image: Tomash Library |
John mentions the 16th century instructional Libro dabaco, a pocket-sized book on practical mathematics for merchants, co-written by Girolamo and Giovanni Tagliente (since only one was the nominal author, I'll use the singular Tagliente for convenience). The book is described in more detail in Paul F Gehl's online book Humanism For Sale ("Making and Marketing Schoolbooks in Italy, 1450-1650") in the section 6.03 Commercial skills.
The problem lies in the book's multiplication "per colonna" ("by columns") system, of which there are some nice images here as part of the The Erwin Tomash Library. Both John and Paul comment that Tagliente doesn't explain the algorithm, because his text says the diagrams are self-explanatory. They're not!
image: Tomash Library |
This example evaluates 9876 x 6789 = 67048164. As John says, you can see where the digits in the individual rows come from: Tagliente multiplies pairs of digits in the two multipliers. But how do those rows produce the final product?
It turns out to be clear when the multiplications are assigned their correct column values: the result is just their sum.
This leaves the question of why the digits are multiplied in that order to produce the rows.
The full method turns out to be very elegant. Modern long multiplication involves carrying at each stage of producing the partial sums. For example 1234 x 5678 requires four multiplications (each with four sub-multiplications), all with carries when done by hand:
1234 x 8 = 9872
1234 x 70 = 86380
1234 x 600 = 740400
1234 x 5000 = 6170000
then you sum the results to get 7006652.
As with ordinary long multiplication, Tagliente's method involves multiplying all permutations of digit pairs taken from the two multipliers - but it does it in such an order that there are no carries within each row. Each single multiplication produces a result in the range 00-99 (i.e. occupying two decimal places) and each pass of the algorithm produces a row of such results, each shifted from the previous by two decimal places, so that they can just be written adjacently.
Applying this to 1234 x 5678, the order of multiplication could go:
1234 x
5678
Row 1: 1x5, 2x6 3x7, 4x8 (vertically aligned)
Row 2: 1x6, 2x7, 3x8 (offset 1 right)
Row 3: 1x7, 2x8 (offset 2 right)
Row 4: 1x8 (offset 3 right)
Row 5: 2x5, 3x6, 4x7 (offset 1 left)
Row 6: 3x5, 4x6 (offset 2 left)
Row 7: 4x5 (offset 3 left)
The results are:
(05)(12)(21)(32)
(06)(14)(24)
(07)(16)
(08)
(10)(18)(28)
(15)(24)
(20)
It looks like this when laid out in columns to show the place values:
I haven't tried it with multiplications between figures with different number of digits, but as long as the place values are kept correcly, the permutation routine would work fine. The order of permutation doesn't really matter as long as all the permutations are covered, and Tagliente's own examples show several different routines (see Addendum 3).
I'm sure Tagliente arranged his calculations according to this column layout, because the final sum doesn't work otherwise. But I can only guess that the woodcut creator for Libro dabaco was more interested in artistic design than mathematical correctness, and broke the layout, obscuring the method. This is further suggested by the errors in the host-and-chalice example for 927 x 789, which gets the correct answer of 731403 despite a number of mistakes in the sub-results (such as having 2x7=16).
Multiplication "per coppa" - image: Tomash Library |
The procedure in the above example is meant to be structured as:
Row 1: 9x8, 2x9
Row 2: 9x7, 2x8, 7x9
Row 3: 2x7, 7x8
Row 4: 7x7
Row 5: 9x9
Results:
(72)(18)
(63)(16)(63)
(14)(56)
(49)
(81)
Corrected for place values:
72180 +
631663
14560
4900
8100
------------
731403
As I said, the order of permutations seems flexible. This third example goes through them in yet another order, and, unlike the other two, is rather more faithful to the column structure.
image: Tomash Library |
Addendum:I just found a reference to the above methods in the 1908 bibliographic catalogue Rara Arithmetica (Internet Archive ID raraarithmeticac00smituoft). Again without explanation of the algorithm, its authors comment on the 'diamond' and 'triangle' plates above, in the 1541 Tagliente:
For two curious forms of multiplication see Fig. 64. Such arrangements of the work in multiplication were quite common, particularly in the early Spanish and Italian arithmetics of the first half of the sixteenth century. That they should have found place in a popular mercantile treatise is, however, rather surprising.
- page 116, Rara Arithmetica; a catalogue of the arithmetics written before the year MDCI, with description of those in the library of George Arthur Plimpton, of New York, Smith, David Eugene, and Plimpton, George Arthur, pub. Boston Ginn, 1908.
Addendum 2 (upgraded from comments): Leon has commented that:
This looks a lot like Lattice Multiplication.
Thanks: yes indeed. I didn't spot this, but it's algorithmically very similar. Lattice multiplication handles the permutations of the multiplier digits in the rather more foolproof way of laying them out along the axes of a rectangular grid. It creates exactly the same array of two-digit numbers as Tagliente's method, but at an angle: you sum them down a diagonal (from top right to bottom left) rather than vertically. For comparison:
Image from interactive app here |
The Tagliente method could be described as a hybrid between the lattice method and regular long multiplication.
Addendum 3: And Thony Christie, at the interesting-looking blog The Renaissance Mathematicus, also solved it, pointing out that Taliente's method is, in effect, a sloppily-structured version of an algorithm called the Diamond (see Multiplying the Renaissance way). This led me to a source: Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (ed. Ivor Grattan-Guinness, 2003) of which pages 203-206, a section by the abacco arithmetic expert Warren Van Egmond, have clear descriptions of such algorithms. It seems they derived from methods originally for the abacus, but adapted to the needs of pen-and-paper calculation. Forms included multiplication in croce (in the form of a cross) which refers to the set patterns for multiplying the digits, as in the Tagliente method; and several forms laid out per campana (a bell shape), per coppa (in the shape of a cup), or as a diamond or circle (see diagram, page 205). Italian mercantile calculation of this period was a kind of 'Burgess Shale' of long multiplication, when diverse forms initially flourished and eventually resolved into the modern form. WVE notes that unlike many mathematical operations, these were not imported from Hindu or Arab mathematics, but "developed independently by the abaccists themselves, with much trial and error".
Victor J. Katz's 2000 Using History to Teach Mathematics: an International Perspective (page 15) has an example of the per coppa calculation.
- Ray (and thanks to John Ptak for raising the topic).
This looks a lot like Lattice Multiplication. There are videos on YouTube. I think it's taught in Japan?
ReplyDeleteThe Encyclopedia is edited by Ivor GG but the article you quote is written by Warren Van Egmond who is the expert for abbaco arithmetic.
ReplyDelete