October i i, i '^94] 



NA TURE 



D-^D 



of ihe fact that one i|uantity is greater or less than 

 another is not measurement. Measurement implies ihe 

 ability to represent numerically, so that ratios can be 

 accurately expressed. Among primitive races measurement 

 by cnumeraton is very restricted. Tribes bordering on 

 savagery at the present time are often found to be unable to 

 enumerate beyontl three or four. This stalcnient is quite 

 positively made by competent authorities, in spite of the f.ict 

 that the abibtv to enumerate the number of fingers on at least 

 one hanil would appear to be necessary to even the lowest order 

 of intelligence. It is cuiious to note in this conneciion that 

 experiment has apparently proved that four is ihe maximum 

 number of objects whose accurate enumeration is possible tit a 

 single glance and without counting, by the most highly culti- 

 vated man. 



As man emerges from savagery his powers of enumeration 

 increase. He soon discovers the necessity for units of a higher 

 order which themselves represent a collection, and easily linds 

 such units provided by nalure in the groups of fingers on his 

 two hands. Thus the decimal system of arithmetic is invented ; 

 not in one place or by one people, but everywhere and when- 

 ever man finds that somewhat extensive enumeration is desirable 

 or necessary. It is a singular exception to this general rule, 

 however, that the Greeks failed to invent a decimal arithmetic. 



With systems of notation capable of indefinite extension, 

 measurement by enumeration becomes rigorously e;.act ; that 

 is, barring blunder^, which can always be discovered and 

 avoided, the number of units in a group, if capable of being 

 counted at all, can be counted with absolute accuracy. Thus, 

 the cash in the Treasury of the United States may be more than 

 a hundred million dollars, that is more than ten thousand 

 million cents, and the exact quantity can be ascertained to a 

 single cent. By simple enumeration, therefore, this quant ily of 

 money is measured so accurately that the error cannot be as 

 much as one part in ten thousand millions, and this might be 

 extended in any degree, if only the cash is there to be counted. 



At a comparatively early stage, therefore, this kind of 

 measurement was perfected, but there are two systems or 

 methods of measurement derived from it that are worthy of brief 

 comment. The first includes that variety of mensuration in 

 which the numerical value of a magnitude cannot be obtained 

 by simple counting, but is derived by calculation based on 

 rigorously exact relationship. This is of a distinctly higher 

 order than that just considered, and it is only found among 

 highly intelligent people, those, in short, who have cultivated 

 a knowledge of pure mathematics. A very simple illustration 

 is the determination of the area of a triangle when its base and 

 altitude are known. In this and similar cases a rigorously 

 accurate result is attainable when the data are absolutely 

 correct, but simple counting would be impossible. There are 

 cases, however, and these constitute another step along ihe line 

 in which we are travelling, in which an absolutely accurate 

 evaluation is impossible, but in which any deiiyabU degree of 

 .accuracy, however high, may be reached. Perhaps the best 

 known example of this is the determination of the circum- 

 ference of a circle when its diameter is known. The ratio of the 

 former to the latter, which cannot be exactly exprcs;e I, has 

 been determined with a degree of approximation by modern 

 computers, which makes it possible to reduce the outstanding 

 error to an inconceivably small quantity. An attempt to 

 illustrate this may not be without interest. 



In a display ol mathematical genius which has perhaps never 

 been surpassed, Archimedes more than two thousand years ago 

 discovered the first real approximation to the value of this con- 

 stant. The accuracy of his result may be shown in the fact 

 that if the diameter of a circle be exactly one inch, its circum- 

 ference as determined by the value of the constant found by 

 Archimedes will not be in error more than the thickness of a 

 human hair. If the value of the constant is more accurately 

 known, it will be possible to compute the circumference of a 

 proportionately larger circle so that the error shall not exceed a 

 hair's-breadth. Let us go at once from the circle one inch in 

 diameter to one having a radius equal to the distance from the 

 earlh to the sun and a circumference of nearly 600 millions of 

 miles. It is difficult to form any adequate conception of the 

 enormous stretch of 9J millions of miles which separates the 

 earth from the sun. The immensily of it is in some degree 

 realised on rellecting that if it were possible for a child to extend 

 an arm across this spice, and plunge his hand into the while 

 hot layer of the sun Irom which light is radiated, he might grow 



NO. 1302, VOL. 50] 



to youth, manhood, old a^e, and unless he 1 ve 1 throigh the 

 almost unprecedented period of 125 years, death woulu come 

 before he would feel the pain of burnin.;, so great is the distance 

 through which the sensation must travel. But even ihis circle, 

 of 600 millions of miles in circumference, is almost immeasur- 

 ably small in comparison with the one for which we are seek- 

 ing. Multiply it by a million; a million million ; a million, 

 million million ; in fact multiiily it by a number expressed by 

 the word million repeated 9S time=, and we reach a circle of 

 utterly inconceivable dimensions ; yet so precisely do we know 

 the ratio of the circumference to the diameter of a circle, that 

 having given the diameter of such a circle, its circumference can 

 be determined within the breadth of a hair. For all ordina y, 

 practical purposes this is sufTicient. 



A very modern and an extremely imporlant species of 

 measurement involving only enumeration is to be found in the 

 statistical metho 1 of treating certain chsses of problems in 

 which the object is to follow the fortunes of a group rather than 

 an individual. It has long been advantageously applied to 

 social, political, and economical questions, and within a few 

 years, in the hands of such men as Clerk Maxwell, Boltzmann, 

 and others, it has proved to be a powerful .agent in physic \1 

 investigations. 



It depends in great measure on what may be called the 

 principle of the " long run," which is, that phenomena of 

 apparently the most accidental and lawless character will, in 

 the long run, occur with regularity and obedience to law, to 

 such an extent as to render their prediction quite possible. .\t 

 least one great railroad system in this country has so tabulated 

 and investigated all accidents happening to its employes and 

 patrons that it is able to foretell with a good degree of accutacy 

 the number of people who will, during the next year, meet with 

 death on its line ; how many will lose a foot, how many an 

 arm, and soon ; and its Board of Directors is ihus always ready 

 to weigh the cost of a new invention to add to the safely of 

 travel, against the probable damages to be paid for fatal and 

 other injuries which said invention might prevent. 



Further argument is unnecessary to show that measurement 

 by enumeration, the first to appear in Ihe evolution of man 

 and his accomplishments, has advanced with man and kept pace 

 with his accomplishments ; that it has contributed greatly 10 

 his advancement, and that at any given period it may fairly 

 stand as an exponent of his condition. 



But in a far greater degree is this true of the second of the 

 two forms of measurement to which men have resorted, namely, 

 lliat ,,in which a conventional unit embodying the particular 

 quality to be measured is compared to the magnitude to be 

 evaluated. Nearly all operations ordinarily called measure- 

 ments belong to this class, and its necessity must have followed 

 closely upon the introduction of measurement by enumeration. 

 Of the three fundamental measures, from which it is convenient 

 to derive all others, namely, length, mass and time, the first 

 and last were undoubtedly the earliest to receive attention, and 

 it is more than likely that some rude system of time measure- 

 ment constituted the earliest contribution to metrology. 

 Nature is lavish in the number and variety of time units which 

 she has furnisheil man, some of which satisfy Ihe most rigorous 

 demands of modern science. In the early sages of chrono- 

 metric development the method of enumeration was alone 

 available. By taking the solar d.ay as the unit, counting the 

 number of days in a lunar period furnished the month. The 

 year was similarly obtained, at first from the mere cycle of 

 the seasons, but in a somewhat more advanced st.age of develop- 

 ment, from more exact observations upon the sun. Before this, 

 there must have existed a demand for the division of the day 

 into smaller units of time. Much ingenuity and often genius of 

 a high order was shown in ihe invenlionof chronometric devices. 

 A remarkably clever determination of the angular diameter 

 of the sun was made by the Chaldeans by ihe use of one 

 of the earliest forms of time-measuring apparatus. .Vt the 

 mouent the sun's disk appeared in the eastern horizon a fine 

 stream of water flowing from the bottom of a vessel in which 

 the level was kept constant, was caught in a small cup, into 

 which it was allowed to How until the lower limb of the sun 

 was visible. The sm.ill cup being instantly withdrawn, anoilur 

 much larger receptacle was substituted for it, and into this the 

 small stream fell during all the d.ay and until the sun ap- 

 peared in the east again on the following morning. It was 

 found that the water in the large vessel w.as 720 times that in 

 the smaller, from which it appeared th.at the apparent diameter 



