= Solution
{c}
The passage belongs to the English reception of <Euclid>'s <number theory>. <Henry Billingsley>'s <The Elements of Geometrie by Henry Billingsley>, published in 1570, brought the <Elements> into English with extensive explanatory material. The original title page advertises translation and annotations; the extract's worked examples show what explanatory prose could add to a compact <definition>. <Henry Billingsley>'s numbering calls this Definition 23 of Book VII; the familiar modern numbering is VII, Definition 22. The difference is editorial, not a difference in the <perfect number> being defined.<The Elements of Geometrie by Henry Billingsley>[Title page and Book VII, Definition 23, in the Text Creation Partnership transcription]{ref}<Euclid's Elements by Euclid/Book VII, Definition 22>[Definition and guide]{ref}
Here a number's parts are its positive <proper divisors>, rather than arbitrary pieces of a <sum> or all smaller <integers>. In modern notation the <definition> is
$$
\boxed{\sum_{\substack{d\mid n\\1\leq d<n}}d=n,\qquad\text{equivalently }\sigma(n)=2n,}
$$
where $\sigma$ is the <sum-of-divisors function>. Thus $12$ is not a <perfect number>: its positive <proper divisors> total $1+2+3+4+6=16$. It is essential to exclude $n$ itself: including it would double the required <sum>. The pairing of a <divisor> $d$ with the fraction $1/(n/d)$ of the whole explains the extract's language of halves, thirds and so on. Its examples are complete checks because they identify every positive <proper divisor>, not merely some convenient decomposition.
The larger mathematical context is the construction in <Euclid's Elements by Euclid/Book IX, Proposition 36>. If $q=2^p-1$ is a <prime number>, then $N=2^{p-1}q$ is a <perfect number>. To reconstruct the argument, <prime factorization> gives all its <divisors> as $2^j$ and $q2^j$, with $0\leq j\leq p-1$. A <finite geometric series> therefore gives
$$
\sigma(N)=(1+q)\sum_{j=0}^{p-1}2^j
=2^p(2^p-1)=2N.
$$
The choices $p=2,3,5,7$ produce $6,28,496,8128$. The <primality> assumption matters: choosing $p=4$ produces $120$, whose <sum-of-divisors function> is $360$, not $240$. The construction gives a reason for the examples and connects <perfect numbers> with <Mersenne primes>.<Euclid's Elements by Euclid/Book IX, Proposition 36>[Statement, construction and guide]{ref}
The extract itself supplies a <definition> and examples, not a classification or a <mathematical proof> that such numbers are rare. In particular, the converse that every even <perfect number> has this form belongs to the later <Euclid-Euler theorem>; it should not be read back into this passage. The historically significant feature is \b[the transmission of deductive arithmetic through vernacular explanation]: a book called geometry also teaches the <divisibility> theory needed for a general construction.
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