30 MATHEMATICS. 
To determine with staves alone, the height of an object whose foot cannot 
be reached, we employ two of unequal lengths, DE and FG (fig. 8). Erect 
the two staves so that the eye, placed at the ground at J, shall see their 
summits, E and F, in the same line with that of the object, B. The staves 
being of known height, measure the distances, JD and DG (together equal 
to JC): with the same staves repeat the operation at another point of the 
line, JA, as at C’ and D’, obtaining the values J’D’ and D’C’ (together 
=J’C’). As the triangles JDE, JGF, and JAB, are similar, and also 
J‘D'E’, J’C’F’, and J’AB, as well as GF=C’F’ and ED=E'D, we will 
have | 
JD: DE::JG:GF 
ID DG TATA 
JG:GF::JA:AB::JG:JA::GF: AB. 
J’D’: DE: : J'C’: GF 
J'D': DE: : J'A: AB 
JiC': J'A::GF: AB. But 
JG JAsevGEeAB 
— Cee 
IGL JC. JA PAs: GF: AB 
JG and J’C’ are, however, known; JA—J’A=JJ’ is also known, conse- 
GE xa" 
quently AB = ree 
The shadow of an object when the sun shines may be used for measuring 
its height, although this method has no claim to great accuracy. Erect a 
perpendicular post or staff, of known length, and measure as nearly as 
possible at the same time, the length of its shadow and that of the object ; 
then the length of the staff will be to that of the object as the lengths of 
their shadows. If in the line of the shadow we erect a post so that the end 
of its shadow coincides with that of the object’s shadow, then the same 
proportions will hold good, and the method is at the same time more 
convenient (pl. 4, fig. 6). 
If the foot of the object to be measured cannot be reached, we may apply 
the preceding method on two different days, when the sun has a decidedly 
different height, best of all at the time of true noon, when the shadow falls 
exactly in the true meridian. If we indicate the length of the object's 
shadow at the two different times, by C and C’, those of the post’s shadow 
by c, c’,and the length of the post by a, then the height of the object will be 
a(C —C’) 
c—c'* 
Instead of the shadow we may use a horizontal reflecting surface (of oil 
or mercury). Erect at any point, D (fig. 7), a staff, DE, of known length, 
not to exceed a few feet ; find a place, C, between the staff and the object, 
where the mirror shall reflect the top of the object to the eye placed at E. 
In this case, the triangles, CDE, ABC, are similar, and if AC can be mea- 
30 
