OF THE
ART
OF
Changes.
These clear dayes of Knowledge, that have ransackt the dark corners of most Arts and Sciences, and freed their hidden mysteries from the bonds of obscurity, have also registred this of Ringing, in the Catalogue of their Improvements; as well the Speculative as the Practick part, which of late years remain’d in Embryo, are now become perfect, and worthy the knowledge of the most ingenious. Although the Practick part of Ringing is chiefly the subject of this Discourse, yet first I will speak something of the Art of Changes, its Invention being Mathematical, and produceth incredible effects, as hereafter will appear. But first, I will premise a word or two, to shew what the nature of those Changes are. Some certain number of things are presupposed to be changed or varied; as 2.3.4.5.6. or any greater number whatsoever; then the number of things to be so varied must have the like number of fixed places assigned them. As if five men were sitting upon five stools in a row; the stools are supposed to be fixed places for the five men, but the men by consent may move or change to each others places at pleasure, yet still sitting in a row as at first: now this Art directs how, and in what order those five men may change places with each other, whereby they may sit sixscore times in a row, and not twice alike. And likewise a Peal of five Bells, being raised up to a fit compass for ringing of Changes, are there supposed to have five fixed places, which time assigns to their notes or strokes; yet the notes of the Bells may change into each others places at pleasure: now this Art also directs the manner and method of changing the five notes in such sort, that they may strike sixscore times round, and not twice alike.
The numbers of Changes are thus to be discovered. Two must first be admitted to be varied two wayes; then to find out the Changes in three, the Changes on two must be multiplied by three, and the product will be six, which are the compleat number of Changes on three.
Those six Changes being multiplied by four, will produce 24, which are the compleat number of Changes on four. The 24 Changes on four, being multiplied by five, will produce 120, which are the compleat number of Changes on five. And in like manner the 120, being multiplied by six, will produce 720, which are the compleat number on six. The 720, being multiplied, by seven, will produce 5040, which are the number of Changes on seven. The 5040, being multiplied by eight, will produce 40320, which are the number of Changes on eight. Those Changes on eight, being multiplied by nine, will produce 362880, which are the number of Changes on nine. Those Changes on nine, being multiplied by ten, will produce 3628800, which are the number on ten. Those on ten, being multiplied by eleven, will produce 39916800, which are the number on eleven. Those also being multiplied by twelve, will produce 479001600, which are the compleat number of Changes on twelve. And if twelve men should attempt to ring all those Changes on twelve Bells, they could not effect it in less than seventy five years, twelve Lunar Months, one week, and three days, notwithstanding they ring without intermission, and after the proportion of 720 Changes every hour. Or if one man should attempt to prick them down upon Paper, he could not effect it in less than the aforesaid space. And 1440 being prickt in a sheet, they would take up six hundred sixty five Reams of Paper, and upwards, reckoning five hundred Sheets to a Ream; which Paper at five shillings the Ream, would cost one hundred sixty six Pounds five Shillings,
The reason of the aforesaid Multiplication, by which the numbers of Changes are discovered, and also that those Products are the true numbers of Changes, will plainly and manifestly appear in these following Demonstrations.
But first, two must be admitted to be varied two ways, thus.——
| 12 | |
| 21 |
And then consequently, three will make three times as many Changes as two; for there are three times two figures to be produced out of three, and not twice two the same figures, which are to be produced by casting away each of the three figures one after another. First, cast away 3, and 1.2 will, remain; cast away 2, and 1.3 will remain; cast away 1, and 2.3 will remain. So that here are three times two figures produced out of the three, and not twice two the same figures, as 12. 13. 23. each two may be varied two ways, as before: then to the changes which each two makes add the third figure which is wanting; as to the two changes made by 1.2 add the 3, to the changes on 1.3 add the 2, and to the changes on 2.3 add the 1, and the three figures will stand six times together, and not twice alike, as here appeareth.