## Friday, November 22, 2013

### Galera Division

Galera division is a technique used in the late Medieval and early Renaissance periods in southern Europe. It was a technique of the abbaco schools, popular schools of reckoning that developed to serve the needs of merchants and navigators. For more about abbaco schools I would recommend two excellent articles in Loci: Randy Schwartz's article on the Pamiers Manuscript, and William Branson's article on solving a cubic equation the way Girolamo Cardano would have done it in the 1500s.

For a bit of background, you must understand that the people doing arithmetic at this time (the 1300s and 1400s) were doing so without the equals sign (=), the plus sign (+), the minus sign (-),  the multiplication sign (x), the division sign ($$\div$$) or the decimal point (.). These were all yet to be invented! Problems were expressed in words. Numbers that were not whole numbers were always expressed in fractions.  (It's somehow comforting to know that they already did use the fraction notation we  use today, a numerator and denominator separated by a bar).  Money and weights involved complex non-decimal fractions, like 24 grossi to a ducat and 32 pizoli to the grosso (Venetian coinage), or 12 ounces to the pound and 6 sazi to the ounce.

For example, here's a problem from the Treviso Arithmetic, a how-to manual of arithmetic written in 1478 in Venetian (rather than Latin -- indicating to us that it was intended for a wide audience outside of universities):

Se lire.100.e $$\frac{1}{4}$$ de seda valisseno ducati 42 g 2.e$$\frac{1}{5}$$ che valerano lire 9816 onze.3.e $$\frac{1}{6}$$[F. 36, v.]

If 100 and $$\frac{1}{4}$$ pounds of silk are valued at 42 ducats, 7 and $$\frac{1}{5}$$ grossi, what will 9816 pounds, 3 and $$\frac{1}{6}$$ ounces be valued at?

But reckoners had tools to navigate these complicated numbers and get the right results. They were not afraid of big numbers, and indeed they regularly did calculations that would choke a modern hand calculator. Here's the author of the Treviso Arithmetic dividing 12,030 into 14,350,278,384 in the process of showing us how to solve the problem above.

 Treviso Arithmetic, 1478, F. 38, r.

It's called galera division because the mass of cancelled digits that proliferates above and below the dividend and divisor in the completed problem resembles a galley (galera) sailing directly at you: narrowing to the waterline below, narrowing to the top sails above. In the Treviso Arithmetic it is called batello or "boat" division.

As Frank Swetz points out in his book Capitalism and Arithmetic, galera division had the advantage, in an age when paper was expensive, of filling a smaller, more compact space on paper than our modern long division technique would.

 1607 divided by 42: long division (left) and galera division (right)

Galera is intriguing because it's a lost technique used by people centuries ago, but it's also worth noting that it involves acts of multiplication and subtraction that are frequently smaller than those you'll do in long division. In long division, one multiplies the new quotient digit by the whole divisor, and then subtracts the result, however big it is, from the relevant part of the dividend. In galera division, as I'll show below, one multiplies the new quotient digit by each digit of the divisor in turn, subtracting these smaller results individually from the relevant piece of the dividend above.

Galera division can get away with this because its process is one of constantly adjusting the dividend, crossing out digits and replacing them with others. So, to know your way around a galera division problem, notice that at any given time the current dividend can be assembled from the uncrossed-out digits at the tops of the columns of figures. Similarly, the divisor is crossed out and re-written at the bottom of the columns. Learn to find these with your eye, and you'll be looking at the problem the way a galera divider did.

 At this stage in the problem the dividend is 347.

So let's get started. We'll divide 42 into 1607.

### Setup

Write the divisor under the dividend, much like a fraction, and place a vertical bar to the right of them.  Align them so that the first digits of the divisor go into the first digits of the dividend.

Let's say however that the problem was 38 into 699. We would observe that 38 is less than 69 – it can go into it –  so it would be lined up like this:

When the first digits of the dividend and divisor are the same, you have to look at the next digit. So, 18 into 1274 would be aligned like this:

Now, back to dividing 42 into 1607.

### Step 1

Consider how many times the divisor (42) will go into just those digits of the dividend that are immediately above and to the left of it (160).  The answer is 3 times, so write a 3 to the right of the vertical line. This is the beginning of the quotient.

### Step 2

Multiply this new quotient digit by the leftmost, that is, most significant, digit in the divisor, and hold that number in your head. (3 times 4 makes 12. Hold 12 in your head.)   Strike out the portion of the dividend that is above and to the left of this leftmost divisor digit (16), and over it write the difference between it and the number in your head (12 from 16 leaves 4). Also strike out leftmost digit of the divisor (4), to indicate we've “used” it.

(Notice that the dividend has been changed now 407, but not all the digits are on the same line.)

### Step 3

Now multiply the new quotient digit by the next most significant digit of the divisor, and hold that number in your head. (3 times 2 makes 6. Hold 6 in your head.)  Strike out the portion of the dividend that is above and to the left of this digit of the divisor (40), and over it write the difference between it and the number in your head (6 from 40 leaves 34). Also strike out this digit of the divisor (2), to indicate we've “used” it. (The dividend is now 347, with all three digits on different lines.)

### Step 4

Write in a fresh copy of the divisor, but shifted one column right. (The 4 will go under the original 2, and the 2 will go to the left of the original 2.)

Now we're ready to guess the next digit of the quotient, and repeat those four steps again.

### Step 1

Consider how many times the divisor (42) will go into just those digits of the dividend that are immediately above and to the left of it (347).  The answer is 8 times, so write an 8 to the right of the vertical line.

### Step 2

Multiply this new quotient digit by the leftmost, that is, most significant, digit in the divisor, and hold that number in your head. (8 times 4 makes 32. Hold 32 in your head.)  Strike out the portion of the dividend that is above and to the left of this leftmost digit of the divisor (34), and over it write the difference between it and the number in your head (32 from 34 leaves 2). Also strike out leftmost digit of the divisor (4), to indicate we've “used” it. (Notice that the dividend has been changed now to 27, but not all the digits are on the same line.)

### Step 3

Now multiply the new quotient digit by the next most significant digit of the divisor, and hold that number in your head. (8 times 2 makes 16. Hold 16 in your head.)  Strike out the portion of the dividend that is above and to the left of this digit of the divisor (27), and over it write the difference between it and the number in your head (16 from 27 makes 11). Also strike out this digit of the divisor (2), to indicate we've “used” it. (The dividend is now 11, with digits on different lines.)

We can't move the divisor any further right, so we're done.  We don't have to write the divisor again. The answer is 38 with a remainder of 11. Done!

## 10's-complement Subtraction

In doing that galera division we performed a lot of subtraction. It's easy to assume that people in the 1400s did subtraction as we do today, but they did not!  In the Treviso Arithmetic, subtraction did not employ the "borrowing" method we use, but used a technique we can call 10's-complement subtraction.

Today, when faced with subtracting a larger digit from a smaller digit, we "borrow." Let's say we are subtracting 16 from 41. We begin with the one's place, and faced with taking 6 from 1 we "borrow" from the ten's place to make 11, decrement the 4 to a 3, take 6 from 11 to get 5, and then move on.

In 10's-complement subtraction you also begin with the one's place, but faced with taking 6 from 1, you pause and note the 10's complement of 6, that is, the number you would add to 6 to get 10. It's 4.

This 4 you add to the number above that you were trying to subtract 6 from. Four plus 1 is 5, so you write a 5 below the line as your answer digit for this column. (Note that this means you don't have to have memorized the subtraction tables for numbers larger than 9!)

Finally, instead of decrementing the next digit of the minuend (upper number), we add one to the next digit of the subtrahend (the lower number). Same result.