On researching log scalers, it seems that many have multiple sets of numbers.
Here are examples of two modern log scale sticks.
The rule used on this stick is the Doyle Rule. First measure the length of the log. Then place the rule against the log along its diameter. It will tell you how may board feet there are in a log of that length. Now the diameter of the log in inches is on the edge of the rule. The first set of numbers after the length of the log represent a diameter of 6 inches (5, 3, 2) and each set of numbers is one inch apart so the next set represent 7 inches diameter (10, 6, 5), etc.
So a 10 foot long log of diameter 7 inches would contain 6 board feet.
The Doyle Rule is quite a simple equation. D is the diameter in inches and L is the length in feet.
The number of Board Feet (BF) is then = ((D-4)^2 * L)/16
So for our 7 inch diameter and 10 foot long log:
BF=((7-4)^2 *10)/16 = (3^2*10)/16 = 90/16 = 5.6 - this is rounded up to 6 board feet.
Now according to THIS report: "This formula says to subtract four inches from the diameter for slabs and edgings, square the result, and adjust for log length. Log taper is ignored. The edgings and slab allowance are too large for small logs and too small for large logs. As a result, the rule seriously under scales small logs and over scales large logs."
But our log scaler seems quite different. There does not seem to be a row of numbers that represent the length of the log and the numbers do not seem to grow at a rate indicative of a length being implied in the equation.
This the data taken from the one in the Museum collection.
Now there are hundreds of rules, though only a handful are used today. The rule used varies by tree type and different states adopt different standards.
I think that this rule is measuring the number of board feet in 1 foot of a log of a diameter D. The three rows are different rules. On examination of the numbers it seems likely the bottom row is probably the Doyle Rule. The middle row seems to be very close to the Spaulding Rule, at least for smaller diameters. This is a rule that is used for redwoods with an adaptation, and in 1878 it was adopted as the statue rule by California and is sometimes called the California Rule.
This is the Spaulding equation:
BF = ((1-0.266)*((pi*D^2)/48) - 2) * L
Now the above equation is just an approximation to the Spaulding Scale. In other words Spaulding created a rule in 1868 - it is based upon diagrams of logs with diameters from 10 to 96 inches, and lengths 12 to 24 feet long. Later, McKenzie fitted an equation to the data. It is the approximate equation above.
You can view the book Volume table for fir, cedar, spruce timber ..., based upon Spaulding log rule by G. G. Johnson HERE.
The top row is likely the Brereton Rule, also used for redwoods and Douglas fir. The Brereton equation is:
BF = 0.6545 Da^2*L
Where Da is the average of the two end diameters.
Here is a table with the comparison of what is actually on the rule on the left, and the results of calculating the three equations on the right for the diameter in inches.
Here is a table comparing the middle row of the log scaler to the entries in Spaulding's Scale Rule table for 16 and 32 foot logs.
You can read more about the different log rules in the technical report A Collection of Log Rules by the USDA Forest Service HERE.
You can read about how inaccurate the rules are in the pamphlet A discussion of Log Rules, prepared by McKenzie, HERE.
From the Santa Cruz Evening Sentinel, July 27, 1897, courtesy of newspapers.com. I had read this article so many times but, until now, had not appreciated the Spaulding Rule.