English translation from the Latin
Chapter XIX of the Preface of
Cogitata Physico-Mathematica
by Marin Mersenne
as quoted in
EDOUARD LUCAS, Lycée Charlemagne, Paris

Translated from the French by
Translation Edited by
DOUGLAS LIND, University of Virginia, Charlottesville, Va.


First published in the American Journal of Mathematics, Vol. 1(1878) pp. 184-240, 289-321


   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 at1 Point 14 of the preface to Hydraulics, add the art which has been discovered2 whereby however many numbers you wish may be found which not only are twice the sum of their aliquot parts3 (such are 120 (the smallest of all), 672, 523776, 1476304896, and 459818240, which, when multiplied by4 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 Harmony6, in which are found7 14182439040 and other quadruples8 of their own aliquot parts) but also are in a given ratio with their aliquot parts. 1This "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.
2What "inventam" ("found") means here is enigmatic. Discovered by whom?
3The literal translation is "which not only have a double ratio with their aliquot parts reduced to one sum."
4Literally, "led into."
5This is an expansion of the literal "whose aliquot parts are triple."
6Literally, "let our Harmony be seen." By this, Mersenne seems to be referring to his book Harmonie universelle, published in 1636-7, seven years before the present Cogitata physico-mathematica. See the Encyclopaedia Britannica article on Mersenne.
7Not in the Latin, but this seems to be what he means.
8The 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 amicable9 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 vice-versa 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. 9A technical expression used by Dickson, Uspensky and others.
10This 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 point11 it will be worth while to note that the 2812 numbers exhibited by Petrus Bungus as perfect in Chapter 28 of his book on numbers13 are not all perfect. Indeed, 20 are imperfect, so that he has only 8 perfect ones, namely 6, 28, 496, 8128, 3355033614, 8589869056, 137438691328, and 2305843008139952128. These are from Bungus' table15, 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. 11Literally, "where."
12Actually, only 24. See Dickson, History of the Theory of Numbers, Vol. 1, p. 12, note.
13Dickson, p. 9, note 42.
14This seems to be Mersenne's correction for Bungus' error 23... . See Dickson, p. 13.
15Literally, "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.
16Mersenne'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  6218, in 1 + 2 + 22 + ...19. The ninth perfect number is the power of the exponent 68 minus 1;20 the tenth, the power of the exponent 128 minus 121; 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 mis-print for "Bungianus."
18Evidently this is an error, since according to Uspensky and Heaslet, Elementary Number Theory, p. 82, 261 - 1 is a prime number, and thus  260(261 - 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 mis-read his correspondent and reported that 67 instead of 61 produces a prime, 267 - 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.
19This is Dickson's rendering of what Mersenne calls "of double progression beginning from 1."
20This evidently in error. See Dickson, p. 13; Uspensky, p. 82.
21Actually, 2126(2127 - 1) is the 12th perfect number since Mersenne has omitted 288(289 - 1) and 2106(2107 - 1) which are perfect. See Uspensky, p. 82.
22Another error. See "Mersenne and Fermat Numbers," by R. M. Robinson Proceedings of the A.M.S. Vol. 5, p. 842-846 for a list of perfect numbers obtained through the use of a modern high-speed computer. The 13th perfect number is 2520(2521 - 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 23It 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 2n-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 high-speed 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 multitude24 of perfect numbers, as also to recognize whether given numbers consisting of 15 or 20 digits25 are prime or not, since not even an entire century is sufficient for this investigation, in any way known up to now. 24The text here is corrupt. Multitudinem. not multidunum must be read.
25Omitting 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, 371-380.

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