trine of Chances, is the ratio of the num ber of chances by which the event may happen, to the number by which it may both happen and fail. So that, if there be constituted a fraction, of which the numerator is the number of chances for the events happening, and the denomi nator the number for both happening and failing, that fraction will properly express the value of the probability of the event's happening. Thus, if an event have 3 chances for happening, and 2 for failing, the sum of which being 5, the fraction will fitly represent the probability of its happening, and may be taken to be the measure of it. The same

thing may be said of the probability of failing, which will likewise be measured by a fraction whose numerator is the number of chances by which it may fail,

and its denominator the whole number of chances both for its happening and failing: so the probability of the failing of the above event, which has 2 chances to fail, and 3 to happen, will be expressed or measured by the fraction 2.

Hence, if there be added together the fractions which express the probability for both happening and failing, their sum will always be equal to unity, or 1; since the sum of their numerators will be equal to their common denominator. And since it is a certainty that an event will either happen or fail, it follows that a certainty, which may be considered as an infinitely great degree of probability, is fitly represented by unity. See CHANCES; LIFE, duration of.

PROBATE of wills, is the exhibiting and proving wills and testaments before the ecclesiastical judges, delegated by the bishop, who is ordinary of the place where the party dies.

By the stamp acts, a very heavy duty is now payable upon these instruments, and a man can entitle himself to personal property only by means of a probate; that is, by having proved the will.

PROBLEM, in logic, a proposition that neither appears absolutely true nor false; and, consequently, may be asserted either in the affirmative or negative. A logical or dialectical problem, according to the school-men, consists of two parts; a subject, about which the doubt is raised; and a predicate, or attribute, which is the thing doubted, whether it be true of the subject or not. Problems may be divided into physical, ethical, and metaphysical; physical, when it is doubted whether such and such properties be

long to certain natural bodies; ethical, when the doubt is, whether or not it be proper to do or omit certain actions; and metaphysical, when the doubt relates to spirits, &c.

PROBLEM, in geometry, is a proposition wherein some operation or construction is required; as, to divide a line or angle, erect or let fall perpendiculars, &c. A problem is said to consist of three parts; the proposition, which expresses what is to be done; the solution, wherein the several steps whereby the thing required is to be effected are rehearsed in order; and, lastly, the demonstration, wherein is shown, that by doing the several things prescribed in the solution, the thing required is obtained.

PROBLEM, in algebra, is a question or proposition, which requires some unknown truth to be investigated, and the truth of the discovery demonstrated. So that a problem is to find a theorem.

PROBLEM, Kepler's, in astronomy, is the determining a planet's place from the time; so called from Kepler, who first proposed it. It was this, to find the position of a right line, which, passing through one of the foci of an ellipsis, shall cut off an area described by its motion, which shall be in any given proportion to the whole area of the ellipsis.

The proposer knew no way of solving the problem but by an indirect method; but Sir Isaac Newton, Dr. Keill, &c. have since solved it directly and geometrically, several ways.

PROBLEM, Deliacal, or a problem for finding two mean proportionals between two given lines, in geometry, is the doubling of the cube; it was so called from the people of Delos, who, upon consulting the oracle for a remedy against a plague, were answered, that the plague should cease when Apollo's altar, which was in form of a cube, should be doubled. See CUBE.

PROCEDENDO, is a writ which lies where a cause is removed out of an inferior to a superior court.

PROCELLARIA, the petrel, in natural history, a genus of birds of the order Anseres. Generic character: bill straight, but hooked at the end; nostrils generally contained in one tube, at the base of the bill; legs naked a little above the knee; back toe little more than a spur. There are twenty-three species, of which the following are the principal.

P. gigantea, or the giant petrel, is more than three feet long, and about se

ven wide. These birds are often seen sailing just above the water without moving their wings for a long time together, and, being particularly alert on the approach of storms, often fill the mariner with apprehension and alarm. They abound most in southern latitudes, and though their principal food is fish, devour also the putrid carcases of seals and whales.

P. capensis, or the pintado petrel, abounds about the coasts of the Cape of Good Hope. These birds are about the size of the kittiwake gull, and are often observed in such numbers that many hundreds have been taken in one night. They are often taken with a rod and line by a hook bated with lard. They frequently discharge oil from their nostrils on those who hold them, spurting it in their faces with great violence.

P. glacialis, or the fulmar petrel, weighs nearly a pound and a half, and is found in the northern coasts of this island, and thence even beyond Iceland and Greenland, where the natives use it for food, though its flesh is highly offensive to those not used to it. The fat is burnt in their lamps. These birds subsist chiefly on fish, but often banquet on the carcases of whales, particularly the fat parts, which they afterwards eject from ther stomachs into the mouths of their young. They often spurt it in the faces of their enemies, and exhibit indeed no other mode of resistance. They are stated to be so amazingly fat, that, on being passed through the hands with great compression, the fat flows off like oil.

P. puffinus, or the shear water petrel, is smaller than the last. These birds are found in vast numbers in the Orkneys, where they are highly valued for their feathers as well as flesh. They are in some places salted and barrelled, especially in the Isle of Man. In Denmark they

sometimes reside in rabbit burrows. See Aves, Plate XIII. fig. 5.

P. pelagica, or the stormy petrel, is of the size of a swallow, and rarely seen but at sea; and in tempestuous weather, numbers are observed frequently following, as if for shelter, in the wakes of vessels. They dive sometimes for half an hour together, and live principally upon fish, but will eat a variety of offal thrown from ships. In the Ferro islands they are so astonishingly fat that the natives are stated to use them as candles after drawing a wick through their bodies. These are the birds so well known to seamen

by the name of "Mother Carey's Chickens;" and are the smallest species of the genus, and are common on the sea coast of the United States. See Aves, Plate XII. fig. 6.

PROCESS, in law, is the manner of proceeding in every cause, being the writs and precepts that proceed, or go forth upon the original upon every action, being either original or judicial.

Es

PROCKIA, in botany, a genus of the Polyandria Monogynia class and order. Natural order of Rosaceæ, Jussieu. sential character: calyx three-leaved, besides two leaflets at the base; corolla none; berry five-cornered, many seeded. There is but one species, viz. P. crucis.

PROCYON, in astronomy, a fixed star of the second magnitude in the constellation called canis minor.

PRODUCING, in geometry, signifies the drawing out a line further, till it have any assigned length.

PRODUCT, in arithmetic and geometry, the factum of two or more numbers, or lines, &c. into one another: thus 5 x 420, the product required. In lines it is always (and in numbers sometimes) called the rectangle between the two lines, or numbers, multiplied by one another.

PROFILE, in architecture, the draught of a building, fortification, &c. wherein are expressed the several heights, widths, and thicknesses, such as they would appear were the building cut down perpendicularly from the roof to

the foundation.

thus

}

Za, a

ba 26a

In numbers 10, 8, 6, 4,

a,

a,

a,

a,

a,

If 1, 3, 5, 7, 9, &c. a, a + b, a + 2b, a+3b, &c. a, a b, a- 2b, a- - 3 b, &c. are in arithmetical progression. Hence it is manifest, that if a be the first term, and a+b the second, a + 26 is the third, a+3b the fourth, &c. and a+n-1b the nth or last lerm.

“The sum of a series of quantities in arithmetical progression is found by multiplying the sum of the first and last terms by half the number of terms."

Let a be the first term, b the common difference, n the number of terms, and s the sum of the series: Then,

a ta+b+a+2b..a+n−1.0=s, or a+n-1.b+a+n-2.b+a+n—3b.+a=s.

Sum, 2a+n-1.b+2a+n-1.b+2a+1.6 + &c. to n terms,=28,

or, 2a+n-1.bxn=2s.

Hence, the series is 14, 11, 8, 5, &c. PROGRESSION, geometrical. Quantities are said to be in geometrical progression, or continual proportion, when the first is to the second, as the second to the third, and as the third to the fourth, &c. that is, when every succeeding term is a certain multiple, or part of the preceding term. If a be the first term, and ar the second, the series will be a, ar, ar2, ar^. art, &c. For a ar ar: ar2 :: ar2: ar3, &c.

The constant multiplier is called the common ratio, and it may be found by dividing the second term by the first.

"If quantities be in geometrical progression, their differences are in geometrical progression.”

Let a, ar, ar, ar3, art, &c. be the quantities; their differences, ara, ar1

ars

·ar2urt — - ar3. &c. form a geometrical progression, whose first term is ur- a, and common ratio r.

"Quantities in geometrical progression are proportional to their differences." For a ar :: ar -- a: ar2 -ar:: ar2 ar: ar3-ar2, &c.

"In any geometrical progression, the

the first to the square of the second."

Let a, ar, ar2, &c. be the progression; then a ar :: a2 : a2 r2.

Hence it appears, that the duplicate ratio of two quantities (Euc. Def. 10. 5.) is the ratio of their squares.

In the same manner it may be shown, that the first term is to the n+1ch term, as the first raised to the nth power, to the second raised to the same power.

"If any terms be taken at equal intervals in a geometrical progression, they will be in geometrical progression."

Let a, a r...rn...........ar3n..............arn....&c. be the progression, then a, arm, a r2n, a r3n, &c. are at the interval of n terms, and form a geometrical progression, whose common ratio is rn.

"If the two extremes, and the number of terms in a geometrical progression be given, the means may be found."

Let a and b be the extremes, n the num

"To find the sum of a series of quantities in geometrical progression, subtract the first term from the product of the last term and common ratio, and divide the remainder by the difference between the common ratio and unity."

Let a be the first term, r the common ratio, n the number of terms, y the last term, and s the sum of the series:

Then atar+ar2....+arn-2+a rn—1 s; and multiplying both sides by r, artar+ar3....+arn-1+arn=rs

Sub. a+artar2+ar3....+arn-1

=8

X S

, any

64.

-64.

41

Ex. 3. Required the sum of 12 terms of the series, 1-3, 9,-27, &c.

In this case, a =

therefore, s =

-

1, r = - 3, n = 12:

3-1 312-1

3-1

=

4

Ex. 4. To find the sum of the series 1 1 1 1

Here a 1,r =

224,)

+&c. in infinitum.

8

It may be observed, in connection with this subject, that the recurring decimals are quantities in geometrical progression, &c. is the common

where

1

1 1 10' 100' 1009

ratio, according as one, two, three, &c. figures recur; and the vulgar fraction, corresponding to such a decimal, is found by summing the series.

Ex. 5. Required the vulgar fraction &c. corresponding to the decimal 123123123,

Let .123123,123, &c.s; then multiply both sides by 1000; and 123.123123 123, &c. = 1000s, and by subtracting the former equation from the latter, 123 = 123 41 9998; therefore s =

999 333* PROHIBITION, in law, is a writ properly issuing only out of the Court of King's Bench, being the King's prerogative writ; but, for the furtherance of justice, it may now also be had in some cases pleas, or Exchequer, directed to the out of the Court of Chancery, Common judge and parties of a suit in an inferior court, commanding them to cease from the prosecution thereof, upon a sugges tion, that either the cause originally, or some collateral matter arising therein, does not belong to that jurisdiction, but the cognizance of some other court. Upon the court being satisfied that the matter alleged by the suggestion is sufficient, the writ of prohibition immediately

issues.

PROJECTILES, are such bodies as, being put in a violent motion by any great force, are then cast off or let go from the place where they received their quantity of motion; as a stone thrown from a sling, an arrow from a bow, a bullet from a gun, &c. It is usually taken for granted, by those who treat of the motion of projectiles, that the force of gravity near the earth's surface is every where the same, and acts in parallel directions: and that the effect of the air's resistance up on very heavy bodies, such as bombs and cannon-balls, is too small to be taken into consideration.

Sir Isaac Newton has shown that the gravity of bodies which are above the superficies of the earth is reciprocally as the squares of their distances from its centre: but the theorems concerning the descent of heavy bodies, demonstrated by Gallileo, and Huygens, and others, are built upon this foundation, that the action of gravity is the same at all distances; and the consequences of this hypothesis are found to be very nearly agreeable to experience. For it is obvious that the error arising from the supposition of gravity's acting uniformly and in parallel lines, must be exceedingly small; because even the greatest distance of a projectile above the surface of the earth is inconsiderable, in comparison of its distance from the centre, to which the gravitation tends. But then, on the other hand, it is very certain, that the resist. ance of the air to very swift motions is much greater than it has been commonly represented. Nevertheless, (in the appli cation of this doctrine to gunnery,) if the amplitude of the projection, answering to one given elevation, be first found by experiment, (which we suppose,) the amplitudes in all other cases, where the elevations and velocities do not very much differ from the first, may be determined, to a sufficient degree of exactness, from the foregoing hypothesis; because, in all such cases, the effects of the resistance will be nearly as the amplitudes themselves and were they accurately so, the proportions of the amplitudes, at different elevations, would then be the very same as in vacuo.

Now, in order to form a clear idea of the subject here proposed, the path of every projectile is to be considered as depending on two different forces; that is to say, on the impellant force, whereby the motion is thus begun, (and would be continued in a right line,) and on the force of gravity, by which the projectile,

during the whole time of its flight, is continually urged downwards, and made to deviate more and more from its first direction. As whatever relates to the track and flight of a projectile, or ball, (neglecting the resistance of the air,) is to be determined from the action of these two forces, it will be proper, before we proceed to consider their joint effects, to premise something concerning the nature of the motion produced by each, when supposed to act alone, independently of the other; to which end we have premised the two following lemmata.

Lemma I. Every body, after the impressed force whereby it is put in motion ceases to act, continues to move uniformly in a right line, unless it be interrupted by some other force or impediment.

This is a law of nature, and has its demonstration from experience and matter of fact.

Corollary. It follows from hence, that a ball, after leaving the mouth of the piece, would continue to move along the line of its first direction, and describe spaces therein proportional to the times of their description, were it not for the action of gravity; whereby the direction is changed, and the motion interrupted

Lemma II. The motion, or velocity, acquired by a ball, in freely descending from rest, by the force of an uniform gravity, is as the time of the descent; and the space fallen through, as the square of that time.

The first part of this lemma is extremely obvious: for since every motion is proportional to the force whereby it is generated, that generated by the force of an uniform gravity must be as the time of the descent; because the whole effort of such a force is proportional to the time A of its action; that is, as the

P

C

time of the descent.

To demonstrate that the d distances descended are proportional to the squares ofthe e times, let the time of falling through any proposed distance AB, be represented by the right line PQ; which conceive to be divided into an indefinite number of very small equal particles, repreIg sented, each by the symbol m; and let the distance descended in the first of them be Ac; and the second cd; in -B the third de; and so on.