Making a Promptuary
by Jim Hansen
John Napier's Promptuary was described in an appendix in his 1617 book Rabdologia, first published in Latin the year of his death. The term rabdology was coined by Napier as meaning "Calculation with Rods" or "Rod reckoning." Promptuary comes from the Latin promptuarium, "a place where things are stored ready for use." This was to be his final contribution to mathematics.
For roughly 21,000 categorized links to Napier and his works (and presented in a form more civilized than Google's), check out this site. Winzy also seems to give better math search results that a raw Google inquiry.
This project is based on information in the first English translation of Rabdology. It was published in 1990 by MIT as Book 15 of the Charles Babbage Institute's Reprint Series for the History of Computing. Easy reading, information on this slim volume can be found at MIT Press, publishers, here.
My interest in the Promptuary follows that of Napier's Bones, which I have used in classroom activities for some time. It was originally brought to my attention by Ian Bruce, a prolific Latin translator of historical mathematics literature.
Both devices operate on the same principle, essentially that of copying multiplication tables on strips of something that can be positioned next to each other. In Napier's time, wood and often ivory was used, hence the term "bones." When the strips are properly aligned, only simple addition is required to solve any multiplication problem regardless of size.
Of the two, the bones are clearly more useful as they can multiply, divide and find square and cube roots. The Promptuary can only multiply. Division is possible but only after first finding the reciprocal of one of the multipliers.
Napier's main mathematical effort was aimed at reducing arithmetic calculations to a simple process, and in this he succeeded brilliantly. It must be remembered that at the time of his death in 1617, few could multiply past 5 times 5, and the main writing instrument was the quill or feather pen. (The pencil had to wait until roughly 1830 before it became a practical tool.) From the scientific and mathematical perspective, it was a time of phenomenal growth as science based on Aristotelian logic was being discarded in favor of observation and experiment.
As astronomy grew into a proper science, so did its demands for calculation, this at a time when mathematics was barely able to support it. Without doubt, John Napier's contributions to mathematics calculation almost singlehandedly enabled the scientific revolution to proceed as it did with increasing speed. In a sense, Napier, almost alone, developed a mathematical infrastructure that enabled others to proceed in their own direction, and he was not by any means a mathematical lightweight.
Napier first described the decimal point, enabling calculations to be made without the use of complex fractions. He discovered what eventually would be called "Pascal's Triangle" and placed it in common use long before Pascal was even born. (If you can read Latin, see Napier's De Arte Logistica, which has not as yet been translated to English.) His concept of "local arithmetic," described in Rabdology, shows a radical departure in arithmetic calculation methodology. Here he converts numbers to what we now call binary form, performs his computation, then converts the binary solution back to decimal notation using a chess board!
And he invented - and was calculating to 18 decimal places - logarithms at a time when his contemporaries were still writing books about arithmetic. There were many more inventions, but Napier's place in history is quite secure without them. Indeed, the 1911 Encyclopedia Brittannica, which has the most inclusive Napierian biography popularly available, puts it this way: "...in the history of British science (Mirifici Logarithmorum Canonis descriptio ..., Napier's publication describing logarithms) can be placed as second only to Newton's Principia."
Napier's Promptuary
The original description of the Promptuary called for 200 wooden "slips" to be made. Half of them were to be inscribed with a coded set of the multiplication tables, and the other set to have holes drilled in them that would reveal only those products involved with the calculation. The main part of the Promptuary is actually just that, a place to store the slips.
It seems clear that Napier developed the Promptuary device as a simplification of his wildly successful "Computing Rods"(more popularly known as "bones") which went on to be commonly used for more than 200 years. The problem was that the bones, while very flexible, reveal every possible result at the same time, and some small skill - or instruction - is needed to make good use of them. "The High Speed Promptuary for Multiplication" as Napier calls it, avoids that difficulty, but performs only multiplication.
I made a modern version of the Promptuary by printing the "table slips" on strips of card stock, and the "mask slips" on overhead transparency stock. Napier gives a clear account of how to make the tables and masks. It would have been a long and difficult process to make them of wood, especially in Napier's time when sandpaper hadn't even been invented.
The Promptuary itself, the box to store the slips, grew from a simple "quick and dirty" project to one that I spent way too much time on. It turns out that finding a means to store multiple copies 20 different kinds of slips - 10 kinds of table slips and 10 kinds of mask strips - is a bigger project than at first it seems.
Making the Slips
The original Promptuary was designed for 10 by 10 multiplication, and so 10 copies of each table and mask must be printed on each slip. I used slips one inch wide and 11 inches long, just barely printable on a standard 8.5in by 11in sheet of copy paper. For this task A4 paper is a better choice.
In Rabdology, Napier provides a single table and mask slip example, but gives thorough instruction on the table and mask design. Remembering that each of these must be printed 10 high on the slips after the following work is done, I'll now go through the layout here.
For a table slip, start by laying out a 3 by 3 cell grid array. Next, draw diagonals through every cell to divide it as shown. The result is an array of 18 triangular cells. Napier named the cells using a single letter for each as shown in the illustration from Rabdology, below. Notice that except for cell "a," there is a left and a right value for each letter. Napier did not explain how he came up with the encoding pattern.
You can follow along with this pattern using the original example (at the left) as given by Napier for slip 4 . First, write the label number at the top of your slip. This is a number between zero and nine, and the slip pattern is dependent on this number. In the following I call this number "x".
The table cells are filled with values as follows:
| Letter | Value |
|---|---|
| a | Slip # = x |
| b left | msb of 2x |
| b right | lsb of 2x |
| c left | msb of 3x |
| c right | lsb of 3x |
| d left | msb of 4x |
| d right | lsb of 4x |
| e left | msb of 5x |
| e right | lsb of 5x |
| f left | msb of 6x |
| f right | lsb of 6x |
Write a zero in any place there is no second
order msb, or where the lsb is a zero.
The above are copies of the original table slip sample and layout grid found in Napier's 1617 edition of Rabdology. Used with permission of MIT Press, publisher. The book was translated from original Latin by William Richardson.
At left is an example of a table filled out for the number five. Ten of these tables must be stacked, one on top another, then a slip number (in this case, 5) added to the top to make the complete slip. When finished It looks something like the image on the right.
The Mask Slips
The mask slips use the same layout and coded lettering scheme as the table slips. Napier drilled holes in the mask slips at the indicated positions. Because I am using a real mask, the positions originally to have holes are now "clear" and are rest are covered or blacked out. The following describes how each cleared hole is identified.
The mask image below and to the left assumes that the mask slip is standing vertical. But in use, mask slips are always turned clockwise 90 degrees, and thus lay horizontally. For that reason the image must be rotated as shown below on the right before being duplicated and stacked to make the slip. This was not mentioned by Napier. At first I was confused because his mask slip example didn't match his lucid explaination. But once I realized that the mask was rotated, it all became clear.
| Slip Number | Cleared Letters |
|---|---|
| 0 | none |
| 1 | a |
| 2 | both b |
| 3 | both c |
| 4 | both d |
| 5 | both e |
| 6 | both f |
| 7 | both g |
| 8 | both h |
| 9 | both i |
Although the logic behind the slips is easy to grasp, physically making them gets rather involved. You can avoid this trauma by using the slip designs that I have provided as .tif files, but you'll probably want to find a way to print more than one file per page to avoid wasting paper.
The Slip Holder
Once the size of the slips are known, the minimum size of the box is also known. I made a two-level "slip holder" using copy paper, cardboard and foam core. The table slips are shown loaded in the front or top level, and the mask slips are in the back or lower level. The rows were deliberately horizontally offset and the back raised up to make both rows of slips easily visible.
Although not visible in these photos, the slip holder is held in place with hangers that allow it to be lifted out of the top cover to stand directly on the base, but still attached to the cover. This places it at a good visual angle and makes it even easier to select and replace the slips.
To make the dividers I folded copy paper using a guide printed on the back. The folded ridges in this paper were designed to be just over an inch wide and about the same height as the foam board that will eventually be placed along the sides and bottom.
Once completed the divider paper was glued onto a cardboard backing, as well as spacers made of foam board. The bottom spacer on this level is made about a half inch wider than the one that will be glued to the top level. This raises the height of the slips of the back row, again to help visibility and selection.
The second level is now glued on top of the first set of dividers, along with another set of spacers around the sides and bottom. Next I made a dress front plate of 1/16th inch birch "aircraft plywood" for the front and sides and glued those into place. And finally, a bottom dress panel was made from a piece of scrap wood and glued on as well. All that remains at this point is to develop the case.
The Promptuary Case
My solution for the Promptuary case was to buy two inexpensive unfinished "filing boxes" from Michael's, an arts and crafts store. The tops of the ones I used measured 13in by 13in and were about 3/4in high, making one of them precisely the right size to hold the slip holder. I did not use the box bottoms.
I lightly sanded the slip holder wood surfaces and the box covers, stained them, then sprayed on several coats of urethane. The two covers were fitted with brass hinges and rubber "stick on" feet. (These are available from Radio Shack and other electronics stores.) I glued concealed neodymium magnets into the case so it snaps shut via magnetic attraction.
The finishing touch came when the operating directions and name plate were added. I printed them on ordinary laser printer paper, then diluted some of the stain I used on the case, and brushed it on. This gave a nice "antique" look to the paper, which after drying I cut to size. This type of labeling is not unprecidented; many old instruments have such labels and I thought these gave the Promptuary a nice feel. After all, it is one of probably a very few built over the last three hundred years!
The instructions and name plate were glued to the case and slip holder using "Aleene's Original Tacky Glue" which I deluted slightly with water. This was a scary process. I desparately wanted to avoid a mistake at this point because it had consumed several weekends that for which I had other plans. In any event, I put some Aleene's into a small bowl, added a teaspoon (or so) of water, then painted the backs of the labels using an acid brush.
As this happens, the paper instantly starts curling, a frightening happening. But not to worry, once they touched the Promptuary they flattened right out as I gently brushed them with a clean paper towel. A word of caution: Aleene's sticks like epoxy: instantly and seemingly permanently. The label must be placed accurately the first try because I'm not sure a second chance for repositioning is possible.
But all went well and the above photo shows the results of which I am quite pleased. I hope someone besides my beloved wife finds an interest in the Promptuary. (She says she's really interested, but I have my doubts!)
Operating instructions given elsewhere provide several computing examples and describe how the Proptuary is used. For those wishing to make their slips from scratch, a detailed guide for making and printing the slips is also supplied, but most will find it more convenient to download the slip files I used. Finally, I've included a copy of the 1911 Britannica entry for John Napier for your perusal. It is a local copy from its source, and completely unedited. You'll find several non-critical OCR errors and that it is a dense, but interesting read.
Jim