ÂÙ¤ÅÔ´ (Euclid)
ÂÙ¤ÅÔ´à»ç¹¹Ñ¡¤³ÔµÈÒʵÃì·ÕèÊÓ¤Ñ
áÅÐà»ç¹·ÕèÃÙé¨Ñ¡¡Ñ¹´Õ
ÂÙ¤ÅÔ´à¡Ô´·ÕèàÁ×ͧÍàÅç¡«Ò¹à´ÃÕÂ
»ÃÐà·ÈÍÔÂÔ»µì àÁ×èÍÃÒÇ 365 »Õ
¡è͹¤ÃÔʵ¡ÒÅ
àÁ×èÍÁÕªÕÇÔµÍÂÙ訹¡ÃзÑ觻ÃÐÁÒ³»Õ
300 ¡è͹¤ÃÔʵ¡ÒÅ
ÊÔ觷ÕèÁÕª×èÍàÊÕ§¤×ͼŧҹàÃ×èͧ
The Elements
ËÅÑ¡°Ò¹áÅÐàÃ×èͧÃÒÇà¡ÕèÂǡѺµÑÇÂÙ¤ÅÔ´Âѧ¤§ÊѺʹ
à¾ÃÒÐÁÕ¼Ùéà¢Õ¹äÇéËÅÒÂÃٻẺ
ÍÂèÒ§äáçµÒÁ¼Å§Ò¹àÃ×èͧ The
Elements
Âѧ¤§ËŧàËÅ×ÍÍÂÙ訹¶Ö§·Ø¡Çѹ¹Õé
¨Ò¡ËÅÑ¡°Ò¹·ÕèÊѺʹ·ÓãËéÊѹ¹ÔÉ°Ò¹·Õèà¡ÕèÂǡѺÂÙ¤ÅÔ´ÁÕËÅÒÂá¹Ç·Ò§
àªè¹ ÂÙ¤ÅÔ´à»ç¹ºØ¤¤Å·Õèà¢Õ¹àÃ×èͧ The
Element
ËÃ×ÍÂÙ¤ÅÔ´à»ç¹ËÑÇ˹éÒ·ÕÁ¹Ñ¡¤³ÔµÈÒʵÃì·ÕèÍÒÈÑÂÍÂÙè·ÕèÍàÅç¡«Ò¹à´ÃÕÂ
áÅÐä´éªèÇ¡ѹà¢Õ¹àÃ×èͧ The Elements
ÍÂèÒ§äáç´ÕÊèǹãËè¡çÁÑè¹ã¨ÇèÒÂÙ¤ÅÔ´ÁÕµÑǵ¹¨ÃÔ§
áÅÐà»ç¹»ÃÒªìÍѨ©ÃÔÂзҧ´éÒ¹¤³ÔµÈÒʵÃì·ÕèÁÕªÕÇÔµã¹Âؤ¡ÇèÒ
2,000 »Õ ËÅÑ¡¡ÒÃËÒ Ë.Ã.Á.·Õè§èÒ·ÕèÊØ´áÅÐÃÙé¨Ñ¡¡Ñ¹´Õ¨¹¶Ö§»Ñ¨¨ØºÑ¹¤×Í ãËé¹ÓµÑÇàÅ¢¨Ó¹Ç¹¹éÍÂËÒõÑÇàÅ¢¨Ó¹Ç¹ÁÒ¡ àÈÉ·ÕèàËÅ×ÍÁÒà·Õº¡ÑºàÅ¢¨Ó¹Ç¹¹éÍ ¨ÑºËÒáѹä»àÃ×èÍÂ æ ·Óàªè¹¹Õ騹ŧµÑÇ ä´é Ë.Ã.Á. à»ç¹àÅ¢·ÕèŧµÑǵÑÇÊØ´·éÒÂ
| a = bq1 + r2 , | 0 < r2 < b ; | 330 = 140 . 2 + 50; | ||
| b = r2q2 + r3 , | 0 < r3 < r2 ; | 180 = 50 . 2 + 40; | ||
| r2 = r3q3 + r4 , | 0 < r4 < r3 ; | 50 = 40 . 1 + 10; | ||
| .......... | .......... | 40 = 10 . 4 | ||
| rn-2 = rn-1qn-1 + rn , | 0 < rn < rn-1 ; | |||
| rn-1 = rnqn | ||||
|
Ë.Ã.Á. ¢Í§ (330, 140) ¤×Í 10 |
||||
¼Å§Ò¹¢Í§ÂÙ¤ÅÔ´ÂѧÁÕÍÕ¡ÁÒ¡ÁÒÂ
â´Â੾ÒÐã¹àÃ×èͧÃÒÇà¡ÕèÂǡѺµÑÇàÅ¢
»ÃÒ¡¯¡Òóì·Ò§¸ÃÃÁªÒµÔ àÃ×èͧ¢Í§áʧ ·Ò§à´Ô¹¢Í§¨Ø´º¹àÊé¹â¤é§áÅмÔÇâ¤é§ ÃÙ»¡ÃÇÂáÅÐÂѧÁÕËÅÑ¡¡Ò÷ҧ´¹µÃÕ
ÍÂèÒ§äáçµÒÁ ËÅÑ¡ÊÙµÃËÅÒÂÍÂèÒ§ä´éÊÙËÒÂ
ä»
ÃÈ. Â×¹ ÀÙèÇÃÇÃó,
ÊӹѡºÃÔ¡ÒäÍÁ¾ÔÇàµÍÃì ÁËÒÇÔ·ÂÒÅÑÂà¡ÉµÃÈÒʵÃì