Divisor then it is a Direct Rule As In the following Questions Quest. 1. If 8 Labourers can do a certain piece of work in 12 dayes in how many dayes will 16 Labourers do the same Answer in 6 dayes Having placed the numbers according to the 6 Rule of the 10th Chapter I consider that if 8 men can finish the work in 12 dayes 16 men will do it in lesser or fewer dayes then 12 therefore the biggest Extream must be the Divisor which is 16 and therefore it is the Rule of 3 Inverse wherefore I multiply the first and second numbers together viz. 8 by 12 and their product is 96 which divided by 16 Quotes 6 days for the Answer and in so many days will 16 Labourers perform a piece of work when 8 can do it in 12 days Quest. 2. If when the measure viz. a peck of wheat cost 2 shillings the peny Loaf weighed according to the Standard Statute or Law of England 8 ounces I demand how much it will weigh when the peck is worth 1 s. 6 d. according to the same Rate or Proportion Answer 10 oz. 13 p.w. 8 grs. Having placed and reduced the given Numbers according to the 6 and 9 rules of the 10th Chapter I consider that at 1 s. 6 d. per Peck the peny Loaf will weigh more then at 2 s. per Peck for as the price decreaseth the weight Increaseth and as the price Increaseth so the weight diminisheth wherefore because the term Requireth more then the second the lesser Extream must be the Divisor viz. 1 s. 6 d. or 18 peâ—â—e and having finished the work I find the Answer to be 10 oz. 13 p.w. 8 gr and so much will the peny Loaf weigh when the peck of wheat is worth 1 s. 6 d. according to the given Rate of 8 ounces when the peck is worth 2 shillings the work is plain in the following operation Quest. 3. How many pieces of money or Merchandize at 20 s. per piece are to be given or Received for 240 pieces the value or price of every piece being 12 shillings Answer 144. For if 12 s. Require 240 pieces then 20 shillings will Require less therefore the biggest Extream must be the Divisor which is the 3 number c. See the work Quest. 4. How many yards of 3 quarters broad are Required to double or be equal in measure to 30 yards that are 5 quarters broad Answer 50 yards For say If 5 quarter wide Require 30 yards long what length will three quarters broad require here I consider that 3 quarters broad will Require more yards then 30 for the narrower the cloth is the more in length will go to make equal measure with a broader piece Quest. 5. At the Request of a friend I lent him 200 l. for 12 moneths promising to do me the like Curtesie at my necessity but when I came to Request it of him he could let me have but 150 l. now I desire to know how long I may keep this money to make plenary Satisfaction for my former kindness to my Friend Answer 16 Moneths I say If 200 l. Require 12 Moneths what will 150 l. Require 150 l. will Require more time then 12 Moneths therefore the lesser extream viz. 150 must be the Divisor Multiply and Divide and you will find the 4th inverted Proportional to be 16 and so many Moneths I ought to keep the 150 l. for satisfaction Quest. 6. If for 24 s. I have 1200 l. weight carried 36 M. how many M. shall 800 l. be carried for the same mony Answer 24 M. Quest. 7. If for 24 s. I have 1200 l. carried 36 Miles how many pound weight shall I have carried 24 miles for the same money Answer 800 l. weight Quest 8. If 100 workmen in 12 dayes finish a piece of work or service how many workmen are sufficient to do the same in 3 dayes Answer 400 workmen Quest. 9. A Colonel is besieged in a Town in which are 1000 Souldiers with provision of Victuals only for 3 Moneths the Question is how many of his Souldâ—ers must he dismâ—ss that his Victuals may last the Remaining Souldiers 6 Moneths Answer 500 he must keep and dismiss as many Quest. 10. If wine worth 20 l. is sufficient for the ordinary of 100 men when the Tun is sold for 30 l. how many men will the same 20 pounds worth suffice when the Tun is worth 24 l. Answer 125 men Quest. 11. How much plush is sufficient to line a Cloak which hath in it 4 yards of 7 quarters wide when the Plush is but 3 quarters wide Answer 9 â…“ yds of Plush Quest. 12. How many yards of Canvas that is Ell wide will be sufficient to line 20 yards of Say that is 3 quarters wide Answer 12 yards Quest. 13. How many yards of matting that is 2 foot wide will cover a floor that is 24 foot long and 20 foot broad Answer 240 foot Quest. 14. A Regiment of Souldiers consisting of 1000 are to have new Coats and each coat to contain 2 yds 2 qrs of Cloth that is 5 qrs wide and they are to be lined with Shalloon that is 3 quarters wide I demand how many yards of Shalloon will line them Answer 16666 â…” yards Quest. 15. A Messenger makes a Journey in 24 dayes when the day is 12 hours long I desire to know in how many dayes he will go the same when the day is 16 hours long Answer in 18 dayes Quest. 16. Borrowed of my friend 64 l. for 8 Moneths And he hath occasion another time for to borrow of me for 12 Months I desire to know how much I must lend to make good his former kindness to me Answer 42 l. 13 s. 4 d. 4. The General Effect of the Rule of 3 Inverse is contained in the Definition of the same that is to find a fourth term in a Reciprocal Proportion inverted to the Proportion given The second Effect is by two prises or values of two several pieces of money or Merchandize known to find how many pieces of the one price is to be given for so many of the other And consequently to Reduce or Exchange one sort of Money or Merchandize into another Or contrariwise to find the price unknown of any piece given to Exchange in Reciprocal Proportion The third Effect is by two dâ—ffering prizes of a measure of wheat bought or sold and the weight of the Loaf of bread made answerable to one of the prises of the measure given to find out the weight of the same Loaf answerable to the other price of the said measure given Or contrariwise by the two several weights of the same prized Loaf and the price of the measure of wheat answerable to one of those weights given to find out the other price of the measure answerable to the other weight of the same Loaf The Fourth Effect is by two lengths and one breadth of two Rectangular planes known to find out another breadth unknown Or by two
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
given the hâ—gher or superiour number is called the major number and the lower or inferiour is called the minor number and the operation of Subtraction being finished the rest or remainder is called the difference of the numbers given 3. In Subtraction place the numbers given respectively the one under the other in such sort as like degrees places or denominations may stand in the same Series viz. Units under units Tens under Tens c. Pounds under Pounds c. Feet under Feet and Parts under Parts c. This being done draw a line underneath as in Addition 4. Having placed the numbers given as is before directed and drawn a line under them Substract the lower number which in this case must alwayes be lesser than the uppermost out of the higher number and subscribe the difference or remainder respectively below the line and when the work is finished the number below the line will give you the Remainder As for Example let 364521 be given to be Subtracted from 795836 I set the lesser under the greater as in the margent and draw a line under them then beginning at the Right hand I say 1 out of 6 and there Remains 5 which I set in order under the line then I proceed to the next saying 2 from 3 rests 1 which I note also under the line and thus I go on untill I have finished the work and then I find the Remainder or difference to be 431315. 5. But if it so happen as commonly it doth that the lowermost number or figure is greater then the uppermost then in this case add ten to the uppermost number and Subtract the said lowermost number from their sum and the remainder place under the line and when you go to the next figure below pay a unit by adding it thereto for the 10 you borrowed before and subtract that from the higher number or figure And thus go on untill your Subtraction be finished As for Example Let 437503 be given from whence it is required to subtract 153827 I dispose of the numbers as is before directed and as you see in the margent then I begin saying 7 from 3 I cannot but adding 10 thereto I say 7 from 13 and there Remains 6 which I set under the line in order then I proceed to the next figure saying 1 that I borrowed and 2 is 3 from 0 I cannot but 3 from 10 and there remains 7 which I likewise set down as before then one that I borrowed and eight is nine from five I cannot but nine from fifteen and there remains six then one I borrowed and three is four from seven and there remains three then five from three I cannot but five from thirteen and there remains eight then one I borrowed and one are two from four and there rests two And thus the work is finished and after these numbers are Subtracted one from another the inequality remainder excess or difference is found to be 283676. Examples for thy further experience may be these that follow From 3475016 From 3615746 Take 738642 Take 5864 Rests 2736374 Rests 3609882 6. If the Sums or Numbers to be Subtracted are of several Denominations place the lesser Sum below the greater and in the same Rank and order as is shewed in Addition of the same Numbers then begin at the Right hand and take the lower number out of the uppermost if it be lesser but if it be bigger then the uppermost then borrow a Unit from the next greater Denomination and turn it into the parts of the lesser Denomination and add those parts to the uppermost Number and from their Sum substract the lowermost noting the remainder below the line then proceed and pay 1 to the next Denomination for that which you borrowed before and proceed in this order untill the work be finishâ—ed An Example of this Rule may be thiâ— that followeth let 375 l. 13 s. 07 d. 1 qrs be given from whence let it be required to Subtract 57 l. 16 s. 03 d. 2 qrs In ordeâ— whereunto I place the numbers as you see iâ— the margent and thus I begin at the leasâ— Denomination saying l. s. d. qrâ— 375 13 07 1 57 16 03 2 317 17 03 â—3 two from one I cannot therefore I borrow one peny from the next denomination and turn it into farthings which is four and adding four 〈◊〉 one which is five I say but two from fivâ— and there remains three which I put undâ—â— the line then going on I say one that I borrowed and three is four from 7 and there Rests three then going on I say sixteen from thirteen I cannot but borrowing one pound and turning it into twenty shillings I add it to thirteen and that is thirty three wherefore I say sixteen from thirty three and there remains seventeen which I set under the line and go on saying one that I borrowed and seven is eight from five I cannot but eight from fifteen and there remains seven then one that I borrowed and five is six from seven there Rests one and nothing from three Rests three and the work is done And I find the remainder or difference to be 317 l. 17 s. 03 d. 3 qrs Another Example of Troy weight may be this I would Subtract 17 l. 10 oz. 11 p.w. 20 gr from 24 l. 05 oz. 00 p.w. 08 gr I place the numbers according to Rule and begin saying twenty l. oz. p.w. gr 24 05 00 08 17 10 11 20 06 06 08 12 from eight I cannot but borrow one peny weight which is twenty four grains and add them to eight and they are thirty two wherefore I say twenty from thirty two Rests twelve then one that I borrowed and eleven is twelve from 00 I cannot but twelve from twenty borrowing an ounce which is twenty peny weight and there remains eight then one that I borrowed and ten is eleven from five I cannot but eleven from seventeen and there rests six then one that I borrowed and seven is eight from four I cannot but eight from fourteen and there Rests six then one that I borrowed and one is two from two and there rests nothing so that I find the Remainder or difference to be 6 l. 6 oz. 8 p.w. 12 gr 7. It many times happeneth that you have many Sums or Numbers to be Subtracted from one number as suppose a man should lend his friend a certain Sum of Money and his friend had paid him part of his debt at several times then before you can conveniently know what is still owing you are to add the several Numbers or Sums of Payment together and Subtract their Sum from the whole Debt and the Remainder is the Sum due to the Creditor as Suppose A lendeth to B 564 l. 13 s. 10 d. and B hath Repaid him 79 l. 16 s. 08 d. at one time and 163 l. 18 s. 11 d. at another time and 241 l. 15 s. 08 d. at another time and you would know how the Accompt