Difference between revisions of "ספר דיני ממונות"

Prologue

The First Chapter Discusses the Addition of Integers to Each Other, as well as the Addition of Coins and the Scales of the Dinar, Since They Have Another Method

• If a person says: sum three numbers, the first is 223, the second 342 and the third 422.

First, the units should be summed with the units; they are 3, 2, 3; the result is 7. Write it beneath.

Then, we sum the tens, which are 2, 4, 2; it is 8. We write it in the rank of tens.

Then, we add up the hundreds, which are 4, 3, 2; it is 9. We write it in the place of the hundreds.

Hence, the result is 987. Write it as this diagram:

• As if you say: sum 203 with 402 and with 809.

First, we add up the units, which are 9, 2, 3; they are 14. Write 4 in the place of the units and keep 1.

Then, add up all the tens. You find only zeros there. So, we write the 1 we got from the units.

We add up the hundreds, which are 14. We write 4 in the place of the hundreds and 1 in the fourth rank, which is the thousands.

The result is 1414, as this diagram:

• If you wish to add up numbers: the first of which has zeros in the units and the tens and 2 in the hundreds; the second has a zero in the units, 4 in the tens and 2 in the hundreds; the third has 3 in the units, zero in the tens and 4 in the hundreds.

Do as follows:

Start from the units. You find 3 there. Write it in the place of the units.

Then, add up the tens, which are 4. Write it in the place of the tens.

Add up the hundreds, which are 2, 2, 4; the result is 8. Write it in the place of the hundreds.

Therefore, the result is 843, as this diagram:

• If you wish to add up four numbers: the first of which has 4 thousands, 2 hundreds and 4; the second has 2 hundreds and 12; the third has 12; and the fourth has 30.

The units, which are 2, 2, 4, should be summed; they are 8. Write 8 in the place of the units.

Then, add up the tens, which are 3, 1, 1; they are 5. Write it in the rank of tens.

Add up the hundreds, which are 2, 2; they are 4. Write it in the rank of hundreds.

Sum the thousands; they are 4.

The result is 4458, as this diagram:

Addition of Coins

• Example: we wish to sum four numbers, each of which includes four different [kind of] coins:

In the first number there are 2 minyanim, 3 carlini, 4 gerot, and 2 dinar.

In the second number there are 5 minyanim, 3 carlini, 2 gerot, and 1 dinar.

In the third number there are 4 minyanim, 2 carlini, 1 gerah, and 3 dinar.

In the fourth number there are 5 minyanim, 9 carlini, 7 gerot, and 2 dinar.

We start to sum the [dinar], which are 2, 3, 1, 2; they are 8.

Know that 6 dinar are worth one gerah.

So, we convert the 6 into a gerah; 2 remain. Write 2 in the place of the dinar.

Then, we add up the gerot, which are 7, 1, 2, 4; they are 14; plus the 1 we have, they are 15.

Know that ten gerot are worth one carlino.

5 gerot still remain. Write 5 in the place of the gerot.

Then, add up the carlini, which are 9, 2, 3, 3; they are 17; plus the one we have, they are 18.

Know that 10 carlini are worth one ducato.

So, we write 8 carlini and keep the ten, which is one magen.

We add up the minyanim, which are 5, 4, 5, 2; they are 16. We add 1 to them; they are 17.

Therefore, the result is 17 minyanim, 8 carlini, 5 gerot, and 2 dinar, as this diagram:

minyanim	carlini	gerot	dinar
2 5 4 5	3 3 2 9	4 2 1 7	2 1 3 2
17	8	5	2

Check: if you want to check it, cast out all the nines from these four numbers; 1 remains.

Then, cast out all the nines from the resulting number.

Take 5 for every gerah into which you have converted the dinar and add it to the number, i.e. the resulting amount; the remainder is one also, as this diagram.

• If you wish to sum three numbers:

In the first number there are 2 minyanim, 3 carlini, 4 gerot, and 4 dinar.

In the second [number] there are 1 magen, 2 carlini, 3 gerot, and 4 dinar.

In the third number there are 1 magen, 5 carlini, 3 gerot, and 5 dinar.

First, sum the dinar, which are 5, 4, 4; they are 13.

We convert them into 4 gerot; 1 dinar remains.

Then, sum the gerot, which are 3, 3, 4; they are 10. Add 2 to them; they are 12. Write 2.

Then, sum the carlini, which are 5, 2, 3; they are 10. Add 1 to them; they are 11.

Then, sum the minyanim, which are 1, 1, 2; they are 4. We add [1] to them; they are 5.

Hence, the total sum is 5 minyanim, 1 carlino, 2 gerot, and 1 dinar, as this diagram:

minyanim	carlini	gerot	dinar
2 1 1	3 2 5	4 3 3	4 4 5
5	1	2	1

Check: If you want to check it: cast out all the nines above; the remainder is 1.

Then, return to the result and cast out all the nines also.

Add with them 5 for every gerah.

The reason for this is that the 6 dinar written above, i.e. in the numbers above and beneath, are worth 1 gerah.

Therefore, we take five for every [gerah] we received when summing the [gerot].

So, we take ten for the 2 gerot and add it to the result; the remainder is also 1, which is equal to the reserved 1, as this diagram.

The Second Chapter Discusses Subtraction as well as Subtraction of Currencies

• Example: we wish to subtract 122 from 234.

Write them in two lines.

Then, start subtracting from the units. Subtract the 2 below from the 4 above; the remainder is 2.

Subtract the 2 tens below from the 3 tens above; the remainder is 1. Write it below in the place of the tens.

Subtract the hundreds: subtract the hundred below from the 2 hundreds above; the remainder is one.

Hence, 112 still remains, as this diagram:

Check: If you want to check it, add the smaller number to the remainder and the greater number will return.

If you wish, cast out all the nines from the greater number; the remainder is a zero.

Then, cast out all the nines from the smaller number and from the remainder; it is also a zero and it is equal to the reserved zero, as this diagram.

• Example: we wish to add up 2 minyanim, 1 tarini, 8 gerot, and 5 dinar, with 2 minyanim, 2 tari, 12 gerot, and 4 dinar.

First, you should add up the dinar, which are 4, [5]; they are 9.

Write 3 in the place of the dinar and we have one gerah.

Then, we add up the gerot, which are 8, 12, and 1; they are 21.

We convert the 20 into one tari; 1 remains. Write it in the place of the gerot.

Then, add up the tari, which are 2, 1, and 1; they are 4.

Write 4 in the place of the tari.

Then, add up the minyanim, which are 2, and 2; they are 4.

Therefore, the total result is 4 minyanim, 4 tari, 1 gerah, and 3 dinar, as this diagram.

Check: If you wish to check it, cast out all the nines from all the numbers.

Then, cast out all the nines from the result. For every gerah you received from the dinar, take 5. For every tari you received from the gerot, take 1. For every magen you received, take 4. You receive a number equal to the upper number and the answer is a zero.

• Example: we wish to subtract 123 from 245.

Write them in two lines, then draw the third line beneath them.

Start subtracting from the units that are 3. Take 3 from the 5; the remainder is two. Write it on the third line in the place of the units.

Subtract the tens, i.e. 2 from 4; the remainder is 2. Write it below in the place of the [tens]

Subtract the hundreds: subtract 1 from 2; the remainder is 1. Write it below on the third line in the place of the hundreds.

Therefore, the remainder is 122 and this is its diagram:

• Another example: we wish to subtract 207 from 349.

Start from the units and subtract the 7 below from the 9 above; the remainder is 2. We write it below on the third line.

Then, subtract the tens: you find a zero below, i.e. in the bottom number. So, you should say: subtract the zero below from the 4 tens that are above in the greater number; the remainder is 4. Write the 4 below, in the place of the tens.

Subtract the hundreds, meaning, 2 from 3; the remainder is 1. Write it below in the place of the hundreds.

Therefore, the remainder is 142, according to this diagram:

• Another example: we wish to subtract 253 from 304.

First, we subtract 3 from 4; 1 remains. Write it below on the third line.

Then, subtract the tens.

Thus, we want to subtract the 5 tens below from the zero above, but we cannot, because we cannot subtract 5 from a zero. So, we draw a dot beneath, next the 200, and this dot is worth ten above int the place of the zero. Hence, subtract 5 from 10; the remainder is 5. Write it in the place of the tens.

Then, subtract the hundreds, which are two, [from the hundreds] above. But we already said that the dot is worth 1. We add it to the 2; they are 3. Subtract 3 from 3; nothing remains.

Therefore, 51 still remains, as this diagram:

• Another example: we wish to subtract 1403 from 2040.

Write them in two rows.

Then, start from the units and subtract from the zero written above, in the place of the units, the three below corresponding to it: we cannot. So, you must draw a dot beneath the place of the tens, next to the zero. It is worth 1 in its place, i.e. in the place of the zero below, and with the zero above, in the place of the units, it is worth ten. Now, subtract the 3 below from the ten; the remainder is 7. Write it one a third line.

Then, subtract the 1 below, which is the dot, from the 4 above, in the place of the tens; the remainder is 3. Write it in the position of its rank.

Then, subtract the hundreds: subtract the 4 below, in the place of the hundreds, from the zero above, in the place of the hundreds; we cannot. So, we draw a dot there beneath the one. This dot is worth 1 in its place, and with the zero above it is worth ten. We subtract the 4 below from the ten above; the remainder is 6.

Then, we subtract the thousands from the thousands: we subtract 2000 below, with the dot that is worth one, from 200 above; nothing remains.

Therefore, the remainder is 637 as this diagram:

• Example: we wish to subtract 135 from 246.

You should start from the hundreds, which is the last rank. Subtract from the 2, which are the hundreds written above, the 1 below that is also in the place of the hundreds; the remainder is 1. Write it below on the third line.

Then, subtract the 3, which are the tens below, from the 4, which are the tens above; the remainder is 1. Write it on the third line.

Subtract the units: subtract the 5 below, in the place of the units, from the 6 above; the remainder is 1. Write it on the third line.

Therefore, 111 still remains and this is its diagram:

• Example: we wish to subtract 382 from 468.

Start from the hundreds and subtract the 3 below from the 4 above; the remainder is 1.

It was appropriate to write it below on the third line, but since the digit above, in the place of the tens, is smaller than the one below that corresponds it, also in the place of the tens, you shift the one back to the 6 above.

Know that this one that is shifted backwards always equals ten in the preceding rank.

Hence, this one in the [rank of] hundreds equals ten with the 6 above in the place of the tens. With the 10 it is 16.

Then, subtract the 8 below from 16; the remainder is 8. Write this 8 in the place of the tens.

Subtract the units: they are 8 above and 2 below. Subtract 2 from 8; the remainder is 6. Write it below on the third line in the place of the units.

Therefore, the remainder is 86 and this is its diagram:

• Example: we wish to subtract 3456 from 4321.

You should start from the thousands: subtract the bottom 3 from the upper 4; the remainder is 1.

We shift back the 1, because the bottom hundreds exceed the upper [hundreds]. The 1 that is shifted back equals a ten in the place of the hundreds above, which are 3, so they are 13.

We subtract 4 from 13; the remainder is 9.

Yet, you do not write all of it. We write only 8 and keep 1, because the upper number in the place of the tens is less than the one corresponding to it below.

We already said that every one equals ten. You find 2 in the place of the tens above, and with 1 that equals 10 they are 12.

Subtract the 5 in the place of the tens below from 12; the remainder is 7.

Yet, we write only 6 of this 7 on the third line, because we have to subtract the units below that are greater than the units above.

You find one above in the place of the units and with the 1 that was shifted back they are 11.

Subtract 6 that is below in the place of the units from 11; the remainder is 5.

Write all of it, as you do not need to shift back any rank, since we have already returned back as much as we could.

Therefore, the remainder after subtracting is 865, according to this diagram:

• Example: we wish to subtract 2304 from 5060.

Start from the thousands: subtract the bottom 2 from the upper 5; the remainder is 3.

But, we write on the third line only 2 of these 3 and keep the 1.

We shift the 1 to the zero above, in the place of the hundreds; it equals 10 as mentioned above. We subtract from it the 3 below corresponding to it, i.e. in the place of the hundreds also; the remainder is 7.

Write the 7 and do not keep anything, because the upper number, i.e. the rank of tens above, is greater than the one corresponding it below. Meaning, the 6 in the place of the tens above is greater than the zero below in the place of the tens also.

Subtract a zero from 6; the remainder is 6.

Yet, we do not write the 6, but keep 1, because the 4, which is in the place of the units below, is greater than the zero above in the place of the units also. So, write 5.

Shift back the 1; it equals ten. Subtract the 4, which is below, in the rank of units, from 10; 6 remains. Write it in the place of units on the third line.

Therefore, the remainder is 2756 and this is its diagram:

Subtraction of Coins

• Example: we wish to subtract 124 zehuvim, 4 tari, 6 gerot and 3 dinar from 233 zehuvim, 3 tari, 4 gerot and 5 dinar.

You should start with the dinar, which is the smallest [type of] currency:

Subtract from the 5 dinar above the 3 below; 2 remain. Write them one a third line.

Subtract the gerot: the 6 below from the 4 above - we cannot. So, we draw a dot below in the place of the 4 tari.

This dot worth twenty gerot above in the place of the gerot.

We add to them the 4 gerot above; they are 24.

We subtract 6 from them; the remainder is 18. Write them on the third line [in the place of] the gerot.

Subtract the tari: subtract the 4 below from the 3 tari above - we cannot.

The dot there is worth 1; with the 4 they are 5. We cannot subtract 5 from 3. So, we draw a dot below in the place of the units of the zehuvim.

Know that this dot is worth 5 above, in the place of the tari, because 5 tari are worth one zahuv, and in its position it is worth only 1.

We add the 5 to the 3; they 8.

We subtract 5 from them; 3 remain. Write the 3 below in the place of the tari, on the third line.

Subtract the zehuvim: subtract the 5 in the place of the units below from the 3 in the place of the units above - we cannot. So, we draw a dot below in the place of the tens and it is worth ten.

Add the 3 to them; they are 13.

Subtract [5] from them; the remainder is [8]. Write it below on the third line, in the place of the units.

Subtract the tens: the 3 lower with the dot there from the 3 upper. Subtract 3 from 3; the remainder is a zero. Write a zero.

Subtract the hundreds: the [lower] 1 from the [upper] 2; the remainder is 1.

Therefore, the remainder from the subtraction is 108 minyanim, 3 tari, 18 gerot, and 2 dinar, as this diagram:

zehuvim	tarini	gerot	dinar
233 124	3 4	4 6	5 3
108	3	18	2

• As if you say: we wish to subtract 145 zehuvim, without tari, gerot and dinar, from 234 zehuvim, 3 tari, 2 gerot, and 1 dinar.

You should start from the lowest [type of] currency: subtract the zero below from the dinar above; the remainder is 1. We write it on the third line, in the place of the dinar.

Subtract the gerot: the zero below from the 2 gerot above; the remainder is 2. Write it below on the third line.

Subtract the tari: the zero below from the 3 above; the remainder is 3. Write it below on the third line.

Subtract the zehuvim: subtract the 5 below in the place of the units from the 4 above in the place of the units also - we cannot. So, we draw a dot below in the place of the tens.

We add it to the upper 4; they are 14.

We subtract five from them; the remainder is 9.

We subtract the lower five from the 3 above in the place of the tens - we cannot. So, we draw a dot below in the place of the hundreds.

We add it to the upper 3; they are 13.

We subtract 5 from them; the remainder is 8. Write it in the place of the tens.

Subtract the hunderds, which are 2 from 2; nothing remains. Write a zero in their place.

Therefore, 89 zehuvim, 3 tari, 2 gerot, and 1 dinar still remain, as this diagram:

zehuvim	tarini	gerot	dinar
234 145	3 0	2 0	1 0
89	3	2	1

• If you wish to subtract 387 zehuvim, 2 tari, 3 gerot, and 4 dinar from 456 zehuvim without any tari, gerah, or dinar.

You should start from the dinar: subtract the 4 below from the zero above - we cannot. So, we draw a dot below, next to the 3 gerot that are there.

This dot is worth six above in the place of the dinar, because 6 dinar are one gerah.

We subtract 4 from 6; the remainder is 2. Write it on the third line, [in the place of] the dinar.

Subtract the gerot: subtract the 4 below with the dot from the zero above - we cannot. So, we draw a dot below, next to the 2 tari.

This dot is worth twenty in the place of the gerot above, because twenty gerah are one shekel, i.e. tari.

We subtract the 4 from 20 above; the remainder is 16. Write it below [in the place of] the gerot.

Subtract the tari: we subtract three from zero - we cannot. So, we draw a dot below, in the place of the units of the zehuvim.

This dot is worth 5, because 1 zahuv is worth 5 tari.

We subtract 3 from 5; the remainder is 2.

Subtract the zehuvim: we subtract 8 from 6 - we cannot. So, you must draw a dot below, in the place of the tens and it is worth ten.

Add it to the 6; they are 16.

Subtract 8 from them; the remainder is 8. We write it in the place of the units below, on the third line.

Subtract the tens: subtract 9 from 5 - we cannot. So, you must draw a dot below, in the place of the 3 hundreds and it is worth ten.

Add it to the 5; they are 15.

We subtract 9 from them; the remainder is 6. Write it below in the place of the tens.

Therefore, the remainder from the subtraction is 68 minyanim, 2 tari, 16 gerot, and 2 dinar, and this is their diagram:

zehuvim	tarini	gerot	dinar
456 387	0 2	0 3	0 4
68	2	16	2

• Example: we wish to subtract 123 minyanim, 2 tari, 4 gerot, and 3 dinar from 234 minyanim, 4 tari, 5 gerot, and 4 dinar.

You should start from the hundreds of the minyanim, which is the greatest [type of] currency. Subtract the 1 below, in the place of the hundreds, from the 2 above, also in the place of the hundred; the remainder is 1. Write it below, in the place of the hundreds.

Subtract the tens of the minyanim: 2 below from 3 above; the remainder is one. Write it below in the place of the tens.

Subtract the units of the minyanim: the lower [3] from the upper 4; the remainder is 1.

We have completed the calculation of the minyanim. Now, we subtract the other [types of] currencies:

We subtract the lower 2 tari from the upper 4 tari; the remainder is 2. Write it below in the place of the tari.

Then, the gerot: subtract the lower 4 from the upper 5; the remainder is 1. Write it below in the place of the gerot.

Then, subtract the dinar: they are 4 above and 3 below. Subtract 3 from 4; the remainder is 1. Write it below, on the third line, in the place of the dinar. The procedure is completed.

Therefore, the remainder is 111 minyanim, 2 tari, 1 gerah, and 1 dinar, as this diagram:

minyanim	tarini	gerot	dinar
234 123	4 2	5 4	4 3
111	2	1	1

• Example: we wish to subtract 159 minyanim, 4 tari, 5 gerot, and 3 dinar from 246 minyanim, 3 tari, 2 gerot, and 1 dinar.

You should start from the hundreds of the minyanim. Subtract the 1 below from the 2 above; the remainder is 1.

But, do not write anything on the third line, because you need to shift it backwards, since the lower digits are greater than the upper [digits].

We shift the 1 backwards to the tens above, which are 4; they are 14.

We subtract 5 from them; the remainder is 9.

We do not write the 9 below on the third line, but keep 1 to shift it backwards. We write 8 below in the rank of tens.

Then, subtract the units of the minyanim: we subtract the lower 9 from the upper 6 - we cannot. So, we add the 1 in our hand that is worth ten to the 6; they are 16.

We subtract 9 from them; the remainder is 7.

We do not write the 7 below, but we write 6, keep 1 in our hand and shift it backwards.

Then, we subtract the tari: we subtract the lower 4 from the upper 3 - we cannot. So, we shift back the magen in our hand to the 3 tari; they are 8, because the magen is worth 5 tari.

We subtract 4 from them; the remainder is 4.

We do not write the 4, but 3, and keep the 1 in our hand to shift it backwards to the gerot.

Then, we subtract the gerot: 5 from 2 - we cannot. So, we shift back the tari to the 2 gerot; they are 22, because the tari is worth twenty gerah.

We subtract 5 from the 22; the remainder is 17.

We do not write all of it, but 16, and keep the 1.

Then, we subtract the dinar: we subtract the lower 3 from the upper 1 - we cannot. So, we shift back the gerah that we kept to the dinar; they are 7 with the 1 dinar, because the gerah is worth 6 dinar.

We subtract 3 from them; the remainder is 4.

The remainder is 86 minyanim, 3 tari, 16 gerot, and 4 dinar; this is their diagram:

minyanum	tarini	gerot	dinar
246 159	3 4	2 5	1 3
86	3	16	4

• Example: we wish to subtract 250 minyanim, 4 tari, no gerah, and 4 dinar from 347 minyanim, no tari, 4 gerot, and no dinar.

You should start from the last digit of the minyanim, i.e. start from the hundreds and subtract the 2 below in the place of the hundreds from the 3 above also in the place of the hundreds; 1 remains.

[Do not] write it below on the third line, but write a wheel there, i.e. a zero, because we need this 1 very much, for the tens of the minyanim below are greater than the upper tens.

Then, subtract the tens: the bottom 5 from the upper 4. But, we cannot. So, we take the one we kept and shift it to the 4; it is 14.

We subtract 5 from it; the remainder is 9.

We write the 9 in the place of the tens below, because we do not need it, since the upper units of the minyanim are 4 and below there is nothing but a wheel.

We subtract a zero from the 7; the remainder is 7.

But, we do not write this 7 in the place of the units of the minyanim, because we need one of them. So, we write 6 below and keep the one.

Then, we subtract the tari: the 4 below from the zero above. We cannot. So, we shift to it the one we have in our hand, it is 5, because you already know that the magen is worth 5 tari. We subtract 4 from it; the remainder is 1.

We write it below in the place of the tari and do not keep it [...] because you find four above in the place of the gerot and below you find a wheel.

Then, subtract the gerot: subtract the wheel below from the upper 4; the remainder is 4.

You do not write the four below in the place of the gerot, but write 3 and keep the 1.

Then, subtract the dinar: you find a wheel above and below you find 4. You cannot subtract from the wheel, so we add the gerah that is left in your hand to the upper wheel; they are 6, as you already know that the gerah is 6 dinar. Subtract the 4 below from the 6; the remainder is two.

Write it below in the place of the dinar.

Therefore, the remainder is 96 minyanim, 1 tari, 3 gerot, and 2 dinar, as is written here:

minyanim	tarini	gerot	dinar
347 250	0 4	4 0	0 4
96	1	3	2

Chapter Three: Discusses the Multiplication in all Possible Ways Found in Integers

Type one

• Example: we wish to multiply 23 by 2[4].

We write the 23 above and the [2]4 below. If you switch between them it does not matter, because everything leads to same result and there is one way for all.

So, first we multiply the 4 [below] by the 3 [above], which is in the rank of units. We say: 4 times 3 is 12.

We write the 2 below, on a third line, and keep the ten.

Then, we multiply the 4 below, in the place of the units, by the 2 above, in the place of the tens; it is 8.

Add to it the ten you got from the units; it is 9. Write it in the place of the tens.

Remove the first digit, i.e. do not multiply anymore by the 4 below, in the place of the units, because we have multiplied it by all the upper ranks.

Then, multiply the second rank below by all the ranks above.

Multiply it first by 3, i.e. the 2 below, in the place of the tens, by the 3 above, in the place of the units; it is 6.

Write the 6 on another line, which we call the fourth line. Write it in the place of the tens.

Then, multiply the 2 below by the 2 above and say: 2 times 2; it is 4.

Write 4 in the rank that follows the tens, i.e. the hundreds.

Then, add up all; the result is 552 and this is it diagram:

• Example: we wish to multiply 234 by 24.

First you should multiply the 4 below, in the place of the units, by the 4 above, in the place of the units also; it is 16.

Write 6 in the place of the units, on a third line, and keep the 10.

Multiply again the 4 in the place of the units below by the 3 above, in the place of the tens; the result is 12. Add to it the ten you have; it is 13.

Write 3 in the place of the tens, on the third line, and keep the ten.

Multiply again the lower 4 by the 2 above, in the place of the hundreds; it is 8. Add to it the ten you have; it is 9.

The multiplication by the first digit below has completed.

Now, we turn to multiply the second digit, in the rank of the tens, by all the upper digits.

We multiply the 2 below, in the rank of tens, by the 4 above, in the rank of units; it is 8.

Write the 8 on a fourth line corresponding to the tens.

Multiply again the 2 below by the 3 above, in the place of the tens; it is 6.

Write it on the fourth line, in the place of the hundreds.

Multiply again the 2 below by the 2 above, in the rank of hundreds; it is 4.

Write the 4 on the fourth line, in the place of the thousands.

Then, add up all these products; the result is 5616, as this diagram:

• Example: we wish to multiply 222 by 333.

Write them in two rows.

Start multiplying the 2 below, in the place of the units, by the 3 above, in the place of the units also; the result is 6.

Write it on a third line, in the place of the units.

Multiply the same 2, i.e. which is below, in the rank of units, by the tens above; it is 6.

Write it on the third line, in the place of the tens.

Multiply the 2 [below] also by the 3 above, in the place of the hundreds; it is also 6.

Write it in the place of the hundreds.

We have completed all the multiplications by the first lower digit.

Now, we want to multiply the second digit, which is 2, in the place of the tens, by all the upper digits.

First, multiply the 2 by the 3 above, in the place of the units; it is 6.

Write it on a fourth line, in the place of the tens.

Multiply again the 2 [below] by the 3 above, in the place of the tens; it is 6.

Write it on the fourth line, in the place of the hundreds.

Multiply again the same 2 by the 3 above, in the place of the hundreds; it is 6.

Write it on the fourth line, in the place of the thousands.

We have completed all the multiplications of the second lower digit by all the upper digits.

Multiply again the third rank, i.e. the third digit below, by all the upper digits.

First, multiply the 2 below by the 3 above, in the place of the units; it is 6.

Write it on a fifth line, in the place of the hundreds.

Multiply again the 2 by the 3 above, in the place of the tens; it is 6.

Write it on the fifth line, in the place of the thousands.

Multiply again the 2 below by the 3 above, in the place of the hundreds; it is 6.

Write it on the fifth line, in the place of the ten thousands.

All nine multiplications have been completed.

Then, add up all these products, each [rank with its corresponding rank]; the result is 73926, according to this diagram:

• Example: we wish to multiply 230 by 324.

First, you should multiply the 4 below, in the rank of units, by the zero above, in the rank of units also. Say: 4 times a zero is zero.

Write a zero in the place of the units.

Multiply again the 4 below by the 3 above, in the place of the tens; the result is 12.

Write 2 in the place of the tens and keep the ten.

Multiply again the 4 by the 2 above, in the place of the hundreds; the result is 8.

Add to it the ten you already have; the result is 9

Write the 9 in the rank of hundreds, on the third line.

Then, [multiply] the 2 below, in the place of the tens, because we have already completed the multiplication of the first lower digit by all the upper digits.

Multiply the 2 by the zero above, in the place of the units; the result is a zero.

Write a zero on a fourth line, in the place of the tens.

Multiply also the lower 2 by the [3] above, in the place of the tens; the result is 6.

Write it corresponding to the hundreds on the fourth line.

Multiply again the lower 2 by the 2 in the place of the hundreds; the result is 4.

Write it on the fourth line, in the place of the thousands.

The multiplication of the digit 2 below has been completed.

Then, multiply the third digit below by all the upper digits.

First, multiply it by the zero; the result is a zero.

Write a zero on a fifth line, in the place of the hundreds.

Multiply also the 3 below by the 3 above, in the second rank; it is 9.

Write the 9 in the rank of thousands, on the fifth line.

Multiply the 3 a third time by the 2 above, in the place of the hundreds; it is 6.

Write it in the place of the ten thousands, on the fifth line.

The multiplication by all three digits has been completed.

Add up all together; the total result is 74520 and this is its diagram.

• Example: we wish to multiply 240 by 368.

First, multiply the 8 below by the zero above, in the place of the units; the result is a zero.

Multiply again the lower 8, in the rank of units, by the 4 in the rank of tens; it is 32.

Write the 2 in the place of the tens and keep the 30.

Multiply the 8 by the 2 above in the rank of hundreds; it is 16.

Add the 3 to it; it is 19.

Write it on the third line.

We have already completed the multiplication of the 8 below, in the place of the units.

Then, multiply the tens below, which is 6:

First, multiply it by the zero above; the result is a zero.

Write a zero on a fourth line, in the place of the tens.

Multiply also the lower tens by the 40 above; the result is 24.

Write the 2 in the place of the hundreds and keep the 20.

Multiply the 6 below by the 2 above; the result is 12.

Add the 2 to it; it is 14.

Then, we multiply the hundreds below by all the digits above:

First, we multiply by the zero above, in the place of the units; it is a zero.

We multiply it by the 4 above, in the rank of tens; it is 12.

Write the 2 in the place of the thousands.

We multiply it by the hundreds above; the result is 6.

Add the one to it; it is 7.

Therefore the result of multiplying the one by the other is 88320 as this diagram:

• Example: we wish to multiply 403 by 230.

First, you should multiply the zero below, in the rank of units, by the 3 above, in the rank of units also; the result is a zero.

Write it below, in the rank of units, on a third line.

Multiply also the zero by the zero above; the result is a zero.

Write it on the third line, in the rank of tens.

Multiply the zero by the upper 4 in the place of the hundreds; the result is a zero.

Write a zero on the third line, in the rank of hundreds.

We have already completed the multiplications of the zero in the lower line by all the upper ranks.

Then, we multiply the 3 below, in the rank of tens, by all the upper ranks:

First, we multiply it by the 3 above, in the rank of units; the result is nine.

Write it on a fourth line, in the rank of tens.

Multiply also the 3 below by the zero above, in the place of the tens; the result is a zero.

Write it in the rank of the hundreds.

Multiply again the 3 below by the upper 4, in the place of the hundreds; the result is 12.

Write the 10 in the rank of the tens of thousands and the 2 in the place of the units of the thousands.

We have completed the multiplication of the 3 below, in the place of the tens, by all the upper ranks.

Now, we multiply the 2 below, in the place of the hundreds, by all the upper ranks:

First, we multiply the 2 by the 3 above, in the place of the units; the result is 6.

We write it on a fifth line, in the place of the hundreds.

Multiply the 2 by the zero above, in the place of the tens; the result is a zero.

Write the zero in the place of the units of the thousands.

Multiply the 2 a third time by the 4 above, in the place of the hundreds; the result is 8.

Write the 8 below on the fifth line, in the place of the tens of the thousands.

We have completed all 9 multiplications.

We add up all; the result is 92690 and this is its diagram:

Type two

• Example: we wish to multiply 24 by 25.

[Illustration of the procedure:]

25		25		25		25
24		24		24		24
		0		0		00

	25
	24
	600

Write them in two rows:

Start multiplying the 4 below in the place of the units by the 5 above in the place of the units also; it is twenty.

Write a zero in the place of the units, because you have nothing left that exceeds the tens.

Multiply again the 4 below in the place of the units by the 2 above in the place of the tens; it is 8.

Add to it the 2 tens you kept; they are 10.

Do not write anything, but multiply the 2 below in the place of the tens by the 5 above in the place of the units; the result is ten.

Add the 10 units to it; it is 20.

Since you have nothing left that exceeds the tens, we write a zero below in the place of the tens and keep the 2.

Multiply again the 2 below in the rank of tens by the 2 above in the rank of tens; it is 4.

Add to it the 2 you have; it is 6.

Write it in the place of the hundreds.

Therefore, the result is 600, as this diagram:

• Example: we wish to multiply 123 by 456.

[Illustration of the procedure:]

123		123		123
456		456		456
		8		88

First, we multiply the 6 below in the rank of units by the 3 above also in the rank of units; it is 18.

We write 8 below on a third line and keep the one.

Multiply again the 6 below in the place of the units by the 2 above in the place of the tens; it is 12.