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March 4th, 2015, 06:31 PM  #1 
Newbie Joined: Jul 2014 From: Hong Kong Posts: 7 Thanks: 0  Difficult text written by Piero della Francesca (14121492)
I have great difficulty in understanding a text written by Piero della Francesa (14121492) who is best known for his painting but he also wrote mathematical works. In one of them, the (Treatise on Calculation), it is stated: (The difficult text begins) When things and squares and cubes and squares of squares are equal to numbers, one should divide the number of things by the number of cubes, square the result and add to the number. Then the thing will be equal to the square root of the square root of the sum minus the root of the result of dividing things by cubes. (The difficult text ends) I was given a page from a book named Solving Cubic Equations. In the page, it also says the text can lead to the following two equations x^4+12x^3+54x^2+108x=175 and x^4+2x^3+3x^2+2x=8 The book also asks "Does Piero's method work?" I have searched in google using the key words "When things and squares and cubes and squares of squares are equal in numbers" and found a page written by Cardano. They shared similarity in using similar words like "things", "cubes" and "squares". I believe that they have some shared understanding in the choice and the usage of words at their times and so some words are possibly omitted. This creates the difficulty in understanding the difficult text. Could anyone kindly offer any help so that I could interpret the difficult text? 
March 4th, 2015, 07:40 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
I interpret "things" as "x", "squares" as "x^2", "cubes" as "x^3", and "squares of squares" as "x^4".

March 4th, 2015, 09:57 PM  #3 
Newbie Joined: Jul 2014 From: Hong Kong Posts: 7 Thanks: 0 
In this case, how to relate the difficult text with the two equations? I mean, how to develop the two equations out of the text?

March 5th, 2015, 05:45 AM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Quote:
Quote:
dx + cx^2 + bx^3 + ax^4 = n (d/b)^2 + n = (s  d/b)^(1/4) for some sum s. Possibly s = n, I don't know. At least this should give you a starting point.  
March 5th, 2015, 12:29 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,921 Thanks: 2203  
March 5th, 2015, 06:19 PM  #6 
Newbie Joined: Jul 2014 From: Hong Kong Posts: 7 Thanks: 0 
I am grateful to CRGreathouse for the contribution. I would try to work on it. Referring to skipjack's suggestion, I am afraid I cannot post the text in its original language. I think that both Piero della Francesca and Cardano are Italian. I work in a high school and my boss gave me a page asking if I could interpret the difficult text. I am quite sure it is from a book named Solving Cubic Equation and it is printed in English. As my job is on a temporary basis, if I can do the task well, I will have a better chance of staying at my position. Thus, I appeal that all who know the text do contribution. 
March 5th, 2015, 08:12 PM  #7 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038  

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