He studied medicine at the Universities of Heidelberg, Freiburg and Leipzig, earning his doctorate in 1879. He served as an assistant to Wilhelm Heinrich Erb (1840-1921) in Heidelberg, Julius Friedrich Cohnheim (1839-1884) in Leipzig, and from 1881 was a clinical assistant to Christian Bäumler (1836-1933) at Freiburg. Here he also worked in the physiological-chemical institute. In 1886 he became an associate professor, followed by a directorship at Eppendorf Hospital in Hamburg (1888). He 1892 he was named professor of internal medicine at the University of Breslau.
With surgeon Theodor Rumpel (1862-1923), he was co-author of an illustrated patho-anatomical atlas called: Pathologisch-anatomische Tafeln nach frischen Präparaten mit erläuterndem anatomisch-klinischem Text (Patho-anatomical panels for fresh preparations with explanatory anatomical-clinical text). Other works by Kast include:
- Ueber die Wirkung des Acetphenetidins. Written with chemist Oscar Hinsberg (1857-1939). Centralblatt für die medicinischen Wissenschaften, Berlin, 1887, 25: 145-148. On the introduction of phenacetin.
- Sulfonal, ein neues Schlafmittel. Berliner klinische Wochenschrift, 1888, 25: 309-314. On the introduction of sulphonal, earlier discovered by Eugen Baumann (1846-1896).
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