Page:Cyclopaedia, Chambers - Supplement, Volume 2.djvu/908

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■ him. They have all fhfewri their inability of demonftrating, ■without taking forr.e axioms or poftulata for granted; and indeed moft of them exprefiy admit this, only contending

' that the principles they a flu me are more evident than thofe of

■ Euclid. But whether they he fq, or not, is of no importance to the prefent queftion, which is, whether we are not often obliged to have recourfe to axioms, that is, to felf-evident, or indemonftrable truths, in the ftridtcft fenfe; and whether thefe can be fupplied by deductions, or fyllogiftic reafoning, from definitions only. We ihould not have taken any no- tice of this queftion, had not Hobbes and other metaphyfi- cians and logicians of repute run counter to the common opinion.

But tho' we are forced to affume axioms and poftulata in geometry, it may be a question, whether any thing of this kind be neceiTary in arithmetic, or the fcience of numbers; and whether the analyfes of our notions may not here be carried up to the notions of unity, and of the act of addition, tamquam prffioUia prima; and whether it was not fome con- sideration of this kind that led Auftotle to fay, that arith- metic was more accurate (ax^&rtpw) than geometry. It is certain at leaft, that the analyfis may be, though it feldom is, carried farther in the former than in the latter of thefe feiences. Becaufe the axioms commonly afTumed, and pe- culiar to arithmetic, fuch as the addition and multiplication tables, are no more than the aggregates of the fimple figns, or fimple notations of numbers; and may eaflly be demon- strated from the definitions of the fimple figns made ufe of; fuch as that i -|- i = 2; 2+1=3; 3 + r — 4-> & c - Bifhop Berkeley a obferves vcryjuftly, that the principles of fcience are neither objects of fenfe nor imagination, but no- tions of relation, that is, acts of the mind. Thus, fpace, time, number b are not objects of fenfe or imagination, altho' the things coextended with fpace or time, or things num- bered, be objects of (enfe. This is moft evident in number, which is plainly different from the perception of the things numbered. Nor can the adt by which we number be taught or exemplified. For inftance, that act of the mind by which-we conceive 1 -\- 1 = 2, cannot be explained or analyfed into others; and fuppofmg it could, we mufr ftill ftop fomewhere, and wherever that be, there Is fomething - a difciple muft have a fe, non a praceptore, as Ariftotle fays. So that, ftrictly (peaking, principles arc not taught. And thofe who maintain them innate, are perhaps not fo abfurd as Locke pretends. [? Reflections on Tar-water, Art. 264. b Ibid. Art. 28S.J

Leibnitz maintained, that the principles of contradiction, and of a fufficient reafon, were the foundations of all fcience; that the fir ft was fufficient for the demonftration of all necef- ' fary, and the other of all contingent, truths- But though it be true, that the principle of contradiction, that is, the reducli '0 ad abjurdum, often occurs exprefly, and is oftner implied in geometry, yet by what has been (aid it appears, that this principle alone is not fufEcient to demonftrate all the other univerfally received principles of that fcience. Far lefs is it true, that we are enabled by the principle of a fufficient rea- fon, which amounts to the exclufion of pure chance out of

■ the univerfe, to demonftrate all phyfics and morals; but addi- tional principles derived from experience, muft be aflumed.

PRIVIES of a Camp, See the article Camp, Append. ■PRIVET (Suppl.) — Mock Privet, a name fometimes given to the Phillyrea of botanifts. See the article Phillyrea,

■ Slippl.

PROCELLARIA, in ornithology, the name ufed in the Stock-

■ holmTranfactions, and elfewhere, for the Jlorm-foik^orjlorm- bird. See Storm, Suppl. and Petrel, Append.

PROBABILITY (Cycl.) — -In the. doctrine of Probability, one important- cbfervation may be made, viz. that if one pre- "ttiifs only of an argument be probable, the conclufion is ne- cefiarily probable. But if two or more premifTes be proba- ble the conclufion will not be neceflarily probable. Thus for ■ inftance, fuppofmg the Probability of each premifs expreffed by T ^, the Probability of the conclufion will be but t \\., which fliews it to be improbable. For we may call any thins improbable, ifthe meafure of the chance for its happening is lefs than 'V If there had been three premifles, and the Probabi- lity of each' equal to T 7 E, the Probability of the conclufion would be ^, which is confidcrably improbable. Again, fuppofmg the Probability of the truth of each premifs to be

-■■■% to 1 or exprefled by I, the probability of the conclufion i-n-the cafe- of the two prcmilfes would be %. Where three premifles are aflumed to infer a conclufion, this would be, 8 7; and in cafe of four premifles, the Probability of the conclu- ■fion would be but >±, which is lefs than \, fo that one might with advantage lay 4 to 1 againft the truth of a conclufion

'" "founded on four probable premifles, for the truth of which feparately taken 2 to 1 might be laid. It is to be obferved in all thefe cafes, thauhe premifTes are fuppofed independent, that is, not neceflarily connected with each' other. Hence it is eafy to account, how it happens, that the moft plauiiblc political and phyiical reafonings, lead fo often to

■ conclulir.ns felfej in fact.

Mr. de iVIoivre hi;s folved two problems, tending to eftablifh

the degree of aflent that ought to be given to experi- ence. He determines from his folutions, that if after takiiv a great number <n experiments, it ihould have been obferved, that the happenings or failings of an event have been very near in a ratio of equality, it may fafely be concluded, that _ the Probabilities of its happening or lading, at any one time afligned, are very near equal.

And if alter taking a great number of experiments, it ihould be perceived, that the happenings and failings have been nearly in a certain proportion, fuch as 2 to 1, it may fafely be con- cluded, that ihc Probabilities of happening or failing at any one time afiigned, will be very near in that proportion j and that the greater the number of experiments has been, fo much the nearer the truth will the conjectures be, that are derived from them.

Chance very little difturbs the events which, in their natu- ral inftitution, were dcfigned to happen or fail according to fome determined law. For if in order to help our concep- tion we image a round piece of metal, with two polifhcd oppofite fac^s, differing in nothing but their colour, whereof one may be fuppofed to be white and the other black, it is plain that this piece may with equal facility exhibit a white or black face; and we may even fuppofe that it was framed with that particular view of (hewing fometimes the one face, fometimes the other; and that confequcntly, if it be tolled up, chance will decide the appearance. But altho' chance may produce ait inequality of appearance, and that a greater inequality, ac- cording to the length of time in which it may exert itlblf, ftill the appearance, either one way or the other, will per- petually tend to a proportion of equality. This is, in like manner applicable to a ratio of inequality; and thus in all cafes it will be found, that although chance produces irre- gularities, ftill the odds will be infinitely great, that in pro- cefs of time, thofe irregularities will bear no proportion to the recurrency of that order which naturally refults from ori- ginal dehgn. See De Moivre's Doctrine of Chances, p. 231 . — 2,13.

PROBLEM (Cycl.)— Kepler's Problem (Cycl.) — As to the folution of this problem, the late -excellent mathematician Mr. Machin obferves, that many attempts have been made, at dif- ferent times, but never yet with tolerable fuccefs, towards the folution of the problem propofed by Kepler : To divide the area of a femicircle into given parts, by a line from a given point of the diameter, in order to find an univerfal rule for the motion of a body in an elliptic orbit, P'or amono- the feveral methods offered, fome are only true in fpcculation, but are really of no fervice. Others are not different from his own, which he judged improper. And as to the reft, they are all fome way or other fo limited and confined to particular conditions and circumftances, as ftill to leave the problem in general untouch'd, To be more particular, it is evident, that all conftructiens by mechanical curves are fceming folutions only,, but in reality unapplicable; that the roots of infinite fe- riefes are, upon account of their known limitations in all re- fpects, fo far from affording an appearance of being fufficient rules, that they cannot well be fuppofed as offered for anv thing more than exercifes in a method of calculation. A.nd then, as to the univerfal method, which proceeds by a conti- nued correction of the errors of a falfe pofition, it is, when duly confidcred, no method of folution at all in itfelf; becaufe unlefs there be fome antecedent rule or hypothefis to begin the operation (as fuppofe that of an uniform motion about the up- per focus, for the orbit of a planet; or that of a motion in a parabola for the perihelion part of the orbit of a comet; or fome other fuch) it would be impoflible to proceed one ftep in it. But as no general rule has ever yet been laid down to aflift this method, foasto make it always operate, it is the fame in effect as if there were no method at all. And accord- ingly in experience it is found, that there is no rule now fub- fifting but what is abfolutely ufelefs in the elliptic orbits' of comets; for in fuch cafes there is no other way to proceed but that which was ufed by Kepler. To compute a tabic for fome part of the orbit, and therein examine if the time to which.tbe place is required, will fall out any wherein that part. So that, upon the whole, it appears evident, that this problem (contra- ry to the received opinion) has never yet been advanced one ftep towards its true folution. Vid. Machin, in Phil. Tranf. N u 447< and Martyn's Abridg. Vol. 8. p. 73. Mr. Machin afterwards proceeds to give his own folution of this pisbfcm, which is particularly neceffary in orbits of a great excentricity; and he illuftratcs his method by examples, for the orbits of Mercury, of Venus, of the Comet of the year 1682, and of the great Comet of the year 1680, all which fhew the univerfality of that method.! See' Phil. 'Tranf Inc. cit,

PROPERTY (Cycl.) — In the law of England, ftrictly fpeak-

{ ing, that which is called an ejlaie in lands and tenements, is

termed a property in perfonal chattels, the law confidering the

firft as permanent, the other as temporary and precarious.

Peer Williams's Reports, p. 3.

PROSCARAB/EUS, in zoology, the name by which fome call the Meloe, a genus of four-winged flies. See the article Meloe, Append.

3 PSEUDO-