of three also by 〈◊〉 things of 〈…〉 are reduced to another 〈…〉 any Number of Integers by the price of the Integer the Product will discover the price of the Quantity or Number of Intâ—gers given In a Râ—ctangular Solid if you multiply the bredâ—h of the base by the depth and that Product by the length this last Product will discover the Solidity or content of the same Solid Some Questions proper to this Rule may be these following Quest. 1. What is the content of a square piece of ground whose length is 28 perches and breadth 13 perches Answer 364 square perches for multiplying 28 the length by 13 the breadth the Product is so much Quest. 2. There is a square battail whose Flank is 47 men and the files 19 deep what Number of men doth that battail contain Facit 893 for multiplying 47 by 19 the Product is 893. Quest. 3. If any one thing cost 4 shillings what shall 9 such things cost Answer 36 shillings for multiplying 4 by 9 the Product is 36. Quest. 4. If a piece of Money or Merchandize be worth or cost 7 shillings what shall 19 such pieces of Money or Merchandize cost Facit 133 shillings which is equal to 6 l. 13 s. Quest. 5. If a Souldier or Servant get or spend 14 s. per moneth what is the Wages or Charges of 49 Souldiers or Servants for the same time multiply 49 by 14 the Product is 686 s. for the Answer Quest. 6. If in a day there are 24 hours how many hours are there in a year accounting 365 dayes to constitute the year Facit 8760 hours to which if you add the 6 hours over and above 365 dayes as there is in a year then it will be 8766 hours now if you multiply this 8766 by 60 the Number of Minutes in an hour it will produce 525960 for the Number of Minuts in a Year CHAP. VII Of Division of whole Numbers 1. DIVISION is the Separation or Parting of any Number or Quantity given into any parts assigned Or to find how often one Number is Contained in another Or from any two Numbers given to find a third that shall consist of so many Units as the one of those two given Numbers is Comprehended or contained in the other 2. Division hath three Parts or Numbers Remarkable viz. First the Dividend Secondly the Divisor and Thirdly the Quotient The Dividend is the Number given to be Parted or Divided The Divisor is the Number given by which the Dividend is divided Or it is the Number which sheweth how many parts the Dividend is to be divided into And the Quotient is the Number Produced by the Division of the two given Numbers the one by the other So 12 being given to be divided by 3 or into three equal parts the Quotient will be 4 for 3 is conâ—ained in 12 four times where 12 is the Dividend and 3 is the Divisor and 4 is the Quotient 3. In Division set down your Dividend and draw a Crooked line at each end of it and before the line at the left hand place the Divisor and behind that on the right hand place the figures of the Quotient as in the margent where it is required to divide 12 by 3 First I set down 12 the Dividend and on each side of it do I draw a crooked line and before that on the left hand do I place 3 the Divisor then do I seek how often 3 is contained in 12 and because I find it 4 times I put 4 behind the Crooked line on the Right hand of the Dividend denoting the Quotient 4. But if the Divisor being a single Figure the Dividend consisteth of two or more places then having placed them for the work as is before directed put a point under the first Figure on the left hand of the Dividend provided it be bigger then or equal to the Divisor but if it be lesser then the Divisor then put a point under the second Figure from the left hand of the Dividend which Figures as far as the point goeth from the left hand are to be Reckoned by themselves as if they had no dependance upon the other part of the Dividend and for distinction sake may be called the Dividual then ask how often the Divisor is contained in the Dividual placing the answer in the Quotient then multiply the Divisor by the Figure that you placed in the Quotient and set the product thereof under the Dividual then draw a line under that product and Subtract the said Product from the Dividual placing the Remainder under the said line then put a point under the next figure in the Dividend on the Right hand of that which you put the point before and draw it down placing it on the Right hand of the Remainder which you found by Subtraction which Remainder with the said Figure annexed before it shall be a new dividual then seek again how often the divisor is contained in this new dividual and put the Answer in the Quotient on the Right hand of the Figure there before then multiply the divisor by the last Figure that you put in the Quotient and subscribe the Product under the dividual and make Subtraction and to the Remainder draw down the next Figure from the grand dividend having first put a point under it and put it on the right hand of the Remainder for a new dividual as before c. Observing this general Rule in all kind of Division first to seek how often the divisor is contained in the dividual then having put the answer in the quotient multiply the Divisor thereby and Subtract the Product from the dividual An Example or two will make the Rule plain Let it be Required to divide 2184 by 6 I dispose of the Numbers given as is before directed and as you see in the margent in order to the work then because 6 the divisor is more then 2 the first Figure of the dividend I put a point under 1 the second Figure which make the 21 for the Dividual then do I ask how often 6 the divisor is contained in 21 and because I cannot have it more then 3 times I put 3 in the Quotient and thereby do I multiply the divisor 6 and the product is 18 which I set in order under the dividual and Subtract it therefrom and the Remainder 3 I place in order under the line as you see in the Margent Then do I make a point under the next Figure of the dividend being 8 and draw it down placing it before the Remainder 3 So have I 38 for a new dividual then do I seek how often 6 is contained in 38 and because I cannot have more than 6 times I put 6 in the quotient and thereby do I multiply the divisor 6 and the product 36 I put under the dividual 38 and Subtract it therefrom and the remainder 2 I put under the line as you see in the Margent Then do I put a point under the
dividend and put it before the said Remainder 286 so have I 2865 for a new dividual and because it consisteth of four places viz. a place more than the divisor I seek how often 3 the first Figure of the divisor is contained in 28 the two first of the dividual and I say there is 9 times 3 in 28 but multiplying my whole divisor 385 therby I find the product to be 3465 which is greater than the dividual 2865 wherefore I choose eight which is lesser by a Unit than nine and thereby I multiply my divisor 385 and the product is 3080 which still is greater then the said dividual wherefore I choose another Number yet a Unit lesser viz. 7 and having multiplyed my divisor thereby the product is 2695 lesser than the dividual 2865 wherefore I put seven in the quotient and Subtract 2695 from the dividual 2865 and there remains 170 then I pull down the last figure 3 in the dividend and place it before the said Remainder 170 and it makes 1703 for a new dividual then for the reason abov-said I seek how often three is contained in 17 the answer is 5 but multiplying the divisor thereby the product is 1925 greater than the dividual wherefore I say it will bear four a Unit lesser and by it I multiply the divisor 385 and the product is 1540 which is lesser than the dividual and therefore I put four in the quotient and Subtract the said product from the dividual and there Remaineth 163 and thus the work is finished and I find that 1183653 being divided by 385 or into 385 equal shares or parts the Quotient or one of those parts is 3074 and besides there is 163 Remaining And thus the Learner being well versed in the method of the foregoing Examples he may be sufficiently quallified for the dividing of any greater summe or number into as many parts as he pleaseth that is he may understand the method of dividing by a Divisor consisting of 4 or 5 or 6 or any greater number of places the method being the same with the foregoing Examples in every Respect Other Examples of Division So if you divide 47386473 by 58736 you will find the Quotient to be 806 and 45257 will Remain after the work is ended In like manner if you would Divide 3846739204 by 483064 the Quotient will be 7963 and the Remainder after Division will be 100572. Compendiums in Division 1. IF any given Number be to be Divided by another Number that hath Cyphers prefixed on the Right side thereof omitting the Cyphers you may cut off so many Figures from the Right hand of the Dividend as there are Cyphers before the Divisor and let the Remaining numbers in the Dividend be divided by the Remaining number or numbers in the Divisor observing this Caution that if after your Division is ended any thing Remain you are to prefix the number or numbers that were cut off from the Dividend before the Numbers Remaining and such new found Number shall be the Remainder As for Example Let it be Required to divide 46658 by 400 now because there are two cyphers before the divisor I cut off as many Figures from before the Dividend viz. 58 so that then there will Remain only 466 to be divided by 4 and the Quotient will be 116 and there will Remain 2 before which I prefix the two Figures 58 which were cut off from the Dividend and it makes 258 for the True Remainder so that I conclude 46658 divided by 400 the Quotient will be 116 and 258 Remaineth after the work is ended as by the work in the Margent 2. And hence it followeth that if the divisor be 1 or a unite with Cyphers prefixed you may cut off so many figures from before the dividend as there are Cyphers in the divisor and then the figure or figures that are on the left hand will be the Quotient and those on the right hand will be the Remainder after the Division is ended as thus if 45783 were to be divided by 10 I cut off the last figure 3 with a dash thus 45783 and the work is done and the quotient is 4578 the number on the left hand of the dash and the Remainder is 3 on the right hand In like manner if the same Number 45783 were to be divided by 100 I cut off 2 figures from the end thus 45783 and the quotient is 457 and the remainder 83. And if I were to divide the same by 1000 I cut off 3 figures from the end thus 45783 and the quotient is 45 and 783 the Remainder c. 6. The General effect of Division is contained in the definition of the same thâ—â— is by having two unequal numbers given to finde a third number in such proportion to the dividend as the divisor hath to unite or 1 It also discovers what reason or proportion there is between numbers so if you divide 12 by 4 it quotes 3 which shews the reason or proportion of 4 to 12 is triple The second effect is by the superficial measure or content and the length of any oblonge Rectangular paralelogram or square plane known to finde out the breadth thereby or contrariwise by having the superficies and breadth of the said figure to finde out the length thereof Also by having the solidity and length of a solid to find the superficies of the base Contra. The third effect is by the contents Reason price vallue buying selling expences wages exchange interest profit or loss of any things be it Money Merchandize or what else to find out the contents reason price value buying selling expence wages exchange interest profit or loss of any one thing of like kinde The fourth effect is to aid to compose and to make other Rules but principally the Rule of proportion called the Golden Rule or Rule of three and the Reduction of Moneys weight and measure of one denomination into another by â—t also fractions are abbreviated by finding a common measurer unto the numerator and denominator thereby discovering commensurable numbers If you Divide the value of any certain quantity by the same quantity the Quotient 〈◊〉 the Rate or value of â—he Integer as 〈◊〉 eight yards of Cloth cost 96 shillings here if you divide 96 the value or price of the given quantity by 8 the same quantity the Quotient will be 12 s which is the value or price of 1 of those yards contra If you divide the value or price of any unknown quantity by the value of the Integer it gives you in the quotient that unknown quantity whose price is thus divided as if 12 shillings were the value of 1 yard I would know how many yards are worth 96 shillings here if you divide 96 the price or value of the unknown quantity by 12 the rate of the Integer or one yard the quotient will be 8 which is the number of yards worth 96 shillings Some Questions answered by Division may be these
following Quest. 1. If 22 things cost 66 shillings what will 1 such like thing cost facit 3 shillings for if you divide 66 by 22 the Quotient is 3 for the Answer so if 36 yards or ells of any thing be bought or sold for 108 l. how much shall 1 yard or ell be bought or sold for facit 3 l. for if you divide 108 l. by 36 yards the Quotient will be 3 l. the price of the Integer Quest. 2. If the expence charges or wages of 7 years amounts to 868 l. what is the expence charges or wages of one year facit 124 l. for if you divide 868 the wages of 7 years by 7 the number of years the Quotient will be 124 l. for the Answer see the work Quest. 3. If the content of a superficial foot be 144 Inches and the breadth of a board be 9 Inches how many Inches of that board in length will make such a foot facit 16 Inches for by dividing 144 the number of square Inches in a square foot by 9 the Inches in the breadth of the board the Quotient is 16 for the number of Inches in length of that board to make a superficial foot Quest. 3. If the content of an Acre of Ground be 160 square Perches and the length of a furlong propounded be 80 Perches how many Perches will there go in bredth to make an Acre facit 2 Perches for if you divide 160 the number of Perches in an Acre by 80 the length of the furlong in Perches the Quotient is 2 Perches and so many in breadth of that furlong will make an Acre Quest. 5. If there be 893 men to be made up into a battail the front consists of 47 men what Number must there be in the File Facit 19 deep in the File For if you divide 893 the Number of men by 47 the number in front the Quotient will be 19 the file in depth the work followeth Quest. 6. There is a Table whose Superficial Content is 72 feet and the breadth of it at the end is 3 feet now I demand what is the Length of this Table Facit 24 feet long for if you divide 72 the content of the Table in feet by 3 the bredth of it the Quotient is 24 feet for the length thereof which was Required See the operation as followeth The proof of Multiplication and Division Multiplication and Division Interchangably prove each other for if you would prove a summe in Division whether the operation be Right or no Multiply the Quotient by the Divisor and if any thing Remain after the Division was ended add it to the Product which Product if your summe was Rightly divided will be equal to the Dividend And Contrariwise if you would prove a summe in Multiplication divide the Product by the Multipliar and if the work was Rightly performed the Quotient will be equal to the Multiplicand See the Example where the work is done and undone Let 7654 be given to be Multiplyed by 3242 the product will be 24814268 as by the work appeareth And then if you Divide the said Product 24814268 by 3242 the Multipliar the Quotient will be 7654 equal to the given Multiplicand In like manner to prove a Summe or Number in Division If 24814268 were Divided by 3242 the Quotient would be found to be 7654 then for proof if you Multiply 7654 the Quotient by 3242 the Divisor the Product will amount to 24814268 equal to the Dividend Or you may prove the last or any other Example in Multiplication thus viz. Divide the Product by the Multiplicand and the Quotient will be equal to the Multipliar see the work From whence ariseth this Corollary that any operation in Division may be proved by Division for if after your Division is ended you divide the Dividend by the Quotient the new Quotient thence ariseing will be equal to the Divisor of the first operation for Tryal whereof let the last Example be again Repeated For proof whereof divide again 24814268 by the Quotient 7654 and the Quotient thence will be equal to the first Divisor 3242 see the work But in proving Division by Division the Learner is to observe this following Caution that if after his Division is ended there be any Remainder before you go about to prove your work Subtract that Remainder out of your Dividend and then work as before as in the following Example where it is Required to divide 43876 by 765 the Quotient here is 57 and the Remainder is 271 See the work following Now to prove this work Subtract the Remainder 271 out of the Dividend 43876 and there Remaineth 43605 for a new Dividend to be divided by the former Quotient 57 and the Quotient thence arising is 765 equal to the given Divisor which proveth the operation to be Right Thus have we gone through the four Species of Arithmetick viz. Addition Subtraction Multiplication and Division upon which all the following Rules and all other operations whatsoever that are possible to be wrought by numbers have their Immediate dependance and by them are Resolved Therefore before the Learner make a further step in this Art let him be well acquainted with what hath been delivered in the foregoing Chapters CHAP. VIII Of Reduction 1. REDUCTION is that which brings together 2 or more numbers of different denominations into one denomination or it serveth to change or alter Numbers Money Weight Measure or Time from one Denomination to another and likewise to abridge fractions to their lowest Termes All which it doth so precisely that the first Proportion Remaineth without the least jot of Error or Wrong Committed So that it belongeth as well to Fractions as Integers of which in its proper place Reduction is generally performed either by Multiplication or Division from whence we may gather that 2 Reduction is either Descending or Ascending 3. Reduction Descending is when it is Required to Reduce a Sum or Number of a greater Denomination into a lesser which Number when it is so reduced shall be equal in value to the Number first given in the greater Denomination as if it were Required to know how many shillings pence or farthings are equall in value to a hundred pounds or how many ounces are contained in 45 hundred weight or how many dayes hours or minutes there are in 240 Years c. And this kind of Reduction is generally performed by Multiplication 4. Reduction Ascending is when it is Required to Reduce or bring a Sum or Number of a smaller Denomination into a Greater which shall be equivalent to the given number As suppose it were Required to find out how many Pence Shillings or Pounds are equal in value to 43785 Farthings or how many Hundreds are equal to or in 3748 l. pounds c. and this kind of Reduction is alwayes performed by Division 5. When any Sum or Number is given to be Reduced into another Denomination you are to consider whether it ought to be Resolved by the