# 1911 Encyclopædia Britannica/Leonardo of Pisa

**LEONARDO OF PISA** (Leonardus Pisanus or Fibonacci),
Italian mathematician of the 13th century. Of his personal
history few particulars are known. His father was called
Bonaccio, most probably a nickname with the ironical meaning
of “a good, stupid fellow,” while to Leonardo himself another
nickname, Bigollone (dunce, blockhead), seems to have been
given. The father was secretary in one of the numerous factories
erected on the southern and eastern coasts of the Mediterranean
by the warlike and enterprising merchants of Pisa. Leonardo
was educated at Bugia, and afterwards toured the Mediterranean.
In 1202 he was again in Italy and published his great work,
*Liber abaci*, which probably procured him access to the learned
and refined court of the emperor Frederick II. Leonardo
certainly was in relation with some persons belonging to that
circle when he published in 1220 another more extensive work,
*De practica geometriae*, which he dedicated to the imperial
astronomer Dominicus Hispanus. Some years afterwards
(perhaps in 1228) Leonardo dedicated to the well-known astrologer
Michael Scott the second edition of his *Liber abaci*, which
was printed with Leonardo’s other works by Prince Bald.
Boncompagni (Rome, 1857–1862, 2 vols.). The other works
consist of the *Practica geometriae* and some most striking
papers of the greatest scientific importance, amongst which the
*Liber quadratorum* may be specially signalized. It bears the
notice that the author wrote it in 1225, and in the introduction
Leonardo tells us the occasion of its being written. Dominicus
had presented Leonardo to Frederick II. The presentation was
accompanied by a kind of mathematical performance, in which
Leonardo solved several hard problems proposed to him by John
of Palermo, an imperial notary, whose name is met with in
several documents dated between 1221 and 1240. The methods
which Leonardo made use of in solving those problems fill the
*Liber quadratorum*, the *Flos*, and a *Letter to Magister Theodore*.
All these treatises seem to have been written nearly at the same
period, and certainly before the publication of the second edition
of the *Liber abaci*, in which the *Liber quadratorum* is expressly
mentioned. We know nothing of Leonardo’s fate after he issued
that second edition.

Leonardo’s works are mainly developments of the results obtained
by his predecessors; the influences of Greek, Arabian, and Indian
mathematicians may be clearly discerned in his methods. In his
*Practica geometriae* plain traces of the use of the Roman *agrimensores*
are met with; in his *Liber abaci* old Egyptian problems reveal
their origin by the reappearance of the very numbers in which the
problem is given, though one cannot guess through what channel
they came to Leonardo’s knowledge. Leonardo cannot be regarded
as the inventor of that very great variety of truths for which he
mentions no earlier source.

The *Liber abaci*, which fills 459 printed pages, contains the most
perfect methods of calculating with whole numbers and with
fractions, practice, extraction of the square and cube roots, proportion,
chain rule, finding of proportional parts, averages, progressions, even
compound interest, just as in the completest mercantile arithmetics
of our days. They teach further the solution of problems leading to
equations of the first and second degree, to determinate and
indeterminate equations, not by single and double position only, but
by real algebra, proved by means of geometric constructions, and
including the use of letters as symbols for known numbers, the
unknown quantity being called *res* and its square *census*.

The second work of Leonardo, his *Practica geometriae* (1220)
requires readers already acquainted with Euclid’s planimetry, who
are able to follow rigorous demonstrations and feel the necessity for
them. Among the contents of this book we simply mention a trigonometrical
chapter, in which the words *sinus versus arcus* occur, the
approximate extraction of cube roots shown more at large than in
the *Liber abaci*, and a very curious problem, which nobody would
search for in a geometrical work, viz.—To find a square number
remaining so after the addition of 5. This problem evidently
suggested the first question, viz.—To find a square number which
remains a square after the addition and subtraction of 5, put to our
mathematician in presence of the emperor by John of Palermo,
who, perhaps, was quite enough Leonardo’s friend to set him such
problems only as he had himself asked for. Leonardo gave as solution
the numbers 1197144, 1697144, and 697144,—the squares of 3512, 4112 and
2712; and the method of finding them is given in the *Liber quadratorum*.
We observe, however, that this kind of problem was not
new. Arabian authors already had found three square numbers of
equal difference, but the difference itself had not been assigned in
proposing the question. Leonardo’s method, therefore, when the
difference was a fixed condition of the problem, was necessarily very
different from the Arabian, and, in all probability, was his own
discovery. The *Flos* of Leonardo turns on the second question set
by John of Palermo, which required the solution of the cubic equation
*x*^{3} + 2*x*^{2} + 10*x* = 20. Leonardo, making use of fractions of the
sexagesimal scale, gives *x* = 1^{0} 22^{i} 7^{ii} 42^{iii} 33^{iv} 4^{v} 40^{vi}, after having
demonstrated, by a discussion founded on the 10th book of Euclid,
that a solution by square roots is impossible. It is much to be
deplored that Leonardo does not give the least intimation how he
found his approximative value, outrunning by this result more than
three centuries. Genocchi believes Leonardo to have been in possession
of a certain method called *regula aurea* by H. Cardan in the
16th century, but this is a mere hypothesis without solid foundation.
In the *Flos* equations with negative values of the unknown quantity
are also to be met with, and Leonardo perfectly understands the
meaning of these negative solutions. In the *Letter to Magister*
*Theodore* indeterminate problems are chiefly worked, and Leonardo
hints at his being able to solve by a general method any problem
of this kind not exceeding the first degree.

As for the influence he exercised on posterity, it is enough to say
that Luca Pacioli, about 1500, in his celebrated *Summa*, leans so
exclusively to Leonardo’s works (at that time known in manuscript
only) that he frankly acknowledges his dependence on them, and
states that wherever no other author is quoted all belongs to
Leonardus Pisanus.

*Fibonacci’s series* is a sequence of numbers such that any term is
the sum of the two preceding terms; also known as *Lamé’s series*. (M. Ca.)