TABLE II. Combinatorics, showing the number of permutations for things in which there is not a distinct variety, but some are similar.

Combinatorial Table.

This table is a seventeenth-century guide for calculating permutations of a multiset (arrangements of objects where some are identical). While the logic is sound, the printed numbers in columns 5 and 10 contain some calculation errors typical of the era's typesetting; for instance, the permutation of 8 items with 4 similar should be 1,680, but is printed here as 1,663.

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USE OF THE TABLE.

Use of the Table.

This table has 10 columns, decreasing in order proportionally. In the first column, 10 numbers are contained signifying the combinable things, or the series of things to be combined. The second column contains combinations of things where every single item differs from the others. Thus, 5 The Latin text mistakenly says "IV" (4), but refers to the row for 120, which corresponds to V (5). things can be changed 120 times; whereas 7 things can be changed 5040 times. The third column contains combinations of things in which two items are the same. The fourth column contains combinations of things in which three are the same, and so on for the other columns in order up to the tenth column, in which combinations of things where nine items are the same are contained.

The use of this table, therefore, is this: if someone desires to know how many times a name of, for example, 8 letters can be changed, which nevertheless has 4 of the same letters—as this invented name MARABANA In the name M-A-R-A-B-A-N-A, the letter 'A' appears four times. shows—since there are 4 identical letters, one asks how many times it can be combined. Therefore, look in the first column for VIII (for the proposed name has that many letters); then look for the column of the combination of things in which four are the same, and the intersecting angle will give 1663; Modern calculation ($8! / 4!$) yields 1,680. and that is how many times the proposed name of 8 letters, of which 4 are the same, can be combined. Likewise, a name of six letters in which three are the same is changeable in 120 ways; you will find this number at the intersecting angle if you move across from the number VI in the first column and ascend from the third column Kircher counts the "similar" columns starting after the "all diverse" column. upward.