Front Page Titles (by Subject) APPENDIX D.: SCALE OF EQUIVALENTS. - An Autobiography, vol. 1
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APPENDIX D.: SCALE OF EQUIVALENTS. - Herbert Spencer, An Autobiography, vol. 1 
An Autobiography by Herbert Spencer. Illustrated in Two Volumes. Vol. I (New York: D. Appleton and Company 1904).
Part of: An Autobiography
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SCALE OF EQUIVALENTS.
[Devised and made in 1841, but not published.]
Having occasion between two and three years since [this was written about 1842] to reduce a long list of distances given in inches and tenths into decimals of feet, it occurred to me that by making use of a geometrical representation of the relative values of the component parts I might be able to read off the results without the aid of any calculation. The method adopted was this. A line, AB, being taken to represent a foot was divided on the one side into 120 divisions representing tenths of inches and on the other into 100 divisions representing decimals of a foot; and the divisions having been numbered as usual, for convenience of reading, the equivalent of each dimension given in inches and tenths was read off on the other side of the scale in the new denomination. The same method was evidently applicable to superficial and solid quantity as well as to linear, and to weights and values as well as to quantities.
On reconsidering the subject some time afterwards, it occurred to me that an apparatus might be made which would be universally applicable to the reduction of quantities, weights, values, &c., in the manner above exemplified. The accompanying plate represents, in a completed form but of half the size, the instrument which I constructed for this purpose. AB is a line divided into equal parts; AC a line drawn at right angles to it; and C a point taken in AC to which convergent lines are drawn from the divisions in AB: the lines beyond BC being drawn to divisions in AB produced, but subsequently cut off.
It follows from a simple geometrical principle that any line drawn parallel to AB, and cutting the converging lines, is divided by them into equal parts; and that, consequently, a line may be moved along parallel to AB, until there is found a place at which it is divided into a desired number of equal parts: supposing that such number falls within the limits of the scale. This fact is taken advantage of thus. A moveable scale, EF, is kept parallel to the line AB, by an arm MD, sliding in a dovetailed groove, shown in section at G. This scale is divided into such number of equal divisions as may be thought most generally useful: one edge being divided decimally and the other duodecimally; or in a different way if required for some special purpose. Of course, the divisions must be in any case so arranged that the zero of the scale may coincide with the line AC.
If now the divisions on the scale be assumed to represent units of any denomination, either of length, surface, bulk, weight, or value; and if, knowing that a certain number of these units is equivalent to a certain number in some other denomination, the scale be slid forward until the divisions representing the equivalent numbers coincide; then any quantity of the one kind will have its corresponding value in the other indicated by the opposite division.
For instance, 51.796 French kilogrammes are equal to 112 lbs. English; and, taking multiples for the sake of accuracy, 880.5 kilogrammes equal 1904 lbs. Then if the scale be moved along towards C until the division at h (880.5) coincides with the division at k (1904) on the converging lines, we shall have the edge of the scale occupying the position shown by the dotted line kl; and we shall then have a line, represented by the edge of the scale, divided on the one side into kilogrammes and on the other into English pounds. Hence any weight short of 2000 lbs., stated either in the French or the English denomination, may have its value in the opposite denomination read off at sight.
In the same manner prices and sums of money stated in some Continental currency, may have their relative values in English money ascertained; and the calculations called for by the varying rates of exchange with foreign countries, may readily be performed.
The system is applicable not only to the reduction of quantities from one denomination into another, but also to the calculation of equivalents of different orders. Thus, if any quantities given in bulk have their values in weight required, the process will be just the same: a certain number of the units of quantity corresponds to some other number of the units of weight, and the scale being adjusted so that these divisions coincide the results are read off as before. Again a list of tons and cwts. of some material charged at per ton, may have the values of the several items found; by using a sliding scale properly divided for the purpose, and assuming the large numbered divisions to represent tons. In short, any calculation coming within the sphere of ordinary proportion, provided it be within the limits of the scale, may be performed by it. The instrument is not intended to be employed in those cases where a single calculation only has to be made: the time required for adjustment would probably be greater than that taken in obtaining the result by the ordinary method. But its advantages are to be gained in cases where a number of operations of the same kind have to be gone through.
It must be understood that the divisions may be used in a variety of ways. Thus the spaces between the black numbered lines may be taken as units and their divisions as decimals, the large ones being tenths and their sub-divisions (not shown in the plate) hundredths. Or each of the divisions in the drawing may represent one and their sub-divisions tenths: each of the great divisions being then read as 10. And again, each of the ultimate divisions may be considered a unit, as instanced in the first example of the application of the instrument. The same variety of assumptions may be made with the moveable scale; and if it be remembered that in addition to the extended application allowed by making these assumptions, we may employ several scales with divisions of different magnitudes, it will be seen that there are few cases in which the instrument may not be advantageously used. To make the apparatus quite complete, the sliding scale may have a vernier attached, as shown at K.
On a first trial with the original instrument which is made of cardboard, and is otherwise incomplete in several points, and therefore takes longer intervals for adjustment, reading off, &c., than a perfect one would do, it was found that the time taken by a fast calculator (after he had found the constant) was just double that required by the instrument. With a perfect apparatus and a little practice the ratio would probably be made much greater.