manuscripts directly or from the translations made by his countrymen, Gerard of Cremona and Plato of Tivoli. Leonardo's Geometry contains an elegant geometrical demonstration of Heron's formula for the area of a triangle, as a function of its three sides. Leonardo treats the rich material before him with skill and Euclidean rigour.

Of still greater interest than the preceding works are those containing Fibonacci's original investigations. We must here preface that after the publication of the Liber Abaci, Leonardo was presented by the astronomer Dominicus to Emperor Frederick II. of Hohenstaufen. On that occasion, John of Palermo, an imperial notary, proposed several problems, which Leonardo solved promptly. The first problem was to find a number x, such that x2+5 and x2 - 5 are each square numbers. The answer is x=35; for (35)2+5=(4,1⁄2)2, (3,1⁄21⁄2)' — 5 =(2) His masterly solution of this is given in his liber quadratorum, a copy of which work was sent by him to Frederick II. The problem was not original with John of Palermo, since the Arabs had already solved similar ones. Some parts of Leonardo's solution may have been borrowed from the Arabs, but the method which he employed of building squares by the summation of odd numbers is original with him.

2

The second problem proposed to Leonardo at the famous scientific tournament which accompanied the presentation of this celebrated algebraist to that great patron of learning, Emperor Frederick II., was the solving of the equation x23 + 2x2+10x = 20. As yet cubic equations had not been solved algebraically. Instead of brooding stubbornly over this knotty problem, and after many failures still entertaining new hopes of success, he changed his method of inquiry and showed by clear and rigorous demonstration that the roots of this equation could not be represented by the Euclidean irrational quantities, or, in other words, that they could not be

constructed with the ruler and compass only. He contented himself with finding a very close approximation to the required root. His work on this cubic is found in the Flos, together with the solution of the following third problem given him by John of Palermo: Three men possess in common an unknown sum of money t; the share of the first is; that of the second,; that of the third, Desirous of

3

t

6

depositing the sum at a safer place, each takes at hazard a certain amount; the first takes x, but deposits only; the second carries y, but deposits only; the third takes z, and deposits Of the amount deposited each one must receive 6 exactly, in order to possess his share of the whole sum. Find x, y, z. Leonardo shows the problem to be indetermiAssuming 7 for the sum drawn by each from the deposit, he finds t=47, x=33, y = 13, z=1.

One would have thought that after so brilliant a beginning, the sciences transplanted from Mohammedan to Christian soil would have enjoyed a steady and vigorous development. But this was not the case. [During the fourteenth and fifteenth centuries, the mathematical science was almost stationary. Long wars absorbed the energies of the people and thereby kept back the growth of the sciences. The death of Frederick II. in 1254 was followed by a period of confusion in Germany. The German emperors and the popes were continually quarrelling, and Italy was inevitably drawn into the struggles between the Guelphs and the Ghibellines. France and England were engaged in the Hundred Years' War (1338-1453). Then followed in England the Wars of the Roses. . The growth of science was retarded not only by war, but also by the injurious influence of scholastic philosophy. The intellectual leaders of those times quarrelled over subtle subjects in meta

physics and theology. Frivolous questions, such as "How many angels can stand on the point of a needle?" were discussed with great interest. Indistinctness and confusion of ideas characterised the reasoning during this period. Among the mathematical productions of the Middle Ages, the works of Leonardo of Pisa appear to us like jewels among quarryrubbish. The writers on mathematics during this period were not few in number, but their scientific efforts were vitiated by the method of scholastic thinking. Though they possessed the Elements of Euclid, yet the true nature of a mathematical proof was so little understood, that Hankel believes it no exaggeration to say that "since Fibonacci, not a single proof, not borrowed from Euclid, can be found in the whole literature of these ages, which fulfils all necessary conditions."

The only noticeable advance is a simplification of numerical operations and a more extended application of them. Among the Italians are evidences of an early maturity of arithmetic. Peacock says: The Tuscans generally, and the Florentines in particular, whose city was the cradle of the literature and arts of the thirteenth and fourteenth centuries, were celebrated for their knowledge of arithmetic and book-keeping, which were so necessary for their extensive commerce; the Italians were in familiar possession of commercial arithmetic long before the other nations of Europe; to them we are indebted for the formal introduction into books of arithmetic, under distinct heads, of questions in the single and double rule of three, loss and gain, fellowship, exchange, simple and compound interest, discount, and so on.

There was also a slow improvement in the algebraic notation. The Hindoo algebra possessed a tolerable symbolic notation, which was, however, completely ignored by the Mohammedans. In this respect, Arabic algebra approached much more closely to that of Diophantus, which can scarcely

be said to employ symbols in a systematic way. Leonardo of Pisa possessed no algebraic symbolism. Like the Arabs, he expressed the relations of magnitudes to each other by lines. or in words. But in the mathematical writings of the monk Luca Pacioli (also called Lucas de Burgo sepulchri) symbols began to appear. They consisted merely in abbreviations of Italian words, such as p for piu (more), m for meno (less), co for cosa (the thing or unknown quantity). "Our present notation has arisen by almost insensible degrees as convenience suggested different marks of abbreviation to different authors; and that perfect symbolic language which addresses itself solely to the eye, and enables us to take in at a glance the most complicated relations of quantity, is the result of a small series of small improvements." 23

We shall now mention a few authors who lived during the thirteenth and fourteenth and the first half of the fifteenth centuries. About the time of Leonardo of Pisa (1200 A.D.), lived the German monk Jordanus Nemorarius, who wrote a once famous work on the properties of numbers (1496), modelled after the arithmetic of Boethius. The most trifling numeral properties are treated with nauseating pedantry and prolixity. A practical arithmetic based on the Hindoo notation was also written by him. John Halifax (Sacro Bosco, died 1256) taught in Paris and made an extract from the Almagest containing only the most elementary parts of that work. This extract was for nearly 400 years a work of great popularity and standard authority. Other prominent writers are Albertus Magnus and George Purbach in Germany, and Roger Bacon in England. It appears that here and there some of our modern ideas were anticipated by writers of the Middle Ages. Thus, Nicole Oresme, a bishop in Normandy (died 1382), first conceived a notation of fractional powers, afterwards re-discovered by Stevinus, and gave rules for operating with them.

His notation was totally different from ours. Thomas Bradwardine, archbishop of Canterbury, studied star-polygons, -a subject which has recently received renewed attention. The first appearance of such polygons was with Pythagoras and his school. We next meet with such polygons in the geometry of Boethius and also in the translation of Euclid from the Arabic by Athelard of Bath. Bradwardine's philosophic writings contain discussions on the infinite and the infinitesimal-subjects never since lost sight of. To England falls the honour of having produced the earliest European writers on trigonometry. The writings of Bradwardine, of Richard of Wallingford, and John Maudith, both professors at Oxford, and of Simon Bredon of Winchecombe, contain trigonometry drawn from Arabic sources.

The works of the Greek monk Maximus Planudes, who lived in the first half of the fourteenth century, are of interest only as showing that the Hindoo numerals were then known in Greece. A writer belonging, like Planudes, to the Byzantine school, was Moschopulus, who lived in Constantinople in the early part of the fifteenth century. To him appears to be due the introduction into Europe of magic squares. He wrote a treatise on this subject. Magic squares were known to the Arabs, and perhaps to the Hindoos. Mediæval astrologers and physicians believed them to possess mystical properties and to be a charm against plague, when engraved on silver plate.

In 1494 was printed the Summa de Arithmetica, Geometria, Proportione et Proportionalita, written by the Tuscan monk Lucas Pacioli, who, as we remarked, first introduced symbols in algebra. This contains all the knowledge of his day on arithmetic, algebra, and trigonometry, and is the first comprehensive work which appeared after the Liber Abaci of Fibonacci. It contains little of importance which cannot be