JF Ptak Science Books Post 1657
Many thanks! to Ray Girvan and Thony Christie for solving this multiplication problem!
Ray Girvan (JS Blog--Journal of a Southern Bookreader) http://jsbookreader.blogspot.com/2011/11/taglientes-multiplication-by-columns.html
Thony Christie (Renaissance Mathematicus) http://thonyc.wordpress.com/2011/11/10/1202/
Giovanni Tagliente was many things, though perhaps he wasn't mathematically inclined, not really. He was however a very capable putting together books and addressing some elementary needs of the working classes that reached out into the mass of the great unwashed offering instructionals on how to do the basics of--in this case-- applied mathematics, as with this book, Libro de abacho...1, this edition printed in Venice in 1564. This was a small, pocket-sized book2, a references on how to do the stuff of daily (and not so daily) calculations, and in principle was a concise, Humanist guideline for getting through the daily bits of the calculating and mercantile life.
To the 21st century eye however the actual explanations of how to conduct Tagliente's approach to actually "doing" the math might not be so evident. They seems to be inspired acts in helping the reader understand the process, say, of multiplication, though it does appear a little mysterious, and explanations for the actual process are--as the author(s) says--self-evident in the woodcut, and are not elaborated in the text.
I've been unsuccessful thus far in determining how the product to the problem of 9876 x 6789 (67.048,164) was reached via adding the individual results of the factors' multiplications, though seeing the individually parts of the calculation is apparent3 (explained below).
I thought that this would be easier, because it was after all easier once upon a time, at least to people not me.
I think that I get a D+ on this one. All I really wanted to say is that Tagliente's diagrams are pretty.
1. The first edition, printed in 1524, is titled so, in full: Title. ' Libro // dabaco che in//segna a fare // ogni ragione mercadantile, & // pertegare le terre
co 1'arte di //la Geometria, e altre no//bilifsime raginoe ftra-//ordinarie co la Ta-//riffa come refpon//deno
li pefi & // Monede de molte terre del mon-//do con la inclita citta di Vene-//gia. Elquel Libro fe chiama //Thefauro vniuerfale.' (F. I, r.)
Colophon. ' Stampato in Milano per lo. Antonio da Borgho.// Nell' anno del. M. D. XLI.' (F. 80, v.)
2. "The Tagliente math books, like others of Tagliente's textbooks, were precisely calibrated to teach discrete skills at a basic level and to exercise students with a limited number of problems to solve. The authors did not pretend to cover an entire field, nor were their books designed for a specific school course. The treatments are highly incomplete even for the limited fields they addressed. It may be that Tagliente had in mind to simplify these skills to the point where how-to manuals could substitute for professional tuition. More likely still, he intended them as pocket-sized books of reference that would remind adults what they had learned in school. Whatever his intent, his pretty little booklets found a ready market."--from the thoughtful and insightful Humanism for Sale blogsite, here.
3. If the factors are multiplied in the following way, the results of the woodcut image above are obtained, as follows.
First, we'll number the 15 steps of the process:
48 (4) 49 (3) 48 (3)
54 (10) 56 (9) 56 (8) 54 (7)
72 (12) 72 (11)
63(15) 64 (14) 63 (13)
And so the key for step # ____ )"lower" referring to the lower factor's digits, upper to the upper):
Step 1: 9 (lower) x9
Step 2: 8 (lower) x6
Step 3: 7 (lower) x 7
Step 4: 6 (lower) x8
and so on, going diagonally across the upper and lower factors, through the 15 steps.
I don't know how they used to get to the product. Perhaps someone out there sees this in a flash--please let me know.