Euclid of Alexandria (∼325–∼265BC) was one of the most prominent mathematicians of the Antiquity and is best known for”gathering all mathematical knowledge” in the treatise The Elements. The Elements is a model of how pure mathematics should be written, with axioms, definitions, theorems, and coherent proofs. The long-lasting influence of The Elements has made Euclid a leading mathematics teacher of all time. Bernt Michael Holmboe (1795–1850) was teacher at Christiania Kathedralskole in Norway from 1818 until 1826, after that he was professor at the university in Christiania. Holmboe wrote several textbooks in mathematics, and he was very influential in the development of school mathematics in the first half of the nineteenth century in Norway. Holmboe’s presentation of the subject matter was known to be traditional and ”Euclidean”. This paper will analyse Holmboe’s presentation of some basic definitions, postulates, and theorems. Both Euclid and Holmboe use a rigorous axiomatic method, and one focus of the paper will be to elaborate on Holmboe’s approach to the problematic topic of parallel lines, and the parallel postulate from the Elements. Holmboe gives a proof of the parallel postulate, and this proof is presented in detail.