The merchants map of commerce wherein the universal manner and matter relating to trade and merchandize are fully treated of, the standard and current coins of most princes and republicks observ'd, the real and imaginary coins of accounts and exchanges express'd, the natural products and artificial commodities and manufactures for transportation declar'd, the weights and measures of all eminent cities and towns of traffick in the universe, collected one into another, and all reduc'd to the meridian of commerce practis'd in the famous city of London / by Lewis Roberts, merchant.

CHAP. CCCCL. Of the Abreviation of Division and Multiplication after the Italian and foreign manner and method.

* 1.1FOr as much as I have in many parts of this particular Tract of Exchanges, followed the Arithmetical method and manner of those rules practised in the calculation of these Ex∣changes by the Bankers and Exchangers of Italy, it will be here needful for the better enlightning of the same, and the easier casting up and calculation thereof, that I shew how the Italian Bro∣kers and Exchangers do abreviate their labour, and shorten their task therein, and the rather I have presumed to add the same here, and in this place, partly in regard that I have not found it published by any of our English Arithmeticians, but princinally to shew the learner the ways how the same are there wrought and Arithmetically calculated.

It is generally confest by all Arithmeticians, that the whole Art of Arithmetick depends upon five principal Rules, now commonly in all Countreys received and taught, that is, by Numerati∣on, Addition, Substraction, Multiplication, and Division, and that no one proposed question in Arith∣metick can be perfected without the help of some of these; for the three former, I find not any disagreement in the common received manner by them and us, and therefore I will omit to speak any thing thereof; but of the two latter, whereby is observed that most Rules and Questions of all Exchanges are perfected and performed, I will here insist upon, induced prin∣cipally, as I said before, to enlighten thereby the precedent Examples that I have handled in the calculations of the Exchanges before-mentioned.

I will then in the first place, contrary to the custom of our English Masters in this Science, begin with that part of Arithmetick which we call Division, and by an example or two of the working thereof, explain the same to such as either shall be desirous to learn it, or such as shall desire to make use of the before-mention'd Tables.

* 1.2A certain Merchant then bought 46 Cloths, which cost him 673 l. and desireth by a brief way to know what one Cloth doth stand him in; To do which, I dispose of the question af∣ter the manner of the Rule of Three, and say, If 46 Clothes cost 673 l. how much doth one Cloth cost?

Now for as much as it would prove to be too dissicult, at first sight after the common man∣ner, to find how often 46 the Divisor is found in 673, it will be more facile and commodious, to take it after their method thus, Take then the first figure, which is 4, and see how often the same is included in the figure 6, which is once, the which 1 I write then under the Divisor, drawing a line between them, and then multiply it by the whole Divisor 46, beginning at 6, saying once 6 is 6, and next coming to the sum that is to be divided 673, I chuse the second figure 7, from whence I take 6 and there remains one, which I place under the said 7, and re∣turning again to the Divisor, I multiply 1 by 4, which giveth 4, which I deduct from the o∣ther figure 6, of the sum to be divided, and there remains 2, the which I write under the 6; so that 46 taken by this means out of 67, there remains 21, from whence I proceed and put this before the figure 3 remaining, which thereby makes 213, for the sum that now remains to be divided by 46, saying in 21, how many times 4? which cannot be but 4 times, for in taking 5 there will remain but 1, which with the following figure doth make 13, (the which number cannot pay 5 times 6, and for this cause I can take but 4,) multiplying the Divisor as at first, saying (beginning always by the last figure of the Divisor) 4 times 6 is 24, and taking the last figure 3 from the sum to be divided 213, the which for payment, of 24 I borrow 3 tens, which I bear in mind, and say 24 from 33 there rests 9, the which I place under the 3, and then come to multiply the other figure 4 of the divisor by 4, and it makes 16, which with 3 tens born in mind, makes 19, which must be deducted from the sum to be divided 21, so there will rest 2, the which I place under the 1; as by the Example appeareth more at large.

So that 673 l. divided by 46 Cloths, the quotient giveth 14 l. and the rest is 29 l. which now is to be divided by 46, which cannot be done, and therefore the same to be reduced to shillings, which multiplied by 20 come to 580 s. which must now be divided by 46, in the manner before shewed, saying, how many times 4 in 5? which is once, the which I write in the quotient at the side of 14 l. proceeding from the first division, multiplying it by 6, and it giveth 6, which taken from 8 the rest is 2, which I put under the 8, and multiply the other figure of the divisor 4 by 1, which giveth 4 taken from 5, there rests 1, then 46 substracted from 58 there rests 12, right with which I put the other figure 0, resting of the sum to be divided, and return to say, how many times 6 in 12, the which I can take but 2, and I place it in the quotient, and multiply it by the last figure of the divisor, saying 2 times 6 is 12, which I deduct from 120, the which to do, I say (borrowing 2 tens, which I bear in mind) 12 from 20 rests 8, which I place under the 0, and multiply the other figure of the Divisor 4 by, making 8 with the 2 born in mind, comes

to be 10, taken from 12 there rests 2, so that 580 s. divided by 46, the product is 12 s. and there remaineth yet 28 s. which must be brought into pence, multiplied by 12, and it makes 336, the which must be divided by 46, saying in 33, how many times 4? which is 7 times, the which I place in the quotient by the shillings, multiplying 7 by 6, which makes 42 from 46, there rests 4, the which I place under the 6, and keep 4 in mind borrowed, adding them with 4 times 7, they make 32, which deducted from 33, there rests 1, the which I place un∣der the 3, so that 336 divided by 46, the product giveth 7, and there rests 14 d. to divide, which is a thing of a small value.

So that if 46 pieces of Cloth cost 673 l. the one will cost 14 l. 12 s. 7 d. as may be seen by the Example here under wrought. 〈 math 〉〈 math 〉

* 1.3To shew the brevity of this manner of dividing, I will shew here another Example, which cannot without much difficulty be performed by the common manner of division in cancelling the figures. and yet is very easily and compendiously performed this way, by observing the order which is before prescribed, and therefore to avoid prolixity, I shall not need here to put down any particular Explication, the Example shall be thus then, to divide 19999100007 by 99999, which by the Product doth give 199993 without any Remainder, as shall appear by the working here underneath: 〈 math 〉〈 math 〉

* 1.4Division which is wrought by the Rule of Practise, is done in this manner by the help of the parts found therein, as for Example, if 72 pieces of Serges cost 169 l. 12 s. how much will the piece stand in, then dividing by 72, I find first the parts thereof; and observe that 8 times 9 makes 72, I take then the ⅛ of the sum to be divided, saying the eighth part of 16 is 2, which I place under a line drawn, and the ⅛ of 9 is 1, there remains one pound, which is 20 s. which, with the 12. makes 32, the ⅛ whereof is 4, so that the eighth of the said 169 l. 12 s. is 21 l. 4 s. of which number I take the 1/9 (which shall be the price of the piece) in this manner, saying, the 1/9 of 21 is 2, the rest is 3 l. which is 60 s. and with the 4 maketh 64 s. of which the 1/9 is 7, then there rests 1, which is 12 pence, and the 1/9 of 12 is 1, so that the ninth part of 21 l. 4 s. is 2 l. 7 s. 1 d. the price of the piece; as by this Example is shewed: 〈 math 〉〈 math 〉

* 1.5But when it happens that any broken numbers fall in the Divisor, the Divisor and the sun•• to be divided, must then be reduced to one and the self same denominator; as for Example, If 13½ pieces should cost 264 l. 17 s. 6 d. what would the piece stand in? to do which I reduce into halfs the pieces 13½ multiplying the same by 2 making 27 halfs, doing the same with the sum to be divided, multiplying it by 2, which comes to be l. 529. 15, which to be divided by 27, must be considered that 3 times 9 is 27, therefore must be taken ⅓ and 1/9 of the said third in this manner, saying ⅓ of 5 is 1, of 22 is 7, and of 19 is 6, and there remains 1 l. which is 20 s. which with the 15 makes 35, the ⅓, of which is 11, and there rests 2 s. which are 24 d.

the ⅓ whereof is 8, and afterward taking of the 1/9 the said ⅓ saying, the 1/9 of 17 is 1, and of 86 is 9, and there rests 5 l. which with the 11 s. is 111 s. the 1/9 whereof is 12, then rests 3 s. which with the 8 d. remaining is 44 d. the 1/9 whereof is 4, so that the product of the said 1/9 giveth 19 l. 12 s. 4 d. the value of the said piece, as by Example. 〈 math 〉〈 math 〉

* 1.6Again, at 34 l. 16s. the 21⅓ yard, how much will the yard amount to? Do this as the preceding rule, putting the yard into thirds, in multiplying them by 3 they make 64 for divisor to 34l. 16 also multiplyed by 3, which make 104 l. 8 s. which to divide by 64, is to be considered that 8 times 8 is 64, and therefore the 1/8 of an eight is the price of a yard, as Example. 〈 math 〉〈 math 〉

Many other divisions are resolved in the same manner as the preceding, which I willingly here omit, and refer them to the occurrences of Traffick that shall happen herein, and now I will proceed to Multiplication abreviated, by which two rules both the Golden Rule of Three, and many other in Arithmetick are wrought and performed, commodious and necessary to this Tract of Exchanges, and this Map of Commerce, as being indeed the proper rules, by which the Exchanges in this Book are cast up and calculated.

* 1.7The method that hath been shewed in Division, may in some sort serve also in Multiplication in this manner, suppose you were to multiply 56 yards by 4l. 18s. 9d. you must consider that 56 is composed of 7 times 8, and therefore you must multiply the said 4. 18. 9. by 7, and its product by 8, beginning with the pence, and saying 7 times 9 is 63, I write 3 pence, and retain 5 s. which I add with 7 times 8, which make 61, write then 1 and retain 6, which added to 1 time 7 makes 13, which is 6 l. 10s. and following the common method of Addition, I put down one ten, and retain 6l. which I add with 4 times 7, and they make 34l. the which product I multiply again by 8, beginning to multiply by the pence which are with the pounds, and then by the s. calculating for 12 d. one s. and for 20 s. 1 pound, they then make 276l. 10s. for the value of 56 yards, as shall be more plainly demonstrated by this Example fol∣lowing 〈 math 〉〈 math 〉

* 1.8Many other questions may be answered as the abovesaid, but yet note, that to multiply by an uneven number, such as is 31, 43, and the like, then do in this manner, posito I demand at 5l. 9 s. 3 d. the Yard what will 43 Yards come unto? Now for as much as 43, hath no dividable parts, I take 42, multiplying it by 6 and by 7, as hath been shewed, and for the Yard that doth remain, I add the last Multiplication 5 l. 9 s. 3 d. which is the cost of 1 Yard and it makes 23 l. 17. 9. d. the cost of 43 Yards, as may be observed by the following Example. 〈 math 〉〈 math 〉

* 1.9But when there is any broken number in the Yards or pieces, do thus by Example at l. 7. 14. 6 the piece what will 81½ cost? Then for 81 I multiply by 9 the cost of the piece, and its product again by 9, because that 9 times 9 is 81, and I find l. 625. 14. 6 for the value of the said 18 pieces; and for the ½ piece I take the ½ of 7 l. 14. 6 d. and adding it thereto, the same comes to be l. 629. 11. s. 9 d. and so much the 81½ cost. Example. 〈 math 〉〈 math 〉

* 1.10But for as much as this may seem difficult, I will here note another way of Multiplication abreviated, serving as well in Exchanges or in Merchandizing posito; I would know what comes 154 Yards unto, at 56 shillings the Yard: To do this, multiply the said Yards, by the half of the said money, which is 28 s and in adding its product, double the last figure, taking that for so many s. and the rest for pounds, as doth appear by this Example following, which I add in this manner, saying, 2. 4. and 9 makes 15, and after the ordinary manner, you must set down 5, and bear 1 ten; but in this method, you must double it, setting down 10 for 5; the which doubled, you must hold as so many shillings, and so proceed in the addition of the rest, and adding the tenth born of 15, it will come to l. 431. 4. s the value of 154 Yards; and this note is to be observed in all other questions of this nature, as by example doth appear. 〈 math 〉〈 math 〉

* 1.11There is yet another brief way of Multiplication, used in France and many parts beyond the Seas, which is done by taking the parts of 10 or of 100 in this manner, I would multiply 113 Yards by 1¼, I note what part 1¼ is in 10, and find it to be ⅛ therefore is 113 Yards to be multiplyed by 10, or else more brief by adding an o, and in taking the ⅛, which shall be the value of the said 113 Yards, and for to multiply by 1⅔ you must take the ⅙ of the sum to be multiplyed, after you have

added thereunto an 0 because that 1 ••/•• is the ⅙ of 10, and for to multiply by 3⅓ you must add an 0 and take ⅓, because that 3⅓ is the ⅓ of 10, and so in many others, in taking always the parts of 10; and note, that the same may be done in taking the parts of 100, as to multiply 137 Yards by 8⅓, you must add two 00 to the sum, then take 1/12 because that 8⅓ is the 2/12 part of 100, and for to multiply by 12½, you must add two 00 and take the ⅛, because that the 〈◊〉〈◊〉 of 100 is 12½, as may be seen by these following Examples. 〈 math 〉〈 math 〉

* 1.12Again, at 3 s. the pound, what will the 100l. come to? to do this in brief, a cypher is to be added to the cost of the pound which is 3 s. and it makes 30, of which sum take the ½ and it makes 15 l. which makes the cost of the hundred, and so for others by these Examples following. 〈 math 〉〈 math 〉

* 1.13Again, at 3 d. the pound, I would know how much 100 l. comes to? multiply the pence (which the pound is worth) by 5, and out of its product take 1/12 which shall be the cost of the quin∣tal; and to find the cost of the pound, you must multiply the value of the hundred by 12, and the ⅕ of the product is the cost of the pound, as followeth by these Examples. 〈 math 〉〈 math 〉

But I have wandered too far, and proceeded farther in this subject than I intended, therefore here will conclude both the calcuations of Exchanges, and these methods of abreviating the rules of Division and Multiplication, referring what is here by me omitted in both the said subjects to the ingenious hand and head of the mysterious Exchanger.

FINIS.

Notes

• * 1.1

  Of Division and Multipli∣cation abrevi∣ated after the Italian man∣ner.
• * 1.2

  Division abre∣viated in Cal∣culations of Accounts.
• * 1.3

  Another Ex∣ample.
• * 1.4

  Another me∣thod of divi∣ding by the Rule of Pra∣ctise.
• * 1.5

  Note, another Example.
• * 1.6

  Another.
• * 1.7

  Multiplication abreviated.
• * 1.8

  Another Ex∣ample.
• * 1.9

  Another Ex∣ample.
• * 1.10

  Another way of Multipica∣tion abrevia∣ted.
• * 1.11

  Another way of abreviated Multiplicati∣on.
• * 1.12

  Another Ex∣ample.
• * 1.13

  Another.