of his work he writes:-"Incipit Pars Tertia de solutione quarundam quæstionum secundum modum Algebræ et Almucabalæ, scilicet oppositionis et restaurationis." This work appears to contain the earliest notice of the introduction of the Hindu Arithmetic and Algebra into Italy. Leonardo, the earliest scholar of the Arabian algebraists, is reported by Targioni Tozzetti to have employed the small letters of the alphabet to denote quantities. But it would appear that he only did so because he represented quantities by straight lines, and designated these lines by letters in the elucidation of his Algebraic Solution of Problems. Leonardo and his disciples translated the Arabic word shai by the Latin res and the Italian cosa. The Arabic word mal was rendered by the Latin census and the Italian censo, as terms of the same import; for it was in the acceptation of amount of property or estate (census, quicquid fortunarum quis habet) that census was used by Leonardo. And it appears that the name of the science was derived from the first and second powers of the unknown quantity, "Ars Rei et Census," and sometimes "Regula Rei et Census," the rule of the thing and the product, or the rule of the root and the square, in allusion to equations of the second degree. Villani, the historian of Florence, writes of Paoli di Dagomari, who died about 1350, as a great geometer and most skilful arithmetician, and who surpassed both the ancients and the moderns in the knowledge of equations, which at that time constituted the most advanced branch of the science. Also, Raffaelo Caracci, of the same century, composed a work entitled "Ragionamento di Algebra," in which he writes of Guglielmo di Lunis, who had translated a treatise on Algebra from Arabic into Italian. This was the work of Mohammed Ben Musa, and was executed not long after the introduction of that science into Italy by Leonardo of Pisa. All these combine to show that the Italians were in possession of the knowledge of the Arabic numerals and the science of Algebra long before the rest of the nations of Europe. Paciolus, and all the Italian writers on arithmetic and algebra, declare that Leonardo of Pisa was the first who introduced the knowledge of algorithm and algebra to his countrymen. Paciolus appears to have taught these sciences at Venice about 1460, and he specially makes mention of his three predecessors, who had in succession filled the office expressly dedicated to the exposition of these sciences. Their names were Paolo della Pergola, Demetrio Bragadini, and Antonio Cornaro, the last of whom had been his fellow-disciple. The first Italian writer on the science of Algebra whose treatise was printed in Italy was Lucas Paciolus, or Lucas de Burgo. This work was published in 1494, with the title, "L'Arte Maggiore ditta dal Volgo la Regola de la Cosa over Alghebra e Almucabala." He adopted the name of "L'Arte Maggiore" to distinguish it from the science of number, which was then called "L'Arte Minore." He generally employs both the terms Algebra and Almucabala, and from the former word devised the adjective Algebratico. He informs his readers that the science is named Algebra wa Almucabala in the Arabic language, and those words are thus explained in the Roman language: “Algebra, id est, restauratio; Almucabala, id est, oppositio vel comparatio et solidatio." These two words being only the names of two preliminary operations in the preparation of equations for solution. The words Algebra and Algorism were introduced at the same time into Europe, and the latter term was always applied to treatises on Arithmetic with the Indian figures. Algorism, or Algorithm, was corrupted into Augrym, and the counters which were used in calculation were called " augrym stones," as may be read in Chaucer's Miller's Tale. After the appearance in print of the works of Lucas de Burgo on arithmetic and algebra, the knowledge of these sciences in the following century, not only in Italy, but also in other countries of Europe, engaged the attention of learned men, who promoted the advancement of them and introduced numerous improvements. It appears from the treatises of the first Italian writers on Algebra that they did not understand how to turn the extension of the science of arithmetic to the purposes for which it has subsequently been found most useful. The application of the different methods of reckoning in common life appeared to them, as a contrast with algebra, an entirely theoretic knowledge. They appear to have held ideas as inadequate as their teachers of the real meaning of the word "equation." Neither they nor the Arabians knew equality to be any other than two or more absolutely positive quantities, and the Italians follow the special rules of their Arabian teachers. As it has been very aptly described-" they worked with an instrument the use of which they did not fully comprehend, and employed a language which expressed more than they were prepared to understand; a language, under the notion first of negative, and then of imaginary quantities, seemed to involve such mysteries as the accuracy of mathematical science must necessarily refuse to admit." The early Italian writers on Algebra introduced some abbreviations of the terms they learned from their Arabian instructors. They employed the initial letters p and m of the words plus and minus to denote a positive and a negative quantity, and used C and q for the first and second unknown quantities. They also adopted the characters C, C, C, to denote the first, second, and third powers respectively, and denominated the biquadrate and the higher powers as relato primo, secondo, tertio, &c. They assumed R, the initial letter of radix, to denote the square root. It appears that the Italian mathematicians adopted the same classification of cubic equations as the Arabic writers; and that in the year 1505 Scipio Ferreo, Professor of Mathematics at Bononia, discovered a solution of one of these classes, which by the Arabic writers was termed "cube and things equal to numbers," and in modern notation would be expressed by x+ax=b. The like solution was also effected by Lewis Ferrari. Scipio Ferreo made known his discovery to Florido, one of his pupils, who challenged Tartaglia to solve the question. Tartaglia effected the solution, and in turn proposed to Ferreo two other cases of cubic equations, the solutions of which he himself had previously discovered. Jerome Cardan was a physician at Bononia, and read public lectures on the mathematical sciences. In the year 1539 he published a work on Algebra in the Latin language, entitled "Ars Magna," and in 1545, to this work he published an addition which contained the recent discoveries on the solution of cubic equations, all of which he claimed as his own, except two cases discovered by Lewis Ferrari. Cardan's great work begins with some remarks on the name and terms employed in the science. In general he expresses the terms of the science by words at length, designating a known quantity or number by the word numerus, the first power of the unknown quantity by res, the second power by quadratum, the third by cubum, &c., employing no symbol for the unknown quantity. He uses the letters p and m for plus and minus, and R, the initial letter of radix, to indicate the square root. He sometimes uses for res the words positio and quantitas ignota. In treating of squares and square roots, he shows that all squarenumbers have two square roots, one positive (vera), and the other negative (ficta). He adds that odd powers have only one root, veraor positive, but none ficta or negative, because a negative number raised to an odd power can never produce a positive number. In quadratic equations he admits both positive and negative (vera et ficta) roots, whether integral, fractional, or surd, but does not allow such quantities as involve the square root of a negative quantity to be roots, which are now called imaginary or impossible roots. Also he considers such quadratic equations as involve the fourthand second powers of the unknown quantity, may have four, two, or no roots at all: as x1+12=6x2 has no roots, all being impossible x+3x=28 has two roots, +2 and -2, the other two being impossible; and +12=7x2 has four roots, +2, −2, +√3, −√3. Among the numerous improvements of the science contained in his works, the most celebrated is the method of solving one class of cubic equations which incorrectly bears his name. He has amply exemplified the methods of dealing with all the forms of cubic equations. He remarks that the equation x+6x= 20 has only one root, +2, the other two being imaginary; but he considers 3+16=12x has two roots, +2 and -4; of the third root, +2, being the same as the first, he takes no account. The equation=2x+21 has only one negative root, the other two being impossible, or involving the square root of a negative quantity. In this manner he shows how a cubic equation may have one, two, or three roots, according to the form of the equation and the relation of the coefficients. He exhibits, by numerous examples, how a cubic equation can have only one root, positive or negative. This constitutes the special case which bears the name of Cardan's rule for the solution of a cubic equation. In modern language Cardan's rule is applicable in all cases where one root is positive or negative, and the other two roots imaginary or impossible. And it has been since discovered that Cardan's rule is also applicable when two roots of a cubic equation are equal. He was not ignorant of the difficulty of that case which has been called the irreducible case of cubic equations, but failed to discover the solution. In the twenty-ninth chapter he has given the method of Lewis Ferrari for the solution of biquadratic equations, and has illustrated the method by numerous examples. And in general, it may be added, he has dealt with the transformation of equations and the nature of their roots, whether integral, surd, or imaginary. Nicolas Tartaglia was born at Brescia, A.D. 1500, and at the capture of that city by the French in 1512 he was wounded and left for dead. By his mother's care he recovered from his wounds, but from the effect of a wound on his lips he stammered so much that he was nicknamed Tartaglia, from the Italian word which signifies that fault. In 1534 he settled in Venice and became Professor of the Mathematical Sciences, an office which he retained for twenty-five years. The first part of his great work, entitled "Trattato di Numeri e Misure," was published at Venice in 1556, and the second part in 1560. The whole work consists of three large volumes. The first volume contains a system of practical and mercantile arithmetic; the other two volumes treat of geometry, mensuration, speculative arithmetic, and algebra. This work was abridged and translated into French by Guillaume Gosselin de Caen in 1578. Tartaglia is chiefly celebrated for the discovery he made of the solution of that class of cubic equations which is commonly, but improperly, attributed to Cardan. He communicated his discovery to Cardan under the promise of secrecy with an oath, which Cardan did not keep, but published' it in 1545. In 1546 Tartaglia printed at 1 Tartaglia wrote to Cardan in the following terms, on learning the latter was about to publish his rules: "M. Hieronime, I have received your letter, in which you write that you understand the rule for the case 23=ax+b, when 462 is greater than; but when a exceeds b2, you cannot resolve the equation, and there Venice "Quesiti e Invenzioni Diverse," and dedicated his work to King Henry VIII. of England. This work contains his correspondence with Cardan, and gives the history of the discovery of his rules for the solution of cubic equations. John Muller, generally called Regiomontanus, Joannes de Regio Monte (Konigsberg), in his work "De Triangulis Omnimodis libri quinque," Norimberg, 1533, writes, in reference to Cardinal Nicolaus, of Casa, "Paucis enim admodum artem Algebræ, sive rei et census satis cognitam scio." A treatise on Algebra was composed in Latin by a native of Spain, with the title of "Arithmetica Practicæ seu Algorismi Tractatus, a Petro Sanchez Teruelo, noviter compilatus explicatusque. Impressus Parisiis per Thomam Rees in Domo Rubra post Carmelitas, Anno Domini 1513." Another treatise was published in 1536 in the Spanish language, bearing the title of "Tractado subtilissimo de Arismetica y de Geometria; compuesto y ordenado por el Reverendo Padre fray Juan de Ortega, de la Orden de los predicadores." An original treatise on arithmetic and algebra entitled, "Arithmetica Integra," was first put forth at Nuremburg, in 1544, by Michael Stifel or Stifelius, with an Introduction1 by Philip Melancthon. The work is fore you request me to send you the solution of the equation 3=9x+10. To which I reply that you have not used a good method in that case, and that your whole process is entirely false. As to resolving you the equation you have sent, I must say that I am very sorry that I have already given you so much as I have done ; for I have been informed by a credible person that you are about to publish another algebraical work, and that you have been boasting through Milan of having discovered some new rules in algebra. But take notice, that if you break your faith with me, I shall certainly keep my word with you; nay, I even assure you to do more than I promised." 1 Philippus Melancthon Lectori, S.: "Non mihi si linguæ centum sint, oraque centum, enumerare possem, quam multis in rebus usus sit numerorum. Et ita sunt in conspectu atque obviæ utilitates non solum numerorum, sed etiam artis, quæ longas et intricatas rationes mirabili dexteritate subducit et explicat, ut neminem quamlibet hebetem esse existimem, qui non et numeros miretur et de arte ipsa præclare sentiat. Quare si prolixum encomium de his utilitatibus instituerim, perinde facerem, ut si accenderem, quemadmodum Græci dicunt, ἐν τῇ μεσημβρίᾳ λυχνόν. Sed hominis studiosi est intelligere, quas utilitates proprie afferat Arithmetica his, qui solidam et perfectam doctrinam in cæteris philosophiæ partibus explicant. Quod enim vulgo dicunt, principium esse dimidium totius, id vel maxime in philosophiæ partibus conspicitur. Unus est aditus ad præstantissimam philosophiæ partem de motibus coelestibus, cognitio arithmetices. Et hæc tantam vim habet, ut mediocriter exercitatus in Arithmetica, facile cætera perspiciat et assequatur. Ita plus quam dimidium totius ejus philosophiæ tenet is, qui mediocriter cognovit Arithmeticen. Hanc tantam utilitatem præditi generosis naturis diligenter considerent, ut et exuscitent animos ad amorem hujus artis, et præparent se ad percipiendas cæteras artes. Quid quod multum etiam in physicis, multum in historiis utimur hac subtiliori doctrina de |