A good resource on hand feeding finches:
http://www.finchniche.com/a-handfeeding.php
http://uniquebeak.homestead.com/Handfeeding.html
http://www.ladygouldianfinch.com/features_handfeeding2.mgi
This little guy got tossed, so have been hand feeding from day 2 or 3. I use hand feeding formula and mix in a little baby 1st foods applesauce at first to make sure there is enough moisture and fiber to keep things 'moving' through the bird. I use a very tiny paintbrush to give him very tiny bits of runny food and gradually reduce the amount of water as the bird gets older. It's better to err on the side of more water than less.
He just started squeaking today. He had two siblings, one which had a parent-inflicted gash on its head. That one survived two days after being rescued by me from the cage floor. I also rescued the last egg, which then hatched. That one survived two days and died also.
Not sure why this pair keeps tossing babies though. :-(
To keep them warm I tuck him in a larger prescription bottle, lined with paper towel scraps, tucked in my shirt (yes this is easier for women than men). The cover of the bottle has many holes drilled in it for ventilation, and i usually place a piece of tissue over the bird too to keep it from getting any body parts stuck in the cover holes and grooves. I could drill holes in the bottom as well to aid ventilation. I will probably eventually do that but he seems to be fine like this.
I can even tuck it in the same place for night time and the bird is kept warm by my body heat and there is enough oxygen under the covers for a creature this small. As long as the ventilation holes aren't covered by your skin, it should be fine. This way I don't have to worry about a brooder being too hot or too cold. He goes to the store and church if necessary, and running errands with me too. In a day or two he will be going into the society nest so they can finish raising him. But they have three larger chicks to care for too right now. I want him to catch up a bit.
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