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 107. 
 
 
A straight line commensurable with that which produces with a medial area a medial whole is itself also a straight line which produces with a medial area a medial whole. 
 
 
Let AB be a straight line which produces with a medial area a medial whole, and let CD be commensurable with AB;  I say that CD is also a straight line which produces with a medial area a medial whole. 
   
   
For let BE be the annex to AB, and let the same construction be made;  therefore AE, EB are straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and further the sum of the squares on them incommensurable with the rectangle contained by them. [X. 78]  Now, as was proved, AE, EB are commensurable with CF, FD, the sum of the squares on AE, EB with the sum of the squares on CF, FD, and the rectangle AE, EB with the rectangle CF, FD;  therefore CF, FD are also straight lines incommensurable in square which make the sum of the squares on them medial, the rectangle contained by them medial, and further the sum of the squares on them incommensurable with the rectangle contained by them. 
       
       
Therefore CD is a straight line which produces with a medial area a medial whole. [X. 78]   
   
   
 
Permanent link
Go to Wiki Documentation
Enhet: Det humanistiske fakultet   Utviklet av: IT-seksjonen ved HF
Login