Chapter 31 - The Boy Tar by Mayne Reid
“Quod Erat Faciendum”
To find the cubic contents of the butt in feet or inches, and afterwards reduce them to liquid measure—to gallons or quarts—would have been easy enough, and only required a simple computation in figures. I knew that I was arithmetician enough to make this computation, even though I possessed neither pen nor paper, slate nor pencil; and if I had, there was no light by which I could have used them. “Ciphering,” therefore, in the ordinary way, was out of the question; but I had often practised myself in casting up accounts by a mental process, and I could add and subtract, multiply or divide a considerable series of figures without the aid of either pen or pencil. The problem I had before me would involve but a limited number of figures, and I felt satisfied I could easily manage it, so far as that was concerned.
I have said that it would have been a simple and easy computation to find the contents of the cask in cubic feet or inches. Would have been supposes that there was a difficulty—and there was one. An important preliminary matter had to be settled before I could enter upon any calculation—a very important one; and that was, that I had not yet reduced my measurements—neither the diameters nor the length—to feet and inches! I had measured the cask with plain pieces of stick, and had registered the dimensions in simple notches; but what of this? I knew not what distance these notches might be from the end, or from each other—how many feet or inches! I might make a rude guess, but that would be of no service to me; so that after all my pains I had as yet no data to go upon, nor could I have any until I had first measured my measuring-rods themselves!
Apparently, here was a difficulty not to be got over. Considering that I had no standard of measurement within reach—neither yard-stick, nor foot rule, nor graduated scale of any kind—you will naturally conclude that I must have abandoned the problem. A computation founded on the mere length of the stick would have been absurd, and could have given me no information whatever upon the point about which I wanted to be informed. To find the cubic and liquid contents of the cask, I must first have its length, with its largest and shortest diameters, expressed in standard terms—that is, either in feet or inches, or some other divisions of a scale.
And how, I ask, was this to be ascertained, when I possessed no standard of measurement about my person? None whatever. I could not make one; for in order to do so, I should have required another for a guide. Of course, I could not guess the length either of feet or inches.
How, then, was I to proceed?
Apparently, the difficulty was not to be got over. The thing seemed impracticable.
To you it may seem so, but it did not to me. I had thought of this before. I should not have proceeded as far as I had done—taking so much pains and trouble with the splitting and splicing of my sticks, and making my measurements so exact—had I not foreseen this difficulty, and thought of a way to surmount it. All this had been prospectively arranged. I knew before-hand that I could measure my sticks, and tell their linear dimensions to the exactness of an inch.
“How?”
Thus, then—
When I said just a little ago that I had no standard about my person, I spoke the truth only literally. Although not exactly about my person, I had one in my person—I was myself that standard! You will now remember my having submitted myself to a measurement, which showed me to be four feet in length. Of what value that knowledge now proved to me!
Knowing, then, my own height to be very nearly four feet, I could notch off that measure upon one of the sticks, which would give me a measuring-rule of four feet in length.
I proceeded to obtain this result without delay. The process was simple and easy. Laying myself horizontally, I planted my feet against one of the great ribs of the ship, and rested the end of the stick between them. I now stretched myself out at full length, and guiding the rod so as to keep it parallel to the axis of my body, I brought it across my forehead, and beyond. With my fingers I could tell the point that was opposite the crown of my head, and carefully marking this point, I afterwards notched it with the knife. I now possessed a four-foot rule, exact enough for my purpose.
But there were difficulties yet to be encountered. With a four-foot rule, I was but little advanced towards my computation. I might make a nearer approach to the measurement of my diameters, but that would not avail. I must know them exactly. I must know them in inches, and even fractions of inches; for, as I have said, an error of half an inch in some of my data would make a difference of gallons in the result. How, then, was I to divide a four-foot stick into inches, and register the inches upon its edge? How was this to be done?
It seems simple enough. The half of my four feet—already ascertained—would give me two feet; and the half of that again would reduce the standard to a foot. This again notched in the middle would make two lengths of six inches each. Then I could subdivide those into lengths of three inches, which, if not small enough for my calculation, could be still further subdivided into three equal parts, each of which would be the desired minimum of an inch.
Yes, all this seems easy enough in theory, but how was it to be put in practice upon a piece of plain straight stick, and in the midst of as perfect darkness as that which surrounds a blind man? How was I to find the exact middle—for it must be exact—of even the four feet, much less divide and subdivide till I got down to the inches?
I confess that I was puzzled for awhile, and had to pause and reflect.
Not very long, however; I was soon able to get over this trifling obstacle.
The plan that first suggested itself was to cut a third piece of stick of a little over two feet in length, which I could easily guess at within a few inches. This I could apply alongside of my four-foot rule, beginning at the end, and proceeding as if I was measuring the latter with the former. Of course, on the first application, two lengths would reach from the end of the rule to the notch that marked the four feet length, and perhaps extend a little beyond. I should then shorten the measure and apply it again. This time its end would have approached nearer to the aforesaid notch. Another bit cut off would bring it still nearer; and the process being repeated, by shaving gradually from the end of the stick, I should at last find that two lengths of it would exactly correspond with the length of my four-foot rule. I should then have a piece exactly two feet in length, and by the help of this I could find the middle part of the longer piece, and could mark it with a “nick.”
By cutting the short piece into two nearly equal parts, I could then take the larger of them, and, by a similar process, obtain the standard of a foot, and mark it also upon my rule; and so on till I had succeeded in arriving at the inches.
Of course, to do all this would require time, patience, and the nicest precision; but I had plenty of time upon my hands, and it was my interest to be both patient and precise.
Although I regarded not the time, just as I was about to carry out the plan described, another suggested itself that promised to lead me sooner to the issue; it would call for less patience, though an equal amount of precision.
This new plan was a sort of corollary of the former one, the only difference being, that instead of a stick I should perform my subdivision and graduation with a string.
The thongs of my buskins came into my mind—the very thing!
I could not have found a better string for the purpose. They were strips of best calfskin, cut with the grain, and could not have been stretched the eighth part of an inch. They would, therefore, measure as accurately as a rule of boxwood or ivory.
One would not be long enough; so I knotted the two together, taking care to make a neat, firm knot of it. They made a string of over four feet, and having laid it along the four-foot rule, I cut it with my knife to that length exactly. I was not satisfied till I had measured it over and over again, each time pulling the thong with all my strength, lest some “kink” might be lurking in it. A slight error would derange my intended scale, though there is less danger in graduating four feet down to inches than in going from the less to the greater standard. In the former, each subdivision naturally lessens the error, while in the latter it is continually doubled.
When convinced that I had got the thong to the precise length, I placed its two ends together, and then drawing it with a firm pull through my fingers, I creased it exactly in the middle. Holding it taut upon the blade of my knife, I cut through at the crease, and thus divided it into two moieties of equal length, each two feet long. The part with the knot I laid aside as being no longer needed, and the remaining half I again doubled, and cut into two. This gave me two pieces each a foot in length.
One of these I next folded in triple, and creased for cutting as before. This was a delicate operation, and required all the skill of my fingers to accomplish, for it is much easier to divide a string into two equal parts than into three. I was a good long time before I could get it trebled to my satisfaction; but I succeeded at length, and then severed the parts.
My object in thus cutting into three, was to get the pieces in even fractions of four inches each, in order that by two more doublings I might arrive more accurately at the inch.
And in two more doublings I found it.
To make sure that I had committed no error, I took up the knotted piece, which I had laid aside, and after placing the other fragments where they could be got at, I reduced the second half of the string as I had done the first.
To my gratification, the inch I obtained from both exactly corresponded. There was not a hair’s breadth of difference.
I was now in possession of a guide to the true graduation of my measuring-stick. I had pieces of one foot, of four inches, of two, and of one; and by the help of these I proceeded to mark my rod after the manner of a draper’s yard-stick.
It occupied some time, for I worked with care and caution; but my patience was rewarded by finding myself in possession of a measure upon which I could rely, even in a calculation involving the question of my life.
I was not much longer in deciding the point. The diameters were now measured by feet and inches, and the mean of the two taken. This was reduced to surface measure by the usual method of squaring the circle (multiplying by eight, and dividing by ten). This gave the base of the hollow cylinder, which would be equal to the frustum of a cone of like altitude; and another multiplication by the length produced the entire cubic content.
Dividing by sixty-nine, I got the number of quarts, and so gallons.
The butt, when full, had contained somewhat above 100 gallons—as near as I could calculate, about 108—and therefore it was in all likelihood an old pipe that had once contained sherry.