Page images
PDF
EPUB
[blocks in formation]

This problem differs from the preceding, in that the game must necessarily end in twenty-three throws, whereas in the former, the play may be unlimited, on account of the reciprocations of loss

and gain, which destroy one another. Let a and represent the proportion of the chance for throwing 11 and 14; and raise a+ to the twentythird power, or to a power whose index is the number of all the counters wanting one, and the 12 first terms of that power will be to the 12 last in the same proportion as the probabilities of winning.

Prob. VIII. Three gamesters A, B, and C, out of a heap of twelve counters, of which four are white and eight black, dracu blindfold one counter at a time, in this manner: A begins to draw, B follows A, C follows B; then A begins again: and they continue to draw in the same order, till one of them, who is to be reputed the winner, draws the first white: what are their respective probabilities of winning? Let n be the number of counters, a the number of white, and b the number of black, and 1 the stake or sum played for.

[blocks in formation]

1. A has a chances for a white counter, and b chances for a black one; and, therefore, the probability of his winning is -; and his expectation on the stake 1, when he begins to draw,

is

n

a

a -: now#

n

a+b

=

[ocr errors]
[blocks in formation]
[ocr errors]

=

[ocr errors]

stake, there will remain, 12. B has a chances for a white counter, and the number of remaining counters is n-1; therefore his probability of winning will be. and his

a n-1

[blocks in formation]
[ocr errors]

b

[ocr errors]

a b nx(n−1)

from,

h

n

ab

Q, R, S, &c. denote the preceding terms, and take as many terms of this series as there are units in b+1, (for b representing the number of black b+1), then the sums of the first, fourth, seventh, counters, the number of drawings cannot exceed &c. terms, of the second, fifth, eighth, &c. terms, and of the third, sixth, &c. terms will be the respective expectations of A, B, C ; or, as the stake is fixed, these sums will be proportional to their respective probabilities of winning. Let n, then, in the case of this problem, =12, a=4, and 6=8; and the general series will become +P+

[subsumed][subsumed][merged small][merged small][merged small][ocr errors][merged small][merged small][subsumed]

Y; or, multiplying the whole by 495 in order to take away the fractions, the series will be 165+ 120 +84 +56 + 35+ 20+ 10+4+1; and A will have 165 +56 +10=231, B, 120+35+4=159, and C, 84+20+1=105; therefore their respect ive probabilities of winning will be proportional to the numbers 231, 159, and 105, or 77, 53, and 35.

Prob. IX. A and B having twelve counters, four of them white, and eight black; A wagers with B, that taking out seven counters, blindfold, three of them shall be white: what is the ratio of their expectancies? 1. Seek how many cases there are for seven counters to be taken out of twelve; they will be found from the doctrine of combinations to be 792.

12 11 10 9 8 7 6
-X - =792.

[merged small][ocr errors]

2. Set aside three white ones, and find all the cases wherein four of the eight black ones may be combined therewith; they will be found to be 70. 8 77 6 5

I

-X. -X- -X =70. 3 4

2

And since there are four cases, in which three white may be taken out of four; multiply 70 by 4: thus the cases, wherein three whites may come out with four blacks, are found to be 280.

3. By the common rules of gaming, he is reputed conqueror, who produces an effect oftener will be than he undertook to do, unless the contrary be expressly agreed on; and therefore, if A take out four whites with three blacks, he wins. Set aside four whites, and then find all the cases wherein three of the eight blacks may be combined with four whites: these cases will appear to be 56.

[ocr errors]

aud

=

; but nb-ab bb; therefore it will

x (n-1) bx (6-1)

be

[ocr errors][merged small]

S. C has a chances for a white counter, and the number of remaining counters is n-2; therefore his probability of winning will be; and his expectation in the remaining stake will be bx (b-1)xa

n× (n−1) × (n−2) ̊

n-2

[blocks in formation]
[blocks in formation]

From the solution of this problem it appears, that, if a BE the number of white counters, b the number of black, n the whole number =a+b, c the number of counters to be taken out of n, and p the number of white counters to be found precisely in c, then the number of chances R+S, &c. in which P, for taking none of the white, or one single white, or two white and no more, or three white and ne

b Wherefore, write down the series + n n-1

a

b-2

b-3

[ocr errors][merged small]

P

[blocks in formation]

b X

411 a-2

-x

[ocr errors]

1

2

3

a-3
4

&c.

6-1 b-2

X

2

--X--X

2

792

[ocr errors][merged small][merged small][ocr errors]

&c. continued till the num

ber of terms in which there is a be equal to p,
and the number of terms in which there is b be
equal to c-p. And the number of all the chances
for taking a certain number c of counters out
of the number n is expressed by the series
N 2-1 n-2
&c. continued to as many terms
as there are units in c, for a denominator. E. gr.
Resume the supposition of the problem, only that
of the seven counters drawn, there shall not be
one white; and let p=0, and c-p=7=b1; then
taking 1 of the first series and 7 terms of the se-
cond, the number of chances will be 1x8; the
ratio of which to all the 7's that can be taken out
8 1
of 12 is
: therefore the odds that there
99
shall be one or more white counters among the 7
which are drawn are 98 to 1. The probability
of drawing one white counter and no more will be
112 14
or the odds 85 to 14: the probability
792 99
of drawing all the 4 white among the 7 will be
56 7
or the odds 92 to 7. If n and c
792 99
were large, the foregoing method would be im-
practicable; and, therefore, the following theorem
may be used. Let n, c, and p be as before; and
make n-c=d. The probability of taking pre-
cisely the number p of white counters will be
exc-1xc-2, &c. xdxd- 1 x d−2, &c. x
a-l a-2

[blocks in formation]

ratio of their expectancies? This problem M. Ber-
noulli solves analytically. Here, calling the
number of gamesters n+1, he finds that the pro-
babilities of any two immediately following
each other in the course of playing, are in the
ratio 1+2" to 2"; and therefore the expectancies
of the several gamesters, A, B, C, D, E, &c. are
in a geometrical progression 1 + 2" : 2" : : a: c
::cd::d: e, &c. Hence it is easy to deter-
mine the state of the probabilities of any two
gamesters, either before the game, or in the course.
thereof. If, e. gr. there be three gamesters, A, B,
C, then n=2 and 1+2′′ : 2′′ :: 5:4::a: c; that
is, their several probabilities of winning, before
A have overcome B, or B, C, are as the num-
bers 5, 5, 4; and therefore their expectancies are
5 5 4
for all of them taken together must
14' 14' 14
make 1, or absolute certainty. After A has over-
come B, the probabilities for A, B, and C, will be

[blocks in formation]

77, as in the answer above. If there be four gamesters, A, B, C, D, their probabilities from the beginning will be as 81, 81, 72, 64. After A has beat B, the several probabilities of B, D, C, A, will be as 25, 32, 36, 56, respectively. Af ter A has beat B and C, the probabilities of C, B, D, A, will be as 16, 18, 29, 87.

Prob. XII. Three gamesters, A, B, and C, whose dexterities are equal, deposit each one piece, and engage apon these terms: that two of them shall begin to play, and that the vanquished party shall give place to the third, who is to take up the conqueror; and the same condition to go round: each person when vanquished, forfeiting a certain sum to the main stake; which shall be all swept by the person who first beats the other two successively. How much, now, is the chance of A, and B, better or worse than that of C? 1. If the forfeiture be to the sum each person first deposited, as 7 to

nxn-in-2 × n-3 × n-4 × u−5 × n − 6xn 6, the gamesters are upon an equal footing. 2. If

-7xn-8, &c.

Note, The first series of the numerator contains as many terms as there are units in p; the second as many as there are units in a-p; and the third as many as there are units in p; and the denominator as many as there are units in a.

To avoid tiresome prolixity in this article, we must desist from farther investigations; which, in the following problems, grow very long, and more perplexed. In the rest, therefore, we shall content ourselves to give the answer, or result, without the process of arriving at it; which may be of use, as it furnishes so many data, from whence, as standards, we may be enabled occasionally to judge of the probability of events of the like kinds; though without letting the mind into the precise manner and reason thereof.

the forfeiture be in a less ratio to the deposit, A
and B are on a better footing than C; if in a
greater ratio, the advantage is on the side of C.
3. If there were no fines, the probabilities of win-
ning would be proportional to the expectations;
and the expectations, after the first game, would
12 6 2
be
the first belonging to B, the se-
7' 7' 7

cond to C, and the third to A; and, therefore,
dividing the sum of the probabilities belonging to
A and B into two equal parts, the probabilities of
winning would be proportional to the numbers 5,
4, 5; and it is 5 to 2 before the play begins that
either A or B win the set, or 5 to 4 that one of
them who shall be fixed upon win it. 4. If three
gamesters, A, B, and C are engaged in a povie,
and have not time to play it out, but agree to di-
vide (s) the sum of the stake and fines, in propor-
4
-s will be the
tion to their respective chances;

Prob. X. A and B play with two dice on this condition, that A shall win, if he throws six: and B, if he throws seven: A to have the first throw, in lieu of rubich B to have two throws; and both to continue with two throws each turn, till one of them wins: What is share of B, who has got one game:s that of C, the ratio of the chance of A to that of B? Ans. As 10355 to 12276.

Prob. X1. If any number of gamesters, A, B, C, D, E, &c. equal in point of dexterity, deposit each one piece of money; and engage, on these conditions, that two of them, A and B, beginning the game, whichever of them shall be overcome, shall give place to the third C, who is to play with the conqueror; and the conqueror here to be taken up by the fourth man D; and thus on, till some one, having conquered them all round, draws the stake: what is the

who should next come in, and the share of A, who was last beat.

M. Bernoulli gives an analytical solution of the same problem, only made more general; as not being confined to three gamesters, but extending to any number at pleasure.

Prob. XIII. A and B, two gamesters of equal dexterity, play with a given number of balls; after some

[blocks in formation]

Prob. XIV. Two gamesters, A and B, of equal dexterity, are engaged in play, on this condition, that as often as A exceeds B, be shall give him one piece of money; and that B shall do the like, as oft as ▲ exceeds him; and that they shall not leave off till one has won all the other's money and now having four pieces, treo by-standers, R and S, lay a wager on the number of turns in which the game shall be finished; viz. R, that it shall be over in ten turns: what is the value of the expectancy of S?

[ocr errors]

560

1024

or

35

64

[blocks in formation]

and subtracting from 1, the remainder will

express the probability of the play ending in ten games; and it is 35 to 29, if two equal gamesters play together, that there will not be four stakes lost on either side in ten games.

if each player has five pieces, and the wager were, that the game shall end in ten turns, and the dexterity of A were double that of B, the expectancy of S would be

3800

6561

If each gamester have four pieces, and the ratio of the dexterities be required to make it an even wager, that the game shall end in four turns, it will be found that the one must be to the other as

3.274 to 1. And if the skill of either be to that of the other as 13 407 to 1, it is a wager of 3 to 1, that the play will be ended in four games.

If each gamester have four pieces, and the ratio of their dexterities be required to make it an even

day, that the game shall be ended in six turns, the

answer will be found to be as 2.576 to 1.

Prob. XV. Two gamesters, A and B, of equal dexterity, having agreed not to leave off playing till ten games are overs a spectator, R, lays a wager with another, S, that by that time, or before, A shall have beat B by three games: what is the value 350 11 of the expectancy of R? of the same 1024' 32 wager; or it is to that of S, as 352 to 672. See BASSET, HAZARD, LOTTERY, PIQUET, QUADRILLE, RAFFLING, WHISK, &c.

or

GAMMARUS, in the Fabrician system of entomology, a tribe of the cancer genus. See CANCER.

GA'MMER. s. The compellation of a woman corresponding to gaffer.

GAMMON. s. (gambone, Italian.) 1. The buttock of a hog salted and dried; the lower end of the flitch (Dryden). 2. A kind of play with dice (Thomson).

GAMMUT, GAMUT, or GAM-UT, in music, the name given to the table or scale laid down by Guido Aretinus, to the notes of which he applied the syllables, ut, re, mi, fa, sol, la. See GUIDO.

The gammut is also called the harmonical hand, because Guido first arranged his notes upon the figure of a hand.

This scale is an improvement upon the diagram of the ancients, which was indeed confined within too narrow limits. Guido added four notes above the note hyperboleon, and

one below the proslambanomenos: the latter he called hypo-proslambanomenos, and denot edit by the letter G or the Greek r gamma, which, being at the foot of the scale, imparted a name to the whole. The gammut was divided into three columns, the first called molle (Hat), the second natural, and the third durum (sharp). It consisted of twenty notes, viz. two octaves and a major sixth. The first octave was distinguished by capital letters, as G, A, B, &c. The second by small letters, g, a, b, &c. and the supernumerary sixth by double letters, as gg, aa, bb, &c. By the word gammut we now generally understand the whole present existing scale; and to learn the names and situations of its different notes is to learn the gammut.

'GAN, for began, from 'gin, for begin (Spenser).

GANACHES, in the manage, the two bones on each side of the hinder part of a horse's head, opposite its onset, or neck, which form the lower jaw, and give it motion. Here it is that those glands are placed which are chiefly affected in the strangles or glanders.

T. GANCH. v. a. (ganciare, Italian.) To drop from a high place upon hooks, by way of punishment; a practice in Turkey.

GA'NDER. s. (zandja, Saxon.) The male of the goose. See ANAS.

GANDERSHEIM, a town of Lower Saxony, in Germany. Lat. 51. 54 N. Lon. 18. 20 E.

To GANG. v. a. (gangen, Dutch.) To go; to walk an old word not now used, except ludicrously (Spenser. Arbuthnot).

GANG. s. (from the verb.) A number herding together; a troop; a company; a tribe (Prior).

GANGES, a river of Asia, which rises by two branches from the mountains of Kentaisse, in the country of Thibet; these two branches take a westerly direction, inclining to the north, for a course of about 300 miles in direct distance, when meeting the great chain or ridge of mount Himmaleh, which extends from Cabul along the north of Hindustan, and through Thibet, the rivers are compelled to turn to the south, in which course they unite their waters, and form what is properly termed the river Ganges. This body of water now forces a passage through the ridge of mount Himmaleh, at the distance, possibly, of 100 miles below the place of its first approach to it, and sapping its foundation, rushes through a cavern, and precipitates itself into a vast bason, which it has worn in the rock, at the hither foot of the mountains. From this second source (as it may be termed) of the Ganges, its course becomes more eastwardly than before, through the rugged country of Sirinagur, until, at Hurdwar, it finally escapes from the mountainous tract, in which it has wandered for about 800 British miles. At Hurdwar it opens itself a passage through mount Sewallick; which is the chain of mountains that borders on the level country, on the north of the province of Delhi. After entering Hindustan, it passes

by Anopsheer, Furruckabad, Canoga, Cawnpour, Allahabad, where it is joined by the Jumna, Merzapour, Chunar, Benares, Patna, thirty-six miles above which it is joined by the Dewah, and sixteen miles above the same town by the Soane, and opposite to it by the Gunduck. After leaving Patna, it passes by Bar, Monghir, forty miles east of which it is joined by the Cosa, it then passes by Rajemal, forty miles below which it is joined by a branch of the Sanpoo, or Teesta, and eighty miles below that by another branch of the same river. Soon after which it divides into a multitude of branches, called the Mouths of the Ganges, which empty themselves into the Bay of Bengal, in Lat. 21. 40. to 22 N. See BURRAM

FOOTER.

The Hindus regard this river as a kind of deity; hold its waters in high veneration, and visit it annually from all parts of Hindustan, in order to perform certain superstitious rites. GANGLION. (ganglion, yayyhuv, a kuot.) In anatomy, is applied to a knot in the course of a nerve. In surgery it is an encysted tumour, formed in the sheath of a tendon, and containing a fluid like the white of an egg. It most frequently occurs on the back of the hand or foot.

GANGRENE. (gangræna, yryyfaive; from ayu, to feed upon.) A mortification of any part of the body, before endowed with vitality. It is known by the insensibility, coldness, lividness, and flaccidity of the part, and by the foetor it exhales.

To GANGRENE. v.a. (gangrener, French.) To corrupt to mortification (Dryden).

To GANGRENE. v. . To become mortified (Wiseman).

GA'NGRENOUS. a. (from gangrene.) Mortified; producing or betokening mortification (Arbuthnot).

GANGUE, or MATRIX, is a general term to express the earthy and stony substances in which metallic ores are generally enveloped. These substances are various; frequently spar, quartz, fluors, hornblend, or sulphat of barytes. By German mineralogists, the word gang is used in a different sense, to denote the metallic vein itself.

GA'NGWAY. s. In a ship, the several ways or passages from one part of it to the other. GA'NGWEEK. s. (gang and week.) Roga

tion week.

GANJAM, a town of the peninsula of Hindustan, in one of the northern Circars, subject to the English. Lat. 19. 22 N. Lon. 85. 20 E.

GANNET, in ornithology. See PELI

CANUS.

GANTELOPE. GA'NTLET. S. (gante lope, Dutch.) A military punishment, in which the criminal running between the ranks receives a lash from each man (Dryden).

GANAMEDE, in fabulous history, son of Tros, king of Troy, was the most beautiful youth that ever was scen. Jupiter was so charmed with him, that he carried him away, and made him his cup-bearer in the room of

Hebe. He deified this youth, and to comfort his father, made him a present of some very swift horses. The abbe le Pluche observes, that Ganymede was the name of the horse or image exposed by the ancient Egyptians, to warn the people before their annual inundations, to raise their terraces to a just and proper height.

GANZA, a kind of wild goose (Hudibras). GAOL (gaola, Fr. geole, i. e. caveala, a cage for birds), is used metaphorically for a prison. It is a strong place or house for keeping of debtors, &c. and wherein a man is restrained of his liberty to answer an offence done against the laws and every county hath two gaols; one for debtors, which may be any house where the sheriff pleases; the other for the peace and matters of the crown, which is the county gaol. If a gaol be out of repair, or insufficient, &c. justices of peace, in their quarter sessions, may contract with workmen for the rebuilding or repairing it; and by their warrant order the sum agreed on for that purpose to be levied on the several hundreds and other divisions in the county by a just rate, 11 & 12 William III. c. 19. See PRISON.

GAOL-DELIVERY. The administration of justice being originally in the crown, in former times our kings in person rode through the realm once in seven years, to judge of aud determine crimes and offences; afterwards justices in eyre were appointed; and since, justices of assise and gaol-delivery, &c. A commission of gaol-delivery is a patent in nature of a letter from the king to certain persons, appointing them his justices, or two or three of them, and authorising them to deliver his gaol at such a place of the prisoners in it: for which purpose it commands them to meet at such a place, at the time they themselves shall appoint; and informs them, that, for the same purpose, the king hath commanded his sheriff of the same county to bring all the prisoners of the gaol and their attachments before them at the day appointed. The justices of gaoldelivery are empowered by the common law to proceed upon indictments of felony, trespass, &c., and to order to execution or reprieve: they may likewise discharge such prisoners as on their trials are acquitted, and those against whoin, on proclamation being made, no evidence has appeared: they have authority to try offenders for treason, and to punish many particular offences, by statute 2 Hawk. 24, 2 Hale's Hist. Placit. Cor. 35.

GAOLER, the keeper of a gaol or prison. Sheriffs are to make such gaolers for whom they will be answerable: but if there be any default in the gaoler, au action lies against him for an escape, &c., yet the sheriff is most usually charged.-2. Inst. 592. Where a gaoler kills a prisoner by hard usage, it is felony.-3. Inst. 52. No fee shall be taken by gaolers, but what is allowed by law and settled by the judges, who may determine petitions against their extortions, &c. 2 Geo. II. c. 22.

GAP. s. (from gape.) 1. An opening in a

broken fence (Tusser). 2. A breach (Knolles). 3. Any passage (Dryden). 4. An avenue; an open way (Spenser). 5. A hole; a deficiency (More). 6. Any interstice; a vacuity (Swift). 7. An opening of the mouth in speech during the pronunciation of two successive vowels (Pope). 8. To stop a GAP. To escape by some mean shift (Swift) 9. To stand in the GAP. To make defence.

GAP, an ancient town of France, in the department of the Upper Alps, and lately a bishop's sec. It is seated on the small river Bene. Lat. 44. 34 N. Lon. 6. 10 E.

GAP-TOOTHED. a. (gap and tooth.) Having interstices between the teeth (Dryden).

To GAPE. v. n. (zeapan, Saxon.) 1. To open the mouth wide; to yawn (Swift). 2. To open the mouth for food, as a young bird (Dryden). 3. To desire earnestly; to crave (Denham). 4. To open in fissures or holes Shakspeare). 5. To open with a breach Dryden). 6. To open; to have an hiatus (Dryden). 7. To make a noise with open throat (Roscommon). 8. To stare with hope or expectation (Hudibras). 9. To stare with wonder (Dryden). 10. To stare irreverently (Job).

GA'PER. s. (from gape.) 1. One who opens his mouth. 2. One who stares foolishly. 3. One who longs or craves (Carew). GAR, in Saxon, signifies a weapon: so Eadgar is a happy weapon (Gibson). To GAR. v. a. (giera, Islandick.) To cause; to make: obsolete (Spenser).

GAR-FISH, in ichthyology. See Esox. GARAMOND (Claude), a very eminent French engraver and letter-founder, was a native of Paris, where he died in 1561. He was the first who banished the Gothic, er black letter, from printing; for which he substituted the Roman letter. His types were very good, and particularly the small Roman, which was called by way of eminence Garamond's small Roman. By the command of Francis I., he cast the three Greek types employed in Robert Stephens's beautiful editions of the New Testament, and various Greek authors.

GARASSE (Francis), a French Jesuit, born at Angoulame, in 1585. He was a man of considerable imagination, but of a bad taste, and scurrilous in his style of writing. In 1625 he published a book entitled, A Summary of the principal Truths of the Christian Religion: this was attacked by the abbot of St. Cyran, and was the principal cause of the controversy between the Jesuits and Jansenists. It was censured by the Sorbonne; and the Jesuits prudently banished Garasse to one of their houses at a distance from Paris. He died of the plague, which he caught at Poictiers, while he was attending persons afflicted with that dreadful disorder; this happened in 1631. GARB. s. (garbe, French.) 1. Dress; clothes; habit (Milton). 2. Fashion of dress (Denham). 3. Exteriour appearance (Shakspeare).

GARBAGE. s. (garbear, Spanish.) The bowels; the offal (Roscommon). VOL. V.

GARBEL. s. A plank next the keel of a

ship.

GA'RBIDGE. GA'RBISH. S. Corrupted from garbage (Mortimer).

To GARBLE. v. a (garbellare, Italian.) To sift; to part; to separate the good from the bad (Locke).

GA'RBLER. s. (from garble.) He who separates one part from another (Swift).

GARCILÁSSO, or GARCIAS LASSO DE LA VEGA, an eminent Spanish poet, was descended from a noble family, and born at Toledo in 1500. He was educated under the eye of the emperor Charles V. who had a par ticular esteem for him. Having accompanied that prince in his expeditions, he received a wound of which he died, at Nice, in his 36th year. Garcilasso is one of those to whom the Spanish poetry owes the greatest obligations: he extended its bounds, and introduced many beauties. His works were printed at Naples in 1664, by the learned Sanctius.

GARCINIA, in botany, a genus of the class dodecandria, order monogynia. Calyx four-leaved, inferior; petals four; berry eightseeded, crowned with the stigma. Four species; all East Indian trees. The two following are the species chiefly worthy of notice.

The

1. G. mangostana. Mangostan. A Java tree, about the size of a crab-apple, with ovate leaves, and one-flowered peduncles. flower like that of a single rose; the fruit round, about the size of an orange, and very delicious. The Japanese suppose it the most elegant tree indigenous to their island, and hence it is largely cultivated in all their pleasure-gardens.

2. G. cambogia. Gamboge-tree. With elliptic leaves; solitary, terininal, and nearly sessile flowers. It is a native of India; and the gum-resin known among ourselves by the name of gamboge is obtained by wounding the bark.

GARÇON, or GARSOON, a French term, literally signifying a boy or male child any time before his marriage. It is also applied to certain inferior officers, among us called grooms.

GARD. s. (garde, French.) Wardship; care; custody.

GARD, a department of France, including part of the late province of Languedoc. Nismes is the episcopal town.

GARD (Pont du), a Roman aqueduct in France, nine miles N. E. of Nismes, erected, it is supposed, by Agrippa, in the time of Augustus. It is 160 feet in height, and consists of three bridges rising above each other, and uniting two craggy mountains. The highest of these bridges has six arches, of great blocks of stone, without cement, the centre one has eleven; and the lowest (under which flows the Gardon, an inconsiderable, but rapid river) has 36. Lewis XIV. when he repaired, in 1699, the damages which this stupendous work had sustained by time, caused a real bridge, over which passengers now pass, to be constructed by the side of the lower range of

« PreviousContinue »