494 



NATURE 



{April 5, 1877 



ihem, and the little heap that it forms on the bottom 

 line. This part of the apparatus is like a harrow with its 

 spikes facing us ; below these are vertical compartments ; 

 the whole is faced with a glass plate. I will pour pellets 

 from any point above the spikes, they will fall against the 

 spikes, tumble about among them, and after pursuing 

 devious paths, each will finally sink to rest in the com- 

 partment that lies beneath the place whence it emerges 

 irom its troubles. 



The courses of the pellets are extremely irregular, it is 

 rarely that any two pursue the same path from beginning 

 to end, yet notwithstanding this you will observe the 

 regularity of the outline of the heap formed by the accu- 

 mulation of pellets. 



Fig. 2. 



This outline is the geometrical representation of the 

 aurve of deviation. If the rows of spikes had been few, 

 the deviation would have been slight, almost all the 

 pellets would have lodged in a single compartment and 

 would then have resembled a column ; if they had been 

 very numerous, they would have been scattered so widely 

 that the part of the curve for a long distance to the right 

 and left of the point whence they were dropped would 

 have been of uniform width, like an horizontal bar. With 

 intermediate numbers of rows of teeth, the curved contour 

 of the heap would assume different shapes, all having a 

 strong family resemblance. I have cut some of these out 

 of cardboard ; they are represented in the diagrams (Figs. 



2 and 3), Theoretically speaking, every possible curve of 

 deviation may be formed by an apparatus of this sort, 

 by varying the length of the harrow and the number of 

 pellets poured in. Or if I draw a curve on an elastic 

 sheet of india-rubber, by stretching it laterally I produce 

 the effects of increased dispersion ; by stretching it ver- 

 tically I produce that of increased numbers. The latter 

 variation is shown by the successive curves in each of the 

 diagrams, but it does not concern us to-night, as we are 

 dealing with proportions, which are not affected by the 

 size of the sample. To specify the variety of curve so far 

 as dispersion is concerned, we must measure the amount 

 of lateral stretch of the india-rubber sheet. The curve 

 has no definite ends, so we have to select and define two 

 points in its base, between which the stretch may be 

 measured. One of these points is always taken directly 

 below the place where the pellets were poured in. This 

 is the point of no deviation, and represents the mean 

 position of all the pellets, or the average of a race. It is 

 marked as 0°. The other point is conveniently taken at 

 the foot of the vertical line that divides either half of the 

 symmetrical figure into two equal areas. I take a half 

 curve in cardboard that 1 have again divided along this 

 line, the weight of the two portions is equal. This distance 

 is the value of 1° of deviation, appropriate to each curve. 



Fig. 3. 



We extend the scale on either side of 0° to as many 

 degrees cis we like, and we reckon deviation as positive, 

 or to be added to the average, on one side of the centre 

 say to the right, and negative on the other, as shown in 

 the dia^ams. Owing to the construction, one quarter or 

 25 per cent, of the pellets will lie between 0° and 1°, and 

 the law shows that 16 per cent, will lie between -f- 1° and 

 -j- 2°, 6 per cent, between + 2° and + 3°> and so on. It 

 is unnecessary to go more minutely into the figures, for 

 it will be easily understood that a formula is capable of 

 giving results to any minuteness and to any fraction of a 

 degree. 



Let us, for example, deal with the case of the Ameri- 

 can soldiers. I find, on referring to Gould's Book, that 

 1° of deviation was m their case i"676 inches. The 

 curve I held in my hand has been drawn to that 

 scale. I also find that their average height was 67"24 

 inches. I have here a standard marked with feet and 

 inches. I apply the curve to the standard, and imme- 

 diately we have a geometrical representation of the statis- 

 tics of height of all those soldiers. The lengths of the 

 ordinates show the proportion of men at and about their 

 heights, and the area between any pairs of ordinates give 

 the propoitionate number of men between those limits. 



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