[Sidenote: p. 79.] But it is easier to work this way. Divide by 9 the roots ascertained from the elements, as we have just found 19 units from the name Hector, and read the remaining root. For example, if I divide the 19 by 9, there remains a unit, for twice 9 is 18, and the remainder is a unit. For if I subtract 18 from the 19, the remainder is a unit. Again, of the name Patroclus[78] these numbers 8, 1, 3, 1, 7, 2, 3, 7, 2 are the roots; added together they make 34 units. The remainder of these units is 7, viz., 3 from the 30 and 4 from the 4. Therefore 7 units are the root of the name Patroclus. Those then who reckon by the rule of 9 take the 9th part of the number collected from the roots and describe the remainder as the sum of the roots; but those who reckon by the rule of 7 take the 7th part. For example, in the name Patroclus the aggregate of the roots is 34 units. This divided into sevens makes 4 sevens, which are 28; the [Sidenote: p. 80.] remainder is 6 units. He says that by the rule of 7, 6 is the root of the name Patroclus.[79] If, however, it be 43, the 7th part, he says, is 42, for 7 times 6 is 42, and the remainder is 1. Therefore the root from the 43 by the rule of 7 becomes a unit. But we must take notice of what happens if the given number when divided has no remainder,[80] as for example, if from one name, after adding together the roots, I find, _e. g._, 36 units. But 36 divided by 9 is exactly 4 enneads (for 9 times 4 is 36 and nothing over). Thus, he says the 9 itself is plainly the root. If again we divide the number 45 we find 9 and no remainder (for 9 times 5 is 45 and nothing over), in such cases we say the root is 9. And in the same way with the rule of 7: if, _e. g._, we divide 28 by 7 we shall have nothing over (for 7 times 4 is 28 and nothing left), [and] they say the root is 7. Yet when he reckons up the names and finds the same letter twice, he counts it only once. For example, the name [Sidenote: p. 81.] Patroclus has the Alpha twice and the Omicron twice,[81] therefore he counts the Alpha only once and the Omicron only once. According to this, then, the roots will be 8, 3, 1, 7, 2, 3, 2, and added together make 27,[82] and the root of the name by the rule of 9 will be the 9 itself and by that of 7, 6.