NUMBER. 
mo 
In like manner, we might combine three, four, five, &c. 
fquares, together ; and in that cafe it would be found, that 
tliree fquares are capable of forming 128 figures, that four 
could form 256, &c. The immenfe variety of compart¬ 
ments which arife, in this manner, from fo fmall.a num¬ 
ber of elements, is really afionifhins. Father Sebaftian 
gives thirty different kinds, feleffed from a hundred; and 
thefe even are only a veryfmall part of thefe which might 
be formed. 
In confequence of Sebaftian’s memoir, father Douat, 
one of his affociates, was induced to purfue this f'ubjeft 
ftill farther, and to publifh, in the year 1722, a large work, 
in which it is confidered in a different manner. In this 
work it may be feen, that four fquares, each divided into 
two triangles of different colours, repeated and changed 
in every manner poffible, are capable of forming 256 dif¬ 
ferent figures; and that thefe figures themfelves, taken 
two and two, three and three, and fo on, will form a pro¬ 
digious multitude of compartments, engravings of which 
occupy the greater part of the book. 
It is rather furprifing that this idea fhould have been 
fo little employed in architecture; as it might furnifh an 
uiexhauftible fource of variety in pavements, and other 
works of the like kind. However this may be, it forms 
the objeft of a paftime, called by the French Jen rln Par¬ 
quet. The. inftrurnent employed for this paftime, confifts 
of a fmall table, having a border round if, and capable of 
receiving,64., or 100, fmall fquares,each divided into two tri¬ 
angles of different colours,with which peopleamufe them¬ 
felves in endeavouring to form agreeable combinations. 
"The poffible permutations, or changes of fituation, in 
a fmall number, as 3 or 4., muft be obvious enough at firft 
fight ; but, where the number is very confiderable, no¬ 
thing fhort of mathematical demonftration is fufficient 
to fatisfy the mind refpefting the wonderful variability of 
thefe numbers. To give this demonftration, the matter 
is now reduced to a tabular fyftem in the annexed pyramid 
of figures ; the rationale of which may he underftood by 
the three firft digits, 1,2, 3, whofl relative placesat prefent 
-are ir. their natural or arithmetical order, but neverthelefs 
are fufceptible of five permutations, confequently they are 
liable to be placed in fix different pofitions, conftantly in- 
creafing, thus, 123, 132, 213, 231, 312, or 321, whofe 
changes can go no farther, becaufe every figure has been 
aflbeiated with all the others in every poflible way. The 
fame thing has been fhown, at p. 314., with the four firft 
digits; and let us repeat, that the changes are to be 
made into funis which are conftantly increafing by indi¬ 
vidual and gradual receffion, until at laft the order of 
them is completely inverted ; and the only fatisfaftory 
way of effecting and infuring all thefe variations without 
danger of repetition is, not by promifeuous changes, but 
by this grartus; from the order of which, the following 
plain Scholinm may be deduced, in permuting of figures : 
That the units and tens (changing alternately) keep their 
places only once; the hundreds twice; thoufands fix times ; 
tens of ditto twenty-four times ; hundreds of ditto one hun¬ 
dred and twenty times ; &c. and, by dividing any permu¬ 
tation by its own multiplying index, this may be found 
to any extent, as a quotient. But all thefe quotients are 
found alfo by infpebtion only, on the left hand of the 
multipliers in that half of the pyramid. 
The fame mode of procedure is to be adopted with any 
greater fum of digits, numbers, or letters; when it will 
be found, that the permutations or changes increafe in the 
compound ratio of the accumulating numbers, combined 
with all the previous ones; whofe increments are conti¬ 
nually multiplied into all the produ&s of all the foregoing- 
numbers, as may be feen by the Table (which is given 
entire, to elucidate this aftonifhing power) ; and where, 
though the firft digit, card, &c. has but one relative place, 
or rather no relativeat all, buta fixed and abfolute place; 
yet two may be tranflccatcd, as 1, 2, or 2, 1; and three, 
as feen above, into fix places. Alfo twelve will admit of 
479001600 changes. Again; if a party of thirteen per¬ 
sons were to be promifeuoufiy feated in a company, it 
would be 6227020800 chances to one, that, at a fubfequent 
meeting, they were all placed in the fame relative fitua¬ 
tion ; and, if a ftranger to their former arrangement ftiould 
attempt to replace them, or they, having forgotten how 
they were fituated, were to undertake to reftore that order, 
it would require 11,847 years to produce all the poflible 
changes to effedt it, at twelve hours per day, and a half- 
minute for each change, and no miftake ; becaufe, the 
number of half-minutes in one year, at twelve hours 
per day, ftand thus, 60 X 2 X 12 X 365= 525600 ; then 
6227020800-7-525600— 11,847 years. Therefore, if a pad¬ 
lock, or indeed any other lock, were conftrudted of 13 
movable pieces, to form fome word or fentence containing 
13 letters; and which, to make the matter more fecure, 
ought to be cryptographic ; it muft be obvious, from the 
above ftatement, that there would be no danger of fuch 
lock being picked, by any perfon unacquainted with the 
fecret, in a lefs time than 11,847 years; which may be 
called perftSl Jecuriti/. For inftance, the anagrammatized 
fentence, Luhrhtsam ytoc, will admit of 6227020800 
permutations by the g;radus, before it will be grammatized 
into the inverted fentence fought, of Copy Master Hul ; 
which alone would open the lock. Such would be the 
immenfe fecurity derived from only thirteen independent 
pieces t And the 24 letters of the alphabet are capable of 
the egregious lum of 620448401733239439360000 changes. 
Confequently, if thele letters were written fo fmall that 
each one occupied no more than the hundredth part of a 
fquare inch, the permutation of the whole 24 letters would 
coverthecntire furface ofthis world(feaand land) 186153-6 
times ; admitting its fuperficies to be 199,256,910 miles ! 
as demonftrated by Mr. Snart, in his Mathematical Synopfis, 
which ftiows all the propertiesand relative proportions that 
fuhfift between fpheres, circles, and cubes, of any dimen- 
fions. To ftiow how many hundredths of an inch are in 
the furface ofthis globe, the fummation would ftand thus : 
199256910X 3097600 X 129600 = 79991479292313600000 ; 
which fum, forming a divifor to the permutations of 24, 
gives for quotient 7756 4; then, becaufe the permutation 
of all the letters is the lum fought, and this quotient is 
but for of an inch, 7756 4 muft be multiplied by 
24=186153-6, the true number of times of the above per¬ 
mutation to cover the whole furface of this globe, if writ¬ 
ten within the preferibed limits. 
The furprife, then, at the numberof languages, all pro¬ 
duced out of the 24 letters, muft give way to rational 
demonftration, which this Table fo amply fupplies, as well 
as the number of different faces daily feen (no two of 
which are exadtly alike) ; becaufe, if the tout enfemble of 
the human phyfiognomy, conftfting of feature, colour, 
age, animation, fize, &c. amount to 30 identities, which 
may fairly be admitted, the fum of the varieties that 
might be made upon that number would be fo enormous, 
as fhown by experience, as well as the Table, that no 
affignable time could ever be expedited to produce one 
that (hould be an abfolute fac-ftmile of another. For the 
fame tea fob, the 68 tones and femitones of a piano-forte 
can be fo varied, as to produce fuch a divernty of tunes 
as can never be exhaufted. And the wonderful pheno¬ 
mena of the kaleidofcope, which have fo aftonifhed and 
amufed the public, may alfo herein be refolved ; and the 
variety of its appearances be both afeertained and ac¬ 
counted for, by taking the permutation of its number of 
objects, and multiplying by the averaged number of ftdes 
to each objedft. Thus, fuppofing them all to be hexahedra ; 
multiply by 6, and that product will be the permutation 
fought. But, when an attempt is made to fcan the per¬ 
mutation of fifty numbers (the maximum produdl of this 
Table), the human mind is itunhed and confounded at 
the immenfity of the fum! Nor can the molt highly- 
wrought metaphor of human ingenuity convey to the 
underftanding any comprehenfible figure equally ftupen- 
dous ! A fum, to which the aftonilhing permutation of 
twenty-four bears not a million-millionth part fo great a 
proportion 
