## Stony Brook University MAT 118 Spring 2013

### Fibonacci on Rabbits

Chapter XII of Fibonacci's Liber Abaci (literally, "the book of the abacus;" the book of calculation) is organized as a nine-part compendium of problems. Part VII is "On the method of false position" and starts with the problem "Two men who ship wool for a fee." The next problem is "On a Merchant Carrying Precious Stones to Constantinople." Then "On Two Men with Fish and the Customs Agent." And so on. The 17th problem is "How Many Pairs of Rabbits Are Created by One Pair in One Year." (Using the titles as given in L. E. Sigler's translation: Fibonacci's Liber Abaci, Springer-Verlag, New York 2002).

Here is Fibonacci's 1202 text with my translation.

 parium 1 primus 2 Secundus 3 tercius 5 Quartus 8 Quintus 13 Sestus 21 Septimus 34 Octauus 55 Nonus 89 Decimus 144 Undecimus 233 Duodecimus 377
Quot paria coniculorum in uno anno ex uno pario germinentur.
How many pairs of rabbits will spring in one year from one pair.

Qvidam posuit unum par cuniculorum in quodam loco, qui erat undique
A person put one pair of rabbits in a certain place, which was on all sides
pariete circundatus, ut sciret, quot ex eo paria germinarentur in uno anno : cum
surrounded by a wall, so as to know, how many would be engendered by that pair in a year: since
natura eorum sit per singulum mensem aliud par germinare ; et in secundo
it is their nature to give birth to another pair after one month; and to give
mense ab eorum natiuitate germinant. Quia suprascriptum par in primo mense
birth during the second month after birth. Because the above-mentioned pair gives birth
germinat, duplicabis ipsum, erunt paria duo in uno mense. Ex quibus unum,
during the first month, you will double it; there will be two pairs after one month. Of these one,
scilicet primum, in secundo mense geminat ; et sic sunt in secundo mense paria
say the first, will give birth during the second month: and so there are after the second month
3 ; ex quibus in uno mense duo pregnantur ; et, gemitiantur in tercio mense
3 pairs; of which during one month two get pregnant; and give birth in the third month to
paria 2 coniculorum ; et sic sunt paria 5 in ipso mense ; ex quibus in ipso
2 pairs of rabbits; and so there are 5 pairs after that month; of which during the same month
pregnantur paria 3 ; et sunt in quarto mense paria 8 ; ex quibus paria 5
3 pairs get pregnant; and after the forth month there are 8 pairs; of which 5 pairs
geminant alia paria 5 : quibus additis cum parijs 8, faciunt paria 13 in quinto
give birth to another 5 pairs: which added to the 8 pairs, make 13 pairs after the fifth
mense ; ex quibus paria 5, que geminata fuerunt in ipso mense, non concipiunt
month; of which 5 pairs, which were born in that same month, do not conceive
in ipso mense ; sed alia 8 paria pregnantur ; et sic sunt in sexto mense paria 21 ;
during that same month; but the other 8 pairs get pregnant; and so there are in six months 21 pairs;
cum quibus additis parijs 13, que geminantur in septimo, erunt in ipso paria
when to those are added the 13 pairs, that are born in the seventh month, there will be after that month
34 ; cum quibus additis parijs 21, que geminantur in octauo mense, erunt in
34 pairs; when to those are added the 21 pairs, that are born during the eighth month, there will be after
ipso paria 55 ; cum quibus additis parijs 34, que geminantur in nono mense,
that month 55 pairs; when to those are added the 34 pairs, that are born during the ninth month,
erunt in ipso paria 89 ; cum quibus additis rursum parijs 55, que geminantur in
there will be after that month 89 pairs; when to those are added in turn the 55 pairs, that are born during
decimo, erunt in ipso paria 144 ; cum quibus additis rursum parijs 89, que
the tenth, there will be after that month 144 pairs; when to those are added in turn the 89 pairs, that
geminantur in undecimo mense, erunt in ipso paria 233. Cum quibus etiam
are born during the eleventh month, there will be after that month 233 pairs. When to those, furthermore,
additis parijs 144, que geminantur in ultimo mense, erunt paria 377 ; et tot
are added the 144 pairs, that are born during the last month, there will be 377 pairs; and all
paria peperit suprascriptum par in prefato loco in capite unius anni. Potes enim
the pairs have been engendered by the abovementioned pair in the described place in the course of one year. You can in fact
uidere in hac margine, qualiter hoc operati fuimus, scilicet quod iunximus
see in this margin how we worked, to wit: we added
primum numerum cum secundo, uidelicet 1 cum 2 ; et secundum cum tercio ;
the first number to the second, that is, 1 to 2; and the second to the third;
et tercium cum quarto ; et quartum cum quinto, et sic deinceps, donec iunximus
and the third to the fourth; and the fourth to the fifth, and thus successively, until we reached
decimum cum undecimo, uidelicet 144 cum 233; et habuimus suprascriptorum
the tenth to the eleventh, that is 144 to 233, and we obtained the sum of
cuniculorum summam, uidelicet 377 ; et sic posses facere per ordinem de
the above mentioned rabbits, that is 377; and you could have followed this rule to treat an
infinitis numeris mensibus.
infinite number of months.

-Tony Phillips
March, 2013