123
213
132
312
231
321

Four will make four times as many changes as three. For there are four times three figures to be had out of four, and not twice three the same figures, which are to be produced by casting away each of the four figures by turns. First cast away 4, and 123 will remain; cast away 3, and 124 will remain; cast away 2, and 134 will remain; and lastly, casting away 1, and 234 will remain; so that here is 123, 124, 134, 234, and not twice three the same figures. Now each three may be varied six ways, according to the preceding Example. Then to the six changes which each three makes, add the fourth figure which is wanting; as to the six changes on 123 add the 4, to the six changes on 124 add the 3, to the six changes on 134 add the 2, and to the six changes on 234 add the 1, which renders the changes compleat; for then the four figures stand twenty four times together, and not twice alike, as here appears.

1234
2134
1324
3124
2314
3214
1243
2143
1423
4123
2413
4213
1342
3142
1432
4132
3412
4312
2341
3241
2431
4231
3421
4321

Five will make five times as many changes as four; for there are five times four figures to be had out of five, and not twice four the same figures, which are to be produced as before, by casting away each of the five figures by turns. Cast away 5, and 1234 will remain; cast away way 4, and 1235 will remain; cast away 3, and 1245 will remain; cast away 2, and 1345 will remain; cast away 1, and 2345 will remain. So that here are five times four figures produced, and not twice four the same figures. Now each four may be varied twenty four ways, as in the preceding example; then to the twenty four changes which each four makes, add the fifth figure which is wanting: as to the twenty four changes on 1234, add the 5; to the twenty four changes on 1235, add the 4. to the changes on 1245, add 3. to the changes on 1345, add 2. and to the changes on 2345, add 1. which renders the changes compleat, for then the five figures stand sixscore times together, and not twice alike.

1234512354124531345223451
2134521354214533145232451
1324513254142531435224351
3124531254412534135242351
2314523154241533415234251
3214532154421534315243251
1243512534125431354223541
2143521534215433154232541
1423515234152431534225341
4123551234512435134252341
2413525134251433514235241
4213552134521435314253241
1342513524145231453224531
3142531524415234153242531
1432515324154231543225431
4132551324514235143252431
3412535124451234513245231
4312553124541235413254231
2341523514245133451234521
3241532514425134351243521
2431525314254133541235421
4231552314524135341253421
3421535214452134531245321
4321553214542135431254321

And in this manner the compleat numbers of changes on six, seven, eight, nine, ten, eleven, twelve, &c. may also be demonstrated.

The numbers of changes will also plainly appear by the methods, whereby they are commonly prickt and rung. Now the nature of these methods is such, that the changes on one number comprehends the changes on all lesser numbers, and that so regularly, that the compleat number of changes on each lesser number are made in a most exact method within the greater; insomuch that a compleat Peal of changes on one number seemeth to be formed by uniting of the compleat Peals on all lesser numbers into one entire body; which will manifestly appear in the 479001600 changes on twelve: for that Peal comprehends the 39916800 changes on eleven; these likewise comprehend the 3628800 changes on ten, these changes on ten comprehend the 362880 on nine, these on nine comprehend the 40320 on eight, these on eight comprehend the 5040 on seven, these likewise the 720 on six, the 720 also comprehend the 120 on five, the 120 comprehend the 24 changes on four, these also comprehend the six changes on three, and the six comprehend the two changes on two. Each of these Peals (viz.) on eleven, ten, nine, eight, seven, six, five, four, three, and two, being made in a most exact method within the changes on twelve. For Example, two are first admitted to be varied two ways, thus——

12
21

Now the figure 3 being hunted through each of those two changes, will produce the six changes on three. The term Hunt, is given to a Bell to express its motion in Ringing, which in figures is after this manner. It must lie behind, betwixt, and before the two figures: first behind them thus, 1 2 3; then betwixt them, thus, 1 3 2; now before them, thus, 3 1 2: this is called a hunting motion, and here it has hunted through the first change of the two, wherein it made three variations, as appears in the figures, standing thus in order.——

123
132
312