Counts to Calculation
Theoretical computations of tree volume began to appear in the 1760s. In the first definitive work of scientific tree measurement (Holzmesskunde ), Carl Christoph Oettelt's Practial proof that mathematics performs indispensable services for forestry , the problem of estimating the quantity of wood on a tree without felling it figured prominently. Oettelt was an experienced surveyor and had held the title of "Forest-Geometer" in the civil service of Saxony-Gotha before taking over the forest department in Ilmenau, where he would later serve under Goethe. In the Practical proof , Oettelt criticized the crude techniques commonly used to estimate the quantity of wood. Most foresters used the so-called Bruststärke , or a stack of wood piled to chest height, to veil their wild guesses as to how many boards a tree had delivered. Estimating in this way, they commonly made the value of a tree proportional to its diameter. Heinrich Wilhelm Döbel, one of the most conscientious writers on forestry around 1750, exemplified the problem. In his influential Gamekeeper's practicum , Döbel struggled to find a simple computation for the problem—in fact, relatively easy—of estimating the volume of a felled trunk. Oettelt invoked geometry: "A tree is the same as a cone with a circular base." With the appropriate formula for the volume of a cone, calculating the volume and mass of trees was not so troublesome.
Oettelt's treatment of wood mass as a mathematical quantity was a radical departure. The holzgerechte Jäger had shown little potential for forest geometry. Döbel argued vehemently that exact calculations of wood mass were unnecessary, "since you don't measure wood like
you do gold." He preferred the simple "farmer's calculation" to disputations and proofs. The mathematically oriented foresters, among them Johann Vierenklee and Carl Wilhelm Hennert, joined Oettelt's cause. They corrected and improved his geometric calculations in a series of books that culminated in 1812 in the definitive work on forest stereometry by Johann Hossfeld. As abstract, mathematics-based forestry gained sway during the 1780s and 1790s, compilations of tables based on controlled measurements replaced the older crude techniques described by Oettelt.
Those who compiled such tables had to bridge the gap between tree conics and precise measurement. Consider the problem of converting from cubic measures of wood mass to Klafter , the unit of stacked cordwood familiar to the forester, and back again. The interstices and warping of real wood might defeat the most exact geometrical analysis of its volume. Since mass or volume constituted the central quantity of the new forestry science, small errors due to branches, warped stocks, and imperfections of nature multiplied rapidly as one reasoned from the tree to the forest. Equating the economic measure—volume of stacks of hardwood—and the computed volume did not work out.
The quantifiers, beginning with Oettelt and Hennert, searched for scientific sandpaper to achieve a greater semblance of precision. Oettelt measured as accurately as possible the volume of the cord, then ordered the wood chopped into small pieces. The volume of each piece could be measured with greater accuracy. He summed these individual measurements, and compared the sum to the original cord. After repeated tests he determined that a typical span of cordwood measuring approximately 110 cubic feet contained 14 to 18 cubic feet of empty space, about 15 percent of its volume. Hennert borrowed Diogenes' barrel: he poured water into a box filled with wood; the volume of the box less the volume of the water yielded the solid content of wood (Derbgehalt ). By 1812 Hossfeld, in his Lower and higher practical stereometry (1812), had replaced Hennert's water with sand and contrived even more accurate xylometers. Such innovations made feasible "measurement and calculation of all regular and irregular bodies, and especially trees in the forest."
In the German tradition, the mathematician's forest was populated not by the creations of undisciplined nature, but by the Normalbaum . Forest scientists planted, grew, and harvested this construct of tables, geometry, and measurements in their treatises and on it based their calculations of inventory, growth, and yield. Writers and instructors gave foresters in the field the tools for reckoning the dimensions of the standard tree. Most treatises contained instructions for averaging measurements made on a test plot, but foresters were happier to use the Normalbaum . Tables of numbers representing measurements and calculations, or Erfahrungstabellen , provided data organized by classes of trees under specified conditions. A small number of variables governed the forester's choice of one or another of these tables. For example, the wood mass of the typical sixty-year-old pine on good soil was given as a function of its height and circumference. These tables, which appeared in every complete manual of rational forestry
practice, generally did not bother with regional variation, the bugaboo of 18th-century agricultural treatises.
By the end of the 18th century, German writers on forest management had worked out steps for determining, predicting, and controlling wood mass. Heinrich Cotta presented the clearest and most widely read exposition of these steps in his Systematic instruction for the assessment of woods , published in 1804; they were elaborated in his Directions for the organization and assessment of the forest , which appeared sixteen years later. Cotta's first book, which consisted of lectures originally prepared for students attending the forestry school under his direction, was an example of systematization induced by the necessity of teaching. In his method, the "geometric survey" of the woods supplied the Taxator with information about the extent of his forests. The next step required calculations of wood mass of individual trees, then of stands, and finally of the forest as a whole; growth rates were computed for each level of organization. Finally, Cotta's forester qua cameralist linked the forest balance sheet to the monetary budget by determining the value of the yield.
If the standing forest is capital and its yield is interest, the forester can complete the chain of conversions from wood to numbers to units of currency: an estimate for the worth of the forest can thus be used to predict income, calculate taxes, assess the worth of the forest, or determine damage to it resulting from a natural disaster. For Cotta, the fundamental problem of forestry management was determining the "standing value" of a forest, given uninterrupted maintenance costs and full harvest some 100 to 150 years hence. Cotta's forestry science thus consisted of sound methods for inventory and prediction: "From summary investigations based entirely on verified
judgment, we go through various stages to more exact investigations, first of individual trees, then of the supply, growth, and yield-determination of individual stands, and finally of whole forests." Similar procedures, from the forest to the tree and back again, also appeared in practical manuals such as Georg Hartig's New instructions for the Royal Prussian forest-geometers and forest-assessors .
In one respect, Cotta differed from Oettelt's line. He preferred careful ocular estimates based on tables to geometrical deduction, which he not only considered impractical in the field, but also inaccurate, since branches and other irregularities confound the comparison of trees to cylinders and cones. For Cotta, the only absolutely sure method was to chop up a tree and measure its volume (or mass) in the same unit of measure to be adopted in the taxation itself. This view did not weaken his allegiance to mathematical forestry. He was skeptical only of geometrical estimates, not of quantification.
The Forstwissenschaftler , and particularly Cotta, championed use of "experience tables." Their use reinforced the notion of a forest filled with standard trees. The forester was to instruct his assistants in the use of these tables so that a mental picture of a tree encountered in a forest corresponded to an entry in the tables. With sufficient repetition, a good forester could make an instant association from the mental picture triggered by the tree to the value of the wood mass contained in the table. The next step was to generalize: every tree of the same height has the same mass (or volume). The standard forester was trained to find the standard tree. For Cotta, the "eyeball measure" could displace the "measuring hand" if every forester learned to see the archetypical. The practiced eye could indeed attain this mechanical perfection, "as subsequent measurements and calculations prove[d]."
The head forester thus trained his assistants to internalize Erfahrungstabellen and become computers of wood mass. He remained at his desk manipulating the Normalbaum and numerical data based on local measurements. He could produce his own tables if necessary; according to Hartig, the Taxator was responsible for all "mathematical preliminaries" of forest assessment—determining growth rates, preparing maps and calculating tables—before delegating to his staff routine measurements and the mechanical application of tables. The assistants marched, tallied, catalogued, and marked under the watchful gaze of their supervisor, who—according to Hartig's directions—never counted with them. Instead, his duty was to "dictate principles, record the results in the Assessment-Register, and make sure that there are no mistakes."
By 1800, the forest assessor trained in the cameral sciences specialized in theoretical principles, mathematical preliminaries, and the cumulation and analysis of data, a far cry from Beckmann with his colored nails and squad of assistants. An array of numbers stood for the quantity of wood in the forest. The forester or cameralist trained in forestry science felt no need to step off every acre with the exactness given to the test plot, the geometrical abstraction, or exact measurements of the volume of cordwood. Instead, he could sample and generalize. The work of the assessment and management of the forest thus required only standard trees and Erfahrungstabellen . As Cotta argued, the crucial quantities of his science were "determined mathematically" from the "premises" of forestry science, not through "direct real measurement ." The scientific forester had abandoned Beckmann's empiricism in favor of "sure mathematical deductions, experiments and experiences in the given and understood units of measure." Under the banner of Wissenschaft , the new breed of qualified forester breathed the quantitative spirit into administrative practice.