Arithmetic, Part 1 · Section 3 – Section 8

Arithmetic · Part 1 · Section 3 – Section 8

p. 10Addition of numbers of diverſe Denominations. 1 Addition of Engliſh Money

The moſt uſual Coyns uſed in England are Pounds, Shillings, Pence, and Farthings, of which Coyns

4 Farthings make 1 Peny thus charactred d.

12 Pence make 1 Shilling thus charactred s.

20 Shillings make 1 Pound. thus charactred li.

for a farthing we uſe q.

THE RƲLE.

In the addition of numbers of divers denominations this order is to be obſerved, viz. Place all numbers of the ſame denomination one directly under another, as Pounds under pounds, ſhillings under ſhillings, pence under pence, and farthings under farthings. Then draw a line under them, and begin your Addition with the leaſt denomination firſt, obſerving how many times the next greater denomination is contained in the leaſt, and for every time carry one unite to the next denomination, as before you did the tens, ſetting down the remainder if any be, then adding the next denomination together, take notice how many times the next greater denomination is contained in that leſſer, carrying for every time one to the next, p. 11thus proceeding till you have gone over all the denominations, be they never ſo many.

Example 1.

Let the numbers to be added together be 37 li. 16 s. 9 d. 3 q. 21 li. 9 s. 8 d. 1 q. 13 li. 12 s. 9 d. 2 q. Place the numbers as in the margine, draw a line under them, and begin with the leaſt denomination (which in this example is farthings,) firſt, ſaying, 2 q. and 1 q. is 3 q. and 3 q. is 6 q. which is one peny and 2 q. remaining, which 2 q. I place under the line, and carry the one peny to the next row which is the place of pence, ſaying, one peny and 9 d. is 10 d. and 8 d. is 18 d. which is 1 s. and 6 d. (Now againſt the 8 I make a prick with my pen for my better remembrance to ſignifie that there is one ſhilling to be carried to the place of ſhillings,) then go on and ſay 6 d. and 9 d. is 15 d. which is 1 s. and 3 d. therefore againſt 9. I make a prick with my pen, and (becauſe that is the laſt number) I ſet down the odde 3 d. under the place of pence, and (becauſe I finde two pricks in the line of pence, therefore) I carry 2 s. to the place of ſhillings ſaying 2 s. which I carryed, and 12 s. is 14 s. and 9 s. is 23 s. which is one pound, and 3 s. remaining, I make a prick againſt 9, and going on, ſay 3 s. and 16 s. is 19 s. which (being there is no more numbers to be added, and being alſo leſſe then 20 s.) I ſet under the line, and finding one prick in the line of ſhillings, I therefore carry p. 12one to the place of pounds, ſaying, one which I carryed and 3 is 4, and 1 is 5, and 7 is 12. ſet down the 2 under the line (as in addition of numbers of one denomination) and carry one to the next row, ſaying one that I carried and 1 is 2, and 2 is 4, and 3 is 7, which being the laſt I ſet down, and ſo the total or groſſe ſum is 72 li. 19 s. 3 d. 2 q.

Example 2

Let the numbers to be added be 29 li. 16 s. 8 d. 32 li. 17 s. 9 d. 81 li. 13 s. 11 d. and let it be required to find the totall or groſſe ſum. Here in this Example the leaſt denomination is pence, therefore I begin with them, and ſay, 11 d. and 9 d. is 20 d. which is 1 s. and 8 d. make a prick againſt the 9 and ſay 8 d. and 8 d. is 16 d. that is 1 s. and 4 d. make a prick againſt the 8 and ſet down the odde 4 d. then (becauſe there are two pricks in the line of pence) you muſt carry 2 s. to the place of ſhillings, ſaying 2 s. which I carry and 13 s. is 15 s. and 17 s. is 32 s. which is 1 l. 12 s. make a prick agianſt 17, and ſay 12 s. and 16 s. is 28 s. make a prick againſt 16, and (becauſe there is no more numbers to be added) ſet down the odde 8 s. under ſhillings, and (being there is two pricks in the line of ſhillings) carry 2 to the place of pounds, ſaying 2 and 1 is 3, and 2 is 5, and 9 is 14, ſet down 4 and carry 1 to the next line, and ſay 1 and 8 is 9, and 3 is 12, and 2 is 14, which (becauſe p. 13it is the laſt) you muſt ſet down, ſo is the totall or groſſe ſum 144 li. 8 s. 4 d.

Other Examples for Practice.

2 Addition of Troy Weight.

Troy weight is a Weight uſed in England, by the which is weighed, Bread, Gold, Silver, Pearl, &c. the moſt uſual denominations of which weight are, Pounds, Ounces Peny weights, and grains, of which

24 Grains make 1 Peny weig. thus charactred pw.

20 Peny weight make 1 Ounce thus charactred ou.

12 Ounces make 1 Pound thus charactred lib

for a grain we write gr.

The Addition of Troy weight (and conſequently of any other weight or meaſure whatſoever either Domeſtique or Forreign) differeth nothing at all from the addition of Coine laſt p. 14taught, if the affinity of one denomination to an other be firſt known, for whereas in money becauſe 1… d. make 1 s. you therefore obſerve how many twelves there are in the addition of your pence, and for every 12 you adde one ſhilling to the place of ſhillings, ſo in the addition of Troy weight, knowing that 24 gr. make one peny weight, you muſt therefore in the addition of Grains of Troy weight obſerve how many times 24 you finde in your line of Grains, and for every 24, carry one to the place of peny weights, likewiſe in the addition of peny weights Troy, you muſt conſider how many times 20 is contained in your line, and for every 20 carry one to the place of ounces, (becauſe 20 peny weights make an ounce.) Alſo in the addition of Ounces Troy, you muſt obſerve how many times 12 you finde in your line of ounces, and for every 12 carry one to the place of pounds then laſtly adde your pounds together, as numbers of one denomination.

Example.

Let the numbers to be added together be 7 li. 11 ou. 13 pw. 19 gr. 6 li 7 ou. 16 pw. 19 gr. 3 li. 7 ou. 9 pw. 6 gr. Place your numbers as in addition of money, each under other according to their reſpective denominations, as in the margine then draw a line under them and begin your Addition with the leaſt denomination firſt, viz. grains, p. 15ſaying 6 gr. and 19 gr. is 25 gr. which is one peny weight and one grain, make a prick againſt 19, and carry the odde grain to the number above, ſaying 1 gr. and 19 gr. is 20 gr. which (becauſe it is leſſe then one peny weight) I ſet under the line, then finding one prick in the line of grains, I (therefore) carry one to the place of peny weights, ſaying 1 and 9 is 10, and 16 is 26, which is one ounce, and 6 pw. make a prick againſt 16, and ſay 6 and 13 is 19, which (being leſſe then an ounce) ſet under the line, then for the one prick, carry 1 to the place of ounces, ſaying 1 and 7 is 8, and 7 is 15, which is one pound and 3 ou. make a prick at 7, and ſay 3 and 11 is 14, which is one pound and 2 ounces, ſet down the 2 ounces, and for the two pricks carry 2 pounds to the place of pounds, ſaying 2 and 3 is 5, and 6 is 11, and 7 is 18, which ſet under the place of pounds, ſo is your addition ended, and the ſum is 18 li. 2 ou. 19 pw. 20 gr.

Other Examples for Practice.

p. 163 Addition of Avoirdupois little Weight.

There is another kind of weight moſt commonly uſed in England called Avoirdupois little weight, by which is weighed all ſorts of wares or merchandiſe Garbable, as Suger, Pepper, Cloves, Mace, &c. This weight is commonly divided into theſe denominations, Pounds, Ounces, and Drams, of which

16 Drams make 1 Ounce thus charact. cu.

16 Ounces make 1 Pound thus charact. li.

for a dram we write dr.

In the addition of Avoirdupois weight, you muſt obſerve the very ſame method and order as in Money and Troy weight, having due reſpect to the quantity of the denomination, as in the addition of drams to make a prick at every 16, ſetting down the remainder and for every prick carrying a unite to the next place. The preceding rules being ſo copious in this particular I ſhall forbear to make any verball illuſtration, but only give you ſome examples ready wrought, together with the moſt uſual parts into which the Weights and Meaſures now uſed in England are divided into: which to the ingenious will be of moſt validity.

p. 17Examples of Addition of Avoirdupois little weight.

4 Addition of Avoirdupois great weight.

There is a weight commonly uſed in England, by which is weighed all commodities that are ſold by the hundred, as Corents, Wool, Fleſh, Butter, Cheeſe, and the like, the which hundred weight containeth 112 pounds, and the hundred weight is divided into quarters, pounds, and ounces, ſo that

16 Ounces makes 1 Pound thus chaeactred li.

28 Pounds makes 1 quarter of a C. thus chaeactred qr.

4 quarters makes 1 Hundred weig. thus chaeactred C.

for an Ounce we write oz.

p. 18Examples of Addition of Avoirdupois great Weight.

I might further proceed to ſhew you Examples of addition of common Engliſh meaſures, viz. of long meaſures, Liquid meaſures, and dry meaſures, as alſo of Time, Motion, &c. but the preceding Examples being of ſufficient extent, I ſhall forbear to trouble either my ſelfe or the Reader with that which I conceive ſuperfluous: Only, before I leave Addition, I will give you a brief view of the moſt uſual meaſures in England, which take as followeth. And

1 Of Liquid Meaſures.

Liquid meaſures are thoſe in which all ſorts of Liquid ſubſtances are meaſred, of which (according to the Statute of 12 Hen. 7. chap. 5.) a Pinte is the leaſt, from which the greater Liquid meaſures are deduced, according as is expreſſed in the Table following.

p. 192 Pintes make 1 Quart

2 Quarts make 1 Pottle

2 Pottles make 1 Gallon

8 Gallons make 1 Firkin of Ale, Sope, or

9 Gallons make 1 Firkin of Beer (Herings

10 ½ Gallons make 1 Firkin of Salmon or Eels

2 Firkins make 1 Kilderkin

2 Kilderkins make 1 Barrel

42 Gallons make 1 Tierce of Wine

63 Gallons make 1 Hogſhed

2 Hogſheads make 1 Pipe or But

2 Pipes or Buts make 1 Tun of Wine.

2 Of Dry Meaſures.

Dry Meaſures are theſe in which all kind of dry ſubſtances are meaſured, as Corn, Salt, Cole, Sand, &c. of which a pinte is the leaſt.

2 Pintes make 1 quart

2 quarts make 1 Pottle

2 Pottles make 1 Gallon

2 Gallons. make 1 Peck

4 Pecks make 1 Buſhel Land meaſure

5 Pecks make 1 Buſhel Water meaſure

8 Buſhels make 1 quarter

4 quarters make 1 Chaldron

5 quarters make 1 Wey.

3 Of Long Meaſures.

Long Meaſure is that by which is meaſured Cloth, Land, Board, Glaſſe, Pavement, Tapeſtry, &c… p. 20of which meaſures (according to the Statute of 33 Ed. 1. and 25 El.) a Barley corn is the leaſt. So that

3 Barley Corns make 1 Inch

12 Inches make 1 Foot

3 Foot make 1 Yard

3 Foot 9 inches make 1 Ell

6 Foot make 1 Fadome

5 ½ Yards, or 16 ½ Foot make 1 Pole or Perch

40 Perches make 1 Furlong

8 Furlongs make 1 Engliſh mile.

4 Of Time.

Time conſiſteth of Years, Moneths, Weeks, Dayes, Houres, and Minutes. So that

60 Minutes make 1 Hour

24 Hours make 1 Day natural

7 Dayes make 1 Week

4 Weeks make 1 MonethMonth of 28 dayes

13 Moneths, one day 6 houres make 1 Year.

5 Of Apothecaries Weights.

The weights uſed by Apothecaries are Grains, Scruples, Drams, and Ounces, of which

p. 2120 Grains make 1 Scruple thus charact. ℥

3 Scruples make 1 Dram thus charact. ʒ

8 Drams make 1 Ounce thus charact. ℈

12 Ounces make 1 Pound thus charact. li.

By help of theſe Tables, and the rules and cautions before expreſſed, any man may make addition of any of the aboveſaid meaſures one with another, and therefore I ſhall forbear to illuſtrate them by Examples, but leave them to every mans own practice, and thus I conclude Addition.

The Proof of Addition.

Having placed your numbers in order, and added them together, and ſet the Total under the line, cut off the upper number by drawing a line with your pen betwixt that and the others, then adde all the numbers together except the uppermoſt, and ſet the Total of them under the Total before found, then adde this laſt Total, and the firſt number which you cut off with your pen together, and if the ſum of thoſe two numbers be equal with your Total ſum firſt found, then is your work right, otherwiſe not.

Example. In the firſt example of whole numbers, the ſums to be added were 7833, 5609, 376, and 8547, theſe numbers placed in due order and added together, the total or groſſe ſum of them is 22364, now to prove whether this p. 22Total be true or not, I cut off the uppermoſt number, (to wit 7832,) with a daſh of the pen, and I adde the other three numbers together, namely, 5609, 376, and 8547, and the Total of them is 14732, which number being added to 7832 (the number cut off) the ſum of them is 22364 exactly agreeing with the Total firſt found, cleerly evidenceth that the addition was truly performed; but if they had diſagreed then the work had been erroneous. The like courſe muſt be taken for the proof of thoſe ſums which have different denominations as in Money and Weight, as by the examples following will appear.

Other Examples proved.

1 Example of Money.

p. 23There are are other wayes to prove Addition, by caſting away of all the nines in numbers of one denomination, and of all the twelves, twenties, and nines, in pounds ſhillings and pence, &c. but this as the moſt certain and eaſie I embrace, and thus much for Addition and the proofe thereof.