AT this point it may not be inopportune to make a break in the record of
Cardan's life and work, and to treat in retrospect of that portion of his
time which he spent in the composition of his treatises on Arithmetic and
Algebra. Ever since 1535 he had been working intermittently at one or
other of these, but it would have been impossible to deal coherently and
effectively with the growth and completion of these two books--really the
most important of all he left behind him--while chronicling the goings and
comings of a life so adventurous as that of the author.
The prime object of Cardan's ambition was eminence as a physician. But,
during the long years of waiting, while the action of the Milanese doctors
kept him outside the bounds of their College, and even after this had been
opened to him without inducing ailing mortals to call for his services, he
would now and again fall into a transport of rage against his persecutors,
and of contempt for the public which refused to recognize him as a master
of his art, and cast aside his medical books for months at a time,
devoting himself diligently to Mathematics, the field of learning which,
next to Medicine, attracted him most powerfully. His father Fazio was a
geometrician of repute and a student of applied mathematics, and, though
his first desire was to make his son a jurisconsult, he gave Jerome in
early youth a fairly good grounding in arithmetic and geometry, deeming
probably that such training would not prove a bad discipline for an
intellect destined to attack those formidable tomes within which lurked
the mysteries of the Canon and Civil Law. Mathematical learning has given
to Cardan his surest title to immortality, and at the outset of his career
he found in mathematics rather than in medicine the first support in the
arduous battle he had to wage with fortune. His appointment to the Plat
lectureship at Milan has already been noted. In the discharge of his new
duties he was bound, according to the terms of the endowment of the Plat
lecturer, to teach the sciences of geometry, arithmetic, and astronomy,
and he began his course upon the lines laid down by the founder. Few
listeners came, however, and at this juncture Cardan took a step which
serves to show how real was his devotion to the cause of true learning,
and how lightly he thought of an additional burden upon his own back, if
this cause could be helped forward thereby. Keenly as he enjoyed his
mathematical work, he laid a part of it aside when he perceived that the
benches before him were empty, and, by way of making his lectures more
attractive, he occasionally substituted geography for geometry, and
architecture for arithmetic. The necessary research and the preparation of
these lectures led naturally to the accumulation of a large mass of notes,
and as these increased under his hand Jerome began to consider whether it
might not be worth his while to use them in the composition of one or more
volumes. In 1535 he delivered as Plat lecturer his address, the _Encomium
Geometriæ_, which he followed up shortly after by the publication of a
work, _Quindecim Libri Novæ Geometriæ_. But the most profitable labour of
these years was that which produced his first important book, _The
Practice of Arithmetic and Simple Mensuration_, which was published in
1539, a venture which brought to the author a reward of ten crowns.[84] It
was a well-planned and well-arranged manual, giving proof of the wide
erudition and sense of proportion possessed by the author. Besides dealing
with Arithmetic as understood by the modern school-boy, it discusses
certain astronomical operations, multiplication by memory, the mysteries
of the Roman and Ecclesiastical Calendars, and gives rules for the
solution of any problem arising from the terms of the same. It treats of
partnership in agriculture, the Mezzadria system still prevalent in
Tuscany and in other parts of Italy, of the value of money, of the strange
properties of certain numbers, and gives the first simple rules of Algebra
to serve as stepping-stones to the higher mathematics. It ends with
information as to house-rent, letters of credit and exchange, tables of
interest, games of chance, mensuration, and weights and measures. In an
appendix Cardan examines critically the work of Fra Luca Pacioli da Borgo,
an earlier writer on the subject, and points out numerous errors in the
same. The book from beginning to end shows signs of careful study and
compilation, and the fame which it brought to its author was well
deserved.
Cardan appended to the Arithmetic a printed notice which may be regarded
as an early essay in advertising. He was fully convinced that his works
were valuable and quite worth the sums of money he asked for them; the
world was blind, perhaps wilfully, to their merits, therefore he now
determined that it should no longer be able to quote ignorance of the
author as an excuse for not buying the book. This appendix was a
notification to the learned men of Europe that the writer of the _Practice
of Arithmetic_ had in his press at home thirty-four other works in MS.
which they might read with profit, and that of these only two had been
printed, to wit the _De Malo Medendi Usu_ and a tract on _Simples_. This
advertisement had something of the character of a legal document, for it
invoked the authority of the Emperor to protect the copyright of Cardan's
books within the Duchy of Milan for ten years, and to prevent the
introduction of them from abroad.
The Arithmetic proved far superior to any other treatise extant, and
everywhere won the approval of the learned. It was from Nuremberg that its
appearance brought the most valuable fruits. Andreas Osiander,[85] a
learned humanist and a convert to Lutheranism, and Johannes Petreius, an
eminent printer, were evidently impressed by the terms of Cardan's
advertisement, for they wrote to him and offered in combination to edit
and print any of the books awaiting publication in his study at Milan. The
result of this offer was the reprinting of _De Malo Medendi_, and
subsequently of the tract on Judicial Astrology, and of the treatise _De
Consolatione_; the _Book of the Great Art_, the treatises _De Sapientia_
and _De Immortalitate Animorum_ were published in the first instance by
these same patrons from the Nuremberg press.
But Cardan, while he was hard at work on his Arithmetic, had not forgotten
a certain report which had caused no slight stir in the world of
Mathematics some three years before the issue of his book on Arithmetic,
an episode which may be most fittingly told in his own words. "At this
time[86] it happened that there came to Milan a certain Brescian named
Giovanni Colla, a man of tall stature, and very thin, pale, swarthy, and
hollow-eyed. He was of gentle manners, slow in gait, sparing of his words,
full of talent, and skilled in mathematics. His business was to bring word
to me that there had been recently discovered two new rules in Algebra for
the solution of problems dealing with cubes and numbers. I asked him who
had found them out, whereupon he told me the name of the discoverer was
Scipio Ferreo of Bologna. 'And who else knows these rules?' I said. He
answered, 'Niccolo Tartaglia and Antonio Maria Fiore.' And indeed some
time later Tartaglia, when he came to Milan, explained them to me, though
unwillingly; and afterwards I myself, when working with Ludovico
Ferrari,[87] made a thorough study of the rules aforesaid. We devised
certain others, heretofore unnoticed, after we had made trial of these new
rules, and out of this material I put together my _Book of the Great
Art_."[88]
Before dealing with the events which led to the composition of the famous
work above-named, it may be permitted to take a rapid survey of the
condition of Algebra at the time when Cardan sat down to write. Up to the
beginning of the sixteenth century the knowledge of Algebra in Italy,
originally derived from Greek and Arabic sources, had made very little
progress, and the science had been developed no farther than to provide
for the solution of equations of the first or second degree.[89] In the
preface to the _Liber Artis Magnæ_ Cardan writes:--"This art takes its
origin from a certain Mahomet, the son of Moses, an Arabian, a fact to
which Leonard the Pisan bears ample testimony. He left behind him four
rules, with his demonstrations of the same, which I duly ascribe to him in
their proper place. After a long interval of time, some student, whose
identity is uncertain, deduced from the original four rules three others,
which Luca Paciolus put with the original ones into his book. Then three
more were discovered from the original rules, also by some one unknown,
but these attracted very little notice though they were far more useful
than the others, seeing that they taught how to arrive at the value of the
_cubus_ and the _numerus_ and of the _cubus quadratus_.[90] But in recent
times Scipio Ferreo of Bologna discovered the rule of the _cubus_ and the
_res_ equal to the _numerus_ (_x^3 + px=q_), truly a beautiful and
admirable discovery. For this Algebraic art outdoes all other subtlety of
man, and outshines the clearest exposition mortal wit can achieve: a
heavenly gift indeed, and a test of the powers of a man's mind. So
excellent is it in itself that whosoever shall get possession thereof,
will be assured that no problem exists too difficult for him to
disentangle. As a rival of Ferreo, Niccolo Tartaglia of Brescia, my
friend, at that time when he engaged in a contest with Antonio Maria
Fiore, the pupil of Ferreo, made out this same rule to help secure the
victory, and this rule he imparted to me after I had diligently besought
him thereanent. I, indeed, had been deceived by the words of Luca
Paciolus, who denied that there could be any general rule besides these
which 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 again for Tartaglia's answers. The
end of the matter was that Tartaglia handed over to the messenger the
questions which Fiore had propounded in the Venetian contest, and
authorized Juan Antonio to get a copy of his own from the notary who had
drawn up the terms of the disputation with Fiore. The date of this
communication is January 2, 1539, and on February 12 Cardan writes a long
letter to Tartaglia, complaining in somewhat testy spirit of the reception
given to his request. He is aggrieved that Tartaglia should have sent him
nothing but the questions put to him by Fiore, thirty in number indeed,
but only one in substance, and that he should have dared to hint that
those which he (Cardan) had sent for solution were not his own, but the
property of Giovanni Colla. Cardan had found Colla to be a conceited fool,
and had dragged the conceit out of him--a process which he was now about
to repeat for the benefit of Messer Niccolo Tartaglia. The letter goes on
to contradict all Tartaglia's assertions by arguments which do not seem
entirely convincing, and the case is not made better by the abusive
passages interpolated here and there, and by the demonstration of certain
errors in Tartaglia's book on Artillery. In short a more injudicious
letter could not have been written by any man hoping to get a favour done
to him by the person addressed.
In the special matter of the problems which he sent to Tartaglia by the
bookseller Juan Antonio, Cardan made a beginning of that tricky and
crooked course which he followed too persistently all through this
particular business. In his letter he maintains with a show of indignation
that he had long known these questions, had known them in fact before
Colla knew how to count ten, implying by these words that he knew how to
solve them, while in reality all he knew about them was the fact that they
existed. Tartaglia in his answer is not to be moved from his belief, and
tells Cardan flatly that he is still convinced Giovanni Colla took the
questions to Milan, where he found no one able to solve them, not even
Messer Hieronimo Cardano, and that the mathematician last-named sent them
on by the bookseller for solution, as has been already related.
This letter of Tartaglia's bears the date of February 13, 1539, and after
reading it and digesting its contents, Cardan seems to have come to the
conclusion that he was not working in the right way to get possession of
this secret which he felt he must needs master, if he wanted his
forthcoming book to mark a new epoch in this History of Mathematics, and
that a change of tactics was necessary. Alfonso d'Avalos, Cardan's friend
and patron, was at this time the Governor of Milan. D'Avalos was a man of
science, as well as a soldier, and Cardan had already sent to him a copy
of Tartaglia's treatise on Artillery, deeming that a work of this kind
would not fail to interest him. In his first letter to Tartaglia he
mentions this fact, while picking holes in the writer's theories
concerning transmitted force and views on gravitation. This mention of the
name of D'Avalos, the master of many legions and of many cannons as well,
to a man who had written a Treatise on the management of Artillery, and
devised certain engines and instruments for the management of the same,
was indeed a clever cast, and the fly was tempting enough to attract even
so shy a fish as Niccolo Tartaglia. In his reply to Jerome's scolding
letter of February 12, 1539, Tartaglia concludes with a description of the
instruments which he was perfecting: a square to regulate the discharge of
cannon, and to level and determine every elevation; and another instrument
for the investigation of distances upon a plane surface. He ends with a
request that Cardan will accept four copies of the engines aforesaid, two
for himself and two for the Marchese d'Avalos.
The tone of this letter shows that Cardan had at least begun to tame the
bear, who now seemed disposed to dance _ad libitum_ to the pleasant music
of words suggesting introductions to the governor, and possible patronage
of these engines for the working of artillery. Cardan's reply of March 19,
1539, is friendly--too friendly indeed--and the wonder is that Tartaglia's
suspicions were not aroused by its almost sugary politeness. It begins
with an attempt to soften down the asperities of their former
correspondence, some abuse of Giovanni Colla, and an apology for the rough
words of his last epistle. Cardan then shows how their misunderstanding
arose chiefly from a blunder made by Juan Antonio in delivering the
message, and invites Tartaglia to come and visit him in his own house in
Milan, so that they might deliberate together on mathematical questions;
but the true significance of the letter appears in the closing lines. "I
told the Marchese of the instruments which you had sent him, and he showed
himself greatly pleased with all you had done. And he commanded me to
write to you forthwith in pressing terms, and to tell you that, on the
receipt of my letter, you should come to Milan without fail, for he
desires to speak with you. And I, too, exhort you to come at once without
further deliberation, seeing that this said Marchese is wonted to reward
all men of worth in such noble and magnanimous and liberal fashion that
none of them ever goes away dissatisfied."
The receipt of this letter seems to have disquieted Tartaglia somewhat;
for he has added a note to it, in which he says that Cardan has placed him
in a position of embarrassment. He had evidently wished for an
introduction to D'Avalos, but now it was offered to him it seemed a burden
rather than a benefit. He disliked the notion of going to Milan; yet, if
he did not go, the Marchese d'Avalos might take offence. But in the end he
decided to undertake the journey; and, as D'Avalos happened then to be
absent from Milan on a visit to his country villa at Vigevano, he stayed
for three days in Cardan's house. As a recorder of conversations Tartaglia
seems to have had something of Boswell's gift. He gives an abstract of an
eventful dialogue with his host on March 25, 1539, which Cardan begins by
a gentle reproach anent his guest's reticence in the matter of the rule of
the _cosa_ and the _cubus_ equal to the _numerus_. Tartaglia's reply to
this complaint seems reasonable enough (it must be borne in mind that he
is his own reporter), and certainly helps to absolve him from the charge
sometimes made against him that he was nothing more than a selfish
curmudgeon who had resolved to let his knowledge die with him, rather than
share it with other mathematicians of whom he was jealous. He told Cardan
plainly that he kept his rules a secret because, for the present, it
suited his purpose to do so. At this time he had not the leisure to
elaborate farther the several rules in question, being engaged over a
translation of Euclid into Italian; but, when this work should be
completed, he proposed to publish a treatise on Algebra in which he would
disclose to the world all the rules he already knew, as well as many
others which he hoped to discover in the course of his present work. He
concludes: "This is the cause of my seeming discourtesy towards your
excellency. I have been all the ruder, perhaps, because you write to me
that you are preparing a book similar to mine, and that you propose to
publish my inventions, and to give me credit for the same. This I confess
is not to my taste, forasmuch as I wish to set forth my discoveries in my
own works, and not in those of others." In his reply to this, Cardan
points out that he had promised, if Tartaglia so desired, that he would
not publish the rules at all; but here Messer Niccolo's patience and good
manners gave way, and he told Messer Hieronimo bluntly that he did not
believe him. Then said Cardan: "I swear to you by the Sacred Evangel, and
by myself as a gentleman, that I will not only abstain from publishing
your discoveries--if you will make them known to me--but that I will
promise and pledge my faith of a true Christian to set them down for my
own use in cypher, so that after my death no one may be able to understand
them. If you will believe this promise, believe it; if you will not, let
us have done with the matter." "If I were not disposed to believe such
oaths as these you now swear," said Tartaglia, "I might as well be set
down as a man without any faith at all. I have determined to go forthwith
to Vigevano to visit the Signor Marchese, as I have now been here for
three days and am weary of the delay, but I promise when I return that I
will show you all the rules." Cardan replied: "As you are bent on going to
Vigevano, I will give you a letter of introduction to the Marchese, so
that he may know who you are; but I would that, before you start, you show
me the rule as you have promised." "I am willing to do this," said
Tartaglia, "but I must tell you that, in order to be able to recall at any
time my system of working, I have expressed it in rhyme; because, without
this precaution, I must often have forgotten it. I care naught that my
rhymes are clumsy, it has been enough for me that they have served to
remind me of my rules. These I will write down with my own hand, so that
you may be assured that my discovery is given to you correctly." Then
follow Tartaglia's verses:
"Quando chel cubo con le cose apresso
Se agualia à qualche numero discreto
Trouan dui altri differenti in esso
Dapoi terrai questo per consueto
Ch'el lor' produtto sempre sia eguale
Al terzo cubo delle cose neto
El residuo poi suo generale
Delli lor lati cubi ben sottratti
Varra la tua cosa principale.
In el secondo de cotesti atti
Quando chel cubo restasse lui solo
Tu osseruarai quest' altri contratti
Del numer farai due tal part 'a uolo
Che luna in l'altra si produca schietto
El terzo cubo delle cose in stolo
Delle qual poi, per commun precetto
Torrai li lati cubi insieme gionti
Et cotal summa sara il tuo concetto
Et terzo poi de questi nostri conti
Se solve col recordo se ben guardi
Che per natura son quasi congionti
Questi trouai, et non con passi tardi
Nel mille cinquecent' e quatro è trenta
Con fondamenti ben sald' è gagliardi
Nella citta del mar' intorno centa."
Having handed over to his host these rhymes, with the precious rules
enshrined therein, Tartaglia told him that, with so clear an exposition,
he could not fail to understand them, ending with a warning hint to Cardan
that, if he should publish the rules, either in the work he had in hand,
or in any future one, either under the name of Tartaglia or of Cardan, he,
the author, would put into print certain things which Messer Hieronimo
would not find very pleasant reading.
After all Tartaglia was destined to quit Milan without paying his respects
to D'Avalos. There is not a word in his notes which gives the reason of
this eccentric action on his part. He simply says that he is no longer
inclined to go to Vigevano, but has made up his mind to return to Venice
forthwith; and Cardan, probably, was not displeased at this exhibition of
petulant impatience on the part of his guest, but was rather somewhat
relieved to see Messer Niccolo ride away, now that he had extracted from
him the coveted information. From the beginning to the end of this affair
Cardan has been credited with an amount of subtle cunning which he
assuredly did not manifest at other times when his wits were pitted for
contest with those of other men. It has been advanced to his disparagement
that he walked in deceitful ways from the very beginning; that he dangled
before Tartaglia's eyes the prospect of gain and preferment simply for the
purpose of enticing him to Milan, where he deemed he might use more
efficaciously his arguments for the accomplishment of the purpose which
was really in his mind; that he had no intention of advancing Tartaglia's
fortunes when he suggested the introduction to D'Avalos, but that the
Governor of Milan was brought into the business merely that he might be
used as a potent ally in the attack upon Tartaglia's obstinate silence.
Whether this may have been his line of action or not, the issue shows that
he was fully able to fight his battle alone, and that his powers of
persuasion and hard swearing were adequate when occasion arose for their
exercise. It is quite possible that Tartaglia, when he began to reflect
over what he had done by writing out and handing over to Cardan his
mnemonic rhymes, fell into an access of suspicious anger--at Cardan for
his wheedling persistency, and at himself for yielding thereto--and packed
himself off in a rage with the determination to have done with Messer
Hieronimo and all his works. Certainly his carriage towards Cardan in the
weeks ensuing, as exhibited in his correspondence, does not picture him in
an amiable temper. On April 9 Jerome wrote to him in a very friendly
strain, expressing regret that his guest should have left Milan without
seeing D'Avalos, and fear lest he might have prejudiced his fortunes by
taking such a step. He then goes on to describe to Tartaglia the progress
he is making in his work with the Practice of Arithmetic, and to ask him
for help in solving one of the cases in Algebra, the rule for which was
indeed contained in Tartaglia's verses, but expressed somewhat obscurely,
for which reason Cardan had missed its meaning.[98] In his reply,
Tartaglia ignores Jerome's courtesies altogether, and tells him that what
he especially desires at the present moment is a sight of that volume on
the Practice of Arithmetic, "for," says he, "if I do not see it soon, I
shall begin to suspect that this work of yours will probably make manifest
some breach of faith; in other words, that it will contain as
interpolations certain of the rules I taught you." Niccolo then goes on to
explain the difficulty which had puzzled Cardan, using terms which showed
plainly that he had as poor an opinion of his correspondent's wit as of
his veracity.
Cardan was an irascible man, and it is a high tribute to his powers of
restraint that he managed to keep his temper under the uncouth insults of
such a letter as the foregoing. The more clearly Tartaglia's jealous,
suspicious nature displays itself, the greater seems the wonder that a man
of such a disposition should ever have disclosed such a secret. He did not
believe Cardan when he promised that he would not publish the rules in
question without his (the discoverer's) consent--why then did he believe
him when he swore by the Gospel? The age was one in which the binding
force of an oath was not regarded as an obligation of any particular
sanctity if circumstances should arise which made the violation of the
oath more convenient than its observance. However, the time was not yet
come for Jerome to begin to quibble with his conscience. On May 12, 1539,
he wrote another letter to Tartaglia, also in a very friendly tone,
reproaching him gently for his suspicions, and sending a copy of the
_Practice of Arithmetic_ to show him that they were groundless. He
protested that Tartaglia might search from beginning to end without
finding any trace of his jealously-guarded rules, inasmuch as, beyond
correcting a few errors, the writer had only carried Algebra to the point
where Fra Luca had left it. Tartaglia searched, and though he could not
put his finger on any spot which showed that Messer Hieronimo had broken
his oath, he found what must have been to him as a precious jewel, to wit
a mistake in reckoning, which he reported to Cardan in these words:
"In this process your excellency has made such a gross mistake that I am
amazed thereat, forasmuch as any man with half an eye must have seen
it--indeed, if you had not gone on to repeat it in divers examples, I
should have set it down to a mistake of the printer." After pointing out
to Cardan the blunders aforesaid, he concludes: "The whole of this work of
yours is ridiculous and inaccurate, a performance which makes me tremble
for your good name."[99]
Every succeeding page of Tartaglia's notes shows more and more clearly
that he was smarting under a sense of his own folly in having divulged his
secret. Night and day he brooded over his excess of confidence, and as
time went by he let his suspicions of Cardan grow into savage resentment.
His ears were open to every rumour which might pass from one class-room to
another. On July 10 a letter came to him from one Maphio of Bergamo, a
former pupil, telling how Cardan was about to publish certain new
mathematical rules in a book on Algebra, and hinting that in all
probability these rules would prove to be Tartaglia's, whereupon he at
once jumped to the conclusion that Maphio's gossip was the truth, and that
this book would make public the secret which Cardan had sworn to keep. He
left many of Cardan's letters unanswered; but at last he seems to have
found too strong the temptation to say something disagreeable; so, in
answer to a letter from Cardan containing a request for help in solving an
equation which had baffled his skill, Tartaglia wrote telling Cardan that
he had bungled in his application of the rule, and that he himself was now
very sorry he had ever confided the rule aforesaid to such a man. He ends
with further abuse of Cardan's _Practice of Arithmetic_, which he declares
to be merely a confused farrago of other men's knowledge,[100] and with a
remark which he probably intended to be a crowning insult. "I well
remember when I was at your house in Milan, that you told me you had never
tried to discover the rule of the _cosa_ and the _cubus_ equal to the
_numerus_ which was found out by me, because Fra Luca had declared it to
be impossible;[101] as if to say that, if you had set yourself to the task
you could have accomplished it, a thing which sets me off laughing when I
call to mind the fact that it is now two months since I informed you of
the blunders you made in the extraction of the cube root, which process is
one of the first to be taught to students who are beginning Algebra.
Wherefore, if after the lapse of all this time you have not been able to
find a remedy to set right this your mistake (which would have been an
easy matter enough), just consider whether in any case your powers could
have been equal to the discovery of the rule aforesaid."[102]
In this quarrel Messer Giovanni Colla had appeared as the herald of the
storm, when he carried to Milan in 1536 tidings of the discovery of the
new rule which had put Cardan on the alert, and now, as the crisis
approached, he again came upon the scene, figuring as unconscious and
indirect cause of the final catastrophe. On January 5, 1540, Cardan wrote
to Tartaglia, telling him that Colla had once more appeared in Milan, and
was boasting that he had found out certain new rules in Algebra. He went
on to suggest to his correspondent that they should unite their forces in
an attempt to fathom this asserted discovery of Colla's, but to this
letter Tartaglia vouchsafed no reply. In his diary it stands with a
superadded note, in which he remarks that he thinks as badly of Cardan as
of Colla, and that, as far as he is concerned, they may both of them go
whithersoever they will.[103]
Colla propounded divers questions to the Algebraists of Milan, and
amongst them was one involving the equation _x^4 + 6x^2 + 36 = 60x_, one
which he probably found in some Arabian treatise. Cardan tried all his
ingenuity over this combination without success, but his brilliant pupil,
Ludovico Ferrari, worked to better purpose, and succeeded at last in
solving it by adding to each side of the equation, arranged in a certain
fashion, some quadratic and simple quantities of which the square root
could be extracted.[104] Cardan seems to have been baffled by the fact
that the equation aforesaid could not be solved by the recently-discovered
rules, because it produced a bi-quadratic. This difficulty Ferrari
overcame, and, pursuing the subject, he discovered a general rule for the
solution of all bi-quadratics by means of a cubic equation. Cardan's
subsequent demonstration of this process is one of the masterpieces of the
_Book of the Great Art_. It is an example of the use of assuming a new
indeterminate quantity to introduce into an equation, thus anticipating by
a considerable space of time Descartes, who subsequently made use of a
like assumption in a like case.
How far this discovery of Ferrari's covered the rules given by Tartaglia
to Cardan, and how far it relieved Cardan of the obligation of secresy, is
a problem fitted for the consideration of the mathematician and the
casuist severally.[105] An apologist of Cardan might affirm that he cannot
be held to have acted in bad faith in publishing the result of Ferrari's
discovery. If this discovery included and even went beyond Tartaglia's, so
much the worse for Tartaglia. The lesser discovery (Tartaglia's) Cardan
never divulged before Ferrari unravelled Giovanni Colla's puzzle; but it
was inevitable that it must be made known to the world as a part of the
greater discovery (Ferrari's) which Cardan was in no way bound to keep a
secret. The case might be said to run on all fours with that where a man
confides a secret to a friend under a promise of silence, which promise
the friend keeps religiously, until one day he finds that the secret, and
even more than the secret, is common talk of the market-place. Is the
obligation of silence, with which he was bound originally, still to lie
upon the friend, even when he may have sworn to observe it by the Holy
Evangel and the honour of a gentleman; and is the fact that great renown
and profit would come to him by publishing the secret to be held as 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 Naudé.[106] In the course of his essay Naudé 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, Naudé 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
Naudé,[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,
Ptolemæus, 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, miseré 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 é
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 Mathématiques_. 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 jurisprudentiæ lauream redeuntem Brixiam
Nicolaus secutus est, cæpitque ex Mathematicis gloriam sibi ac divitias
parare, æque paupertatis impatiens, ac fortunæ melioris cupidus, quam dum
Brixiæ tuetur, homo morosæ, et inurbanæ rusticitatis prope omnium civium
odia sibi conciliavit. Quamobrem alibi vivere coactus, varias Italiæ urbes
incoluit, ac Ferrariæ, Parmæ, Mediolani, Romæ, Genuæ, arithmeticam,
geometricam, ceteraque quæ ad Mathesim pertinent, docuit; depugnavitque
scriptis accerrimis cum Cardano ac sibi ex illis quæsivit 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 præmiis 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 quæ
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'è tra noi, che spero durara fin che viveremo, che mi mandati
sciolta questa questione. 1 cubo piu 3. cose egual à 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
Magnæ_ (_Opera_, tom. iv. p. 222). "Deceptus enim ego verbis Lucæ
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 quærere 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, impudentiæ tamen ac
stultitiæ suæ non aliud testimonium quæras, quam ipsos illius libros, in
quibus nominatim splendidiorem unumquemque e civibus suis proscindit: adeò
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 matter briefly: "Ex
hoc ad artem magnam, quam collegi, dum Jo. Colla certaret nobiscum, et
Tartalea, à quo primum acceperam capitulum, qui maluit æmulum habere, et
superiorem, quam amicum et beneficio devinctum, cum alterius fuisset
inventum."--ch. xlv. p. 175.
[106] Prefixed to the _De Vita Propria_.
[107] In a question of broken faith, Cardan laid himself open especially
to attack by reason of his constant self-glorification in the matter of
veracity.
[108] Leonardo knew that quadratic equations might have two positive
roots, and Cardan pursued this farther by the discovery that they might
also have negative roots.
[109] "Caput xxviii. De capitulo generali cubi et rerum æqualium numero,
Magistri Nicolai Tartagliæ, Brixiensis--Hoc capitulum habui à prefato viro
ante considerationem demonstrationum secundi libri super Euclidem, et
æquatio hæc cadit in [Symbol: Rx]. cu v binomii ex genere binomii secundi
et qunti [m~]. [Symbol: Rx]. cuba universali recisi ejusdem
binomii."--_Opera_, tom. iv. p. 341.
[110] Montucla, who as a historian of Mathematics has a strong bias
against Cardan, gives him credit for the discovery of the _fictæ radices_,
but on the other hand he attributes to Vieta Cardan's discovery of the
method of changing a complete cubic equation into one wanting the second
term.--Ed. 1729, p. 595.
[111] _Opera_, tom. i. p. 66.