From chemistry class, or from making home preserves, at one time you may have become acquainted with the fact that corks and rubber stoppers for flasks are made in sizes such as 5, 4, 3, 2, 1, 0, 00, and 000. Thus, 00 can be thought of as an old way to write -1, and 000 as an old way to write -2.
The names used for the sizes of model railroad trains originated in this fashion:
size gauge 3 2 1/2 inch 2 2 inch 1 1 3/4 inch O 1 1/4 inch OO 16.5 mm OOO 9.5 mm
although "originated" is perhaps a strong word; Märklin, a firm still in the model railroad business (having pioneered the Z gauge/scale and electronic control of individual trains, even if the very first attempt along that line was made by General Electric in the United States back in the 'sixties, known as ASTRAC) originally brought out several numbered sizes of toy trains as follows:
V 117mm IV 75mm III 67mm II 54mm I 48mm
with a 0 gauge of 35mm later added to that series. Unlike the other gauges discussed here, these measurements were between the centers of the two rails, leading to confusion in the early history of toy train manufacturing.
The gauge is the spacing between the inner edges of the two tracks, or the outer edges of the flanges of the two wheels on an axle.
Commonly, in line with practice in other fields of model making, one tends to want to refer to different scales; one might buy a model kit in a 1/48 scale, for example. But in model railroading, the gauge is the incontrovertible fact which determines if a given train will run on a given track.
Note that 0, 00, and 000 became O, OO, and OOO, the numbers changing to letters. (This can easily be seen from the typesetting of any book on model railroading, since in almost any typeface, the letter O is visibly wider than the number 0. It certainly is true that these scales originated in a numeric form, but it is also true that their names are letters at present.) The gauge shown here as OO is also that for HO, or "half O". There was also a gauge half the size of size 1 which was called HI, the 1 changing to an I. This gauge was essentially identical to what later became known as S.
Railroad tracks in the United States, and several other countries, including most other English-speaking nations, have a gauge in real life of 4 feet, 8 1/2 inches. This corresponded to the ruts left in the ground by wheeled vehicles used in the Roman empire.
Thus, one can immediately derive the correct scale for any model railroad gauge by dividing this size by the gauge, right?
Unfortunately, not. Model railroad trains, for various reasons, are not entirely true to scale. Thus, "HO scale", preferred in the United States, and OO, preferred in the United Kingdom, share the same gauge of 16.5 mm, but are nominally designed to reflect two different scales: 3.5 millimeters to the foot for HO, and 4 millimeters to the foot for OO.
The most well-known gauges and scales are:
name gauge scale G 45 mm 1:22.5 O 1 1/4 " 1:48 S 7/8 " 1:64 OO 16.5 mm 1:76.2 (exactly) (4 mm to the foot) HO 16.5 mm 1:87.0857... (3.5 mm to the foot) TT 12 mm 1:101.6 (exactly) (3 mm to the foot) N 9 mm 1:160 Z 6.5 mm 1:200
The S and TT sizes are no longer common in current use, but at one time they were popular in the United States, and thus they are still relatively well remembered. Also, in the United States, for a time, OO was used as the designation for trains that had the same 4 mm to the foot scale as associated with that designation in Britain, but with a 19mm gauge.
Since N gauge trains are more expensive than HO scale trains, because of the more exacting precision required in their manufacture, it would seem that the TT scale would provide greater compactness than HO scale along with lower cost than that of either HO or N, and, thus, I personally feel that it is regrettable that what appears to be the ideal scale, or at least the least expensive scale, has fallen into disuse. It may also be noted that while the TT size originated with a scale of 3 mm to the foot, a more accurate 1:120 scale has come to be associated with it as it became less popular.
Trains made for specialized purposes, such as hauling ore from mines, often had a much narrower gauge than the 4' 8 1/2 " of ordinary railway trains. For convenience, instead of having trains and track manufactured to the exact gauge of such trains when taken to scale, railroad modellers have used track designed to represent conventional railway trains in a smaller scale, and used it as the track for narrow-gauge trains in a layout using one of the other conventional track sizes for regular trains.
While less serious model railroaders can treat the gauge as the given, and mix HO and OO equipment with abandon, when it comes to narrow-gauge equipment, it is clearly the scale that is the given; it is the equipment that would be found in the same layout that should be grouped together in a chart.