Henry Sutton's Spiral Calculator

Introduction

This is the second calculator I've published that was developed with the help of Claude AI (Claude). I will talk more about Claude in a separate section.

I began this replica after a sudden realization I had while looking at the device on the British Science Museum page.

I'd seen this instrument many times, and had it on my bucket list for a replica, but I'd been paying no attention to the arms, which I admit I should have immediately noticed, are strangely arranged.

The arms do NOT run radially outward from the centre, as required in a divider-based circular slide rule, as seen on the Oughtred Circle of Proportions and the John Brown Spiral slide rule.

The arms touch only at the inner and outer edges; there's no line or edge along the center. However, the photos show that Sutton included small holes at either end of the arms. Clearly, Sutton intended these holes to hold a thread or hair to provide the radial line for reading the scales.

The two arms are actually offset from the center, allowing an operator to read the scales on either side of the hairline, as shown in the replica image above.

As far as I know, this is the earliest use of a hairline in such a calculator, linear or circular. It dates from 1663, less than 30 years after Oughtred.

In addition to the customary PDF files, I have attached 3D models of the cursors for anyone wishing to attempt them.

The disk has two styles of scales: Circular and spiral.

Circular scales (outer edge)

Locating the CIRCULAR scales

The outermost area of the slide rule features 3 circular scales.

1) The linear L scale (E for 'Equal parts' would be more appropriate) is on the outer rim and has 1000 divisions, offering 3 decimal places before interpolation.

2) The next inward circular scale is a 2-cycle natural sine scale. By natural, I mean the scale is not logarithmic.

3) The next inward circular scale is a 2-cycle 'meridional parts' scale (may have been called the scale of latitudes), used for navigation calculations using the Mercator projection. The units are minutes of arc of latitude, also called meridional minutes or meridional parts.

Spiral Scales (center to outer edge)

Locating the SPIRAL scales

The spiral band has 12 turns, which is 2 cycles of 5 turns and 2 extra turns on the outer edge.

The spiral band has three scales for the first 5 turns, representing a single cycle. We can't actually see the first cycle because it is beneath the arms' center pivot. We can see the second cycle as it spirals outward.

The spiral band expands to four scales on the fifth turn to accommodate the N or "Number" scale (Log x). The four scales run for one move cycle of 5 turns, then the N scale drops out, and the 3 scales continue for up to 2 turns after that.

From the innermost to outermost, the scales are ALL logarithmic, e.g., the sin(x) scale is actually log10(sin (x)), etc.

So for the entire spiral length the scales are:

Sin (x) runs for 10 turns and 2 cycles ( 34' to 90 degrees). At the end of turn 10, the spiral line switches from a sin(x) scale to a cos(x) scale, starting at 1 degree and ending after about 66 degrees.

Tan (x) runs for 12 turns, with 2 cycles to 45 degrees at turn 10; it ends a bit after 68 degrees on turn 12.

Versin(x) runs for 11.5 turns and stops at 180 degrees. The Versine function is: Versin(x) = 1-cos(x).

N runs from 10 to 100 in 5 turns.

The versine scale is laid out differently than the other 3 scales. It is narrower and uses dots rather than the 'railroad' track style used in the N, tan and sine scales.

The PDF file

For any home builder: First, the calculator is rendered full size (340 mm in diameter) because the scales are so dense that printing them smaller isn't feasible. I included images of about 1/2 disk size for 11x17 inch paper, Normally, those would need to go to a printer.

I found the scales so hard to locate that I added pages with coloured scales to help me familiarize myself. They remain in the PDF, so...some pages may look just a bit lurid.

sutton_disk.pdf

The 3D printed cursors with hairline

The picture above shows how the cursor (threads) on the dividers are attached. I experienced a little difficulty arranging the thread so that it followed the inside bottom edge of the dividers. The thread really wants to run straight between the centres of the 2 holes on the arms.

With a little patience, I persuaded the threads to lie close to the page and on the inside edge of the arm ends. I needed a couple of loops through the hole, as shown above. I used superglue to lock the threads, which means they aren't coming off now. I suppose I'd have to reprint them to change the thread. A period-correct alternative to my glue might have been a drop of wax.

To put the SR together, I used a map pin to prick the left and right pages in the center of the 2 small holes (top and middle of the disk). The centre hole of the cursors is 3 mm in diameter.

I then aligned the two pages with 2 map pins before I glued the two sides down to a substrate (foam board from a dollar store).

3D STL files for the cursor arms

Download the ZIP file containing STL files of the arms HERE .

Other Comments

I found the Sutton device quite hard to operate when I initially tried out my PDF version. This is largely because the 4 scales in the spiral band aren't clearly separated (e.g., no white space between scales INSIDE the spiral). Without separators, it can be difficult to locate the proper scale in the band.

Additionally, the relative position of the scales 'seems' to be different depending on whether you are looking at the top of the disk or the bottom (left and right). E.g., the outer edge of the band would be at the bottom or top, depending on where you look on the disk.

Prior examples of this device:

Sutton may have been experimenting with the form developed by Thomas Brown and described by his son, John Brown (John's book on the Joint Rule). Alternately, Sutton may have developed his spiral by independently adapting the same source Thomas used: Gunter.

These excerpts from John Browne's book (taken from Cajori's essay on the Gunter scale) cite elements in Thomas Browne's spiral that are also found on the Sutton device.

"...Thirdly, you have in 5 turns the lines of numbers, sines, Tangents, in three margents in divisions, and the line of versed sines in pricks, under the line of Tangents, according to Mr. Gunter’s cross-staff ",
"... also a line of natural sines, for finding of differences in sines, for finding hour and Azimuth...",
"... (on the) outermost edge is a line of equal parts to get the Logarithm of any number..."

So, both Brown and Sutton used a central band of 5 turns with the same 4 scales. They both used an equal parts scale on the outer circumference. All the scales on the Sutton calculator, except the meridional parts scale, were described as being on the Brown device.

I'll presently find Gunter's original work and update this section if I find anything.

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