# Popular Science Monthly/Volume 44/November 1893/Mathematical Curiosities of the Sixteenth Century

 MATHEMATICAL CURIOSITIES OF THE SIXTEENTH CENTURY.
By M. V. BRANDICOURT.

IN the great intellectual revival of the sixteenth century, mathematics as well as letters and the arts were recuperated first from the pure sources of antiquity. Casting away poor Latin translations, second-hand versions through the Arabic, on which the Middle Ages had fed, geometricians emulated one another in zeal for learning the Greek language, in order that they might read in the original text the works of Euclid, Archimedes, Ptolemy, and Diophantus. Most of the works published at this epoch were only translations from Grecian authors. "The great thought of that time," says Montucla, "was simply to refine the minds of students and cause them to taste of a learning almost unknown till then. This could not be done all at once, and the human mind, like a weak stomach which too solid food would tire out, had to be brought by degrees to considerations of a higher order."

One of the earliest translations of Euclid is found in the Margarita philosophica of G. Riesch, prior of La Chartreuse at Friborg—a Latin book printed in Gothic characters at Heidelberg in 1496. It is a sort of encyclopædia of the science of the beginning of the sixteenth century, and certifies to the very extensive knowledge of the author. Each of the scientific treatises contained within it is adorned with very curious engravings of a naïve character.

Memmius, a noble of Venice, made a translation of the works of Apollonius in 1537, which was published after his death by one of his sons.

The mathematical sciences were then cultivated with most success in Italy; and when Francis I, of France, sent across the Alps for architects, painters, and sculptors to construct and adorn the magnificent châteaux of Chambord and Chenonceaux, he was thus also able to ask for his colleges algebraists who were certainly the first mathematicians in Europe. Algebra was not then what it has since become, a science employing only letters, signs, and symbols, having a well-defined significance and serving as the characters of a very clear and very precise language, which the initiated could understand as well as they could their mother tongue. The unknown quantity was then called "the thing" (res, coser; from which algebra was for some time named the art of the thing), and it was often represented by R. The square of the unknown quantity was called census (2). The signs ${\displaystyle \times }$ and ${\displaystyle =}$ were not known, but the initials of the words for which they stand were used. The sign—was not required, for the fruitful theory of negative quantities was not as yet known. In equations the coefficients of the unknown quantities were always figures, which became combined with the other factors during the operations, and of which no trace appeared in the final result. "We may conceive," says M. Chasles, in his History of Geometrical Methods, "that this cramped condition of imperfection did not constitute an algebraic science like that of our days, the power of which resides in those combinations of the signs themselves which assist the reasonings of intuition and lead by a mysterious way to the results sought."

Tartaglia Nicolo was an illustrious figure among the mathematicians of Italy. Born at Brescia in 1500, he was terribly mutilated at an early age, when his native city was captured by Gaston de Foix. His skull was broken in three places and his brain exposed, his jaws were split by a wound across his face, and he could not speak or eat. He nevertheless recovered, but always stammered, whence his name (tariagliare, to stammer). He was his own schoolmaster, and, after he had learned to read and write, devoted himself to the study of the ancient geometricians. At thirty-five years of age he taught mathematics in Venice. There he accepted a challenge which Fiori sent him, to solve twenty problems, all of which depended upon a particular case of cubic equations. Tartaglia solved them in less than two hours, and to commemorate his triumph composed mnemotechnic verses containing the solution. He was also the author of the ingenious formula for finding directly the area of a triangle of which all three of the sides are known.

Cardan Jerome, who was born in Paris, of Italian parents, September 24, 1501, was one of the most extraordinary men of his time. At twenty-two years of age, when he had just terminated his studies at the University of Pavia, he taught Euclid publicly. He also taught medicine, traveled in Scotland, Germany, and the Low Countries, and returning established himself in Rome as a pensioner of Pope Gregory XIII, and died there in 1576. Scaliger and De Thou assert that he had calculated the day of his death by astrology, and then starved himself to secure the fulfillment of his predictions.[1]

Such was the final eccentricity of this mathematician, who believed firmly in astrology and had visions, and he professed that he had been informed in a dream of all that was to happen to him. His costume, his bearing, corresponded with his strange character. He appeared sometimes in rags, sometimes splendidly dressed; ran through the streets at night, and the next day was drawn in a three-wheeled carriage. Yet he published a treatise on mathematics, Ars magna, which was remarkable for the age. Pertinently to the publication of this work he had controversies with Tartaglia, of which something should be said, for the curious picture they offer of the manners of the learned world in the sixteenth century.

Tartaglia, as we have said, discovered the solution of cubic equations. Cardan employed toward him all the persuasions in his power to obtain a communication to himself of the famous discovery. "I swear to you on the holy gospels," he promised, "that if you teach me your discoveries I will never publish them, and will, besides, record them for myself in cipher, so that no one shall be able to understand them after my death." Tartaglia, trusting in Cardan's good faith, communicated to him his rules summarized in twenty-seven mnemotechnic verses, in three strophes of nine verses each. Cardan, assisted by his pupil Ferrari, succeeded in extending the rules, solved equations of the fourth degree, and published the whole in the Ars magna. Tartaglia, irritated at the algebraist astrologer's violation of his word, fell into a violent rage. He sent to his enemy, according to the fashion of the time, several challenges, and in one of them went so far as to threaten Cardan and his pupil that he would wash their heads together and at the same time, "a thing which no barber in Italy could do." Cardan finally agreed to attend a disputation, which was to be held in a church in Milan on the 10th of August, 1548. He did not appear, but sent his pupil Ferrari. Ferrari bore his part in the contest alone, and the affair would have resulted in favor of Tartaglia if the hostile attitude of Cardan's friends had not caused him to leave Milan by a byroad. "These mathematical jousts," says M. Victorien Sardou, "these challenges proclaimed by heralds and trumpets, with great parade of pompous words and swelling eulogies, were more becoming to charlatans than to really learned men; but charlatanism was then in fashion; a discovery was the finder's secret, and a method of calculating was speculated upon as if it was a new medicinal powder." We do not wholly agree with M. Sardou. We see an example of intellectual activity and find a proof of the importance that was attached to algebraic discoveries in these scientific tournaments in which all classes of society are interested as formerly, in ancient Greece, they applauded the challenges of poets and the contests of athletes.

Leaving the Italian mathematicians and crossing the Alps, we find in Paris Pierre de la Ramée (better known by bis Latinized name Ramus) occupying at the Collége Saint-Gervais a chair of Mathematics which he had founded and which was subsequently made illustrious by Roberval. Ramus was born in 1515, at the little village of Cutry, and, a simple domestic at the Collége de Navarre, he found time to study all alone. He had the audacity at one-and-twenty years of age to sustain in the open Sorbonne, which swore by Aristotle alone, that all that the Stagyrite philosopher had said was false. Stranger still, "he seems to have convinced his judges, who conferred the degree of Master of Arts upon the bold innovator. Teaching philosophy, he continued to decry Aristotle. The Sorbonne was moved by his course to bring him before a tribunal, which declared him rash, arrogant, and impudent for having presumed to condemn the course and art of logic received by all nations." He was prohibited from writing and teaching contrary to Aristotle, "under penalty of corporeal punishment." He translated Euclid; and his Scholæ mathematicæ, in thirty-one books, was long used as a guide in the teaching of mathematics.

A mathematician of far superior merit to these was Viète, who expounded for the first time some of the most profound and most abstract theories that the human mind has ever invented. Born in 1540, in Poitou, he was appointed in 1580 maître des requétes in Paris. His time was thenceforth divided between the duties of his office and the study of mathematics. He had an extraordinary power of labor. De Thou, his historian, relates that he sometimes spent three days in his study, taking no more food and rest than were absolutely necessary, and not leaving his chair or desk for them. He was commissioned by Henri IV to decipher some dispatches which the court of Madrid had sent to the Governor of the Low Countries. He acquitted himself very well of this difficult task so well, indeed, that the Spaniards accused him of sorcery. He also solved in a few moments and in the presence of Henri IV a problem that had been proposed by Adrien Romain to all the mathematicians in the world. It was a problem extemporized as a diversion—an equation in the forty-fifth degree. The great analyst demonstrated that the equation depended upon the division of an arc into forty-five parts. He was the one who first in equations represented all the quantities by letters, with which all operations were performed which it had been usual to perform with numbers.

Viète published trigonometrical tables, in which he enunciated for the first time the law according to which the series of multiple or submultiple arcs increase. An enumeration of all his labors would require more space than we can spare. By his learned labors of analysis this man, the creator of modern algebra, prepared the way in which, were to follow with giant steps, making themselves illustrious, Descartes, Fermat, Pascal, and finally Newton.—Translated for The Popular Science Monthly from La Nature.

1. In one of his excursions to England he cast the horoscope of Edward VI, for whom he predicted a long life. Unfortunately, the king died in the next year. Having become used to such accidents, he was not disconcerted, but revised his calculations, rectified some of the figures, and found that the king had died in full accordance with the rules of astrology.