by Mayne Reid
CHAPTER THIRTY ONE.
"QUOD ERAT FACIENDUM."
To find the cubic contents of the butt in feet or inches, and afterwardsreduce them to liquid measure--to gallons or quarts--would have beeneasy enough, and only required a simple computation in figures. I knewthat I was arithmetician enough to make this computation, even though Ipossessed neither pen nor paper, slate nor pencil; and if I had, therewas no light by which I could have used them. "Ciphering," therefore,in the ordinary way, was out of the question; but I had often practisedmyself in casting up accounts by a mental process, and I could add andsubtract, multiply or divide a considerable series of figures withoutthe aid of either pen or pencil. The problem I had before me wouldinvolve but a limited number of figures, and I felt satisfied I couldeasily manage it, so far as that was concerned.
I have said that it _would have been_ a simple and easy computation tofind the contents of the cask in cubic feet or inches. _Would havebeen_ supposes that there was a difficulty--and there _was_ one. Animportant preliminary matter had to be settled before I could enter uponany calculation--a very important one; and that was, that I had not yetreduced my measurements--neither the diameters nor the length--to feetand inches! I had measured the cask with plain pieces of stick, and hadregistered the dimensions in simple notches; but what of this? I knewnot what distance these notches might be from the end, or from eachother--how many feet or inches! I might make a rude guess, but thatwould 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 mymeasuring-rods themselves_!
Apparently, here was a difficulty not to be got over. Considering thatI had no standard of measurement within reach--neither yard-stick, norfoot rule, nor graduated scale of any kind--you will naturally concludethat I must have abandoned the problem. A computation founded on themere length of the stick would have been absurd, and could have given meno information whatever upon the point about which I wanted to beinformed. To find the cubic and liquid contents of the cask, I mustfirst have its length, with its largest and shortest diameters,expressed in _standard_ terms--that is, either in feet or inches, orsome other divisions of a scale.
And how, I ask, was this to be ascertained, when I possessed no standardof 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. Ofcourse, 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 seemedimpracticable.
To you it may seem so, but it did not to me. I had thought of thisbefore. I should not have proceeded as far as I had done--taking somuch pains and trouble with the splitting and splicing of my sticks, andmaking my measurements so exact--had I not foreseen this difficulty, andthought of a way to surmount it. All this had been prospectivelyarranged. I knew before-hand that I _could measure_ my sticks, and telltheir 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, Ispoke the truth only literally. Although not exactly _about_ my person,I had one in my person--I was myself that standard! You will nowremember my having submitted myself to a measurement, which showed me tobe 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 notchoff that measure upon one of the sticks, which would give me ameasuring-rule of four feet in length.
I proceeded to obtain this result without delay. The process was simpleand easy. Laying myself horizontally, I planted my feet against one ofthe great ribs of the ship, and rested the end of the stick betweenthem. I now stretched myself out at full length, and guiding the rod soas to keep it parallel to the axis of my body, I brought it across myforehead, and beyond. With my fingers I could tell the point that wasopposite the crown of my head, and carefully marking this point, Iafterwards 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-footrule, I was but little advanced towards my computation. I might make anearer approach to the measurement of my diameters, but that would notavail. I must know them _exactly_. I must know them in inches, andeven fractions of inches; for, as I have said, an error of half an inchin 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 registerthe 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 thestandard to a foot. This again notched in the middle would make twolengths of six inches each. Then I could subdivide those into lengthsof three inches, which, if not small enough for my calculation, could bestill further subdivided into three equal parts, each of which would bethe desired minimum of an inch.
Yes, all this seems easy enough in theory, but how was it to be put inpractice upon a piece of plain straight stick, and in the midst of asperfect darkness as that which surrounds a blind man? How was I to findthe exact middle--for it must be exact--of even the four feet, much lessdivide 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 triflingobstacle.
The plan that first suggested itself was to cut a third piece of stickof a little over two feet in length, which I could easily guess atwithin 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 latterwith the former. Of course, on the first application, two lengths wouldreach from the end of the rule to the notch that marked the four feetlength, and perhaps extend a little beyond. I should then shorten themeasure and apply it again. This time its end would have approachednearer to the aforesaid notch. Another bit cut off would bring it stillnearer; and the process being repeated, by shaving gradually from theend of the stick, I should at last find that two lengths of it wouldexactly correspond with the length of my four-foot rule. I should thenhave a piece exactly two feet in length, and by the help of this I couldfind 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 thentake the larger of them, and, by a similar process, obtain the standardof a foot, and mark it also upon my rule; and so on till I had succeededin arriving at the inches.
Of course, to do all this would require time, patience, and the nicestprecision; but I had plenty of time upon my hands, and it was myinterest to be both patient and precise.
Although I regarded not the time, just as I was about to carry out theplan described, another suggested itself that promised to lead me soonerto the issue; it would call for less patience, though an equal amount ofprecision.
This new plan was a sort of corollary of the former one, the onlydifference being, that instead of a _stick_ I should perform mysubdivision 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 werestrips of best calfskin, cut with the grain, and could not have beenstretched the eighth part of an inch. They would, therefore, measure asaccurately as a rule of boxwood or ivory.
One would not be long enough; so I knotted the two together, taking careto 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 tothat length exactly. I was not satisfied till I had measured it overand over again, each time pulling the thong with all my strength, lestsome "kink" might be lurking in it. A slight error would derange myintended scale, though there is less danger in graduating four feet downto inches than in going from the less to the greater stan
dard. In theformer, each subdivision naturally lessens the error, while in thelatter it is continually doubled.
When convinced that I had got the thong to the precise length, I placedits two ends together, and then drawing it with a firm pull through myfingers, I creased it exactly in the middle. Holding it taut upon theblade of my knife, I cut through at the crease, and thus divided it intotwo moieties of equal length, each two feet long. The part with theknot I laid aside as being no longer needed, and the remaining half Iagain doubled, and cut into two. This gave me two pieces each a foot inlength.
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 fingersto accomplish, for it is much easier to divide a string into two equalparts than into three. I was a good long time before I could get ittrebled to my satisfaction; but I succeeded at length, and then severedthe parts.
My object in thus cutting into three, was to get the pieces in evenfractions of four inches each, in order that by two more doublings Imight 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 theycould be got at, I reduced the second half of the string as I had donethe 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 mymeasuring-stick. I had pieces of one foot, of four inches, of two, andof one; and by the help of these I proceeded to mark my rod after themanner of a draper's yard-stick.
It occupied some time, for I worked with care and caution; but mypatience was rewarded by finding myself in possession of a measure uponwhich I could rely, even in a calculation involving the question of mylife.
I was not much longer in deciding the point. The diameters were nowmeasured by feet and inches, and the _mean_ of the two taken. This wasreduced to surface measure by the usual method of squaring the circle(multiplying by eight, and dividing by ten). This gave the base of thehollow cylinder, which would be equal to the frustum of a cone of likealtitude; and another multiplication by the length produced the entirecubic 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 nearas I could calculate, about 108--and therefore it was in all likelihoodan old pipe that had once contained sherry.