Page:Cyclopaedia, Chambers - Supplement, Volume 2.djvu/907

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fion, fuch as lines, furfaces and folids ; fo that in this fenfe of the word pojlulatum, Euclid, befides axioms, or thofe prin- ciples which are common tu all kinds of quantity, has affumed ■ certain pojiutata to be granted him, peculiar to extenfive mag- nitude. Hence feveral of the principles affumed in his ele- ments, and ranked among the axioms by the moderns, are by Proclus ranked among the pojlidata ; which has induced Dr. Wallis to judge, that the lalt of the two fenfes given to the term pojiulatum is. molt agreeable to the meaning of the an- tient geometers. And thofe who contend for this fenfe of the word, add, that Euclid, in populating to draw a right line from one point to another, does not mean that any man can actually do this, but only that it may be conceived as pulli- ble. So that poftulaia are axioms no lefs than the other prin- ciples affumed in the elements of geometry, but axioms re- lating to a particular fubject, and not common to all. Jffid- lis's Open Vol. i. p. 667, 668. See the article Prin- ciple, append,

POTATOES, the Engliih name of the tuberofe-rooted, efcu- Jent Lycoperftcon or Sslanum of botanical writers. See the ar- ticle Solanum, Suppl.

The Engliih name feems evidently formed from Batatas, the Indian name of the fame plant. See the article Potatoes, Suppl.

Spanijh Potatoe, the name by which fome call feveral fpe- cies of Convolvulus, or bindweed. See the article Convol-. vulus, Suppl,

POTENT! LL A, a name ufed by fome for feveral (pedes of Cinque/oil. See the article Cinquefoil, Suppl.

VOUCH- (Suppl-) — Shepherd' s-P ouch, a name fometimes given to the Alyjpm, or mad-wort. Seethe article Alysson, Suppl.

POWER (Cyd. and Suppl.) — Arithmetical Power is ufed by Mr. Mach'tn, for compofite numbers or quantities whofe factors are in arithmetical progreffion. See Phil. Tranf. N° 447, and Dr. Marty n's Abridg. Vol. 8. p. 78. Mr. Machin ufes a particular notation for quantities of this kind. The quantity exprefled by this notation has a double index ; that at the head of the root at the right hand, but feparated by a hook to diftinguilh it from the common index, denotes the number of factors ; and that above, within the hook on the left hand, denotes the common difference of the factors proceeding in a decreafmg or mcreafmg arithmetical progreffion.

Thus the quantity n -jr- a denotes by its index m on the light hand, that it is a compofite quantity, confifting of fo many factors as there are units in the number ,R ; and the index a above on the left, denotes the common difference of the factor's decreasing in an arithmetical progression, if it be poSitive ; or increafing, if it be negative ; and fo fignifies, in the common notation, the common, number or quantity,

a-t" a • n -fc- a ~* n • n th a w~ 2 a • & c * '

For example, n + $ is = n -J- 5 j n + 3. n -+■ 1. n—i

n — 3. n — 5, consisting of fix factors, whofe common differ- ence is 2. After the fame manner, K-J-4 (S — jfri.4 . n-\-% n. n — 2 . n — 4, confifting of five factors. According to which method it will eafily appear, that if a be an integer,

2 ptf-f-2 then n 4. 2 aA-t will hr — ^Z~ f ^ZT g. mi— 2s

P R I

PRECESSION (Cycl) — The fmcffion of the equinoaial points varies ; nor are aftronotners entirely agreed as to the quantity of the variation, fo as to eflablifh what the mean preceffion is. Dr. Bradley affumes the mean precision to be one degree in feventy one years and an half. See Phil. Tranf N° 485. p. 22.

According to this eftimate, the platonic or great year would be equal to 25740 folar years.

Sir Ifaac Newton, in determining the quantity of the annual prtceffion from the theory of gravity, upon fuppofition that the equatorial is to the polar diameter of the earth, as 230 is to 229, finds the fun's adion fufficient to produce a prc- cejfwn of 9 " \ only ; and collecting from the tides the proportion between the fun's force and the moon's to be as 1 to 4 J, he fettles the mean preajjien refuking from their joint afiions, at 50". But fince the difference between the polar and equatorial diameter is found, by the late obferva- tions of the gentlemen of the Royal Academy of Sciences at Paris, to be greater than what Sir Ifaac had computed it to be ; the precejfion arifing from the fun's action mufl likewife be greater than what he has ftated it at, nearly in the fame proportion. From whence it will follow, that the moon's force mufl bear a lefs proportion to the fun's, than 4 i to 1. See Dr. Bradley in Philofoph. Tranf. N° 485; P a & 37-

PREVENTERS, on board a (hip of war, are ropes of differ- ent iizes, cut into fhort lengths, and knotted at each end, to be ready in cafe a fhroud mould be ihot or broke, that they may be feized to them. Blamkley's Naval Expofitor, p. 123.

PRICKET, among fportfmen. See the article SpiTTi-Ri

continued to fuch a number of double fafiors as are exprefled ty a + i> °r half the index, which in this cafe is an even

rmmber. _Thus alfoa-j-M will be equal to n.

nn—^. nn — 16. "—36, and fo on, where there are to be fo many double factors, as with one fin<r,le one it, will make up the index 2*4-1, which is an odd"'number. If the common dif ferenc e a be an unit, it is omitted : Thus n (' = n. n — 1 . n — 2 . 2—3 . n — 4 . n — 5, contain- ing fix factors. So 6 (« =6. 5. 4. 3. 2. 1. and the like for others.

If the common difference a be nothing, the hook is omitted, and it becomes the fame with the geometrical power : Thus

M" _-

n-\-a = «-f-<?| according to the common notation. The learned author above quoted applies this do6trine of a- rithmetical powers to the investigation of the principal rule in the method of fluxions, and its inverfe, which is, that if the

ordinate y = m z, then will the area, or rather the

form of the quantity for the area, be — % m ; or vice verfa, that if the area be % '", the ordinate will he m * m — * ; which occafion he obferves, that the fymbol *, confidered as a component part of the rectangle » , may bear a plain interpretation, viz. that it is the meafure according to which the quantity z is meafured. See Phil. Tranf. foe. cit, in the poftfeript. See alfo Fluxion, Suppl, %

PRICK- Madam, the name of a fpecies of Sedum ; the fame with the yellow houfe-leek, with fharp-pointed leaves. See the article Sedum, Suppl. P^icK-Timber, a name Sometimes given to the Euonymus, or

fpindle-tree. See the article Euonymus, Suppl. PRIEST'S Pintle, in botany, a name by which Arum, or wake-robin is fometimes called. See the article Arum* Suppl. PRHV1KOSE, the Englifti name of a genus of plants, called by botaniils Primula veris. See the article Primula verisj Suppl. Primrose-*™?, or A%£*-Primrose, the Engliih name of a genus of plants, known among authors by that of Onagra^ See the article Onagra, Suppl. PRINCIPLE (Cycl.) — Philofophers and mathematicians are generally agreed in admitting that there are axioms, that is» certain indemonftrable truths, which mufl: be reckoned a- mong the principles of human knowledge in the ftridteft fenfe. Hobbes feems to advance the contrary, when he fays, in his logic, or firft part of his book de Corpore, that defini- tions, or their parts, are the only primary proportions (pro- pofitiones prima) that is, principles in an abfolute and ftrict fenfe. But this doctrine of Hobbes cannot be admit- ted. Per however true it may be in itfclf, and with refpect to the divine mind, it feems certain, that the human mind has never yet attained, and perhaps never will attain to a perfect analyfis. of its own notions in all cafes. And wherever this analyfis ceafes, definitions ccafe with it ; and where defini- tions ceafe, we are forced, if we would demonftrate any thing of the undefined fubject, to have recourle to axioms, or to indemonftrable truths admitted by all men, tho' never yet demonftrated by any one. This will appear evident to thofe who attentively confider Euclid's Element?. This great geometer, it is known, does not, ftrictly fpeakmg, de- fine a right line ; becaufe he could not probably, analyfe the notion of rectitude \ for that which is commonly called his definition of a right line, in the beginning of his elements, is no definition, nor is it ever applied afterwards, as the defini- tions of an equilateral triangle, a Square, and a circle are. But to fupply the place of a definition, he has affumed the axioms, that two right lines cannot comprehend a fpace ; and that they cannot have a common fegment ; and thefe axioms be- come of ufe more than once afterwards. No one has as yet been able to fupply with fuccefs what Euclid omitted to do ; for tho' fome, both ancients and moderns, have demonftrated feveral very plain axioms, as that the whole is greater than its part, yet none have demonftrated all the principles of ex- tenfive magnitude affumed by Euclid, which Proclus and others chofe to diiiingui{h by the name of Pojiulate. See the ar- ticle Postulate, Append.

Hobbes brings this very inffance of the demonstration that a whole is greater than its part, to prove* that the propositions commonly called axioms, are not, ftrictly ("peaking, primary, but only fecondary propositions, and really deducible from definitions. Had he attempted the demonstration of all Eu- clid's axioms and poltulata, he would foon have found that he had made a rafh induction ; and what he gives for a de- monstration of the properties of parallels, evidently (hews* how little able he was to fupply what Euclid had omitted. What is here faid of Hobbes, may be applied to others who have attempted to refine upon Euclid, and who have been fond of carrying the analyfes of their demonflfations beyond

him.