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an additional reason for keeping silence, or as a justification for speech? In forming a judgment after a lapse of three and a half centuries as to Cardan's action, while having regard both to the sanctity of an oath at the time in question, and to the altered state of the case between him and Tartaglia consequent on Ludovico Ferrari's discovery, an hypothesis not overstrained in the direction of charity may be advanced to the effect that Cardan might well have deemed he was justified in revealing to the world the rules which Tartaglia had taught him, considering that these isolated rules had been developed by his own study and Ferrari's into a principle by which it would be possible to work a complete revolution in the science of Algebra.
In any case, six years were allowed to elapse before Cardan, by publishing Tartaglia's rules in the _Book of the Great Art_, did the deed which, in the eyes of many, branded him as a liar and dishonest, and drove Tartaglia almost wild with rage. That his offence did not meet with universal reprobation is shown by negative testimony in the _Judicium de Cardano_, by Gabriel Naude.[106] In the course of his essay Naude lets it be seen how thoroughly he dislikes the character of the man about whom he writes. No evil disposition attributed to Cardan by himself or by his enemies is left unnoticed, and a lengthy catalogue of his offences is set down, but this list does not contain the particular sin of broken faith in the matter of Tartaglia's rules. On the contrary, after abusing and ridiculing a large portion of his work, Naude breaks out into almost rhapsodical eulogy about Cardan's contributions to Mathematical science. "Quis negabit librum de Proportionibus dignum esse, qui cum pulcherrimis antiquorum inventis conferatur? Quis in Arithmetica non stupet, eum tot difficultates superasse, quibus explicandis Villafrancus, Lucas de Burgo, Stifelius, Tartalea, vix ac ne vix quidem pares esse potuissent?" It seems hard to believe, after reading elsewhere the bitter assaults of Naude,[107] that he would have neglected so tempting an opportunity of darkening the shadows, if he himself had felt the slightest offence, or if public opinion in the learned world was in any perceptible degree scandalized by the disclosure made by the publication of the _Book of the Great Art_.
This book was published at Nuremberg in 1545, and in its preface and dedication Cardan fully acknowledges his obligations to Tartaglia and Ferrari, with respect to the rules lately discussed, and gives a catalogue of the former students of the Art, and attributes to each his particular contribution to the mass of knowledge which he here presents to the world. Leonardo da Pisa,[108] Fra Luca da Borgo, and Scipio Ferreo all receive due credit for their work, and then Cardan goes on to speak of "my friend Niccolo Tartaglia of Brescia, who, in his contest with Antonio Maria Fiore, the pupil of Ferreo, elaborated this rule to assure him of victory, a rule which he made known to me in answer to my many prayers." He goes on to acknowledge other obligations to Tartaglia:[109] how the Brescian had first taught him that algebraical discovery could be most effectively advanced by geometrical demonstration, and how he himself had followed this counsel, and had been careful to give the demonstration aforesaid for every rule he laid down.
The _Book of the Great Art_ was not published till six years after Cardan had become the sharer of Tartaglia's secret, which had thus had ample time to germinate and bear fruit in the fertile brain upon which it was cast. It is almost certain that the treatise as a whole--leaving out of account the special question of the solution of cubic equations--must have gained enormously in completeness and lucidity from the fresh knowledge revealed to the writer thereof by Tartaglia's reluctant disclosure, and, over and beyond this, it must be borne in mind that Cardan had been working for several years at Giovanni Colla's questions in conjunction with Ferrari, an algebraist as famous as Tartaglia or himself. The opening chapters of the book show that Cardan was well acquainted with the chief properties of the roots of equations of all sorts. He lays it down that all square numbers have two different kinds of root, one positive and one negative,[110] _vera_ and _ficta_: thus the root of 9 is either 3. or -3. He shows that when a case has all its roots, or when none are impossible, the number of its positive roots is the same as the number of changes in the signs of the terms when they are all brought to one side. In the case of _x^3 + 3bx = 2c_, he demonstrates his first resolution of a cubic equation, and gives his own version of his dealings with Tartaglia. His chief obligation to the Brescian was the information how to solve the three cases which follow, _i.e. x^3 + bx = c. x^3 = bx + c._ and _x^3 + c = bx_, and this he freely acknowledges, and furthermore admits the great service of the system of geometrical demonstration which Tartaglia had first suggested to him, and which he always employed hereafter. He claims originality for all processes in the book not ascribed to others, asserting that all the demonstrations of existing rules were his own except three which had been left by Mahommed ben Musa, and two invented by Ludovico Ferrari.
With this vantage ground beneath his feet Cardan raised the study of Algebra to a point it had never reached before, and climbed himself to a height of fame to which Medicine had not yet brought him. His name as a mathematician was known throughout Europe, and the success of his book was remarkable. In the _De Libris Propriis_ there is a passage which indicates that he himself was not unconscious of the renown he had won, or disposed to underrate the value of his contribution to mathematical science. "And even if I were to claim this art (Algebra) as my own invention, I should perhaps be speaking only the truth, though Nicomachus, Ptolemaeus, Paciolus, Boetius, have written much thereon. For men like these never came near to discover one-hundredth part of the things discovered by me. But with regard to this matter--as with divers others--I leave judgment to be given by those who shall come after me. Nevertheless I am constrained to call this work of mine a perfect one, seeing that it well-nigh transcends the bounds of human perception."[111]
FOOTNOTES:
[84] It was published at Milan by Bernardo Caluschio, with a dedication--dated 1537--to Francesco Gaddi, a descendant of the famous family of Florence. This man was Prior of the Augustinian Canons in Milan, and a great personage, but ill fortune seems to have overtaken him in his latter days. Cardan writes (_Opera_, tom. i. p. 107):--"qui cum mihi amicus esset dum floreret, Rexque cognomine ob potentiam appellaretur, conjectus in carcerem, misere vitam ibi, ne dicam crudeliter, finivit: nam per quindecim dies in profundissima gorgyne fuit, ut vivus sepeliretur."
[85] There is a reference to Osiander in _De Subtilitate_, p. 523. Cardan gives a full account of his relations with Osiander and Petreius in _Opera_, tom. i. p. 67.
[86] November 1536.
[87] Ferrari was one of Cardan's most distinguished pupils. "Ludovicus Ferrarius Bononiensis qui Mathematicas et Mediolani et in patria sua professus est, et singularis in illis eruditionis."--_De Vita Propria_, ch. xxxv. p. 111. There is a short memoir of Ferrari in _Opera_, tom. ix.
[88] _Opera_, tom. i. p. 66.
[89] Fra Luca's book, _Summa de Arithmetica Geometria Proportioni e Proportionalita_, extends as far as the solution of quadratic equations, of which only the positive roots were used. At this time letters were rarely used to express known quantities.
[90] The early writers on Algebra used _numerus_ for the absolute or known term, _res_ or _cosa_ for the first power, _quadratum_ for the second, and _cubus_ for the third. The signs + and - first appear in the work of Stifelius, a German writer, who published a book of Arithmetic in 1544. Robert Recorde in his _Whetstone of Wit_ seems first to have used the sign of equality =. Vieta in France first applied letters as general symbols of quantity, though the earlier algebraists used them occasionally, chiefly as abbreviations. Aristotle also used them in the _Physics.--Libri. Hist. des Sciences Mathematiques_. i. 104.
[91] _Opera_, tom. iv. p. 222.
[92] In the conclusion of the Treatise on Arithmetic, Cardan points out certain errors in the work of Fra Luca. Fra Luca was a pupil of Piero della Francesca, who was highly skilled in Geometry, and who, according to Vasari, first applied perspective to the drawing of the human form.
[93] Tartaglia, _i.e._ the stutterer.
[94] Papadopoli, _Hist. Gymn. Pata._ (Ven. 1724).
[95] "Balbisonem post relatam jurisprudentiae lauream redeuntem Brixiam Nicolaus secutus est, caepitque ex Mathematicis gloriam sibi ac divitias parare, aeque paupertatis impatiens, ac fortunae melioris cupidus, quam dum Brixiae tuetur, homo morosae, et inurbanae rusticitatis prope omnium civium odia sibi conciliavit. Quamobrem alibi vivere coactus, varias Italiae urbes incoluit, ac Ferrariae, Parmae, Mediolani, Romae, Genuae, arithmeticam, geometricam, ceteraque quae ad Mathesim pertinent, docuit; depugnavitque scriptis accerrimis cum Cardano ac sibi ex illis quaesivit nomen et gloriam. Tandem domicilium posuit Venetiis, ubi non a Senatoribus modo, ut mos Venetus habet eruditorum hominum studiosissimus, maximi habitus est, at etiam a variis Magnatum ac Principum legatis praemiis ac muneribus auctus sortem, quam tamdiu expetierat visus sibi est conciliasse. Ergo ratus se majorem, quam ut a civibus suis contemneretur, Brixiam rediit, ubi spe privati stipendii Euclidis elementa explanare coepit; sed quae illum olim a civitate sua austeritas, rustica, acerba, morosa, depulerat, eadem illum in eum apud omnes contemptum, et odium iterum dejicit, ut exinde horrendus ac detestabilis omnibus fugere, atque iterum Venetias confugere compulsus fuerit. Ibi persenex decessit."--Papadopoli, _Hist. Gymn. Pata.,_ ii. p. 210.
[96] This work is the chief authority for the facts which follow. The edition referred to is that of Venice, 1546. There is also a full account of the same in Cossali, _Origine dell' Algebra_ (Parma, 1799). vol. ii. p. 96.
[97] _Quesiti et Inventioni_, p. 115.
[98] Cardan writes: "Vi supplico per l'amor che mi portati, et per l'amicitia ch'e tra noi, che spero durara fin che viveremo, che mi mandati sciolta questa questione. 1 cubo piu 3. cose egual a 10." Cardan had mistaken (1/3 _b_)^3 for 1/3 _b_^3, or the cube of 1/3 of the co-efficient for 1/3 of the cube of the co-efficient.--_Quesiti et Inventioni_ p. 124.
[99] _Quesiti et Inventioni_, p. 125.
[100] "Non ha datta fora tal opera come cose composto da sua testa ma come cose ellette raccolte e copiate de diverse libri a penna."--_Quesiti et Inventioni_, p. 127.
[101] Cardan repeats the remark in the first chapter of the _Liber Artis Magnae_ (_Opera_, tom. iv. p. 222). "Deceptus enim ego verbis Lucae Paccioli, qui ultra sua capitula, generale ullum aliud esse posse negat (quanquam tot jam antea rebus a me inventis, sub manibus esset) desperabam tamen invenire, quod quaerere non audebam." Perhaps he wrote them down as an apology or a defence against the storm which he anticipated as soon as Tartaglia should have seen the new Algebra.
[102] Subsequently Tartaglia wrote very bitterly against Cardan, as the latter mentions in _De Libris Propriis_. "Nam etsi Nicolaus Tartalea libris materna lingua editis nos calumniatur, impudentiae tamen ac stultitiae suae non aliud testimonium quaeras, quam ipsos illius libros, in quibus nominatim splendidiorem unumquemque e civibus suis proscindit: adeo ut nemo dubitet insanisse hominem aliquo infortunio."--_Opera_, tom. i. p. 80.
[103] _Quesiti et Inventioni_, p. 129.
[104] Montucla, _Histoire de Math._ i. 596, gives a full account of Ferrari's process.
[105] In the _De Vita Propria_, Cardan dismisses the
In any case, six years were allowed to elapse before Cardan, by publishing Tartaglia's rules in the _Book of the Great Art_, did the deed which, in the eyes of many, branded him as a liar and dishonest, and drove Tartaglia almost wild with rage. That his offence did not meet with universal reprobation is shown by negative testimony in the _Judicium de Cardano_, by Gabriel Naude.[106] In the course of his essay Naude lets it be seen how thoroughly he dislikes the character of the man about whom he writes. No evil disposition attributed to Cardan by himself or by his enemies is left unnoticed, and a lengthy catalogue of his offences is set down, but this list does not contain the particular sin of broken faith in the matter of Tartaglia's rules. On the contrary, after abusing and ridiculing a large portion of his work, Naude breaks out into almost rhapsodical eulogy about Cardan's contributions to Mathematical science. "Quis negabit librum de Proportionibus dignum esse, qui cum pulcherrimis antiquorum inventis conferatur? Quis in Arithmetica non stupet, eum tot difficultates superasse, quibus explicandis Villafrancus, Lucas de Burgo, Stifelius, Tartalea, vix ac ne vix quidem pares esse potuissent?" It seems hard to believe, after reading elsewhere the bitter assaults of Naude,[107] that he would have neglected so tempting an opportunity of darkening the shadows, if he himself had felt the slightest offence, or if public opinion in the learned world was in any perceptible degree scandalized by the disclosure made by the publication of the _Book of the Great Art_.
This book was published at Nuremberg in 1545, and in its preface and dedication Cardan fully acknowledges his obligations to Tartaglia and Ferrari, with respect to the rules lately discussed, and gives a catalogue of the former students of the Art, and attributes to each his particular contribution to the mass of knowledge which he here presents to the world. Leonardo da Pisa,[108] Fra Luca da Borgo, and Scipio Ferreo all receive due credit for their work, and then Cardan goes on to speak of "my friend Niccolo Tartaglia of Brescia, who, in his contest with Antonio Maria Fiore, the pupil of Ferreo, elaborated this rule to assure him of victory, a rule which he made known to me in answer to my many prayers." He goes on to acknowledge other obligations to Tartaglia:[109] how the Brescian had first taught him that algebraical discovery could be most effectively advanced by geometrical demonstration, and how he himself had followed this counsel, and had been careful to give the demonstration aforesaid for every rule he laid down.
The _Book of the Great Art_ was not published till six years after Cardan had become the sharer of Tartaglia's secret, which had thus had ample time to germinate and bear fruit in the fertile brain upon which it was cast. It is almost certain that the treatise as a whole--leaving out of account the special question of the solution of cubic equations--must have gained enormously in completeness and lucidity from the fresh knowledge revealed to the writer thereof by Tartaglia's reluctant disclosure, and, over and beyond this, it must be borne in mind that Cardan had been working for several years at Giovanni Colla's questions in conjunction with Ferrari, an algebraist as famous as Tartaglia or himself. The opening chapters of the book show that Cardan was well acquainted with the chief properties of the roots of equations of all sorts. He lays it down that all square numbers have two different kinds of root, one positive and one negative,[110] _vera_ and _ficta_: thus the root of 9 is either 3. or -3. He shows that when a case has all its roots, or when none are impossible, the number of its positive roots is the same as the number of changes in the signs of the terms when they are all brought to one side. In the case of _x^3 + 3bx = 2c_, he demonstrates his first resolution of a cubic equation, and gives his own version of his dealings with Tartaglia. His chief obligation to the Brescian was the information how to solve the three cases which follow, _i.e. x^3 + bx = c. x^3 = bx + c._ and _x^3 + c = bx_, and this he freely acknowledges, and furthermore admits the great service of the system of geometrical demonstration which Tartaglia had first suggested to him, and which he always employed hereafter. He claims originality for all processes in the book not ascribed to others, asserting that all the demonstrations of existing rules were his own except three which had been left by Mahommed ben Musa, and two invented by Ludovico Ferrari.
With this vantage ground beneath his feet Cardan raised the study of Algebra to a point it had never reached before, and climbed himself to a height of fame to which Medicine had not yet brought him. His name as a mathematician was known throughout Europe, and the success of his book was remarkable. In the _De Libris Propriis_ there is a passage which indicates that he himself was not unconscious of the renown he had won, or disposed to underrate the value of his contribution to mathematical science. "And even if I were to claim this art (Algebra) as my own invention, I should perhaps be speaking only the truth, though Nicomachus, Ptolemaeus, Paciolus, Boetius, have written much thereon. For men like these never came near to discover one-hundredth part of the things discovered by me. But with regard to this matter--as with divers others--I leave judgment to be given by those who shall come after me. Nevertheless I am constrained to call this work of mine a perfect one, seeing that it well-nigh transcends the bounds of human perception."[111]
FOOTNOTES:
[84] It was published at Milan by Bernardo Caluschio, with a dedication--dated 1537--to Francesco Gaddi, a descendant of the famous family of Florence. This man was Prior of the Augustinian Canons in Milan, and a great personage, but ill fortune seems to have overtaken him in his latter days. Cardan writes (_Opera_, tom. i. p. 107):--"qui cum mihi amicus esset dum floreret, Rexque cognomine ob potentiam appellaretur, conjectus in carcerem, misere vitam ibi, ne dicam crudeliter, finivit: nam per quindecim dies in profundissima gorgyne fuit, ut vivus sepeliretur."
[85] There is a reference to Osiander in _De Subtilitate_, p. 523. Cardan gives a full account of his relations with Osiander and Petreius in _Opera_, tom. i. p. 67.
[86] November 1536.
[87] Ferrari was one of Cardan's most distinguished pupils. "Ludovicus Ferrarius Bononiensis qui Mathematicas et Mediolani et in patria sua professus est, et singularis in illis eruditionis."--_De Vita Propria_, ch. xxxv. p. 111. There is a short memoir of Ferrari in _Opera_, tom. ix.
[88] _Opera_, tom. i. p. 66.
[89] Fra Luca's book, _Summa de Arithmetica Geometria Proportioni e Proportionalita_, extends as far as the solution of quadratic equations, of which only the positive roots were used. At this time letters were rarely used to express known quantities.
[90] The early writers on Algebra used _numerus_ for the absolute or known term, _res_ or _cosa_ for the first power, _quadratum_ for the second, and _cubus_ for the third. The signs + and - first appear in the work of Stifelius, a German writer, who published a book of Arithmetic in 1544. Robert Recorde in his _Whetstone of Wit_ seems first to have used the sign of equality =. Vieta in France first applied letters as general symbols of quantity, though the earlier algebraists used them occasionally, chiefly as abbreviations. Aristotle also used them in the _Physics.--Libri. Hist. des Sciences Mathematiques_. i. 104.
[91] _Opera_, tom. iv. p. 222.
[92] In the conclusion of the Treatise on Arithmetic, Cardan points out certain errors in the work of Fra Luca. Fra Luca was a pupil of Piero della Francesca, who was highly skilled in Geometry, and who, according to Vasari, first applied perspective to the drawing of the human form.
[93] Tartaglia, _i.e._ the stutterer.
[94] Papadopoli, _Hist. Gymn. Pata._ (Ven. 1724).
[95] "Balbisonem post relatam jurisprudentiae lauream redeuntem Brixiam Nicolaus secutus est, caepitque ex Mathematicis gloriam sibi ac divitias parare, aeque paupertatis impatiens, ac fortunae melioris cupidus, quam dum Brixiae tuetur, homo morosae, et inurbanae rusticitatis prope omnium civium odia sibi conciliavit. Quamobrem alibi vivere coactus, varias Italiae urbes incoluit, ac Ferrariae, Parmae, Mediolani, Romae, Genuae, arithmeticam, geometricam, ceteraque quae ad Mathesim pertinent, docuit; depugnavitque scriptis accerrimis cum Cardano ac sibi ex illis quaesivit nomen et gloriam. Tandem domicilium posuit Venetiis, ubi non a Senatoribus modo, ut mos Venetus habet eruditorum hominum studiosissimus, maximi habitus est, at etiam a variis Magnatum ac Principum legatis praemiis ac muneribus auctus sortem, quam tamdiu expetierat visus sibi est conciliasse. Ergo ratus se majorem, quam ut a civibus suis contemneretur, Brixiam rediit, ubi spe privati stipendii Euclidis elementa explanare coepit; sed quae illum olim a civitate sua austeritas, rustica, acerba, morosa, depulerat, eadem illum in eum apud omnes contemptum, et odium iterum dejicit, ut exinde horrendus ac detestabilis omnibus fugere, atque iterum Venetias confugere compulsus fuerit. Ibi persenex decessit."--Papadopoli, _Hist. Gymn. Pata.,_ ii. p. 210.
[96] This work is the chief authority for the facts which follow. The edition referred to is that of Venice, 1546. There is also a full account of the same in Cossali, _Origine dell' Algebra_ (Parma, 1799). vol. ii. p. 96.
[97] _Quesiti et Inventioni_, p. 115.
[98] Cardan writes: "Vi supplico per l'amor che mi portati, et per l'amicitia ch'e tra noi, che spero durara fin che viveremo, che mi mandati sciolta questa questione. 1 cubo piu 3. cose egual a 10." Cardan had mistaken (1/3 _b_)^3 for 1/3 _b_^3, or the cube of 1/3 of the co-efficient for 1/3 of the cube of the co-efficient.--_Quesiti et Inventioni_ p. 124.
[99] _Quesiti et Inventioni_, p. 125.
[100] "Non ha datta fora tal opera come cose composto da sua testa ma come cose ellette raccolte e copiate de diverse libri a penna."--_Quesiti et Inventioni_, p. 127.
[101] Cardan repeats the remark in the first chapter of the _Liber Artis Magnae_ (_Opera_, tom. iv. p. 222). "Deceptus enim ego verbis Lucae Paccioli, qui ultra sua capitula, generale ullum aliud esse posse negat (quanquam tot jam antea rebus a me inventis, sub manibus esset) desperabam tamen invenire, quod quaerere non audebam." Perhaps he wrote them down as an apology or a defence against the storm which he anticipated as soon as Tartaglia should have seen the new Algebra.
[102] Subsequently Tartaglia wrote very bitterly against Cardan, as the latter mentions in _De Libris Propriis_. "Nam etsi Nicolaus Tartalea libris materna lingua editis nos calumniatur, impudentiae tamen ac stultitiae suae non aliud testimonium quaeras, quam ipsos illius libros, in quibus nominatim splendidiorem unumquemque e civibus suis proscindit: adeo ut nemo dubitet insanisse hominem aliquo infortunio."--_Opera_, tom. i. p. 80.
[103] _Quesiti et Inventioni_, p. 129.
[104] Montucla, _Histoire de Math._ i. 596, gives a full account of Ferrari's process.
[105] In the _De Vita Propria_, Cardan dismisses the
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