WKIOHTS AND MEASURES. 



WEIGHTS AND MEASURES. 



14 



4. Th weight of the average of observations U the mm of the 

 weight* of the oompoiMot observation*. If * observations, A,, A,, Ac . 

 be node, all of the aura weight r, the average U 3A-r, the weight 

 of the average is nc, and iU probable error is -476036-7- V(*r). 

 the weights be different, say r , t v Ac., then Sc A-=-2c is the average, 

 2<- is iU weight, and -4769S6-*- v'(3c) >t> probable error. In the 

 former ease the probable error of the average nioy be directly found 

 from the nun of the squares of the reputed errors,* by the formula 



67449 



5. Cattrii paribut, the probable error of an average will not be in- 

 versely as the number of observations, but as the square root of that 

 number. If p be the probable error of an observation, and r that of 

 the average of H such observations, then j>= V* P. An observer who 

 takes such a mode as gives the probable error of an observation firi< e 

 as great as it need be, must not hope to indemnify himself for hia 

 carelessness by making ttcite as many observations as would otherwise 

 be necessary, but must make /our lima as many. 



6. If p be the probable error of an observation, an average, or other 

 result, the following table will be sufficient to connect the probable 

 error with other errors, for any rough purpose of estimation : 



This table is to bo interpreted as follows : If p be the probable 

 error above mentioned, it is 1 J to 1, or 8 to 2, against the error turning 

 out less than 79 *p, and it is 1J to ] for the error turning out less 

 than 1*25 x p. It is 8 to 1 against the error being leas than "21 x />, 

 and 8 to 1 for its being less than 2'36 x p. It is 1000 to 1 againat the 

 error being less than -002 x p, and 1000 to 1 for the error beiug less 

 than 4-90 x p. 



WEIGHTS AND MEASURES. The subject of weights and 

 measures is one the actual state of which is prosperous in the inverse 

 ratio of the number of books or the length of articles which are written 

 upon it. There is nothing in it which might not, if the most natural 

 and simple system were adopted, be described in a very few pages. 

 "We aru speaking of course only with reference to a possible time ; for, 

 let that time arrive when it may, the history of the past must be a 

 confused and repulsive subject. 



| In the article WJEIOHT, Ac., STANDARD OF, we shall give some idea of 

 the recent history of the attempts which have been made in England 

 .to secure a permanent measure of length. These have only succeeded, 

 at least until very recently, somewhat farther than to the extent of 

 making it possible to restore to the merchant a system sufficiently 

 near to that which now exists, if the latter should be lost ; but they 

 have all confessedly failed in perpetuating sufficient exactness for 

 scientific purposes. The same may be said of the French endeavour to 

 create a recoverable standard by the measurement of the earth. 

 [TRIGONOMETRICAL SURVEY.] So that in fact we are now come back 

 again to the old notion, that the true way to maintain a measure U to 

 construct accurate copies out of durable material, and to preserve those 

 copies with care. 



The measures of time (of which we speak more particularly in YEAB, 

 TIME, PERIODS OF REVOLUTION) are the only usual ones in which a 

 natural standard exists ; to which we may add, that in the kindred 

 operation of counting there is something of thn same kind. The 

 phenomena of the daily revolution of the earth, and the ten fingers on 

 the two hands, have secured to the whole human race, above the degree 

 of the lowest savages, one mode of assigning periods of duration and 

 large collections of number. But even in these two subject* details 

 have differed considerably in different times and countries ; and much 

 more has this happened with respect to measures In which the choice 

 of a standard is |mrcly arbitrary, as in the case of length, surface, 

 capacity, and weight The angle U another magnitude which has a 

 natural measure [AHOLE] ; and, as this has neverbeen out of the hands 

 of geometers, a greater uniformity has prevailed in the measurement 

 of angular magnitude than of any other whatsoever. The measures of 



. 



length obviously regulate those of surface and capacity. There is no 

 other way of denning an area or a solidity, except by describing, for 

 the area, lengths, and for the solidity, surfaces, by which the area or 

 solid may be bounded in a given manner. Measures of weight may be 

 obtained by defining, as standards, given bulks of given substances ; 

 and as water is the most common and most easily purified of all 

 substances, U has been chosen by common consent as the referee for 



The departure! from the average store mentioned : the averse* being 

 taken for the trnlb, the dqiartam arc taken for the errors. 



such standards. A measure of length then is all that is wanted in the 

 first instance ; and most nations, ancient and modern, have been in the 

 habit of referring all the resulting measures to those of length alone. 

 Nevertheless, there is no small difficulty in obtaining a oompari 

 a measure of weight deduoed from length with one already existing, in 

 such a manner as to perpetuate the Utter, if the utmost accuracy be 

 required. (Kater, Construction and Adjustment,' &o., ' Phil. Trans.,' 

 IBM.) So that the commissioners who last reported on the subject 

 advise that the standard of weight shall no longer be deduoed from 

 that of length, but shall be simply a piece of metal or other durable 

 substance. 



It i not our object in this article to consider weights and measures 

 in a scientific point of view, but simply to give some historical account 

 of the measures actually in use, and some tables of the principal ones, 

 ancii-nt and modem. There is no subject whose history is more dis- 

 tinctly divided into three periods, ancient, middle, and modern, than 

 that of weights and measures. The ancient period, ending with the 

 decline of the Roman empire, during which the classical standards 

 were preserved and employed; the middle period, during which, while 

 the names and relations of the classical measures were preserved among 

 the learned, the standards were lost, and the various difference* of 

 national measures began to exist among the people ; the modern period, 

 which hardly begins before the 17th century, in which the discrepan- 

 cies of national measures were noted, and the attempts at a system 

 founded upon natural philosophy began to be made. 



The origin of measures of length is unquestionably to be found in 

 the ports of the human body ; both their usual lengths, roughly 

 speaking, and their names, establish this beyond a doubt Ti. 

 the digit, the palm, the span, the cubit, etc., are in all languages derived 

 from the same source ; nor, in the popular view of measurement, do 

 they materially differ in length : the yard is but a variety of the word 

 rod, and has no intrinsic meaning. It is also unquestionable that in 

 former times, when authentic measures were not so easily to be 

 obtained, the hands, arms, and feet were much more frequently used 

 than they are at present, when every workman, however humble, is in 

 possession of a measure. George Agricola, presently named, says that 

 in his time (the beginning of the 16th century) the French workmen 

 commonly measured a foot by joining the extremities of the thumbs, 

 clenching the fingers, and keeping the thumbs as widely extended as 

 they could: " vulgo pednu metiuntur opifices manibus in pugnos 

 controctis et porrectis pollicibus oltrinsecusque obverais : " nor is this a 

 bad measure of a French foot At what period the slightly variable 

 measures derived from the living man were first exchanged for a fixed 

 and legal average or other conventional value, whether amui, 

 Oreeks or Romans, is unknown. All that can be said is, that none of 

 the earlier writers enter otherwise than incidentally -upon the question, 

 and that the fixed and legal measures were of early date. Moat authors 

 give some little information upon the subject; even the poets are 

 frequently cited for their allusions. Fixing the end of the ancient 

 period about the middle of the 6th century (simply because the chain 

 of writers who are cited on ancient weights and measures ends there), 

 and omitting names as well known as Homer or Virgil, Hesychius or 

 Suidas, Pliny or Vitruvius, there is direct information on the subject 

 in the works or fragments of Cato, Celsus, ColumeUa, Dioscori.les, 

 (ialcn, Hero, Julius Frontinus, Julius Pollux, Martianua Capella, 

 Modestinus, Oribasius, Palladium, Paulus, Pomponius, Priscian (who 

 wrote expressly on the subject), Proclus, Rhemnius Fannius (who 

 wrote a poem on the subject, often attributed to Priacian), Scribonius, 

 Itoetius, Festus Pouipcius, Ulpianus, Volusiua Mfocianus, and Varro. 



It may bo convenient to end the middle period and commence the 

 modern with the work of Lucas Paetus (1573), as being the earliest of 

 the writers who are frequently cited for success in their attempts to 

 restore the almost forgotten values of the Roman measures. But this 

 middle period may be divided into that which preceded and followed 

 the invention of printing. All that took place in the former part of it 

 is a blank ; we know but the result, namely, the (probably gradual) 

 introduction of measures differing from those of Rome in magnitude, 

 though retaining the same names. Nevertheless the writers, as we 

 have seen in MILE, retained, besides a uniformity of expression, an 

 intended uniformity of meaning : if they had not the Roman foot and 

 mile, they thought they had. When the German mile was introduced, 

 which was about four Roman miles, the latter were called Italian 

 miles. An abundance of passages might be cited from writers of 

 different countries about the beginning of the 16th century, when 

 books began to be plentiful, all coinciding in requiring the following 

 explanation, namely, that the learned hod among th,-i:i, . l\c, or 

 believed they had, a system of measures in terms of which I h. . 

 mnnicatod with each other, not recognising nor in any way alluding to 

 the common or vernacular measures. It is our supposition that this 

 system began in ignorance that the national measures really did differ 

 from one another at all, and was continued under the impression that 

 a common system was desirable, attainable, and, by keeping to the 

 Roman measures, attained. 



As this point in the history of measures is not alluded to by any 

 metrologist, and as some of its consequences are remarkable, it will bo 

 desirable to state some proofs of our assertion. So far as we can 

 it was hardly thought necessary, oven after the Iflth century ha-i 

 menccd, and certainly not before, to mention the scale of men 



