palace of Westminster: he disposes likewise of the sword of state, to be carried before the king, to what lord he pleases.

The office of Lord Great Chamberlain of England is hereditary; and where a person dies siezed in fee of this office, leaving two sisters, the office belongs to both, and they may execute it by deputy, but such deputy must be approved of by the king, and must not be of a degree inferior to a knight. To the Lord Chamberlain the keys of Westminster Hall, and the Court of Requests, are delivered upon all solemn occasions. He goes on the right hand of the sword, next the king's person. The Gentleman Usher of the Black Rod, Yeoman Usher, &c. are under his authority.

CHAMBERLAIN, Lord of the Household, an officer who has the oversight and direction of all the officers belonging to the king's chambers, except the precinct of the king's bed-chamber.

He has the oversight of the officers of the wardrobe at all his Majesty's houses, and of the removing wardrobes, or of beds, tents, revels, music, comedians, hunting, messengers, &c. retained in the king's service. He moreover has the oversight and direction of the serjeants at arms, of all physicians, apothecaries, surgeons, barbers, the king's chaplains, &c. and administers the oath to all officers above stairs.

CHAMBERLAIN of London, keeps the city money, which is laid up in the chamber of London: he also presides over the affairs of masters and apprentices, and makes free of the city, &c. His of fice lasts only a year, but the custom usually obtains to re-choose the same person, unless charged with any misdemeanor in his office.

CHAMBERS, (EPHRAIM,) author of the dictionary of sciences, called the Cyclopædia." He was born at Milton, in the county of Westmoreland, where he received the common education for qualifying a youth for trade and commerce. When he became of a proper age, he was put apprentice to Mr. Senex, the globe-maker, a business which is connected with literature, especially with geography and astronomy. It was during Mr. Chamber's residence with this skilful artist, that he acquired that taste for literature which accompa nied him through life, and directed all his pursuits. It was even at this time that he formed the design of his grand work, the Cyclopædia; some of the first articles of which were written behind VOL. III

the counter. To have leisure to pursue this work, he quitted Mr. Senex, and took chambers at Grey's Inn, where he chiefly resided during the rest of his life. The first edition of the Cyclopædia, which was the result of many years intense application, appeared in 1728, in 2 vols. folio. The reputation that Mr. Chambers acquired by the execution of this work procured him the honour of being elected F. R. S. November 6, 1729. In less than ten years time, a second edition became necessary; which accordingly was printed, with corrections and additions, in 1738: and this was followed by a third edition the very next year.

Mr. Chambers's close and unremitting attention to his studies at length impaired his health, and obliged him occasionally to take a country lodging, but without much benefit; he afterwards visited the south of France, but still with little effect; he therefore returned to England, where he soon after died, at Islington, May 15, 1740, and was buried at Westminster Abbey.

After the author's death, two more editions of his Cyclopædia were published. The proprietors afterwards procured a supplement to be compiled, by Mr. Scott and Dr. Hill, but chiefly by the latter, which extended to two volumes more; and the whole has since been reduced into one alphabet, in four volumes, by Dr. Rees, forming a very valuable body of the sciences.

A new edition of the same work, or rather a new work under the title of the "New Cyclopædia," is now publishing by the same learned Editor. This work, of which Dr. Rees has published already nine volumes, will probably extend to thirty volumes quarto. It will, when complete, be unquestionably the most comprehensive body of science ever presented to the world.

CHAMELEON. See LACERTA.

CHAMPION, a person who undertakes a combat in the place or quarrel of another; and sometimes the word is used for him who fights in his own

cause.

It appears that champions, in the just sense of the word, were persons who fought instead of those, that, by custom, were obliged to accept the duel, but had a just excuse for dispensing with it, as being too old, infirm, or being ecclesiastics, and the like. Such causes as could not be decided by the course of common law were often tried by single combat ; and he who had the good fortune to con S

quer was always reputed to have justice on his side. Champions, who fought for interest only, were held infamous: these hired themselves to the nobility, to fight for them in case of need, and did homage for their pension.

When two champions were chosen to maintain a cause, it was always required that there should be a decree of the

judge to authorize the combat: when the judge had pronounced sentence, the accused threw a gage or pledge, originally a glove or gauntlet, which being taken up by the accuser, they were both taken into safe custody, till the day of battle appointed by the judge.

Before the champions took the field, their heads were shaved to a kind of crown or round, which was left at the top: then they made an oath, that they believed the person who retained them to be in the right, &c. They always engaged on foot, and with no other weapon than a club and a shield, which weapons were blessed in the field by the priest, with a world of ceremonies; and they always made an offering to the church, that God might assist them in the battle.

The action began with railing, and giv. ing each other ill language; and at the sound of a trumpet, they went to blows. After the number of blows or encounters expressed in the cartel, the judges of the combat threw a rod into the air, to advertise the champions that the combat was ended. If it lasted till night, or ended with equal advantage on both sides, the accused was reputed the victor. If the conquered champion fought in the cause of a woman, and it was a capital offence, the woman was burnt, and the champion hanged. If it was the champion of a man, and the crime capital, the vanquished was immediately disarmed, led out of the field, and hanged, together with the party whose cause he maintained. If the crime was not capital, he not only made satisfaction, but had his right hand cut off the accused was close confined in prison, till the battle was

over.

CHAMPION of the king, a person whose office it is, at the coronations of our kings, to ride armed into Westminster-hall, while the king is at dinner there, and, by the proclamation of a herald, make challenge to this effect, viz. "That if any man shall deny the king's title to the crown, he is there ready to defend it in single combat, &c." Which done, the king drinks to him, and sends him a gilt cup, with a cover, full of wine, which the

champion drinks, and has the cup for his fee. This office is hereditary.

CHANCE, in a general sense, a term applied to events not necessarily produc ed as the natural effects of any proper foreknown cause. We certainly mean no more in saying that a thing happened by chance, than that its cause is unknown to us for chance itself is no natural agent or cause; it is incapable of producing any effect, and is no more than a creature of man's own making; for the things done in the corporeal world are really done by the parts of the universal matter, acting and suffering, according to the laws of motion established by the author of nature.

Chance is also confounded with fate and destiny.

The

CHANCES, doctrine of, in mixed mathematics, a subject of great importance, especially as applied to the doctrine of life annuities, assurance, &c. in a great commercial country like this. writers on this branch of science have been comparatively few. In our own language the principal treatises are, a large quarto by De Moivre, and a very small work by the celebrated' Mr. Thomas Simpson, in which, however, there are some problems never before attempt. ed, or, at least, never before communicated to the public. In the year 1753, Mr. Dodson rendered this subject more accessible to persons not far advanced in analytical studies, by publishing, in his se cond volume of the "Mathematical Repository," a number of questions, with their several solutions, with an express reference to the doctrine of life annuities. We shall give his first problem.

Suppose a round piece of metal, equally formed, having two opposite faces, one white, the other black, be thrown up, in order to see which of those faces will be uppermost after the metal has fallen to the ground, when, if the white face appears uppermost, a person is to be entitled to 57. it is required to determine, before the event, what chance or probability that person has of receiving the 5. and what sum he may expect should be paid to him in consideration of his resigning his chance to another.

Solution. Since there is nothing in the form of the metal that can incline it to shew one face rather than the other, and since it must shew one, it will follow, that there is an equal chance for the appear ance of either face, or there is one chance out of two for the appearance of the white face, and consequently the probability of it may be expressed by the frac

tion; if, therefore, any other person should be willing to purchase his chance, he must give for it the half of 51. or 21. 108. This is one of the most simple cases: before, however, we proceed, it may be proper to give some definitions introductory to the doctrine.

Def. 1. The probability of an event is the ratio of the chance for its happening to all the chances for its happening or failing: thus, if out of six chances for its happening or failing, there were only two chances for its happening, the probability in favour of such an event would be in the ratio of two to six; that is, it would be a fourth proportional to 6, 2, and 1, or 4. For the same reason, as there are four chances for its failing, the probability that the event will not happen will be in the ratio of 4 to 6, or, in other words, it will be a fourth proportional to 6, 4, and 1, or. Hence, if the fractions expressing the prbabilities of an event's both happening or failing be added together, they will always be found equal to unity. For let a be the number of chances for the event's happening, and 6 the number of chances for its failing, the probability in the first case being a and in the se

cond case

a+b a+b

b

a+b

a+b

their sum will be =

1. Having therefore determined the probability of any event's either happening or failing, the probability of the contrary will always be obtained by subtracting the fraction expressing such probability from unity.

Def. 2. The expectation of an event is the present value of any sum or thing, which depends either on the happening or on the failing of such an event. Thus, if the receipt of one guinea were to depend on the throwing of any particular face on a die, the expectation of the person entitled to receive it would be worth 38. 6d.; for since there are six faces on a die, and only one of them can be thrown to entitle the person to receive his money, the probability that such a face will be thrown being (according to Def. 1.) it follows, that the value of his interest before the trial is made, or, which is the same thing, that his expectation is equal to one-sixth of a guinea, or 3s. 6d. Were his receiving the money to depend on his throwing either of two faces, his expectation would be equal to two-sixths of a guinea, or 7s. And, in general, supposing

A b according as it depends either. on a+b' the event's happening, or on its failing.

Def. 3. Events are independent, when the happening of any one of them does neither increase nor lessen the probability of the rest. Thus, if a person undertook with a single die to throw an ace at two successive trials, it is obvious (however his expectation may be effected) that the probability of his throwing an ace in the one is neither increased nor lessened by the result of the other trial.

Theor. The probability that two subsequent events will both happen, is equal to the product of the probabilities of the happening of those events considered separately.

Suppose the chances for the happening and failing of the first event to be denoted by b, and those for its happening only to be denoted by a. Suppose, in like manner, the chances for the second event's happening and failing to be denoted by d, and those for its happening only by c; then will the probability of the happening of each of those events, separately considered, be (according to Def. 1) and respectively. Since it is necessary that the first event should happen before any thing can be determined in regard to the second, it is evident that the expectation on the latter must be lessened in proportion to the improbability of the former. Were it certain that the first event would happen, in other

failing n times successively.

Rem. It should be observed, that in some instances the probability of each subsequent event necessarily differs from that which preceded it, while in others it continues invariably the same through any number of trials. In the one case the probabilities are expressed, as in the theorem, by fractions, whose numerators and denominators continually vary; in the other they are expressed, as in the corollary, by one and the same invariable fraction. But this perhaps will be better understood by the following examples.

1. Suppose that out of a heap of cour ters, of which one part of them are white and the other red, a person were twice successively to take out one of them, and that it were required to determine the probability that these should be red counters. If the number of the white be 6, and the number of the red be four, it is evident, from what has already been shown, that the probability of taking out a red one the first time will be but the probability of taking it out the 2d time will be different; for since one counter has been taken out, there are now only nine remaining; and since, in order to the 2d trial, it is necessary that the counter taken out should have been a red one, the number of those red ones must have been reduced to 3. Consequently, the chance of drawing out a red counter the 2d time will be 3, and the probability of drawing it out the first and 4 x 3 2d time will (by this theorem) be

2

15

10 x 9

2. Suppose next, that with a single die a person undertook to throw an ace twice successively in this case the probability of throwing it the first does not in the least alter his chance of throwing it the second time, as the number of faces on the die is the same at both trials. The probability, therefore, in each will be expressed by the same fraction, so that the probability, before any trial is made, will,

by the preceding corollary, be }=}%. On these conclusions depend all the com putations, however complicated and laborious, in the doctrine of chances. But this, perhaps, will be more clearly exemplified in the two following problems, which will serve to explain the principles on which every other investigation is founded on this subject.

Prob. 1. To determine the probability that an event happens a given number of times, and no more, in a given number of trials.

Sol. 1. Let the probability be required of its happening only once in two trials, and let the ratio of its happening to that of its failing be as a to b. Then, since the event can take place only by it happening the first, and failing the second time, the probability of which is

a

+6

X

a+b a+b or by its failing the first and happening the second time, the probability of which is

b

the sum of

a+b2 these two fractions, or

the probability required.

a+b)

2. Let the probability be required of its happening only twice in three trials. In this case, the event, if it happens, must take place in either of three different ways: 1st, by its happening the first two, and failing the third time, the probability of which is ; 2dly, by its a + b3 failing the first, and happening the other two times, the probability of which is

baa

at bl

a ab

; or, 3dly, by its happening the first and third, and failing the second time, the probability of which is

aba

a+b13. The sum of these fractions, therefore, or 3 ba a

39

will be the required probabili. a+b ty. By the same method of reasoning, the probability of its happening only once in three trials, or, which is the same thing, of its failing twice in three 3bba trials, may be found equal to 3. a+b

3. Let the probability of the event's happening only once in four trials be required. In this case it must either happen the first and fail in the three succeeding trials; or happen the second and

46 a3 will be.

4.

a+b) 4. Let the probability be required of its happening twice and failing twice in four trials: here the event may be determined in either of six different ways: 1st, by its happening the first and second, and failing in the third and fourth trials; 2dly; by its happening the first and third, and failing the second and fourth trials; 3dly, by its happening the first and fourth, and failing the second and third trials; 4thly, by its happening the second and third, and failing the first and fourth trials; 5thly, by its happening the second and fourth, and failing the first and third trials; or, 6thly, by its happening the third and fourth, and failing the first and second trials. Each of these probabilia2 bi

ties being expressed by

that the sum of them, or

a+b 6 a' b2

4, it follows

49

a+b)

press the probability required.

4

will ex

By proceeding in the same manner, the probability in any other case may be determined. But if the number of trials be very great, these operations will become exceedingly complicated, and therefore recourse must be had to a more general method of solution.

n-d

n.

Supposing n to be the whole number of trials, and d the number of times in which the event is to take place, the probability of the event's happening d times successively, and failing the remaining n d times, ad will bead. bn--d a + b a+b) But as there is the same probability of its happening any other d assigned trials and failing in the rest, it is evident that this probability ought to be repeated as often as d things can be combined in n things, which, by the known rules of combina

X a+b

Sol. It will be observed that this problem materially differs from the preceding, in as much as the event in that problem was restrained, so that it should happen neither more or less often than a given number of times, while in this problem the event is determined equally favourable by its happening either as often or oftener than a given number of times, so that in the present case there is no further restriction than that it should not fall short of that number.

1. Let the probability be required of an event happening once at least in two trials. If it happens the first and fails the second time, or fails the first and happens the second time, or happens both times, the event will have equally succeeded. The

probability in the first case is

probability in the second is

a a

ab

a+b ba

a+b

the

2,and the

probability in the third is; hence the