Difference between revisions of "ספר האלזיברא"

Introduction

Definitions of algebraic terms

They wanted to signify two meanings by this word: one whole thing and an unknown thing, which we do not know

• X (variable) = thing

The author asked the grammarians of their language about the meaning of this word and they told him that it indicates a fixed number. They meant by this an unknown fixed number.

• X² = square

• (X²)² square of a square

• X³ = cube

• (X³)³ cube of a cube

• number (constant)

First Section: Preliminary Study of the Foundation of Algebra

1) If you wish to multiply a root of a known number by a root of a number

to bring it closer to perception an example is given:

2) If you wish to multiply a root of a known number by a known number

אחר תכפול המרבע האחר בחברו ושרש העולה הוא מה שרצית לדעת

Example:

Therefore,

3)

Example:

4)

Example:

5)

Example:

according to the mentioned way

The proof for this is clear to the one who understands from the proofs of the previous teachings.

6)

Example:

7)

Example:

8)

Example:

9)

כפלהו בו' יעלה ק"נ והנה שרש ק"נ הוא העולה מכפל מספר ה' בשרש מספר ו‫'

10)

11)

• number × additive = additive:

• number × subtractive = subtractive:

• additive × additive = additive:

• additive × subtractive = subtractive:

• subtractive × subtractive = additive:

ABGD□:

• AB =

• AG =

AB - AH =

AG - AW =

drawing line HZ from point H, parallel to AG, BD

drawing line WC from point W, parallel to AB, GD

These two lines are intersect in ABGD□ at point T

They divide ABGD□ into four areas:

the gnomon of ABGD□ = TG□ + TA□ + TB□

TD□ = the sought-after, because its area is equal to the requested number, that is the product of the mentioned numbers, as one can see

• ABGD□ =

• – (TA□ + TB□) =

• – (TA□ + TG□) =

(TA□ + TB□) + (TA□ + TG□) =

TD□ – TA□ = AD□ – [((TA□ + TB□) + (TA□ + TG□)] =

• TA□ =

TD□ = AD□ – [((TA□ + TB□) + (TA□ + TG□)] + TA□

[TD□ = ]

12)

• the quadrilateral surface that is generated from both segments =

• double the quadrilateral surface that is generated from both segments =

13)

14)

greatest common divisor

finding the measure of each of the :

15)

16)

17)

18)

19)

20)

21)

22)

23)

24)

• line AB is cut randomly at point G

• AZ is cut from line AG, so that it equals the smaller segment GB: AZ = GB

• ZG = AG - AZ = the excess of the larger segment over the smaller segment

constructing:

• AGDH□ = AG²

• GBCW□ = GB²

drawing line ZI from point Z, parallel to AD and GH.

extending line WC straight until it meets line ZI at point K.

GB = AZ

ZB = [GZ + GB = GZ + AZ] = AG the larger segment

BW = GB the smaller segment

BK□ = [ZB × BW] = [AG × GB]

AD = AG

AZ = GB

ZD□ = [AD × AZ] = AG × GB

KB□ + ZD□ = 2·(AG × GB)

KH□ = [2·(AG × GB)] - [AG² + GB²] = ZG²

KC = ZG

HC = ZG

HC = GH - GC = the excess of the larger segment over the smaller segment

AG² + GB² = BK□ + ZD□ + KH□ = (AG × GB) + (AG × GB) + ZG²

• A[B] = the side of a square whose area is

• AG = the side of a square whose area is

• AH□ = AG²

the area of the rectangle encompassed by the two segments =

double the area of the rectangle encompassed by the two segments =

(A[B]² + AG²) - [2·(A[B] × AG)] = = (A[B] - AG)²

Second Section: Algebra

Introduction

The number of units in the thing is equal to the number of things in the square

And to the number of squares in the cube

And to the number of cubes in the square of the square

The six canonical equations

1) Things that are equal to numbers

• Divide a Number Problem: I want to divide the number ten into two parts, so that when the one part is divided by the other part the result is five
  

• the divisor = one thing =

• the dividend = five things, as the result of division =

the sum of the two parts =

2) Squares that are equal to numbers

• Find a Number Problem: I want to find a number such that when its third is subtracted from it the square of the remainder is 20
  

3) Squares that are equal to things / roots

Based on the preliminary rule:

Example:

• Find a Number Problem: I want to find a number such that when its third is subtracted from it the remainder is the root of the original number
  

4) Things / roots and numbers that are equal to squares

Normalization:

= AB =

AB is cut randomly on point Z

AZ =

constructing: ABGD□ = AB²

drawing line ZC from point Z, parallel to AG and BD

AC□ =

each measure of AZ occupies one thing in AC□

ZD□ =

AD□ = AC□ + ZD□

[AB²= = AC□ + ZD□]

AZ =

AZ is halved on point T

ZD□ + TZ² == TB²

• Relying on Euclid, Elements, Book II, proposition 6:

TB =

= AB = TB + AT =

• Find a Number Problem: we want to find a number such that when we add to it 28 the sum is twice its square

Normalization:

5) Squares and numbers that are equal to things

Normalization:

AG =

AG is cut into two equal segments at point Z

• GZ = ½AG =

AG is cut into two unequal segments at point B

• GB =

constructing:

• ABHW□ = AB²

extending line HW to C

CH = CG

drawing line GH

GW□ =

AH□ = ABHW□ + GW□ =

AZ² = BH□ + ZB² =

• Relying on Euclid, Elements, Book II, proposition 5:

ZB² = AZ² – BH□ =

= AB = ZB + AZ =

• How Much Problem: a trader went trading with a certain amount in his hand and he earned six. Then he returned with the amount and earned again as he earned on the first time, and it turned out that he had 27. You want to know how much the original amount was

• Euclid, Elements, Book VI, proposition 17:

Normalization:

6) Squares and things that are equal to numbers

Normalization:

describing:

• AGHD□ = AG²

adding to it:

BGHW□ =

AGHD□ + BGHW□ =

GB = the side of GBHW□ =

finding the side of the square AGHD□ = AG = []

BG is halved at point Z [= Z midpoint of GB]

AZ² = AW□ + GZ² =

• Relying on Euclid, Elements, Book II, proposition 6:

AZ =

= AG = AZ ‒ ½GB = AZ – GZ =

Equations of the higher degrees that can be reduced to the canonical equations of the second degree

7) Cubes that are equal to numbers

Normalization:

The reason is clear

8) Cubes that are equal to things

Normalization:

• Based on proposition 2 above and the preliminary rule:

Example:

9) Squares of squares that are equal to numbers

Normalization:

The reason is clear

10) Squares of squares that are equal to things

Normalization:

• Based on proposition 7 above and the preliminary rule:

Example:

11) Squares of squares that are equal to squares

Normalization:

• Based on proposition 2 above and the preliminary rule:

Example:

12) Squares of squares that are equal to cubes

Normalization:

• Based on proposition 1 above and the preliminary rule:

13) Cubes and squares that are equal to things

Normalization:

• Based on proposition 6 above and the preliminary rule:

14) Cubes and things that are equal to squares

Normalization:

• Based on proposition 5 above and the preliminary rule:

15) Squares and things that are equal to cubes

Normalization:

• Based on proposition 4 above and the preliminary rule:

Epilogue

• Relying on the wisdom of his readers

• The author is busy in other matters

Notes

1. ↑ משלי ד, יא
2. ↑ ישעיהו כח, י
3. ↑ ישעיהו נג, י
4. ↑ איוב לג, כג
5. ↑ בראשית א, ב
6. ↑ תהילים כ, ו
7. ↑ משלי ה, טז
8. ↑ ישעיה יב, ג
9. ↑ שמואל א כב, יד

Additional excerpt

• Exchange Problem - Currencies - a man exchanged 23 peraḥim – some for liṭra of Rome and some for liṭra of Marciana. One peraḥ is worth of liṭra of Marciana twice as much as of liṭra of Rome minus a quarter of a liṭra. It turned out that he had 30 liṭra of Rome and 30 liṭra of Marciana. How many peraḥim did he exchange for liṭra of Rome; how many peraḥim did he exchange for liṭra of Marciana; how many liṭra of Rome and how many liṭra of Marciana is one peraḥ worth?