ON THE STRENGTH OF PILLARS. 
407 
the ratio according- to the theory of Euler. In similar pillars the length is to the 
diameter in a constant ratio. Calling the length n d, where n is a constant quantity, 
we have, in these different cases, the strength as 
rf 3 ' 76 d 3-55 d 4 
n l / x d 1 ' 7 ’ n‘ x d V/> rd dt 
or, making the divisions, these become 
d 2,06 d 1 ' 85 a f 
n 7 ’ n 7 ? ri 
In the first of these cases, the strength varies as a power of the diameter somewhat 
higher than the square; in the second, somewhat lower; and in the third, as the square. 
We may therefore conclude, that in similar pillars the strength is nearly as the square 
of the diameter, or of any other linear dimension*: and as the area of the section is as 
the square of the diameter, the strength is nearly as the area of the transverse section. 
The experiments in Tables I. and II. give for the strength of similar pillars, powers 
of their linear dimensions as below. 
Diameters of pillars 
compared. 
Length of pillars 
compared. 
Breaking weight 
of pillars. 
Powers of the dimen- 
sions. 
Pillars with rounded 
ends. 
inch. 
r -497 
•99 
inches. 
7*5625 
15*125 
lbs. 
526 2 
19752 
1-908 ~ 
V" 
Mean from the powers 1-865. 
•76 
1-52 
15-125 
30-25 
9223 
32531 
1-819 
•99 
L 1*97 
30-25 
60-5 
6105 
25403 
2-057 
' # - .-V j 
Pillars with flat ends. 
f *51 
1-56 
20-166 
60-5 
3830 
28962 
1-841 
•50 
•997 
30-25 
60-5 
1662 
6238 
1-9081 
•51 
1-02 
15-125 
30-25 
6764 
21844 
1-6913 
•50 
1-022 
10-083 
20-166 
8931 
31804 
1-8323^ 
In the Table above, the pillars compared were from models which were similar. 
I have, therefore, neglected slight deviations from similarity in the castings from these 
models. It appears then that the power of the lineal dimensions, according to which 
their strengths vary, is somewhat lower than the second. 
45. If pillars be so formed as equally to resist being crushed (as shown in Art. 6.) 
by the breaking weight, they will be similar. 
We have seen, that when pillars require a force to break them by flexure, which ex- 
ceeds a certain portion of the force which would crush them without flexure, the pillar 
* In deducing this conclusion, Euler remarks, that if, of two similar pillars of the same material, one be 
double the linear dimensions of the other, the larger will but bear four times as much as the smaller, though 
its weight is eight times as great. Berlin Memoirs, 1757. 
