Babylonian Geometry

Leyte Normal UniversityLeyte Normal University Jonas P. VillasJonas P. Villas

Babylonian Geometry

Babylonians Contribution to Mathematics:

• they develop procedures for determining areas and volume of various kind of figures

•They work out algorithms to determine square roots.

•They solve mathematical problems that modern mathematics would interpret in terms of linear and quadratic equations.


Procedures for Determining Procedures for Determining Lengths and AreasLengths and Areas

Note: Formulas were Note: Formulas were presented in terms of presented in terms of what are today called what are today called coefficients lists, lists coefficients lists, lists of constants that of constants that embody mathematical embody mathematical relationship between relationship between certain aspects of certain aspects of various geometrical various geometrical figures. figures.

Say: 0; 52 ; 30 (=7/8) as the coefficient of the height of a triangle means that the altitude of an equilateral triangle is 7/8 of the base while the number 0;26;15 (=7/16) as the coefficient for area means that the area of an equilateral triangle is 7/16 times the square of the side.

(Note) that this is just approximately correct, and that they both approximate square root of 3 by 7/4)

the defining component for the triangle is the side.


Babylonians took the circumference as the Babylonians took the circumference as the defining component of the circle.defining component of the circle.

we gave two coefficients for the circle:we gave two coefficients for the circle:

0;20 (=1/3) for the diameter and 0;05 (=1/12) for the area.0;20 (=1/3) for the diameter and 0;05 (=1/12) for the area.

The meaning of the first coefficient is that the diameter is one The meaning of the first coefficient is that the diameter is one third of the circumference.third of the circumference.

The second means that the area is one-twelfth of the square The second means that the area is one-twelfth of the square of the circumferenceof the circumference

Procedures for Determining Procedures for Determining Lengths and AreasLengths and Areas

example


Based on the tablet: Based on the tablet:

YBC 7302 (Yale Babylonian YBC 7302 (Yale Babylonian Collection)Collection)

459

3Interpretation:

The circle has a circumference 3 and the area is found by dividing 9 = 3^2 by 12 to get 0;45 (=3/4)


Haddad 104 Circle calculations virtually always use the circumference.

Example:

Finding the area of the cross section of a log of diameter 1;40 (= 1 2/3)

•The scribe first multiplies by 3 to find the circumference equal to 5.

•5 is then squared and multiply it to ½ to get the area 2;05 (= 2 1/12).

Note: The Babylonian value of , the ratio of circumference to diameter is 3;

4 = 12 is produced and is the constant by which to divide the square of the circumference to give the area.


Other Babylonian Other Babylonian Coefficients for other Coefficients for other Figures Bounded by Circular Figures Bounded by Circular ArcsArcs

Babylonians calculated areas of two different Babylonians calculated areas of two different double bows:double bows: The bargeThe barge

- made up of two- quarter circle arcs- made up of two- quarter circle arcs Bulls eyeBulls eye

- Composed of two third-circle arcs.- Composed of two third-circle arcs.


This figure were analogous to the circle and This figure were analogous to the circle and that their defining component was the arc that their defining component was the arc making up the side.making up the side.

Thus areas of the two figures:Thus areas of the two figures:

2/9 a2 9/32 a2

Where a is the length of the arc.

Note:

These results are accurate under the assumption that the area of the circle is C2/12 and that square root of 3 = 7/4.


Volume of SolidsVolume of Solids

Babylonians realized that the volume V of a Babylonians realized that the volume V of a rectangular block is rectangular block is VV = = ll x x ww x x hh

They also knew how to calculate the volume of They also knew how to calculate the volume of a prism given the area of a base.a prism given the area of a base.

Like Egyptians and though Babylonians are Like Egyptians and though Babylonians are creating pyramidal structure there exist no creating pyramidal structure there exist no document that give explicitly the volume of a document that give explicitly the volume of a pyramid.pyramid.


BM 96954 (British BM 96954 (British Museum)Museum)

A problem involving a grain pile in the shape A problem involving a grain pile in the shape of a rectangular pyramid with an elongated of a rectangular pyramid with an elongated apex, like a pitched roof.apex, like a pitched roof.

w

l

t

h

)(23

tl

hwV

Where:

l = length of the solid

w = width

h = the height

t = length of the arc


Derivation of the formulaDerivation of the formula

w

l

t

h


)(

)(

2

36

36

23332

32

tl

hwlhwt

hwlhwthwt

hwthwlhwt

tlhwhwtV

3hw

This implies that the Babylonians were aware of the correct formula for the value of a pyramid

l -t

h w

w

t

h


A formula for the Volume of A formula for the Volume of a truncated Pyramida truncated Pyramid

Square base aSquare base a22, square top b, square top b22 and a height in h. and a height in h.

hbaba

V )(22

23

1

2

Setting b = 0, gives rise to the complete pyramid formula

haa

V )(22

23

1

2


There are other tablets where in volume is There are other tablets where in volume is calculated by the rule:calculated by the rule:

hbaV 22

2

1A simple but incorrect

generalization for the rule of the area of a trapezoid.

Remarks:

Although this formula is incorrect, the calculated answer would not differ much from the correct ones.

Likewise, since there are no accurate empirical method for measuring the volume, it is difficult to see how anyone would realize that the answer were wrong.

Finally, the problems wherein this formulas occurred are that of practical ones often related to the numbers of workmen needed to build a particular structure and hence the slight inaccuracy would ha\vfe little effect on the final answer.


1.2.4 Square Roots and 1.2.4 Square Roots and the Pythagorean the Pythagorean TheoremTheorem

THE SQUARE ROOT ALGORITHM

PYTHAGOREAN THEOREM


THE SQUARE ROOT THE SQUARE ROOT ALGORITHMALGORITHM

When Babylonians needed a square root When Babylonians needed a square root to solve a problem they usually refer to a to solve a problem they usually refer to a table of square roots and is a rational table of square roots and is a rational number. number.

There are cases that occurs and There are cases that occurs and thus the results is generally written as thus the results is generally written as 1:25 (=1 5/12). 1:25 (=1 5/12).

2


YBC 7289YBC 7289

This is an interesting tablet where it This is an interesting tablet where it includes a drawing of a square with includes a drawing of a square with side indicated as 30 and two numbers, side indicated as 30 and two numbers, 1:24,51,10 and 42;25,35, written on the 1:24,51,10 and 42;25,35, written on the diagonal.diagonal.


YBC 7289YBC 7289 The product of 30 and The product of 30 and 1:24,51,10 is precisely 1:24,51,10 is precisely 42;25,35.42;25,35.

It is a reasonable assumption It is a reasonable assumption that the last number that the last number represents the length of the represents the length of the diagonal and that the other diagonal and that the other number represents number represents 2

1:24,51,101:24,51,10

42;25,3542;25,35

3030

Remarks: Whether is given as 1;25 or as 1:24,51,10, there is no record of how the value was calculated. Moreover, the scribes were surely aware that the square of neither of these was exactly 2, or that these values were not exactly the length of the side of the square of area 2 and further they must have known that these values were approximations.

2


How were the values How were the values determined?determined?

(x + y)(x + y)2 2 = x= x22 + 2xy + y + 2xy + y22

N

Find: Side Square root of N

Choose a regular value a close to but less then the desired result.

Setting b = N - a2

The next step is to find c so that 2ac + c2 is as close as possible to b.

If a2 is close enough to N, then c2 will be small in relation to 2ac and so c can be chosen to equal (1/2)b(1/a)

a2

c

ac

ac c2 ac

b

a

That is : ).a/(b)/(abaN 1212


Similarly,Similarly,

)a/(b)/(aba 1212

In the particular case of square root of 2, begin with a = 1;20 (=4/3). Then a2 = 1;46,40, b=0;13,20, and 1/a = 0;45, so square root of 2 = the square root of 1;46,40 +0;13,20 approximately 1;20 + (0;30)(0;13,20)(0;4) = 1;20 + 0;05 = 1;25 (or 17/12

(1;25)2 differs from 2 by only 0;0,25 = 1/144

Main Slide 2


Pythagorean TheoremPythagorean Theorem

In any right triangle the sum of the areas In any right triangle the sum of the areas of he squares of the legs equals the area of he squares of the legs equals the area of the square of the hypotenuse.of the square of the hypotenuse.

Named after the sixth century BCE Greek Named after the sixth century BCE Greek philosopher-mathematician.philosopher-mathematician.

Arguably the most important elementary Arguably the most important elementary theorem in mathematics.theorem in mathematics.


Pythagorean Triples from Plimton 322

(d/y)2 x d # y

1.9834028 119 169 1 120

1.9491586 3367 4825 2 3456

1.9188021 4601 6649 3 4800

1.8862479 12,709 18,541 4 13,500

1.8150077 65 97 5 72

1.7851929 319 481 6 360

1.7199837 2291 3541 7 2700

1.6845877 799 1249 8 960

1.6426694 481 769 9 600

1.5861226 4961 8161 10 6480

1.5625 45 75 11 60

1.4894168 1679 2929 12 2400

1.4500174 161 289 13 240

1.4302388 1771 3229 14 2700

1.3871605 28 53 15 45


(d/y)2 x d # y

1.9834028 119 169 1 120

1.9491586 3367 4825 2 3456

1.9188021 4601 6649 3 4800

1.8862479 12,709 18,541 4 13,500

1.8150077 65 97 5 72

1.7851929 319 481 6 360

1.7199837 2291 3541 7 2700

1.6845877 799 1249 8 960

1.6426694 481 769 9 600

1.5861226 4961 8161 10 6480

1.5625 45 75 11 60

1.4894168 1679 2929 12 2400

1.4500174 161 289 13 240

1.4302388 1771 3229 14 2700

1.3871605 28 53 15 45

The columns headed with x and d can be translated as square-side of the short side and the square side of the diagonal.

Try subtracting a square of column x from the square of column d.

This gives rise to a result which is a perfect square.

It is indicated in the added column y

The first column of the left represents the quotient of (d/y)2


How and why are these How and why are these triples derived? triples derived?

Considering that these triples where Considering that these triples where written at a particular time and place written at a particular time and place probably in Larsa around 1800 BCE, an probably in Larsa around 1800 BCE, an understanding of its construction and understanding of its construction and meaning must come from an meaning must come from an understanding of the context of the time understanding of the context of the time and how mathematical tablets were and how mathematical tablets were written.written.


How and why are these How and why are these triples derived?triples derived?

Note: The first column in the Babylonian table is Note: The first column in the Babylonian table is virtually always written in numerical order (either virtually always written in numerical order (either ascending or descending), while subsequent columns ascending or descending), while subsequent columns depend on those to their left.depend on those to their left.

The label in the top (d/y)The label in the top (d/y)22 means the holding square of means the holding square of the diagonal from which 1 is torn out so that the short the diagonal from which 1 is torn out so that the short side comes up.side comes up.

The 1 in the heading indicates that the scribe is dealing The 1 in the heading indicates that the scribe is dealing with reciprocal pairs. with reciprocal pairs.


Relating reciprocals to Relating reciprocals to Pythagorean TriplesPythagorean Triples

Finding integer solution to the equationFinding integer solution to the equation xx22 + y + y22 = d = d22 (dividing by y)(dividing by y) (x/y)(x/y)22 + 1 = (d/y) + 1 = (d/y)22

(Setting u to x/y and v to d/y)(Setting u to x/y and v to d/y) uu22 + 1 = v + 1 = v22

(v+u) (v-u) = 1(v+u) (v-u) = 1

That is we can think of v+u and v-u as the sides of the That is we can think of v+u and v-u as the sides of the rectangle whose area is 1.rectangle whose area is 1.

illustration


Relating reciprocals to Relating reciprocals to Pythagorean TriplesPythagorean Triples

Split of from this rectangle one with Split of from this rectangle one with sides u and v – u and move it to the sides u and v – u and move it to the bottom left after the rotation of 90 bottom left after the rotation of 90 degreesdegrees

The Resulting figure is an L-Shaped The Resulting figure is an L-Shaped figure usually called a gnomon, with figure usually called a gnomon, with long sides both equal to v, a figure long sides both equal to v, a figure that is the difference of vthat is the difference of v22 – u – u22 = 1 of = 1 of two squares.two squares.

Note: This square whose area, Note: This square whose area, vv22 = (d/y) = (d/y)22, is the entry in the left most , is the entry in the left most column in the table from tablet column in the table from tablet Plimton 322 has a gnomon area 1 Plimton 322 has a gnomon area 1 torn out so that the remaining square torn out so that the remaining square is the square on the short side, as the is the square on the short side, as the original column heading says.original column heading says.

v + u

v - u uu

v - u

v

u

u


Calculating the entries in Calculating the entries in the Tablethe Table

It is possible that:It is possible that: The scribe began with the value v + uThe scribe began with the value v + u Then he found its reciprocal v – u in a tableThen he found its reciprocal v – u in a table He then solve for u = ½ [(v+u) – (v-u)]He then solve for u = ½ [(v+u) – (v-u)]

The first column in the table is then the value 1 + uThe first column in the table is then the value 1 + u22

Since u, 1, and v satisfy the Pythagorean identity, Since u, 1, and v satisfy the Pythagorean identity, the scribe could find a corresponding Pythagorean the scribe could find a corresponding Pythagorean triple by multiplying each of these values by a triple by multiplying each of these values by a suitable number y, (one chosen to eliminate suitable number y, (one chosen to eliminate fractional values.fractional values.

example


ExampleExample

v + u = 2;15 (=2 ¼) v + u = 2;15 (=2 ¼) (the reciprocal)(the reciprocal) v – u = 0;26,40 (=4/9) v – u = 0;26,40 (=4/9) (the scribe then find u)(the scribe then find u) u = 0;54,10 = 65/72u = 0;54,10 = 65/72

We can find u by taking half the sum of v + u and v – u, but the scribe We can find u by taking half the sum of v + u and v – u, but the scribe found b as square root of 1 + ufound b as square root of 1 + u22 = the square root of 1;48,54,01,40 = the square root of 1;48,54,01,40 and is then equal to 1;20,50 or square root of 1 + uand is then equal to 1;20,50 or square root of 1 + u22 = to 1.8150077 = to 1.8150077 = 1 25/72. Multiplying the values for u, v and 1 by 1, 12 = 72 gives = 1 25/72. Multiplying the values for u, v and 1 by 1, 12 = 72 gives the values 65 and 97 for x and d respectively. (shown in line 5 of the values 65 and 97 for x and d respectively. (shown in line 5 of the Plimton Table as well as the value 72 for y. Conversely, the the Plimton Table as well as the value 72 for y. Conversely, the value of v + u for line 1 of the Plimton Table can be found by adding value of v + u for line 1 of the Plimton Table can be found by adding 169/120 (=1;24,30) and 119/120 (=0;59,30) to get 288/120 (=2;24)169/120 (=1;24,30) and 119/120 (=0;59,30) to get 288/120 (=2;24)


Why were the particular Why were the particular Pythagorean triples on Pythagorean triples on Plimton 322 chosen?Plimton 322 chosen?

Knowing the answer is definitively impossible.Knowing the answer is definitively impossible. It is noticeable that if we calculate the values of v + u It is noticeable that if we calculate the values of v + u

for every line of the tablet they form a decreasing for every line of the tablet they form a decreasing sequence of regular sexagesimal numbers of no more sequence of regular sexagesimal numbers of no more than four places from 2;24 to 1;48.than four places from 2;24 to 1;48.

Notice as well that not all numbers are included there Notice as well that not all numbers are included there are five missing. Probably because that scribe may are five missing. Probably because that scribe may have decided that the table is long enough without have decided that the table is long enough without them.them.

Remarkably it is evident that the scribes were well Remarkably it is evident that the scribes were well aware of the Pythagorean relationship.aware of the Pythagorean relationship.


BM 85196BM 85196

A beam of length 30 stands against a wall. The A beam of length 30 stands against a wall. The upper end slipped down a distance 6. How far upper end slipped down a distance 6. How far did the lower end move?did the lower end move?

This implies d = 30 and y = 24 are the given. This implies d = 30 and y = 24 are the given. We are suppose find for x.We are suppose find for x.

The scribe calculates x using the theorem x = The scribe calculates x using the theorem x = square root of 30square root of 3022 -24 -2422 = square root of 324 = = square root of 324 = 1818


A tablet found in Susa in A tablet found in Susa in modern Iranmodern Iran

The problem is to calculate the radius The problem is to calculate the radius of a circle circumscribed about in an of a circle circumscribed about in an isosceles triangle with altitude 40 and isosceles triangle with altitude 40 and base 60.base 60.

Considering the circumscribed triangle Considering the circumscribed triangle ABC, whose hypotenuse is the desired ABC, whose hypotenuse is the desired radius, the scribed derived the equation radius, the scribed derived the equation rr2 2 = 30= 3022 + (40 – r ) + (40 – r )22 from the from the Pythagorean Theorem. He then Pythagorean Theorem. He then calculated that 1,20r = 30calculated that 1,20r = 3022 + 40 + 4022 = = 41,40 and, using reciprocals, that 41,40 and, using reciprocals, that r – (0;0,45)(42,40) = 31;15.r – (0;0,45)(42,40) = 31;15.

40 A

BC

r

END

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Babylonian Geometry