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.

A Geometry Puzzle: Alternating Hexagons and Squares In a Ring

Background

This intriguing pattern is an alternating sequence of six hexagons and six squares. Both basic shapes have the same edge length, and they pack perfectly around a centre.

Furthermore, this fairly complex construction can be done with only a ruler and straightedge. Typically, you start with a circle, within which you construct what will be the dodecagonal centre of the figure above. The alternating hexagons and squares then sit on the sides of this dodecagon.

Because we will have to create other hexagons along the way, I'll call the hexagons in the final pattern around the outside ring hexagons.

The construction goes like this. (I will assume you know how to construct a hexagon within a circle, and how to bisect a line segment.)

1. Draw a circle.

2. Construct a hexagon within it.

3. Bisect one of the sides of the hexagon and draw a ray from the circle's centre through it. Construct another hexagon beginning with a vertex that falls where this ray intersects the circle.

4. Connect the successive vertices of the two hexagons to make a dodecagon, a 12-sided regular polygon.

5. Using the side length of the dodecagon as radius, draw circles around each vertex of the dodecagon.

6. Using every other intersection of these circles as a centre, draw six more circles of the same radius.

7. Construct hexagons within these last six circles. One side of each will coincide with a side of the dodecagon.

8. Connect hexagons to form squares.

9. The final figure without the construction lines. Rotating it 15° counterclockwise will make it look like the one at the very top of the post.

The Puzzle

OK, now for the puzzle. If we connect the centres of the six hexagons in the ring, we get another, larger hexagon.

This hexagon, which I'll call the master hexagon, can be used as the repeating frame for tiling a larger area with the pattern.

But if you were to do this, you would draw the pattern of master hexagons first, and would then construct the ring pattern based on it.

So here's the puzzle: how do you construct the ring pattern of hexagons and squares, given only the master hexagon?

Solution

The basic problem is to locate the dodecagon that forms the inside of the ring. Once we have that, we can construct  the ring, as above. But how do we get from the master hexagon to the dodecagon? 

10. Begin with the master hexagon.

11. Locate its centre by connecting vertices, and construct the circumscribing circle.

12. Bisect one side of the hexagon, and construct a second hexagon, much as you did in step 3 above, starting from the point where this bisector meets the circle.

13. Connect vertices of one hexagon to make a six pointed star.

14. Connect the vertices of the other hexagon in a similar fashion.

15. Connect the intersections of those two six-pointed stars, to make a dodecagon. This is the dodecagon that will form the inside of the ring.

16. Using steps 5, 6 and 7 above, construct ring hexagons from the sides of the dodecagon.

17. And, as in step 8 above, connect the ring hexagons to form squares.

It's interesting to compare the construction lines one uses when beginning with a circle to those drawn when beginning with the master hexagon.

18. "Forward" construction lines (that is, those beginning from the circle that circumscribes the dodecagon) are black. "Reverse" construction lines (beginning with the master hexagon) are blue.

Commentary

Why does using this method to construct the dodecagon within the master hexagon work?

Well, we know the master hexagon has a concentric dodecagon within it somewhere. But which dodecagon?

Because each dodecagon has a different edge length, each implies a different size of hexagons arrayed around it. We want the dodecagon where the hexagon's centre will fall at a vertex of the master hexagon (the red one, below, in this case).

The "right" dodecagon will have vertices that are 60° apart when viewed from a vertex of the master hexagon. Necessarily then, these dodecagon vertices will fall somewhere on the sides of equilateral triangles drawn within the master hexagon.

Drawing the other equilateral triangle within this hexagon gives us a general idea of where these dodecagon vertices will fall, but nothing precise. As well, this pattern so far only has 6-fold rotational symmetry.

If we add another master hexagon, rotated 15°, and its inner triangles, we get a pattern with the necessary 12-fold rotational symmetry.

The set of four equilateral triangles has 3 sets of common intersections, all of which will make dodecagons (red, orange and yellow, below). But only the outermost set (red) creates dodecagon sides that subtend a 60° angle when viewed from the vertices of the master hexagon.

A Dado from the Alhambra's Hall of Justice, via Owen Jones and Eric Broug

This art project, on the doors of my garden shed, began with a pattern in Owen Jones' 1856 Grammar of Ornament.

Jones' Plate XLIII, No. 11, as shown in the online version of the Grammar Of Ornament (rotated 90°)

Jones notes that this pattern is a dado from the Hall of Justice in the Alhambra, Granada, Spain. I had to look this up: a dado is the lower half of a wall, below a dividing rib called the dado rail. This is a horizontal feature sticking out of the middle of a wall, something your grandmother might have called the "chair rail."

My project was to paint this pattern on a wall. I wasn't going to paint it on as a dado, however: I would cover the whole the wall of my garden shed with it, using the rotated version shown above.

Although we have wonderful computer software now that allows us to make fabulous patterns and to experiment with colouring them in different ways, and we have copiers and scanners to facilitate this, and indeed we have a cheap and bountiful supply of paper, markers, rulers and compasses, none of this equipment allows us to get a design anywhere other than on paper. What are the challenges involved with getting it onto a wall?

Construction method

If you look at this dado pattern, you will see that it is centred on 12-pointed stars, around which radiate 12 pairs of parallel lines. I had Eric Broug's book Islamic Geometric Patterns, so I looked through it to see if he reproduced something like this. As luck would have it, I found the same pattern, without the colouring.

The same pattern in Broug, Notice the hexagonal symmetry.

Broug gives an elegant method of construction for this pattern using only a compass and straightedge. In essence, one constructs hexagons within a circle, and then connects various intersection points. Once the host of construction lines are drawn, a few segments (only a few!) are selected to make the pattern.

Illustration from Broug showing the construction lines (faint) and the pattern (red). The heavier black line indicates the bounds of the basic hexagonal unit

Paper Test

Doing these things on paper is relatively easy. Following Broug's instructions, I made the construction lines and then inked in the pattern on a small sheet of paper. The radius of the master circle was 8.5cm.

Broug's design realized on paper with compass and straightedge. Construction lines are in pencil, with inked-in segments showing the actual pattern.

By tracing, I transferred this hexagonal cell to another sheet, and then traced in a couple of other copies at the edges. I experimented with colouring it.

Repeated pattern traced onto other paper and coloured with felt tip pen.

One of the advantages of doing this small-scale proof before going to the wall is that you become familiar with the various shapes that made up the pattern. There are the blue stars (A), and connecting them are blue or grey darts (B), pointed shapes with convex backs. Between the lines of darts we have petals (C), pointed shapes with concave backs, coloured in trios of orange or green. All other space, including the hexagons (D), the rays (E), and the winged darts (F), is yellow.

The basic repeating unit in both Jones' pattern and Broug's pattern is the hexagonal cell. However, in Jones' pattern the hexagons are packed in horizontal rows, whereas in Broug's they are packing in vertical rows.

Repetition in Jones' version uses vertical packing (black hexagons), whereas Broug's version uses horizontal packing (red hexagons).

So, to build Jones' pattern using Broug's construction method I'd have to rotate it 30°.

Paper master

My basic strategy here was to work on the wall using a stencil. The stencil would be made from a paper version that I would draw by hand.

In this full-size pattern, the master circle was 40 cm in radius. The hexagon within it was 35 cm from centre to mid-side, a measurement that would become important later.

I drew in horizontal and vertical axis lines to remind me how this master hexagon would need to be placed on the wall to create the vertical packing in Jones' pattern.  

Large pattern on paper: vertical axis in green, horizontal axis in blue.

To channel or not to channel

Before making a stencil, I had to make a decision about the white channels.

Are they there between the blocks of colour in Jones's reproduction? They are definitely there, although whoever drew the illustration for the Grammar of Ornament wasn't consistent about the width of these channels. Broug suggests that in Islamic design you generally do want channels separating polygons (as opposed to polygons touching one another), and that you strive to have channels of consistent width.

I chose a channel width of 16 mm, or 8 mm on either side of each pattern line. The 8 mm was not chosen because of an eye for design, or as the result of a mathematical calculation: it was simply that I had a clear plastic ruler with a second line 8 mm from the edge.

Stencil

I transferred the vertices of the pattern lines from paper to foam core using a pin; then I drew between pin holes to reproduce the pattern on the foam core.  I marked channels out around the centrelines with the clear plastic ruler, and then cut out the stencil.

This stencil covered only part of the design, one quarter of the figure. Although in many ways it would be ideal to cut a stencil of the entire hexagonal cell, it would have taken a long time and required a bigger piece of foam core than I had. Practical considerations!

By clipping the channels at the outer hexagon and the horizontal and vertical axes running through the figure, I made a stencil that could abut itself and repeat in all directions.  It included three full rays, a dart, a petal and three points of the central star. The hexagon and the winged dart were there as partial edge figures.  I could flip it over and work with either side, and four repeats should cover an entire figure.

The one-quarter pattern stencil. Yellow lines indicate where stencil was trimmed to both the outer hexagon and the quartering lines.

I also cut a flexible stencil out of cardstock. This is useful where walls meet ceilings and floors. You can press this kind of stencil into corners.

Notations on the stencil

A stencil is a remarkable tool, and it can carry all sorts of extra information. Before I got to drawing on the wall, I added two kinds of notations to my stencil: matching zones, and alignment marks.

Matching zones were places where the stencil would abut against another repeat of itself. When drawing along the edge of the stencil, you can skip these zones.  I flagged them with felt tip pen on both sides of the stencil.

Alignment marks showed where I could expect the stencil to align with horizontal and vertical lines drawn through the centre of central star.

A's indicates matching zones; line B-B connects alignment marks

Picking Colours

At this point I was curious about the colours in the original pattern, not in Jones but in the Alhambra. I was suspicious about Jones' reproduction because all the patterns on that plate had the same four or five colours. As well, the image from his book online was somewhat different in colour from the Dorling Kindersley reprint of the Grammar of Ornament that I owned. I went searching on the web for pictures of the Hall of Justice at the Alhambra, pictures that might show the dado.

It turns out this room is known by several different names: the Hall of Justice, the Hall of Kings, Sala de Justicia, and Sala de Reyes. Detailed large images of the lower walls are not very common, because people are mostly photographing the ceiling. However on Wikimedia Commons I found a nice large high-res image showing the dado, taken by José Luiz Ribeiro in 2013 and released under a Creative Commons attrubution share-alike license.

Hall of Kings, Alhambra. © José Luiz Bernardes Ribeiro

Let's zoom in on the dado on the left side of the photograph and compare it with the image in Jones.

Photograph of the dado from the Alhambra (left) and Plate XLIII #11 from the present Dorling Kindersley edition of Jones's Grammar of Ornament (right)

Although the pattern is recognizably the one that Jones reproduced, the differences in colour are appalling. Where the original has black stars and darts, Jones's were blue. Where light blue darts ran across the pattern, Jones used a warm grey. Dark, jade-green petals in the original became a leaf green in Jones. The pale orange petals in the Alhambra were changed into rich orange petals. A white background in the original became gold.

And importantly, here are no channels in the original. There are simply tiles placed next to each other. (Although it's not clear what the material is: it could be semi-precious stones or it could be tiles. I haven't been there.)

So my wall painting was not going to be a reproduction of this dado in any sense. It would be a pattern inspired (distantly) by a dado in the Alhambra, recoloured and presented to the English-speaking world by Owen Jones in the nineteenth century, and then further altered by that idea that I got from Eric Broug's book, of having channels between the shapes.

Final result

Using the stencil I sketched the outlines of all shapes on the wall. The flexible stencil was indeed handy where the wall met the ceiling and floor.

I painted it using leftover house paint we had, but trying to approximate the colours in Jones's plate.

It is in fact even difficult to see where the doors are now.