# A History of Mathematics/Middle Ages/Europe During the Middle Ages

EUROPE DURING THE MIDDLE AGES.
With the third century after Christ begins an era of migration of nations in Europe. The powerful Goths quit their swamps and forests in the North and sweep onward in steady southwestern current, dislodging the Vandals, Sueves, and Burgundians, crossing the Roman territory, and stopping and recoiling only when reaching the shores of the Mediterranean. From the Ural Mountains wild hordes sweep down on the Danube. The Roman Empire falls to pieces, and the Dark Ages begin. But dark though they seem, they are the germinating season of the institutions and nations of modern Europe. The Teutonic element, partly pure, partly intermixed with the Celtic and Latin, produces that strong and luxuriant growth, the modern civilisation of Europe. Almost all the various nations of Europe belong to the Aryan stock. As the Greeks and the Hindoos—both Aryan races—were the great thinkers of antiquity, so the nations north of the Alps became the great intellectual leaders of modern times.
Introduction of Roman Mathematics.
We shall now consider how these as yet barbaric nations of the North gradually came in possession of the intellectual ​treasures of antiquity. With the spread of Christianity the Latin language was introduced not only in ecclesiastical but also in scientific and all important worldly transactions. Naturally the science of the Middle Ages was drawn largely from Latin sources. In fact, during the earlier of these ages Roman authors were the only ones read in the Occident. Though Greek was not wholly unknown, yet before the thirteenth century not a single Greek scientific work had been read or translated into Latin. Meagre indeed was the science which could be gotten from Roman writers, and we must wait several centuries before any substantial progress is made in mathematics.
After the time of Boethius and Cassiodorius mathematical activity in Italy died out. The first slender blossom of science among tribes that came from the North was an encyclopaedia entitled Origines, written by Isidorus (died 636 as bishop of Seville). This work is modelled after the Roman encyclopædias of Martianus Capella of Carthage and of Cassiodorius. Part of it is devoted to the quadrivium, arithmetic, music, geometry, and astronomy. He gives definitions and grammatical explications of technical terms, but does not describe the modes of computation then in vogue. After Isidorus there follows a century of darkness which is at last dissipated by the appearance of Bede the Venerable (672-735), the most learned man of his time. He was a native of Ireland, then the home of learning in the Occident. His works contain treatises on the Computus, or the computation of Easter-time, and on finger-reckoning. It appears that a finger-symbolism was then widely used for calculation. The correct determination of the time of Easter was a problem which in those days greatly agitated the Church. It became desirable to have at least one monk at each monastery who could determine the day of religious festivals and could compute the calendar. ​Such determinations required some knowledge of arithmetic. Hence we find that the art of calculating always found some little corner in the curriculum for the education of monks.
The year in which Bede died is also the year in which Alcuin (735-804) was born. Alcuin was educated in Ireland, and was called to the court of Charlemagne to direct the progress of education in the great Frankish Empire. Charlemagne was a great patron of learning and of learned men. In the great sees and monasteries he founded schools in which were taught the psalms, writing, singing, computation (computus), and grammar. By computus was here meant, probably, not merely the determination of Easter-time, but the art of computation in general. Exactly what modes of reckoning were then employed we have no means of knowing. It is not likely that Alcuin was familiar with the apices of Boethius or with the Roman method of reckoning on the abacus. He belongs to that long list of scholars who dragged the theory of numbers into theology. Thus the number of beings created by God, who created all things well, is 6, because 6 is a perfect number (the sum of its divisors being $\scriptstyle {1+2+3=6}$); 8, on the other hand, is an imperfect number ($\scriptstyle {1+2+4<8}$); hence the second origin of mankind emanated from the number 8, which is the number of souls said to have been in Noah's ark.
There is a collection of "Problems for Quickening the Mind" (propositiones ad acuendos iuvenes), which are certainly as old as 1000 A.D. and possibly older. Cantor is of the opinion that they were written much earlier and by Alcuin. The following is a specimen of these "Problems": A dog chasing a rabbit, which has a start of 150 feet, jumps 9 feet every time the rabbit jumps 7. In order to determine in how many leaps the dog overtakes the rabbit, 150 is to be divided by 2. In this collection of problems, the areas of triangular and quadrangular pieces of land are found by the same formulas of ​approximation as those used by the Egyptians and given by Boethius in his geometry. An old problem is the "cistern-problem" (given the time in which several pipes can fill a cistern singly, to find the time in which they fill it jointly), which has been found previously in Heron, in the Greek Anthology, and in Hindoo works. Many of the problems show that the collection was compiled chiefly from Roman sources. The problem which, on account of its uniqueness, gives the most positive testimony regarding the Roman origin is that on the interpretation of a will in a case where twins are born. The problem is identical with the Roman, except that different ratios are chosen. Of the exercises for recreation, we mention the one of the wolf, goat, and cabbage, to be rowed across a river in a boat holding only one besides the ferry-man. Query: How must he carry them across so that the goat shall not eat the cabbage, nor the wolf the goat? The solutions of the "problems for quickening the mind" require no further knowledge than the recollection of some few formulas used in surveying, the ability to solve linear equations and to perform the four fundamental operations with integers. Extraction of roots was nowhere demanded; fractions hardly ever occur.
The great empire of Charlemagne tottered and fell almost immediately after his death. War and confusion ensued. Scientific pursuits were abandoned, not to be resumed until the close of the tenth century, when under Saxon rule in Germany and Capetian in France, more peaceful times began. The thick gloom of ignorance commenced to disappear. The zeal with which the study of mathematics was now taken up by the monks is due principally to the energy and influence of one man,—Gerbert. He was born in Aurillac in Auvergne. After receiving a monastic education, he engaged in study, chiefly of mathematics, in Spain. On his return he taught ​school at Rheims for ten years and became distinguished for his profound scholarship. By King Otto I. and his successors Gerbert was held in highest esteem. He was elected bishop of Rheims, then of Ravenna, and finally was made Pope under the name of Sylvester II. by his former pupil Emperor Otho III. He died in 1003, after a life intricately involved in many political and ecclesiastical quarrels. Such was the career of the greatest mathematician of the tenth century in Europe. By his contemporaries his mathematical knowledge was considered wonderful. Many even accused him of criminal intercourse with evil spirits.
Gerbert enlarged the stock of his knowledge by procuring copies of rare books. Thus in Mantua he found the geometry of Boethius. Though this is of small scientific value, yet it is of great importance in history. It was at that time the only book from which European scholars could learn the elements of geometry. Gerbert studied it with zeal, and is generally believed himself to be the author of a geometry. H. Weissenborn denies his authorship, and claims that the book in question consists of three parts which cannot come from one and the same author. This geometry contains nothing more than the one of Boethius, but the fact that occasional errors in the latter are herein corrected shows that the author had mastered the subject. "The first mathematical paper of the Middle Ages which deserves this name," says Hankel, "is a letter of Gerbert to Adalbold, bishop of Utrecht," in which is explained the reason why the area of a triangle, obtained "geometrically" by taking the product of the base by half its altitude, differs from the area calculated "arithmetically," according to the formula $\scriptstyle {{\frac {1}{2}}a(a+1)}$, used by surveyors, where a stands for a side of an equilateral triangle. He gives the correct explanation that in the latter formula all the small squares, in which the triangle is ​supposed to be divided, are counted in wholly, even though parts of them project beyond it.
Gerbert made a careful study of the arithmetical works of Boethius. He himself published two works,—Rule of Computation on the Abacus, and A Small Book on the Division of Numbers. They give an insight into the methods of calculation practised in Europe before the introduction of the Hindoo numerals. Gerbert used the abacus, which was probably unknown to Alcuin. Bernelinus, a pupil of Gerbert, describes it as consisting of a smooth board upon which geometricians were accustomed to strew blue sand, and then to draw their diagrams. For arithmetical purposes the board was divided into 30 columns, of which 3 were reserved for fractions, while the remaining 27 were divided into groups with 3 columns in each. In every group the columns were marked respectively by the letters C (centum), D (decem), and S (singularis) or M (monas). Bernelinus gives the nine numerals used, which are the apices of Boethius, and then remarks that the Greek letters may be used in their place. By the use of these columns any number can be written without introducing a zero, and all operations in arithmetic can be performed in the same way as we execute ours without the columns, but with the symbol for zero. Indeed, the methods of adding, subtracting, and multiplying in vogue among the abacists agree substantially with those of to-day. But in a division there is very great difference. The early rules for division appear to have been framed to satisfy the following three conditions: (1) The use of the multiplication table shall be restricted as far as possible; at least, it shall never be required to multiply mentally a figure of two digits by another of one digit. (2) Subtractions shall be avoided as much as possible and replaced by additions. (3) The operation shall proceed in a purely mechanical way, without requiring trials. ​That it should be necessary to make such conditions seems strange to us; but it must be remembered that the monks of the Middle Ages did not attend school during childhood and learn the multiplication table while the memory was fresh. Gerbert's rules for division are the oldest extant. They are so brief as to be very obscure to the uninitiated. They were probably intended simply to aid the memory by calling to mind the successive steps in the work. In later manuscripts they are stated more fully. In dividing any number by another of one digit, say 668 by 6, the divisor was first increased to 10 by adding 4. The process is exhibited in the adjoining figure. As it continues, we must imagine the digits ${\overset {\widehat {\vert {\widehat {C~\vert ~D~}}\vert {\widehat {~S}}\vert }}{\begin{array}{|c|c|c|}&&\scriptstyle {6}\\&&\scriptstyle {4}\\\hline \scriptstyle {\not 6}&\scriptstyle {\not 6}&\scriptstyle {\not 8}\\\hline \scriptstyle {\not 6}&\scriptstyle {\not 6}&\scriptstyle {\not 8}\\\scriptstyle {\not 2}&\scriptstyle {\not 4}&\scriptstyle {\not 6}\\\scriptstyle {\not 1}&\scriptstyle {\not 8}&\scriptstyle {\not 8}\\\scriptstyle {\not 1}&\scriptstyle {\not 4}&\scriptstyle {\not 8}\\&\scriptstyle {\not 2}&\scriptstyle {2}\\&\scriptstyle {\not 4}&\\&\scriptstyle {\not 6}&\\&\scriptstyle {\not 2}&\\&\scriptstyle {\not 2}&\\\hline &\scriptstyle {\not 6}&\scriptstyle {\not 6}\\&\scriptstyle {\not 2}&\scriptstyle {\not 8}\\&\scriptstyle {\not 1}&\scriptstyle {\not 2}\\\scriptstyle {1}&\scriptstyle {1}&\scriptstyle {1}\\\end{array}}}$which are crossed out, to be erased and then replaced by the ones beneath. It is as follows: $\scriptstyle {600\div 10=60}$, but, to rectify the error, $\scriptstyle {4\times 60}$, or 240, must be added; $\scriptstyle {200\div 10=20}$, but $\scriptstyle {4\times 20}$, or 60, must be added. We now writs for $\scriptstyle {60+40+60}$, its sum 180, and continue thus: $\scriptstyle {100\div 10=10}$; the correction necessary is $\scriptstyle {4\times 10}$, or 40, which, added to 80, gives 120. Now $\scriptstyle {100\div 10=10}$, and the correction $\scriptstyle {4\times 10}$, together with the 20, gives 60. Proceeding as before, $\scriptstyle {60\div 10=6}$; the correction is $\scriptstyle {4\times 6=24}$. Now $\scriptstyle {20\div 10=2}$, the correction being $\scriptstyle {4\times 2=8}$. In the column of units we have now $\scriptstyle {8+4+8}$, or 20. As before, $\scriptstyle {20\div 10=2}$; the correction is $\scriptstyle {2\times 4=8}$, which is not divisible by 10, bet only by 6, giving the quotient 1 and the remainder 2. All the partial quotients taken together give $\scriptstyle {60+20+10+10+6+2+2+1=111}$, and the remainder 2.
Similar but more complicated, is the process when the divisor contains two or more digits. Were the divisor 27, ​then the next higher multiple of 10, or 30, would be taken for the divisor, but corrections would be required for the 3. He who has the patience to carry such a division through to the end, will understand why it has been said of Gerbert that "Regulas dedit, quæ a sudantibus abacistis vix intelliguntur." He will also perceive why the Arabic method of division, when first introduced, was called the divisio aurea, but the one on the abacus, the divisio ferrea.
In his book on the abacus, Bernelinus devotes a chapter to fractions. These are, of course, the duodecimals, first used by the Romans. For want of a suitable notation, calculation with them was exceedingly difficult. It would be so even to us, were we accustomed, like the early abacists, to express them, not by a numerator or denominator, but by the application of names, such as uncia for $\scriptstyle {\frac {1}{12}}$, quincunx for $\scriptstyle {\frac {5}{12}}$, dodrans for $\scriptstyle {\frac {9}{12}}$.
In the tenth century, Gerbert was the central figure among the learned. In his time the Occident came into secure possession of all mathematical knowledge of the Romans. During the eleventh century it was studied assiduously. Though numerous works were written on arithmetic and geometry, mathematical knowledge in the Occident was still very insignificant. Scanty indeed were the mathematical treasures obtained from Roman sources.
Translation of Arabic Manuscripts.
By his great erudition and phenomenal activity, Gerbert infused new life into the study not only of mathematics, but also of philosophy. Pupils from France, Germany, and Italy gathered at Rheims to enjoy his instruction. When they themselves became teachers, they taught of course not only the use of the abacus and geometry, but also what they had ​learned of the philosophy of Aristotle. His philosophy was known, at first, only through the writings of Boethius. But the growing enthusiasm for it created a demand for his complete works. Greek texts were wanting. But the Latins heard that the Arabs, too, were great admirers of Peripatetism, and that they possessed translations of Aristotle's works and commentaries thereon. This led them finally to search for and translate Arabic manuscripts. During this search, mathematical works also came to their notice, and were translated into Latin. Though some few unimportant works may have been translated earlier, yet the period of greatest activity began about 1100. The zeal displayed in acquiring the Mohammedan treasures of knowledge excelled even that of the Arabs themselves, when, in the eighth century, they plundered the rich coffers of Greek and Hindoo science.
Among the earliest scholars engaged in translating manuscripts into Latin was Athelard of Bath. The period of his activity is the first quarter of the twelfth century. He travelled extensively in Asia Minor, Egypt, and Spain, and braved a thousand perils, that he might acquire the language and science of the Mohammedans. He made the earliest translations, from the Arabic, of Euclid's Elements and of the astronomical tables of Mohammed ben Musa Hovarezmi. In 1857, a manuscript was found in the library at Cambridge, which proved to be the arithmetic by Mohammed ben Musa in Latin. This translation also is very probably due to Athelard.
At about the same time flourished Plato of Tivoli or Plato Tiburtinus, He effected a translation of the astronomy of Al Battani and of the Sphœrica of Theodosius. Through the former, the term sinus was introduced into trigonometry.
About the middle of the twelfth century there was a group of Christian scholars busily at work at Toledo, under the ​leadership of Raymond, then archbishop of Toledo. Among those who worked under his direction, John of Seville was most prominent. He translated works chiefly on Aristotelian philosophy. Of importance to us is a liber algorismi, compiled by him from Arabic authors. On comparing works like this with those of the abacists, we notice at once the most striking difference, which shows that the two parties drew from independent sources. It is argued by some that Gerbert got his apices and his arithmetical knowledge, not from Boethius, but from the Arabs in Spain, and that part or the whole of the geometry of Boethius is a forgery, dating from the time of Gerbert. If this were the case, then the writings of Gerbert would betray Arabic sources, as do those of John of Seville. But no points of resemblance are found. Gerbert could not have learned from the Arabs the use of the abacus, because all evidence we have goes to show that they did not employ it. Nor is it probable that he borrowed from the Arabs the apices, because they were never used in Europe except on the abacus. In illustrating an example in division, mathematicians of the tenth and eleventh centuries state an example in Roman numerals, then draw an abacus and insert in it the necessary numbers with the apices. Hence it seems probable that the abacus and apices were borrowed from the same source. The contrast between authors like John of Seville, drawing from Arabic works, and the abacists, consists in this, that, unlike the latter, the former mention the Hindoos, use the term algorism, calculate with the zero, and do not employ the abacus. The former teach the extraction of roots, the abacists do not; they teach the sexagesimal fractions used by the Arabs, while the abacists employ the duo-decimals of the Romans.
A little later than John of Seville flourished Gerard of Cremona in Lombardy. Being desirous to gain possession of ​the Almagest, he went to Toledo, and there, in 1175, translated this great work of Ptolemy. Inspired by the richness of Mohammedan literature, he gave himself up to its study. He translated into Latin over 70 Arabic works. Of mathematical treatises, there were among these, besides the Almagest, the 15 books of Euclid, the Sphœrica of Theodosius, a work of Menelaus, the algebra of Mohammed ben Musa Hovarezmi, the astronomy of Dshabir ben Aflah, and others less important.
In the thirteenth century, the zeal for the acquisition of Arabic learning continued. Foremost among the patrons of science at this time ranked Emperor Frederick II. of Hohenstaufen (died 1250). Through frequent contact with Mohammedan scholars, he became familiar with Arabic science. He employed a number of scholars in translating Arabic manuscripts, and it was through him that we came in possession of a new translation of the Almagest. Another royal head deserving mention as a zealous promoter of Arabic science was Alfonso X. of Castile (died 1284). He gathered around him a number of Jewish and Christian scholars, who translated and compiled astronomical works from Arabic sources. Rabbi Zag and Iehuda ben Mose Cohen were the most prominent among them. Astronomical tables prepared by these two Jews spread rapidly in the Occident, and constituted the basis of all astronomical calculation till the sixteenth century. The number of scholars who aided in transplanting Arabic science upon Christian soil was large. But we mention only one more. Giovanni Campano of Novara (about 1260) brought out a new translation of Euclid, which drove the earlier ones from the field, and which formed the basis of the printed editions.
At the close of the twelfth century, the Occident was in possession of the so-called Arabic notation. The Hindoo methods of calculation began to supersede the cumbrous ​methods 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 Sphœrica 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 Casso, "forsooth a great mathematician," by calling him a "counter-caster." 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.
In 1220, Leonardo of Pisa published his Practica Geometriœ, 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 $\scriptstyle {x^{2}+5}$ and $\scriptstyle {x^{2}-5}$ are each square numbers. The answer is $\scriptstyle {x=3{\frac {5}{12}}}$; for $\scriptstyle {(3{\frac {5}{12}})^{2}+5=(4{\frac {1}{12}})^{2},~(2{\frac {5}{12}})^{2}-5=(2{\frac {7}{12}})^{2}}$. 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.
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 $\scriptstyle {x^{3}+2x^{2}+10x=20}$. 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 $\scriptstyle {\frac {t}{2}}$; that of the second, $\scriptstyle {\frac {t}{3}}$; that of the third, $\scriptstyle {\frac {t}{6}}$. Desirous of depositing the sum at a safer place, each takes at hazard a certain amount; the first takes x, but deposits only $\scriptstyle {\frac {x}{2}}$; the second carries y, but deposits only $\scriptstyle {\frac {y}{3}}$; the third takes z, and deposits $\scriptstyle {\frac {z}{6}}$. Of the amount deposited each one must receive exactly $\scriptstyle {\frac {1}{8}}$, 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 $\scriptstyle {t=47}$, $\scriptstyle {x=33}$, $\scriptstyle {y=13}$, $\scriptstyle {z=1}$.
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 ​metaphysics 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 quarry-rubbish. 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."
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. Mediaeval 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 ​found in Fibonacci's great work, published three centuries earlier.
Perhaps the greatest result of the influx of Arabic learning was the establishment of universities. What was their attitude toward mathematics? The University of Paris, so famous at the beginning of the twelfth century under the teachings of Abelard, paid but little attention to this science during the Middle Ages. Geometry was neglected, and Aristotle's logic was the favourite study. In 1336, a rule was introduced that no student should take a degree without attending lectures on mathematics, and from a commentary on the first six books of Euclid, dated 1536, it appears that candidates for the degree of A.M. had to give an oath that they had attended lectures on these books. Examinations, when held at all, probably did not extend beyond the first book, as is shown by the nickname "magister matheseos," applied to the Theorem of Pythagoras, the last in the first book. More attention was paid to mathematics at the University of Prague, founded 1384. For the Baccalaureate degree, students were required to take lectures on Sacro Bosco's famous work on astronomy. Of candidates for the A.M. were required not only the six books of Euclid, but an additional knowledge of applied mathematics. Lectures were given on the Almagest. At the University of Leipzig, the daughter of Prague, and at Cologne, less work was required, and, as late as the sixteenth century, the same requirements were made at these as at Prague in the fourteenth. The universities of Bologna, Padua, Pisa, occupied similar positions to the ones in Germany, only that purely astrological lectures were given in place of lectures on the Almagest. At Oxford, in the middle of the fifteenth century, the first two books of Euclid were read.
Thus it will be seen that the study of mathematics was ​maintained at the universities only in a half-hearted manner. No great mathematician and teacher appeared, to inspire the students. The best energies of the schoolmen were expended upon the stupid subtleties of their philosophy. The genius of Leonardo of Pisa left no permanent impress upon the age, and another Renaissance of mathematics was wanted.