The Gnomish Circler Device

The Gnomish Circler Device

The Gnomish Circler is a tool, most frequently used by the gnomes, which allows the user to calculate precisely features of circles and spheres. The circular ratio - the relationship between a circle’s diameter and its circumference, as well as being important in other calculations - can be used to adjust measurements accordingly - for example, to calculate the circumference when give the diameter or vice versa.

Description

The Circler is almost invariably made of metal and consists of some fraction of a circle - usually a quarter, although they are often a half or even a full circle instead - with a single straight pointer that can be rotated about the centre and two dials running around it. One dial runs around the edge of the instrument, and the other is positioned a very precise distance from the centre - this distance is not known precisely, but the nature of the device’s construction makes such knowledge unnecessary. For our purposes here, it will be enough to say that it is slightly less than a third of the way to the outer dial. The common shape of Circlers as a quarter circle has led to the adoption by the gnomes of a small quarter circle as their symbol for the circular ratio - shorthand for “use a Circler here”, perhaps.

These dials are marked out very accurately in nailsbreadths, and the central pointer reaches all the way to the outer dial. On all but the very lowest quality Circlers, the pointer, being a sturdy piece of metal, has a little “window” - often with small internal arrows pointing out the precise point of measurement - over the inner dial, and some have a similar way of indicating the number in the larger ring. Better made example also often have small runners or even tiny wheels to allow the pointer to swing around easily and without scratching the face of the device, and sometimes there is a glass cover over the entire device, much like on a pocket watch, to prevent damage to the pointer - if it is bent, the instrument is rendered nigh on useless. This cover can usually be opened, but some have tried to cut down on the hassle of opening and closing with every use by introducing an internal system of cogs to allow the dial to be turned; most regard this as actually producing worse results, as it is harder to position precisely, and so the idea has largely been abandoned.

There are sometimes decorations of some sort or another, but most functional examples are austere and designed solely for use rather than for display. Far more likely to be decorated is the box in which it is kept; for those without a glass cover, this is vital in order to preserve the instrument, and those who do use a cover often have a box as well.

Construction

One of the most important pieces of equipment for the manufacture of the Circler is a wheel, usually metal, as close as possible to a perfect circle; this is used for calibrating the device precisely, and is why the ratio does not need to be known for the Circler to be made. This wheel also has a clearly marked point on its circumference, in order that the calibration is as accurate as possible. This can be made simply with the use of a pair of compasses to mark out a circle on bronze, brass or iron before cutting and filing with great precision.

In the creation of the device itself, the first thing to be done is the cutting of the shape of the dial - a half, quarter or full circle - but larger than the final item is intended to be. Using a pair of compasses set to match the diameter of the calibration wheel, the inner dial is marked out at that distance from the centre of the device. This is when the calibration wheel is used. Its marked point is placed precisely on the centre of the device, and it is carefully rolled out through one complete revolution - moving a distance equal to the circular ratio applied to the inner distance. This point is marked, but several more measurements are made in order to have the device calibrated precisely. Only when four or five consecutive measurements give the same point will it finally be marked, have a pair of compasses calibrated to that distance and have the outer circle marked out from it. A few more measurements are taken and if all is well the Circler is cut down to a reasonable size before the dials are fully marked out in nailsbreadths and grains and sometimes even smaller units.

The pointer is then added; this is attached to a pin at the centre and swings around to indicate which numbers pair with which. If it is going to run on runners - one or two small ones, set so as to keep the load of its own weight off the pointer - these are now added and their tracks formed. When the pointer is complete, it is placed on a pin through the centre and oiled well to allow it to swing smoothly. Any casing with a glass lid and so on is now made and affixed to the device, and all the finishing aesthetic touches are given to it. It is polished, oiled, and placed in a well-made box to protect it from malign chance.

Usage

Our readers will appreciate that if one goes around a certain fraction of a circle - say, a quarter or a third - one will have gone a certain distance; and if one goes the same fraction of the way around the edge of a circle twice as far across, one will have gone twice the distance that one went around the smaller circle. Therefore, by moving from a circle of one size to another, the ratio of the distances moved will be the same as the ratios of the diameters of the two circles. This is the key principle behind the operation of the Circler, for while the precise ratio of the two circles is not known, if the pointer is placed over the desired value on the smaller circle the result can simply be read off the larger one. Likewise, if one desires the reverse, one uses the outer dial for input and reads off the inner.

The ratio of the outer and inner dials on the Circler is, as has been mentioned, known as the circular ratio, and occurs in a wide variety of geometric calculations relating to circles, cones, cylinders, spheres and, in fact, virtually every calculation to do with the circular shapes. For the circumference, as has been mentioned, one inputs the diameter of the circle; for the area of a circle, one inputs the area of a square with sides the length of the radius of the circle; and so on and so forth.

History/Origin

It has long been known, logically, that there is a relationship between the diameter of a circle and its circumference; however, it was not until the year 1496 a.S. that a gnomish geometer by the name of Fundle Foranaft thought of trying to calculate what it is in order to use it in calculations. He assumed that it was an easy to find and simple ratio; however, after a few months research into the problem, he found that all his approximations failed to be accurate. Refusing to be put down, he simply started to speak of the circular ratio. His symbol for it, which crops up over and again in his work, was fairly complex and involved all sorts of lines and circles; nor was he particularly consistent with it. However, it sufficed until he came to need an actual result. Determined, this time, that the problem would not elude him, he had a small disc constructed with a diameter equal to the value he wanted to convert. By rolling this disc along a small track, he was able to come up with an answer.

However, he was not satisfied with this, and soon afterwards the first sketches of the Circler appeared, when he realised that he could simulate a disk of any diameter by using a small circle inside a larger one, both precisely marked out and only requiring one disc for calibration. The device, which he had made and began to use, proved very popular with other geometers and swiftly spread throughout the Kingdom of Santharia, becoming widely used by the year 1500 a.S. and even finding its way into an elaborate display piece, the making overseen by Foranaft himself, that was sent to the Santhran. Perhaps a courtier with an interest in geometry uses it, or perhaps it merely sits on its ornate bronze stand and impresses visiting dignitaries.

Myth/Lore

Some say that it was in fact Foranaft’s daughter, a young Gnome with various names depending on whom one listens to, who came up with the idea of using his original disc to calibrate a device that would allow him to apply the circular ratio to a wide range of numbers; but the extent to which she understood the issues involved and the problems of the numbers, and thus the precision of her suggestion, remain very much in doubt. Many academics, of course, believe that it is nothing more than hearsay; what is certain is that the Circler is a useful and accurate device, although reading it and recording the result unfortunately necessitates some inaccuracy.