We prove that a binary geometry of rank
n
(
n
⩾
2
)
n\;(n \geqslant 2)
not containing
M
(
K
5
)
M({K_5})
and
F
7
{F_7}
(respectively,
M
(
K
5
)
M({K_5})
and
C
10
{C_{10}}
) as a minor has at most
3
n
−
3
3n - 3
(respectively,
4
n
−
5
4n - 5
) points. Here,
M
(
K
5
)
M({K_5})
is the cycle geometry of the complete graph on five vertices,
F
7
{F_7}
the Fano plane, and
C
10
{C_{10}}
a certain rank
4
4
ten-point geometry containing the dual Fano plane
F
7
∗
F_7^{\ast }
as a minor. Our technique is elementary and uses the notion of a bond graph. From these results, we deduce upper bounds on the critical exponents of these geometries.