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
as in the Margent and the work is finished and I find the sum of the said numbers to amount to 132 l. 02 oz. 09 p.w. 22 gr This is sufficient for the understanding of the following Examples or any other that shall come to thy view The way of proving these or any sums in this Rule is shewed Immediately after the ensuing Examples Addition of English money l. s. d. qrs l. s. d. qrs 436 13 07 1 48 15 11 1 184 09 10 3 76 10 07 3 768 17 04 2 18 00 05 3 564 11 11 0 24 19 09 2 1954 12 09 2 168 06 10 1 Addition of Troy weight l. oun p.w. gr l. oun p.w. gr 15 07 13 12 145 09 12 18 18 06 04 20 726 08 14 10 11 10 16 18 380 07 06 13 09 04 10 22 83 10 16 20 19 11 18 04 130 00 10 12 22 00 00 00 74 07 15 00 97 05 04 04 1541 08 16 01 Addition of Apothecaries weights l. oun dr scr gr l. oun dr scru gr 48 07 1 0 14 60 03 4 0 10 74 05 5 2 10 48 10 6 0 14 64 10 7 1 16 34 08 2 1 15 17 08 1 0 11 18 11 2 2 11 34 09 6 1 09 160 07 1 2 15 Â Â Â Â Â 35 02 5 1 07 240 05 6 1 00 358 07 7 0 12 Addition of Averdupois weight Tun C. qrs l. i. oun dr 75 13 1 15 36 10 12 48 07 3 21 22 11 13 60 11 1 17 11 07 04 21 07 0 25 15 04 10 12 16 0 11 20 00 09 218 16 0 05 106 03 00 Addition of Liquid Measure Tun Pipe hhd gall Tun hhds gall pts 45 1 1 48 30 3 40 4 15 0 1 17 12 0 28 6 38 0 0 47 47 5 60 5 12 1 0 56 57 3 22 3 21 1 1 18 17 0 00 0 133 1 1 60 166 1 26 2 Addition of Dry Measure Chald. qrs bush pec qrs bush pec gall 48 3 7 3 17 3 1 1 13 1 4 0 50 1 3 0 54 0 6 2 14 5 3 1 16 3 6 1 40 2 0 1 40 1 0 1 30 0 3 0 173 3 0 3 152 5 3 1 Addition of Long Measure yds qrs na els qrs na 35 3 3 56 1 3 14 1 2 13 3 2 74 2 3 48 2 1 48 0 1 50 1 0 30 1 0 74 0 2 15 0 0 17 1 0 218 1 1 260 2 0 Addition of Land Measure Acre Rood per. Acr. Rood Perch 12 3 18 86 1 36 14 0 24 47 3 24 30 2 19 73 2 18 48 3 30 60 0 07 28 1 38 04 2 08 50 3 26 14 1 14 185 3 35 286 3 27 The proof of Addition 6. Addition is proved after this manner when you have found out the sum of the Numbers given then separate the uppermost line from the rest with a stroke or dash of the pen and then add them all up again as you did before leaving out the uppermost line and having so done add this new invented Sum to the uppermost line you separated and if the Sum of those two lines be equal to the Sum first sound out then the work was performed true otherwise not As for Example let us prove the first example of Addition of money whose sum we found to be 265 l. 9 s. 5 d. 2 qrs and which we prove thus having separated the l. s. d. qts 136 13 04 2 79 07 10 3 33 18 09 1 15 09 05 0 265 09 05 2 128 16 01 0 265 09 05 2 uppermost number from the rest by a line as you see in the margent then I add the same together again leaving out the said uppermost line and the sums thereof I set under the first Sum or true sum which doth amount to 128 l. 16 s. 01 d. 0 qrs then again I add this new Sum to the uppermost line that before was separated from the rest and the Sum of these two is 265 l. 9 s. 05 d. 2 qrs the same with the first Sum and therefore I conclude that the operation was rightly performed 7. The main end of Addition in Questions Resolvable thereby to know the sum of several debts parcels Integers c. some Questions may be these that follow Quest. 1. There was an old man whose age was required to which he replyed I have seven sons each having two years between the birth of each other and in the 44 year of my age my eldest son was born which is now the age of my youngest I demand what was the old mans age Now to Resolve this Question first set down the fathers age at the birth of his first child which was 44 then the difference between the eldest and the youngest which is 12 years and then the age of the youngest which is 44 and then add them all together and their sum is 100 the compleate age of the Father Quest. 2. A man lent his friend at severall tâ—mes these several sums viz. at one time 63 l. at another time 50 l. at another time 48 l. at another time 156 l. now I desire to know how much was lent him in all Set the sums lent one under another as you see in the margent and then add them together and you wâ—ll find their sum to amount to 317 l. whâ—ch is the Total of all the several sums lent and so much is due to the Creditor Quest. 3. From London to Ware is 20 miles thence to Huntington 29 miles thence to Standford 21 thence to Tuxford 36 miles thence to Wentbridge 25 miles from thence to York 20 miles Now I desire to know how many miles it is from London to York according to this Reckoning Now to answer this Question set down the several distances given as you see in the margent and add them together and you will finde their sum to amount to 151 which is the true distance in miles between London and York Quest. 4. There are 2 numbers the least whereof is 40 and their difference is 14 I desire to know what is the greater number and also what is the sum of them both First set down the least viz. 40 and 14 the difference and add them together and their sum is 54 for the greatest number then I set 40 the least under 54 the greatest and add them together and their sum is 94 equal to the greatest and least numbers CHAP. V. Of Subtraction of whole Numbers 1. Subtraction is the taking of a lesser number out of a greater of like kind whereby to find out a third number being or declaring the Inequality excess or difference between the numbers given or Subtraction is that by which one number is taken out of another number given to the end that the residue or remainder may be known which remainder is also called the rest or difference of the numbers given 2. The number out of which Subtraction is to be made must be greater or at least equal with the other number
I begin saying seven times four is 28 then I set down 8 and Carry 2 then say 7 times 6 is 42 and 2 that I carried is 44 that is 4 and go 4 then 7 times 7 is 49 and 4 that I carry is 53 which I set down because I have not another figure to Multiply Thus have I done with the 7 then I begin with the 2 saying 2 times 4 is 8 which I set down under the 4 the second figure or place of tens in the line above it as you may see in the margent Then I proceed saying 2 times 6 is 12 that is 2 and carry one then two times seven is fourteen and one that I carry is fifteen which I set down because 't is the product of the last figure so that the product of 764 by 7 is 5348 and by 2 is â—528 which being placed the one under the other as before is directed and as you see in the margent and a line drawn under them and they added together Respectively make 20628 the true product Required being equal to 27 times 764. Another Example may be this Let it be Required to Multiply 5486 by 465 I dispose of the Multiplicand and Multiplier according to Rule and begin Multiplying the first figure of the Multiplier which is five into the whole Multiplycand and the product is 27430 then I proceed and Multiply the second figure 6 of the Multiplier into the Multiplicand and find the product to amount to 32916 which is subscribed under the other product Respectivly then do I Multâ—ply the third and last figure 4 of the Multiplier into the Multiplicand and the product is 21944 which is likewise placed under the second line Respectively then I draw a line under the said products being placed the one under the other according to Rule and add them together and the sum is 2550990 the true product sought being equal to 5486 times 465 or 465 times 5486. More Examples in this Rule are these following 7. Although the foormer Rules are sufficient for all Cases in multiplication yet because in the work of multiplication many times great labour may be saved I shall acquaint the Learner therewith viz. If the multiplicand or multiplier or both of them end with Cyphers then in your multiplying you may neglect the Cyphers and multiply only the significant figures and to the product of those significant figures add so many Cyphers as the Numbers given to be multiplyed did end with that is annex them on the Right hand of the said product so shall that give you the true product Required As if I were to multiply 32000 by 4300 I set them down in order to be multiplyed as you see in the margent but neglecting the Cyphers in both numbers I only multiply 32 by 43 and the product I find to be 1376 to which I annex the 5 Cyphers that are in the multiplicand and multiplier and then it makes 137600000 for the true product of 32000 by 4300. 8. If in the multiplier Cyphers are placed between significant figures then multiply only by the significant figures neglecting the cyphers but here special notice is to be taken of the true placing of the first figure after the neglect of such Cypher or Cyphers and therefore you must observe in what place of the multiplier the figure you multiply by standeth and set the first figure of that product under the same place of the product of the first figure of your multiplier As for Example let it be Required to multiply 371568 by 40007 first I multiply the multiplicand by seven and the product is 2600976 then neglecting the Cyphers I multiply by 4 and that product is 1486272 now I consider that four is the fifth figure in the multiplier therefore I place two the first figure of the Product by four under the fifth place of the first Product by seven and the rest in order and having added them together the total product is found to be 14865320976. other Examples in this Rule are these following 9. If you are to multiply any Number by a unit with Cyphers viz. by 10 100 1000 c. Then prefix so many Cyphers before the multiplicand and that Numbeâ— when the Cyphers are prefixed is the Proâ—duct Required as if you would multiply 428 by 100 annex two Cyphers to 428 and it is 42800 If it were Required tâ— multiply 102 by 10000 annex 4 Cypherâ— and it gives 1020000 for the Product Required The Proof of Multiplication 10. Multiplication is Proved by Division and to speak truth all other wayes are false and therefore it will be most convenient in the first place to learn Division and by that to prove Multiplication There is a way at this day generally used in Schools to Prove multiplication which is this first add all the Figures in the multiplicand together as if they were simple Numbers casting away the Nines as often as it comes to so much and noting the Remainder at last which in this case cannot be so much as 9 Cast likewise the Nines out of the multiplier as you did out of the multiplicand and note that Remainder then multiply the Remainders the one by the other and cast the Nines out of that Product observing the Remainder and lastly Cast the Nines out of the total product and if this Remainder be equal to the Remainder last found then they conclude the work to be Rightly performed but there may be given a thousand nay infinite false Products in a multiplication which after this manner may be Proved to be true and therefore this way of Proving doth not deserve any Example but we shall deferr the Proof of this Rule till we come to Prove Division and then we shall Prove them both together 11. The general effect of Multiplication is contained in the definition of the same which is to find out a third Number so often containing one of the two given Numbers as the other containeth unit The second effect is by having the length and breadth of any thing as a pararellogram or long plain to find the superficial content of the same and by having the superficial content of the base and the length to find the solidity of any parallelepipedon Cylinder ar other solid figures The third Effect is by the contents price vallue buying selling expence wages exchange simple Interest gain or loss of any one thing be it Money Merchandise c. to find out the value price expence buying selling exchange or Interest of any Number of things of like Name Nature and Kind The fourth Effect is not much unlike the other by the Contents Vallue or price of one part of any thing Denominated to find out the Content Vallue or Price of the whole thing all the parts into which the whole is divided multiplying the price of one of those parts The Fifth Effect is to aid to compound and to maâ—e 〈◊〉 Rules as chiefly the Rule of 〈◊〉 called thâ— Golden Rule 〈…〉
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