Of the Great Art of Consonance and Dissonance

Page 4

If, however, you wish to change 5 things, place the numbers in their natural order The "natural order" refers to the sequence of counting numbers: 1, 2, 3, 4, 5., as follows: 1. 2. 3. 4. 5. Then multiply all of them among themselves This is the calculation for a "factorial" (5!), written as 1 × 2 × 3 × 4 × 5.; the result will be 120. I say that the items can be exchanged among themselves that many times, as is shown in the following example, where we demonstrate that five vowels can be changed one hundred and twenty times. For each individual letter out of the five can be placed in the first position twenty-four times original: "vigesies quater", while the remaining letters are shuffled among themselves, as is clear in the following example.

Example of the permutation of five things.

	1		2		3		4		5
	a		e		i		o		u
---	---	---	---	---	---	---	---	---	---
1	aeiou	25	eaiou	49	iaeou	73	oaeiu	97	uaeio
2	aeiuo	26	eaiuo	50	iaeuo	74	oaeui	98	uaeoi
3	aeoiu	27	eaoiu	51	iaoeu	75	oaieu	99	uaieo
4	aeoui	28	eaoui	52	iaoue	76	oaiue	100	uaioe
5	aeuio	29	eauio	53	iaueo	77	oauei	101	uaoei
6	aeuoi	30	eauoi	54	iauoe	78	oauie	102	uaoie
---	---	---	---	---	---	---	---	---	---
7	aieou	31	eiaou	55	ieaou	79	oeaiu	103	ueaio
8	aieuo	32	eiauo	56	ieauo	80	oeaui	104	ueaoi
9	aioeu	33	eioau	57	ieoau	81	oeiau	105	ueiao
10	aioue	34	eioua	58	ieoua	82	oeiua	106	ueioa
11	aiuoe	35	eiuao	59	ieuao	83	oeuai	107	ueoai
12	aiueo	36	eiuoa	60	ieuoa	84	oeuia	108	ueoia
---	---	---	---	---	---	---	---	---	---
13	aoeiu	37	eoaiu	61	ioaeu	85	oiaeu	109	uiaeo
14	aoeui	38	eoaui	62	ioaue	86	oiaue	110	uiaoe
15	aoieu	39	eoiau	63	ioeau	87	oieau	111	uieao
16	aoiue	40	eoiua	64	ioeua	88	oieua	112	uieoa
17	aouei	41	eouai	65	iouae	89	oiuae	113	uioae
18	aouie	42	eouia	66	iouea	90	oiuea	114	uioea
---	---	---	---	---	---	---	---	---	---
19	aueio	43	euaio	67	iuaeo	91	ouaei	115	uoaei
20	aueoi	44	euaoi	68	iuaoe	92	ouaie	116	uoaie
21	auieo	45	euiao	69	iueao	93	oueai	117	uoeai
22	auioe	46	euioa	70	iueoa	94	oueia	118	uoeia
23	auoei	47	euoai	71	iuoae	95	ouiae	119	uoiae
24	auoie	48	euoia	72	iuoea	96	ouiea	120	uoiea

In no different way, numbers arranged under any natural series and multiplied by each other will give the mutations of however many things are proposed. Truly, to understand this, nothing else seems to be required except the following table, from which everything said thus far shines forth clearer than the sun original: "sole clarius elucescunt," a Latin idiom meaning something is perfectly obvious or self-evident..