?a ?a??

1
?a ?a??µat??? st?? ???a?a ????pt?
?. ?ast???

• ? p??te? µ???µe? e??atast?se?? ???? ap? t? ?e???
  ????a? t?? pe???d? 8.000-6.000 p.?.
• Sta µ?sa t?? 4?? ????et???da? d?µ?????????a? d??
  ßas??e?a t? ???e?? st?? ??t? ????pt? ?a? t?
  ??t?? st?? ??? ????pt?.
• G??? st? 3.000 p.?. ? ßas????? ????? ? ???e? ap?
  t?? ??? ????pt? ????e?se t? ßas??e?? t?? ??t?
  ????pt?? ?a? e??p???se t?? pe????? s???a
  ßas??e??. ? ???a µetat??p??e se µ?a s???e?t??t???
  desp?te?a
• ?e t?? ???e? ????se ? ???µa??µe?? ep??? t??
  d??aste???, ? ep??? t?? Fa?a? (µ???? t? 341
  p.?.).

2
?st???a t?? ???a?a? ????pt??
??apa??stas? t?? ????a???a? t?? ???e?
3
? ???a?a ????pt??
4
? ??af? st?? ???a?a ????pt?

• ?? p??t? e?d?? ??af?? st? ???a?a ????pt?
  eµfa??st??e t? 3.100 p.?. pe??p?? ?a? ?ta? ?
  ?e?????f???.
• G??? st? 2.000 p.?. ep????t?se ??a ???? e?d??
  ??af?? p?? ???µ??eta? ?e?at??? ??af?.
• ?a? ???? st? 700 p.?. eµfa??st??e ?a? ? d?µ?t???
  ??af?.

5
??d? ??af?? st?? ???a?a ????pt?
6
?? ??afe?? st? ???a?a ????pt?
7
?? p???? t?? a???pt?a??? ?a??µat????.

• ??p???? Rhind, e??a? µ?a s?????? 84 p??ß??µ?t??
  p?? a?t????ft??e pe??p?? t? 1650 p.?. ap? ??a
  p??t?t?p? t?? 1850 p.?.
• ??p???? t?? ??s?a?, ???ft??e ???? st? 1850 p.?.
  ???a? µ?a s?????? 25 p??ß??µ?t??.
• ? de?µ?t???? ?????d???, p?? ???ft??e ???? st?
  1650 p.?. ?a? pe????e? 26 a????sµata µ??ad?a???
  ??asµ?t??.
• ?p?s?? ?p???e? ? p?p???? Kahun ?a? ? p?p???? t??
  ?e???????, p?? e??a? t?? 1850 p.?. pe??p?? ?a?
  pe??????? µa??µat???? p???e?? ?a? p??ß??µata.

8
????pt?a??? p?p???? µe ?a??µat???
??p???? t?? ??s?a?
??p???? Rhind
9
?? pa?ast?se?? t?? a???µ?? st?? ???a?a ????pt?
????µ?? st?? ?e?at??? ??af?
????µ?? st?? ?e?????f??? ??af?
10
????µ?t???? p???e?? st?? ????t?sµ? t?? ???a?a?
????pt??
???s?es?
11
?p? t? p??ß??µa 79 t?? p?p???? Rhind
?eta??af? t?? ?e?at???? ??af?? t?? a???µ?? se
s??????? ap?d?s?. ???ta ap? de??? sta a??ste??
?a? st? s????e?a s?µf??a µe t?? s?µe???? t??p?.
12
????µ?t???? p???e?? st?? ????t?sµ? t?? ???a?a?
????pt??
????ap?as?asµ??
16 . / 160 10 / 80 5 / 256
ap?t??esµa
16?16
13
????ap?as?asµ??
12?12
17 1/ 34 2 68 4/ 136 8/ ----------------
---------------- 221 13
17?13
14
??a??es?
1 80 2 160 4 / 320 / 10
/ 800 / -------------------------------------
---------------- 14 1120
297 12
1 12 2 24 4
48 8 / 96 / 16 /
192 / ½ / 6 / ¼ /
3 / ----------------------------------------
-------------------------------- 24 ½ ¼ 297
1120 80
15
???sµata
16
???a?a? t?? ??asµ?t??
µe n2?1. ?p? t?? ??p??? Rhind
17
???ß??µata
?p? t?? p?p??? Rhind ?a µ???ast??? 6 ??µ?? se 10
a????p???. ?p??t?s?
?pa???e?s? 1
??a??es? 1 10
2
1
5 /
1 /
4
2
8
4
18
?p?????sµ?? a??
? ???? a?? ? ?a s?µa??e? p?s?t?ta ? s????
??p???? p?a?µ?t??.
???ß??µa 26 t?? p?p???? Rhind ??a p?s?t?ta ?a?
t? t?ta?t? µ???? a?t?? ?????? µa?? 15. ???a e??a?
? p?s?t?ta ?st? 4. ??te t? 4 ?a? 1 (t? t?ta?t?
µ???? t?? 4) ???e? 5 ?a? ??? 15. G?a ?a ß?e?e? t?
s?st?, p??ta ?p??????eta? ? ap????s? ?a?
s?µe???eta? ?t? t? 15 e??a? t??p??s?? t?? 5. ?ts?
? d?????s? t?? a?????? a??a??et?? pa?ad????
???eta? µe t??p?as?asµ? t??, 3 f???? t? 4, d??.
12. ??ap?st??eta? ?t? a?t? e??a? s?st?, ??at? 12
?a? 3 (t? t?ta?t? t?? 12) ???e? 15. ? µ???d??
a?t?, p?? ???s?µ?p??????e ?a? se
µeta?e??ste?e? ep????, ???µ??eta? µ???d?? t??
a??a??et?? pa?ad???? ? µ???d?? t?? ?a??asµ????
??s??.
19
Ge?µet????? ???se?? t?? ????pt???
?p? t?? ?p?????se? µa?t???e? d?ap?st??eta? ?t? ??
????pt??? ??????a? ?a ?p????????? ta eµßad?
?????????? pa?a????????µµ??, ??????????
t???????, t?ape??e?d??, ??????. ?p?s?? ??????a?
?a ?p????????? ?????? ?????d???, p??aµ?d?? ?a?
???????? p??aµ?d??.
???ß??µa 51 t?? p?p???? Rhind ?a ?p?????ste? ?
ep?f??e?a e??? t????????? ???af??? µe ß?s? 4
(µ???de? µ?t??s??) ?a? p?e??? 10.
?? ap?t??esµa ß??s?eta? µe t?? p???ap?as?asµ? t??
µ?s?? t?? 4 ep? 10, d??. 20.
20
???ß??µa 50 t?? p?p???? Rhind ??a ??????? ????f?
??e? d??µet?? 9 (µ???de? µ?t??s??). ??s? e??a?
t? eµßad?? t??
? ?p?????sµ?? ???eta? ?? e??? ???ta ß??s?eta? t?
??a ??at? t?? 9, p?? e??a? 1. ??t? afa??e?ta? ap?
t? 9 ?a? ???eta? 8. ?a? ?p??????eta? t? ????µe??
t?? 8 ep? 8. 1 8 2 16
4 32 \ 8 64 / ?? 64
e??a? t? ap?t??esµa.
? a??????µ?? ??a t?? ?p?????sµ? t?? eµßad??
?????? a?t?st???e?, µe ta s?µe???? ded?µ??a, st??
t?p?
21
???ß??µa 14 t?? p?p???? t?? ??s?a? ?p?????sµ??
t?? ????? ???????? p??aµ?da?, µe ???? 6 (µ???de?
µ?????) ? µ?a ß?s? 4 ?a? ? ???? 2.
???????µ?? ????ap?as?asµ?? t?? 4 ep? 4, ???e?
16. ????ap?as?asµ?? t?? 4 ep? 2, ???e?
8. ????ap?as?asµ?? t?? 2 ep? 2, ???e? 4. ???s?es?
t?? 16 ?a? 8 ?a? 4, ???e? 28. ?? ??a t??t? t?? 6,
???e? 2. ????ap?as?asµ?? t?? 28 ep? 2, ???e?
56. ?? 56 e??a? t? ap?t??esµa.
?e ta s?µe???? ded?µ??a a?t?st???e? st?? t?p?