The study discusses problems as related to the existence of a so-called A. carotis externa (= ventralis) in the Anamnia, and is based on the analysis of old and new literature. The facts presented are illustrated by several figures, demonstrating the organization pattern of branchial arteries and their various branches in adult animals as well as during the development of all types of branchial vessels. Summarizing the aspects involved, it can be concluded that an A. carotis ventralis does not exist, neither in reality nor as the first stage of an A. carotis externa. All of the efferent vessels coming from the 6 original branchial arteries, always arise from the respective A. branchialis efferens and contain oxygenated blood. As A. hypobranchialis lateralis and medialis these vessels supply the entire hypobranchial region, the floor of the mouth, and the heart. Moreover, a rostral prolongation of the Aorta ventralis is also not proven to exist during the ontogenesis of branchial arches.