An Episode in the History of Multiplication

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.

The above image is a woodcut diagram "explaining" how to do multiplication.

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. 

 Notes:

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:

81 (1)

48 (4) 49 (3) 48 (3)

42(6) 42(5)

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. 

Comments

[Ray Girvan]

This wasn't "in a flash" but I see how it works. When those multiplications are assigned their proper column values, they simply sum to the answer. I've put a scanned image in a backdated blog post: http://jsbookreader.blogspot.com/2005/07/taglientes-multiplication.html

Maybe the woodcut creator didn't realise the column positions were important?

[Ray Girvan]

Update: fully analysed.

http://jsbookreader.blogspot.com/2011/11/taglientes-multiplication-by-columns.html

Thanks for a very interesting problem!

[John F. Ptak]

RAY! Thanks so much for figuring this out. I tried adding up the columns in different arrangements but didn't see this. It actually comes "close" by adding the numbers in their place order: the eight-places added, then the seventh, and so on. Nope. I appreciate this and added your comments and links to the post.