Introduction: Genaille's Rods
Many people are familiar with Napier's Bones, a manual calculating system invented by John Napier in 1617. This reduces multiplication and division to addition and subtraction using tables printed on the 'bones'.
Fewer people have heard of a development of Napier's Bones invented in 1891 called Genaille-Lucas Rulers (aka Genaille's Rods). These were invented by French railway Engineer Henri Genaille in response to an arithmetic problem posed by French mathematician Édouard Lucas.
So it could be said that this calculating system was an entry in an early version of the Instructables Made with Math contest!
Unlike Napier's Bones, calculation using Genaille's Rods does not require any mental arithmetic.
In this Instructable I describe the form and use of Genaille-Lucas Rulers, and the design and 3D printing of a set of the rods.
To complete the circle (rectangle?), this was my entry in the Made with Math contest - and it won a First Prize! Many thanks to everyone that took an interest in this project.
On 17th November 2021 I added an extra step describing a 3D printed storage box for the rods.
Step 1: The Form of Genaille's Rods
The full set of Genaille's Rods consists of an Index rod and ten calculating rods. Each of the calculating rods has a digit from 0 to 9 at the top - see picture.
A workable set of the rods can be produced by printing out the drawing and cutting it into individual rods. If desired, the printout can be glued to cardboard before cutting.
I produced the initial drawing of the rods using AutoCAD LT 2022, referring to pictures online.
I attach the DWG file for interest.
I also attach the drawing in PDF format.
Step 2: The Use of Genaille's Rods
The set of rods is used to carry out multiplication as follows:
As an example consider 21879 x 5
Lay the Index rod at the left of the work area and arrange the calculation rods 2,1,8,7,9 against it as shown in the picture.
To multiply by 5 the row labelled 5 on the Index rod is used. Start with the uppermost number in that row on the rightmost rod. In this case it is 5. Write this digit down. Working to the left, follow the black triangle to the digit it is pointing at on the next rod to the left - in this case it is 9. Write this to the left of the first digit.
Repeat this following the triangles to the left, noting the digit at each triangle apex, until the final small digit on the Index rod is reached and written down to the left of the others. This process is indicated by the red arrows on the diagram.
Reading the noted digits from left to right gives the value 109395 which is indeed 5 x 21879.
A few minutes practice will allow the rods to be used without the need to write the digits down.
Step 3: 3D Printing a Set of Genaille's Rods
I decided to 3D print a set of Genaille's Rods. These rods would be more durable than paper or card versions.
Initially the full set was to be printed in one operation. However, the maximum print area of my Prusa i3 MK 2 is nominally 200mm x 250mm. Initial experiments showed that the numbers on the rods were too small to print successfully on rods 250mm long using the only size nozzle I had available (0.4mm). The first picture shows a screenshot of the full set model in Microsoft 3D Builder. I also attach the 3mf file in case anyone would like to try printing this with a smaller nozzle.
The answer was to print each rod individually lying diagonally across the print bed. This allowed a rod length of 30cm and the small digits were much clearer.
Step 4: Creating the 3D Models
This step describes the creation of the 3D model of one of the rods using Microsoft 3D Builder.
Insert a cube and adjust its dimensions to X:300 Y:20 Z:2 as shown in the first picture.
Adjust the position of the base to X:0 Y:0 Z:1 as shown in Picture 2.
If desired the base can be coloured. In my case I was going to use 'Bone White' printing filament as a nod to Napier's Bones, so I coloured the base with an approximation of this shade.
It would have been extremely time consuming and tedious to model the digits, frame and triangles using the basic objects in 3D Builder. Fortunately there is a useful tool built in that can import a 2D graphic and extrude it vertically to create a 3D object.
Starting with the AutoCAD drawing of the full set I isolated one rod and zoomed it as large as possible on my monitor. Using Microsoft Snip & Sketch (free in Windows 10) I screen grabbed this image and saved it as a png file - see Picture 3.
In 3D Builder import the graphic into the model of the base created above as follows:
Click The Insert menu and select + Add - Picture 4
Click Load Image at the bottom left of the dialogue. Navigate to the graphic of the rod saved above and click Open.
A 3D version of the graphic will appear - picture 5. Note that this an inverted version. To correct this slide the Inverse switch at the top of the dialogue to Off. The object will now look like Picture 5.
Some adjustment of the Levels slider may make the digits, frame and triangles sharper. When satisfied, click Import Image at the top left of the dialogue.
Adjust the dimensions of the imported object to X:300 Y:20 Z:2 to match the base.
Finally adjust the position to X:0 Y:0 Z:2.5
This will exactly position the Rod Upper at the correct 3D location.
If desired, the rod upper can be coloured black.
The complete 3D model of rod 1 is shown in the final picture.
Save the model in 3mf format with a suitable name such as Rod 1 30cm
Repeat the process to create the 3D models and 3mf files for the other rods and Index.
I attach the 3mf files for all of the rods and index.
Step 5: Slicing the 3D Model
The next step is to produce the g-code files needed to print the rods.
I use the excellent free Prusa Slicer program to convert 3mf files to the g-code files required by my Prusa i3 MK 2 printer. It is available here: PrusaSlicer
Open the rod 3mf file in Prusa Slicer.
The rod will be coloured blue indicating that the ends are outside the print area - first picture.
Click on the rotate icon at the left of the screen. Rotate the rod using the blue handle until it turns green indicating that it is completely within the print area. It will be laying diagonally as shown in Picture 2. Click on the rotate icon to turn it off.
Next click on the Variable Layer Height icon at the top of the screen. This allows thinner layers where there is more detail.
Click on Adaptive, then Smooth. A vertical graph appears with a blue line indicating the layer thickness. The further to the right the blue line is, the thicker the layer. If desired the line can be adjusted using the mouse. See Picture 3.
Click the Slice Now button. The result is shown in Picture 4.
A change of filament colour is required so that the base is white and the Rod Upper is black. To insert an instruction to pause the printer (to allow the filament to be changed) move the small orange triangle at the top of the vertical height bar downwards until it is just above the point where the digits, triangles and frame begin to be printed. A height of about 2.3mm is ideal - Picture 5.
Click the small orange hexagon containing a + sign next to the triangle to insert a colour change instruction. The top part of the vertical bar will change colour. Move the small triangle up and down to check the 3D model and the colour change.
If you need to remove the colour change position, click the small hexagon that is now grey with a cross within it. This will clear the instruction and you are free to set it at another height.
Re-slice the model if required. When all is well, export the g-code to an SD Card.
Step 6: Printing the Rods
Insert the SD Card from the previous step into the printer.
Preheat the printer for the material to be used. I print with PLA.
When the printer is up to temperature, load the filament for the rod base. In my case I used Bone White.
Select 'Print from SD Card' from the menus and select the correct g-code file for the rod to be printed.
The base will then be printed - Picture 1 shows the base under construction.
At the point where a filament change from white to black is required the printer will pause and a beep will be emitted.
Change the filament reel and load the black PLA into the extruder. Follow the instructions on the screen.
Press the control knob to restart the print. Picture 2 shows the print just after the filament change.
The triangles, frame and digits forming the rod upper will be printed.
A complete rod took about 3 hours to print on my machine. Pictures 3 and 4 show the completed rod.
Congratulations! You now have the first rod of the set! Only ten more to go...
Step 7: Storage Box
(This step added on 17th November 2021)
I decided to 3D print a storage box for my set of Genaille's Rods.
This was printed in two parts. No supports required if printed as shown in the attached 3MF files.
When pushed together the two halves of the box are a tight fit. However, care should be taken when the box contains the rods as it may come apart if lifted by one end. A future design will include a latch!
Step 8: Final Thoughts
The Genaille Lucas rulers described in this Instructable are an ingenious method of carrying out multiplication without the need for mental arithmetic.
Multiplications involving more than one instance of a given digit will require duplicate rulers (e.g. 34145 x 6 will need two number 4 rods).
A modified version of the rulers was invented to carry out division - see Wikipedia link below.
Unfortunately the rulers were soon made redundant by the introduction of mechanical calculators. Genaille-Lucas Rulers are, however, an interesting chapter in the history of mathematics.
For more information on Genaille-Lucas rulers see Genaille–Lucas rulers - Wikipedia
Information on Napier's Bones can be found here: Napier's bones - Wikipedia
If you are interested in learning more about using Microsoft 3D Builder see my previous Instructable:
I hope you have found this Instructable interesting. I look forward to your comments and ideas.
This was my entry in the Made with Math contest - and it won a First Prize! Many thanks to everyone that took an interest in this project.
First Prize in the
Made with Math Contest