You are here: BP HOME > BPG > Euclid: Elementa > fulltext
Euclid: Elementa

Choose languages

Choose images, etc.

Choose languages
Choose display
  • Enable images
  • Enable footnotes
    • Show all footnotes
    • Minimize footnotes
Search-help
Choose specific texts..
Euclid: Elementa
    Click to Expand/Collapse Option Complete text
Click to Expand/Collapse OptionTitle
Click to Expand/Collapse OptionPreface
Click to Expand/Collapse OptionBook I
Click to Expand/Collapse OptionBook ΙI
Click to Expand/Collapse OptionBook IΙΙ
Click to Expand/Collapse OptionBook IV
Click to Expand/Collapse OptionBook V
Click to Expand/Collapse OptionBook VI
Click to Expand/Collapse OptionBook VII
Click to Expand/Collapse OptionBook VIII
Click to Expand/Collapse OptionBook ΙΧ
Click to Expand/Collapse OptionBook Χ
Click to Expand/Collapse OptionBook ΧI
Click to Expand/Collapse OptionBook ΧIΙ
Click to Expand/Collapse OptionBook ΧIΙΙ
Back to library
PROPOSITION 19. 
 
 
The rectangle contained by rational straight lines commensurable in length is rational. 
 
 
For let the rectangle AC be contained by the rational straight lines AB, BC commensurable in length;  I say that AC is rational. 
   
   
For on AB let the square AD be described;  therefore AD is rational. [X. Def. 4]  And, since AB is commensurable in length with BC, while AB is equal to BD, therefore BD is commensurable in length with BC.  And, as BD is to BC, so is DA to AC. [VI. 1]  Therefore DA is commensurable with AC. [X. 11]  But DA is rational;  therefore AC is also rational. [X. Def. 4] 
             
             
Therefore etc. 
 
 
Permanent link
Go to Wiki Documentation
Enhet: Det humanistiske fakultet   Utviklet av: IT-seksjonen ved HF
Login