My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
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 (1412-1492)

I have great difficulty in understanding a text written by Piero della Francesa (1412-1492) 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?
SherlockFogg is offline  
 
March 4th, 2015, 07:40 PM   #2
Global Moderator
 
CRGreathouse's Avatar
 
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".
CRGreathouse is offline  
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?
SherlockFogg is offline  
March 5th, 2015, 05:45 AM   #4
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Quote:
Originally Posted by SherlockFogg View Post
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.
Quote:
Originally Posted by SherlockFogg View Post
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?
I'm not an expert, but I get something like this:

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.
CRGreathouse is offline  
March 5th, 2015, 12:29 PM   #5
Global Moderator
 
Joined: Dec 2006

Posts: 20,469
Thanks: 2038

Quote:
Originally Posted by SherlockFogg View Post
. . . a text written by Piero della Francesa
Can you post the relevant portion of the text in its original language?
skipjack is offline  
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.
SherlockFogg is offline  
March 5th, 2015, 08:12 PM   #7
Math Team
 
Joined: Oct 2011
From: Ottawa Ontario, Canada

Posts: 14,301
Thanks: 1023

Quote:
Originally Posted by SherlockFogg View Post
x^4+12x^3+54x^2+108x=175
and
x^4+2x^3+3x^2+2x=8
Well, regardless of where these come from,
x=1 is obvious as one of the solutions.
Denis is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
della, difficult, francesca, piero, text, written



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Why in the answer written all discharge r-soy Physics 0 March 1st, 2012 07:06 AM
Numbers written on a board =) =) =) (3) Sara so Algebra 4 January 5th, 2011 06:32 PM
Number written on a board =) =) Sara so Algebra 4 January 4th, 2011 06:29 AM
Numbers written on a board =) Sara so Algebra 3 January 3rd, 2011 09:27 PM
Help with Written Proofs mathmajorintrouble Abstract Algebra 11 May 4th, 2009 11:41 AM





Copyright © 2019 My Math Forum. All rights reserved.