An Institution of General History (1680) William Howell - by William Howell (sometimes spelled Howel)

By William Howell (sometimes spelled Howel)

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But then 4 Arthur Gittleman, History of Mathematics, Merrill Publishing Company, 1975. , 45 b c = . e. the area of the square ACHI equals that of the rectangle ADJK as before. Similarly, CBF G equals KJEB and a2 + b2 = nc + (c − n)c = c2 . This proof is a bit more memorable than Euclid’s and today, with our willingness to multiply and divide arbitrary real numbers, the proof is valid as soon as we remove the step yielding integers. We just set a = BC, b = AC, c = AB, d = AK, c − d = KB, cite the similarity of the triangles, and conclude c b = , d b a c = c−d a whence b2 = cd, a2 = c(c − d), and the areas of the two rectangular pieces of the large square are those of the corresponding small squares.

Naber cites the Dutch writer Multatuli (real name: Douwes Dekker) for what the latter declared to be the simplest possible proof and which I reproduce as Figure 3, below. ✚❙ ✚ ✚ ✓❩*❩ ❙ ✚ ✓ ❙ ❩ ✚ § ✓ ❩ ❙ ❙ ❙ † ❩ ✓ ✚ ❩ ❙ ✚ * ✓ ✚ ‡ ✓ ❙ C ✚ ✓ A✓ ❙ ❩ ❙✚ ✓ ❩ § ❩ ✓ † ❩ ❩ ✓ ‡ ❩✓ Figure 3 B Figure 2 Naber also notes that this proof was known to the Hindu mathematicians. The most famous Indian proof, however, has got to be that of Bhaskara, which I reproduce in full in Figure 4. ❙ ❙ ✚ ✚ ✚ ❙✚❙ ❙✚❙ ✚ ❙ ✚ ✚ Figure 4 3 Behold!

Constance Reid has written a number of popular biographies of mathematicians. She is not a mathematician herself, but had access to mathematicians, in particular, Julia Robinson and her husband Raphæl. N. David, Games, Gods and Gambling; A History of Probability and Statistical Ideas, Dover. Jacob Klein, Greek Mathematical Thought and the Origin of Algebra, Dover. 36 2 Annotated Bibliography I haven’t seen these books which are listed on Dover’s website. Petr Beckmann, A History of π, Golem Press, 1971.

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