Jerome Cardan, William George Waters [reading list TXT] 📗
- Author: William George Waters
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he had published, so I was not moved to seek that which I despaired of finding; but, having made myself master of Tartaglia's method of demonstration, I understood how many other results might be attained; and, having taken fresh courage, I worked these out, partly by myself and partly by the aid of Ludovico Ferrari, a former pupil of mine. Now all the discoveries made by the men aforesaid are here marked with their names. Those unsigned were found out by me; and the demonstrations are all mine, except three discovered by Mahomet and two by Ludovico."[91]
This is Cardan's account of the scheme and origin of his book, and the succeeding pages will be mainly an amplification thereof. The earliest work on Algebra used in Italy was a translation of the MS. treatise of Mahommed ben Musa of Corasan, and next in order is a MS. written by a certain Leonardo da Pisa in 1202. Leonardo was a trader, who had learned the art during his voyages to Barbary, and his treatise and that of Mahommed were the sole literature on the subject up to the year 1494, when Fra Luca Pacioli da Borgo[92] brought out his volume treating of Arithmetic and Algebra as well. This was the first printed work on the subject.
After the invention of printing the interest in Algebra grew rapidly. From the time of Leonardo to that of Fra Luca it had remained stationary. The important fact that the resolution of all the cases of a problem may be comprehended in a simple formula, which may be obtained from the solution of one of its cases merely by a change of the signs, was not known, but in 1505 the Scipio Ferreo alluded to by Cardan, a Bolognese professor, discovered the rule for the solution of one case of a compound cubic equation. This was the discovery that Giovanni Colla announced when he went to Milan in 1536.
Cardan was then working hard at his Arithmetic--which dealt also with elementary Algebra--and he was naturally anxious to collect in its pages every item of fresh knowledge in the sphere of mathematics which might have been discovered since the publication of the last treatise. The fact that Algebra as a science had made such scant progress for so many years, gave to this new process, about which Giovanni Colla was talking, an extraordinary interest in the sight of all mathematical students; wherefore when Cardan heard the report that Antonio Maria Fiore, Ferreo's pupil, had been entrusted by his master with the secret of this new process, and was about to hold a public disputation at Venice with Niccolo Tartaglia, a mathematician of considerable repute, he fancied that possibly there would be game about well worth the hunting.
Fiore had already challenged divers opponents of less weight in the other towns of Italy, but now that he ventured to attack the well-known Brescian student, mathematicians began to anticipate an encounter of more than common interest. According to the custom of the time, a wager was laid on the result of the contest, and it was settled as a preliminary that each one of the competitors should ask of the other thirty questions. For several weeks before the time fixed for the contest Tartaglia studied hard; and such good use did he make of his time that, when the day of the encounter came, he not only fathomed the formula upon which Fiore's hopes were based, but, over and beyond this, elaborated two other cases of his own which neither Fiore nor his master Ferreo had ever dreamt of.
The case which Ferreo had solved by some unknown process was the equation _x^3 + px = q_, and the new forms of cubic equation which Tartaglia elaborated were as follows: _x^3 + px^2 = q_: and _x^3 - px^2 = q_. Before the date of the meeting, Tartaglia was assured that the victory would be his, and Fiore was probably just as confident. Fiore put his questions, all of which hinged upon the rule of Ferreo which Tartaglia had already mastered, and these questions his opponent answered without difficulty; but when the turn of the other side came, Tartaglia completely puzzled the unfortunate Fiore, who managed indeed to solve one of Tartaglia's questions, but not till after all his own had been answered. By this triumph the fame of Tartaglia spread far and wide, and Jerome Cardan, in consequence of the rumours of the Brescian's extraordinary skill, became more anxious than ever to become a sharer in the wonderful secret by means of which he had won his victory.
Cardan was still engaged in working up his lecture notes on Arithmetic into the Treatise when this contest took place; but it was not till four years later, in 1539, that he took any steps towards the prosecution of his design. If he knew anything of Tartaglia's character, and it is reasonable to suppose that he did, he would naturally hesitate to make any personal appeal to him, and trust to chance to give him an opportunity of gaining possession of the knowledge aforesaid, rather than seek it at the fountain-head. Tartaglia was of very humble birth, and according to report almost entirely self-educated. Through a physical injury which he met with in childhood his speech was affected; and, according to the common Italian usage, a nickname[93] which pointed to this infirmity was given to him. The blow on the head, dealt to him by some French soldier at the sack of Brescia in 1512, may have made him a stutterer, but it assuredly did not muddle his wits; nevertheless, as the result of this knock, or for some other cause, he grew up into a churlish, uncouth, and ill-mannered man, and, if the report given of him by Papadopoli[94] at the end of his history be worthy of credit, one not to be entirely trusted as an autobiographer in the account he himself gives of his early days in the preface to one of his works. Papadopoli's notice of him states that he was in no sense the self-taught scholar he represented himself to be, but that he was indebted for some portion at least of his training to the beneficence of a gentleman named Balbisono,[95] who took him to Padua to study. From the passage quoted below he seems to have failed to win the goodwill of the Brescians, and to have found Venice a city more to his taste. It is probable that the contest with Fiore took place after his final withdrawal from his birthplace to Venice.
In 1537 Tartaglia published a treatise on Artillery, but he gave no sign of making public to the world his discoveries in Algebra. Cardan waited on, but the morose Brescian would not speak, and at last he determined to make a request through a certain Messer Juan Antonio, a bookseller, that, in the interests of learning, he might be made a sharer of Tartaglia's secret. Tartaglia has given a version of this part of the transaction; and, according to what is there set down, Cardan's request, even when recorded in Tartaglia's own words, does not appear an unreasonable one, for up to this time Tartaglia had never announced that he had any intention of publishing his discoveries as part of a separate work on Mathematics. There was indeed a good reason why he should refrain from doing this in the fact that he could only speak and write Italian, and that in the Brescian dialect, being entirely ignorant of Latin, the only tongue which the writer of a mathematical work could use with any hope of success. Tartaglia's record of his conversation with Messer Juan Antonio, the emissary employed by Cardan, and of all the subsequent details of the controversy, is preserved in his principal work, _Quesiti et Inventioni Diverse de Nicolo Tartalea Brisciano_,[96] a record which furnishes abundant and striking instance of his jealous and suspicious temper. Much of it is given in the form of dialogue, the terms of which are perhaps a little too precise to carry conviction of its entire sincerity and spontaneity. It was probably written just after the final cause of quarrel in 1545, and its main object seems to be to set the author right in the sight of the world, and to exhibit Cardan as a meddlesome fellow not to be trusted, and one ignorant of the very elements of the art he professed to teach.[97]
The inquiry begins with a courteously worded request from Messer Juan Antonio (speaking on behalf of Messer Hieronimo Cardano), that Messer Niccolo would make known to his principal the rule by means of which he had made such short work of Antonio Fiore's thirty questions. It had been told to Messer Hieronimo that Fiore's thirty questions had led up to a case of the _cosa_ and the _cubus_ equal to the _numerus_, and that Messer Niccolo had discovered a general rule for such case. Messer Hieronimo now especially desired to be taught this rule. If the inventor should be willing to let this rule be published, it should be published as his own discovery; but, if he were not disposed to let the same be made known to the world, it should be kept a profound secret. To this request Tartaglia replied that, if at any time he might publish his rule, he would give it to the world in a work of his own under his own name, whereupon Juan Antonio moderated his demand, and begged to be furnished merely with a copy of the thirty questions preferred by Fiore, and Tartaglia's solutions of the same; but Messer Niccolo was too wary a bird to be taken with such a lure as this. To grant so much, he replied, would be to tell everything, inasmuch as Cardan could easily find out the rule, if he should be furnished with a single question and its solution. Next Juan Antonio handed to Tartaglia eight algebraical questions which had been confided to him by Cardan, and asked for answers to them; but Tartaglia, having glanced at them, declared that they were not framed by Cardan at all, but by Giovanni Colla. Colla, he declared, had sent him one of these questions for solution some two years ago. Another, he (Tartaglia) had given to Colla, together with a solution thereof. Juan Antonio replied by way of contradiction--somewhat lamely--that the questions had been handed over to him by Cardan and no one else, wishing to maintain, apparently, that no one else could possibly have been concerned in them, whereupon Tartaglia replied that, supposing the questions had been given by Cardan to Juan Antonio his messenger, Cardan must have got the questions from Colla, and have sent them on to him (Tartaglia) for solution because he could not arrive at the meaning of them himself. He waved aside Juan Antonio's perfectly irrelevant and fatuous protests--that Cardan would not in any case have sent these questions if they had been framed by another person, or if he had been unable to solve them. Tartaglia, on the other hand, declared that Cardan certainly did not comprehend them. If he did not know the rule by which Fiore's questions had been answered (that of the _cosa_ and the _cubus_ equal to the _numerus_), how could he solve these questions which he now sent, seeing that certain of them involved operations much more complicated than that of the rule above written? If he understood the questions which he now sent for solution, he could not want to be taught this rule. Then Juan Antonio moderated his demand still farther, and said he would be satisfied with a copy of the questions which Fiore had put to Tartaglia, adding that the favour would be much greater if Tartaglia's own questions were also given. He probably felt that it would be mere waste of breath to beg
This is Cardan's account of the scheme and origin of his book, and the succeeding pages will be mainly an amplification thereof. The earliest work on Algebra used in Italy was a translation of the MS. treatise of Mahommed ben Musa of Corasan, and next in order is a MS. written by a certain Leonardo da Pisa in 1202. Leonardo was a trader, who had learned the art during his voyages to Barbary, and his treatise and that of Mahommed were the sole literature on the subject up to the year 1494, when Fra Luca Pacioli da Borgo[92] brought out his volume treating of Arithmetic and Algebra as well. This was the first printed work on the subject.
After the invention of printing the interest in Algebra grew rapidly. From the time of Leonardo to that of Fra Luca it had remained stationary. The important fact that the resolution of all the cases of a problem may be comprehended in a simple formula, which may be obtained from the solution of one of its cases merely by a change of the signs, was not known, but in 1505 the Scipio Ferreo alluded to by Cardan, a Bolognese professor, discovered the rule for the solution of one case of a compound cubic equation. This was the discovery that Giovanni Colla announced when he went to Milan in 1536.
Cardan was then working hard at his Arithmetic--which dealt also with elementary Algebra--and he was naturally anxious to collect in its pages every item of fresh knowledge in the sphere of mathematics which might have been discovered since the publication of the last treatise. The fact that Algebra as a science had made such scant progress for so many years, gave to this new process, about which Giovanni Colla was talking, an extraordinary interest in the sight of all mathematical students; wherefore when Cardan heard the report that Antonio Maria Fiore, Ferreo's pupil, had been entrusted by his master with the secret of this new process, and was about to hold a public disputation at Venice with Niccolo Tartaglia, a mathematician of considerable repute, he fancied that possibly there would be game about well worth the hunting.
Fiore had already challenged divers opponents of less weight in the other towns of Italy, but now that he ventured to attack the well-known Brescian student, mathematicians began to anticipate an encounter of more than common interest. According to the custom of the time, a wager was laid on the result of the contest, and it was settled as a preliminary that each one of the competitors should ask of the other thirty questions. For several weeks before the time fixed for the contest Tartaglia studied hard; and such good use did he make of his time that, when the day of the encounter came, he not only fathomed the formula upon which Fiore's hopes were based, but, over and beyond this, elaborated two other cases of his own which neither Fiore nor his master Ferreo had ever dreamt of.
The case which Ferreo had solved by some unknown process was the equation _x^3 + px = q_, and the new forms of cubic equation which Tartaglia elaborated were as follows: _x^3 + px^2 = q_: and _x^3 - px^2 = q_. Before the date of the meeting, Tartaglia was assured that the victory would be his, and Fiore was probably just as confident. Fiore put his questions, all of which hinged upon the rule of Ferreo which Tartaglia had already mastered, and these questions his opponent answered without difficulty; but when the turn of the other side came, Tartaglia completely puzzled the unfortunate Fiore, who managed indeed to solve one of Tartaglia's questions, but not till after all his own had been answered. By this triumph the fame of Tartaglia spread far and wide, and Jerome Cardan, in consequence of the rumours of the Brescian's extraordinary skill, became more anxious than ever to become a sharer in the wonderful secret by means of which he had won his victory.
Cardan was still engaged in working up his lecture notes on Arithmetic into the Treatise when this contest took place; but it was not till four years later, in 1539, that he took any steps towards the prosecution of his design. If he knew anything of Tartaglia's character, and it is reasonable to suppose that he did, he would naturally hesitate to make any personal appeal to him, and trust to chance to give him an opportunity of gaining possession of the knowledge aforesaid, rather than seek it at the fountain-head. Tartaglia was of very humble birth, and according to report almost entirely self-educated. Through a physical injury which he met with in childhood his speech was affected; and, according to the common Italian usage, a nickname[93] which pointed to this infirmity was given to him. The blow on the head, dealt to him by some French soldier at the sack of Brescia in 1512, may have made him a stutterer, but it assuredly did not muddle his wits; nevertheless, as the result of this knock, or for some other cause, he grew up into a churlish, uncouth, and ill-mannered man, and, if the report given of him by Papadopoli[94] at the end of his history be worthy of credit, one not to be entirely trusted as an autobiographer in the account he himself gives of his early days in the preface to one of his works. Papadopoli's notice of him states that he was in no sense the self-taught scholar he represented himself to be, but that he was indebted for some portion at least of his training to the beneficence of a gentleman named Balbisono,[95] who took him to Padua to study. From the passage quoted below he seems to have failed to win the goodwill of the Brescians, and to have found Venice a city more to his taste. It is probable that the contest with Fiore took place after his final withdrawal from his birthplace to Venice.
In 1537 Tartaglia published a treatise on Artillery, but he gave no sign of making public to the world his discoveries in Algebra. Cardan waited on, but the morose Brescian would not speak, and at last he determined to make a request through a certain Messer Juan Antonio, a bookseller, that, in the interests of learning, he might be made a sharer of Tartaglia's secret. Tartaglia has given a version of this part of the transaction; and, according to what is there set down, Cardan's request, even when recorded in Tartaglia's own words, does not appear an unreasonable one, for up to this time Tartaglia had never announced that he had any intention of publishing his discoveries as part of a separate work on Mathematics. There was indeed a good reason why he should refrain from doing this in the fact that he could only speak and write Italian, and that in the Brescian dialect, being entirely ignorant of Latin, the only tongue which the writer of a mathematical work could use with any hope of success. Tartaglia's record of his conversation with Messer Juan Antonio, the emissary employed by Cardan, and of all the subsequent details of the controversy, is preserved in his principal work, _Quesiti et Inventioni Diverse de Nicolo Tartalea Brisciano_,[96] a record which furnishes abundant and striking instance of his jealous and suspicious temper. Much of it is given in the form of dialogue, the terms of which are perhaps a little too precise to carry conviction of its entire sincerity and spontaneity. It was probably written just after the final cause of quarrel in 1545, and its main object seems to be to set the author right in the sight of the world, and to exhibit Cardan as a meddlesome fellow not to be trusted, and one ignorant of the very elements of the art he professed to teach.[97]
The inquiry begins with a courteously worded request from Messer Juan Antonio (speaking on behalf of Messer Hieronimo Cardano), that Messer Niccolo would make known to his principal the rule by means of which he had made such short work of Antonio Fiore's thirty questions. It had been told to Messer Hieronimo that Fiore's thirty questions had led up to a case of the _cosa_ and the _cubus_ equal to the _numerus_, and that Messer Niccolo had discovered a general rule for such case. Messer Hieronimo now especially desired to be taught this rule. If the inventor should be willing to let this rule be published, it should be published as his own discovery; but, if he were not disposed to let the same be made known to the world, it should be kept a profound secret. To this request Tartaglia replied that, if at any time he might publish his rule, he would give it to the world in a work of his own under his own name, whereupon Juan Antonio moderated his demand, and begged to be furnished merely with a copy of the thirty questions preferred by Fiore, and Tartaglia's solutions of the same; but Messer Niccolo was too wary a bird to be taken with such a lure as this. To grant so much, he replied, would be to tell everything, inasmuch as Cardan could easily find out the rule, if he should be furnished with a single question and its solution. Next Juan Antonio handed to Tartaglia eight algebraical questions which had been confided to him by Cardan, and asked for answers to them; but Tartaglia, having glanced at them, declared that they were not framed by Cardan at all, but by Giovanni Colla. Colla, he declared, had sent him one of these questions for solution some two years ago. Another, he (Tartaglia) had given to Colla, together with a solution thereof. Juan Antonio replied by way of contradiction--somewhat lamely--that the questions had been handed over to him by Cardan and no one else, wishing to maintain, apparently, that no one else could possibly have been concerned in them, whereupon Tartaglia replied that, supposing the questions had been given by Cardan to Juan Antonio his messenger, Cardan must have got the questions from Colla, and have sent them on to him (Tartaglia) for solution because he could not arrive at the meaning of them himself. He waved aside Juan Antonio's perfectly irrelevant and fatuous protests--that Cardan would not in any case have sent these questions if they had been framed by another person, or if he had been unable to solve them. Tartaglia, on the other hand, declared that Cardan certainly did not comprehend them. If he did not know the rule by which Fiore's questions had been answered (that of the _cosa_ and the _cubus_ equal to the _numerus_), how could he solve these questions which he now sent, seeing that certain of them involved operations much more complicated than that of the rule above written? If he understood the questions which he now sent for solution, he could not want to be taught this rule. Then Juan Antonio moderated his demand still farther, and said he would be satisfied with a copy of the questions which Fiore had put to Tartaglia, adding that the favour would be much greater if Tartaglia's own questions were also given. He probably felt that it would be mere waste of breath to beg
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