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 11. 
 
 
Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side. 
 
 
Let A, B be square numbers, and let C be the side of A, and D of B;  I say that between A, B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to D. 
   
   
For let C by multiplying D make E.  Now, since A is a square and C is its side,  therefore C by multiplying itself has made A.  For the same reason also D by multiplying itself has made B.  Since then C by multiplying the numbers C, D has made A, E respectively,  therefore, as C is to D, so is A to E. [VII. 17]  For the same reason also, as C is to D, so is E to B. [VII. 18]  Therefore also, as A is to E, so is E to B.  Therefore between A, B there is one mean proportional number. 
                 
                 
I say next that A also has to B the ratio duplicate of that which C has to D.  For, since A, E, B are three numbers in proportion,  therefore A has to B the ratio duplicate of that which A has to E. [V. Def. 9]  But, as A is to E, so is C to D.  Therefore A has to B the ratio duplicate of that which the side C has to D.  Q. E. D. 
           
           
 
Permanent link
Go to Wiki Documentation
Enhet: Det humanistiske fakultet   Utviklet av: IT-seksjonen ved HF
Login