"Destination Host Unreachable" on local network
 

I am running Red Hat 8.0 on my computer on witch I have 2 Ethernet adapters. One connected on the internet, and the other connected to another computer I have in my room and I use for all sorts of tests. Pinging from one another is not possible, yet the green light from the adapter is on. What can be wrong ?
I looked in the routing table and it looks like this :
Kernel IP routing table
Destination Gateway Genmask Flags Metric Ref Use Iface
193.226.83.128 * 255.255.255.128 U 0 0 0 eth0
192.168.0.0 * 255.255.255.0 U 0 0 0 eth1
127.0.0.0 * 255.0.0.0 U 0 0 0 lo
default a-docsis-C51-83 0.0.0.0 UG 0 0 0 eth0

I see that it has the route for network 192.168.0.0 , so what's wrong ?

And one more thing, where is my route towards the internet ? My gateway is 193.226.83.129, as set from the "Network Device Control" on eth0, but I don't see it in the table. Can someone please clear me up ?