Current hierarchical embeddings are inaccurate in both reconstructing the original taxonomy and answering reachability queries over Direct Acyclic Graph. In this paper, we propose a new hierarchical embedding, the Euclidean Embedding (EE), that is correct by design due to its mathematical formulation and associated lemmas. Such embedding can be constructed during the visit of a taxonomy, thus making it faster to generate if compared to other learning-based embeddings. After proposing a novel set of metrics for determining the embedding accuracy with respect to the reachability queries, we compare our proposed embedding with state-of-the-art approaches using full trees from 3 to 1555 nodes and over a real-world Direct Acyclic Graph of 1170 nodes. The benchmark shows that EE outperforms our competitors in both accuracy and efficiency.