Page:Cyclopaedia, Chambers - Volume 1.djvu/421

 COM

The

( Z67 ) COM

^ r ^^A,^ Rmmi ,, g>mefilmSy anges and Alternations, w Hch thofe » llmt, m

combmd m all the Manners fofftble, can ZdeHo '

Suppofe two Quantities, <tand£; their Variations will be 2 ; confequently, as each of thofe may be comllZj ™ h \W» Aefe there muft be idded ™ V it tlT The whole Number, therefore, will be 1 +, = , TS; were three Quantities, and the ExponenTof the VarS

The Combatants, who were called Atbltti, prepar'd them- felves for it from their youth, by conftant Exercife and a very rigid Regimen : they only eat certain Things, and at certain Hours ; drank no Wine, had no commerce with Wo- men : both their Labour and their Reft were regulated. See Athietje, Gladiator, £$c.

COMBATANT, is the Heralds Word for two Lions &c born in a Coat of Arms in a fighting pofture, rampant •' and their Paces towords each other. '

COMBINATION, is properly underftood of an Affemblage of feveral Things by two, and two. °

Combination is alfo ufed for the Variation, or Alterna- tion of any Number of Quantities, Letters, Sounds, or the like, in all the different Manners poffible. See Changes

9. Merfenne gives us the Combinations of all the Notes a „d Sounds in Mufick, as far as 24; the Sum whereof amounts to 90 Figures, or Places.

The Number of Combinations of the 24 Letters of the Alphabet, taken firft two by two, then three by three Igc M. Treftet has calculated to be 1391724288S8725299942 512849340^00. See Letter, and Alphabet

The Words in the following Verfe may be combin'd a thoufand twenty two feveral ways.

lU tibifunt Dotes, Virgo, quot fitter a Cailo.

Dotlrine of Combinations.

Any Xwnber of giiantities being given, together with the number m each Combination ; to find the Number of Combinations. J

One Quantity, we obferve, admits of no Combination ; two, a and b, of one; of three a b c, there are three Com- binatmn, viz a b, a c, be ; of four, fix, ab,ac,bc,ad, bi, i ; of five, ten, a b, a c, b c, a d, b d, cd, a e, be, ce, de.

Whence it appears, that the Numbers of Combinations proceed as r, , 6, 10, &c i. c. are triangular Numbers, whofe Side differs by Unity from the Number of given Quantities : if that, v.g. be q, the Side of the Number of Combinations will^be q —, ; a „a therefore the Number of Cmiinations L'/lt^ See Triangular Number. /If three Quantities are to be combin'd, and the Number in each Combination : be three, there will only be one Combi- mmi.abc; if a fourth be added, the Combinations will be found abc, abd, bed, aed; if a fifth, ten, abc, lid, bed, acd, abe,bde, bee, ace,ade; if a fixth twenty {<?,:. The Numbers of Combinations, therefore proceed as 1, 4, to, 20 ie. are the firft pyramidal triangu- lar Numbers, whofe Sides differ by two' Unites from the Number of given Quantities. See Pvr am.oal Number

Hence, if the Number of given Quantities be a, the Side will be £-2; and therefore, the Number of Cimbi^Z

were would

1 ■ 2 3 '

Hence is eafily deduced a general Rule of determining the £ umber of Combinations in any Cafe : For, fuppofe the Num- ber of Quantities to be combined, q, the Exponent of the Combination will be the Number of Combinations ? ~" + I

f — «+' ?~ "+3 ?— « + 4 9— »+5,l£c. * "

2 • 3 • 4 ~. j — tUl the

Number to be added be equal to n.

_Suppofe, v. g. the Number of Quantities to be combined Exponent of the Combination 4; the Number of 6 ~ 4+1 6 — 4+2 6 — 4+ ;

Combinations will be

«-4 + 4

5—3 6— :

6 — 4+ 2

<?+o ».

=15.

4 1 • - -

hW'i 1{k be ^ eflr ' d "'° have'all rhe poffible Combina- m'l If u g c VC " Quantities boning with the Combina- ">"s ot the feveral Twos, proceeding to threes, &c. there mult be added ?~'. ? + ° ? — 2 q — i q + o q— 3

tL> ?-, ? -fo,'ci?/ ' ' ' ' ' 3 ' X •
 * • ~^~ . Whence the Number of Combina-

3 4

! '">m poffible will be tl^Zb. + 1-l ~ '• f — * + ?■? — i -

^ i-f~" ' + 7.g — iy — 2. ?-

•? — 4,

l$c. which

wdabrTl°M r C ["' : ^ of the Bi™™ 1 "'. rais'd to the Power q, aondg d ot the Exponent of the Power increas'd by Unity,

£ .he K J ' h - e PoWer ? ; »nd fince 1 -f-, = 2 ;2?- * '_ J

Nurab„ f „ °' a " the P° ffible Combinations, v.g. If the

l '»«ioL !v„S: am " ieS be u the N^ber of poffible Com- 'ions will be 2 — «= 32 5 = 26".

"■7 ^ //wfo '- of Quantities being given, to find the Num-

-- — ^ — . — „„, „..„ lllo i.jt^uuenr or tfte 1 = . i the Combinations will be 3, and the Changes 9 1 to which it the three Combinations of each Quantity rtith it felf a a bb ec be added, we ftall have tat Number of Changed 3 -f- 3 -f- 3 = 9. 6 •

/ !, n n^| C m tf nnCr ' '" S CT i deHt - if the S ivcn Quantities were 4, and I the Exponent 2 the Number of Changes would be lS i 'I 5. -5, ci?r. and, in general, if K, „' Suppofe the Quantities 3, and the Exponent of Variation

«*«, baa, abb,aae, aea, eaa, abc, bac, bca acb, cab, cba, ace, eac, oca, bba, bab, bbb, bbe, ebb, vcv, bec, cbc, CCc. 9

After the fame manner, it will appear, if the Quantities -e 4, and the Exponent 3, the Number of Changes n, „„,v i+ — * ' ! and, m general, if the Number of Quantities be ;=*, and the Exponent 3, tho Number of rha ff 7 4 be f ■ fl, thus Foceeding, it will be found, ' ri,«w ' e Number of Quantities be », \ni the Exponent n the Number of Changes will be w : wherefore, if all ffie Antecedents be added, where the Exponent is lefs, the Number of poffible Changes will be found n « -f- n « — 'A- ii«-*+n'>-3-\-n<'-i-\-n»-,J r nn-s tf c. Till at length, the Number fubtrafled from n, leaves 1 ; becaufe the beginning is from fingle Quantities taken once.'

Since then the Number ot poffible Changes is a geome- trical Progreffion, whofe firft or fmalleft Term is I" the

greateft n», and the Denominator?/ ; it will be = Cn"-4-'

n) :{n — 1.) k ~

Suppofe, v. g. «= 4. the Number of poffible Changes

l Tt : L 4 ~r = ^? 2 ° : 3 ~ 5 4 - - Suppofe, again, ^ = 24, the Number of poffible Changes will be (24= <—- 4 ) ■ (24 —

^ ~ 320005586-4440,581898^7779553482726-00 : 23 = 1391724288887252999425128493402200. In fo many va- rious manners therefore, may the 24 Letters of the Alpha- bet be vatied and combined among thcmfelves,

F. Truchet, in the Memoirs of the French Academy ihews, that two fquare Pieces, each divided diagonally by two Colours, may be arranged and combin'd Cx different ways, fo as ro form fo many different Kinds of Chequer- work ; which appears furprbing enough, when one confiders
 * «,two Letter*, or Figures, can only be combined twice.

This Note mayy be of ufe to Mafons, Paviours, &c See Pavement.

COMBING of Wool, in Commerce, the drawing, or paf- fing it acrofs the Teeth of a kind of Card, called iWtg

TnATDc r P lnnin »;- See WoOL. Cloth, Spinning, fgc.

CUMBS. See Honey-combs.

COMBUST, in Aftronomy. When a Planet is not above b Degrees and 30 Minutes diftant from the Sun, cither be

™ r „- him ' '" is faid to teCombuft, or in Combufflion.

COME. The fmall Fibres or Tails of Malt, upon it firft Jhooting forth is thus called. See Malt.

COMEDY, in its proper Senfe, a Dramatic Piece, reprefent- lnglome agreeable and diverting Tranfaflion : or, it is anally goncal Reprcfentation of fomcthing in private Life • for the Amufement and Inttruftion of the Speftators. See Dkaih

In this Senfe, Comedy is oppos'd to Tragedy ■ the Sub- jects whereof are grave, and violent; and the Perfons of the firft Rank. Sec Tragedy.

Scaligcr defines Comedy a Dramatic Poem, very bufy pleafant in the Conclufion, and writ in a popular Style.

Jrifiotle calls it an Imitation of the ivor/t, or, rather of the lo-veft Clafs of 'Perfons, by way of ridicule : This Defi- nition Corneille finds fault with, and maintains, that the Actions of Kings themfelves may enter Comedy ; provided they be luch as are not vety momentous, nor attended with any confiderable danger. He adds, that a Poem wherein the greateft Peril is the Lois of a Miftrefs, has no ri»hr to any higher Appellation than that of Comedy : But then he makes a Diftmction in Comedies, and dignifies thofe where great Perlonagcs are introdue'd, with the Epithet of Heroic Comedies, to diftinguilh them from ordinary ones.

Mr. Cougrcve feems pretry much of the fame Sentiment: he undcrftands Jnftotle's Definition of the worfl Men • on which bottom, 'tis fufficient to conftitute a Comedy, that the Action rcprefented be that of fome ill Man broughr on the Stage to be cxpos'd..

MDacier is of a contrary Opinion : He maintains, that Comedy allows of nothing grave, or ferious, unlefs it be turn d to ridicule ; and that Raillery and Ridicule are its only proper and genuine Charaaeriftics: In which Opinion he is warmly feconded by Mr. Dennis.

Thus different are Critics and Comic Authors on the Na- ture of Comedy : lome diftinguifhing it from Tragedy by the

low-