ods inherited from Rome. Algebra, with its rules for solving linear and quadratic equations, had been made accessible to the Latins. The geometry of Euclid, the Sphærica of Theodosius, the astronomy of Ptolemy, and other works were now accessible in the Latin tongue. Thus a great amount of new scientific material had come into the hands of the Christians. The talent necessary to digest this heterogeneous mass of knowledge was not wanting. The figure of Leonardo of Pisa adorns the vestibule of the thirteenth century.

It is important to notice that no work either on mathematics or astronomy was translated directly from the Greek previous to the fifteenth century.

The First Awakening and its Sequel.

Thus far, France and the British Isles have been the headquarters of mathematics in Christian Europe. But at the beginning of the thirteenth century the talent and activity of one man was sufficient to assign the mathematical science a new home in Italy. This man was not a monk, like Bede, Alcuin, or Gerbert, but a merchant, who in the midst of business pursuits found time for scientific study. Leonardo of Pisa is the man to whom we owe the first renaissance of mathematics on Christian soil. He is also called Fibonacci, i.e. son of Bonaccio. His father was secretary at one of the numerous factories erected on the south and east coast of the Mediterranean by the enterprising merchants of Pisa. He made Leonardo, when a boy, learn the use of the abacus. The boy acquired a strong taste for mathematics, and, in later years, during his extensive business travels in Egypt, Syria, Greece, and Sicily, collected from the various peoples all the knowledge he could get on this subject. Of all the methods of calculation, he found the Hindoo to be unquestionably the

best. Returning to Pisa, he published, in 1202, his great work, the Liber Abaci. A revised edition of this appeared in 1228. This work contains about all the knowledge the Arabs possessed in arithmetic and algebra, and treats the subject in a free and independent way. This, together with the other books of Leonardo, shows that he was not merely a compiler, or, like other writers of the Middle Ages, a slavish imitator of the form in which the subject had been previously presented, but that he was an original worker of exceptional power.

He was the first great. mathematician to advocate the adoption of the "Arabic notation." The calculation with the zero was the portion of Arabic mathematics earliest adopted by the Christians. The minds of men had been prepared for the reception of this by the use of the abacus and the apices. The reckoning with columns was gradually abandoned, and the very word abacus changed its meaning and became a synonym for algorism. For the zero, the Latins adopted the name zephirum, from the Arabic sifr (sifra-empty); hence our English word cipher. The new notation was accepted readily by the enlightened masses, but, at first, rejected by the learned circles. The merchants of Italy used it as early as the thirteenth century, while the monks in the monasteries adhered to the old forms. In 1299, nearly 100 years after the publication of Leonardo's Liber Abaci, the Florentine merchants were forbidden the use of the Arabic numerals in book-keeping, and ordered either to employ the Roman numerals or to write the numeral adjectives out in full. In the fifteenth century the abacus with its counters ceased to be used in Spain and Italy. In France it was used later, and it did not disappear in England and Germany before the middle of the seventeenth century." Thus, in the Winter's Tale (iv. 3), Shakespeare lets the clown be embarrassed by

a problem which he could not do without counters. Iago (in Othello, i. 1) expresses his contempt for Michael Cassio, "forsooth a great mathematician," by calling him a "countercaster." So general, indeed, says Peacock, appears to have been the practice of this species of arithmetic, that its rules and principles form an essential part of the arithmetical treatises of that day. The real fact seems to be that the old methods were used long after the Hindoo numerals were in common and general use. With such dogged persistency does man cling to the old!

The Liber Abaci was, for centuries, the storehouse from which authors got material for works on arithmetic and algebra. In it are set forth the most perfect methods of calculation with integers and fractions, known at that time; the square and cube root are explained; equations of the first and second degree leading to problems, either determinate or indeterminate, are solved by the methods of 'single' or 'double position,' and also by real algebra. The book contains a large number of problems. The following was proposed to Leonardo of Pisa by a magister in Constantinople, as a difficult problem: If A gets from B 7 denare, then A's sum is five-fold B's; if B gets from A 5 denare, then B's sum is seven-fold A's. How much has each? The Liber Abaci contains another problem, which is of historical interest, because it was given with some variations by Ahmes, 3000 years earlier 7 old women go to Rome; each woman has 7 mules, each mule carries 7 sacks, each sack contains 7 loaves, with each loaf are 7 knives, each knife is put up in 7 sheaths. What is the sum total of all named? Ans. 137,256.3

:

In 1220, Leonardo of Pisa published his Practica Geometric, which contains all the knowledge of geometry and trigonometry transmitted to him. The writings of Euclid and of some other Greek masters were known to him, either from Arabic

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 a2 - 5 are each square numbers. The answer is x=35; for (351⁄2)2+5=(411⁄2)2, (3,1⁄21⁄2)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.

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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 x+2x2+10x = 20. 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 depositing the sum at a safer place, each takes at hazard a certain amount; the first takes æ, but deposits only; the second carries y, but deposits only; the third takes z, and

3

t

6

x

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 indeterminate. Assuming 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