1944 The Reafon of this Rule is plain; for the Divifor and Dividend expreffing Things of the fame Value and Name, it is evident the Operation is to be managed as with abftract Numbers, by the General Rule; i. e. The Quantity expreffed by the Divifor is contained in the Quantity expreffed by the Dividend, as oft as the Divifor is contained in the Dividend, taken purely as Numbers. Again, If the Divifor and Dividend exprefs Things of the fame general Nature, which can be faid to contain one another, then tho they are not of one particular Species or Name, yet the Queftion is poffible, only it requires that they be reduced to Numbers that exprefs Quantities of one Species or Name; and then it is manifeft, that the Divifion of thefe Numbers by the General Rule, gives the true Quote. So in Exam. 1. 3. is contained in 24 as oft as 3 in 24. But in Examp. 2. 35. is oftner contained in 127. than 3 in 12; for it is as oft as 3 in 240, the Number of Shillings equal to 12 And, because the Divifor and Dividend are then only in a State to be managed as pure Numbers, when they are both fimple Numbers of one Name, this fhews the Reafon of reducing mix'd Numbers. CASE 2. The Dividend being Applicate, and the Diviför Abstract. Rule. If the Dividend is a fimple Number, greater than the Divifor, divide it by the General Rule; the Quote is a Number of the fame things as the Dividend and if there is no Remainder, the Operation is finished; but if there is a Remainder, reduce it to the next Denomination, and divide; and fo on, as long as there is a Remainder, and any lower Denomination, and make a Fraction of the laft Remainder. Thus you have the Answer in one Species or feveral; which is an Applicate Number contained in the Dividend as oft as the Divifor expreffes, or is fuch a part of the Dividend as the Divisor denominates. (See Ex. 1.) ~~ Again, If the Dividend is a fimple Number, less than the Divifor, you must first reduce it to a lower Species, till it be equal to, or greater than the Divifor, and then divide and proceed with the Remainder as before. (See Ex. 3, 4.) If it's not equal to the Divifor in any Species, then the Anfwer is a Fraction of the given Species, whofe Numerator is the given Dividend. (Ex. 6.) Laftly, If the Dividend is a mix'd Number, you may do the Work two ways: Either (1.) Reduce it to a fimple Number of the loweft Species, and then divide, so you'll bave the Anfwer in that Species, (which may be reduced again to fuperiour Species by Divifion, as has been formerly explained, and will be more particularly by and by.) But it will be better to proceed in this manner: (2.) Begin with the Number of the highest Species in the Dividend; divide it, and reduce the Remainder to the next Species, taking in the given Number of that next Species; then divide; and fo go on. (See Ex. 2.) But if the Number of the higheft Species is lefs than the Divifor, reduce it, taking in the given Number of the next Species, and fo on, till you have a Number equal to, or greater than the Divifor: (Ex. 5.) And if that be not in any of the known Species, then the Answer is only a Fraction, whofe Numerator is the Dividend, reduced to the lowest Species, and refers to an Unit of that Species. (See Examp. 7.) The Reason of this Practice is plain: For, if we find any propofed Part of all the Members of which any Number or Quantity is compofed, we have the like Part of the Whole; fince the Whole is nothing elfe but all the Parts. And if the Dividend is lefs than the Divifor, yet if reduced to another Species, it is equal to, or greater than the Divifor; it's plain that the equivalent Number in another Species being divided by the fame Divifor, gives the true Answer: Thus 187.=360s. therefore the 24th part of 18/ is the 24th Part of 360s. SCHOLIUM. What's already done fhews the Practice of Divifion, or the Solution of fimple Questions, where the Propofition is directly and fimply to divide one Number by another, (in any of the Senfes above explained.) And as to the Solution of mix'd Queftions, all the further Direction can be given for knowing when Divifion is to be applied, is to confider well the Effect of Divifion as above explained: Which may be reduced to two principal Views, viz. Finding how oft one Number is contained in another, or finding a propofed aliquot Part of a Number. Then, when the Reafon and Nature of a Question Queftion requires, that you find how oft one Number, fimple or mix'd, of any kind of Thing, is contained in another Number of the fame kind of thing; or, that you find fuch a Part of any Number of things, as another Number denominates, or as an Unit of any Species of Things is of a certain given Number of the fame Things; then is Divifion your Work, as in the following Examples. MIX'D PRACTICAL QUESTIONS for DIVISION. A fpecial Clafs of fuch Questions is comprehended under Title of, REDUCTION from a lower to a higher Species, i. e. To find a Number of things of a higher Species, equal in Value to a given Number of a lower Species; or, at least, to find the greatest Number of the higher contained in the Number of the lower, with what remains over of that lower Species: Suppofing always, that an Unit of the lower Species is an aliquot Part of an Unit of the higher. For which this is the Rule. Divide the given Number by that Number which expreffes how many Units of the Species to be reduced, are contained in an Unit of the Species to which it is to be reduced. The Quote is the Number fought of that higher Species, and the Remainder is a Number of the Species reduced. Thus you may reduce gradually from the loweft to the higheft Species; or all at once to the higheft, if the Number of Units of the loweft, which make one of the higheft, is known. (As you may know by the Reduction Tables, explained in §. 4.) In all the following Queftions, I have performed the Divifions by the contracted Methods, explained in Chap. vi. §. 2. Examp. 1. In Money. 4174608395 f. 12 18652098d: 3 f. 20 15543415: 6 d. 77717 1: 1 s. Here 74608395 f. being divided by 4, Quote 18652098 d. and 3 f. over; thefe d. divided by 12, Quote 1554341s. and 6 d. over; thefe fb. divided by 20, Quote 777171. and is. over. Wherefore 74608395 f. 18652098 d. 3f.=1554341s. 6d. 3f=77717%. 15. 6d. 3f. = The Reason of Dividing in these Cafes is obvious: For Ex. Since 4 Farthings=1 d. therefore as many times as 4f. are contained in any Number of f. fo many d. is that Number of f. equal to. In which obferve, that the immediate Effect of the Divifion is an abstract Number, fhewing how oft 4f. is contained in a greater Number of f. and we apply the Name of d. to the Quote, from the Reason of the Question, as now explain'd. Queft. 1. 8 Men have equal Shares of a Stock of 146 l. 16 s. what is each Man's Share? Anfwer, 181. 75. viz. the 8th Part of 1461. 16s. For the Nature of the Queftion plainly directs us to take an 8th Part. Where obferve, that tho' 8 expreffes a Number of Men in the mixt Propofition, yet in the Operation it is confidered abftractly as the Denomination of that Part of 1467. 16s. which the Nature of the Question requires to be taken. 8) 1467. 16s. 18: 7 O Qreft A certain Number of Perfons, each of whofe Ages is 15 Years 3 Months (reckoning 13 Months to 1 Year) make up in all 182 Years, io Months; how many Perfons are there? Anfwer, 12. viz. The Number of times that 15 Years, 3 Months, are contained in 182 Years, 10 Months. I Queft. 3. What is the Value of 1 Yard of Cloth, whereof 48 Yards coft 15. 10s. 4 d. 48. 1. S. d. f. 15 10 4:0 82: 11: 8:24 : 6:5:21 or Anfwer, 6s. 5d. 2 f. Part of 15 l. 10s. 4 d. For it's plain, the Value of 1 Yard of 48 Yards, muft be the 48th Part of the Value of the whole 48 Yards, which directs us to Divifion, or taking a 48th Part of 15. 10s. 4d. And fo this 48, which in the mix'd Propofition is applied to Yards, is con fidered abstractly in the Operation; which is therefore not a Divifion of Money by Yards, which cannot be made in any Senfe, but taking fuch a Part of the Money as 1 Yard is of 48 Yards, viz. a 48th Part. Obferve, Had it been propofed, in the last Question, to find the Value of 1 Quarter of a Yard, we may do it either by finding firft the Value of 1 Yard, and then the 4th Part of this is the Value of 1 Quarter; or, by reducing 48 Yards to Quarters, which make 192; and taking the 1924 Part of the given Money. The Reafon for which is the fame as for the other Cafe. The following Question requires all the four Operations of Arithmetick. Quest. A Father left among 5 Sons an Eftate, confifting of 500l. in Cafh; with 5 Bills, each of 48 10s. 65. He ordered 201. to be beftowed upon his Burial, and his Debts to be paid, amounting to 1647. Then his free Eftate to be divided in this manner, viz. The eldest Son to have the 3d Part, and the other 4 Sons to have equal Shares. What is the Share of each Son? Anfwer 1861. 4s. 2d. to the eldeft; and 931. 2s. 1 d. to each of the rest. Operation. 481.: 10s. 6d. 20 5 Bills. 164 242 12: 6 184 500: 00 : o Cash. 742 12: 6 Total 184 00 o deduced. 3)558 12: 6 Free Estate. SCHOLIUM. As Queftions may be varioufly mix'd, fo the Solution will depend upon a due Confideration of the feveral Parts of the Question, and what Operation each may require, according to the Senfe and Effect of the different Operations in Arithmetick. But there are mix'd Applications of Multiplication and Divifion, which require other Rules and Directions, to know when and how they are to be made; thefe you'll learn in Book 4. and especially in Book 6. under the Name of Proportion. What has been already done in this Book, being fufficient for explaining the Nature of the fundamental Operations, and their more fimple Applications in whole Numbers. For the Doctrine of 186: 4: 2 Eldeft Son. 4)372: 8: 4 Remains. 93: 2 the Share of each of the other 4. Fractions, you have it in the next Book. СНАР. CHAP. VIII. Containing the more particular Rules of the LITERAL ARITHMETICK, necessary in the following Parts of this Work. C I. For ADDITION or SUBTRACTION. ASE I. If the Numbers to be added or fubtracted are expreffed all by the fame Letter, multiplied by certain Numbers, as, 34, or 5 b, add or fubtract the Coefficients (i.e. the Numbers by which the Letters are multiplied) and multiply the fame Letter into the Sum or Difference, it is the Sum or Difference fought. Examp. 3 a +5a=8a. Ex. 2b+3b+b=6b. Ex. 5n-21=zn. SCHOLIUM. In order to understand the other Cafes, we must premise this Obfervation, viz. After the Addition or Subtraction of one Number to, or from another, we may suppose another added to, or fubtracted from the preceding Sum or Difference; and another added to, or fubtracted from the last Sum or Difference, and fo on: Then is this whole Work expreffed by fetting the Numbers, or Letters representing them, in the fame order, with the proper Signs of the feveral Operations betwixt them. Thus, if b is added to a, and from the Sume is fubtracted, and from this Difference d fubtracted, and to this laft Difference e added; the Refult of all this is expreffed thus, a 4b-c-de. But again, Obferve, That if the fame Numbers can be added or fubtracted in any other order, the final Refult or Effect will still be the fame; which, in all the poffible Orders wherein the Operations can be made, is plainly the Difference betwixt the Sum of all thefe Terms that are added in the feveral Steps, and the Sum of all these that are fubtracted: Because in whatever order any Numbers are added and fubtracted, it's evident there is fo much in whole added, as the Sum of all these that are added in the feveral Steps, and as much fubtracted in whole, as the Sum of all that are fubtracted in the feveral Steps: Wherefore, the final Refult is the Difference of thefe Sums. Whence, again, this follows, That 'tis no matter in what order we place the feveral Terms of a mix'd Expreffion, if we always prefix the fame Signs to the fame Letters, and alfo take the Meaning of the Expreffion to be univerfally the fubtracting all these Terms that have - prefix'd, out of the Sum of all these that have prefix'd: So that when the Operations can be performed in a propofed Order, we may explain the Expreffion either according to that Order, or in the preceding general way, (if that is not the propofed Order.) And if they cannot be performed in the propofed Order, then we explain it after the general way, as the univerfal Meaning of all fuch Expreffions; for tho' fome may be explain'd another way, yet the final Refult is always equal to this. For Example, Ifb is greater than a, then abc can't be explained in the Order of thefe Letters and Signs; and if it is at all poffible, it is fo in this Order, a+c-b; yet it may reprefent the fame thing ftanding thus, abc, while we do not fo much regard O 2 the. |