Page:Cyclopaedia, Chambers - Volume 1.djvu/601

 Div

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DIV

^Dividend. 2. That whereby the Dividend is to be divided, call'd the Divifor. 3. That which cxpreiTes how often the Divifor is contain'd in the Dividend ; or the Number re- fusing from the Divifion of the Dividend by the Divifor, call'd the ghiotisnt. See Quotient, $Sc.

There are divers Ways oi performing Divifion^ one call'd the Englijb, another the Flemijh, another the Italian, another the Spanift, another the German^ and another the Indian Way, all equally good, and juft, as finding the Quotient with the fame Certainty, and only differing in the manner of ar- ranging, and difpofing the Numbers,

We have likewife -Divifion in Integers 5 Divifion in Frac- tions ; and Divifion in Species, or Algebra.

Divifion is perform'd by feeking how often the Divifor is contain'd in the Dividend 5 and when the latter confifts of a greater Number of Figures than the former, the Dividend mufi be taken into Parts, beginning from the left, and proceed- ing to the Right, and feeking how often the Divifor is found in each of thofe Parts. For Example, 'tis required to divide 67 59 by g ; I firft feck how oft 3 is contain'd in 6", viz. twice 5 then, how oft in 7, which is likewife twice, with one remaining. This 1, therefore, is joyn'd to the next Figure 5, which makes 1 5, and I feek how oft 3 in 1 5 5 and laftly, how oft 3 in 9. All the Numbers exprefling how oft 3 is contain'd in each of thofe Parts, I write down according to the Order of the Parts of the Dividend, that is, from Left to Right, and feparatc 'cm from the Dividend itfclf, by a Line, thus. Divifor. Dividend. Quotient.

3), *75S> C**53-

It appears, therefore, that 3 is contain'd 225 3 times in 6755); or that 15759 being divided into 3, eachPart will be 22.53. ^ there be any Remainder, that is, if the Divifor repeated a certain Number of times is not equal to the Dividend, what remains is wrote over the Divifor fraclion-wife. Thus, if in- ftead of 6759 the Dividend were only 6758, the Quotient will be the fame as in the former Cafe, except for the lafl Figure 8 j for 3 being only contain'd twice in 8, the lafi Number in the Quotient will be 2 5 and as twice 3 is only 6', there remains 2 of the Dividend; which I write after the Quotient, with the Divifor underneath it, and a Line to fe- parate the two 5 thus,

3) 6758 (2252 t

SPfoof of Division.

Divifion is proved by multiplying the Quotient by the Di- vifor, or the Divifor by the Quotient 5 and adding what re- mains of the 'Divifion, if there be any thing. If the Sum be found equal to the Dividend, the Operation is juft, other- wife there is a Miftake.

r- in Decimal FraBions. ~y r- Decimal.

Division < in Vulgar Fractions. C See-5 Fraction. C of Proportion. 3 c Proportion.

Division, in Species, or Algebra, is perform'd by reducing the Dividend and Divifor into the Form of a Fraction : This Fraction being the Quotient.

Thus, if ab were to be divided by cd 7 it muft be placed

thus, —. and that Fraction is the Quotient : tho' others

c a, ^~

chufe to write it thus, cd) ab, or cd-.ab, or ab'~cd, which laft Mark "- is the moft common Character for Di- vifion. See Character.

To perform the Work of Divifion algebraically, thefe Rules are to be obferved : 1. When the Dividend is equal to the Divifor, the Quotient is Unity, and muft be placed in the Quotient, becaufe every Thing contains itfelf once.

2. When the Quotient is exprefs'd Fraction-wife (as in fimple Divifion^) if the fame Letters are found equally re- peated in each Member of the Numerator, and Denomina- tor; caft away thofe Letters, and the Remainder is the Quo- tient : Thus, ^ {a, and ^ O, &c.

3. When there are any Co-efficients, divide them as in com- mon Arithmetic, and to the Quotients annex the Quantities

exprefs'd by Letters : Thus, - -r- (=15 a.

4. The general Way of Divifion of compound Quantities, is like the ordinary Way in common Arithmetic, refpect. be- ing had to the Rules of.Algcbraic Addition, Subftaftion, and Multiplication ; as alfo that like Signs give +, and unlike — in the Quotient: taking Care to divide every Part of the Dividend by its correfponding Divifor, (that is, that whofe Letters /hew it of the fame Kind with the other) to pre- vent a Fraction, which would otherwife arife : Thus, a-\-b) aa-\-ab — ca — cb (a—c»

aa-\~ab

— ca~cb — ca — cb

Q

That for the fame reafon like Signs giving a Pofitive, and unlike a Negative Quotient, do hold in Divifion, as well as in Multiplication, is clear from confidering the Nature of

Djvifion$ (which is only refolving the Thing into its Parts) confequently, fince every Dividend is nothing clfe but the Pro- duel of the Divifor, and Quotient multiplied^by each other the Quotient muft confift of fuch Signs, which could produce' the Dividend; therefore, if the Dividend be divided by a Quan- tity, that has a fimilar Sign with it, the Quotient mull be pofitive ; if by a Quantity having a diffimilar Sign, the Quotient muft be negative. It may be a general Rule in compound Divifion in Algebra, always to place fuch a Let- ter in the Quotient, as will, when multiplied into the Divi- for, produce the Dividend, for that is always a Re5t-angle under the Divifor, and the Quotient : As for Example, aa— 16) z 6 — 8^—124 zz— 64 C^+S^-h^

128 ss

Division

H£

4 sz— 64 4 zz — 64.

Nepairs "Bones. \

SW

Nepairs Sous,

Division, in Lines, or Geometrical Division, is alfo call'd Application ; the Defign of which, when it is em- ploy 'd about the Couftruflion of plain Problems, is this, viz. a Rectangle being given ; as alfc a right Line ; to find an- other right Line, the Rectangle contain'd under which with the right Line given, mail be equal to the Rectangle firft given : Such Effeftion, or Conftruction is call'd the Appli- cation of a given Reaangle to a right Line given ; and the right Line arifing by fuch Application, is call'd the 'Pa- rabola, or Geometrical Quotient.

This is found by the Rule of Three, by making, As the Line given : is to one Side of the Recc-angle ::'fo is the other Side : to the Line fought. Not unlike to which is des Cartel's Way of working Divifion in Lines, by Scale, and Compafs : Thus fuppofe a c (=<r) were to be di tided by a4U=%) make any Angle at Pfeafure, and therein fet off firft ad (= 3) the Divifor, and then on the fame Leg au== to Unity : Then on the other Leg of the Angle fit ac (= 6} the Dividend, and joyn dc, and to it, through K, draw ub parallel to dc, which fhall cut off a b the Quotient fought ; for as ad : au :: ac : ab ; that is, as the Divifor : is to Unity :: fo is the Dividend : to the Quotient ; on which Reafon depends all Divifion.

Division, in Mufic, the dividing of the Interval of an Octave into a Number of lcffer Intervals. See Octave, and System.

The 4th, and 5th, each of 'em, divide, or meafurc the Oc- tave perfectly, tho' differently. When the 5th is below, and ferves as aBafe to the 4th, the Divifion is call'd Harmoni- cal; When the 4th is below, the Divifion is call'd Authentic. See Scale. See alfo Concinnous.

Division, in War, is a Batallion, or other Body of Forces in March.

The Lieutenant commanded the firft Divifion of Mufque- teers ; and the fecond Lieutenant, the fecond Divifion.

Division, in Sea-Language, the third Part of a Naval Army, or Fleet, or of one of the Squadrons thereof, under the Command of fome General Officer.

Naval Battals are ufually rang'd in three Lines, according to their three Divifions.

DIVISOR, is the dividing Number ; or that which fhewa how many Parts the Dividend is to be divided into. See Dividend, &c.

DIVORCE, a Breach, orDiffolution of the Bond ofMar- riage. See Marriage.

In our Law Divorce is of two Kinds : The one, a Vin- culo Matrimonii 5 which alone is properly Divorce : The other, a Men/a (3 "There ; a Separation from Bed and Board.

The Woman divorced a Vinculo Matrimonii, receives all again that me brought with her : The other has a fuit- able feparatc Maintenance allow'd her out of her 'Hus- band's Effects.

The firft only happens through fome effential Impediment, as Confanguinity or Affinity within the Degrees forbidden, precontract, Impotency, Adultery, c5c. Of which Impedi- ments the Canon Law allows fourteen, comprehended in thefe Verfes :

Error, Conditio, Votum, Cognatio, Crimen, Cultus, Difparitas, Vis, Ordo, Ligamen, Ifonsfias t Sifis affinis, fi forte coire ne./uibis, Si parochi duplicis defit prcefentia Tefiis, Raptave fit Mulier, nee parti reddita Putce.

Divorce is a Spiritual Judgment, and, therefore, is pafs'd in the Spiritual Court. Under the old Law, the Woman di~ vorced was to have of her Husband a Writing, as St. ■Jerom and Jofiephus teftify, to this Effect : Ipromife, that here after I will lay no Claim to thee j Which was call'd a Sill of Divorce,