Notes

Online Versions

• First Edition - a good scan
• Leiden Edition (Lugd. Batatorum) Typis Petri Rammasenij. 1628. But fold out diagrams are not folded out, so it's not clear whether this includes the double-ended promptuary strips.

Versions of Napier's book

• Rabdology - only one copy seems to have this title: the English translation of Napier's First Edition, published by The MIT Press (1990) and translated by William Frank Richardson. ISBN 0-262-14046-2.
• Rabdologia:
  
  • Rabdologia, or, The art of numbring by rods, Seth Partridge (1648). This is not Napier's book. It was written by a surveyor who knew how to work the bones. In English.
  • Verhandeling over de Rabdologia of rekening met staafjes - published 1770 Amsterdam by Jean Morterre, Boekverkoper op den Nieuwendyk. Title means "Treatise on the Rabdologia or Reckoning with Rods". Not sure is this Napier's book translated into Dutch.
• Rabdologiae:
  
  • Rabdologiæ, sev Numerationis per Virgulas, Ioanne Nepero (John Napier). Published Edinburgh (1617). In Latin. Available online. I have a copy of this from the Forgotten Book company - it is a bad scan of the original, with unreadable versions of the fold-out pages, some text missing due to cropping and the last page missing.
  • Rabdologiae, seu Numerationis per Virgulas, John Napier (1626). In Latin. Published in Leiden, the Netherlands (sometimes given as Lugd. or Lugdunum). Also has name Pieter Rammazuyn on the cover - author? editor? This appears to be the original text of the First Edition reset, so that the pages are laid out differently, but I haven't found any differences in the text.

Note: in writing Latin, it's a matter of style whether the ligature æ or the separate letters a e are used, and also whether u or v is used, so the two Latin titles above are equivalent.

According to Theodora.com's encyclopedia, there were editions after the first one published in:

• Italian, Verona, 1623
• Latin, Leiden, 1626 and 1628 - Leiden is known as Lugdunum Batatoria in Latin.
• Dutch, Gouda, 1626

(https://theodora.com/encyclopedia/n/john_napier.html)

The First Edition (as translated into English) describes the promptuary with 10 number strips and 10 mask strips for each digit. It clearly does not allow for a mark strip being used for two different digits by rotating it by 180 degrees. The Wikipedia article about the Promptuary says "The final form described by Napier took advantage of symmetries to compact the rods, and used the materials of the day to hold system of metal plates, placed inside a wooden frame." It cites a book "radley, Michael John (2006), The Age of Genius: 1300 to 1800". But Napier's first edition did not take advantage of symmetries to compact the rods. Is this referring to one of the later editions, with additions by another author?

There's a mistake in the English translation - the promptuary mask strip example (for digit 7) is shown vertical with the number 7 at the top. In the original book, it was shown horizontal across two pages, with the digit 7 at the right end and oriented so that it is right way up when the strip is horizontal. This makes it more obvious that the strip is to be used horizontally (although it was not clear in Napier's description).

Versions of the Bones

The site Science Museum Group has photos of the following sets of bones: (collection.sciencemuseumgroup.org.uk)

Napier's original version of the bones, as decribed in the first edition of his book, are available from Armstrong Metalcraft. You can see some pictures on their website.

The Science Museum, London, has on display an example of the bones the following differences from Napier's originals:

• There is a frame with the numbers from 1 to 9 written down the left and right sides, so the square/cube rod is only two rods wide rather than 3 - it does not need to repeat the numbers from 1 to 9.
• The rods are 9.5x1x1 and do not have the half-square blank space at the top of the rod. At the bottom of the rod, the half-square is divided in two and contains the simples of the left and right faces to make it easier to find the correct rods.
• The first square, containing the simple, is not divided by a diagnoal, and the simple is written much bigger than the other numbers on the rod.
• Although the rods (of which there are 20) have a different table on each face and appear to use Napier's system of having opposite faces adding up to 9, they don't have the same arrangement of faces on the rods, and don't have the tables for the numbers from 5 to 9 going in the opposite direction, so the 9s complement test can't be done.

The Science Museum in Edinburgh has an exmple of the bones in which each rod is flat with two tables, one on each side. These seem to be arranged with opposite sides adding to 9. Again the first square is not divided and contains a big simple.

Bones for Sale

Grand Illusions make a set with 10 single-sided strips, one for each digit. There is a frame. The top square is undivided with a much bigger digit. All zeroes are written into the triangles. This can multiply only numbers that have no repeated digits.

Armstrong Metalcrafts make a beautiful set of bones to Napier's original specification, with 1 x 1 x 10 rods, a half-square header and footer, tables above 4 inverted etc. They are made of polished aluminium. There are twenty rods so that all numbers of 8 digits and any number of 9 digits less than 111,111,111 can be multiplied. There's also the 1 x 3 x 10 strip for square and cube roots.

Design of the Promptuary

The promptuary consists of four parts:

• a set of number strips, engraved with a large digit at one end and with many small digits along the strip
• a set of mask strips, which Napier called 'excised or perforated' strips. Each has a single digit engraved at one end and various triangular holes cut in it
• a board on which to place the strips when performing a calculation
• a box to store the strips. In Napier's design, the top of the box was the board on which the calculations were performed.

The dimensions of the strips depends on the maximum number of digits in the numbers to be multiplied. For a device capable of multiplying two N-digit numbers together, the strips should be (N+1) times as long as they are wide, and there should be 10N number strips and 10N mask strips. So for example, for a promptuary capable of multiplying two five-digit numbers together, the strips should 6 times as long as they are wide, with 50 number strips and 50 mask strips. Napier's example specified strips 1 finger (19mm) wide and 11 fingers (209mm) long, enabling the device to multiply two 10-digits numbers to produce a 20-digit result.

Napier specified that the number strips should be a different thickness from the mask strips - one quarter of a finger (5mm) versus one eight of a finger (2.5mm). This is not necessary for the operation of the device.

The promptuary contains a lot more pieces than a set of Napier's Bones. A set of Napier's Bones with 20 rods is capable of multiplying numbers of up to 8 digits. An equivalent promptuary needs 160 strips.

In the examples and illustrations below, N is set to 5 - that is, the illustrated promptuary can multiply numbers of up to 5 digits.

Number Strips

The number strips are divided into five squares, with about half a square free at each end. A large digit, known as the 'simple', is marked in the space at the top of the strip. A multiplication table is placed in each of the five squares. Each of these multiplication tables is identical - it lists the multiples of the simple and is laid out in a particular way:

The square is divided into nine smaller squares in a 3×3 arrangement. Each of these is divided into two triangles by a diagonal line running from lower left to upper right. The multiples of the digit at the top of the strip, the 'simple', are marked in the table as in the diagram.

[[File:Promptuary diagram 1.png|thumb|Napier's Promptuary: Placing multiples on promptuary grid]]

• The simple itself is marked in the triangle labelled ×1.

• The number that is two times the simple is marked in the two triangles marked ×2. If the number is two digits, the first digit is placed to the left of the main diagonal (marked in red) and the second digit to the right of the diagonal. If the number is a single digit, it is marked to the right of the diagonal,

• The number that is 3 times the simple is written in the triangles marked ×3 in the same way as the multiple of 2

• The other multiplies of the simple are marked in the other triangles in the same way.

• Zeroes may be written in or left blank. This does not affect the operation of the device.

• The triangle in the bottom left of the table is always blank.

The following diagram shows the multiplication table for the simple 7:

[[File:Promptuary diagram 2.png|thumb|Napier's Promptuary: multiple diagram for digit 7]]

Complete number rods for the simple 7 and the simple 2 are shown in the following diagram. The lines delineating the triangles have been omitted.

[[File:Promptuary diagram 3.png|thumb|Napier's Promptuary: two number strips from the promptuary, for digits 7 and 2]]

The Mask Strips

Mask strips are placed horizontally across the calculating, that is, from left to right rather than from top to bottom. They have a large digit written in the space at one end and the rest of the strip contains five squares. Each square has triangular holes cut in it according to the patterns given in the following diagram.

[[File:Promptuary diagram 4.png|thumb|Napier's Promptuary: the mask patterns for digits 0 to 9]]

So for example the mask strips for the simples 3, 6 and 9 will look as follows:

[[File:Promptuary diagram 5.png|thumb|Napier's Promptuary: Three mask strips, for digits 3, 6 and 9]]

The guide lines in the patterns are for positioning the holes. They do not need to appear on the strips. The main diagonal line of each mask pattern, however, shown here is red, is marked on the mask strip. It is an important part of the device. The pattern given here for the simple 0 is from later editions of Napier's book. The version of the 0 strip in the first edition had no holes in it. .

Performing a Multiplication

Number sStrips for the first of the numbers to be mutiplied, the 'multiplicand', are placed on the calculating board side by side, running from top to bottom of the board. In the example shown here, the muliplicand is 772.

Mask strips for the second number, the 'multiplier', are placed horizontally on top of the number strips. In the example, the multiplier it 396.

[[File:Promptuary diagram 6.png|thumb|Napier's Promptuary: calculating 772 times 396]]

The result of the multiplication is read from the device by examining the digits visible through the triangular holes in the mask strips. Those parts of the number strips that are not covered by the mask strips are ignored. The diagonal lines on the mask strips divide the device into diagonal bands containing digits visible through the holes.

• Starting at the right, the first band with any visible digits contains just one digit, a 2. This is written down as the right-most digit of the result.
• The next band from the right has three digits, 2, 1 and 8. These are added together to get 11. The 'units' digit of this addition, 1, is written down as the next digit of the multiplication result. The 'tens' digit, which is 1, is carried into the next band.
• The third band from the right has five digits, 2, 4, 3, 1 and 6 plus the carried 1. These are all added to produce 17. The units digit of this, 7, is written as the next digit of the result. The tens digit, 1, is carried into the next band.
• This process is repeated for each diagonal band from right to left until all the digits have been processed.

The full result has now been written down as 305712. This is the result of multiplying 772 by 396. The multiplication process required only addition, and no intermediate results needed to be written down.