E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
The horns of a dilemma. (or Syllogismum cornutum); at my wits and; a puzzling question. Dulcarnein is the Arabic dhulkarnein (double-horned, having two horns). Hence the 47th proposition of the First Book of Euclid is called the Dulcarnon, as the 5th is the pons asinorum. Alexander the Great is called Iscander Dulcarnein, and the Macedonian æra the æra of Dulcarnein. Chaucer uses the word in Troylus and Cryseyde, book iii. 126, 127.
The horns of the 47th proposition are the two squares which contain the right angle.
To be in Dulcarnon. To be in a quandary, or on the horns of a dilemma.