Arithmetic, Geometry, and Music (1492) - Page 18

On a certain useful accident for the knowledge of superparticulars. Chapter 25.

This, moreover, is found to be admirable and most profound in these orders: that the first and the first follower are coupled to each other with no intermission of a number. For the first pass by with nothing placed in the middle; the second interpose 1, the third two, the third 3, and subsequently they increase with an intermission always one less than they are themselves. And this must necessarily be found in sesqualterals, or in sesquitertials, or in other parts of the superparticular. For since 4 is compared against 3, we have skipped none; for after 3, immediately there are 4. But truly 6 against 8, in the second sesquitertial, one intermission has been made. For between 6 and 8, there is only the septenary, which is the skipped number. Again, so that we compare 9 against 12, which are in the disposition of the third, a skip of two middle numbers has been made. For between 9 and 12 are 10 and 11. According to this mode, the fourth disposition skips 3, the fifth 4.

Description through which it is taught that the multiple is older than the other species of inequality. Chapter 26.

Since we have naturally and according to the proper consequence of order placed the multiple species of inequality before all others, and have shown it to be the first species, although this becomes clear to us in the order of a later work, let us here also briefly and most plainly teach that which we proposed. Let there be such a description in which the natural number is placed in order up to the denary, and in the second verse, the double order is iterated; in the third, the triple; in the fourth, the quadruple; and this up to the decuple. For so we shall know how the species of the multiple will be the prince over the superparticular, and the superpartient, and all others, and we shall simultaneously look at certain other things, both toward the most subtle sharpness, and toward most useful knowledge, and toward the most pleasant exercise of the mind.

Tetragona (Square). Length. Second unit.

First unit.

Width.

Third unit.

A 10x10 multiplication table, also known as a Boethian table or table of Pythagoras. A diagonal line descends from the top-left to the bottom-right corner, crossing through the square numbers (1, 4, 9, 16, 25, 36, 49, 64, 81, 100), which are each marked with a cross or asterisk symbol.

1*	2	3	4	5	6	7	8	9	10
2	4*	6	8	10	12	14	16	18	20
3	6	9*	12	15	18	21	24	27	30
4	8	12	16*	20	24	28	32	36	40
5	10	15	20	25*	30	35	40	45	50
6	12	18	24	30	36*	42	48	54	60
7	14	21	28	35	42	49*	56	63	70
8	16	24	32	40	48	56	64*	72	80
9	18	27	36	45	54	63	72	81*	90
10	20	30	40	50	60	70	80	90	100*

Second unit. Length. Tetragona (Square).

On the reason and exposition of the digested formula. Chapter 27.

If, therefore, the two first sides of the proposed formula that make the angle are looked at from the first up to the 10 and 10 preceding, and if the lower orders are compared, which, beginning the angle from 4, place a limit in the twenties, the double, that is, the first species of multiplicity, is shown, such that the first exceeds the first by only a unit, as 2 [exceeds] 1; the second exceeds the second by a binary, as 4 [exceeds] 2; the third [exceeds] the third by three, as 6...