Cryptology Through History and Inquiry

Sect. 2: Of Secret Writing by Altering the Places of the Letters, Where the Powers Remain the Same

Part 1: Of the Combinations of Three or More Letters

The first remarkable improvement I find of this kind of cryptography by altering the places of letters, is by the regular combinations of three, four, or more letters: I had it of a gentleman, who (I am fully satisfied) would put it to no bad use; but since it may fall of bad hands, I have his leave to provide against its harm. And that we may proceed regularly therein it is necessary, first to inquire, “how many several ways any given number of letters may be combined?” (i.e.) How many differnt positions they can regularly admit of. And for that end I have hereunto subjoined the following table:

Construction of the Table.

1. At the left-hand of your table stands a rank of figures expressing the number of letters to be combined, encreasing in an arithmetical proportion from unity, or 1 and by common excess of 1, or unity.
2. To the right hand of these stands the number of Combinations,¹ or several ways they can be combined.
3. As for 1, being it has but one positions, I set 1 opposite to it in the rank of combinations.
4. I multiply 1, in the column of combinations, by two, in the column of letters, and set the product, viz. 2, opposite to it, which shows how often two letters or things can be combined, viz. two times.
5. I multiply the 2 in the row of combinations, by 3 in the column of letters; and over against it I set 6 for the product, which shows that three letters have sixe regular positions or combinations.
6. ...

Demonstration.

1. It is manifest, that one letter or thing has but one position, and two letters have twice the position of 1, viz. once before and once after it. e.g. AB, BA.
2. From the combination of two letters we find that of 3, for the new Letter added is three times applicable to the former positions, viz. in the beginning, middle, and end, e.g. the letter C being joyned with AB, the first combination may be CAB, and the second CBA. Again put C in the middle, and it yields other differences, viz. ACB, BCA. And when C is put in the lat place, you have a fifth and sixth difference, viz. ABC, BAC.
3. From the combination of three letters arises that of four letters; viz. ABCD: for D can be four several times applied to each of the former six differences; e.g. in the first, second, third, and fourth or last place.
4. From the twenty four combinations arising from four letters, you have that of 5; e.g. if the letter E is added, it is five times applicable to each of the former twenty four regular positions: and so for any other number in infinitum.

I have said regular positions, because any number of letters or things may be irregularly varied in their positions very much above this order; for example, AB is capable of these irregular variations, AA, AB, BA, BB; and at this rate, three letters have no less than 27 positions in all, viz. six regular (as above) and twenty one irregular positions. ... - John Falconer [4, pp. 37-40 (65-68)]

A New Method How to Write Secretly by the Art of Combinations

1. To write by the method proposed, a certain number of letters are combined to lock and unlock the epistle. The differences of writing down the positions [of the letters] ... may be varied to a vast number; ...
2. The order of rows is agreed upon in parting.
3. The number of letters combined, which is the key, may be expressed in the epistle by some mathematical figure, as \(\scriptsize\triangle\) for three letters, \(\Box\) for 4, etc. or by some private mark.
4. They [the individuals communicating] frame a rectangular table of as many columns as there are letters combined.
5. The letters so combined are placed in their natural order along the top of the table.
6. Having determined of how many lines the table shall consist, the order of combinations agreed upon is set down in a row in the first column towards the left hand; as you may see in the following table.
7. The table being thus prepared for writing, they observe the order of the combinations, and write according to its direction.
8. When they have placed one letter of every column of all the lines, they begin a new, and so go on until the writing be finished.
9. And lastly, they take the letters out of the table according to their partitions, as so many barbarous words, upon paper apart and send it to the confidant. - John Falconer [4, pp. 68-72 (40-43)]