I am confused about a certain sentence in the book The works of Archimedes where a method of extracting a square root by Ptolemy is explained. I will write here only the last step because the sentence I am confused with concerns its context.
We have then to subtract $$2(67 + {4\over60}){55\over60^2} + ({55\over60^2})^2,$$ or $$ {442640\over60^3} + {3025\over60^4}$$ from the remainder ${7424\over60^2}$ above found. The subtraction of ${442640\over60^3}$ from ${7424\over60^2}$ gives ${2800\over60^3}$, or ${46\over60^2} + {40\over60^3}$; but Theon does not go further to subtract the remaining ${3025\over60^4}$, instead of which he merely remarks that the square of ${55\over60^2}$ approximates to ${46\over60^2} + {40\over60^3}$. As a matter of fact, if we deduct the ${3025\over60^4}$ from ${2800\over60^3}$, so as to obtain a correct remainder, it is found to be ${164975\over60^4}$.
I am mostly confused about the sentence where he says that the square of ${55\over60^2}$ approximates to ${46\over60^2} + {40\over60^3}$. This is not really a correct approximation so I want to know if maybe something else is meant by this sentence. Also if someone could explain what is the reason of rewriting a fraction as a sum of two fractions?
It is my best understanding that when Theon says the square of $\frac{55}{60^2}$ approximates to $\frac{46}{60^2}+\frac{40}{60^3}$, that he means they are both small enough numbers that he finds it fine to treat their difference as zero, so this works as a fine spot to end the calculation.
As you point out, it’s really not a great approximation relative to the size of the numbers, but it’s not relative to their size that Theon means, it’s relative to the size of the number he’s trying to extract the square root of.