English translation from the
Latin
Chapter XIX of the Preface of
Cogitata PhysicoMathematica
by Marin Mersenne
as quoted in
THE THEORY OF SIMPLY PERIODIC
NUMERICAL FUNCTIONS
by
EDOUARD LUCAS, Lycée Charlemagne, Paris
Translated from the French by
SIDNEY KRAVITZ, Dover, N.J.
Translation Edited by
DOUGLAS LIND, University of Virginia, Charlottesville, Va.
FIBONACCI ASSOCIATION
1969
First published in the American Journal of Mathematics, Vol. 1(1878) pp. 184240, 289321
XIX.
Ad ea quæ de Numeris ad calcem
prop. 20. de Ballist. & puncto 14 Præfationis ad Hydraul.
dicta sunt, adde inuentam artem, quâ numeri, quotquot volueris,
reperiantur qui cum suis partibus aliquotis in vnicam summam redactis,
non solum duplam rationem habeant, (quales sunt 120, minimus omnium,
672, 523776, 1476304896, & 459818240, qui ductus in 3, numerum efficit
1379454720, cuius partes aliquotæ triplæ sunt, quales
etiam sequentes 30240, 32760, 23569920, & alij infiniti, de quibus
videatur Harmonia nostra, in qua 14182439040, & alij suarum partium
aliquotarum subquadrupli) sed etiam sint in ratione data cum suis
partibus aliquotis. 
To what has been said concerning
numbers at the end of Proposition 20 on Ballistics and at^{1}
Point 14 of the preface to Hydraulics, add the art which has been
discovered^{2}
whereby however many numbers you wish may be found which not only
are twice the sum of their aliquot parts^{3}
(such are 120 (the smallest of all), 672, 523776, 1476304896, and
459818240, which, when multiplied by^{4}
3 produces the number 1379454720, the sum of whose aliquot parts
is triple the number itself, ^{5}
as is also the case in the following: 30240, 32760, 23569920, and
limitless others, concerning which refer to our Harmony^{6},
in which are found^{7}
14182439040 and other quadruples^{8}
of their own aliquot parts) but also are in a given ratio with their
aliquot parts. 
^{1}This
"at" is one interpretation. The Latin is here carelessly written
and might equally well mean "on Ballistics and Point 14 of the Preface
to Hydraulics," That is, Mersenne might have written about Ballistics
and about Point 14 of the Preface to Hydraulics when he was writing
proposition 20.
^{2}What
"inventam" ("found") means here is enigmatic. Discovered by whom?
^{3}The
literal translation is "which not only have a double ratio with
their aliquot parts reduced to one sum."
^{4}Literally,
"led into."
^{5}This
is an expansion of the literal "whose aliquot parts are triple."
^{6}Literally,
"let our Harmony be seen." By this, Mersenne seems to be referring
to his book Harmonie universelle, published in 16367, seven
years before the present Cogitata physicomathematica.
See the Encyclopaedia Britannica
article on Mersenne.
^{7}Not
in the Latin, but this seems to be what he means.
^{8}The
Latin has "subquadruples" which means no more than "quadruples."
Perhaps it is a misreading of the manuscript by the printer. 
Sunt etiam alij numeri, quos vocant
amicabiles, quod habeant partes aliquotas à quibus mutuò
reficiantur, quales sunt omnium minimi 220, & 284; huius enim aliquotæ
partes illum efficiunt, vicéque versa partes illius aliquotæ
hunc perfectè restituunt. Quales & 18416 & 17296; nec non
9437036, & 4363584 reperies, aliosque innumeros. 
There are also other numbers which
they call amicable^{9}
because each is the sum of the other's aliquot parts.^{10}
Such are the smallest of all, 220 and 284; for the aliquot parts
of the latter produce the former, and viceversa the aliquot parts
of the former render the latter perfectly. Such also are 18416 and
17296; you will further find 9437036, 4363584, and numberless others.

^{9}A
technical expression used by Dickson, Uspensky and others.
^{10}This
is a paraphrase of the Latin, which says, literally, "because they
have aliquot parts from which they are mutually remade." 
Vbi fuerit operæ pretium aduertere
XXVIII numeros à Petro Bungo pro perfectis exhibitos, capite
XXVIII. libri de Numeris, non esse omnes Perfectos, quippe 20 sunt
imperfecti, adeo vt solos octo perfectos habeat videlicet 6, 28,
496, 8128, 33550336, 8589869056, 137438691328, & 2305843008139952128;
qui sunt è regione tabulæ Bungi, 1, 2, 3, 4, 8, 10,
12, & 29: quique soli perfecti sunt, vt qui Bungum habuerint, errori
medicinam faciant. 
At this point^{11}
it will be worth while to note that the 28^{12}
numbers exhibited by Petrus Bungus as perfect in Chapter 28 of his
book on numbers^{13}
are not all perfect. Indeed, 20 are imperfect, so that he has only
8 perfect ones, namely 6, 28, 496, 8128, 33550336^{14},
8589869056, 137438691328, and 2305843008139952128. These are from
Bungus' table^{15},
lines 1,2,3,4,8,10,12, and 19;^{16}
and these alone are perfect, so that those who have Bungus may remedy
the error. 
^{11}Literally,
"where."
^{12}Actually,
only 24. See Dickson, History of the Theory of Numbers, Vol. 1,
p. 12, note.
^{13}Dickson,
p. 9, note 42.
^{14}This
seems to be Mersenne's correction for Bungus' error 23... . See
Dickson, p. 13.
^{15}Literally,
"from the region of Bungus' table." These "regions" are lines marked
1, 2, etc., where each line number indicates the number of digits.
Dickson, ibid.
^{16}Mersenne's
or the printer's error since there are only 19 digits in the last
named perfect number. 
Porrò numeri perfecti adeo
rari sunt, vt vndecim dumtaxat potuerint hactenus inueniri: hoc
est, alij tres à Bougianis differentes: neque enim vllus
est alius perfectus ab illis octo, nisi superes exponentem numerum
62, progressionis duplæ ab 1 incipientis. Nonus enim perfectus
est potestas exponentis 68 minus 1. Decimus, potestas exponentis
128, minus 1. Vndecimus denique, potestas 258, minus 1, hoc est
potestas 257, vnitate decurtata, multiplicata per potestatem 256.

Further, perfect numbers are so
rare that up to now only eleven have been able to be found, that
is, three others differing from those of Bungus;^{17}
for there is no other perfect number outside of those eight, unless
you go beyond the exponent 62^{18},
in 1 + 2 + 2^{2} + ...^{19}.
The ninth perfect number is the power of the exponent 68 minus 1;^{20}
the tenth, the power of the exponent 128 minus 1^{21};
the eleventh, finally, the power 258 minus 1, that is, the power
257, decreased by unity, multiplied by the power 256.^{22}

^{17}"Bougianis"
in the text is an obvious misprint for "Bungianus."
^{18}Evidently
this is an error, since according to Uspensky and Heaslet, Elementary
Number Theory, p. 82, 2^{61}  1
is a prime number, and thus 2^{60}(2^{61}  1)
is perfect. W. W. R. Ball speculated that the printer had made an
error and had printed a 7 for a 1. R. C. Archibald dismissed
this as "ridiculous." (Scripta
Mathematica, Vol. 3, p. 112). However, if Dickson
is correct in saying that Mersenne was reporting information that
he got from correspondence with Frenicle and Fermat, it is possible
that Mersenne misread his correspondent and reported that
67 instead of 61 produces a prime, 2^{67}  1.
This would explain both errors, and would also account for the peculiar
oversight on his part since he was certainly acute enough to realize
that Bungus included several numbers in his list which are not perfect.
^{19}This
is Dickson's rendering of what Mersenne calls "of double progression
beginning from 1."
^{20}This
evidently in error. See Dickson, p. 13; Uspensky, p. 82.
^{21}Actually,
2^{126}(2^{127}  1)
is the 12^{th} perfect number since
Mersenne has omitted 2^{88}(2^{89}  1)
and 2^{106}(2^{107}  1)
which are perfect. See Uspensky, p. 82.
^{22}Another
error. See "Mersenne and Fermat Numbers," by R. M. Robinson Proceedings
of the A.M.S. Vol. 5, p. 842846 for a list of
perfect numbers obtained through the use of a modern highspeed
computer. The 13^{th} perfect number
is 2^{520}(2^{521}  1).

Qui vndecim alios repererit, nouerit
se analysim omnem quæ fuerit hactenus, superasse: memineritque
interea nullum esse perfectum a 17000 potestate ad 32000; & nullum
potestatum interuallum tantum assignari posse, quin detur illud
absque perfectis. Verbi gratia, si fuerit exponens 1050000, nullus
erit numerus progressionis duplæ vsque ad 2090000, qui perfectis
numeris serviat, hoc est qui minor vnitate, primus existat. 
The person who finds eleven others
will know that he has surpassed every analysis previously made and
will remember meanwhile that there is no perfect number from the
power 17000 to 32000, and that no interval of powers can be assigned
so great but that it may be given without perfect numbers. E. G.,
if there is an exponent 1050000, all the way to 2090000 there will
be no number of double progression such as to serve perfect numbers,
that is, such as to be a prime minus a unity.^{23}

^{23}It
is puzzling that Mersenne puts in all those zeroes since he was
certainly aware that the exponent must be a prime. Is it possible
that he wrote 17...? It also seems that it should be 33 not 32 since
(2 × 17)  1 = 33. Unfortunately
33 is not prime. Similar objections apply to 105 and 209. Eventually
somebody conjectured that if 2^{n}1 is prime, then
this number used as an exponent will produce another prime of the
same form. Robinson reports in his article that D. J. Wheeler disproved
this conjecture in 1953 using a highspeed computer. 
Vnde clarum est quàm rari
sint perfecti numeri, & quàm meritò viris perfectis
comparentur; esseque vnam ex maximis totius Matheseos difficultatibus,
præscriptam numerorum perfectorum multitudinum exhibere; quemadmodum
& agnoscere num dati numeri 15, aut 20 caracteribus constantes,
sint primi necne, cùm nequidem sæculum integrum huic
examini, quocumque modo hactenus cognito, sufficiat. 
From this, it is clear how rare
perfect numbers are, and how deservedly they are compared to perfect
men; and (it is clear) that one of the greatest difficulties in
all mathematics is to show a prescribed multitude^{24}
of perfect numbers, as also to recognize whether given numbers consisting
of 15 or 20 digits^{25}
are prime or not, since not even an entire century is sufficient
for this investigation, in any way known up to now. 
^{24}The
text here is corrupt. Multitudinem. not multidunum must be read.
^{25}Omitting
the comma after 15  an obvious error. 
*Only 24
were given by Bungus. While his table has 28 lines, one for each number
of digits, there are no entry of numbers of 5, 11, 17, 23 digits.
^{42}
Mysticae nvmerorvm significationis liber in dvas
divisvs partes, R. D. Petro Bongo Canonico Bergomate avctore. Bergomi.
Pars prior, 1583, 1585. Pars altera, 1584.
Petri Bungi Bergomatis Numerorum mysteria, Bergomi,
1591, 1599, 1614, Lutetiae Parisiorum, 1618, all four with the same text
and paging. Classical and biblical citations on numbers (400 pages on
1, 2, .., 12). On the 1618 edition, see Fontés, Mém. Acad.
Sc. Toulouse, (9), 5, 1893, 371380. 