MEASUREMENT OF VARIABLE PROPERTIES 145 



§ 108.— TWELFTH EXAMPLE : THE MEASUREMENT 

 OF THE DENSITY OF A SOLID BODY a.— In order to 

 obtain the required value as exactly as possible, the density 

 of a (for instance, a metal) is determined a number of times. 

 We observe (this is a matter of fact) that the successively obtained 

 values are more or less different from each other. We find, 

 for instance, the following figures, each of them being obtained 

 several times : — 



4-84, 4-85, 4-86, 4-87, 4'88, 4-89, 4-90, 4-91, 4-92, 4*93, 4-94, 4*95, 



4-96, 4-97, 4-98, 4-99, 5-00, 5-01, 5-02, 5-03, 5-04, 5-05, 5-06, 5-07, 



5-08, 5-09, 5-10, 5-11, 5-12, 5-13, 5-14, 5-15, 5'i6 



Which is in reality the density of « ? In order to answer 

 this question we add up all the figures and divide the sum by 

 the number of observations : the quotient is the arithmetic 

 mean or mean value. I suppose that the mean value is 5*00 : 

 this figure is, among all the observed figures, the most probable 

 value of the density of a. 



Most of the observed figures deviate from the mean, because 

 of errors committed in the determinations. These errors 

 depend on numerous causes (forces, factors), certain of which 

 tend to produce a positive error, whereas others result in a 

 negative one. In each determination these causes (or certain of 

 them) act simultaneously : their combination (resultant) brings 

 about a more or less important positive or negative error. 

 This is expressed by saying that each error depends on 

 chance } 



We may construct in the following way the curve of errors, 

 or, using biological language, the variation curve of the measure- 

 ments. The distance between the extremes (4*84 and 5*16) is 

 divided arbitrarily into a certain number of equal intervals, 

 for instance, 11 — viz. 



4-84-4-86, 4-87-4-89, 4-90-4-92, 4-93-4-95, 4-96-4-98, 4-99-5-01, 

 502-504, 5'05-507, 5'o8-5-io, 5*ii-5*i3, 5'i4-5*i6 



The observed figures included in each interval are counted. 

 A horizontal line is divided into 11 equal segments, each of 

 which corresponds to an interval. From the middle of each 

 segment a vertical ordinate is erected, proportionate to the 

 number of figures (measurements) in the corresponding interval. 

 The extremities of the ordinates are joined by straight lines 

 (Fig. 18). The curve of errors obtained in this way is a diagram- 

 matic representation of facts, without any calculation. 



We SEE that the observations (measurements) which deviate 

 the least from the mean (interval VL) are the most numerous : 



* Compare with the second example (one toss with one coin), § 92, p. 1 16. 



K 



