Encyclopaedia Britannica, or, a Dictionary of arts, sciences, and miscellaneous literature : enlarged and improved. Illustrated with nearly six hundred engravings > Volume 18, RHI-SCR

AMUSEMENTS OF SCIENCE.
thtneti- Moil of our readers are well acquainted with the
Reerea-qyeftion in multiplication refpedting the price of a horfe
turns. from fucceflively doubling a farthing as often as there
”~v " are nails in the horfe’s tlioes. (See Montucla’s Recrea¬
tions by Hutton, vol. i. or Samtfordand Merton, vol. i.).
The following queftion is of a iimilar nature, but ap¬
pears Hill more furprifing.
ji A courtier having performed fame very important fer-
vice to his fovereign,- the /otter wijhing to confer on him
a fuitahle reward, defred him to afk whatever he thought
proper, promiftng that it fhould be granted. The cour¬
tier, who was well acquainted with the fcience of num¬
bers, requejled onhj that the monarch would give him a
quantity of wheat equal to that which would arfe from
one grain doubled 63 times fucceffvely. What was the
Value of the reward ?
The origin of this problem is related in fo curious a
manner by Al-Sephadi, an Arabian author, that it de¬
fences to be mentioned. A mathematician named SelTa,
fays he, the fon of Daher, the fubjecl of an Indian
prince, having invented the game of chefs, hisfovereign
was highly pleafed with the invention, and wifhing to
confer on him fome reward worthy of his magnificence,
defired him to afk whatever he thought proper, affuring
him that it fhould be granted. The mathematician,
however, afked only a grain of wheat for thefirft fquare
of the chefs-board, two for the fecond, four for the
third, and fo on to the laft or 64th. The prince at firft
was almofl incenfed at this demand, conceiving that it
■was ill fuited to his liberality, and ordered his vizir to
comply with Seffa’s requeft $ but the minifler was much
aftonilhed when, having caufed the quantity of corn ne-
ceflary to fulfil the prince’s order to be calculated, he
found that all the grain in the royal granaries, and that
even of all his fubjedls, and in all Alia, would not be
fufficient. He therefore informed the prince, who fent
for the mathematician, who candidly acknowledged his
inability to comply with his demand, the ingenuity of
which afhmifhed him flill more than the game which he
had invented.
To find the amount of this prodigious reward, to pay
which even the treafury of a mighty prince was infuffi-
cient, we (hall proceed moft eafily by way of geometri¬
cal progreffion, though it might be difeovered by com¬
mon multiplication and addition. It will be found by
calculation, that the 64th term of the double progrefTion,
beginning with unity, is 9,223,372,036,854,775,808.
But the fum of all the terms of a double progreffion,
beginning with unity, may be obtained by doubling the
laft term and fubtradling from it unity. The number,
therefore, of the grains of wheat equal to Sefla’s de¬
mand, will be 18,446,744,073,709,551,615. Now, if
a ftandard Englifh pint contain 9216 grains of wheat,
a gallon will contain 73,728 •, and, as eight gallons
make one bufhel, if we divide the above refult by 8
times 73,728, we fliall have 31,274,997,412,295 for
the number of the bufhels of wheat neceffary to dif-
549
charge the promife of the Indian king 5 and if we fup- Arithmeti-
pofe that one acre of land be capable of producing inca* l{ecrea*
one year, 30 bulhels of wheat, to produce this quantity . ^ ‘ .
would require 1,042,499,913,743 acres, which make
more than 8 times the iurface of the globe ; for the dia¬
meter of the earth being fuppofed equal to 7930 miles,
its whole furface, comprehending land and water, will
amount to very little more than 126,437,889,177 fquare
acres.
If the price of a buftiel of wheat be eftimated at
ICS. (it is at prefent, Auguft 1809, I 2S. 6d. per bulh-
el), the value of the above quantity will amount to
15,637,498,706,147b 1 os. ; a fum which, in all proba¬
bility, far furpafies all the riches on the earth *. * Hutton's
Recrca-
To difeover any Number thought of. tions, vol. i.
Of this problem there are feveral cafes, differing j0 t(4j a
chiefly in complexity of operation. number
I. Defire the perfon who has thought of a number, thought oF.’
to triple it, and to take the exaft half of that triple if it
be even, or the greater half if it be odd. Then defire
him to triple that half, and alk him how many times
that product contains 9 ; for the number thought of
will contain double the number of nines, and one more-
if it be odd.
Thus, if 4 has been the number thought of, its triple
will be 12, which can be divided by 2 without a re¬
mainder. The half of 12 is 6, and if this be multiplied
by 3, we {ball have 18, which contains 9 twice, the
number will therefore be 4 equal twice 2, the number
of nines in the laft produdl.
II. Bid the perfon multiply the number thought of
by itfelf; then defire him to add unity to the number
thought of, and to multiply that fum alfo by itfelf j in
the laft place, afk him to tell the difference of thofe two
produdls, which will certainly be an odd number, and
the leaft half of it will be the number required.
Let the number thought of be w, which multiplied
by itfelf gives 100 5 in the next place loincreafed by I
is 11, which multiplied by itfelf makes 121, and the
difference of thefe two fquares is 21, the leaft half of
which being 10, is the number thought of.
This operation might be varied in the fecond ftep by
defiring the perfon to multiply the number by itfelf, af¬
ter it has been diminiftied by unity, and then to tell the
difference of the two fquares, the greater half of which
will be the number thought of.
Thus, in the preceding example, the fquare of the
number thought of is 100, and that of the fame num¬
ber, fubtrafting I, is 81 ; the difference of thefe is 19,
the greater half of which, or 10, is the number thought
of.
III. Defire the perfon to add to the number thought
of its exaft half if it be even, or its greater half if it be
odd, in order to obtain a firft fum ; then bid him add
to this fum its exa£t half, or its greater half, according
as
firft right-hand column produces 22, and that of all the reft 20, which, with the addition of the 2 carried, fupplies
the other 2’s in the line. From this it is evident, that though, for more eafy illuftration, we have given a que*
IHon containing bnly five lines j feven, nine, or any unequal number may be employed, conftrufling the feventh^
mnth, &c. on fimilar principles.