Note 143 — Chapter IV

It will be found that, taking Euclid’s first book, proposition 6 is obtainable by contraposition from proposition 18, and 19 from 5 and 18 combined; or that 5 can be obtained by contraposition from 19, and 18 from 6 and 19. Similar relations subsist between propositions 4, 8, 24, and 25; and, again, between axiom 12 and propositions 16, 28, and 29. Other examples might be taken from Euclid’s later books. In some of the cases the logical relations in which the propositions stand to one another are obvious; in other cases some supplementary steps are necessary.