Past exam of the history of mathematics course of the University of Oxford / 2023 / 1

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.^[ref][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 iswhere $\sigma$ 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 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 $q{2}^j$, with $0\le j\le p-1$. A finite geometric series therefore givesThe 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.^[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 the transmission of deductive arithmetic through vernacular explanation: a book called geometry also teaches the divisibility theory needed for a general construction.