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Classical Arithmetic, Lesson 17. Book I, Chapter 17

The following tasks are required to complete this lesson:

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

Lesson

Again, the multiple is the first part of greater inequality, being more ancient than all the others, and naturally more excellent, as we shall shortly demonstrate. This number however is such, that when compared with another, it contains the number with which it is compared more than once.

This multiple inequality also is first seen in the natural series of number. For all the numbers that follow unity have the relation of multiples to it. Thus 2 with relation to unity is duple; 3 is triple; 4 quadruple; and thus proceeding in order, all multiple quantities are produced.

The inequality however, which is contra-distinguished to this, is called submultiple, and this also is the first species of less quantity. But this number is such that when compared with another, it measures the sum of the greater number more than once. If therefore, the less number measures the greater only twice, it is called subduple: but if three times, subtriple: if four times, subquadruple; and so on ad infinitum. And they are always denominated with the addition of the preposition sub.

Since, however, multiplicity and submultiplicity are naturally infinite, the proper generations of the species also admit of infi nite speculation. For if numbers are arranged in a natural series, and the several even numbers are selected in a conxequent order, these even numbers will be double of all the even and odd numbers from unity, that follow each other, and this ad infinitum. For let this natural series of numbers be given, viz. I. 2. 3. 4. .5. 6. 7- 8. 9- 10. 11. 12. 13. 14. 15. 16. 17. 18. 19- 20. If therefore, in this series, the first even number is assumed, i. e. 2, it will be the double of the first, i. e. of unitv. But if the following even number 4 is assumed, it will be the double of the second, i. e. of 2. If the third even number 6 is assumed, it will be the double of the third number in the natural series, i. e. of 3. But if the fourth even number is assumed, i. e. 8, it will be the double of the fourth number 4. And the same thing will take place without any impediment in the rest of the series ad infinitum.

Triple numbers also are produced, if in the same natural Series two terms are always omitted, and those posterior to the two are compared to the natural number, 3 being excepted, which as it is triple of unity passes over 2 alone. After 1 and 2 therefore, 3 follows which is triple of 1 . Again, 6 is immediately after 4 and 5, and is triple of the second number 2. The number 9 follows 7 and 8, and is triple of the third number 3. And the like will take place ad infinitum.

But the generation of quadruple numbers begins by the omission of 3 terms. Thus after 1. 2. and 3, follows 4, which is quadruple of the first term I. Again, by omitting 5, 6, and 7, the number 8, which is the fourth following term, is quadruple of the second term 2. And after 8, by omitting the three terms 9, 10, and 1 1, the following number 12, is quadruple of the third term 3. This also must necessarily be the case in a progression to infinity : and if the addition always increases by the omission of one term, different multiple numbers will present themselves to the view in admirable order. For by the omission of four terms a quintuple multiple, of five a sextuple, of six aseptuple, of seven an octuple, and so on, will be produced, the name of the multiple being always one more than – that of the terms which are omitted.

And all the double terms indeed are always even. But of the triple terms, one is always found to be odd, and another even. -Again, the quadruple terms always preserve an even quantity; and they are formed from the fourth number, one of the prior even numbers being omitted in order ; first the even number 2 ; then 6 being omitted, 8 follows as the next quadruple term ; and after 8, the quadruple number 12 follows, the even number 10 being omitted. And so on in the rest. But the quintuple resembles the triple multiple ; for in this the even and odd terms have an alternate arrangement.

Assessment

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