Difference between revisions of "ספר ציפרא"

Prologue

Introduction

The Positional Decimal System

ואין לך חשבון שהוא גדול מן ט‫'

• Units: the first rank is called the rank of units, for as the digit is named so is its numerical value, meaning: 1 is one; 2 is two; and so on until 9, which is nine.

• Tens:

Zero – placeholder digit: when you calculate ten you should write two ranks - 1 in the second rank

and before it a numeral, which is a zero [lit. wheel], like this 0, that has no substance but is a place holder and a remnant of a thing like a wheel, [like stubble] before the wind [Psalms 83, 14].

• א‎0 is 10.

• אא is 11.

For the first א of the units is in the first rank and the second א is of the tens, hence 11.

• בא is 12.

• גא is 13.

• and so on until טא which is 19.

• ב‎0 is twenty.

• אב is twenty-one.

• בב is twenty-two.

• and so on until twenty-nine, which is טב.

• Hundreds: the third rank is the rank of the hundreds, like this:

• א‎00 which is 100.

• ב‎00 which is 200.

• ג‎00 which is 300.

• and so on until 9 hundred, which is ט‎00.

• אאא is 111.

• א‎0א is 101.

• Thousands: the fourth rank is the rank of thousands.

• א‎000 is one thousand.

• ב‎000 is two thousand.

• and so on until 9 thousand, which is ט‎000.

• Tens of thousands: the fifth rank is the rank of tens of thousands.

• Hundreds of thousands: the sixth rank is the rank of hundreds of thousands.

• Thousands of thousands: the seventh rank is the rank of thousands of thousands [= millions].

• Example: if one asks you: how much is 3020?

Note that the zero has no numerical value but is a place holder, so the number is three thousand and twenty.

Chapter One: Multiplication of Units - Shortcuts

• If you wish to know how much is 9 times 9

Write like this:

9 9

Think by how much it exceeds over ten, it is 8.

Write [it] next to the upper 9 to the left, like this:

8 9   9

Then, think by how much the upper 9 exceeds over the 8, you will find only 1.

Write this 1 next to the bottom 9:

8 9   9   1

Then, think by how much the bottom 9 exceeds over the 8, it is 1.

Write this 1 also next to the bottom 9, like this:

8 9   9 1 1

Think how much is 1 times 1, it is 1.

Write 1 before the 8, like this:

8 1   9 1 1

Erase the bottom and the upper 9, so they are eighty-one.

[Illustration of the procedure:]

9		89		89		81
9		9		9		9
				11		11

• Also if you want to know how much is 9 times 8

Write like this:

9 8

Think by how much 9+8 exceeds over ten, it is 7.

Write the 7 next to the 9, like this:

7 9   8

Think by how much the 9 exceeds over the 7, it is 2.

By how much the 8 exceeds over the 7, [it is] 1.

Write like this:

7 9   8 1 2

Multiply 1 by 2, it is 2.

Write this 2 before the 7, like this:

7 2   8 1 2

We find that 9 times 8 is 72.

[Illustration of the procedure:]

9		79		79		72
8		8		8		8
				12		12

• Example: if you wish to know how much is 6 times 6

Write the 6 beneath the 6, like this:

6 6

Say: by how much the two 6 exceed over 10, it is 2.

Write this 2 after the 6, like this:

2 6   6

Then, say: by how much the 6 exceeds over 2, it is 4.

The second 6 also exceeds over 2 by 4.

Multiply 4 times 4, it is 16, meaning sixteen.

Add the 1 to the 2, because the 2 is in the second rank, it is 3.

There is also 6, write the 6 before the 3, like this:

3 6   6

We find that 6 times 6 is 36.

[Illustration of the procedure:]

6		26		[26]		36
6		6		[6]		6
				[44]

8	2	7	3	6	4
9	1	8	2	7	3
72		56		42

• Example: if you wish to know [how much is] 7 times 8.

Write like this: 70

Subtract from it the product of the difference of the greater numeral from ten by the smaller numeral.

As what is infront of you: say 7 times 2 are 14.

Subtract 14 from 70, the remainder is 56.

ותקח ד"א מן 0"ז ונשאר ו"ה

• Such as these:

8	0	6	0	4	0	5	0
8	2	7	3	9	1	8	2
64		42		36		40

• Example: 4 times 4.

Like this:

4 4

Lend to it 3, so that it becomes 11.

Subtract ten, 1 remains in your hand.

Write it like this:

1 4   4

Think by how much the first 4 exceeds over the 1, that is to say 3.

So, the second 4 also exceeds by 3.

1 4   4 3 3

Multiply 3 by 3, it is 9.

Write the 9 before the 1, like this:

1 9   4

Subtract 3 that you have lent to it and 16 remain, which are sixteen.

You will find that your calculation is correct.

1 6   4

[Illustration of the procedure:]

4		14		14		19		16
4		4		4		4		4
				33

• I will teach you an example: if you want to multiply 4 times 4.

Write it like this:

4 4

Think by how much the first 4 is less than 10, it is 6.

Think also by how much the second [4] is less than 10, you find it is also 6.

Multiply 6 by 6, the result is 36.

Think by how much the two 4 are less than ten, it is 2.

Subtract 2 in the second rank from 3, you are left with 16

We find that 4 times 4 is 16.

[Illustration of the procedure:]

4		[4]		[36]		16
4		[4]		[4]
		[66]		[66]

The same for all; note well

• Example: if you want to multiply 4 by 3.

Take the difference of each from 10, i.e. from 4 to 10, it is 6 and from 3 to 10, it is 7

6 times 7 is 42.

ופ"ז ב"ד

Write this 42 for remembrance.

[בד]

Sum 4 and 3 together, it is 7.

Write this 7 next to 42 to the left, like this:

[בדז]

Subtract 10 from [the sum of] the numerals on the left, 12 remains, which is the number that you want to know.

[Illustration of the procedure:]

42

742

12

Deduce from this.

I wrote this even though it is an axiom for the calculation that you wanted, in order to inform you that in this way [...]

קח המרחק עד י’ מן כל אחד ואחד שתרצה לכפול יחד וחשוב אותו ב’ מרחקים זה על זה ומה שיעלה בידך כתוב לזכרון

• Example: you wish to multiply 7 times 9.

Take the difference from 7 to 10, i.e. 3, and the difference from 9 to 10, i.e. 1, and say: 3 times 1 are 3.

Write this 3 for remembrance as this:

ג

• Then, add 7 to 9, the result is 16, meaning sixteen, subtract the 10 and take the 6.

Write it to the left, next to the remaining 3, so it will be sixty three, like this:

גו
ששי' ושלש

[Illustration of the procedure:]

3

63

Look carefully.

• Another example: you wish to multiply 6 times 8.

Take the difference from 6 to 10, i.e. 4, and the difference from 8 [to 10], i.e. 2, and say: 2 times 4 are 8.

Write this 8 for remembrance as this:

[ח]

Then, add 6 to 8, the result is 14, subtract 10 and take 4.

Write 4 to the left, next to the 8 that you have already wrote, the result is 48, i.e. 8 times 6.

[חד]

[Illustration of the procedure:]

[8]

[48]

Look carefully.

• Another example: if you want to know 5 times 6.

Take the difference from 5 to 10, it is 5.

Sum 5 and 6 together, it is 1[1], subtract ten and you are left with 1.

Add it to the [2] and it is 3, so we find it 30. Note well.

[Illustration of the procedure:]

[20]

[30]

• multiplication table of 1 to 9

• half multiplication table [MS Oxford 440, 115r; Oxford 60, 175r]

Chapter Two: Addition

• If you wish to sum three thousand three hundred and seventy-two with three thousand three hundred ninety-two.

Write as follows:

3 3 7 2 3 3 9 2

Add the bottom 2 to the upper 2; it is 4.

3 3 7 2 3 3 9 2       4

Add the bottom 9 to the upper 7; it is 16.

Write the 6 in the second rank, after the 4 that you wrote, like this:

6 4

Add the remaining 1 to the threes that follow it in the third rank, like this:

3 3

So, the total is 7 in the third rank, like this:

:

7 6 4

Then, add the two last threes in the fourth rank; their sum is 6.

Write this 6 in the fourth rank, after 764 that you wrote, like this:

6 7 6 4

The procedure is complete and the sum is six thousand seven hundred and sixty-four, i.e. as follows:

6 7 6 4

[Illustration of the procedure:]

3372		[3372]		[3372]		[3372]		[3372]
3392		[3392]		[3392]		[3392]		[3392]
		4		64		764		6764

• Example: we want to sum 3372 with 9892.

Here is how you write it and its diagram is as follows:

3 3 7 2 9 8 9 2

We add the bottom 2 to the upper 2; it is 4.

We write the 4 corresponding to the bottom 2.

We add also the bottom 9 to the upper 7; the result is 16.

We write 6 beneath the 9.

We add 1 to the next digit in the top row: there is a 3 there. We add 1 to it; it is as if we write 4 in the place of the 3.

We add the upper 4 to the bottom 8; the result is 12.

We write 2 beneath the 8.

We add 1 to the digit next to the upper 4: there is a 3 there. We add 1 to it; it is 4.

We add the upper 4 to the bottom 9; the result is 13.

We write 3 beneath the 9 and 1 after it; it is like this:

3 3 7 2   9 8 9 2 1 3 2 6 4

כי הא' הנותרת לי היא כמו כן מן העשיריות נגד שיט' ראשונ' לכך אני מוסיף אות' תדיר עליה

We find that the result of the above addition is 13 thousand, two hundred and 64.

[Illustration of the procedure:]

3372		3372		3472		4472		3372
9892		9892		9892		9892		9892
		4		64		264		13264

7 8 3 1 2   8 7 5 4 7	6 8 9 7 5   8 7 4 9 6	3 7 0 6 4 5 2 0 5 6
1 6 5 8 5 9	1 5 6 4 7 1	8 9 1 2 0

Checking Methods

1) Casting out 9 by 9:

The first: cast out the nines as many as you can from the two upper lines and take what remains that is less that 9 as the weights of the scales.

Then, cast out the nines in the bottom line also, and if the scales of what remains is as the first scales, know that your calculation is correct.

2) Subtraction

The second scale is that one subtracts the two upper lines from the bottom line, and if all is gone as required no more and no less, then your calculation is correct.

Chapter Three: Subtraction

• We wish to subtract 9 hundred and ninety-nine from one thousand

Here is how you write it:

1 0 0 0   9 9 9

We find that the 1 that is in the upper line, in the fourth rank, is ten for the 9 that is third in the bottom line.

Now, we subtract 9 from this ten; you are left with 1.

Write the 1 in the place of the third zero corresponding to the 9 that you have subtracted and erase the 9, like this:

1 0 0   9 9

Now, the 1 in the third rank of the upper line is also ten for the 9 in the bottom line.

We subtract the second 9; 1 remains.

Write the 1 in the place of the second zero and erase the second 9; erase also the upper 1 that is in the third rank, from which we subtracted. It will be as follows:

1 0   9

Again, we subtract 9 that is beneath the zero from 1 that is above in the second rank, since it is also ten for the nine as above.

So, when we subtract 9 from 10, 1 remains.

Write the 1 in the place of the zero and erase the upper 1 that is in the second rank; erase also the 9.

Then only one remains.

We find that when we subtract 999 from one thousand, 1 remains.

1000		100		10		1
999		99		9

• Another example: if we wish to subtract 321 from 654.

Write as follows:

6 5 4 3 2 1

We subtract the bottom 3 from the corresponding upper 6; 3 remain.

Erase the 6 and write the remaining 3 instead; erase also the 3 that is left beneath.

It is like this:

3 5 4   2 1

Again, we subtract the bottom 2 from the corresponding upper 5; 3 remain.

Write the 3 instead of the 5 and erase the bottom 3 as well as the upper 5.

It is like this:

3 3 4     1

Again, we subtract the bottom 1 from the corresponding upper 4; 3 remain.

Erase the upper 4 as well as the bottom 1 and write 3 instead of the upper 4.

It is like this:

3 3 3

We find that when we subtract 321 from 654, only 333 remains.

654		354		334		333
321		21		1

• If the digit of the subtrahend is larger than the digit of the subtracted in the corresponding rank

• As if you wish to subtract 867 from 956, like this:

9 5 6 8 6 7

How do you subtract larger from smaller, like this: 7 from 6, you cannot and likewise 6 from 5.

This is how you do it: take 1 from 5, which is next to 6, erase the 5 and write 4.

How much does this 1 add up to with the 6? 16, because it is in the rank that is second to 6; so it is 16. Subtract the 7 on the bottom row from 16; 9 remains.

Then, you need to subtract 6 from 4. Do the same as I instructed you: erase 1 from 9; it becomes 14 in the preceding rank. Subtract 6 from it; 8 remains.

956		[949]		[889]		[89]
867		[86 ]		[8  ]

I will teach you how to do it, because you cannot subtract 7 from 6 and also 6 from 5. This is how you do it: first, subtract the bottom 8 from the upper 9; 1 remains. Erase the 9 and write 1 in its place. Erase also the bottom 8. It is like this:

1 5 6   6 7

Now we need to subtract the bottom 6 from the upper 5 but we cannot. Therefore, go to the 1, which is to the left of the upper 5, and add together 1 and 5; it is 15. Subtract the 6 from it; 9 remains. Erase the upper 1 and the upper 5 also and write 9 in their place. Erase the bottom 6 as well. It remains like this:

9 6   7

Again, we need to subtract the bottom 7 from 6, but we cannot. Now, go and take one from the 9 next to the 6 in the upper row and add together the 6 and the 1, which is ten to the 6; it is 16. Now subtract 7 from 16; 9 remains. Erase the upper 6 and 9 and write 9 and 8 in their place, as I instructed you. Erase the bottom 7 as well. It is like this:

8 9

956		156		96		89
867		67		7

Checking Methods

• Addition

• If you want to know if you calculated correctly or not, add 321 to 333 according to the addition chapter: if you find 654 the subtraction is correct.

• If you want to know if you calculated correctly, add 867 to the remaining 89: if you find 956 you are right.

Chapter Four: Multiplication

• If you wish to multiply 1234 by 4321.

Here is its diagram, when you write one number next to the other, the first digit of the lower number corresponds to the last digit of the upper number, like this:

4 3 2 1 1 2 3 4

We multiply the lower 1 by the upper 4, meaning multiply 1 times 4; it is only 4.

We write 4 corresponding to the lower 1 and erase the lower 1, like this:

4     4 3 2 1   2 3 4

We multiply also the lower 2 by the upper 4; the result is 8.

We write 8 corresponding to the lower 2 and erase the lower 2, like this:

4 8   4 3 2 1     3 4

We multiply also the lower 3 by the upper 4; the result is 12.

Now, we write 2 corresponding to the lower 3 and 1 with the 8 that follows it, because it is ten, so it is one rank higher than the 2 before it; the result is 9.

Erase the lower 3, like this:

4 9 2 4 3 2 1       4

We multiply also the lower 4 by the upper 4; the result is 16.

Now, we write 6 corresponding to the lower 4, instead of the upper 4, since we have multiplied the upper 4 by the whole [multiplicand].

We write the 1 with the 2 that follows it, so it becomes 3, like this:

4 9 3 6 3 2 1

[Illustration of the procedure:]

4321		4  4321		48 4321		4924321
1234		234		34		4

4936321

We shift 1234 one rank backwards, beneath the upper 3, in order to multiply the upper 3 by the whole [multiplicand] 1234, like this:

4 9 3 6 3 2 1   1 2 3 4

We multiply the lower 1 by the upper 3, meaning 3 times 1; it is only 3.

Add 3 to the 9 above the lower 1; the result is 12.

We write 2 instead of the 9.

Add the 1, which is ten, to the second rank, meaning the 4; the 4 becomes 5.

We erase the lower 1, like this:

5 2 3 6 3 2 1     2 3 4

We multiply also the lower 2 by the upper 3, 2 times 3; it is 6.

Add the 6 to the 3 above it; it is 9.

We erase the lower 2, like this:

5 2 9 6 3 2 1       3 4

We multiply also the lower 3 by the upper 3, 3 times 3; it is 9.

Add 9 to the 6 above it; the result is 15.

Write 5 instead of the 6 and the 1 in the second rank with the 9 that follows; it becomes 10.

However, we should not write 10, because 10 is not written in the numeral calculation, only up to 9. So, we write a wheel, like this 0 and we write 1 in the following rank, meaning with the 2, so it becomes 3.

We erase the lower 3, like this:

5 3 0 5 3 2 1         4

We multiply also the lower 4 by the upper 3; the result is 12.

Now, we write 2 instead of the upper 3, since we multiplied the 3 by the whole [multiplicand] and we write 1 in the second rank, with the 5; it becomes 6.

We erase the lower 4, like this.

5 3 0 6 2 2 1

[Illustration of the procedure:]

4936321		5236321		5296321		5305321
1234		234		34		4

5306221

Now, we shift 1234 one rank backwards, beneath the upper 2, like this:

5 3 0 6 2 2 1     1 2 3 4

We multiply the lower 1 by the upper 2, 1 times 2; it is 2.

We write 2 corresponding to the lower 1, instead of the zero.

We erase the lower 1, like this:

5 3 2 6 2 2 1       2 3 4

We multiply again the lower 2 by the upper 2, 2 times 2; it is 4.

We write 4 above, corresponding to the lower 2, meaning with the 6; it is 10.

Write a zero instead of the 6 and add 1 after the zero, meaning to the 2; it is like this:

5 3 3 0 2 2 1         3 4

We multiply again the lower 3 by the upper 2, 2 times 3; it is 6.

We write 6 corresponding to the lower 3, meaning with the upper 2; it becomes 8.

We erase the lower 3, like this:

5 3 3 0 8 2 1           4

We multiply again the lower 4 by the upper 2, 2 times 4; it is 8.

We write 8 instead of the upper 2, since we have multiplied the 2 by the whole [multiplicand] and we erase the lower 4, like this:

5 3 3 0 8 8 1

[Illustration of the procedure:]

5306221		5326221		5330221		5330821
1234		234		34		4

5330881

We shift 1234 another rank backwards, in order to multiply it by the first 1, like this:

5 3 3 0 8 8 1       1 2 3 4

We multiply the lower 1 by the upper 1, 1 times 1; it is 1.

We write 1 corresponding to the lower 1, instead of the zero, and erase the lower 1, like this:

5 3 3 1 8 8 1         2 3 4

We multiply again the lower 2 by the upper 1, 2 times 1; it is 2.

We write 2 above the lower 2, meaning with the 8: we add 2 to 8; it is 10.

Write a zero instead of the 8.

We add 1 to the 1; it is 2.

We erase the lower 2, like this:

5 3 3 2 0 8 1           3 4

We multiply again the lower 3 by the upper 1, 3 times 1; it is 3.

We write 3 corresponding to the lower 3, meaning with 8; it is 11.

We erase the 8. We write 1 instead of it and 1 instead of the zero.

We erase the lower 3, like this:

5 3 3 2 1 1 1             4

We multiply also the lower 4 by the upper 1, 4 times 1; it is 4.

We write 4 instead of the corresponding 1, but we do not add 1 to the 4, as I have instructed you before, because we just multiplied 1234 by this 1. This 1 only indicated the number of times, not for adding.

We erase the lower 4, like this:

5 3 3 2 1 1 4

We find that 4321 times 1234 is 5332114, meaning 5 hundred and 33 tens of thousands, 2 thousandths, one hundred and 14.

[Illustration of the procedure:]

5330881		5331881		5332081		5332111
1234		234		34		4

5332114

• Another example: we wish to multiply 209 by 3030.

This is how you write it diagram:

3 0 3 0 2 0 9

We multiply the lower 2 by the upper 3, 2 times 3; it is 6.

We write 6 corresponding to the lower 2 and erase the lower 2.

Erase also the zero before the lower 2 and write a zero before the 6, because we multiply a zero times [zero]; it is a zero, like this:

6 0 3 0 3 0     9

We multiply again 9 by 3 above it, 9 times 3; it is 27.

We write 27 above instead of the last 3; it is 7 instead of the 3 and 2 instead of the zero.

We erase the 9, like this:

6 2 7 0 3 0

[Illustration of the procedure:]

3030		603030		627030
209		9

We shift 209 two ranks backwards, so that the lower 9, which is the first digit in the lower line will correspond the upper 3, like this:

6 2 7 0 3 0     2 0 9

If you want to know why we do not write 209 beneath the upper zero, of course we do so, but in any case it does not yield more than a zero, because if there are digits written and you need to add a zero to them it does not add to them, as will be explained below.

We multiply the lower 2 by the upper 3, 2 times 3; it is 6.

We write 6 corresponding to the lower 2, meaning with the 7; it is 13. and erase the lower 2.

We erase the 7 and write 3 instead of it.

We add 1 to the 2 that follows it, because it is ten, as I wrote above.

We erase 2 and we erase also the zero before the 2.

We do not need to write a zero above, as you already did, since there is already a zero corresponding to it. All the more so if there were a non-zero digit [in the upper result line] corresponding the zero, we would not need not erase this digit in order to write a zero, because, if we erase a non-zero digit, in order to write a zero there, we decrease [the numerical value of the rank], and there is no need [for the zero] to hold the place, since the a non-zero digit is already holding the place.

In short: we never write the lower zero above, unless there is nothing that corresponds it.

But, if there is a zero or a non-zero digit corresponding to the lower zero, we do not need to write a zero [instead], since it is drawn only as a place holder, as I wrote above, because the zero indicates that digit that follows it is of one rank higher than it would have been, had you not written a zero there.

Returning to the above calculation, when you do as I explained, it is like this:

6 3 3 0 3 0         9

We multiply again the lower 9 by the upper 3, 9 times 3; it is 27.

We write 7 instead of the upper 3, then 2 instead of the zero. We erase the lower 9 and also the upper 3, like this:

6 3 3 2 7 0

We find that two hundred and 9 times 3 thousand and 30 is 63 tens of thousands, 3 thousand, two hundred and seventy.

[Illustration of the procedure:]

627030		633030		633270
209		9

• If you wish to know how much are 15 times 1080.

Write 1080 in the first line and 15 beneath it - 5 beneath the 1 and 1 to its left, like this:

1 0 8 0 1 5

Next, multiply the 1 by the entire upper line, then the 5 by the entire upper line:

Say: 1 times 1 is 1.

Write 1 corresponding to the bottom 1 of the multiplier, so that it will be above the 1080, corresponding to the bottom 1. The diagram is like this:

1           1 0 8 0 1 5

Then, multiply 1 times zero; it is zero.

Write a zero before the 1 you wrote above the 1080, which corresponds to the bottom 1 in the first row. Like this:

1 0         1 0 8 0 1 5

Next, multiply 1 times 8; the result is 8.

Write 8 before the zero you wrote in the upper row, corresponding to the zero in the middle row. Like this:

1 0 8       1 0 8 0 1 5

Then, multiply 1 times 0; the result is 0.

Write it before the 8 you wrote in the upper row, which corresponds to the zero in the middle row.

Erase the 1, as it has been multiplied by the whole 1080. It is like this:

1 0 8 0     1 0 8 0   5

You have multiplied the 1 by the whole 1080.

[Illustration of the procedure:]

		1		10		108		1080
1080		1080		1080		1080		1080
15		15		15		15		15

Next, multiply the 5 by the whole 1080.

Start multiplying it by it right above: as you started multiplying the 1 by 1, so multiply the 5.

5 times 1 is 5.

Write 5 corresponding to the 1, above it, that is instead of the zero, like this:

1 5 8 0     1 0 8 0   5

Then, say: 5 times 0 [is 0].

If nothing was written above the zero, you would write a zero there. Now that 8 is written above it, you do not do anything with the zero, because it holds its place.

Then, say: 5 times 8; the result is forty.

Do not write it instead of the zero, because you need the zero, but write 4 above the 8 to the left, then you will find 8 and 4 on top of each other. Erase both of them and write 2 in their place; on the 5 that you wrote to the left write 1; it is 6, like this:

1 6 2 0     1 0 8 0   5

Then, say: 5 times zero is a zero.

Write a zero above the zero. It is like this:

1 6 2 0 0

[Illustration of the procedure:]

1080		1580		1580		1620		16200
1080		1080		1080		1080
15		15		15		15

Understand that the upper row that corresponds to the bottom row is not complete until you multiply all the digits of the multiplier by the whole number.

Checking Methods

• Casting out 9 by 9

First take the number you wanted to multiply in the upper row, cast out the nines and keep the remainder.

If you do not find 9 even once, take it all.

Then, take the bottom number and cast out the nines also.

Take what you have left and multiply it according to the multiplication chapter by what you have left from the upper row after you cast out the nines from it. Cast out the nines from the result you get as well and keep the remainder that is less than 9.

Then, take the [original] result you got when you multiplied the two numbers together and cast out the nines from its as well.

If what remains in your hand is the same as what you have left, know that you calculated correctly, otherwise you must have been mistaken.

Now, we will check the above calculation, so that the one who understands will understand for sure and increase in learning [Proverbs 1, 5]:

• The example: 3030 times 209

First, we sum up 2 times 3, i.e. 6; there is no 9 even once.

Then, we sum up the bottom number, i.e. 209: we sum 9 and 2; it is 11. Subtract 9; 2 remains.

השלך ט' וישאר ב‫'

We multiply 2 by 6: 2 times 6 is 12. Subtract 9; 3 remains. This is the reserved.

השלך ט' וישאר ג' וזה יהיה לך לזכרון

This is the rule: see what is left for you from the product, as now when we were required to multiply 2 times 6, which is 12. Write 12 in digits; it is 12. The excess over 9 is 3.

Then, see the result you got when you multiplied 3030 by 209, i.e. 633270: we sum them up; it is 21. Cast out the nines; 3 remains, which is the same as what you already have left, so the calculation is correct.

נחבר אותם יחד ויהיה כ"א
השלך ט'ט' וישאר ג' כמו שנשאר לך כבר וא"כ החשבון מכוון ודו"ק

Chapter [Five]: Multiplication in a Table [Gelosia]

If you multiply a number of three ranks by a number of three ranks, draw three cells horizontally and vertically in the table.

If you multiply a number of three ranks by two ranks, draw three cells horizontally and two cells vertically.

And vice versa, from top to bottom, one larger and one smaller, provided that each digit is arranged in its place according to the diagram.

• Example: if you wish to know how much is 3 hundred and 21 times 654, written 321 by 654.

Write above the cells, as you see in the table I wrote, and write the multiplicand on the side, one [digit] above the other, as illustrated.

First line

First, multiply 4 time 1; the result is 4. Write it in the half of the first cell, beneath the 1, next to the 4.

Because every cell is divided in half: its upper half is for the units [of the interim product] and the bottom [half] is for the tens.

Then, multiply 4 times 2; the result is 8. Write it in the half of the [second] upper cell, beneath the 2.

Next, multiply 4 times 3; the result is 12. Write it in the upper cell beneath the 3: write 2 in the upper half and 1 in the other half.

Now, erase the 4, because it has been multiplied by all the parts.

Second line

Now, multiply 5 time 1; the result is 5. Write it in the cell next to the 5 beneath the 1.

Then, multiply 5 times 2; the result is 10. Write it in the half of the fifth cell.

Next, multiply 5 times 3; the result is 15. Write it in the sixth cell: 5 in the upper half and 1 in the lower half.

Now, erase the 5.

Third line

Multiply 6 times 1; the result is 6. Write it next to it in the column beneath the 1.

6 times 2 is 12. Write it in the eighth cell.

6 times 3 is 18. Write it in the last cell.

We have multiplied all of it.

I will teach you how much is the total result:

Take the first digit in the first cell, which is 4, and write it in a special place.

Then, calculate the digits on the second diagonal from 8 to 5: add 5 to 8; the result is 13.

Write the 3 next to the 4 that is already written, like this:

3 4

Add the 1, which is ten, to the next diagonal: add it to the 2 at the beginning of the third diagonal; the result is 3. Then, add also the 6 at the end of the diagonal; it is 9.

Write it next to the 4 and the 3, like this:

9 3 4

Then, sum up the digits on the fourth diagonal from the top to the end; the result is 9. Write is next to 9, 3, 4 like this:

9 9 3 4

Sum up the digits on the fifth diagonal; the resulting sum is ten.

I already said that the tens are added to the diagonal below and are not written in the special place.

But, you should write a zero in the place of the ten in order to hold the rank, since there is not even one left to the indicate the rank.

When you add the ten to the 1 on the last diagonal, the result is 2. Write it next to the special 9, 9, 3, 4, like this: 209934.

This is the correct calculation of multiplying 321 by 654.

Its scales are as I wrote above in the multiplication chapter.

• 464 times 464: the result is 215296.

Multiplication - Check

• As 464 times 464 once, which is 215296, or twice: is it correct or not.

Then, take the sum of 4, 6, 4; we find it is 14.

Say: how much greater than 9 is it? We find it is 5. Cast out the 9.

Say: how much is 5 times 5? We find it is 25.

Cast out all the nines that are in 25; you are left with 7.

Then, cast out the nines found in 215296 as well; you are left with 7.

Chapter Six: Division

• If one asks: how many times 7 is in two hundred and 18?

Here is how you do it:

Write two hundred and 18 like this:

2 1 8

The 7 should have been written beneath the 2, in order to divide the 2 by 7, meaning to count how many times 7 is in 2, but 2 cannot be divided by 7. Therefore the 7 is shifted backwards, so that the 2 is in the second rank to it and it becomes twenty-one corresponding to it, like this:

2 1 8   7

Divide twenty-one by 7, meaning count how many times 7 is in twenty-one. We find that there are 3 times 7 in it. Write 3 above the 1 in the upper row, corresponding the 7 in the lower row, like this: