A Source Book for Rule Collectors


  1. Introduction
  2. The E and M Scales
  3. The Sector
  4. Gunter's Line
  5. Interesting Curiosities
  6. References


Philip Stanley's Source Book for Rule Collectors is a delightful and fascinating look at the history of small length-measuring instruments. It is of interest not only to collectors, but to anyone who wishes to know more about practical measurement, materials and manufacturing processes, and the historical details of trades in which measurement played an important role. It is a handsome, soft-bound, 8-1/2" x 11", 286-page book packed with information. Included are reprints of articles, an excellent bibliography, a thorough discussion of all kinds of rules, and a remarkable table of pre-metric European length measurements in the range of 1 in. to 6 ft., thoroughly illustrated with line drawings and black-and-white cuts. Of course, I had used rules myself, and knew vaguely about the history of the slide rule and other calculating devices, but this book was like landing on a fascinating island to explore it after only seeing the island from the sea. I was disappointed, however, not to find an index, which would have been especially useful since important topics are treated in several places. Computers now make the preparation of indexes much easier than in the past, and I hope that later editions may include one. In what follows, "the book" refers to Stanley's book. I'll describe some features that I found interesting to show some of the treasures that the reader will find in it, but the book contains much more of the same.

The general term for the devices discussed in the book is rule, and fundamentally the scope of the subject includes all devices suggested by this name, as well as related objects, such as rolling measures, planimeters and even steam engine indicators. However, specialized areas are only mentioned in passing, such as the familiar calculating slide rule, machinist's rules, surveying instruments and precision scientific instruments. The principal emphasis is on the inexpensive folding rule made from boxwood or ivory that was so widely used, but now has passed from view. This book also treats principally American rules, but also includes some information on British rules. Both were, of course, based on the inch, and the American industry was originated after 1800 by immigrants from Britain, and grew under tariff protection. It was a New England industry, later concentrated in Western Connecticut. Many famous names have now vanished, but Stanley and Lufkin (originating in Cleveland rather than in New England, and originally specializing in timber measurement) are still extant.

A rule may have three functions: (1) measuring linear distances; (2) guiding a pencil, pen or knife in a straight line; and (3) performing numerical calculations. The common desk or school rule is the only one that should be called a ruler. It is usually 6, 12 or 24 inches long and performs only the first two functions. It has generally been of wood, but plastic is now the dominant material. The wooden form is generally equipped with a metal tear edge for cutting paper, which also serves the function of preventing ink from spreading under the ruler. The graduated edge is usually beveled. The graduations are for 1/16, 1/8, 1/4, 1/2 and 1 inch, and are of increasing length. An embellishment line parallel to the edge defines the length of the 1/8 graduation; the 1/16 is shorter, and the others are longer. Since the Civil War, metric millimeter graduations have also been included, normally on the other edge. The draftsman refers to his rule as a scale, and uses it exclusively for the first purpose. It is never used for guiding a pencil or pen, for which the triangles and T-square are employed. The draftsman only uses a steel rule for guiding a knife.

Rules were also made to serve related useful functions, often specific to a trade. Rules could be adapted as protractors, to measure and construct angles, or as levels when equipped with a spirit level vial. An additional leg could serve as a bevel, to transfer angles, or to make the rule into an inclinometer. Caliper heads to measure round objects are often found. Tables of useful information could also be included. One Stanley rule consisted entirely of tables, with no other function. Other rules were specialized to determine the weights of geometrical solids of various substances. The book does not discuss the Vernier scale familiar on surveying and scientific instruments, but also appearing on the familiar Vernier caliper, since these fall under the classification of machinist's rules.

Precision rules have always been made in metal, usually brass or steel. Common rules, however, were made from wood or ivory before plastics were introduced. Wood is cheaper and easier to work than metals, and ivory is very attractive. The wood traditionally used for rules was Turkey boxwood, a hard, yellowish wood with an obscure fine grain. This wood was not only durable, but had the essential property of not expanding or contracting with changes in humidity. Thermal expansion is of little consequence in rules, since the normal range of environmental temperatures would cause little change in length. Much more important is humidity, which does vary greatly. Ivory is very subject to humidity, and was used purely on account of its whiteness, which was attractive and made a good contrast with black graduations; it was a very inferior material for an accurate and durable rule. Fortunately, it was early partly replaced by white celluloid, and now completely by plastics, which are far superior materials in every way. Drafting scales have been made in paper, so that they would expand and contract at the same rate as the drawing paper. Turkey boxwood came from Turkey and (mainly) southern Russia, and appears to have become exhausted or inaccessible by 1900, so that substitutes were required. In the United States, maple made a good substitute, while hickory was used only when strength was required, since it cannot take fine graduation. Mahogany was used when the graduations could be put on white celluloid facings; otherwise they would be hard to read due to the darkness of the wood.

There is, unfortunately, no information on Japanese or Chinese rules. It would be interesting to know how they were graduated, and if the orient's wonder material, bamboo, was used. This would seem to be an interesting subject for investigation.

Boxwood rules were accompanied by brass joints and fittings, while ivory rules were fitted with German silver (nickel silver), since brass stains ivory, as does iron. German silver is just a nickel brass, containing no silver, and is a very satisfactory and attractive white metal. The purpose of folded rules was to make them convenient to carry. A one-foot, two-fold rule had a joint in the middle and folded to a length of 6 inches. A two-foot, four-fold rule was also 6 inches long when folded. A one-foot, four-fold rule was easy to carry in a shirt pocket, since it was only 3 inches long. A few six-fold rules were manufactured, but only for a limited time.

The information on hand manufacture of rules was interesting. Rules (and other graduated things) are hand graduated following a master pattern, using a graduating square and a graduating knife, which cuts when pulled, making a single graduation. Graduation was a rapid procedure, and surprisingly accurate. A boxwood rule could be hand graduated in about 10 minutes. The numbers and other legends were stamped. Graduations were filled with lampblack and linseed oil rubbed into them. Hand manufacture always competed successfully with machines, though now all rules are made and graduated by machine. I also discovered that thermometers were graduated every 10 or 15 degrees from a master (the graduations are not equally spaced), and the intervening spaces filled with evenly-spaced graduations.

The modern alternatives to folding boxwood rules are the retractable metal tape and the zigzag rule, the latter extendable to 6 feet. The metal tape has much to recommend it, especially for longer distances, but is inconvenient in the shop, while the zigzag rule is just nasty. I'd certainly find a one-foot, two-fold rule very convenient, but they are no longer available. Apparently, they were manufactured into the 1950's, but I was not lucky enough to obtain one then.

The E and M scales

Rules also served as calculational devices, very useful for workmen of very limited mathematical ability. A simple example is multiplication by π, the conversion between diameter and circumference, which proved astonishingly difficult for the average Hoosier, even when π was taken as 22/7, because the advanced concepts of multiplication and division were involved. A scale in which 1" was represented by 1/π = 0.3183" could be used for this purpose instead. A small square was used to correlate readings on the regular 1:1 scale (diameter) and on this 1:0.3183 scale (circumference). The square served the function of the cursor on a slide rule, incidentally. There are many examples of the use of dividers and a small square in connection with rules.

A more difficult computation, whose theory involved algebra, and whose execution involved radicals, was cutting a square timber into an octagon. This was the first step in the very frequent operation of making a ship's mast, in which a square timber was transformed into a tapered cylinder. If x is the distance from the edge of the log to the point where the cut began, algebra gives 2(a - x) = √2 x, where the timber is d = 2a wide. From this, it follows that d/x = 2 + √2 = 3.4142. The corresponding distance from the middle is y = a - x, or d/y = 2(1 + √2) = 4.8284. To find the distances x and y, new scales were added similar to the 1/π scales in the preceding paragraph, but with ratios 1:3.4142 and 1:4.8284 instead. These were called, respectively, the E (edge) and M (middle) scales. These scales appeared on carpenter's rules until the 1930's. They were abandoned because no one made masts any more, not because the average carpenter could do the calculation some other way.

The Sector

This book encouraged me to find out how the sector worked, since a very good description of it is given in two places. The sector was a calculational device, a predecessor to the slide rule, that was in common use for over two centuries but is now almost totally forgotten. The sector was invented by Thomas Hood in 1597, and independently by Galileo in 1606, and used for the solution of specific problems. Also in 1606, it was proposed by Edmund Gunter as a general calculational tool. Like the later slide rule, it performs multiplication and division.

A sector is constructed like a two-fold rule, with linear scales radiating from the center of rotation, where there is a gauge point. The scales radiate because otherwise there would be no room for them. The sector is used in conjunction with dividers. The scales are graduated identically, so that it is easy to find equal distances along them to construct isosceles triangles. Some of the scales are often trigonometric and are graduated in angles. An actual sector would have many other scales for particular purposes, some of them like those on any rule.

In the diagram, AB and AC are corresponding scales of the sector. The normal method of use uses the similarity of the isosceles triangles EAD and CAB. The proportionality of the sides gives the relations shown. The sector essentially solves the Rule of Three problem graphically. This problem is to find the unknown quantity in a proportion when the other three quantities are known. Suppose AE and ED are known, and either AC or CB is to be found. The first step is to set the sector for the ratio AE/ED by adjusting the dividers to ED, and then opening the sector until the distance between points E and D at equal distances from A (as shown by the graduations) is equal to the divider setting ED. Then, if AC is known and CB is to be found, simply find the distance between points C and B with the dividers. If CB is known, set the dividers for this magnitude, and then find the points C and B at equal distances from A, again using the graduations. Then, AC is the desired quantity. All calculations with the sector are variations on this procedure.

The upper relation is used for division, the lower for multiplication. To perform either operation, one of the side is given a length of 1. For example, to multiply AE and CB, ED is made equal to 1, so that AC = AE CB. Suppose we wish to multiply a and b. We start by setting the dividers at length 1 on either scale. With one leg of the dividers at, say, point B such that AB = a, we open the sector until the other leg of the dividers falls on corresponding point E, where AE = a. Now, we set the dividers at distance b, and find the corresponding points C and B on the scales at equal distances from A. Then, we can read off the distance AC, which will be equal to the product ab. The reader can easily modify this procedure to find a/b instead.

Gunter's Line

Among his other many accomplishments, Edmund Gunter added a new scale, called a Gunter's Line, to the rule. Distances on this scale were proportional to the logarithms of numbers from 1 to 10, and the scale was repeated, so that there were two cycles. The reader can easily construct a similar scale and experiment with it. Multiplication and division were performed by adding or subtracting distances proportional to the logarithms by means of dividers. To multiply a and b, for example, the dividers were first set to the mantissa of a on the left half of the scale, one leg of the dividers on the gauge point (a brass plug to avoid wear on the wood of the rule) and the other on the graduation for a. Now the dividers were moved so that one leg fell on the point corresponding to the mantissa of b, and the other then fell at some point on the right half of the scale, which was the mantissa of the product ab. The exponents were determined as usual with logarithms. Note that repeating the scale meant that this would always work. I was very surprised to discover that Gunter's line was used, especially in navigation, to a very late date, and did not finally disappear until around 1900.

William Oughtred soon eliminated the use of the dividers by introducing a second Gunter's line on a slide, so that the logarithms could be added and subtracted directly by moving the slide instead of by transferring a distance with dividers. This, of course, was the beginning of the slide rule, to which the sliding cursor was later added to increase the flexibility of operation, and to which Mannheim added the trigonometric scales to make the modern engineer's slide rule. Many rules, however, continued to include Oughtred's slide. These original slide rule scales were designated A and B, as they still are on modern slide rules. The reason for the use of the first two letters of the alphabet, as well as for the double repetition, for these scales is now clear. On a modern slide rule, they are used for squaring and finding square roots, while the main job of calculation has passed to the C and D scales, which are single logarithmic cycles. Because of the cursor, they do not have to be double length, and the precision of a slide rule of a given length is doubled.

Other scales were added to Gunter's line to perform trigonometric operations, especially in connection with navigation (long before the Mannheim slide rule). The TAN scale gave natural tangents, for example, that could be read off with a square. The scale labeled S*T seems merely to be a tangent scale graduated with double angles, so that tan (θ/2) rather than tan θ is given. On all the illustrations in the book, the graduations on one scale are just double or half those on the other. The statement that they give "sines of tangents" (which would be meaningless in ordinary trigonometry), as on p. 165 and p. 217, is probably erroneous.

The Gunter's Line, Oughtred's slide and the sector were all used much later than I ever expected. Even early slide rules show distinct evidence of having been used with dividers. These all survived into the 19th century, used in their peculiar niches. Now the electronic calculator has swept all away.

Interesting Curiosities

I have always been accustomed to the numbering of a scale beginning on the left and increasing to the right (when the numbers are seen upright), and had hardly given it any thought, since it corresponds with the direction of writing. This, indeed, has always been the usual practice for British scales. However, remarkably, American scales were originally numbered in the reverse sense! This often serves as a criterion for the provenance of a certain rule, since American and British practices are otherwise extremely similar. The reason for this inversion is not known, it appears, though simple perversity is suspected. It is the stranger in that early American rule makers came from Britain, bringing their craft with them. The practice seems to have disappeared in the 1940's in boxwood rules, and earlier in other applications. When I looked at my Craftsman 39568 adjustable square, I found to my surprise that the scale on one side was right to left (the old American way) and on the other was left to right.

One of the trades in which special rules were used is that of gauging the liquid contents of a container, especially of the once widely-used wooden barrel. The capacity of a barrel was judged by inserting a pointed, graduated stick, the gauging rod, in the bung until the point was at the chime (where the side and end met). This simple method, using only one measurement, worked only for barrels of similar shape and had to be calibrated for the barrel type. A typical gauging rod was square, and had scales for four types of barrels on its sides. Unlike modern metal drums, wooded barrels were filled through a hole at the widest point of the side of the barrel, the bilge. This bung was stopped by a wooden plug hammered in. On p. 196, the chime or chimb, strictly the rim around a barrel head, is called the "chine" instead, which is really a backbone. On p. 131, where gauging is also being discussed, the word should be "frustum," not "frustrum," in the illustrations of barrel shapes (this is from the original article, not our author's fault).

It is not as easy to calculate the capacity of a barrel as it might seem. Hand-made barrels differed enough to make the measurement of capacity a good idea, which was the reason for the gauging rod. Generally, the formula V = πr2h is used, where r is an average radius usually estimated from the radii of the ends and the bilge as the radius of the end plus some fraction of the difference of the bilge and end radii. For an ordinary wooden barrel, this fraction seems to be about 0.7. At one time, this subject received a great deal of attention, which the book describes.

The amount a barrel lacks of being full is called ullage (from the French ouiller, "to fill") in Britain, and wantage in the United States. A wantage rod was simply a dipstick, inserted vertically in the bung. Similar rules were used for milk cans. This is only one example of many in the book of a trade whose study is encouraged by its use of rules.

I discovered that buttons were measured in units of 1/40", so that a No. 45 button would be 1-1/8" in diameter. These days, I fear, these traditional units are being replaced by the monotonous millimetre. Naturally, there were rules designed to measure buttons, and to show how large to make buttonholes. Watch crystals were also sized in unique units, and rules were made to measure in these units. A Geneva unit was 0.0864", and a size was expressed as units and sixteenths. This is another victim of the millimetre.

A Masonic rule is a two-foot, three-fold rule opening up flat that is graduated on one side from 0 to 24 to represent the hours of a day, and on the other side from 0 to 8 three times, to represent the division of the day into hours of sleep, work and leisure. It was not designed for measurement, but as symbolism.

The yard stick, once used in the sale of yard goods, is now just a metre stick with inch graduations on one side. The defining feature of a yard stick was its division into common fractions of a yard (as well as in inches).


Philip E. Stanley, A Source Book for Rule Collectors (Mendham, NJ: Astragal Press, 2003) ISBN 1-931626-17-0. Enquire at Astragal Press, 5 Cold Hill Road, Suite 12, P.O. Box 239, Mendham, New Jersey, 07945-0239, or at Astragal Press. The book is accompanied by a Concordance and Value Guide, of interest mainly to collectors. The current price is $45.00.

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Composed by J. B. Calvert
Created 22 April 2004
Last revised