r 3 3 a+1) -a and next a second approximation (5) with rl =a-al. 608) the first approximation (4) was al ready known to al-Nasawi.

Five of these are preserved: 1. über abbaci (1202, revised 1228), 2. Practica geometriae (1220), 3. a book entitled "Flos" (1225), 4. a letter to the philosopher Theodorus, who lived in Sicily at the court of the Hohenstaufen emperor Frederick II, 5. Liber quadratorum (1225). 34 Chapter 2. Algebra in Italy A treatise on Book X of Euclid's "Elements", contammg a numerical treatment of the irrationalities which Euclid had demonstrated by lines and areas, is unfortunately lost. Leonardo's importance was recognized at the court of Frederick H.

In the "casus irreducibilis" one has to extract a square root from a negative number. Such square roots, which we now call "imaginary", oceur also in another chapter of the "Ars Magna". In Chapter 37, Cardano poses the problem: To divide 10 in two parts, the produet of whieh is 40. He writes: vi - It is dear that this case is impossible. Nevertheless, we will work thus: We divide 10 into two equal parts, making each 5. These we square, making 25. Subtract 40, if you will, from the 25 thus produced, as I showed you in the chapter on operations in the sixth book, leaving a remainder of -15, the square root of wh ich added to or subtracted from 5 gives parts the product of wh ich is 40.