Surveyors’ Calculation Methods Used BC

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In the dark days BC (Before Calculator), surveyors invented and adapted methods which would lessen their calculations. Here are some interesting relationships, known only to old-guy surveyors.

To Find D on a Railroad Curve
"Stringlining" in railroad surveying refers to a method to realign the tracks. Due to unbalanced forces, the tracks in curves tend to creep and shift, making the middle ordinates (m) for equal-length chords (c) along the curve not equal, as they should be for a circular curve. (See the sketch.) So, the tracks can be "stringlined" or literally moved to make the curve circular, by forcing the m’s to be equal for the same c. To see the amazing process of how this is physically done, go to YouTube and search for "Gandy Dancer." (There’s a similar process for the spiral-curve portion of a track, but that’s for another day.)

For the chord definition of Degree of Curvature used by the railroads, a well-known and very close approximation is: For a 62-foot chord, 1° Degree of Curvature yields 1 inch of Middle Ordinate (distance, m, from the chord midpoint to the curve midpoint). You remember, m = (c2D)/45,840 (m and c in feet). And, m and D are proportional.

So, using this relationship, should you need to measure the D of a railroad curve, here’s the procedure: On the inside of the rail, measure and mark two points (A and B in the sketch) 61.80 feet apart (a chord). Most references say measure a chord of 62 feet. Admittedly this 0.20 foot chord difference will yield a near-zero difference in m, especially for these "small-D" railroad curves, so, there’s not much wrong with the old-school railroad dimension of 62 feet.

Now tightly stretch a string between A and B and measure m (midpoint chord to midpoint curve) in inches. That dimension is the Degree of Curvature. (2.4 inches, for example would mean 2°24′ Degree of Curvature.)

Remember, most railroad curves use spirals (or, transition curves) to "introduce" the main, circular portion of the curve. In the curve system, the sequence will be: Tangent, spiral, circular curve, spiral, tangent. Of course this method would be to determine D of the circular part. See sketch.

Use of Versine
Remember versine, or, versed sine? (Not to be confused with its cousins coversine, covercosine, hacovercosine and excosecant!) Well, versine  = 1 – cosine . To illustrate it, I’m sure you will remember the "Unit Circle," and versine being sort of like the "Middle Ordinate" on that circle. No matter. To surveyors, its use is in taping slope distances.

We can apply versine in two ways to this slope measurement (see the sketch): To compute the horizontal distance taped; and, to compute how much must be added at B to lay out a distance of 100.00 feet. All you do is look up versine 8° in the tables (0.0097319), move the decimal place two to the right (or, multiply by 100.00) and that is the amount, 0.97 to be subtracted from the slope distance to get the horizontal distance, or, add at B to reach 100.00 on the horizontal. See sketch.

The Chain and Acre
The English clergyman and mathematician Edmund Gunter (1581­1626) invented the Gunter’s chain (along with other measuring instruments), being a dimension of 66 feet, and further divided into 100 links. So, the chain unit is decimal. (One chain is also 4 poles or 4 rods or 4 perches.) English and American surveyors have been using it ever since. Our Public Land Survey System (USPLSS) is rooted in the chain. And, the unit, the chain, contains a convenient conversion factor because length in chains times width in chains divided by 10 equals area in acres.

An example: A perfect section is a square, 80.00 chains on a side. Its area: 640 acres. By the way, in our USPLSS, there’s no such thing as a "perfect section," and the GLO draftsman took liberties with their calculations. In my neck of the woods (GLO plats drawn in 1820’s­1840’s from field work conducted 1816­1840’s), the acreages shown are mostly a nominal "640 acres," although the measured GLO dimensions shown on the plat for the section do not support that number. Yes, sometimes when the east-west GLO dimensions differ widely from 80.00, the acreage shown has been calculated by averaging the two east-west dimensions and multiplying by 80 and moving the decimal point one place to the left. But, this is not always the case. In some instances, the "640 acres" shown is far from what could have been calculated from the GLO dimensions shown. (But, I digress.)

Using dimensions measured in Gunter’s chains to compute the area in acres sure was an improvement on an approximate method then in use. That method for irregular-shaped tracts was to sum the sides of the tract (compute its perimeter). Then divide the perimeter by 4 and square that number. This method is exact for a square, but the further the length to width ratio for the figure deviates from 1.0 the more approximate is the result. Or, said another way, the further away from a square the figure departs, the more approximate the result. As a learned observer of the period put it: "It is the way all Surveyors do; whether it originates in idleness, inability or want of sufficient pay…." Sound familiar? Have you ever employed a shortcut for idleness, inability or want of sufficient pay? Tell the truth….

Is there a modern use of the unit, the chain? Yes. As a surveyor, you may get a call from the local cricket club asking you to lay out its "stumps." Take your chain. (You can Google what that means.)

To read about Mr. Gunter and his chain and much more, I recommend: "English Land Measuring to 1800: Instruments and Practices" by A.W. Richeson (1966) and "Measuring America" by Andro Linklater (2002).

Number of Square Feet in an Acre
Can’t remember the number of square feet in an acre? No problem, here’s how to compute it: Write down any three different integers. Reverse their order. Calculate the difference between the two numbers. Now reverse the order of that number and add them. Multiply that number times 40. The resulting answer will always be 43,560. No matter what three integers you chose initially. Here’s an example: See graphic.

Geez! I think I’ll just remember 43,560. But, it’s a pretty good math trick, used to impress your friends and neighbors!

Know of any other "surveyor math tricks?" If so, I’d like to hear of them:

Dick Elgin, PhD, LS, PE is practitioner (sealed about 15,000 surveys), educator and seminar-giver, author (five books), researcher (coauthor "Sokkia Ephemeris") and veteran (Vietnam). Now semiretired, he works for ArcherElgin Surveying in Rolla, Missouri. He rides a Moots bicycle and drives an Alfa Romeo GT 1600 Junior. Current project: A monograph on riparian boundaries.

A 6.155Mb PDF of this article as it appeared in the magazine—complete with images—is available by clicking HERE