How to set out the double guilloche border pattern on a circle. A few calculations are needed but I've tried to make them as simple as possible.
To demonstrate this scaled down version, all my calculations are in millimeters but I just put down the numbers. It doesn't matter whether you use inches, centimetres whatever, the calculations are the same.

1. Place a dot in the middle of your base board. Draw a circle which is the minimum space needed for your motif (in this case my wonderful drawing of a hare). 2. Draw a straight line from the centre of the circle out. 3. Using your staff which has been marked from your double guilloche pattern, place it on the line with one of the width marks on your first circle. Now mark where your centre tesserae (CT) mark is on the line. This is the point past which the centre line of your pattern cannot go any further in, i.e. this is the smallest circle your border pattern can be. 4. The first CT mark we did in image 3. makes a radius of 112mm. 112 (radius) x 2 = 224, the diameter.
Multiply this by 3.14 (Pi). 224 x 3.14 = 703mm. This is the circumference.

We need the pattern to fit on the circle so there is equal distance between all of the centre points (CT). The distance between each CT (a section) is 48mm.
So 703mm divide by 48mm = 14.6. This is no use as it must be a whole number. We can't decrease in size as this is the minimum circumference, any smaller and the pattern will encroach on the area needed for the central motif.

Go larger from here, make the circumference to consist of 15 sections. Now we make the calculation in reverse, find the radius from the circumference.
48mm (the distance between each CT) x 15 = 720mm. This is the circumference needed.

720mm divide by 3.14 (Pi) = 229mm (diameter).
229mm divide by 2 = 115mm, this is the radius needed to make a circumference of 15 'sections' of pattern. (I've rounded this answer up from 114.5mm).
This is now the new A and marks the new radius of 115mm 5. On the mark of 115mm out from the centre draw a circle. This is the centre line for the double guilloche pattern. 6. From this line you need to mark each CT on the circle. From the calculations with photo 4 we know this circle accommodates 15 sections of pattern, we need to mark 15 points where the CT will be. Divide 360 degrees by 15 and we get 24. Divide the circle up into
24 degree segments and mark on the cirlce, B. This is where each centre tesserae need to be. 7. Now use your staff, place the CT mark onto the new centre line circle, D.
C & E show where the inside and outside line of the pattern goes. 8. Draw in circles on C and E.The first circle, inside C can be scrubbed out to avoid confusion. B shows the CT marks. 9. Now start to draw in the pattern. Just do this in any way which works best for you. Here I've gone from directly out from the CT (shown by the red line) down in a sloping arc to the next CT to its right. 10. Once I've done that all the way round then I connect the rest of that line shown here by the red arrow. 11. Do the same as photo 9. but on the inside.
Please note; it's VERY important that when you are drawing in the second line of any strand that the distance remains the same at the beginning and end of each strand, in this case 5 tesserae. F shows where there should be a 5 tesserae space between the CT and the outside line of the strand. Ignore where you think the line should go, just concentrate on getting the strands the correct width at these points. 12. Finishing off the pattern. 13. After telling you in photo 11. to ensure you have the lines marked in properly I now want you to ignore them! Notice how pointed some of the strands on this pattern have become. This is unavoidable drawing out a small circle like this. When you set the tesserae just keep in mind the pattern you are creating, understand which are the important reference points and keeping a width of 5 tesserae set out a more flowing mosaic. 