We have assumed, that all instrmnents are formed at one point of time, and exhausted at another.This is the case with but very few.The period of formation almost always spreads over a large space of time, and that of exhaustion, over another.It is evidently, however, possible to fix on a point, to be determined by a consideration of all the periods at which the labor going to the fortnation was expended, which shall represent the true period of formation; and on another point, determined from a consideration of similar ch'cnmstances regarding the times when the capacity was exhausted, which shall represent the true period of exhaustion.
Thus, suppose a small field in some new settlement in North America, were formed by twelve days labor, it would, were it of the order A, return in one year an equivalent to twenty-four days labor, and then be completely exhausted and worthless.It might, however, be, that it belonged to this order, although it neither yielded so much as twenty-four days labor, nor was exhausted at the end of the year.Say,.that the crop sown is wheat, and, that one bushel wheat is equivalent to one day's labor.Were it at once exhausted, it ought to yield twenty four bushels wheat; it however only yields eighteen, and is not then exhausted.There is consequently a deficiency of six bushels.Now, six bushels at the end of the second year, at the same rate of doubling in a year, ought to produce twelve.
Let us suppose that the next crop is hay, and that the net hay yielded the second year is one ton, equal to eight bushels wheat, then 12 - 8 =4, there is still a deficiency of four bushels, equivalent, at the end of the third year, to eight.If, therefore, the next crop of hay the third year, be equal to what it was the second, that is to eight bushels wheab the deficiency will then be made up.Let us suppose that it is so, and that the field is at that time totally exhausted and useless.It is evident, that such a field, though not producing, or being exhausted as by the supposition, yet producing and being exhausted, in a manner equivalent to the supposition, might, with propriety, be said to belong to the order A.
But, it is farther probable, that such a field, might not produce quite so much grain, or hay, as we have even by the last hypothesis supposed, and would not even at the end of the third year, or for a much longer period, be exhausted; still, if the deficiency in the one, were equivalent to the farther supply in the other, it would evidendy properly belong also to the same order.
Again, by the suppositions we have made, the labor, or its equivalent, was expended exactly at the commencement of the period done year.It might however have been, that some part of the expenditure, going to the formation of this instrument, was made several months before the commencement of the year, and some several months after.But, had what was expended before, been proportionably less, and what was expended after, proportionably greater, the change would not make any alteration to the relation existing between the time and the expenditure, or, consequently, to the place of the instrument.
The spaces over which the several points of time, at which the formation of any instrument is effected, extend, and those over which the several points of time at which its capacity is exhausted also extend, frequently run into each other.Thus according to our system a riding-horse is an instrument.The space of time over which the whole period of his formation extends, commences when his dam is put apart for breeding, continues as long as any thing is laid out for the purpose of giving efficiency and durability to him as an instrument, and probably therefore only terminates a few days before the death of the animal.There would be a number of points all along that space, at each of which something had been expended on his account, and from the date of which, and the amount expended at each, data would be furnished, to ascertain the whole expense of his formation, and the precise point from whence it might be dated.The whole period of his exhaustion would also extend over a large space.It would commence when he was first ridden for pleasure, or business, and would terminate shortly after his death, when his hide went to the tanner, and his flesh to the dogs.An account.pf the several items expended, and the times when they were expended, and of the several items yielded, and the times at which they were yielded, would furnish data for determining the total cost of formation and capacity and the points to be fixed on as the periods of formation and exhaustion, and thus the place of the instrument could be determined.
Calculations of this sort would be intricate, and could not be well effected without having recourse to methods, not usually employed in investigations like the present.In point of fact, there is in practice, as we will afterwards see, a system of notation of instruments, which enables us pretty accurately, and very easily to determine their place in such a series as we have supposed.
It is sufficient for the end here aimed at, to perceive that when all particulars are known, concerning the formation and exhaustion of any instrument, and the periods intervening between these, data are then furnished for placing it in some part of such a series as we have described; and that it may consequently be assumed that every instrument does, in reality, belong to some one order in the series A, B, C, D, etc., or to an order that may be interposed between some two proximate orders of that series.
