The discourse made before the Royal Society the 26. of November, 1674, concerning the use of duplicate proportion in sundry important particulars together with a new hypothesis of springing or elastique motions / by Sir William Petty, Kt. ...

The Second Instance Is in the Strength of Timber, &c.

LEt there be Square Rods or Pieces made of any Clean Timber, or other Ma∣terials, whose Ends let be supported with conve∣nient Blocks or Fulcra: These Rods in Experience will bear weight hung in

the middle of them, ac∣cording to the proportion of their lengths or distance, between the Fulcra; that is to say, a Rod A. being of double length to the Rod B. will bear ½ the weight which B can bear; and be∣ing of triple length, it will bear one third; & sic de caeteris. Again, let two of those equal and alike square Rods be placed one upon the other (so as to touch and sit,) then the two together shall bear 4 times

as much as one alone, and three of them, placed as a∣fore-said, shall bear nine times as much, and so on in proportion of Roots to Squares. Again, lay the same two Rods side by side, to each other, then they shall bear but double, three shall bear triple, and so forward, in Arithmetical proportion. From whence it follows, that four of them placed square, shall bear eight times as much as one alone. But if the same four

Rods taken as One, being of double length making an Octuple quantity to One, they shall bear but four times the weight of One alone. So as two like pie∣ces of Timber, that are in cubical or triplicate pro∣portion of their Sides, are strong but according to duplicate proportion, or the Squares of their respe∣ctive Sides; and conse∣quently, to have like Vessels (differing in Content as the Cubes of their like

Sides) equally strong, the Timber of which they con∣sist must be Quadrato-qua∣dratic; that is to say, a Ship of 400 Tuns, equally strong with one of 50, must have not only 8 times as much Timber in it, but 16 times; which is seldom or never done. Which de∣fect is the true Reason, why great Shipping is both Dearer and Weaker than small Shipping, (no Ship in the world being so strong as a Nutshel;) I say,

Weaker, for what is here said; and Dearer, for what shall be said hereafter in the sixteenth Instance of Masts, Diamonds, &c. And on the other hand, if the Timbers were Quadrato∣quadratic, then the Ship of 400 Tuns would be loaden with her own Ma∣terials; if the Ship of 50 Tuns were not over-tim∣bered.

Now, for not well un∣derstanding these matters, many men designing En∣gines

of strength, do make Models of such Machina∣ments by a Scale (suppose wherein an inch represents a foot,) by which the Mo∣del is the 1/1728 part of the En∣gine intended: And there∣upon they conceive, that if the Model be strong e∣nough to bear 1/1728 part of what the great Machina∣ment is intended to bear, that then the said great Ma∣chinament will be strong enough. Whereas indeed the Model must bear the

full 1/144 of what is intended for the great Machinament; otherwise great mischiefs will appear in the Work. Wherefore the Square of the Linear Difference be∣tween the Model and En∣gin, is the measure and way of trying the strength and sufficiency sought for: The ignorance whereof hath made many a poor Proje∣ctor. Upon these Princi∣ples, a Cask which will hold a Tun, ought to have 16 times as much Timber in

it, as the Cask which holds onely a Barrel, or ⅛ of a Tun; provided one be as strong as the other (which is not usually seen.) For the bigger Vessels, Carts, &c. they are usually the weaker compar'd with the strength of the lesser; which appears also in Animals, whose strength is as the Square Roots of their weights and substance, viz. if 1728. Mice were equiponderate to one Horse, the said Horse is but 1/144 part as strong as

all the said Mice.

From these considerati∣ons the Scantlings of Tim∣ber in Buildings must be adjusted; as for example, Let the Walls of any Room be infinitely, that is, suffi∣ciently strong; let the length and the breadth of the Room be given: Next, suppose the Room is to be made so strong, as that eve∣ry foot and a half square shall bear a Man, and so, that 31½ square feet should bear a Tun weight, (rec∣koning

14 men to the Tun:) Lastly, let the strength of the Timber be also given. Now the Que∣stions are, to find the Scantlings of the Girders, Gise, &c. first in square pieces, and afterwards by altering the Squares into more advantageous ablong Sizes; as for example, Let the Room be supposed 26 foot long and 20 broad, viz. 520 foot in the Area, and able to receive about 250 men, and to bear a∣bout

16 Tuns. Suppose the Timber be such, as whereof a Rod of an inch square, and 20 foot long, will bear 1/20 part of an hun∣dred weight; or, that 20 such Rods, or a Board of 20 inches broad, and 20 foot long within the walls, an whole hundred weight; and so the whole Floor con∣sisting of about 16 such Boards, but 1600. Now if the same Board were planck of 4 inches thick, it would bear 16 times 1600 or 256

hundred weight: If 5 inch∣es, 400 hundred weight: But the whole weight de∣signed being but 325 hun∣dred, some size between 4 and 5 inches thick will suffice in this case, where we suppose the Floor to be of planck without Gise or Girder. Next, suppose in∣stead of this Planck there be used Gise of double thickness to the said Planck, and placed at qua∣druple distance; I say, the Effect and Strength will be

the same with half the stuff. And I also say, that one Girder alone of 18 inches square, and 20 foot long, is near Equivalent to the 17 Gises of 9 inches deep, and 4½ broad-abovemen∣tioned; which Girder has but half the stuff which the Gise had; as the Gise did contein but half the stuff, which the 4½ inch-Planck first mentioned did con∣tein. Which saving of stuff is the reason of divi∣ding Plank into Girders,

Gise, and Board. Where note, that these Proporti∣ons and Scantlings are not offered as exact and best for practice, but onely to intimate the method of in∣quiring into these matters so useful in the world.