Difference between revisions of "ספר האלזיברא"

:The root of 63 is the required.

:The root of 63 is the required.

:<math>\scriptstyle{\color{blue}{\sqrt{7}\times3=\sqrt{63}}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{7}\times3=\sqrt{63}}}</math>

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:<math>\scriptstyle{\color{blue}{\left[ABGD\right]=TG+TA+TB}}</math>

:<math>\scriptstyle{\color{blue}{\left[ABGD\right]=TG+TA+TB}}</math>

|style="text-align:right;"|לשטח ט"ג ושטח ט"א ושטח ט"ב שלשתם יחד נקראם {{#annot:term|815,1823|9D4Q}}רושם התמונה{{#annotend:9D4Q}}<br>

|style="text-align:right;"|לשטח ט"ג ושטח ט"א ושטח ט"ב שלשתם יחד נקראם {{#annot:term|815,1823|9D4Q}}רושם התמונה{{#annotend:9D4Q}}<br>

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:You should add it to the reserved in order to complete the required surface TD.

:You should add it to the reserved in order to complete the required surface TD.

:<math>\scriptstyle{\color{blue}{TD=AD-\left[\left(TA+TB\right)+\left(TA+TG\right)\right]+TA}}</math>

:<math>\scriptstyle{\color{blue}{TD=AD-\left[\left(TA+TB\right)+\left(TA+TG\right)\right]+TA}}</math>

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:{{#annot:√12+√48|738|PyOq}}12) If you wish to add a root of 12 to a root of 48, for example.

:{{#annot:√12+√48|738|PyOq}}12) If you wish to add a root of 12 to a root of 48, for example.

:<math>\scriptstyle\sqrt{12}+\sqrt{48}</math>

:<math>\scriptstyle\sqrt{12}+\sqrt{48}</math>

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:Add to it the two squares, which are 12 and 48; the result is 108.

:Add to it the two squares, which are 12 and 48; the result is 108.

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:The root of 108 is the required.

:The root of 108 is the required.

:<math>\scriptstyle{\color{blue}{\sqrt{12}+\sqrt{48}=\sqrt{12+48+48}=\sqrt{108}}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{12}+\sqrt{48}=\sqrt{12+48+48}=\sqrt{108}}}</math>

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:<span style=color:Green>Geometric proof (no figure is given)</span>: Its proof is that we attach the side of a square of 12 with the side of a square of 48 straightly, so they are two parts of one straight line.

:<span style=color:Green>Geometric proof (no figure is given)</span>: Its proof is that we attach the side of a square of 12 with the side of a square of 48 straightly, so they are two parts of one straight line.

:<math>\scriptstyle{\color{blue}{\sqrt{12}+\sqrt{48}}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{12}+\sqrt{48}}}</math>

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:{{#annot:reference|252|Eeue}}It was already clarified in [[ספר_היסודות_לאקלידס#Elements_II_4|<span style=color:blue>'''Euclid, Elements, Book II, proposition 4'''</span>]] that:

:{{#annot:reference|252|Eeue}}It was already clarified in [[ספר_היסודות_לאקלידס#Elements_II_4|<span style=color:blue>'''Euclid, Elements, Book II, proposition 4'''</span>]] that:

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:{{#annot:definition|252|JPnC}}When a straight line is cut randomly into two segments, the square on the whole line equals the sum of the two squares that are generated from the two segments plus twice the rectangle encompassed by the two segments.

:{{#annot:definition|252|JPnC}}When a straight line is cut randomly into two segments, the square on the whole line equals the sum of the two squares that are generated from the two segments plus twice the rectangle encompassed by the two segments.

:<math>\scriptstyle{\color{OliveGreen}{\left(a+b\right)^2=a^2+b^2+2ab}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\left(a+b\right)^2=a^2+b^2+2ab}}</math>

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:When we multiply the two squares by each other, the resulting root, which is 576, is equal to the right-angled surface formed by the two segments.

:When we multiply the two squares by each other, the resulting root, which is 576, is equal to the right-angled surface formed by the two segments.

:<math>\scriptstyle{\color{blue}{\sqrt{12\sdot48}=\sqrt{576}}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{12\sdot48}=\sqrt{576}}}</math>

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:When we take its double we get double the right-angled surface that is encompassed by both segments and when we add the two squares to it, we get the square of the whole line, whose root is the sought-after.

:When we take its double we get double the right-angled surface that is encompassed by both segments and when we add the two squares to it, we get the square of the whole line, whose root is the sought-after.

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:Multiply 8 by 19; the result is 152, which has no root.

:Multiply 8 by 19; the result is 152, which has no root.

:<math>\scriptstyle{\color{blue}{8\sdot19=152}}</math>

:<math>\scriptstyle{\color{blue}{8\sdot19=152}}</math>

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:<math>\scriptstyle{\color{blue}{2\sdot\sqrt{152}=\sqrt{4\sdot152}=\sqrt{608}}}</math>

:<math>\scriptstyle{\color{blue}{2\sdot\sqrt{152}=\sqrt{4\sdot152}=\sqrt{608}}}</math>

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:Extract the cube root of 8 parts of 27; it is 2-thirds.

:Extract the cube root of 8 parts of 27; it is 2-thirds.

:<math>\scriptstyle{\color{blue}{\sqrt[3]{\frac{8}{27}}=\frac{2}{3}}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt[3]{\frac{8}{27}}=\frac{2}{3}}}</math>

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:<math>\scriptstyle{\color{blue}{5=\sqrt[3]{125}}}</math>

:<math>\scriptstyle{\color{blue}{5=\sqrt[3]{125}}}</math>

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:Take one part of the five mentioned parts; it is half a cube root of 96.

:Take one part of the five mentioned parts; it is half a cube root of 96.

:<math>\scriptstyle{\color{blue}{\frac{1}{3}\sdot\sqrt[3]{324}=\frac{1}{2}\sdot\sqrt[3]{96}}}</math>

:<math>\scriptstyle{\color{blue}{\frac{1}{3}\sdot\sqrt[3]{324}=\frac{1}{2}\sdot\sqrt[3]{96}}}</math>

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:So, the cube of one part is an eighth of 96; which is 12.

:So, the cube of one part is an eighth of 96; which is 12.

:<math>\scriptstyle{\color{blue}{\frac{1}{3}\sdot\sqrt[3]{324}=\sqrt[3]{\frac{1}{8}\sdot96}=\sqrt[3]{12}}}</math>

:<math>\scriptstyle{\color{blue}{\frac{1}{3}\sdot\sqrt[3]{324}=\sqrt[3]{\frac{1}{8}\sdot96}=\sqrt[3]{12}}}</math>

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:The cube root of 1500 is the required.

:The cube root of 1500 is the required.

:<math>\scriptstyle{\color{blue}{\sqrt[3]{96}+\sqrt[3]{324}=\sqrt[3]{125\sdot12}=\sqrt[3]{1500}}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt[3]{96}+\sqrt[3]{324}=\sqrt[3]{125\sdot12}=\sqrt[3]{1500}}}</math>

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:According to my way, when "I have led you in paths of uprightness" to calculate this calculation, "I have taught you" the proof "in the way of wisdom" [Proverbs 4, 11].

:According to my way, when "I have led you in paths of uprightness" to calculate this calculation, "I have taught you" the proof "in the way of wisdom" [Proverbs 4, 11].

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:{{#annot:√30÷√6|740|4Ayu}}15) If you wish to divide a root of 30 by a root of 6.

:{{#annot:√30÷√6|740|4Ayu}}15) If you wish to divide a root of 30 by a root of 6.

:<math>\scriptstyle\sqrt{30}\div\sqrt{6}</math>

:<math>\scriptstyle\sqrt{30}\div\sqrt{6}</math>

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:Divide 30 by 6; the result is 5 and its root is the required.

:Divide 30 by 6; the result is 5 and its root is the required.

:<math>\scriptstyle{\color{blue}{\frac{\sqrt{30}}{\sqrt{6}}=\sqrt{\frac{30}{6}}=\sqrt{5}}}</math>

:<math>\scriptstyle{\color{blue}{\frac{\sqrt{30}}{\sqrt{6}}=\sqrt{\frac{30}{6}}=\sqrt{5}}}</math>

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:This is because {{#annot:definition|811|QcHW}}the ratio of a square to a square is as the ratio of its side to its side duplicated. [ [[ספר_היסודות_לאקלידס#Elements_VIII_11|<span style=color:blue>'''Euclid, Elements, Book VIII, proposition 11'''</span>]] ]

:This is because {{#annot:definition|811|QcHW}}the ratio of a square to a square is as the ratio of its side to its side duplicated. [ [[ספר_היסודות_לאקלידס#Elements_VIII_11|<span style=color:blue>'''Euclid, Elements, Book VIII, proposition 11'''</span>]] ]

:<math>\scriptstyle{\color{OliveGreen}{a^2:b^2=\left(a:b\right)^2}}</math>

:<math>\scriptstyle{\color{OliveGreen}{a^2:b^2=\left(a:b\right)^2}}</math>

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:<math>\scriptstyle{\color{blue}{\frac{20}{\sqrt{10}}=\sqrt{\frac{20^2}{10}}=\sqrt{\frac{400}{10}}=\sqrt{40}}}</math>

:<math>\scriptstyle{\color{blue}{\frac{20}{\sqrt{10}}=\sqrt{\frac{20^2}{10}}=\sqrt{\frac{400}{10}}=\sqrt{40}}}</math>

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|I present a teaching, if you look carefully, you will understand from it the proof of the two teachings that follow it:

|I present a teaching, if you look carefully, you will understand from it the proof of the two teachings that follow it:

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:{{#annot:(√8+√4)×(√8-√4)|737|Gjqg}}17) If you wish to multiply a root of 8 minus a root of 4 by a root of 8 plus a root of 4, for example.

:{{#annot:(√8+√4)×(√8-√4)|737|Gjqg}}17) If you wish to multiply a root of 8 minus a root of 4 by a root of 8 plus a root of 4, for example.

:<math>\scriptstyle\left(\sqrt{8}-\sqrt{4}\right)\times\left(\sqrt{8}+\sqrt{4}\right)</math>

:<math>\scriptstyle\left(\sqrt{8}-\sqrt{4}\right)\times\left(\sqrt{8}+\sqrt{4}\right)</math>

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:Subtract 4 from 8; 4 remains and the remaining 4 is the required.

:Subtract 4 from 8; 4 remains and the remaining 4 is the required.

:<math>\scriptstyle{\color{blue}{\left(\sqrt{8}-\sqrt{4}\right)\times\left(\sqrt{8}+\sqrt{4}\right)=8-4=4}}</math>

:<math>\scriptstyle{\color{blue}{\left(\sqrt{8}-\sqrt{4}\right)\times\left(\sqrt{8}+\sqrt{4}\right)=8-4=4}}</math>

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:Now, we subtract 4 plus a root of 32 that should be subtracted from the reserved 8 plus a root of 32; the remainder is 4 as we have said and this is the sought-after, or we can say that a root of 16 is the required.

:Now, we subtract 4 plus a root of 32 that should be subtracted from the reserved 8 plus a root of 32; the remainder is 4 as we have said and this is the sought-after, or we can say that a root of 16 is the required.

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:18) If you wish to multiply a root of 8 minus a root of 4 by two other roots, so that the result is a root of 64, not a root of 16.

:18) If you wish to multiply a root of 8 minus a root of 4 by two other roots, so that the result is a root of 64, not a root of 16.

:<math>\scriptstyle\left(\sqrt{8}-\sqrt{4}\right)\times\left(\sqrt{a}+\sqrt{b}\right)=\sqrt{64}</math>

:<math>\scriptstyle\left(\sqrt{8}-\sqrt{4}\right)\times\left(\sqrt{a}+\sqrt{b}\right)=\sqrt{64}</math>

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:<math>\scriptstyle{\color{blue}{4\sdot8=32}}</math>

:<math>\scriptstyle{\color{blue}{4\sdot8=32}}</math>

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:Multiply it also by 4; the result is 16.

:Multiply it also by 4; the result is 16.

:<math>\scriptstyle{\color{blue}{4\sdot4=16}}</math>

:<math>\scriptstyle{\color{blue}{4\sdot4=16}}</math>

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:Hence, it should be multiplied by a root of 32 plus a root of 16, so the result is a root of 64 and this is self-explanatory.

:Hence, it should be multiplied by a root of 32 plus a root of 16, so the result is a root of 64 and this is self-explanatory.

:<math>\scriptstyle{\color{blue}{\sqrt{a}+\sqrt{b}=\sqrt{32}+\sqrt{16}}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{a}+\sqrt{b}=\sqrt{32}+\sqrt{16}}}</math>

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:Multiply it also by 4, whose root you want to subtract from a root of 8; the result is 16.

:Multiply it also by 4, whose root you want to subtract from a root of 8; the result is 16.

:<math>\scriptstyle{\color{blue}{4\sdot4=16}}</math>

:<math>\scriptstyle{\color{blue}{4\sdot4=16}}</math>

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:So, a root of 32 plus a root of 16 summed together is the sought-after.

:So, a root of 32 plus a root of 16 summed together is the sought-after.

:<math>\scriptstyle{\color{blue}{\sqrt{64}\div\left(\sqrt{8}-\sqrt{4}\right)=\sqrt{32}+\sqrt{16}}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{64}\div\left(\sqrt{8}-\sqrt{4}\right)=\sqrt{32}+\sqrt{16}}}</math>

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:<math>\scriptstyle{\color{OliveGreen}{\frac{a}{b}\sdot b=a}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\frac{a}{b}\sdot b=a}}</math>

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:So, a root of 32 minus a root of 16 is the result of division.

:So, a root of 32 minus a root of 16 is the result of division.

:<math>\scriptstyle{\color{blue}{\sqrt{64}\div\left(\sqrt{8}+\sqrt{4}\right)=\sqrt{32}-\sqrt{16}}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{64}\div\left(\sqrt{8}+\sqrt{4}\right)=\sqrt{32}-\sqrt{16}}}</math>

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:You can see that this teaching follows the technique of the previous teaching exactly.

:You can see that this teaching follows the technique of the previous teaching exactly.

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:Only that instead of your saying in the previous teaching that the sought-after is a root of 32 plus a root of 16 summed together, in the present teaching you say that it is a root of 32 minus a root of 16.

:Only that instead of your saying in the previous teaching that the sought-after is a root of 32 plus a root of 16 summed together, in the present teaching you say that it is a root of 32 minus a root of 16.

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:Square 2; it is 4.

:Square 2; it is 4.

:<math>\scriptstyle{\color{blue}{2^2=4}}</math>

:<math>\scriptstyle{\color{blue}{2^2=4}}</math>

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:Cube 6; it is 216.

:Cube 6; it is 216.

:<math>\scriptstyle{\color{blue}{6^3=216}}</math>

:<math>\scriptstyle{\color{blue}{6^3=216}}</math>

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:The square root of the cube root of 2 and 4 parts of 25 is the required.

:The square root of the cube root of 2 and 4 parts of 25 is the required.

:<math>\scriptstyle{\color{blue}{\sqrt{6}\div\sqrt[3]{10}=\sqrt{\sqrt[3]{2+\frac{4}{25}}}}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{6}\div\sqrt[3]{10}=\sqrt{\sqrt[3]{2+\frac{4}{25}}}}}</math>

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:Square 5 twice; it is 625.

:Square 5 twice; it is 625.

:<math>\scriptstyle{\color{blue}{\left(5^2\right)^2=625}}</math>

:<math>\scriptstyle{\color{blue}{\left(5^2\right)^2=625}}</math>

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:Divide 625 by 51[2]; the result is one and 113 parts of 51[2].

:Divide 625 by 51[2]; the result is one and 113 parts of 51[2].

:<math>\scriptstyle{\color{blue}{\frac{625}{51{\color{red}{2}}}=1+\frac{113}{51{\color{red}{2}}}}}</math>

:<math>\scriptstyle{\color{blue}{\frac{625}{51{\color{red}{2}}}=1+\frac{113}{51{\color{red}{2}}}}}</math>

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:The square root of the square root of the cube root of 1 and 113 parts of 51[2] is the required.

:The square root of the square root of the cube root of 1 and 113 parts of 51[2] is the required.

:<math>\scriptstyle{\color{blue}{\sqrt[3]{5}\div\sqrt{\sqrt{8}}=\sqrt{\sqrt{\sqrt[3]{1+\frac{113}{51{\color{red}{2}}}}}}}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt[3]{5}\div\sqrt{\sqrt{8}}=\sqrt{\sqrt{\sqrt[3]{1+\frac{113}{51{\color{red}{2}}}}}}}}</math>

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:Subtract 24 from 26; 2 remains and the root of 2 is what you want.

:Subtract 24 from 26; 2 remains and the root of 2 is what you want.

:<math>\scriptstyle{\color{blue}{\sqrt{18}-\sqrt{8}=\sqrt{26-24}=\sqrt{2}}}</math>

:<math>\scriptstyle{\color{blue}{\sqrt{18}-\sqrt{8}=\sqrt{26-24}=\sqrt{2}}}</math>

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:{{#annot:definition|255|cvaS}}When a straight line is cut randomly into two segments, the sum of the squares on both segments equals twice the rectangle encompassed by both segments plus the square that is generated from the excess of the larger segment over the smaller segment. [ [[ספר_היסודות_לאקלידס#Elements_II_7|<span style=color:blue>'''Euclid, Elements, Book II, proposition 7'''</span>]] ]

:{{#annot:definition|255|cvaS}}When a straight line is cut randomly into two segments, the sum of the squares on both segments equals twice the rectangle encompassed by both segments plus the square that is generated from the excess of the larger segment over the smaller segment. [ [[ספר_היסודות_לאקלידס#Elements_II_7|<span style=color:blue>'''Euclid, Elements, Book II, proposition 7'''</span>]] ]

:<math>\scriptstyle{\color{OliveGreen}{a^2+b^2=2ab+\left(a-b\right)^2}}</math>

:<math>\scriptstyle{\color{OliveGreen}{a^2+b^2=2ab+\left(a-b\right)^2}}</math>

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!<span style=color:Green>Geometric illustration</span>

!<span style=color:Green>Geometric illustration</span>

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:Let line AB be cut randomly at point G.

:Let line AB be cut randomly at point G.

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:We also cut segment AZ from line AG, so that it equals the smaller segment GB.  

:We also cut segment AZ from line AG, so that it equals the smaller segment GB.  

:<math>\scriptstyle{\color{blue}{AZ=GB}}</math>

:<math>\scriptstyle{\color{blue}{AZ=GB}}</math>

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:<span style=color:Green>Supposition:</span> I say that double the right-angled surface encompassed by lines AG and GB, summed with the square that is formed by ZG, equals the two squares that are formed by AG and GB, when they are summed together.

:<span style=color:Green>Supposition:</span> I say that double the right-angled surface encompassed by lines AG and GB, summed with the square that is formed by ZG, equals the two squares that are formed by AG and GB, when they are summed together.

:<math>\scriptstyle{\color{blue}{\left[2\sdot\left(AG\times GB\right)\right]+ZG^2=AG^2+GB^2}}</math>

:<math>\scriptstyle{\color{blue}{\left[2\sdot\left(AG\times GB\right)\right]+ZG^2=AG^2+GB^2}}</math>

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:We construct square AGDH from line AG, and square GBKW from line GB.

:We construct square AGDH from line AG, and square GBKW from line GB.

:<math>\scriptstyle{\color{blue}{AGDH=AG^2\quad GBKW=GB^2}}</math>

:<math>\scriptstyle{\color{blue}{AGDH=AG^2\quad GBKW=GB^2}}</math>

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:We draw line ZI from point Z, parallel to AD and GH.

:We draw line ZI from point Z, parallel to AD and GH.

:<math>\scriptstyle{\color{blue}{ZI\parallel AD,\;GH}}</math>

:<math>\scriptstyle{\color{blue}{ZI\parallel AD,\;GH}}</math>

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:We extend line WK straight until it meets line ZI at point C.

:We extend line WK straight until it meets line ZI at point C.

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:Therefore, surface BC is equal to the right-angled surface encompassed by the two lines AG and GB that are the two parts of the whole line.

:Therefore, surface BC is equal to the right-angled surface encompassed by the two lines AG and GB that are the two parts of the whole line.

:<math>\scriptstyle{\color{blue}{BC=\left[ZB\times BW\right]=\left(AG\times GB\right)}}</math>

:<math>\scriptstyle{\color{blue}{BC=\left[ZB\times BW\right]=\left(AG\times GB\right)}}</math>

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:So, the two surfaces CB and ZD are equal to double the right-angled surface encompassed by the two lines AG and GB.

:So, the two surfaces CB and ZD are equal to double the right-angled surface encompassed by the two lines AG and GB.

:<math>\scriptstyle{\color{blue}{CB+ZD=2\sdot\left(AG\times GB\right)}}</math>

:<math>\scriptstyle{\color{blue}{CB+ZD=2\sdot\left(AG\times GB\right)}}</math>

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:Because, line CK is equal to line ZG; and line HK, which is its other side is equal to line ZG.

:Because, line CK is equal to line ZG; and line HK, which is its other side is equal to line ZG.

:<math>\scriptstyle{\color{blue}{CK=ZG,\;HK=ZG}}</math>

:<math>\scriptstyle{\color{blue}{CK=ZG,\;HK=ZG}}</math>

וקו ה"ח שהוא צלעו השני שוה לקו ז"ג

וקו ה"ח שהוא צלעו השני שוה לקו ז"ג

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:Therefore, the two squares formed by AG and GB summed together are equal to the two surfaces BC and ZD, each of which equals the surface encompassed by lines AG and GB that are the two segments of the line, summed with CH that is equal to the square formed by ZG, which is the excess of the larger segment over the smaller segment.

:Therefore, the two squares formed by AG and GB summed together are equal to the two surfaces BC and ZD, each of which equals the surface encompassed by lines AG and GB that are the two segments of the line, summed with CH that is equal to the square formed by ZG, which is the excess of the larger segment over the smaller segment.

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:Which is the square AH.

:Which is the square AH.

:<math>\scriptstyle{\color{blue}{AH=AG^2=8}}</math>

:<math>\scriptstyle{\color{blue}{AH=AG^2=8}}</math>

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:When we multiply 8 by 18, the root of the product is equal to the area of the rectangle encompassed by the two segments.

:When we multiply 8 by 18, the root of the product is equal to the area of the rectangle encompassed by the two segments.

:<math>\scriptstyle{\color{blue}{AB\times AG=\sqrt{8\times18}}}</math>

:<math>\scriptstyle{\color{blue}{AB\times AG=\sqrt{8\times18}}}</math>

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:When we take its double, we receive double the area of the rectangle encompassed by the two segments.

:When we take its double, we receive double the area of the rectangle encompassed by the two segments.

:<math>\scriptstyle{\color{blue}{2\sdot\left(AB\times AG\right)=2\sqrt{8\times18}}}</math>

:<math>\scriptstyle{\color{blue}{2\sdot\left(AB\times AG\right)=2\sqrt{8\times18}}}</math>

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:When we subtract it from 26, which is [the sum of] the areas of the two square, we are left with the square of the excess of the greater [segment] over the smaller [segment].

:When we subtract it from 26, which is [the sum of] the areas of the two square, we are left with the square of the excess of the greater [segment] over the smaller [segment].

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|Before I begin, I offer a clarified introduction:

|Before I begin, I offer a clarified introduction:

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*I say that you must learn and know that the ratio of a square of a square to the cube is the same as the ratio of the cube to the square; and as the ratio of the square to the thing; and as the ratio of the thing to the unit

*I say that you must learn and know that the ratio of a square of a square to the cube is the same as the ratio of the cube to the square; and as the ratio of the square to the thing; and as the ratio of the thing to the unit

:<math>\scriptstyle{\color{OliveGreen}{\left(x^2\right)^2:x^3=x^3:x^2=x^2:x=x:1}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\left(x^2\right)^2:x^3=x^3:x^2=x^2:x=x:1}}</math>

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:This is because the number of units in the thing is the same as the number of things in the square; and as the number of squares in the cube; and as the number of cubes in the square of the square

:This is because the number of units in the thing is the same as the number of things in the square; and as the number of squares in the cube; and as the number of cubes in the square of the square

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|You should keep this introduction in mind, because you will need it for the proofs of the teachings below.

|You should keep this introduction in mind, because you will need it for the proofs of the teachings below.

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|Here I start:

|Here I start:

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:<math>\scriptstyle\begin{cases}\scriptstyle a+b=10\\\scriptstyle\frac{b}{a}=5\end{cases}</math>

:<math>\scriptstyle\begin{cases}\scriptstyle a+b=10\\\scriptstyle\frac{b}{a}=5\end{cases}</math>

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:Say: the divisor by which it is divided is one thing [<math>\scriptstyle{\color{blue}{x}}</math>].

:Say: the divisor by which it is divided is one thing [<math>\scriptstyle{\color{blue}{x}}</math>].

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:The dividend is necessarily five things, as a number resulting from the division [<math>\scriptstyle{\color{blue}{5x}}</math>].

:The dividend is necessarily five things, as a number resulting from the division [<math>\scriptstyle{\color{blue}{5x}}</math>].

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:The sum of the two parts is six things and they are equal to ten.

:The sum of the two parts is six things and they are equal to ten.

:<math>\scriptstyle{\color{blue}{6x=10}}</math>

:<math>\scriptstyle{\color{blue}{6x=10}}</math>

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:According to the method mentioned in this teaching:

:According to the method mentioned in this teaching:

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:2) When squares are equal to numbers.

:2) When squares are equal to numbers.

:<math>\scriptstyle ax^2=c</math>

:<math>\scriptstyle ax^2=c</math>

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:Divide the numbers by [the number of] the squares; the root of the quotient is the thing.

:Divide the numbers by [the number of] the squares; the root of the quotient is the thing.

:<math>\scriptstyle{\color{OliveGreen}{x=\sqrt{\frac{c}{a}}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x=\sqrt{\frac{c}{a}}}}</math>

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:<math>\scriptstyle\left(a-\frac{1}{3}a\right)^2=20</math>

:<math>\scriptstyle\left(a-\frac{1}{3}a\right)^2=20</math>

|style="text-align:right;"|<big>שאלה</big> רציתי למצא מספר כאשר חסר ממנו שלישיתו מרבע הנשאר הוא מספר כ&#x202B;'{{#annotend:xJ4O}}

|style="text-align:right;"|<big>שאלה</big> רציתי למצא מספר כאשר חסר ממנו שלישיתו מרבע הנשאר הוא מספר כ&#x202B;'{{#annotend:xJ4O}}

:Multiply its 2-thirds by themselves; it is 4-ninths of the square of the whole number you wish to find.

:Multiply its 2-thirds by themselves; it is 4-ninths of the square of the whole number you wish to find.

:<math>\scriptstyle{\color{blue}{\left(x-\frac{1}{3}x\right)^2=\left(\frac{2}{3}x\right)^2=\frac{4}{9}x^2=20}}</math>

:<math>\scriptstyle{\color{blue}{\left(x-\frac{1}{3}x\right)^2=\left(\frac{2}{3}x\right)^2=\frac{4}{9}x^2=20}}</math>

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:According to the method mentioned in this teaching:

:According to the method mentioned in this teaching:

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:<math>\scriptstyle{\color{OliveGreen}{x^2:x=x:1}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x^2:x=x:1}}</math>

|style="text-align:right;"|זה הלמוד הולך בדרך הלמוד הראשון מפני כי יחס המרבע אל הדבר כיחס הדבר אל האחד כאשר אמרנו בהצעה

|style="text-align:right;"|זה הלמוד הולך בדרך הלמוד הראשון מפני כי יחס המרבע אל הדבר כיחס הדבר אל האחד כאשר אמרנו בהצעה

:So, if one square equals 3 things, for instance, one square necessarily equals 3 units.

:So, if one square equals 3 things, for instance, one square necessarily equals 3 units.

:<math>\scriptstyle{\color{blue}{x^2=3x\longrightarrow x=3}}</math>

:<math>\scriptstyle{\color{blue}{x^2=3x\longrightarrow x=3}}</math>

דבר אחד ישוה לג' אחדים בהכרח

דבר אחד ישוה לג' אחדים בהכרח

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:<math>\scriptstyle a-\frac{1}{3}a=\sqrt{a}</math>

:<math>\scriptstyle a-\frac{1}{3}a=\sqrt{a}</math>

|style="text-align:right;"|<big>שאלה</big> רציתי למצא מספר כאשר {{#annot:term|181,1362|rg2F}}חסר ממנו{{#annotend:rg2F}} שלישית הנשאר הוא שרש המספר כלו{{#annotend:HmbP}}

|style="text-align:right;"|<big>שאלה</big> רציתי למצא מספר כאשר {{#annot:term|181,1362|rg2F}}חסר ממנו{{#annotend:rg2F}} שלישית הנשאר הוא שרש המספר כלו{{#annotend:HmbP}}

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:<math>\scriptstyle{\color{blue}{x=\frac{1+\frac{1}{2}}{1}=1+\frac{1}{2}}}</math>

:<math>\scriptstyle{\color{blue}{x=\frac{1+\frac{1}{2}}{1}=1+\frac{1}{2}}}</math>

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:4) Things and numbers that are equal to squares

:4) Things and numbers that are equal to squares

:<math>\scriptstyle bx+c=ax^2</math>

:<math>\scriptstyle bx+c=ax^2</math>

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:<span style=color:Green>Normalization:</span> Divide the things and the numbers by [the number of] the squares.

:<span style=color:Green>Normalization:</span> Divide the things and the numbers by [the number of] the squares.

:<math>\scriptstyle{\color{OliveGreen}{\frac{b}{a}x+\frac{c}{a}=x^2}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\frac{b}{a}x+\frac{c}{a}=x^2}}</math>

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:Halve the result of division of the things.

:Halve the result of division of the things.

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:Add the result to the result of division of the numbers.

:Add the result to the result of division of the numbers.

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:Add it to half [the number of] the things resulting from the division; the sum is the thing.

:Add it to half [the number of] the things resulting from the division; the sum is the thing.

:<math>\scriptstyle{\color{OliveGreen}{x=\left(\frac{1}{2}\sdot\frac{b}{a}\right)+\sqrt{\left(\frac{1}{2}\sdot\frac{b}{a}\right)^2+\frac{c}{a}}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x=\left(\frac{1}{2}\sdot\frac{b}{a}\right)+\sqrt{\left(\frac{1}{2}\sdot\frac{b}{a}\right)^2+\frac{c}{a}}}}</math>

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!<span style=color:Green>Geometric illustration</span>

!<span style=color:Green>Geometric illustration</span>

:Cut it randomly at point Z, so segment AZ is 8 measures.

:Cut it randomly at point Z, so segment AZ is 8 measures.

:<math>\scriptstyle{\color{blue}{AZ=8}}</math>

:<math>\scriptstyle{\color{blue}{AZ=8}}</math>

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:We construct square ABGD on AB.

:We construct square ABGD on AB.

:<math>\scriptstyle{\color{blue}{ABGD=AB^2}}</math>

:<math>\scriptstyle{\color{blue}{ABGD=AB^2}}</math>

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:We receive surface AC that is eight things by the measures of line AZ.

:We receive surface AC that is eight things by the measures of line AZ.

:<math>\scriptstyle{\color{blue}{AC=8x}}</math>

:<math>\scriptstyle{\color{blue}{AC=8x}}</math>

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:We cut it in half at point T.

:We cut it in half at point T.

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:<math>\scriptstyle{\color{OliveGreen}{\left[\left(a+b\right)\sdot b\right]+\left(\frac{1}{2}\sdot a\right)^2=\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\left[\left(a+b\right)\sdot b\right]+\left(\frac{1}{2}\sdot a\right)^2=\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2}}</math>

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|colspan=2|

|colspan=2|

:Add half [the number of] the things to it, which is 4, as the number of measures of line AT; the result is 10, as the size of the whole line AB that is the side of the square, which is the thing.

:Add half [the number of] the things to it, which is 4, as the number of measures of line AT; the result is 10, as the size of the whole line AB that is the side of the square, which is the thing.

:<math>\scriptstyle{\color{blue}{x=AB=TB+AT=6+\left(\frac{1}{2}\sdot8\right)=6+4=10}}</math>

:<math>\scriptstyle{\color{blue}{x=AB=TB+AT=6+\left(\frac{1}{2}\sdot8\right)=6+4=10}}</math>

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:<math>\scriptstyle a+28=2a^2</math>

:<math>\scriptstyle a+28=2a^2</math>

|style="text-align:right;"|<big>שאלה</big> רצינו למצא מספר כאשר הוספנו עליו כ"ח {{#annot:term|429,1247|hkKx}}יהיה שוה ל{{#annotend:hkKx}}שני דמיוני מרבעו{{#annotend:AZ3B}}

|style="text-align:right;"|<big>שאלה</big> רצינו למצא מספר כאשר הוספנו עליו כ"ח {{#annot:term|429,1247|hkKx}}יהיה שוה ל{{#annotend:hkKx}}שני דמיוני מרבעו{{#annotend:AZ3B}}

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:According to the method mentioned in this teaching:

:According to the method mentioned in this teaching:

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:<span style=color:Green>Normalization:</span> One thing plus 28 units should be divided by two, which is the number of the squares; the result of division is half a thing plus 14 units.

:<span style=color:Green>Normalization:</span> One thing plus 28 units should be divided by two, which is the number of the squares; the result of division is half a thing plus 14 units.

:<math>\scriptstyle{\color{blue}{\frac{1}{2}x+\frac{28}{2}=\frac{1}{2}x+14=x^2}}</math>

:<math>\scriptstyle{\color{blue}{\frac{1}{2}x+\frac{28}{2}=\frac{1}{2}x+14=x^2}}</math>

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:Add it to the number of units resulting from division, which is 14; the result is 14 and one part of 16.

:Add it to the number of units resulting from division, which is 14; the result is 14 and one part of 16.

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:Extract its root; it is 3 and 3-quarters.

:Extract its root; it is 3 and 3-quarters.

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:Add it to half [the number of] the things resulting from division, which is a quarter of a thing; the result is 4 and this is the thing.

:Add it to half [the number of] the things resulting from division, which is a quarter of a thing; the result is 4 and this is the thing.

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|colspan=2|

|colspan=2|

:5) Squares and numbers that are equal to things

:5) Squares and numbers that are equal to things

:<math>\scriptstyle ax^2+c=bx</math>

:<math>\scriptstyle ax^2+c=bx</math>

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:Subtract the result of division of the numbers from the product.

:Subtract the result of division of the numbers from the product.

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:Extract the root of the remainder.

:Extract the root of the remainder.

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:Add it to half [the number] resulting from the division of the things; the sum is the thing.

:Add it to half [the number] resulting from the division of the things; the sum is the thing.

:<math>\scriptstyle{\color{OliveGreen}{x=\left(\frac{1}{2}\sdot\frac{b}{a}\right)+\sqrt{\left(\frac{1}{2}\sdot\frac{b}{a}\right)^2-\frac{c}{a}}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x=\left(\frac{1}{2}\sdot\frac{b}{a}\right)+\sqrt{\left(\frac{1}{2}\sdot\frac{b}{a}\right)^2-\frac{c}{a}}}}</math>

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!<span style=color:Green>Geometric illustration</span>

!<span style=color:Green>Geometric illustration</span>

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:To show you a proof of it we describe a geometric illustration and present a numerical example:

:To show you a proof of it we describe a geometric illustration and present a numerical example:

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:We cut into two equal segments at point Z, so line GZ is 4 measures.

:We cut into two equal segments at point Z, so line GZ is 4 measures.

:<math>\scriptstyle{\color{blue}{GZ=\frac{1}{2}\sdot AG=4}}</math>

:<math>\scriptstyle{\color{blue}{GZ=\frac{1}{2}\sdot AG=4}}</math>

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:We cut into two unequal segments at point B, so line GB is 2 measures.

:We cut into two unequal segments at point B, so line GB is 2 measures.

:<math>\scriptstyle{\color{blue}{GB=2}}</math>

:<math>\scriptstyle{\color{blue}{GB=2}}</math>

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:<math>\scriptstyle{\color{blue}{CH=CG}}</math>

:<math>\scriptstyle{\color{blue}{CH=CG}}</math>

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:We also draw line GH.

:We also draw line GH.

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:<math>\scriptstyle{\color{blue}{AH=ABHW+GW=x^2+12=8x}}</math>

:<math>\scriptstyle{\color{blue}{AH=ABHW+GW=x^2+12=8x}}</math>

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:<math>\scriptstyle{\color{OliveGreen}{\left(a\sdot b\right)+\left[\left[\frac{1}{2}\sdot\left(a+b\right)\right]-b\right]^2=\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\left(a\sdot b\right)+\left[\left[\frac{1}{2}\sdot\left(a+b\right)\right]-b\right]^2=\left[\frac{1}{2}\sdot\left(a+b\right)\right]^2}}</math>

:<math>\scriptstyle{\color{blue}{AZ^2=BH+ZB^2=\left(\frac{1}{2}\sdot8\right)^2=4^2=16}}</math>

:<math>\scriptstyle{\color{blue}{AZ^2=BH+ZB^2=\left(\frac{1}{2}\sdot8\right)^2=4^2=16}}</math>

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:We subtract surface BH, which is 12, from the square formed by AZ, which is 16; the remaining square formed by ZB is known.

:We subtract surface BH, which is 12, from the square formed by AZ, which is 16; the remaining square formed by ZB is known.

:<math>\scriptstyle{\color{blue}{ZB^2=AZ^2-BH=16-12=4}}</math>

:<math>\scriptstyle{\color{blue}{ZB^2=AZ^2-BH=16-12=4}}</math>

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:Q.E.D.

:Q.E.D.

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:The ratio of one thing to one thing plus 6 is as the ratio of one thing plus 6 to 27 units.

:The ratio of one thing to one thing plus 6 is as the ratio of one thing plus 6 to 27 units.

:<math>\scriptstyle{\color{blue}{x:\left(x+6\right)=\left(x+6\right):27}}</math>

:<math>\scriptstyle{\color{blue}{x:\left(x+6\right)=\left(x+6\right):27}}</math>

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:{{#annot:definition|813|KlWg}}It is already explained in {{#annot:reference|813|cH1a}}[[ספר_היסודות_לאקלידס#Elements_VI_17|<span style=color:blue>'''Euclid, Elements, Book VI, proposition 17'''</span>]] that the product of the first by the last is as the product of the mean by its similar.

:{{#annot:definition|813|KlWg}}It is already explained in {{#annot:reference|813|cH1a}}[[ספר_היסודות_לאקלידס#Elements_VI_17|<span style=color:blue>'''Euclid, Elements, Book VI, proposition 17'''</span>]] that the product of the first by the last is as the product of the mean by its similar.

:<math>\scriptstyle{\color{OliveGreen}{a_1:a_2=a_2:a_3\longrightarrow a_1\sdot a_3=\left(a_2\right)^2}}</math>

:<math>\scriptstyle{\color{OliveGreen}{a_1:a_2=a_2:a_3\longrightarrow a_1\sdot a_3=\left(a_2\right)^2}}</math>

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:<span style=color:Green>Restoration:</span> Now, subtract 12 things from these two equal measures.

:<span style=color:Green>Restoration:</span> Now, subtract 12 things from these two equal measures.

:<math>\scriptstyle{\color{blue}{x\sdot27=\left(x+6\right)^2=x^2+12x+36 /-12x}}</math>

:<math>\scriptstyle{\color{blue}{x\sdot27=\left(x+6\right)^2=x^2+12x+36 /-12x}}</math>

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:<span style=color:Green>Normalization:</span> According to the way we stated in this teaching, 15, which is the number of the things, and 36, which is the number of the units, should be divided by one, which is the number of the squares; the result is 15 things and 36 units.

:<span style=color:Green>Normalization:</span> According to the way we stated in this teaching, 15, which is the number of the things, and 36, which is the number of the units, should be divided by one, which is the number of the squares; the result is 15 things and 36 units.

:<math>\scriptstyle{\color{blue}{15x=\frac{15}{1}x=x^2+\frac{36}{1}=x^2+36}}</math>

:<math>\scriptstyle{\color{blue}{15x=\frac{15}{1}x=x^2+\frac{36}{1}=x^2+36}}</math>

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:Then, halve [the number of] the things; it is 7 and a half.

:Then, halve [the number of] the things; it is 7 and a half.

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:Multiply it by itself; the result is 56 and a quarter.

:Multiply it by itself; the result is 56 and a quarter.

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:Subtract 36 units from it; 20 and a quarter remains.

:Subtract 36 units from it; 20 and a quarter remains.

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:<span style=color:Green>Normalization:</span> Divide the things and the numbers by [the number of] the squares.

:<span style=color:Green>Normalization:</span> Divide the things and the numbers by [the number of] the squares.

:<math>\scriptstyle{\color{OliveGreen}{x^2+\frac{b}{a}x=\frac{c}{a}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x^2+\frac{b}{a}x=\frac{c}{a}}}</math>

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:We describe square AGHD.

:We describe square AGHD.

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:Both together, i.e. the square plus the area, are equal to 48.

:Both together, i.e. the square plus the area, are equal to 48.

:<math>\scriptstyle{\color{blue}{AGHD+BGHW=48}}</math>

:<math>\scriptstyle{\color{blue}{AGHD+BGHW=48}}</math>

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:The size of side GB of surface GBHW is known and it is 2 measures, as the number of the things.

:The size of side GB of surface GBHW is known and it is 2 measures, as the number of the things.

:<math>\scriptstyle{\color{blue}{GB=2}}</math>

:<math>\scriptstyle{\color{blue}{GB=2}}</math>

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:We cut line BG in half at point Z [= Z midpoint of GB].

:We cut line BG in half at point Z [= Z midpoint of GB].

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:So we have line GB is cut in half at point Z and line GA is added to it.

:So we have line GB is cut in half at point Z and line GA is added to it.

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:It is already explained in {{#annot:reference|254|BJrm}}[[ספר_היסודות_לאקלידס#Elements_II_6|<span style=color:blue>'''Euclid, Elements, Book II, proposition 6'''</span>]] that the right-angled surface encompassed by the whole line with the addition and the addition, which is equal to surface AW, whose area is known as 48, plus the square of half the line, whose area is known, which is 1, both together are 49, equals the square of the line formed by half the line with the addition, which is line AZ.

:It is already explained in {{#annot:reference|254|BJrm}}[[ספר_היסודות_לאקלידס#Elements_II_6|<span style=color:blue>'''Euclid, Elements, Book II, proposition 6'''</span>]] that the right-angled surface encompassed by the whole line with the addition and the addition, which is equal to surface AW, whose area is known as 48, plus the square of half the line, whose area is known, which is 1, both together are 49, equals the square of the line formed by half the line with the addition, which is line AZ.

:<math>\scriptstyle{\color{OliveGreen}{\left[\left(a+b\right)\sdot b\right]+\left(\frac{1}{2}\sdot a\right)^2=\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\left[\left(a+b\right)\sdot b\right]+\left(\frac{1}{2}\sdot a\right)^2=\left[\left(\frac{1}{2}\sdot a\right)+b\right]^2}}</math>

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|colspan=2|

|colspan=2|

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:Q.E.D.

:Q.E.D.

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:7) When cubes are equal to numbers:

:7) When cubes are equal to numbers:

:<math>\scriptstyle ax^3=c</math>

:<math>\scriptstyle ax^3=c</math>

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|style="text-align:right;"|זה מובן בעצמו

|style="text-align:right;"|זה מובן בעצמו

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:<math>\scriptstyle{\color{OliveGreen}{x^3=\frac{b}{a}x}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x^3=\frac{b}{a}x}}</math>

|style="text-align:right;"|תחלק הדברים למעקבי&#x202B;'

|style="text-align:right;"|תחלק הדברים למעקבי&#x202B;'

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|style="text-align:right;"|זה הלמוד הולך בדרך הלמוד השני

|style="text-align:right;"|זה הלמוד הולך בדרך הלמוד השני

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:Since the ratio of the cube to the thing is the same as the ratio of the square to the unit.

:Since the ratio of the cube to the thing is the same as the ratio of the square to the unit.

|style="text-align:right;"|מפני כי יחס המעקב אל הדבר הוא כיחס המרבע אל האחד

|style="text-align:right;"|מפני כי יחס המעקב אל הדבר הוא כיחס המרבע אל האחד

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|style="text-align:right;"|וזה יובן מן ההצעה

|style="text-align:right;"|וזה יובן מן ההצעה

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:9) When squares of squares are equal to numbers:

:9) When squares of squares are equal to numbers:

:<math>\scriptstyle ax^4=c</math>

:<math>\scriptstyle ax^4=c</math>

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:<math>\scriptstyle{\color{OliveGreen}{x^4=\frac{c}{a}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x^4=\frac{c}{a}}}</math>

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:10) When squares of squares are equal to things:

:10) When squares of squares are equal to things:

:<math>\scriptstyle ax^4=bx</math>

:<math>\scriptstyle ax^4=bx</math>

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:<math>\scriptstyle{\color{OliveGreen}{x^4=\frac{b}{a}x}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x^4=\frac{b}{a}x}}</math>

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:Extract the cube root of the result and this is the thing.

:Extract the cube root of the result and this is the thing.

:<math>\scriptstyle{\color{OliveGreen}{x=\sqrt[3]{\frac{b}{a}}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x=\sqrt[3]{\frac{b}{a}}}}</math>

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|style="text-align:right;"|וזה יובן מן ההצעה

|style="text-align:right;"|וזה יובן מן ההצעה

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:11) When squares of squares are equal to squares:

:11) When squares of squares are equal to squares:

:<math>\scriptstyle ax^4=bx^2</math>

:<math>\scriptstyle ax^4=bx^2</math>

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:<math>\scriptstyle{\color{OliveGreen}{x^4=\frac{b}{a}x^2}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x^4=\frac{b}{a}x^2}}</math>

|style="text-align:right;"|תחלק המרבעי' למרבעי המרבעים

|style="text-align:right;"|תחלק המרבעי' למרבעי המרבעים

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:Since the ratio of the square of the square to the square is the same as the ratio of the square to the unit.

:Since the ratio of the square of the square to the square is the same as the ratio of the square to the unit.

:<math>\scriptstyle{\color{OliveGreen}{\left(x^2\right)^2:x^2=x^2:1}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\left(x^2\right)^2:x^2=x^2:1}}</math>

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|style="text-align:right;"|וזה יובן מן ההצעה

|style="text-align:right;"|וזה יובן מן ההצעה

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:12) When squares of squares are equal to cubes:

:12) When squares of squares are equal to cubes:

:<math>\scriptstyle ax^4=bx^3</math>

:<math>\scriptstyle ax^4=bx^3</math>

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:<math>\scriptstyle{\color{OliveGreen}{x^4=\frac{b}{a}x^3}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x^4=\frac{b}{a}x^3}}</math>

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|style="text-align:right;"|זה הלמוד הולך בדרך הלמוד הראשון

|style="text-align:right;"|זה הלמוד הולך בדרך הלמוד הראשון

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:<math>\scriptstyle{\color{OliveGreen}{\left(x^2\right)^2:x^3=x:1}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\left(x^2\right)^2:x^3=x:1}}</math>

|style="text-align:right;"|מפני כי יחס מרבע המרבע אל המעקב הוא כיחס הדבר אל האחד

|style="text-align:right;"|מפני כי יחס מרבע המרבע אל המעקב הוא כיחס הדבר אל האחד

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|style="text-align:right;"|כאשר הקדמנו בהצעה

|style="text-align:right;"|כאשר הקדמנו בהצעה

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:13) When cubes and squares are equal to things:

:13) When cubes and squares are equal to things:

:<math>\scriptstyle ax^3+bx^2=cx</math>

:<math>\scriptstyle ax^3+bx^2=cx</math>

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:<math>\scriptstyle{\color{OliveGreen}{x^3+\frac{b}{a}x^2=\frac{c}{a}x}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x^3+\frac{b}{a}x^2=\frac{c}{a}x}}</math>

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:<math>\scriptstyle{\color{OliveGreen}{x=\sqrt{\left(\frac{1}{2}\sdot\frac{b}{a}\right)^2+\frac{c}{a}}-\left(\frac{1}{2}\sdot\frac{b}{a}\right)}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x=\sqrt{\left(\frac{1}{2}\sdot\frac{b}{a}\right)^2+\frac{c}{a}}-\left(\frac{1}{2}\sdot\frac{b}{a}\right)}}</math>

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|style="text-align:right;"|זה הלמוד הולך בדרך הלמוד הששי

|style="text-align:right;"|זה הלמוד הולך בדרך הלמוד הששי

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:<math>\scriptstyle{\color{OliveGreen}{\begin{cases}\scriptstyle x^3:x=x^2:1\\\scriptstyle x^2:x=x:1\end{cases}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\begin{cases}\scriptstyle x^3:x=x^2:1\\\scriptstyle x^2:x=x:1\end{cases}}}</math>

|style="text-align:right;"|מפני כי יחס המעקב והמרבע כל אחד מהם אל הדבר כיחס המרבע והדבר כל אחד מהם אל האחד

|style="text-align:right;"|מפני כי יחס המעקב והמרבע כל אחד מהם אל הדבר כיחס המרבע והדבר כל אחד מהם אל האחד

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|style="text-align:right;"|כמבואר בהצעה

|style="text-align:right;"|כמבואר בהצעה

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|style="text-align:right;"|תחלק הדברי' והמרבעי' למעקבים

|style="text-align:right;"|תחלק הדברי' והמרבעי' למעקבים

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:<math>\scriptstyle{\color{OliveGreen}{x=\left(\frac{1}{2}\sdot\frac{b}{a}\right)+\sqrt{\left(\frac{1}{2}\sdot\frac{b}{a}\right)^2-\frac{c}{a}}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x=\left(\frac{1}{2}\sdot\frac{b}{a}\right)+\sqrt{\left(\frac{1}{2}\sdot\frac{b}{a}\right)^2-\frac{c}{a}}}}</math>

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:<math>\scriptstyle{\color{OliveGreen}{\begin{cases}\scriptstyle x^3:x^2=x^2:x\\\scriptstyle x:x^2=1:x\end{cases}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\begin{cases}\scriptstyle x^3:x^2=x^2:x\\\scriptstyle x:x^2=1:x\end{cases}}}</math>

|style="text-align:right;"|מפני כי יחס המעקב והדבר כל אחד מהם אל המרבע כיחס המרבע והאחד כל אחד מהם אל הדבר

|style="text-align:right;"|מפני כי יחס המעקב והדבר כל אחד מהם אל המרבע כיחס המרבע והאחד כל אחד מהם אל הדבר

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:15) Squares and things that are equal to cubes

:15) Squares and things that are equal to cubes

:<math>\scriptstyle bx^2+cx=ax^3</math>

:<math>\scriptstyle bx^2+cx=ax^3</math>

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:<math>\scriptstyle{\color{OliveGreen}{x=\left(\frac{1}{2}\sdot\frac{b}{a}\right)+\sqrt{\left(\frac{1}{2}\sdot\frac{b}{a}\right)^2+\frac{c}{a}}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{x=\left(\frac{1}{2}\sdot\frac{b}{a}\right)+\sqrt{\left(\frac{1}{2}\sdot\frac{b}{a}\right)^2+\frac{c}{a}}}}</math>

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|style="text-align:right;"|זה הלמוד הולך בדרך הלמוד הרביעי

|style="text-align:right;"|זה הלמוד הולך בדרך הלמוד הרביעי

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:<math>\scriptstyle{\color{OliveGreen}{\begin{cases}\scriptstyle x^2:x^3=x:x^2\\\scriptstyle x:x^3=1:x^2\end{cases}}}</math>

:<math>\scriptstyle{\color{OliveGreen}{\begin{cases}\scriptstyle x^2:x^3=x:x^2\\\scriptstyle x:x^3=1:x^2\end{cases}}}</math>

|style="text-align:right;"|מפני כי יחס המרבע והדבר כל אחד מהם אל המעקב כיחס הדבר והאחד כל אחד מהם אל המרבע

|style="text-align:right;"|מפני כי יחס המרבע והדבר כל אחד מהם אל המעקב כיחס הדבר והאחד כל אחד מהם אל המרבע

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|The end.

|The end.

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|style="text-align:right;"|אמנם קצרתי בהם לשתי {{#annot:term|311|vxTd}}סבות{{#annotend:vxTd}}

|style="text-align:right;"|אמנם קצרתי בהם לשתי {{#annot:term|311|vxTd}}סבות{{#annotend:vxTd}}

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|style="text-align:right;"|'''שמעון בכמה"ר משה יצ"ו בכמה"ר שמעון מטוט זלה"ה'''

|style="text-align:right;"|'''שמעון בכמה"ר משה יצ"ו בכמה"ר שמעון מטוט זלה"ה'''

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*{{#annot:two measures|867|cPgg}}Question: 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?

*{{#annot:two measures|867|cPgg}}Question: 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?

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|style="text-align:right;"|עשה על הדרך הזאת

|style="text-align:right;"|עשה על הדרך הזאת

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|style="text-align:right;"|אמור סכום הפרחי' אשר נחלפו בליט' רומניולי הוא דבר אחד

|style="text-align:right;"|אמור סכום הפרחי' אשר נחלפו בליט' רומניולי הוא דבר אחד

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|style="text-align:right;"|כל אחד מהם נחלף במספר ליט' נעלם

|style="text-align:right;"|כל אחד מהם נחלף במספר ליט' נעלם

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|style="text-align:right;"|והיו ל' ליט&#x202B;'

|style="text-align:right;"|והיו ל' ליט&#x202B;'

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|style="text-align:right;"|נשאר הסכום אשר נחלף בליט' מרקיאני הוא כ"ג פחות דבר אחד

|style="text-align:right;"|נשאר הסכום אשר נחלף בליט' מרקיאני הוא כ"ג פחות דבר אחד

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|style="text-align:right;"|ועתה תכפול מ"ו מספרי' נעלמי' ורביע דבר בדבר יעלה אלף וש"פ אחדים ורביע מרבע

|style="text-align:right;"|ועתה תכפול מ"ו מספרי' נעלמי' ורביע דבר בדבר יעלה אלף וש"פ אחדים ורביע מרבע

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|style="text-align:right;"|וגם כן תכפל צ"ה אחדי' וג' רביעי אחד בדבר יעלה צ"ה דברי' וג' רביעי דבר

|style="text-align:right;"|וגם כן תכפל צ"ה אחדי' וג' רביעי אחד בדבר יעלה צ"ה דברי' וג' רביעי דבר

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|style="text-align:right;"|וכבר ידעת כאשר היו האחדים והמרובעים שוים לדברי' איך תדע מספר הדבר ומידיעת הדבר תדע הכל

|style="text-align:right;"|וכבר ידעת כאשר היו האחדים והמרובעים שוים לדברי' איך תדע מספר הדבר ומידיעת הדבר תדע הכל

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|style="text-align:right;"|אציע

|style="text-align:right;"|אציע

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|style="text-align:right;"|הצעה

|style="text-align:right;"|הצעה

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