Arithmetic, Part 1 · Section 11 – Section 14
Multiplication.
MUltiplication is that part of Arithmetick which teacheth how to encreaſe one number by another, ſo that the number produced by their Multiplication ſhall contain one of the numbers multiplied ſo many times as there are unites contained in the other. Multiplication may fitly be termed a Compendium of Addition, for that it performeth at one operation the ſame which to effect by Addition would require many. For inſtance, if it were required to know how much 7 times 5 is, to perform this by addition, I muſt ſet ſeven fives, or five ſevens, one under another, and adding them together, I ſhall finde that either of their Totalls ſhall contain 35, but this by multiplication is performed with far more brevity, as by examples hereafter ſhall appear.
Before you enter upon the practice of Multiplication, it is neceſſary to remember the product produced by the multiplication of any one of the nine digits, by any other of the ſame, as readily to know; that 4 times 5 is 20, 6 times 7 is 42, 2 times 9 is 18, 7 times 9 is 63. 8 times 9 is 72, &c. Which this Table following, willplainly declare, and muſt be perfectly learned by heart before you attempt to multiply great numbers.
Multiplication Table.
2 times 2 makes 4
2 times 3 makes 6
2 times 4 makes 8
2 times 5 makes 10
2 times 6 makes 12
2 times 7 makes 14
2 times 8 makes 16
2 times 9 makes 18
3 times 3 makes 9
3 times 4 makes 12
3 times 5 makes 15
3 times 6 makes 18
3 times 7 makes 21
3 times 8 makes 24
3 times 9 makes 27
4 times 4 makes 16
4 times 5 makes 20
4 times 6 makes 24
4 times 7 makes 28
4 times 8 makes 32
4 times 9 makes 36
5 times 5 makes 25
5 times 6 makes 30
5 times 7 makes 35
5 times 8 makes 40
5 times 9 makes 45
6 times 6 makes 36
6 times 7 makes 42
6 times 8 makes 48
6 times 9 makes 54
7 times 7 makes 49
7 times 8 makes 56
7 times 9 makes 63
8 times 8 makes 64
8 times 9 makes 72
9 times | 9 | makes 81.
The uſe of the Table of Multiplication, and the manner how it is to be read.
This Table ſheweth what the Summe of any two digits multiplyed one by another doth amount unto, and is thus to be read, 2 times 2 makes 4, 2 times 3 makes 6, 2 times 4 makes 8: Alſo 6 times 4 makes 24, 7 times 8 makes 56, 8 times 8 makes (or is) 64, 9 times 9 is 81, &c.
In Multiplication there are three terms commonly uſed, that is to ſay,
The Multiplicand,
The Multiplier, and
The Product.
The Multiplicand is the number to be multiplied.
The Multiplier is the number by which the Multiplicand is multiplied, and
The Product is the number which is produced by the multiplication of the Multiplicand and the Multiplier together.
Thus, if it were required to multiply 8 by 7, here 8 is the Multiplicand, 7 the Multiplier, and 56 is the Product, for 8 times 7, or 7 times 8, is 56.
In Multiplication it mattereth not which of the two numbers given is made the Multiplicand, or which the Multiplier, for the Product produced by either will be the ſame, but the uſual way is to make the greater number the Multiplicand; and the leſſer number the Multiplier.
THE RƲLE.
The numbers to be multiplied muſt be ſet one under another. viz. the Multiplicand (or greater number). above, and the Multiplier, (or leſſer number) below, the laſt figure of the Multiplier under the laſt figure of the Multiplicand, then draw a line under them, and (having learned the preceding Table perfectly by heart) multiplie every digit of the Multiplier, into every digit of the Multiplicand, ſetting the ſeveral products under the line, then having finiſhed your Multiplication, draw a line, and adde all the products together, and the Sum of thoſe products is the general product of the whole multiplication, as by the following examples will oppear.
Example 1. … Let it be required to multiply 736 by 7. Firſt, I write down 736 the Multiplicand, and under it 7, the Multiplyer, and under them I draw a line, then I multiply 7 into every digit of the Multiplicand, ſaying 7 times 6 is 42, place 2 under the line under 7, and for the four tens keep 4 in mind, then ſay again, 7 times 3 is 21, and 4 which I kept in mind is 25, place 5 under the line, and keep the two tens in mind, then ſay again 7 times 7 is 49, and … which I kept in mind is 51, place 1 under the line, the 5 tens, (becauſe there is no more figures to be multiplied) I ſet down under the line alſo, ſo is the work ended, and the product of this multiplication is 5152.
Example 2. … Let it be required to multiply 2417 by 5, place the numbers one under another, and draw a line under them as in the margine, then begin your multiplication, ſaying 5 times 7 is 35, place 5 under the line, and keep the three tens in mind, then ſay again, 5 times 1 is 5, and 3 which I kept in minde is 8, place 8 under the line, and (becauſe it is leſſe then 10, I keep nothing in mind) then ſay again, 5 times is 20, place a cypher under the line, and keep the two tens in mind; laſtly, ſay 5 times 3 is 15, and 2 which I kept in mind is 17, which 17 being the laſt number I place under the line, and ſo is my Multiplication ended, and the product is 17085.
¶ You may be ſatisfied of the truth of this work if you will take the pains to ſet down the Multiplicand 3417, five times one under another, and adde them together, as ſo many ſeveral ſums, ſo ſhall you find the Total of that addition, to be 17085, exactly the ſame with the product of this Multiplication.
Example 3. … In the two fore-going examples, the Multiplyer conſiſted but of one digit, we are now to ſhew how multiplication is performed when the Multiplier conſiſts of more then one figure, therefore in this example, let it be required to multiplie 5704 by 37, place your numbers and draw a line under them as you ſee in the margine, then begin your Multiplication in this manner, ſaying, 7 times 4 is 28, ſet 8 under the line, and keepe the two tens in mind, then ſay 7 times nothing is nothing, but the two tens in mind is 2, ſet 2 under the line, then ſay 7 times 7 is 49, ſet 9 under the line, and keep 4 in mind, then laſtly, ſay 7 times 5 is 35, and 4 in mind is 39, which being the laſt number to be multiplied I ſet down under the line, ſo is the mulplication of one of the digits (namely 7) finiſhed.
Then begin to multiply the ſecond digit, ſaying 3 times 4 is 12, place 2 in the ſecond line one place towards the left hand, and keep 1 in mind, then ſay 3 times nothing is nothing, but 1 in mind is 1, ſet down 1 by the 2 in the ſecond line, thirdly, ſay 3 times 7 is 21, place 1 in the ſecond line, and keep the two tens in mind; Laſtly, ſay 3 times 5 is 15, and 2 in mind is 17, which 17, (becauſe there is no more figures to be multiplied) I place in the ſecond line alſo.
Having thus done I draw a line under them, and adde theſe two lines together, as in common Addition of numbers of one denomination, ſaying 8 is 8, place 8 under the line, then ſay 2 and 2 is 4, place 4 under the line, then ſay 1 and 9 is 10, place a cypher under the line, and carry 1 to the next place ſaying 1 and 1 is 2 and 9 is 11, place 1 under the line and carry 1 to the next row, ſaying 1 and 7 is 8, and 3 is 11, place 1 under the line, and carry 1 to the next place, ſaying, 1 which I carry and 1 is 2, place 2 under the line and ſo is your multiplication ended, and the product is 211048.
Example 4. … Let it be required to multiply 57325 by 4032, place the multiplicand and multiplierone under another, & draw a line asbefore then proceed to the multiplication as formerly, ſaying, firſt, 2 times 5 is 10, ſet down a cypher and keep 1 in mind: then 2 times 2 is 4, and 1 in mind is 5, place 5 under the line then 2 times 3 is 6, ſet 6 under the line: then 2 times 7 is 14, ſet down 4, and keep 1 in mind, then 2 times 5 is 10, and 1 mind is 11, which 11 (being the laſt) I ſet down.
The multiplication of one of the digits being finiſhed, proceed to the multiplication of the next, ſaying 3 times 5 is 15, ſet down 5 in the ſecond line a place more towards the left hand, and keep 1, then 3 times 2 is 6, and 1 kept is 7, ſet down 7, then 3 times 3 is 9, ſet down 9, then 3 times 7 is 21, ſet down 1 and keep 2 in mind, then 3 times 5 is 15, and 2 in mind is 17, which being the laſt ſet down alſo.
Two of the figures of the multiplier being finiſhed proceed to the third, which (in this example) being a cypher, you may wholly neglect, and proceed to the multiplication of the fourth figure, onelyonly remember to remove the product of the fourth figure one place more to the left hand, as in the example you may ſee, for the cypher, though it be not written down, yet it muſt keep its place, and the figures following muſt be removed a place farther.
Then for the Multiplication of the fourth and laſt digit, ſay 4 times 5 is 20, ſet down a cypher (under 9) and keep 2 in mind, then 4 times 2 is 8 and 2 in mind is 10, ſet down a cypher and keep 1 in mind, then 4 times 3 is 12, and 1 is 13, ſet down 3 and keep 1, then 4 times 7 is 28, and 1 kept is 29, ſet down 9 and keep 2, then 4 times 5 is 20, and 2 kept is 22, which 22 (becauſe the multiplicatiou is ended) ſet down alſo.
Having thus multiplied all the digits ſeverally, draw a line under their products, and adde them altogether as in the former example, ſo ſhall you find their general product to be 231134400.
Other Examples for Practice.
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Compendiums in Multiplication.
1 If the Multiplier conſiſt of cyphers in the laſt place or places, … you may omit the multiplcation of them, and place the former figures of the Multiplier under the Multiplicand, thus if it were required to multiply 3257 by 2600. place the numbers as you ſee in the margent, then multiplying 3257 by 26, the Product will be 84682, to which if you add two cyphers, (becauſe there were two cyphers in the Multiplier) it will be 8468200, which is the true product of the multiplication.
2 If it be required to multiply any number by 10, 100, 1000, 10000, &c, You have no more to do but to add ſo many cyphers to the multiplicand as there are cyphers in the multiplier, thus if you were to multiply 365 by 10, the product will be 3650, or by 100, it would be 36500, or by 1000, it would be 365000, or by 10000, it would be 3650000, &c.
3 If any number given were to be multiplied by 5, you may abreviate your worke thus, adde a cypher to the Multiplicand, take halfe that number, and it ſhall be the product required, thus if it were required to multiply 8627 by 5, adde a cypher to the multiplicand, then it is 86270 the halfe whereof is 43135, which is the product required.
The Proof of Multiplication.
The moſt certain proof of Multiplication is by Diviſion, but becauſe Diviſion is not yet known, I will here ſhew a neer way by which Multiplication may be proved. Which is thus,
THE RƲLE.
Firſt, take a Croſſe as in the Margine, then, any ſums being multiplied, you may prove the truth of your work in this manner, (1) Caſt away all the nines which you can find in the multiplicand, what remaineth ſet on the right ſide of the Croſſe. (2) Caſt away alſo the nines in the Multiplyer, and what remains ſet on the left ſide of the Croſſe. (3) Multiply the figure on the right ſide of the Croſſe by that on the left ſide of the Croſſe, and out of that product caſt away the nines, ſetting the figure remaining over the Croſſe, then (4) Caſt away all the nines in the product, and if the figure remaining be the ſame with that which ſtandeth over the Croſſe, then is your multiplication truly performed, otherwiſe not.
1 Caſt away all the nines in the Multiplicand, ſaying 4 and 3 is 7, and 2 is 9, which being rejected there remains 4, which I ſet on the right ſide of the croſſe, then
2 Caſt away all the nines in the Multiplier, ſaying 2 and 3 is 5, which (being leſſe then 9) I ſet on the left ſide of the croſſe, then
3 Multiply 4 by 5, ſaying 4 times 5 is 20 from which caſt all the nines, and there remain 2, place 2 over the croſſe, and
4 Caſt away all the nines in the Product, ſaying, 2 and 5 is 7, and 4 is 11, caſt away 9, and there remains 2, which exactly agrees with the figure over the croſſe, and demonſtrates that the multiplication is truly performed.
To multiply by any of the nine Digits without charging the memory.
To multiply any number by 2, Either double the number in your mind, or adde it, by ſetting it down twice, ſo 57325 produceth 115750.
To multiply any number by 3, To the number given, adde the double thereof, the ſum is the product, ſo 57325 produceth 171975.
To multiply any number by 4, Double the duplication in your mind, ſo 57325 produceth 229290.
To multiply any number by 5, Conceive a cypher added to the given number, and in your mind half thereof is the product, thus a cypher added to 57325, maketh it 573250, the halfe whereof is 286625.
To multiply any number by 6, Take half adding a cypher, and adde to the half, the figure ſtanding next before, thus 57325|0 produceth 343950.
To multiply any number by 7, Take half and adde it to the double of the former figure, ſuppoſing a cypher added as before, ſo 57325 thus ordered, produceth 401275.
To multiply any number by 8, Double each former figure, and ſubtract it from the following, ſo 57325 produceth 458600.
To multiply any number by 9, Suppoſe the number multiplyed by 1, then ſubtract each former figure from the following, beginning with that next before the cypher, the remainer is the product, ſo 57325 produceth 515925.
Queſtions performed by Multiplication only.
Queſtion 1. If a piece of land be 236 perches long, and 182 perches broad, how many ſquare perches are contained therein? multiply 236 the length, by 182 the breadth, the product is 42952, and ſo many ſquare perches are contained in ſuch a ſquare piece of land.
Queſtion 2. In a year there are 365 dayes natural, and in every day 24 houres, how many houres be there in a year? Multiply 365 the number of dayes, by 24 the number of houres, the product is 8760, and ſo many houres be there in a year.
Queſtion 3. From London to Coventry it is accounted 76 miles, how many yards therefore is it from London to Coventry? Multiply 1760 (which are the number of yards contained in one mile) by 79, the product is 133760, and ſo many yards are between London and Coventry.