NUM
(641)
whereby comparing the feveral Pieces, we bring them into one Sum, as ten : So that Number depends altogether on the Mind of theFerfon that numbers $ whence changing the Idea at pleafure, an hundred Men /hall only be call'd one, or it fhall be two, or four, &c.
Hence, fay they, the Form of a Number t is nor any thing added to the things number' d ; for the Idea is a mere Mode of the Mind, not any thing fuperadded to the things. And hence, tho there may be fome Efficacy in Number, confi- der'd with refpefl to the Matter, as when we fay A triple Rope is not eafily broke ; yet there is none in refpeft to Form : For what Alteration fhould my Idea make ? And hence the Folly of the l'hilofophy of Numbers,
The fame Philofophers call Number a difcrete Quantity : Quantity^ as it admits of more and lefs 5 and difcrete, fihee the feveral Units it confifts of are not united but remain dittinft.
Far the Manner of deftgnmg, or characterizing Numbers^ fee Notation.
i'or that of exfrefjing or reading thofe already charaileriz'd, fee Numeration.
Mathematicians, confidering Number under a great many Circurjiflances, different Relations, and Accidents, make many Kinds of Numbers.
A determinate Number, is that refer'd to fome given Unit ; as a Ternary, or three, which is what we properly call a Number.
An indeterminate Number, that refer'd to Unity in the general ; which is what we caU Quantity. See Quan- tity.
Homogenezl Numbers, are thofe refer'd to the fame Unit.
Beterogeneal Numbers, thofe refer'd to different ones. For every dumber fuppofes fome determinate Unit, which is determined by the Notion to which we have regard in Numbering. £•$?. 'tis a diitinguifhing Property of a Sphere, that the feveral Points of its Surface are equidiitant from its Centre: If then, this be laid down as a Note of Unity, all Bodies to which it agrees will have the nature ot'Unity 5 and are the fame Units, quatenus contain'd under this No- tion. But if Spheres be ditlinguifhed, e. gr. with regard to the Matter they are compofed of; then thofe which be- fore were the fame Units, commence different. Thus, fix golden Spheres and three golden Spheres are homogejieal Numbers among themfelves ; and three brafs Sphere's and four filver ones, are heterogeneous Numbers.
We Numbers, call'd alfo natural Numbers, and Inte- gers, or (imply Numbers, are all the various Affemblages of mains nothing Unity, or the Ideas we have of feveral Multitudes 5 or, according to Wolfus, all thofe which, in the manner of expreffing, refer to Unity, as a Whole does to a Part.
Broken Numbers, or B-aBions, are thofe confining of fe- veral parts of Unity, or thofe which refer to Unity as a Part to the Whole. See Fraction.
Rational ~Nvmbkr, is that commenfurable with Unity. — Rational whole Number^ is that whereof Unity is an Attentat
part. Rational broken Number, that equal to fome slli-
jK.ctpart or parts of Unity. Rational mixt Number, that
confining of a whole Number and a broken one, orofUnity and a Fraction. See Fraction.
NUM
IfTfltwra/NuMRER, or Surd, & Number incommenfurable with Unity. See Surd.
Even Number, that which may be divided into two equal parts, or without Remainder or Fraction ; as 4, 6, 8 ic, lie. The Sum, as alfo the Difference, and the FaBum, or Produce of any Number of even Numbers, is always an even Number,
An even Number multiply'd by an even Number, produces an evenly even Number.
An even Number is faid to be evenly even, when it may be meafur'd or divided without any Remainder by another even Number.
Thus, twice four being eight, eight is an evenly even Number.
Prime Numbers hmong themfelves, are thofe which have no common Meafurc bend: Unity ; as 12 and 19.
Compound Nvmber, is that divifible by fome other Num. her belides Unity ; as 8, divifible by 4, and by 5.
Compound Numbers amour themfelves, thofe which have fome common Meafure befiries Unity ; as in and 15.
FerfcB Number, that whole aliauot Parts addcd t0 „_ ther, make the whole Number i as 6, ig, £Tc. The alifmi 1 arts of 6, being 3, », and I = d. And thofe of ig, being ■4. 7.4. -, '■ which toeether make iS.
ImperfeS Numbers, thofe whofe altqmt Parts added to- gether, make either more or lefs than the Whole whereof they are Parts.
lmperfeB Number,, are diflinguiih'd into Aubniant and Vefemvc.
Abundant Numbers, are thofe whofe aliquot Parts added together, make more than the Number whereof they are Parts; as 12, whofe aliquot Parts, 5,4, 3, 2, ,, ma (. e l6 .
DefeBive Numbers, are thofe whofe aliquot Parts added together, make lefs than the Number whole Parts they are } as i<5, whofe aliquot Parts, 8,4,2, and 1. only make 15.
Flane Number, thatariling from the Multiplication of tV/oNumbersi ex.gr. 6, which is theProdudl of 5 multiply'd by ». T hettoim which thus multiply'd produce a Plane Number, as here, 2 and e, are call'd the Side: of the Plane.
Squire Number, the l'rudufl of any Number multiply'd by itlelr ; thus 4, the Failum of 2, by 2, is ifquare Number. oec Square.
Every fquare Number added to its Root, makes an even Number,
Cubic Number, the Produfl of s. fqttare Number, multi- ply d by its Root ; ex.gr. 8, the Product of the fqttare Num- ber 4, multiply'd by its Root 3. See Cube.
All Cubic Number:, whofe Root is lefs than 6, v. 7. 8 2?, 6-4, 125, being divided by 6, the Remainder is their Root itfelf. 1 hus 8 being divided by 6, 2, the Remainder of the Divifion, is the Cube Root of 8. For the Cubic Num- ber: beyond 125 ; 2i«, the Cube of 6, divided by i, leaves no Remainder; 245, the Cube of 7, leaves a Remainder 1, which added to 6, gives the Cube Root of 343. And 512, the Cube of S, divided by 6, leaves 2, which added to 6, makes the Cube Root of <u. So that the Remain- ders of the Divificns of the Cubes above 216, divided by 6, being added to 6, always give the Root of the C*b,c Number divided ; till that Remainder be 5, and of confe- rence 1 1 the Cube Root of the Number divided : But the Cubic Number above this, being divided by 6, there re- mains nothing, the Cube Root being 12. Thus, if you continue to divide the higher Cubes by <f, you muft not add the Remainder of the Divifion to C, but to 12, the firft Multiple of fi; and thus coming to the Cube of 18, the Remainder of the Divifion mutt not be added to S, nor to 12, but to 18 : and thus in infinitum.
Monf. dc la Hire, from onfidering this Property of the Number 6, with regard to Cubic Numbers, found that all 0- ther Number: rais'd to any Power whatfoever, had each their Divifor, which had the fame effefl wiih regard to them, that 6 has with regard to Cubes. And the general Rule he has difcorcr'd, is this : If the Exponent of the Power of a Number be even, ;'. e. if that Power be rais'd to the 2d, 4th, 6th, (gc Power, it mutt be divided by 2 } and the Remainder, if there be any, added to 2, or to a Multiple of 2, gives the Root of the Number corre'fponding to its Power, i. e. the 2d or tfth Root, Sfc. But if the Ex- ponent of the Power of the Number be uneven, i. e. if it be rais'd to the ;d, 5 th, 7th, lie. Power, the Duple of that Ex- ponent will be the Divifor which fhall have the Property here requir'd. r 3
Polygonous Numbers, the Sums of Arithmetical Progref- fions, beginning with Unity. Thefe, where the Difference of Terms
is 1, are call'd Triangular Number: ; where 2
Square Number: ; where 3, Pentagonal Numbers ; where 4'
Hexagonal Number: ; whete •},Heptagonal,Hc. See Polygon.
Pyramidal Numbers. The Sumsof Polygonous Numbers,
ANumbern ftid to be unevenly even, when it may be collefled alter the fame manner as the Polygons themfelves
as 20, which may are gather^ out of Arithmetical Progreffions, are call'd
equally divided by an uneven Number be divided by 5.
Uneven Number, that which exceeds an even Number at leaf! by Unity ; or which cannot be divided into two equal Parts. Such are 3, 5, 9, tr, lie.
The Sum, or the Difference, of two uneven Numbers, makes an even Number, but the FaSum of two makes an uneven one.
If an even Number be added to an uneven one, or if the one be fubftraSed from the other ; in the former Cafe, the Sum, in the latter the Difference, is an uneven Number. But the FaBum of an even and an uneven Number, is even.
The Sum of any even Number of uneven Numbers, is an even Number', and the Sum of any uneven Number ofuneven Numbers, is an uneven Number.
Primitive or prime Number, is that which is only divifi- ble by Unity ; as, 5,7,11, He
firft Pyramidal Numbers,
The Sums of the firll Pyramidals, are call'd fecond Pyra* miials. The Sums of the fecond Pyramidals, are call'd
tbird Pyramidal:, $£c.
In particular, they are call'd Triangular Pyramidal Num- bers, if rhey arifc out of Triangular Numbers. Firft Pentago- nal Pyramidals, if they arife out of Pentagons, tie See Pyramid.
Cardinal Numbers, thofe which exprefs the Quantity of Units ; as 1, 2, tic
Ordinal Numbers, thofe which exprefs the Order 01 Rank; as iff, 2d, 3d, Sic.
Golden Number, in Chronology, a Period of 10 Tears invented by 'Meton the Atbenian ; at the end whereof, the fame Lunations return in the fame Days, tho' not precifely in the fame Hour and Minute of the Day. See Period and Lunation. 8 A Hence
