One source of these "independent animal cells" was the discarded bandages from surgical wards of hospitals. In days before antibiotics, wounds oozed yellow pus cells, as the damaged flesh healed. These pus cells were absorbed by the bandages, and Miescher could wash them out, and collect reasonable numbers, with relatively little effort.
When either whole pus cells, or later, when pure nuclei were treated with weakly alkaline solutions followed by acid neutralization, a precipitate was obtained which "cannot belong among any of the protein substances known hitherto". Miescher was the first person to see DNA!
At first this substance was called nuclein, was found in cells from yeast, kidney, liver, and nucleated blood cells, and had none of the classic properties of proteins; what was it?
As well as containing the elements hydrogen, carbon, nitrogen and oxygen, Miescher also found phosphorus and that the ratio of phosphorus and nitrogen in this substance was unique. So Miescher thought that his "nuclein" was nothing more than a storehouse of phosphorus waiting to be used by the cells for more interesting purposes!
His boss, Hoppe-Seyler was so unimpressed that he delayed publishing these discoveries and the war of 1870 further delayed matters. After checking Miescher's results, and extending the discovery of nuclein to other substances, a paper announcing it's discovery was finally published in 1871. But no one understood what this material was or what it did.
Those discoveries were to take a lot longer.