**LEONARDUS PISANUS** (Leonardo of Pisa), an Italian mathematician of the 13th century who has left the stamp of his name on a whole period in the history of the science. Of Leonardo's 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. Where Euro-pean and Arab merchants met was at that time the right place for learning arithmetic, and it was certainly with a view to this that the father had Leonardo sent to Bugia to continue his education. But Leonardo aimed at some-thing higher than to make himself an accomplished clerk, and during his travels round the Mediterranean he obtained such erudition as would have gained him the name of a great scholar in much later times. In 1202 Leonardo Fibonacci (i.e., son of Bonaccio) was again in Italy and published his great work Liber Abaci, which probably pro-cured 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 Prac-tica Geometric, which he dedicated to the imperial astro-nomer Dominicus Hispanus. Some years afterwards (perhaps in 1228, as is related by an author on the authority of a manuscript only once seen by him) Leonardo dedicated to another courtier, the well known astrologer Michael Scott, the second edition of his Liber A.baci, which has come down to our times, and has been printed as well as Leonardo's other works by Prince Bald. Boncompagni (Rome, 1857-62, 2 vols.). The other works consist of the Practica Geometrise 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 intro-duction Leonardo himself 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, and we might compare him to a meteor flashing up suddenly on the black back-ground of the midnight sky, and vanishing as suddenly, were it not that his influence was too deep and lasting to allow of his being likened to a phenomenon passing quickly by.

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me Pisis duceret praesentandum" have always been taken to mean that Leonardo was presented to the emperor at Pisa, but the date of

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To explain this influence and the whole importance of Leonardo's scientific work, we must rapidly sketch the state of mathematics about the year 1200. The Greeks, the most geometrical nation on the earth, had attained a high degree of scientific perfection, when they were obliged to yield to the political supremacy of Rome. From this time mathematics in Europe sunk lower and lower, till only some sorry fragments of the science were still preserved in the cell of the studious monk and behind the counting-board of the eager merchant. Geometry was nearly forgotten ; arithmetic made use of the abacus with counters, or with the nine characters the origin of which is still a matter of controversy (see NUMERALS); the zero was still unknown. Among the Arabs it was quite other-wise. Greek mathematics found amongst them a second home, where the science was not only preserved but came to new strength, and was recruited from India, whence in particular came the symbol "zero" and its use, which alone renders possible numerical calculation in the modern sense of the word. Ancient astronomy as well as ancient mathematics reappeared in Europe, from the beginning of the 12th century onwards, in an Arabian dress. Two men especially recognized the worth of these sciences and made it the task of their life to propagate them amongst their contemporaries,—the German monk Jordanns Nemorarius and the Italian merchant Leonardus Pisanus. Leonardo, as we have said, travelled all round the Mediterranean gathering knowledge of every kind. He studied the geometry of Euclid, the algebra of Moham-med ibn Musa Alchárizmi; he made himself acquainted with Indian methods; he found out by himself new theories. So runs his own account; and an exact comparison of Leonardo's works with older sources not only confirms the truth of his narrative, but shows also that he must have studied some other authors,—for instance, Alkarchi. In his Practica Geometrías plain traces of the use of the Roman " agrimensores " are met with; in his Liber Abaci old Egyptian problems occur revealing 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 now be regarded (as Cossali regarded him about 1800) as the inventor of that very great variety of truths for which he mentions no earlier source. But even were the pre-decessors to whom he is indebted more numerous than we are inclined to believe, were he the Columbus only of a territory the existence of which was unknown to his century, the historical importance of the man would be nearly the same. We must remember the general ignorance of his age, and then fancy the sudden appearance of a work like the Liber Abaci, which fills 459 printed pages. These pages set forth 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 inde-terminate 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. We may well wonder, not that the impression caused by a work of such overwhelming character was so deep, but that it made any impres-sion at all, and that the unprepared soil could receive the seed. The second work of Leonardo, his Practica Geometrías (1220), is still more remarkable, since it requires readers already acquainted wdth Euclid's planimetry, who are able to follow rigorous demon-strations 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 geome-trical work, viz., to find a square number which remains a square number when 5 is added to it. This problem evidently suggested the first question 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. The problem was :—To find a square number remain-ing so after the addition as well as the subtraction of 5. Leonardo gave as solution the numbers ll^j, 16I0i!'T, and S-fii,—the squares of 3T\, i^, and 2^; and the Liber Quadratorum gives the method of finding them, which we cannot discuss here. We observe, how-ever, that the 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's + 2x2 + iOx = 20. Leonardo, making use of fractions of the sexagesimal scale, gives x=V> 22' 7" 42 33ly 4y 40vi, 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 posses-sion of a certain method called regula áurea by 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 wdth, and Leonardo perfectly under-stands 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. We have enumerated the main substance of wdiat appear to be Leonardo's own discoveries, and the experienced reader will not hesitate to conclude that they prove him to have been one of the greatest algebraists of any time. 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. (M. CA.)

**Footnotes**

125-1 The words " cum Magister Dominicus pedibus celsitudinis vestrae

1225 excludes this interpretation, as Frederick II. certainly never was in Pisa before July 1226. The translation, therefore, ought;to be— " when Magister Dominicus brought me from Pisa, &c.," the place where Leonardo met the emperor remaining unknown.

**The above article was written by:** Prof. Moritiz Cantor.