For hours I tried various means of reduction, each time getting a littlenearer to the ultimate simplicity; till at last I felt that I hadmastered the principle.
Take the Baconian biliteral cipher as he himself gives it and knock outrepetitions of four or five aaaaa: aaaab: abbbb: baaaa: bbbba: andbbbbb. This would leave a complete alphabet with two extra symbols foruse as stops, repeats, capitals, etc. This method of deletion, however,would not allow of the reduction of the number of symbols used; therewould still be required five for each letter to be infolded. We havetherefore to try another process of reduction, that affecting thevariety of symbols without reference to the number of times, up to five,which each one is repeated.
Take therefore the Baconian Biliteral and place opposite to each itemthe number of symbols required. The first, (aaaaa) requires but onesymbol "a," the second, (aaaab) two, "a" and "b;" the third (aaaba)three, "a" "b" and "a;" and so on. We shall thus find that the 11th(ababa) and the 22nd (babab) require five each, and that the 6th, 10th,12th, 14th, 19th, 21st, 23rd and 27th require four each. If, therefore,we delete all these biliteral combinations which require four or fivesymbols each--ten in all--we have still left twenty-two combinations,necessitating at most not more than two changes of symbol in addition tothe initial letter of each, requiring up to five quantities of the samesymbol. Fit these to the alphabet; and the scheme of cipher is complete.
If, therefore, we can devise any means of expressing, in conjunctionwith each symbol, a certain number of repeats up to five; and if we can,for practical purposes, reduce our alphabet to twenty-two letters,we can at once reduce the biliteral cipher to three instead of fivesymbols.
The latter is easy enough, for certain letters are so infrequently usedthat they may well be grouped in twos. Take "X" and "Z" for instance.In modern printing in English where the letter "e" is employed seventytimes, "x" is only used three times, and "z" twice. Again, "k" is onlyused six times, and "q" only three times. Therefore we may very wellgroup together "k" and "q," and "x" and "z." The lessening of theElizabethan alphabet thus effected would leave but twenty-two letters,the same number as the combinations of the biliteral remaining afterthe elision. And further, as "W" is but "V" repeated, we could keep aspecial symbol to represent the repetition of this or any other letter,whether the same be in the body of a word, or if it be the last ofone word and the first of that which follows. Thus we give a greaterelasticity to the cipher and so minimise the chance of discovery.
As to the expression of numerical values applied to each of thesymbols "a" and "b" of the biliteral cipher as above modified, suchis simplicity itself in a number cipher. As there are two symbolsto be represented and five values to each--four in addition to theinitial--take the numerals, one to ten--which latter, of course, couldbe represented by 0. Let the odd numbers according to their values standfor "a":
a=1 aa=3 aaa=5 aaaa=7 aaaaa=9
and the even numbers according to their values stand for "b":
b=2 bb=4 bbb=6 bbbb=8 bbbbb=0
and then? Eureka! We have a Biliteral Cipher in which each letter isrepresented by one, two, or three, numbers; and so the five symbols ofthe Baconian Biliteral is reduced to three at maximum.
Variants of this scheme can of course, with a little ingenuity, beeasily reconstructed.
APPENDIX C
THE RESOLVING OF BACON'S BILITERAL REDUCED TO THREE SYMBOLS IN A NUMBERCIPHER
Place in their relative order as appearing in the original arrangementthe selected symbols of the Biliteral:
a a a a a a a a a b &c
Then place opposite each the number arrived at by the application of oddand even figures to represent the numerical values of the symbols "a"and "b."
Thus aaaaa will be as shown 9 aaaab will be as shown 72 aaaba will be as shown 521
and so on. Then put in sequence of numerical value. We shall then have:0. 9. 18. 27. 36. 45. 54. 63. 72. 81. 125. 143. 161. 216. 234. 252. 323.341. 414. 432. 521. 612. An analysis shows that of these there are twoof one figure; eight of two figures; and twelve of three figures. Nowas regards the latter series--the symbols composed of three figures--wewill find that if we add together the component figures of each of thosewhich begins and ends with an even number they will tot up to nine;but that the total of each of those commencing and ending with an oddnumber only total up to eight. There are no two of these symbols whichclash with one another so as to cause confusion.
To fit the alphabet to this cipher the simplest plan is to reserve onesymbol (the first--"0") to represent the repetition of a foregoingletter. This would not only enlarge possibilities of writing, but wouldhelp to baffle inquiry. There is a distinct purpose in choosing "0" asthe symbol of repetition for it can best be spared; it would invitecuriosity to begin a number cipher with "0," were it in use in anycombination of figures representing a letter.
Keep all the other numbers and combinations of numbers for purelyalphabetical use. Then take the next five--9 to 45 to represent thevowels. The rest of the alphabet can follow in regular sequence, usingup of the triple combinations, first those beginning and ending witheven numbers and which tot up to nine, and when these have beenexhausted, the others, those beginning and ending with odd numbers andwhich tot up to eight, in their own sequence.
If this plan be adopted, any letter of a word can be translated intonumbers which are easily distinguishable, and whose sequence can beseemingly altered, so as to baffle inquisitive eyes, by the addition ofany other numbers placed anywhere throughout the cipher. All of theseadded numbers can easily be discovered and eliminated by the scribe whoundertakes the work of decipheration, by means of the additions of oddor even numbers, or by reference to his key. The whole cipher is sorationally exact that any one who knows the principle can make a key ina few minutes.
As I had gone on with my work I was much cheered by certain resemblancesor coincidences which presented themselves, linking my new constructionwith the existing cipher. When I hit upon the values of additions ofeight and nine as the component elements of some of the symbols, I feltsure that I was now on the right track. At the completion of my work Iwas exultant for I felt satisfied in believing that the game was now inmy own hands.
APPENDIX D
ON THE APPLICATION OF THE NUMBER CIPHER TO THE DOTTED PRINTING
The problem which I now put before myself was to make dots in a printedbook in which I could repeat accurately and simply the setting forthof the biliteral cipher. I had, of course, a clue or guiding principlein the combinations of numbers with the symbols of "a" and "b" asrepresenting the Alphabetical symbols. Thus it was easy to arrange that"a" should be represented by a letter untouched and "b" by one witha mark. This mark might be made at any point of the letter. Here Ireferred to the cipher itself and found that though some letters weremarked with a dot in the centre or body of the letter, those both aboveand below wherever they occurred showed some kind of organised use. "Whynot," said I to myself, "use the body for the difference between "a" and"b;" and the top and bottom for numbers?"
No sooner said than done. I began at once to devise various ways ofrepresenting numbers by marks or dots at top and bottom. Finally Ifixed, as being the most simple, on the following:
Only four numbers--2, 3, 4, 5--are required to make the number of timeseach letter of the symbol is repeated, there being in the originalBaconian cipher, after the elimination of the ten variations alreadymade, only three changes of symbol to represent any letter. Marks at thetop might therefore represent the even numbers "2" and "4"--one markstanding for "two" and two marks for "four"; marks at the bottom wouldrepresent the odd numbers "3" and "5"--one mark standing for "three" andtwo marks for "five."
Thus "a a a a a" would be represented by "[a:]" or any other letter withtwo dots below: "a a a a b" by ae b, or any other letters similarlytreated. As any letter left plain would represent "a" and any letterdotted in the body would represent "b" the cipher is complete forapplication to any printed or written matter. As in the number cipher,the repetition of a letter could be represented by a symbol which inthis variant would be the same as the symbol for ten or "0." It would beany letter with one dot in the body and two under it, thus--[t:].
For the purpose of adding to the difficulty of discovery, where twomarks were given either above or below the letter, the body mark(representing the letter as "b" in the Biliteral) might be placed at theopposite end. This would create no confusion in the mind of an adviseddecipherer, but would puzzle the curious.
On the above basis I completed my key and set to my work of decipheringwith a jubilant heart; for I felt that so soon as I should have adjustedany variations between the systems of the old writer and my own, workonly was required to ultimately master the secret.