Rhind Mathematical Papyrus (4 of 8)

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RMP 21, 22 and 23 - Re had many names

By rolling a finely carved and polished stone disk on a carefully prepared ground one may find that the ratio circumference to diameter is less than 4, a little more than 3, less than 3 '6, and even a little less than 3 '7. These empirical values or boundaries can be used for generating many more approximate values for the number of the circle, which I call re, for the hieroglyph of the supreme sun god Re was a small circle, and while he had many names, no one knew his true one ...

You can easily see what I do:

4 (plus 3) 7 10 13 16 19 22 25 28

1 (plus 1) 2 3 4 5 6 7 8 9

If the diameter of a circle measures 7 palms or 1 royal cubit, the circumference measures practically 22 palms or 3 cubits 1 palm.

6 (plus 19) 25 44 63

2 (plus 6) 8 14 20

Solve RMP 37 using the value '20 of 63 = 3 '10 '20.

9 (plus 19) 28 47 66 ... 256

3 (plus 6) 9 15 21 ... 81

A famous formula of the Rhind Mathematical Papyrus says that a square with a side length of 8 royal cubits and a circle with a diameter of 9 royal cubits have roughly the same area. This formula is based on a value of 256/81 for pi.

3 (plus 22) 25 47 69 91 113 135 157 179 201 223 245

1 (plus 7) 8 15 22 29 36 43 50 57 64 71 78

267 289 311 333 355 377 399 421 443 465 487 509 531

85 92 99 106 113 120 127 134 141 148 155 162 169

If the side of a square measures 10 royal cubits or 70 palms, the diagonal measures practically 99 palms, and the circumference of the circle around the square measures practically 311 palms. - If you wish to know the circumference of a circle, multiply the diameter by one of the following series:

'99 of 311 = 3 '9 '33 '120 x 377 = 3 '8 '60 = 3 '10 '24

Yet another sequence:

6 (plus 22) 28 50 72 ... 424 ... 600

2 (plus 7) 9 16 23 ... 135 ... 191

191/600 = '600 x 191 = '4 '30 '40 '100

If you know the circumference of a cylinder and wish to know its diameter, multiply the circumference by '4 '30 '40 '100.

21 (plus 22) 43 65 ... 2463

7 (plus 7) 14 21 ... 784 = 28x28

286 (+311)… 132 (+333)… 333 (+355)… 201 (+377)… 2463

91 (+ 99)… 42 (+106)… 106 (+113)… 84 (+120)… 784 = 28x28

If the radius of a circle measures 1 royal cubit or 28 fingers, the area of the circle measures practically 2463 square fingers.

Now for RMP 21-23:

"3 '15 plus '5 '15 equals 1

"3 '30 plus '5 '10 equals 1

'4 '8 '10 '30 '45 plus '9 '40 equals "3

I multiply the first equation by a factor of 135, the second one by a factor of 50, and the third one by a factor of 360:

99 + 36 = 135 35 + 15 = 50 161 + 49 = 240

Consider these numbers as diameters of nine circles. How long are their circumferences? Consulting the above number sequences you will derive the following solutions and values of re:

c 47 22x5 113 22x7 157 311 424 600 377x2

d 15 7x5 36 7x7 50 99 135 191 120x2

These values can easily be given as unit fraction series:

'15 of 47 = 3 '10 '30 '7 of 22 = 3 '7

'36 of 113 = 3 '9 '36 '50 of 157 = 3 '10 '25

'99 of 311 = 3 '9 '33 '135 of 424 = 3 '15 '27 '135

'120 of 377 equals 3 '10 '24 or 3 '8 '60

'600 of 191 equals '4 '30 '40 '100

RMP 36 - a pair of granaries

3 '3 '5 times '4 '53 '106 '212 equals 1

Imagine a square whose diagonal measures 2 royal cubits. Its area measures 2 square cubits.

Imagine a circle whose diagonal measures 3 royal cubits. Using the value '135 of 424 for re, the area of the circle measures 7 '15 square cubits.

Build a granary on the square (inner diagonal 2 royal cubits). Build a granary on the circle (inner diameter 3 royal cubits).

Fill the round granary to a height of 1 royal cubit. You will need 53 quadruple-hekat or 212 hekat of barley. Fill 212 hekat of grain in the square granary. The barley will reach a height of 3 '3 '5 royal cubits.

Fill the square granary to a height of 1 royal cubit. You will need 2 cubic cubits or 3 khar or 15 quadruple-hekat or 60 hekat of barley. Fill 60 hekat of barley in the round granary. It will reach a height of '4 '53 '106 '212 royal cubits.

RMP 31 - a granary on a ring

33 divided by 1 "3 '2 '7 equals 14 '4 '56 '194 '388 '679 '776

Imagine a regular hexagon whose side measures 66 fingers. Inscribe and circumscribe a circle. The two circles form a ring. The radius of the outer circle measures 66 fingers. How long is the radius of the inscribed circle in palms?

1 1 3

2 4 6

1 2 3

3 5 9

8 14 24

4 7 12

11 19 33

30 52 90

15 26 45

41 71 123

112 194 336

56 97 168

The side of the hexagon measures 66 fingers. Multiply 66 fingers by 168/97 and you obtain the diameter of the inscribed circle in fingers. Multiply 33 fingers by 168/97 and you obtain the radius in fingers. Divide 33 fingers by 97/42 = 1 "3 '2 '7 and you obtain the radius of the inscribed circle in palms:

33 fingers divided by 1 "3 '2 '7 equal 14 '4 '56 '97 '194 '388 '679 '776 palms

The area of the ring is given by the difference

area circumscribed circle minus area inscribed circle

The area of the circumscribed circle is found as follows:

radius x radius x re

66 fingers x 66 fingers x '99 x 311 = 13,684 square fingers

Now for the area of the inscribed circle. It measures

14 '4 '56 '194 '388 '679 '776 palms times 14 '4 '56 '194 '388 '679 '776 palms times re

Is anyone prepared to carry out that multiplication ???

Ahmes would smile and offer a much simpler solution based on a fine theorem:

Imagine a regular polygon of 3, 4, 5, 6, 7 ... equal sides. The circumscribed circle and the inscribed circle form a ring. Draw a circle around a side of the polygon. Its area equals the area of the ring.

The side of the regular hexagon measures 66 fingers, the radius of the circle around a side measures 33 fingers, and the area of the ring measures

33 fingers x 33 fingers x '99 x 311 = 3,421 square fingers

The area of the outer circle measures 13,684 square fingers, the area of the ring measures 3,421 square fingers, and the area of the inner circle measures 13,684 - 3,421 = 10,263 square fingers. Comparing these areas reveals the following proportions:

area inner circle / area ring / area outer circle = 3 / 1 / 4

Build a granary on the ring. If the height measures 5 '2 royal cubits or 154 fingers, the volume of the wall measures practically 24 cubic cubits, and the capacity 72 cubic cubits or 108 khar or 540 quadruple-hekat or 2160 hekat.

RMP 7 to 20 - spheres holding rectangles

RMP 7 '4 '28 times 1 '2 '4 equals '2

Imagine a rectangle measuring '4 '28 by 1 '2 '4 royal cubits or 8 by 49 fingers. Its area measures half a square cubit. Transform this area into a square, Its diagonal measures 1 royal cubit or 7 palms or 28 fingers. Draw a circle around the square. How long is its circumference?

1 royal cubit x '7 x 22 equals 3 '7 royal cubits

7 palms x '7 x 22 equal 22 palms

28 fingers x '7 x 22 equal 88 fingers

The circumference measures 3 '7 royal cubits or 22 palms or 88 fingers (mistake less than one millimeter).

Partition the length of the rectangle 8 by 49 fingers according to the number 1 '2 '4. Thus you obtain three rectangles measuring

'4 '28 by 1 royal cubit or 8 by 28 fingers

'4 '28 by '2 royal cubit or 8 by 14 fingers

'4 '28 by '4 royal cubit or 8 by 7 fingers

Imagine a sphere holding the rectangle 8 by 28 fingers and calculate the surface of the sphere using the formula: area circle = diameter x diameter x re. Choose a handy value of re from the number sequence

3 (plus 22) 25 47 69 91 113 ... 311 333 355 377

1 (plus 7) 8 15 22 29 36 ... 99 106 113 120

The diameter squared is found as follows:

8x8 plus 28x28 = 64 plus 784 = 848 = 8 x 106

Using the value '106 of 333 for re we obtain

8 x 106 x '106 x 333 = 2,664 square fingers

The surface of the sphere around the rectangle 8 by 28 fingers measures 2,664 square fingers (mistake 25 square millimeters). Imagine a sphere holding the rectangle 8 by 7 fingers and calculate its surface in the same way:

8x8 plus 7x7 equals 113 113 x '113 x 355 equals 355

The surface of the sphere around the rectangle 8 by 7 fingers measures 355 square fingers (mistake '94 square millimeter).

Imagine a sphere holding the long rectangle 8 by 49 fingers and calculate the surface again, this time using the values '99 of 311 = 3 '9 '33 and '120 of 377 = 3 '8 '60. Round all the numbers:

8x8 plus 49x49 equals 2465

2465 x 3 '9 '33 -- 7395 274 75 = 7744 square fingers

2465 x 3 '8 '60 -- 7395 308 41 = 7744 square fingers

The surface of the sphere holding the rectangle 8 by 49 fingers measures practically 7744 square fingers (mistake 9 square mm). Thus we found a new value for re: '2465 of 7744

289 (plus 355) 649 999 ... 2419 ... 5969 ... 7389 7744

92 (plus 113) 205 318 ... 770 ... 1900 ... 2352 2465

Transform the area 7744 square fingers into a square:

7744 square fingers = 88 fingers by 88 fingers

The side of the square measures 88 fingers (mistake '362 mm). The sphere around the rectangles 8 by 49 fingers holds another rectangle that measures 23 by 44 fingers. Diameter squared:

8x8 plus 49x49 = 23x23 plus 44x44 = 2465 square fingers

The square 44 by 44 fingers and a circle around the rectangle 23 by 44 fingers have practically the same area:

44x44 = 23x23 plus 44x44 x '2465 x 7744 = 1936 square fingers

If you wish to transform a square into a circle of the same area you may shorten the square from 44 to 23 equal parts and draw a circle around the resulting rectangle. - A fairly good approximation is obtained by simply bisecting the square. This solution is based on the value '81 of 256 for re.

RMP 8 '4 times 1 "3 '3 equals '2

A rectangle measures 7 by 56 fingers. The imaginary circle around this rectangle circumscribes another rectangle that measures 28 by 49 fingers or 1 by 1 '2 '4 royal cubits.

7x7 plus 56x56 = 28x28 plus 49x49 (diameter squared)

The periphery of the rectangle 28 by 49 fingers measures 154 fingers. Transform the periphery into the circumference of a circle. Use the value '7 x 22 for re. You will obtain 49 fingers or 1 '2 '4 royal cubits.

154 f x '22 x 7 = 49 fingers or 1 '2 '4 royal cubits

RMP 9 '2 '14 times 1 '2 '4 equals 1

A rectangle measures 16 by 49 fingers. Imagine a sphere holding this rectangle. The sphere also holds a right parallelepiped that measures 28 by 28 by 33 fingers. The face 28 by 28 fingers represents a square cubit and corresponds to the area of the long rectangle measuring 16 by 49 fingers.

16x16 + 49x49 = 28x28 + 28x28 + 35x35 (diameter squared)

RMP 10 '4 '28 times 1 '2 '4 equals '2 (see RMP 7)

A rectangle measures 8 by 49 fingers. The sphere around this rectangle holds a right parallelepiped that measures 1 royal cubit x 10 palms x 9 fingers = 28 by 40 by 9 fingers. The narrow faces measure 9 by 40 fingers each; their diagonals measure exactly 41 fingers, according to the triple 9-40-41.

8x8 + 49x49 = 9x9 + 28x28 + 40x40 9x9 + 40x40 = 41x41

Imagine a sphere holding the rectangle 9 by 40 fingers and calculate its surface using the series 3 '9 '33 and 3 '8 '60:

9x9 plus 40x40 = 41x41 = 1681

1681 x 3 '9 '33 -- 5043 187 51 = 5281

1681 x 3 '8 '60 -- 5043 210 28 = 5281

A new value for the number re: '1681 of 5281

311 (plus 355) 666 1021 ... 4926 (2463) 5281 5636

99 (plus 113) 212 325 ... 1568 (784) 1681 1794

RMP 11 '7 times 1 '2 '4 equals '4

A rectangle measures 4 by 49 fingers. The imaginary sphere around this rectangle holds a right parallelepiped that measures 14 by 14 by 45 fingers. Volume or capacity 12 '28 '56 hekat. Using the value '20 of 63 for re, this volume can be transformed into a cylinder whose diameter, circumference and height measure 5 palms, 63 fingers and 1 royal cubit.

RMP 12 '14 times 1 '2 '4 equals '8

A rectangle measures 2 by 49 fingers. The sphere around it holds a right parallelepiped that measures 20 by 22 by 39 fingers. The numbers 22 and 39 provide a good value for the square root of 3 '7 or '7 of 22.

39 x 39 x 7 plus 1 = 22 x 22 x 22

RMP 13 '16 '112 times 1 '2 '4 equals '8

A rectangle composed of six rectangles measures 2 by 49 fingers (see RMP 12). The smallest rectangle measures '112 by '4 royal cubit and has an area of 1 '2 '4 square fingers. The largest rectangle measures '16 by 1 royal cubit and has an area of 49 square fingers (RMP 14).

RMP 14 '28 times 1 '2 '4 equals '16

A rectangle measures 1 by 49 fingers. A square that measures 7 by 7 fingers has the same area. Imagine a sphere around the square and calculate its surface. Using the value 3 '7 for re you will obtain 308 square fingers.

7x7 plus 7x7 times 3 '7 equals 308

RMP 15 '32 '224 times 1 "3 '3 equals '14

A rectangle measures 1 by 56 fingers. The rectangles 4 by 14 fingers and 8 by 7 fingers have the same area. Draw a circle around the rectangle 4 by 14 fingers and imagine a hemisphere standing on the circle. Its surface measures practically 333 square fingers. Now imagine a sphere holding the rectangle 8 by 7 fingers. The surface measures practically 355 square fingers.

RMP 16 '2 times 1 "3 '3 equals 1

A rectangle measures 14 by 56 fingers and has the same area as a square that measures 7 by 7 palms or 28 by 28 fingers. Imagine a sphere around the square and calculate its surface. Using the value 3 '7 for re, the surface measures 308 square palms. Using the value '784 of 2463 for re, the surface measures 2463 square fingers.

RMP 17 '3 times 1 "3 '3 equals "3

A triple square, a double square and a square measure '6 by 1 square cubit, '6 by "3 square cubit, and '6 by '3 square cubit. The diagonals of the square, of the double square and of the triple square can be approximated by means of my number columns that begin with the lines 1-1-2 and 1-1-5. Draw a circle around each figure. Their areas will maintain the ratio 2:5:10.

RMP 18 '6 times 1 "3 '3 equals '3

The area of a rectangle measures '3 square cubit. Transform this area into a square. Imagine a sphere holding the square. Calculate its surface using the value '784 of 2463 for re. The diameter squared measures '3 plus '3 = "3 square cubits. Multiply this area by 2463 and you obtain 1642 square fingers.

RMP 19 '12 times 1 "3 '3 equals '6

The area of a rectangle measures '6 square cubit. Transform this area into a square. Imagine a sphere holding the square. Calculate the surface of the sphere:

'6 '6 square cubits x 2463 = 821 square fingers

A highly demanding task, solvable in a very simple manner (mistake only one square millimeter).

RMP 20 '24 times 1 "3 '3 equals '12

The area of a rectangle measures '12 square cubit. Transform it into a square. Imagine a sphere holding the square. Transform the surface of the sphere into a circle. Imagine a second sphere holding the circle. It will be the sphere from RMP 18, whose surface measures practically 1642 square fingers.

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