Part 8 (2/2)

From all which evidence it is tolerably clear that the earliest mode of conveying the idea of any number of things, was by holding up as many fingers as there were things; that is--using a symbol which was _equal_, in respect of multiplicity, to the group symbolized. For which inference there is, indeed, strong confirmation in the recent statement that our own soldiers are even now spontaneously adopting this device in their dealings with the Turks. And here it should be remarked that in this recombination of the notion of equality with that of multiplicity, by which the first steps in numeration are effected, we may see one of the earliest of those inosculations between the diverging branches of science, which are afterwards of perpetual occurrence.

Indeed, as this observation suggests, it will be well, before tracing the mode in which exact science finally emerges from the merely approximate judgments of the senses, and showing the non-serial evolution of its divisions, to note the non-serial character of those preliminary processes of which all after development is a continuation. On re-considering them it will be seen that not only are they divergent growths from a common root,--not only are they simultaneous in their progress; but that they are mutual aids; and that none can advance without the rest. That completeness of cla.s.sification for which the unfolding of the perceptions paves the way, is impossible without a corresponding progress in language, by which greater varieties of objects are thinkable and expressible. On the one hand it is impossible to carry cla.s.sification far without names by which to designate the cla.s.ses; and on the other hand it is impossible to make language faster than things are cla.s.sified.

Again, the multiplication of cla.s.ses and the consequent narrowing of each cla.s.s, itself involves a greater likeness among the things cla.s.sed together; and the consequent approach towards the notion of complete likeness itself allows cla.s.sification to be carried higher. Moreover, cla.s.sification necessarily advances _pari pa.s.su_ with rationality--the cla.s.sification of _things_ with the cla.s.sification of _relations_. For things that belong to the same cla.s.s are, by implication, things of which the properties and modes of behaviour--the co-existences and sequences--are more or less the same; and the recognition of this sameness of co-existences and sequences is reasoning. Whence it follows that the advance of cla.s.sification is necessarily proportionate to the advance of generalizations. Yet further, the notion of _likeness_, both in things and relations, simultaneously evolves by one process of culture the ideas of _equality_ of things and _equality_ of relations; which are the respective bases of exact concrete reasoning and exact abstract reasoning--Mathematics and Logic. And once more, this idea of equality, in the very process of being formed, necessarily gives origin to two series of relations--those of magnitude and those of number: from which arise geometry and the calculus.

Thus the process throughout is one of perpetual subdivision and perpetual intercommunication of the divisions. From the very first there has been that _consensus_ of different kinds of knowledge, answering to the _consensus_ of the intellectual faculties, which, as already said, must exist among the sciences.

Let us now go on to observe how, out of the notions of _equality_ and _number_, as arrived at in the manner described, there gradually arose the elements of quant.i.tative prevision.

Equality, once having come to be definitely conceived, was readily applicable to other phenomena than those of magnitude. Being predicable of all things producing indistinguishable impressions, there naturally grew up ideas of equality in weights, sounds, colours, &c.; and indeed it can scarcely be doubted that the occasional experience of equal weights, sounds, and colours, had a share in developing the abstract conception of equality--that the ideas of equality in size, relations, forces, resistances, and sensible properties in general, were evolved during the same period. But however this may be, it is clear that as fast as the notion of equality gained definiteness, so fast did that lowest kind of quant.i.tative prevision which is achieved without any instrumental aid, become possible.

The ability to estimate, however roughly, the amount of a foreseen result, implies the conception that it will be _equal to_ a certain imagined quant.i.ty; and the correctness of the estimate will manifestly depend upon the accuracy at which the perceptions of sensible equality have arrived. A savage with a piece of stone in his hand, and another piece lying before him of greater bulk but of the same kind (a fact which he infers from the _equality_ of the two in colour and texture) knows about what effort he must put forth to raise this other piece; and he judges accurately in proportion to the accuracy with which he perceives that the one is twice, three times, four times, &c. as large as the other; that is--in proportion to the precision of his ideas of equality and number. And here let us not omit to notice that even in these vaguest of quant.i.tative previsions, the conception of _equality of relations_ is also involved. For it is only in virtue of an undefined perception that the relation between bulk and weight in the one stone is _equal_ to the relation between bulk and weight in the other, that even the roughest approximation can be made.

But how came the transition from those uncertain perceptions of equality which the unaided senses give, to the certain ones with which science deals? It came by placing the things compared in juxtaposition. Equality being predicated of things which give us indistinguishable impressions, and no accurate comparison of impressions being possible unless they occur in immediate succession, it results that exactness of equality is ascertainable in proportion to the closeness of the compared things. Hence the fact that when we wish to judge of two shades of colour whether they are alike or not, we place them side by side; hence the fact that we cannot, with any precision, say which of two allied sounds is the louder, or the higher in pitch, unless we hear the one immediately after the other; hence the fact that to estimate the ratio of weights, we take one in each hand, that we may compare their pressures by rapidly alternating in thought from the one to the other; hence the fact, that in a piece of music, we can continue to make equal beats when the first beat has been given, but cannot ensure commencing with the same length of beat on a future occasion; and hence, lastly, the fact, that of all magnitudes, those of _linear extension_ are those of which the equality is most accurately ascertainable, and those to which by consequence all others have to be reduced. For it is the peculiarity of linear extension that it alone allows its magnitudes to be placed in _absolute_ juxtaposition, or, rather, in coincident position; it alone can test the equality of two magnitudes by observing whether they will coalesce, as two equal mathematical lines do, when placed between the same points; it alone can test _equality_ by trying whether it will become _ident.i.ty_. Hence, then, the fact, that all exact science is reducible, by an ultimate a.n.a.lysis, to results measured in equal units of linear extension.

Still it remains to be noticed in what manner this determination of equality by comparison of linear magnitudes originated. Once more may we perceive that surrounding natural objects supplied the needful lessons.

From the beginning there must have been a constant experience of like things placed side by side--men standing and walking together; animals from the same herd; fish from the same shoal. And the ceaseless repet.i.tion of these experiences could not fail to suggest the observation, that the nearer together any objects were, the more visible became any inequality between them. Hence the obvious device of putting in apposition, things of which it was desired to ascertain the relative magnitudes. Hence the idea of _measure_. And here we suddenly come upon a group of facts which afford a solid basis to the remainder of our argument; while they also furnish strong evidence in support of the foregoing speculations. Those who look sceptically on this attempted rehabilitation of the earliest epochs of mental development, and who more especially think that the derivation of so many primary notions from organic forms is somewhat strained, will perhaps see more probability in the several hypotheses that have been ventured, on discovering that all measures of _extension_ and _force_ originated from the lengths and weights of organic bodies; and all measures of _time_ from the periodic phenomena of either organic or inorganic bodies.

Thus, among linear measures, the cubit of the Hebrews was the _length of the forearm_ from the elbow to the end of the middle finger; and the smaller scriptural dimensions are expressed in _hand-breadths_ and _spans_.

The Egyptian cubit, which was similarly derived, was divided into digits, which were _finger-breadths_; and each finger-breadth was more definitely expressed as being equal to four _grains of barley_ placed breadthwise.

Other ancient measures were the orgyia or _stretch of the arms_, the _pace_, and the _palm_. So persistent has been the use of these natural units of length in the East, that even now some of the Arabs mete out cloth by the forearm. So, too, is it with European measures. The _foot_ prevails as a dimension throughout Europe, and has done since the time of the Romans, by whom, also, it was used: its lengths in different places varying not much more than men's feet vary. The heights of horses are still expressed in _hands_. The inch is the length of the terminal joint of _the thumb_; as is clearly shown in France, where _pouce_ means both thumb and inch. Then we have the inch divided into three _barley-corns_.

So completely, indeed, have these organic dimensions served as the substrata of all mensuration, that it is only by means of them that we can form any estimate of some of the ancient distances. For example, the length of a degree on the Earth's surface, as determined by the Arabian astronomers shortly after the death of Haroun-al-Raschid, was fifty-six of their miles. We know nothing of their mile further than that it was 4000 cubits; and whether these were sacred cubits or common cubits, would remain doubtful, but that the length of the cubit is given as twenty-seven inches, and each inch defined as the thickness of six barley-grains. Thus one of the earliest measurements of a degree comes down to us in barley-grains.

Not only did organic lengths furnish those approximate measures which satisfied men's needs in ruder ages, but they furnished also the standard measures required in later times. One instance occurs in our own history.

To remedy the irregularities then prevailing, Henry I. commanded that the ulna, or ancient ell, which answers to the modern yard, should be made of the exact length of _his own arm_.

Measures of weight again had a like derivation. Seeds seem commonly to have supplied the unit. The original of the carat used for weighing in India is _a small bean_. Our own systems, both troy and avoirdupois, are derived, primarily from wheat-corns. Our smallest weight, the grain, is _a grain of wheat_. This is not a speculation; it is an historically registered fact.

Henry III. enacted that an ounce should be the weight of 640 dry grains of wheat from the middle of the ear. And as all the other weights are multiples or sub-multiples of this, it follows that the grain of wheat is the basis of our scale. So natural is it to use organic bodies as weights, before artificial weights have been established, or where they are not to be had, that in some of the remoter parts of Ireland the people are said to be in the habit, even now, of putting a man into the scales to serve as a measure for heavy commodities.

Similarly with time. Astronomical periodicity, and the periodicity of animal and vegetable life, are simultaneously used in the first stages of progress for estimating epochs. The simplest unit of time, the day, nature supplies ready made. The next simplest period, the mooneth or month, is also thrust upon men's notice by the conspicuous changes const.i.tuting a lunation. For larger divisions than these, the phenomena of the seasons, and the chief events from time to time occurring, have been used by early and uncivilized races. Among the Egyptians the rising of the Nile served as a mark. The New Zealanders were found to begin their year from the reappearance of the Pleiades above the sea. One of the uses ascribed to birds, by the Greeks, was to indicate the seasons by their migrations.

Barrow describes the aboriginal Hottentot as denoting periods by the number of moons before or after the ripening of one of his chief articles of food.

He further states that the Kaffir chronology is kept by the moon, and is registered by notches on sticks--the death of a favourite chief, or the gaining of a victory, serving for a new era. By which last fact, we are at once reminded that in early history, events are commonly recorded as occurring in certain reigns, and in certain years of certain reigns: a proceeding which practically made a king's reign a measure of duration.

And, as further ill.u.s.trating the tendency to divide time by natural phenomena and natural events, it may be noticed that even by our own peasantry the definite divisions of months and years are but little used; and that they habitually refer to occurrences as ”before sheep-shearing,”

or ”after harvest,” or ”about the time when the squire died.” It is manifest, therefore, that the more or less equal periods perceived in Nature gave the first units of measure for time; as did Nature's more or less equal lengths and weights give the first units of measure for s.p.a.ce and force.

It remains only to observe, as further ill.u.s.trating the evolution of quant.i.tative ideas after this manner, that measures of value were similarly derived. Barter, in one form or other, is found among all but the very lowest human races. It is obviously based upon the notion of _equality of worth_. And as it gradually merges into trade by the introduction of some kind of currency, we find that the _measures of worth_, const.i.tuting this currency, are organic bodies; in some cases _cowries_, in others _cocoa-nuts_, in others _cattle_, in others _pigs_; among the American Indians peltry or _skins_, and in Iceland _dried fish_.

Notions of exact equality and of measure having been reached, there came to be definite ideas of relative magnitudes as being multiples one of another; whence the practice of measurement by direct apposition of a measure. The determination of linear extensions by this process can scarcely be called science, though it is a step towards it; but the determination of lengths of time by an a.n.a.logous process may be considered as one of the earliest samples of quant.i.tative prevision. For when it is first ascertained that the moon completes the cycle of her changes in about thirty days--a fact known to most uncivilized tribes that can count beyond the number of their fingers--it is manifest that it becomes possible to say in what number of days any specified phase of the moon will recur; and it is also manifest that this prevision is effected by an opposition of two times, after the same manner that linear s.p.a.ce is measured by the opposition of two lines.

For to express the moon's period in days, is to say how many of these units of measure are contained in the period to be measured--is to ascertain the distance between two points in time by means of a _scale of days_, just as we ascertain the distance between two points in s.p.a.ce by a scale of feet or inches: and in each case the scale coincides with the thing measured--mentally in the one; visibly in the other. So that in this simplest, and perhaps earliest case of quant.i.tative prevision, the phenomena are not only thrust daily upon men's notice, but Nature is, as it were, perpetually repeating that process of measurement by observing which the prevision is effected. And thus there may be significance in the remark which some have made, that alike in Hebrew, Greek, and Latin, there is an affinity between the word meaning moon, and that meaning measure.

This fact, that in very early stages of social progress it is known that the moon goes through her changes in about thirty days, and that in about twelve moons the seasons return--this fact that chronological astronomy a.s.sumes a certain scientific character even before geometry does; while it is partly due to the circ.u.mstance that the astronomical divisions, day, month, and year, are ready made for us, is partly due to the further circ.u.mstances that agricultural and other operations were at first regulated astronomically, and that from the supposed divine nature of the heavenly bodies their motions determined the periodical religious festivals. As instances of the one we have the observation of the Egyptians, that the rising of the Nile corresponded with the heliacal rising of Sirius; the directions given by Hesiod for reaping and ploughing, according to the positions of the Pleiades; and his maxim that ”fifty days after the turning of the sun is a seasonable time for beginning a voyage.”

As instances of the other, we have the naming of the days after the sun, moon, and planets; the early attempts among Eastern nations to regulate the calendar so that the G.o.ds might not be offended by the displacement of their sacrifices; and the fixing of the great annual festival of the Peruvians by the position of the sun. In all which facts we see that, at first, science was simply an appliance of religion and industry.

After the discoveries that a lunation occupies nearly thirty days, and that some twelve lunations occupy a year--discoveries of which there is no historical account, but which may be inferred as the earliest, from the fact that existing uncivilized races have made them--we come to the first known astronomical records, which are those of eclipses. The Chaldeans were able to predict these. ”This they did, probably,” says Dr. Whewell in his useful history, from which most of the materials we are about to use will be drawn, ”by means of their cycle of 223 months, or about eighteen years; for at the end of this time, the eclipses of the moon begin to return, at the same intervals and in the same order as at the beginning.” Now this method of calculating eclipses by means of a recurring cycle,--the _Saros_ as they called it--is a more complex case of prevision by means of coincidence of measures. For by what observations must the Chaldeans have discovered this cycle? Obviously, as Delambre infers, by inspecting their registers; by comparing the successive intervals; by finding that some of the intervals were alike; by seeing that these equal intervals were eighteen years apart; by discovering that _all_ the intervals that were eighteen years apart were equal; by ascertaining that the intervals formed a series which repeated itself, so that if one of the cycles of intervals were superposed on another the divisions would fit. This once perceived, and it manifestly became possible to use the cycle as a scale of time by which to measure out future periods. Seeing thus that the process of so predicting eclipses, is in essence the same as that of predicting the moon's monthly changes by observing the number of days after which they repeat--seeing that the two differ only in the extent and irregularity of the intervals, it is not difficult to understand how such an amount of knowledge should so early have been reached. And we shall be less surprised, on remembering that the only things involved in these previsions were _time_ and _number_; and that the time was in a manner self-numbered.

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