From Thomas Paine
I enclose you a Problem not about Bridgs but Trees, and to explain my meaning I begin with a fountain. The Idea seems far fetched, but fountains and Trees are in my walk to Challiot.1
Suppose Fig. 1st a fountain. It is evident 1st. That no more water can pass thro the branching Tubes than pass thro the trunk.
2d. That admitting all the water to pass with equal freedom, the sum of the squares of the diameters of the two first branches must be equal to the square of the diameter of the Trunk, also the sum of the squares of the four Branches, must be equal to the two, and the sum of the squares of the 8 Branches must be equal to the four, and therefore the 8, 4, 2, and the Trunk being reciprocally equal the solid content of the whole will be equal to the Cylinder Fig 2d of the same diameter of the trunk and height of the fountain.
Carry the Idea of a fountain to a Tree growing, Consider the sap ascending in Capillary tubes like the Water in the fountain, and no more sap will pass thro the Branches than pass thro the Trunk.
2dly. Consider the Branches as so many divisions and subdivisions of the Trunk as they are in the fountain, and that their contents are to be found by some rule, with the difference only of a Pyrimidical figure instead of a Cylindrical one.
Therefore to find the quantity of Timber (or rather loads,) in the Tree, figure 3d, draw a Pyramid equal to the height of the Tree as Fig. 4th, taking, for the inclination of the Pyramid, the diameter of the Bottom, and at any discretionary height above it which in this is as 3 & 2.
As sensible Men should never guess, and as it is impossible to judge without some point to begin at, this appears to me to be that point, and by which a person may ascertain near enough the quantity of Tember and loads of Wood on any quantity of Land, and he may distinguish them into Timber, Wood and faggots.
Yours &c. T P
RC (DLC); endorsed. On the problem of dating this memorandum, see note to preceding letter.
1. Thus in MS. Paine meant Chaillot, a fact which establishes his presence in Paris at the time of writing this letter.