Classical Arithmetic, Lesson 18. Book I, Chapter 18

The following tasks are required to complete this lesson:

1. 1. Study the Lesson carefully.
   2. Complete the lesson Assessment.

Lesson

When one number contains the whole of another in itself, and some part of it besides, it is called superparticular. And if the part of the less which it contains is the half, it is called sesquialter; if the third part, sesquitertian; if the fourth, sesquiquartan; if the fifth, sesquiquintan. And the like names being employed to infinity, the form of superparticular numbers will also proceed infinitely. And the greater numbers, indeed, are thus denominated. But of the less, the wholes of which are contained in the greater, and a certain part of them besides; one is called subsesquialter; another subsesquitertian; another subsesquiquartan; another subsesquiquintan; and so on according to the rule and multitude of the greater numbers. The greater numbers also are called leaders, but the less attendants. Of superparticular numbers likewise, the multitude is infinite ; because the progression of their species is boundless. For the sesquialter has far its leaders all the numbers that are naturally triple after 5; but for its attendants, all the numbers that are naturally even after 2. For let the series of natural, of triple, and of double numbers be described in three rows as follows:

The first row therefore contains the series of natural numbers; the second the triple; and the third, the double of them. Hence, if 3 is compared to 2, or 6 to 4, or 9 to 6, or if all the superior triple are opposed to all the inferior double numbers, sesquialter ratio will be produced. For 5 contains in itself 2, and 1 the half of two. Six also contains in itself 4, and 2 the half of 4. And y contains in itself 6, and the half of 6 which is 3. And in a similar manner in the rest. It is likewise requisite to show the method of discovering the sesquitertian, or second species of the superparticular number. And the definition indeed of this comparison is as follows: the sesquitertian is that which when compared to the less number, contains it once, and a third part of it besides. But these numbers are found, if all the terms in a continued series from 4 being made quadruple,’ are compared with all the numbers that are made triple from 3. And the leaders in this case will be quadruple; but the attendants triple. For let there be a series of numbers in a natural order, and under these a quadruple, and under the quadruple a triple series. Let the first triple there fore, be placed under the first quadruple number; the second under the second; the third under the third ; and let all the triples be arranged under all the quadruples after the same manner, as follows:

Hence, if the first number is compared with the first, a sesquitertian ratio will be formed. For 4 contains the whole of 3 in itself, and a third part of 3 besides, i. e. 1. In a similar manner 8 contains the whole of 6, and a third part of it 2. And the same consequence will take place in the rest ad infinitum. It must also be observed, that 3, 6, 9, 12, &c. are attendants, and 4, 8, 12, 16, &c. leaders ; and that the ratio of the former to the latter is subsesquitertian, but of the latter to the former sesquitertian. This also is admirable and most profound in the orders of these numbers, that the first leader and the first attendant are conjoined to each other without the intervention of any other number. But between the second leader, and the second attendant, one number intervenes. Between those in the third rank, two numbers intervene. Between those in the fourth, three. And the intervening numbers are always less by one than the rank of the numbers themselves. But it is necessary » that this should take place in sesquialter, sesquitertian, or other superparticular parts. Thus when 4 is compared to 3, no number intervenes ; for 4 succeeds immediately to 3. But when 8 is compared to 6, which forms the second sesquitertian ratio, one number intervenes; for 7 comes between 6 and 8. Again, when 12 is compared to 9, which forms the third sesquitertian ratio, two numbers intervene, viz. 10 and 11. After the same manner, between those in the fourth order, three numbers intervene; between those in the fifth, four numbers; and so on ad infinitum.

Assessment

• Comprehension Questions