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A Beast this way cometh

                                                             Published Aug 6, 2009

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Daniel Mansfield holds the 3700-year-old Babylonian tablet that he and colleagues used to make their case. UNSW/ANDREW KELLY

This ancient Babylonian tablet may contain the first evidence of trigonometry

By Ron Cowen 

Aug. 24, 2017 , 2:00 PM

 

Trigonometry, the study of the lengths and 

angles of triangles, 

sends most modern high schoolers scurrying to their cellphones to look up angles, sines, and cosines. Now, a fresh look at a 3700-year-old clay tablet suggests that 

Babylonian mathematicians not only developed the first trig table, beating the Greeks to the punch by more than 1000 years, but that they also figured out an entirely new way to look at the subject.

However, other experts on the clay tablet, known as Plimpton 322 (P322), say the new work is speculative at best.

 

Consisting of four columns and 15 rows of numbers inscribed in cuneiform,

the famous P322 tablet was discovered in the early 1900s in what is now southern Iraq 

by archaeologist, antiquities dealer, and diplomat Edgar Banks, the inspiration for the fictional character Indiana Jones.

 

Now stored at Columbia University, the tablet first garnered attention in the 1940s,

when historians recognized that its cuneiform inscriptions contain a series of numbers echoing the Pythagorean theorem,

***

 

In mathematics, the Pythagorean theorem, or Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]

 

{\displaystyle a^{2}+b^{2}=c^{2},}a^{2}+b^{2}=c^{2},

where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. 

The theorem, whose history is the subject of much debate, is named for the Greek thinker Pythagoras,

born around 570 BC.

***

 

 

which explains the relationship of the lengths of the sides of a right triangle. (The theorem: The square of the hypotenuse equals the sum of the square of the other two sides.) But why ancient scribes generated and sorted these numbers in the first place has been debated for decades.

 

 

The cuneiform inscriptions on Plimpton 322 suggest the Babylonians used a form of trigonometry based on the ratios of the sides of a triangle, rather than the more familiar angles, sines, and cosines. UNSW/ANDREW KELLY

Mathematician Daniel Mansfield of the University of New South Wales (UNSW) in Sydney was developing a course for high school math teachers in Australia when he came across an image of P322. Intrigued, he teamed up with UNSW mathematician Norman Wildberger to study it. “It took me 2 years of looking at this [tablet] and saying ‘I’m sure it’s trig, I’m sure it’s trig, but how?’” 

Mansfield says. The familiar sines, cosines, and angles used by Greek astronomers and modern-day high schoolers were completely missing. Instead, each entry includes information on two sides of a right triangle: the ratio of the short side to the long side and the ratio of the short side to the diagonal, or hypotenuse.

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Mansfield realized that the information he needed was in missing pieces of P322 that had been reconstructed by other researchers. “Those two ratios from the reconstruction really made P322 into a clean and easy-to-use trigonometric table,” he says. He and Wildberger concluded that the Babylonians expressed trigonometry in terms of exact ratios of the lengths of the sides of right triangles, rather than by angles, using their base 60 form of mathematics, they report today in Historia Mathematica. 

“This is a whole different way of looking at trigonometry,” Mansfield says. “We prefer sines and cosines … but we have to really get outside our own culture to see from their perspective to be able to understand it.”

 

If the new interpretation is right, P322 would not only contain the earliest evidence of trigonometry, but it would also represent an exact form of the mathematical discipline, rather than the approximations that estimated 

numerical values for sines and cosines provide, notes Mathieu Ossendrijver, a historian of ancient science at Humboldt University in Berlin.

 

The table, he says, contains exact values of the sides for a range of right triangles. That means that—as for modern trigonometric tables—someone using the known ratio of two sides can use information in the tablet to find the ratios of the two other sides.

 

What’s still lacking is proof that the Babylonians did in fact use this table, or others like it, for solving problems in the manner suggested in the new paper, Ossendrijver says.

 

And science historian Jöran Friberg, retired from the Chalmers University of Technology in Sweden, 

blasts the idea. The Babylonians “knew NOTHING about ratios of sides!” 

he wrote in an email to Science.

He maintains that P322 is “a table of parameters needed for the composition of school texts and, [only] incidentally, a table of right triangles with whole numbers as sides.”

But Mansfield and Wildberger contend that the Babylonians, expert surveyors, could have used their tables to construct palaces, temples, and canals.

 

Mathematical historian Christine Proust of the French National Center for Scientific Research in Paris, an expert on the tablet, 

calls the team’s hypothesis “a very seductive idea.” But she points out that no known Babylonian texts suggest that the tablet was used to solve or understand right triangles.

The hypothesis is “mathematically robust, but for the time being, it is highly speculative,” she says. A thorough search of other Babylonian mathematical tablets may yet prove their hypothesis, Ossendrijver says. “But that is really an open question at the moment.”

 

Posted in: ArchaeologyAsiaMath

doi:10.1126/science.aap7690

 

Ron Cowen

 

The Amorite XI

 

ap-par

 (-pir)v. intr. ap-peared, ap-pear-ing, ap-pears. 1. To become visible: a plane appearing in the sky. 2. To come into existence: New strains of viruses appear periodically. 3. To seem or look to be: appeared unhappy. 4. To seem likely: They will be late, as it appears. 5. To come before the public: has appeared in two plays; appears on the nightly news. 6. Law. To present oneself formally before a court as defendant, plaintiff, or counsel.[Middle English aperen, from Old French aparoir, aper-, from Latin apparere : ad-, ad- + parere, to show.]

 

*****

re-nown

(ri-noun)n. 1. The quality of being widely honored and acclaimed; fame. 2. Obsolete. Report; rumor.[Middle English renoun, from Anglo-Norman, from renomer, to make famous : re-, repeatedly (from Latin. See RE-) + nomer, to name, from Latin nominare, from nomen, nomin-, name. See no-men-.]

 

*****

ar-ray

(-ra)v. tr. ar-rayed, ar-ray-ing, ar-rays. 1. To set out for display or use; place in an orderly arrangement: arrayed the whole regiment on the parade ground. 2. To dress in finery; adorn.n. 1. An orderly, often imposing arrangement: an array of royal jewels. 2. An impressively large number, as of persons or objects: an array of heavily armed troops; an array of spare parts. See Synonyms at display. 3. Splendid attire; finery. 4.   Mathematics. A rectangular arrangement of quantities in rows and columns, as in a matrix. Numerical data linearly ordered by magnitude. 5. Computer Science. An arrangement of memory elements in one or several planes.[Middle English arraien, from Anglo-Norman arraier, from Vulgar Latin *arredare. See reidh-.]

 

*****

Naught also nought

 (not)n. 1. Nonexistence; nothingness. 2. The figure 0; a cipher; a zero.pron. Nothing: All their work was for naught.adj. 1. Nonexistent. 2. Insignificant.[Middle English, from Old English nawiht : na, no. See ne. + wiht, thing. See wekti-.]

 

*****

bow2             

bou)v. bowed, bow-ing, bows.v. intr. 1. To bend or curve downward; stoop. 2. To incline the body or head or bend the knee in greeting, consent, courtesy, acknowledgment, submission, or veneration. 3. To yield in defeat or out of courtesy; submit. See Synonyms at yield.v. tr. 1. To bend (the head, knee, or body) to express greeting, consent, courtesy, acknowledgment, submission, or veneration. 2. To convey (greeting, for example) by bending the body. 3. To escort deferentially: bowed us into the restaurant. 4. To cause to acquiesce; submit. 5. To overburden: Grief bowed them down.

 

******

en-treat            

en-tret)    also in-treat (in-)v. en-treat-ed, en-treat-ing, en-treats.v. tr. 1. To make an earnest request of. 2. To ask for earnestly; petition for. 3. Archaic. To deal with; treat.v. intr. To make an earnest request or petition; plead. See Synonyms at beg.[Middle English entreten, from Anglo-Norman entreter : en-, causative pref. See EN-1 + treter, to treat. See TREAT.]--en-treat'ing-ly adv. --en-treat'ment n.

 

 

Excerpted from American Heritage Talking Dictionary

******************

 

 

 

Now

it

came to pass,

when

 

Adonizedec

king

of

Jerusalem

had heard

how Joshua

had taken Ai,

and had utterly destroyed it; . . .

 

***

And Melchizedek

king

 

of

Salem

 

brought forth

bread

and

wine:

**

 

 

and the

darkness

he

called Night.

*

and

he

was the priest

of

the

 

most high God.

****

 

So God

created

man

in his

own image,

 

**

 

And

he

blessed

 

him,

**

 

O

earth,

cover not

thou my blood,

*

And God

said,

 

Let

the

waters under

the heaven

be

gathered

 together unto

 

one place,

**

 

and

let

my cry

have

no place.

 

Also

now,

 

behold,

 

my

witness

isin

heaven,

 

 

 and my record

ison

 high.

**

 

and said,

Blessed

be

Abram

of

the

most high God,

 

possessor

of

heaven

and earth:

 

And blessed

be

the

most high God,

 

which hath

delivered

thine

enemies

into

thy

hand.

**

 

These are

the

generations

of

the

heavens

 

and

of

the

earth

when

 

they

were created,

***********

 

 

 

in

the

day

that the

 

Lord God

 

made

 

the earth

and the heavens,

    

And every

plant

of

the field

before

it

was

in

the earth,

 

 

and

every herb

of the

field

before

it grew:

 

for

the

Lord God

had

not caused

it

to

rain

 

upon the earth,

 

and there

was

not a man

to  

till

the ground.

*

And the

evening

and

the

morning

were

the

first day.

 

*******

 

And when

they

were come

into

the house,

 

and fell down,

 

and worshipped

him:

**

 

And

he

gave

him

tithes of all.

******

 

And Adam

knew

Eve

his wife;

and said,

I

have gotten

a

man

from

the

Lord.

 

*

And

I

will bring

the

blind

 

by a way

 

that they

knew not;

 

I

will lead

them

in

paths

that they

have not known:

I

will make

darkness

 

light

before them,

and crooked things straight.

 

These things

will

I

do

unto them,

 

and not

forsake

them.

*

 

 

And God

made

two great lights;

 

the

greater light

to

rule

the

day,

 

 

and the lesser light

to

rule the night:

 

he

made

the

stars also.

 

*

ninety years,

and

begat

 

Cainan:

 

And Enos

lived

after

 

he

begat Cainan

 

eight hundred

and fifteen years,

 

and begat

sons

and daughters:

*

 

That

the

sons

of

God

saw

 

the

daughters

of

men

 

that they

were

fair;

and they

took

them

wives

of

all

which they chose.

 

And

the

Lord said,

 

My spirit

 

shall not

 

always strive

with

man,

 

for that

he

also

 

is flesh:

 

and also

after that,

 

when the

sons

of

God

came in unto

the daughters

of

men,

 

and

they

 bare

children

to

them,

the same

became

 

mighty men

which were

of

old,

men

of

renown.

 

***

 

And Abram

was

fourscore

and six years old,

 

when

Hagar bare

Ishmael

to

Abram.

****

 

 

And the

Lord

said

unto

Cain,

 

Where

is

Abel

thy brother?

 

And he said,

I

know not:

 

Am I

my

brother's keeper?

***

and

he

called

his

name

Enos:

 

then

began men

to

 call upon

the name

of

the

Lord.

**

 

And Seth

lived after

he

begat

Enos

 

eight hundred

and seven years,

 

and begat sons

and daughters:

 

And all

the

days

of

Seth

were

nine hundred

and twelve years:

and

he

died.

 

And Enos

lived

ninety years,

and

begat

 

Cainan:

 

And Enos

lived

after

 

he

begat Cainan

 

eight hundred

and fifteen years,

 

and begat

sons

and daughters:

*

 

That

the

sons

of

God

saw

 

the

daughters

of

men

 

that they

were

fair;

and they

took

them

wives

of

all

which they chose.

 

And

the

Lord said,

 

My spirit

 

shall not

 

always strive

with

man,

 

for that

he

also

 

is flesh:

 

and also

after that,

 

when the

sons

of

God

came in unto

the daughters

of

men,

 

and

they

 bare

children

to

them,

the same

became

 

mighty men

which were

of

old,

men

of

renown.

 

***

 

*

And

when

Abram

was

ninety

years

old

and nine,

**

 

the

Lord

 

appeared

to

Abram,

 

**

 

and said

 

unto

 

him,

 

I am

the

 

Almighty

God;

 

walk before

me,

 

**

 

There

was

a

man

in the land

of

Uz,

whose name

was

Job;

 

and

that man

was perfect

 

and upright,

 

and one that feared

God,

and eschewed evil.

 

**

walk

 

before

me

 

And

I

will

make my covenant

 

between

 

me and thee,

**

 

And he

said,

Who

told thee

that

thou wast

naked?

 

Hast thou

eaten

of

the

tree,

 

where

of

I

commanded

thee

 

that thou

 

shouldest

not eat?

 

**

And

he

 

entreated

 

 

Abram well

For

 her

sake:

and

he

had sheep,

and oxen,

and

he

asses,

and menservants,

and maidservants,

and she asses,

and camels.

**

 

And Sarai Abram's wife

took

Hagar

her maid

the

Egyptian,

 

 after

Abram

had dwelt ten years

in

the

 land of Canaan,

and gave

her to her

husband

Abram

to

be

his wife.

 

And

he

 went in unto

Hagar,

and

she

conceived:

and when

she

saw that

she

had conceived,

 

her

mistress

was

despised

 

in

her eyes.

**

 

And the

man

said,

The

woman

whom thou

gavest

to

be with me,

 

she

gave me

of

the

tree,

 

and

I

did eat.

**

 

and will

multiply

thee

exceedingly.

 

And Abram

fell

 

on

his face:

 

 

and God

talked

with

him,

 

saying,

As for me,

behold,

my

covenant

is

with thee,

 

and thou

shalt

be a

father

of

many nations..

**

 

I

will make

darkness

light

before them,

 

**

And

I

will make

my

covenant

 

between

 

me

 

and thee,

 

and

will

multiply

thee

exceedingly.

**

 

Unto the woman

he said,

I

will greatly multiply

thy

sorrow

and

thy

conception;

in

sorrow

thou

shalt bring forth children;

 

and

thy

 desire shall

be

to

 thy husband,

and he

shall

rule over thee.

**

Yea,

hath God said,

Ye shall not eat

of

every tree of the garden?

 

*

God

hath said,

Ye

shall

not eat

of

it,

neither shall

ye touch

it,

lest ye die.

*

 

Yea,

they are

greedy dogs

which

can

never have enough,

 

and they are

shepherds

that

cannot understand:

 

they all

look

to

their

own way,

every one

for

his gain,

from his quarter.

 

Come ye,

 

say they,

I

will fetch wine,

and we

will fill

ourselves

with

strong drink;

 

and tomorrow shall be as this day,

 

and much more abundant.

 

But your iniquities

have

separated

 

between

 

you

 

and

your

God,

**

 

Your

country is desolate,

your cities

are burned

with

fire:

 

 your land,

strangers devour

it in

your

presence,

and

it is

desolate,

as

overthrown

by

strangers.

 

and your

sins

have

hid

his

face

from you,

 

that

he

will not

hear.

**

 

the voice

of

thy

 brother's blood

 

crieth

unto me

from

the

ground.

***

 

 

Behold,

I

cry out

of

wrong,

 

but

I am

not

heard:

 

I

cry aloud,

 

but there

is no

judgment.

 

He

hath fenced

up

my way

that

I

 

cannot pass,

 

and

he

hath

set darkness

in

my

paths.

 

**

For

your

hands

are

defiled

with

blood,

**

 

Their

land

also

is

full of idols;

 

they

 worship

the

work

of

their

own hands,

 

that

which

their own fingers

have

made:

***

 

and your

fingers

with

inequity;

 

your lips

have

spoken lies,

 

your tongue

hath

muttered perverseness.

 

None

calleth

for justice,

 

nor any

pleadeth

for

truth:

**

 

 

Noah

was a

just man

and

perfect

in his

generations,

and Noah

walked with God.

***

And when Abram

was

ninety years old

and nine,

the

Lord

appeared

to

Abram,

 

and

said unto

him,

 

I am

the

Almighty God;

 

walk before

me,

 

and

be

thou

perfect.

 

And

I

will make

my

covenant

 

between

 

me

 

and thee,

and will multiply

thee

exceedingly.

 

And Abram

fell

on

his face:

and

God

talked with

him,

saying,

    

As for me,

behold,

my

covenant

is

with thee,

and

thou

shalt

be

a

father of many nations.

 

Neither shall

Thy

 name

any more

be called

Abram,

 

but thy

name shall

be

Abraham;

 

for

a

father

of

many nations

have

I

made thee.

*

 

And Abram

fell

on

his

face:

**

and

his

countenance

fell.

************

 

And Joseph

was

brought down

to

Egypt;

 

and Potiphar,

an

officer

of

Pharaoh,

 

captain

of the

guard,

 

an Egyptian,

bought

him

 of the hands

of the

Ishmeelites,

which had

brought

him down

thither.

 

And the

Lord

was with

Joseph,

 

and

he

was

a

prosperous

 

man;

*

and the

darkness

he

called Night.

*

 

 

and

he

was

in

the house

 

of

his

master

 

the

Egyptian.

 

 

ISamuelYea