POW
wherein, it is fuppofed, are contained all the principal Vir- tues thereof: whence the Name.
Power, in the Feodal Jurifprudence, a Right which the Lord has to re-unite to his Fief, a Dependant Fee held of him; when the Vaffal has alienated it 5 upon reimburfing t^ie Money given for it, with the Legal Colts. See Fee.
The Lord is to exercife his Tower over the Fee, within a "Year after he has notice of the Fall ; other wife he lofes it.
The Word is alfo ufed for the Right a Lord has to feize a Dependant Fee, to compel the Payment of all Dues, Ser- vices, &c.
Powers, in Theology, a Term ufed among the Fathers, t$c for the fixth Order in the Hierarchy of Angels ; com- mencing from Seraphim. See Seraphim.
Theie they fuppofe to be the Spirits who bridle and re- train the Tower of the Devils; prefide over inferior Cau- fes; and prevent contrary Qualities from disturbing the Oeconomy of the "World. See Hierarchy.
Power, in Optics. The Tower of a Glafs, is the Di- itance of the Convexity, from its folar Focus. Sje Focus.
Power, in Arithmetic, the Produce of a Number mul- tiply'd into itfelf. Sec Number.
Thus the Produce of the Number 3, multiply'd by itfelf, viz. 9, is the fecond Tower of 3 ; the Factum of 9, mul- tiply'd by 3, viz. 27, is the third Towers and the Product of 27, again multiply'd by 3, viz. 8i, is the fourth Tower ; iind fo on to Infinity. In refpecl hereof, the firft. Number, 5, is called the Root, or firjl Tower. See Root.
The fecond Tower is call'd the Square; with refpecl to which, 3 is the Square-Root. See Square.
The third Tower, 27, is call'd the Cube, with refpecT: to which, the 3 is the Cube-Root. See Cube.
The fourth Tower, 81, is call'd the "Biqitadrate, or Jgua- drato-giiadratvw; with refpeel to which, 3 is the Biqua- dratic Root. See Biquadratic.
The Number which Ifiews how oft the Root is multi- ply'd into itfelf, to form the Tower ; or how oft the Tower is to be divided by its Root, to come at the Root, is call'd the Exponent of the Tower. See Exponent.
The Moderns, after Zfef Cartes, are contented to diirin- guifh moft of their Towers by the Exponents; as, firfi, fe- cond, third, &e.
The particular Names of the feveral Towers were intro- duced by the Arabs ; viz. Square, Cube, ^uadrato-^ua- dratum or Siquadrate, Surdefolid, Square of the Cube, fe- cond Surdcfolid, ghiadrato-quadrato quadrature, Cube of the Cube, Square of the Surdefolid, third Surdefolid, cVc.
The Names given by Diophantus, follow'd by Viet a and Ovghtred, are, the Side or Root, Square, Cube, §{uadrato- quadratum, Sgttadrato-cuhus, Cubo-cubus, £>uadrato-qua- draio-cubus, guadrato-ctt&o-cuhus, Cubo-aibo-cubu.s, &c.
The Characters wherewith the feveral Towers are de- noted, both in the AtefylG and Cartefian Notation, areas follow.
2 4- 8 i5 32 54 128 %<yC jra 1024 Arab. R q c bq s qc ff tq be fr Cartef. a a* a s a* a* a b a 7 a* a* & ia
Hence, to raife a Quantity to a given 'Power, or Digni- ty, is the fame as to find the Factum anting upon its be- ing multiply'd a given Number of Times into itfelf: E. gr. to raife 2 to the 3d Tower, is the fame as to find the Faflura 8 ; whofe Factors are 2, 2, 2. Sec Square, Cube, 2?C.
Towers of the fame Degree, are to one another in a Ratio of the Roots as manifold as their Exponent contains Units : Thus, Squares are in a duplicate Ratio ; Cubes in a triplicate Ratio; Quad rate-quadrat a, or fourth Towers, in a quadruple Ratio. Sec Ratio.
The Towers of proportional Quantities, are alfo propor- tional to one another. See Proportion.
From a given Tower to" extrafl the Root, or Side ; is the fame as to find a Number, e.gr. 2, which multiply'd any IS! umber of times, e.gr. twice, produces the given Tower, e. gr. the 3d Tower, or 8. See Root.
-To multiply or divide any Tower by another of the fame Root. i Q - For Multiplication, add the Exponents of the Factors; the Sum is the Exponent of the Factum. Thus:
(8?? )
P05C
Faflors -
X 4 J"
a'
Prod.
y-m 1 m -[-» fi™-\-* X»-\- s .
a . For Division, fubflracl the Exponent of the Fciver of the Divifor, from the Exponent of the Dividend ; the Re- mainder is the Exponent of the Quotient. Thus :
V\v\&.XTfxl\\y<°-\-
Divif. «v !!>'»
Wi.de la Hire gives us a very odd Property common to all 'pi/xers: M. Carre had obferv'd, with regard to the
qxzq
Number 6, (hat all the natural cubic Numbers, 8,17, 64., 125, whofe Root is lefs than 6, being divided by 6", the Remainder of the Divifion is the Root itfelf: And if we go further, aitf, the Cube of 6, being divided by 6 leaves no Remainder ; but the Divifor 6, is itfelf the Root* Again, 343, the Cube of 7, being divided by tf, leaves 1 ; which, added to the Divifor 6, makes 7 the Roof, $$C.
M. de la Hire, on considering this, has found that all Numbers, rais'd to any fPower whatever, have Divifors which have the fame efrcfl with regard thereto, that 6 has with regard to Cubic Numbers.
For the finding of thefe Divifors, he difcover'd the fal- lowing general Rule :
If the Exponent of the Tower of a Number be even, i.e. if the Number be rais'd to the- ad, 4th, 6xh Tower., &C it muft be divided by a ; the Remainder of the Di- vifion, in cafe there be any, added to 2, or to a Multi- ple of a, gives the Root of this Number, correfpondir.g to its Tower, i.e. the 2d, tfth, &c. Root.
If the Exponent of the Tower be an uneven Number, i. e. if the Number be rais'd to the 3d, 5th, 7th, g?c. Tower; the Double of this Exponent will be the Divifor, which has the Property mention'd.
Thus is it found in 6, double of 3, the Exponent of the Tower of all the Cubes : Thus, alfo, 10 is the Divifor of all Numbers rais'd to the 5th Tower, &?c.
Power, of an Hyperbola, in Conies, is the Square of the right Line CI, or A B (Tab. Conics, fig. 2 7.)
The Tower of the Hyperbola, is the Sixteenth part of the Squares of the conjugate Semi-Axes; or the fourth pai'tof the Squares of the conjugate Axes. See Hyperbola* Power of the County. See Posse Comitatus. POX, in Medicine, a Difeafe, whereof the Phyficians admit feveral kinds ; as the Small Tox, French Tox, Chicken Tox, and Swine Tox.
Small POX, Variola, is a contagious Difeafe appearing on the Cutis, which it covers with Putties or Ulcerous Eruptions, that leave Efchars behind 'em. Or, it is a ge- neral Eruption of particular Putties tending to Suppuration. SeePusTLE and Suppuration.
The Origin of the Difeafe is uncertain : We find no mention of it before the Arabic Vhydchns. See Disease. The Small Tox bear a great refemblance to the Mealies; fo that for the two or three firtt. days 'tis difficult to diitin- guifli cm : They both arife from an impure Blood, and corrupt Humours ; with this difference, that in the Small Tox, the peccant Matter is more thick and vifcid ; in the Mealies more fubtile, hot, and bilious ; and neither of 'em are known to return after having pafs'd 'em once. See Measles.
'DcUus fays, that the Caufe of the Small Tox is brought into the World with us ; and lies hid till it find an oppor- tunity of burlting forth : He adds, that there is fcarce one in. many thoulands that efcapes it all his Life.
Dr. 'Drake obferves, that the Small Tox not being founded in any permanent habitual Difpolition, has its Period within a limited time neceffary for the extrufion of the peccant Matter out of the Pores of the Skin. For the fait Serum of the Blood being in this Difeafe, by an accidental Fever, thrown out in great quantities on the Glands of the Skin, afts much after the manner of the Lepra Arabum ; but - when the Blood is defpumated, the Scales dry, and fill off". So that he thinks it would be no great Impropriety to call the Small 'Pox, a temporary critical Lepra. See Lepra.
The Small Tox are of two kinds ; the fDifihift, where the Putties Hand a-part ; and the Confluent, where they run into one continued Cake.
The diftinB, or regular Small Pox, Sydenham obfervesj begin with a Shuddering and Chillinefs, which isfucceeded by an intcnle Heat, violent Pain of the Head and Back, Vomiting, Drowfinefs, efpecially in Children, and fome- tirr.es Epileptic Fits; which mew the Tox to be ready to burft forth, and that they will be mild.
The Eruptions are ufually on the fourth day ; upon which the feveri/h Symptoms vanilh, except that Adults are prone to fweat. The Putties firtt: appear in the Face, then the Neck, £S?c. They are at firft reddiih, by degrees fwell and grow whiter ; on the eleventh day the Swelling and Inflammation of the Face vanifh ; and the Putties be- gin to wither. If ever this kind kill, 'tis on the fourteenth or fifteenth day.
The Symptoms of the mftintt Small Tox, as enumerated by Dr. Shaw, are, 1. A Pain in the Head, Back, and Scro- bictdum Cordis. 2. A Fever, which decreafes as the Erup- tions increafe, with rednefs of the Eyes. 3. Naufea and Retchings. 4. Little reddi/h Spots, or beginning Puttules, appearing on the Neck, Face, Breaft, Sfa about the third or fourth day inelufive, from the beginning of the Illnefs. 5. Reltleffnefs. 6. About the feventh or eighth day, other little red Spots ufually appear between the growing Puflules. 7. The Puitules about the ninth day are at their State ; being then generally as big as a large Pea, the
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