As November draws to a close, it seemed an opportune moment to conclude my investigation of Stanley’s planimeter, at least so far as the instrument itself is concerned. One of the great benefits of having a physical specimen to hand is the chance to study it experimentally, so it’s high time we did some actual area measurement with this particular example. However, before getting stuck in, I need to address the elephant in the room: What did the first edition of Stanley’s book have to say about the polar planimeter?
With the same image and very similar descriptions having appeared in every edition from the third of 1868 to the seventh in 1900, I was not honestly expecting any surprises. So when I did eventually get around to consulting the original self-published work of 1866, what I found there came as a bit of a shock.
For a start, the image is missing, even though the engravings of the computing scale and opisometer that bookend chapter 30 are both the same. At first I wondered if the planimeter had been omitted altogether from the text, the chapter being four pages shorter than its counterpart in the third edition, but on closer inspection it was still there – if only just. Whereas in 1868 Stanley had described the polar planimeter as “a very scientific little instrument” (and in subsequent editions “a very exact scientific little instrument”) and given a comprehensive description of its construction and use over five pages, just two years earlier he could manage only a single barbed paragraph:
“There have been many other instruments made for computing areas; the one best known is the Planimeter, perhaps the subject of this chapter would scarcely appear complete without some mention of this scientific but practically useless and complicated instrument, a description of which would occupy considerable space, and might be considered unnecessary, as it has been described by several authors. The principle of the instrument is that of a roller made to register, by rolling in contact with the surface of the paper, the distance over which a point passes in describing the boundary of a given figure; the roller, by calculating machinery, is made to give data from which the area may be calculated with a little less trouble than by trigonometry.”
And that’s it. Compared to the five pages he lavishes on the computing scale, it would almost have been kinder to omit the planimeter entirely. What could have prompted this spasm of barely contained vitriol? It is true that Elliott Bros still had sole agency for Amsler’s planimeter at the time, besides which it was an instrument that represented a direct business challenge to Stanley’s improved computing scales (which he may have attempted to patent). However, Stanley always had an eye for the future, and it would not have escaped his attention that Amsler’s patent was due to expire a few short years from the publication of his book, opening up a potential new market.
There appears to be something more personal at play here, a bitterness that cannot be explained away as the outcome of pure competitive spirit. This is where our handmade polar planimeter comes in, quite possibly fabricated by Stanley himself during the late 1850s in the white heat of its novelty.
As it happens, the W.C. engraving is not the only modification that was made to Stanley’s planimeter after completion. Alongside the scale mark for 10 square inches on the tracer arm, additional lines have been scratched into the brass that appear to be the result of an ad hoc recalibration. Several different positions are marked on both the side and the top of the tracer arm (the top lines cutting across the first digit of the engraved constant), two of which have subsequently been crossed out. Alongside the marks on top of the bar is the scratchy notation “IN”, presumably for inches.
Who made these alterations and why? The second question is the more easily answered, and one I set out to investigate experimentally. After adjusting the planimeter as precisely as possible to the original 10 square inch scale mark, I measured a carefully constructed 5 x 2 inch rectangle, which under ideal conditions should be recorded by the planimeter as 10 square inches (i.e. 1.000 on the combined counter disc, roller and vernier readings). The same rectangle was then measured with the planimeter adjusted to the two most distinct additional lines inscribed on the top and side of the bar, one being around the middle of the range and one at the extreme right.
For additional accuracy I employed the technique suggested by Amsler in his 1856 publication, using a short ruler to guide the tracing point along the straight lines. At each setting of the tracer arm, the desired scale mark was aligned to the edge of the sleeve with the assistance of the micrometer screw and a magnifying loupe (the equipment used is shown in the photo at the top of this post). Similar care was taken to reset the vernier before commencing each measurement.
Unsurprisingly, the first “official” mark proved inaccurate, the vernier scale and counter disc giving a reading of 9.49 square inches. This substantial shortfall was evidently what had led to recalibration becoming necessary in the first place. The second (middle) scratched mark was closer at 9.72, but still some way short. Finally, the third (right hand) mark yielded a creditable 9.95, very close but not exact.
Proportionally greater margins of error were found when a 10 x 10 inch rectangle was measured with the pole inside the figure, a calculation that requires the use of the constant engraved on top of the bar. The original mark gave the result 92.80 sq. in. while the third mark was closer at 102.88 sq. in., but it seems that the accuracy of the engraved constant may also have been compromised by the underlying error.
These experiments confirmed that the scratched marks were indeed the result of a recalibration process, although there seemed no obvious way of knowing when this was done or who did it. As to why it was needed, could the instrument simply have gone “out of whack”? The reasons for this might conceivably be due to a change in the axis of the recording wheel, or else a small alteration in the wheel’s diameter or its frictional resistance, from either corrosion or wear. More importantly, was this recalibration done by Stanley or by somebody else, possibly much later?
It seemed at first that there was no way of ever knowing the answer to these questions, but then it occurred to me that it might be worth testing the other scale marks on the instrument. If all of these were similarly inaccurate, it would point to a fault having developed some time after manufacture, for which an attempt had been made to correct a particular scale (square inches being likely the most used).
To properly test this theory, not only did I remeasure the rectangle from the previous experiment, but I also constructed two new squares to reflect the native units of the two other planimeter scales that were accompanied by constants on top of the bar, namely 1 square decimetre and 0.1 square foot. On this occasion, the outcome was not what I had expected: both measurements were absolutely spot on (1.000 square decimetres and 0.1004 square feet to be precise, as seen below). The inescapable conclusion is that Stanley had made a single mistake during the construction of this instrument, which resulted in the incorrect placement of the 10 square inch mark. Everything else worked exactly as it should.
Anyone with even a superficial knowledge of Stanley’s personality might guess that he would have been furious at this. It is tempting to read this frustration in every line scratched into the lacquered brass and in every withering word on the page of his debut publication. One can imagine that Stanley was ready to hurl the infernal contraption out of the window and return to the comfort of his smooth and reassuringly solid computing scales. He evidently didn’t – or at the very least retrieved the errant instrument from the pavement of Great Turnstile after the event – because the same planimeter was in front of him when he was making revisions for the 1868 edition of his book, and in front of Thomas Collings when he drew the illustration.
The scale error was doubtless due to a botched measurement during initial calibration, easily done if each test area is not traced at least twice to detect possible slip-ups. To be fair, this also reflects my own experience with the planimeter. In my description of the experiments, I have not troubled you with all the nudged tracers, skipped wheels and yanked out needles, not to mention the misplaced pole arms that caused the recording wheel to run off the paper, all while attempting to keep an eye on the counter disc in case it completed a full revolution. These are all foibles of Amsler’s original design, and quite possible to iron out with sufficient practice under the right conditions.
This is not to say that Stanley’s instrument does not have its own flaws. I found the tracer point so sharp that after a couple of passes it began to scratch out the ink lines of my first test area, making repeat measurements extremely difficult unless particular care was taken.
Furthermore, the configuration of the sleeve carrying the tracer arm makes it difficult to accurately set the scale marks. This is partly due to parallax, as – unlike Amsler’s early instruments – Stanley did not bevel the reading edge. Amsler was ultimately to remedy this problem by the addition of a projecting bevel with scribed alignment mark that can be found on most of his later adjustable models.
At times it felt that Stanley’s initial verdict of a “practically useless and complicated instrument” was well deserved! But when the polar planimeter works, it works like a dream, with the astonishing accuracy seen in the later trials. Sometimes it seems almost incomprehensible that the half-rolling, half-slipping motion of a small wheel is able to produce such exact results, and that is before we even get to the geometric wizardry that underpins its operation (seen below in the faint trace of the roller left on the paper).
Once the epiphany of the polar planimeter has been experienced, its wonder never really fades. As Stanley wrote in the 1878 fifth edition of his book:
“The instrument is perfect, subject to the possibility of the hand following exactly the outline of the figure to be computed”
Perhaps over the intervening years, by way of his own planimeter-making experience, Stanley had learnt something about human fallibility.
