ALGEBRA

, a general method of resolving mathematical problems by means of equations. Or, it is a method of performing the calculations of all sorts of quantities by means of general signs or characters. At first, numbers and things were expressed by their names at full length; but afterwards these were abridged, and the initials of the words used instead of them; and, as the art advanced farther, the letters of the alphabet came to be employed as general representations of all sorts of quantities; and other marks were gradually introduced, to express all sorts of operations and combinations; so as to entitle it to different appellations— universal arithmetic, and literal arithmetic, and the arithmetic of signs.

The etymology of the name, Algebra, is given in various ways. It is pretty certain, however, that the word is Arabian, and that from those people we had the name, as well as the art itself, as is testisied by Lucas le Burgo, the first European author whose treatise was printed on this art, and who also refers to former authors and masters, from whose writings he had learned it. The Arabic name he gives it, is Alghebra e Almucabala, which is explained to signify the art of | restitution and comparison, or opposition and comparison, or resolution and equation, all which agree well enough with the nature of this art. Some however derive it from various other arabic words; as from Geber, a celebrated philosopher, chemist, and mathematician, to whom also they ascribe the invention of this science: some likewise derive it from the word Geber, which with the particle al, makes Algeber, which is purely Arabic, and signifies the reduction of broken numbers or fractions to integers.

But Peter Ramus, in the beginning of his Algebra, says “the name Algebra is Syriac, signifying the art and doctrine of an excellent man. For Geber, in Syriac, is a name applied to men, and is sometimes a term of honour, as master or doctor among us. That there was a certain learned mathematician, who sent his Algebra, written in the Syriac language, to Alexander the Great, and he named it Almucabala, that is, the book of dark or mysterious things, which others would rather call the doctrine of Algebra. And to this day the same book is in great estimation among the learned in the oriental nations, and by the Indians who cultivate this art it is called Aljabra, and Alboret; though the name of the author himself is not known.” But Ramus gives no authority for this singular paragraph. It has however on various occasions been distinguished by other names. Lucas Paciolus, or de Burgo, in Italy, called it l'Arte Magiore: ditta dal vulgo la Regola de la Cosa over Alghebra e Almucabala; calling it l'Arte Magiore, or the greater art, to distinguish it from common arithmetic, which is called l'Arte Minore, or the lesser art. It seems too that it had been long and commonly known in his country by the name Regola de la Cosa, or Rule of the Thing; from whence came our rule of coss, cosic numbers, and such like terms. Some of his countrymen followed his denomination of the art; but other Italian and Latin writers called it Regula rei & census, the rule of the thing and the product, or the root and the square, as the unknown quantity in their equations commonly ascended no higher than the square or second power. From this Italian word census, pronounced chensus, came the barbarous word zenzus, used by the Germans and others, for quadratics; with the several zenzic or square roots. And hence [scruple], [dram], , which are derived from the letters r, z, c, the initials of res, zenzus, cubus, or root, square, cube, came to be the signs or characters of these words: like as ℞ and √, derived from the letters R, r, became the signs of radicality.

Later authors, and other nations, used some the one of those names, and some another. It was also called Specious Arithmetic by Vieta, on account of the species, or letters of the alphabet, which he brought into general use; and by Newton it was called Universal Arithmetic, from the manner in which it performs all arithmetical operations by general symbols, or indeterminate quantities.

Some authors define algebra to be the art of resolving mathematical problems: but this is the idea of analysis, or the analytic art in general, rather than of algebra, which is only one particular species of it.

Indeed algebra properly consists of two parts: first, the method of calculating magnitudes or quantities, as represented by letters or other characters: and secondly the manner of applying these calculations in the solution of problems.

In algebra, as applied to the resolution of problems, the first business is to translate the problem out of the common into the algebraic language, by expressing all the conditions and quantities, both known and unknown, by their proper characters, arranged in an equation, or several equations if necessary, and treating the unknown quantity, whether it be number, or line, or any other thing, in the same way as if it were a known one: this forms the composition. Then the resolution, or analytic part, is the disentangling the unknown quantity from the several others with which it is connected, so as to retain it alone on one side of the equation, while all the other, or known, quantities, are collected on the other side, and so giving the value of the unknown one. And as this disentangling of the quantity sought, is performed by the converse of the operations by which it is connected with the others, taking them always backwards in the contrary order, it hence becomes a species of the analytic art, and is called the modern analysis, in contradistinction to the ancient analysis, which chiefly respected geometry, and its applications.

There have arisen great controversies and sharp disputes among authors, concerning the history of the progress and improvements of Algebra; arifing partly from the partiality and prejudices which are natural to all nations, and partly from the want of a closer examination of the works of the older authors on this subject. From these causes it has happened, that the improvements made by the writers of one nation, have been ascribed to those of another; and the discoveries of an earlier author, to some one of much later date. Add to this also, that the peculiar methods of many authors have been described so little in detail, that our information derived from such histories, is but very imperfect, and amounting only to some general and vague ideas of the true state of the arts. To remedy this inconvenience therefore, and to reform this article, I have taken the pains carefully to read over in succession all the older authors on this subject, which I have been able to meet with, and to write down distinctly a particular account and description of their several compositions, as to their contents, notation, improvements, and peculiarities; from the comparison of all which, I have acquired an idea more precise and accurate than it was possible to obtain from other histories, and in a great many instances very different from them. The full detail of these descriptions would employ a volume of itself, and would be far too extensive for this place: I must therefore limit this article to a very brief abridgment of my notes, remarking only the most material circumstances in each author; from which a general idea of the chain of improvements may be perceived, from the first rude beginnings, down to the more perfect state; from which it will appear that the discoveries and improvements made by any one single author, are scarcely ever either very great or numerous; but that, on the contrary, the improvements are almost always very slow and gradual, from former writers, successively made, not by great leaps, and after long intervals of time, but by gradations which, viewed in succession, become almost imperceptible. |

As to the origin of the analytic art, of which Algebra is a species, it is doubtless as old as any science in the world, being the natural method by which the mind investigates truths, causes, and theories, from their observed effects and properties. Accordingly, traces of it are observable in the works of the earliest philosophers and mathematicians, the subject of whose enquiries most of any require the aid of such an art. And this process constituted their Analytics. Of that part of analytics however which is properly called Algebra, the oldest treatise which has come down to us, is that of Diophantus of Alexandria, who flourished about the year 350 after Christ, and who wrote, in the Greek language, 13 books of Algebra or Arithmetic, as mentioned by himself at the end of his address to Dionysius, though only 6 of them have hitherto been printed; and an imperfect book on multangular numbers, namely in a Latin translation only, by Xilander, in the year 1575, and afterwards in 1621 and 1670 in Greek and Latin by Gaspar Bachet. These books however do not contain a treatise on the elementary parts of Algebra, but only collections of difficult questions relating to square and cube numbers, and other curious properties of numbers, with their solutions. And Diophantus only prefaces the books by an address to one Dionysius, for whose use it was probably written, in which he just mentions certain precognita, as it were to prepare him for the problems themselves. In these remarks he shews the names and generation of the powers, the square, cube, 4th, 5th, 6th, &c, which he calls dynamis, cubus, dynamodinamis, dynamocubus, cubocubus, according to the sum of the indices of the powers; and he marks these powers with the initials thus dn_, kn_, ddn_, dkn_, kkn_, &c: the unknown quantity he calls simply ariqmos, numerus, the number; and in the solutions he commonly marks it by the final thus s_; also he denotes the monades, or indefinite unit, by mo_. Diophantus there remarks on the multiplication and division of simple species together, shewing what powers or species they produce; declares that minus (leiyis) multiplied by minus produces plus (nparcin); but that minus multiplied by plus, produces minus; and that the mark used for minus is <*> namely the y inverted and curtailed, but he uses no mark for plus, but a word or conjunction copulative. As to the operations, viz. of addition, subtraction, multiplication, and division of compound species, or those connected by plus and minus, Diophantus does not teach, but supposes his reader to know them. He then remarks on the preparation or simplifying of the equations that are derived from the questions, which we call reduction of equations, by collecting like quantities together, adding quantities that are minus, and subtracting such as are plus, called by the moderns Transposition, so as to bring the equation to simple terms, and then depressing it to a lower degree by equal division when the powers of the unknown quantity are in every term: which preparation, or reduction of the complex equation, being now made, or reduced to what we call a final equation, Diophantus goes no farther, but barely says what the root or res ignota is, without giving any rules for finding it, or for the resolution of equations; thereby intimating that such rules were to be found in some other work, done either by himself or others. Of the body of the work, Lib. 1 contains 43 questions, concerning one, two, three, or four unknown numbers, having certain relations to each other, viz. concerning their sums, differences, ratios, products, squares, sums and differences of squares, &c, &c; but none of them concerning either square or cubic numbers. Lib. 2 contains 36 questions. The first five questions are concerning two numbers, though only one condition is given in each question; but he supplies another by assuming the numbers in a given ratio, viz, as 2 to 1. The 6th and 7th contain each two conditions: then in the 8th question he first comes to trcat of square numbers, which is this, to divide a given square number into two other squares; and the 9th is the same, but performed in a different way: the rest, to the end, are, almost all, about one, two, or three squares. Lib. 3 contains 24 questions concerning squares, chiefly including three or four numbers. Lib. 4 begins with cubes; the first of which is this, to divide a given number into two cubes whose sides shall have a given sum: here he has occasion to cube the two binomials 5+n and 5-n; the manner of doing which shews that he knew the composition of the cube of a binomial; and many other places manifest the same thing. Only part of the questions in this book are concerning cubes; the rest are relating to squares. Two or three questions in this book have general solutions, and the theorems deduced are general, and for any numbers indefinitely; but all the other questions, in all the four books, find only particular numbers. Lib. 5 is also concerning square and cube numbers, but of a more difficult kind, beginning with some that relate to numbers in geometrical progression. Lib. 6 contains 26 propositions, concerning right-angled triangles; such as to make their sides, areas, perimeters, &c, &c, squares or cubes, or rational, &c. In some parts of this book it appears, that he was acquainted with the composition of the 4th power of the binomial root, as he sets down all the terms of it; and, from his great skill in such matters, it seems probable that he was acquainted with the composition of other higher powers, and with other parts of Algebra, besides what are here treated of. At the end is part of a book, in 10 propositions, concerning arithmetical progressions, and multangular or polygonal numbers. Diophantus once mentions a compound quadratic equation; but the resolution of his questions is by simple equations, and by means of only one unknown letter or character, which he chooses so ingeniously, that all the other unknown quantities in the question are easily expressed by it, and the final equation reduced to the simplest form which it seems the question can admit of. Sometimes he substitutes for a number sought immediately, and then expresses the other numbers or conditions by it: at other times he substitutes for the sum or difference, &c, and thence derives the rest, so as always to obtain the expressions in the simplest form. Thus, if the sum of two numbers be given, he substitutes for their difference; and if the difference be given, he substitutes for their sum: and in both cases he has the two numbers easily expressed by adding and subtracting the half sum and half difference; and so in other cases he uses other similar ingenious notations. In short, the chief excellence in this collection of questions, which seems to be only a set of exercises to some rules which had been given elsewhere, is the neat mode of substitution or notation; which being once made, the reduc- | tion to the final equation is easy and evident: and there he leaves the solution, only mentioning that the root or ariqmos is so much. Upon the whole, this work is treated in a very able and masterly manner, manifesting the utmost address and knowledge in the solutions, and forcing a persuasion that the author was deeply skilled in the science of Algebra, to some of the most abstruse parts of which these questions or exercises relate. However, as he contrives his assumptions and notations so as to reduce all his conditions to a simple equation, or at least a simple quadratic, it does not appear what his knowledge was in the resolution of compound or affected equations.

But although Diophantus was the first author on Algebra that we now know of, it was not from him, but from the Moors or Arabians that we received the knowledge of Algebra in Europe, as well as that of most other sciences. And it is matter of dispute who were the first inventors of it; some ascribing the invention to the Greeks, while others say that the Arabians had it from the Persians, and these from the Indians, as well as the arithmetical method of computing by ten characters, or digits; but the Arabians themselves say it was invented amongst them by one Mahomet ben Musa, or son of Moses, who it seems flourished about the 8th or 9th century. It is more probable, however, that Mahomet was not the inventor, but only a person well skilled in the art; and it is farther probable, that the Arabians drew their first knowledge of it from Diophantus or other Greek writers, as they did that of Geometry and other sciences, which they improved and translated into their own language; and from them it was that we received these sciences, before the Greek authors were known to us, after the Moors settled in Spain, and after the Europeans began to hold communications with them, and that our countrymen began to travel amongst them to learn the sciences. And according to the testimony of Abulpharagius, the Arithmetic of Diophantus was translated in Arabic by Mahomet ben-yahya Ba<*>iani. But whoever were the inventors and first cultivators of Algebra, it is certain that the Europeans first received the knowledge, as well as the name, from the Arabians or Moors, in consequence of the close intercourse which subsisted between them for several centuries. And it appears that the art was pretty generally known, and much cultivated, at least in Italy, if not in other parts of Europe also, long before the invention of printing, as many writers upon the art are still extant in the libraries of manuscripts; and the first authors, presently after the invention of printing, speak of many former writers on this subject, from whom they learned the art.

It was chiefly among the Italians that this art was first cultivated in Europe. And the first author whose works we have in print, was Lucas Paciolus, or Lucas de Burgo, a Cordelier, or Minorite Friar. He wrote several treatises of Arithmetic, Algebra, and Geometry, which were printed in the years 1470, 1476, 1481, 1487, and in 1494 his principal work, intitled Summa de Arithmetica, Geometria, Proportioni, et Proportionalita, is a very masterly and complete treatise on those sciences, as they then stood. In this work he mentions various former writers, as Euclid, St. Augustine, Sacrobosco or Halifax, Boetius, Prodocimo, Giordano, Biagio da Parma, and Leonardus Pisanus, from whom he learned those sciences. The order of the work is, 1st Arithmetic, 2d Algebra, and 3d Geometry. Of the Arithmetic the contents, and the order of them, are nearly as follow. First, of numbers figurate, odd and even, perfect, prime and composite, and many others. Then of Common Arithmetic in 7 parts, namely numeration or notation, addition, subtraction, multiplication, division, progression, and extraction of roots. Before him, he says, duplation and mediation, or doubling and halving, were accounted two rules in Arithmetic; but that he omits them, as being included in multiplication and division. He ascribes the present notation and method of Arithmetic to the Arabs; and says that according to some the word Abaco is a corruption of Modo Arabico, but that according to others it was from a Greek word. All those primary operations he both performs and demonstrates in various ways, many of which are not in use at present, proving them not only by what is called casting out the nines, but also by casting out the sevens, and otherwise. In the extraction of roots he uses the initial ℞ for a root; and when the roots can be extracted, he calls them discrete or rational; otherwise surd, or indiscrete, or irrational. The square root is extracted much the same way as at present, namely, dividing always the last remainder by double the root found; and so he continues the surd roots continually nearer and nearer in vulgar fractions. Thus, for the root of 6, he firsts finds the nearest whole number 2, and the remainder 2 also; then 2/4 or 1/2 is the first correction, and 2 1/2 the second root: its square is 6 1/4, therefore 1/4 divided by 5, or 1/20 is the next correction, and 2 1/2 minus 1/20, or 2 9/20 is the 3d root: its square is 6 1/400, therefore 1/400 divided by 4 9/10, or 1/1960, is the 3d correction, which gives 2 881/1960 for the 4th root, whose square exceeds 6 by only 1/3841600: and so on continually: and this process he calls approximation. He observes that fractions, which he sets down the same way as we do at present, are extracted, by taking the root of the denominator, and of the denominated, for so he calls the numerator: and when mixed numbers occur, he directs to reduce the whole to a fraction, and then extract the roots of its two terms as above: as if it be 12 1/4; this he reduces to 49/4, and then the roots give 7/2 or 3 1/2: in like manner he finds that 4 1/2 is the root of 20 1/4; 5 1/2 the root of 30 1/4; “and so on (he adds) in infinitum;” which shews that he knew how to form the series of squares by addition. He then extracts the cube root, by a rule much the same as that which is used at present; from which it appears that he was well acquainted with the co-efficients of the binomial cubed, namely 1, 3, 3, 1; and he directs how the operation may be continued “in infinitum” in fractions, like as in the square root. After this, he describes geometrical methods for extracting the square and cube roots instrumentally: he then treats professedly of vulgar fractions, their reductions, addition, subtraction, and other operations, much the fame as at present: then of the rule-of-three, gain-and-loss, and other rules used by merchants.

Paciolus next enters on the algebraical part of this work, which he calls “L'Arte Magiore; ditta dal vulgo la Regola de la Cosa, over Alghebra e Ahnucabala:” which last name he explains by restauratio & oppositio, and | assigns as a reason for the first name, because it treats of things above the common affairs in business, which make the Arte Minore. Here he ascribes the invention of Algebra to the Arabians, and denominates the series of powers, with their marks or abbreviations, as n°, or numero, the absolute or known number; co. or cosa, the thing or 1st power of the unknown quantity; ce. or censo, the product or square; cu. or cubo, the cube, or 3d power; ce. ce. or censo de censo, the square-squared, or 4th power; p°. r°. or primo relato, or 5th power; ce. cu. or censo de cubo, the square of the cube, or 6th power; and so on, compounding the names or indices according to the multiplication of the numbers 2, 3, &c, and not according to their sum or addition, as used by Diophantus. He describes also the other characters made use of in this part, which are for the most part no more than the initials or other abbreviations of the words themselves; as ℞ for radici, the root; ℞. ℞. radici de radici, the root of the root; ℞ u. radici universale, or radici legata, or radici unita;cu. radici cuba; and ―q[dram]<*> quantita, quantity; p for piu or plus, and m for meno or minus; and he remarks that the necessity and use of these two last characters are for connecting, by addition or subtraction, different powers together; as 3 co. p. 4 ce. m. 5 cu. p. 2 ce. ce. m. 6 ni. that is, 3 cosa piu 4 censa meno 5 cubo piu 2 censa-censa meno 6 numeri, or, as we now write the same thing, 3x+4x2 - 5x3 + 2x4 - 6. He first treats very fully of proportions and proportionalities, both arithmetical and geometrical, accompanied with a large collection of questions concerning numbers in continued proportion, resolved by a kind of Algebra. He then treats of el Cataym, which he says, according to some, is an Arabic or Phenician word, and signifies the Double Rule of False Position: but he here treats of both single and double position, as we do at present, dividing the el Cataym into single and double. He gives also a geometrical demonstration of both the cases of the errors in the double rule, namely when the errors are both plus or both minus, and when the one error is plus and the other minus; and adds a large collection of questions, as usual. He then goes through the common operations of Algebra, with all the variety of signs, as to plus and minus; proving that, in multiplication and division, like signs give plus, and unlike signs give minus. He next treats of different roots in infinitum, and the extraction of roots; giving also a copious treatise on radicals or surds, as to their addition, subtraction, multiplication and division, and that both in square roots and cube roots, and in the two together, much the same as at present. He makes here a digression concerning the 15 lines in the 10th book of Euclid, treating them as surd numbers, and teaching the extraction of the roots of the same, or of compound surds or binomials, such as of 23 p ℞ 448, or of ℞ 18 p ℞ 10; and gives this rule, among several others, namely: Divide the first term of the binomial into two such parts that their product may be 1/4 of the number in the second term; them the roots of those two parts, connected by their proper sign p or m, is the root of the binomial; as in this 23 p ℞ 448, the two parts of 23 are 7 and 16, whose product, 112, is 1/4 of 448, therefore their roots give 4 p ℞ 7 for the root ℞ u. 23 p ℞ 448. He next treats of equations both simple and quadratic, or simple and compound, as he calls it; and this latter he performs by completing the square, and then extracting the root, just as we do at present. He also resolves equations of the simple 4th power, and of the 4th combined with the 2d power, which he treats the same way as quadratics; expressing his rules in a kind of bad verse, and giving geometrical demonstrations of all the cases. He uses both the roots or values of the unknown quantity, in that case of the quadratics which has two positive roots; but he takes no notice of the negative roots in the other two cases. But as to any other compound equations, such as the cube and any other power, or the 4th and 1st, or 4th and 3d, &c, he gives them up as impossible, or at least says that no general rule has yet been found for them, any more, he adds, than for the quadrature of the circle. —The remainder of this part is employed on rules in trade and merchandise, such as Fellowship, Barter, Exchange, Interest, Composition or Alligation, with various other cases in trade. And in the third part of the work, he treats of Geometry, both theoretical and practical.

From this account of Lucas de Burgo's book, we may perceive what was the state of Algebra about the year 1500, in Europe; and probably it was much the same in Africa and Asia, from whence the Europeans had it. It appears that their knowledge extended only to quadratic equations, of which they used only the positive roots; that they used only one unknown quantity; that they had no marks or signs for either quantities or operations, excepting only some few abbreviations of the words or names themselves; and that the art was only employed in resolving certain numeral problems. So that either the Africans had not carried Algebra beyond quadratic equations, or else the Europeans had not learned the whole of the art, as it was then known to the former. And indeed it is not improbable but this might be the case: for whether the art was brought to us by an European, who, travelling in Africa, there learned it; or whether it was brought to us by an African; in either case we might receive the art only in an imperfect state, and perhaps far short of the degree of perfection to which it had been carried by their best authors. And this suspicion is rendered rather probable by the circumstance of an Arabic manuscript, said to be on cubic equations, deposited in the Library of the university of Leyden by the celebrated Warner, bearing a title which in Latin signifies Omar Ben Ibrahim al'Ghajamæi Algebra cubicarum æquationum, sive de problematum solidorum resolutione; and of which book I am in some hopes of procuring either a copy or a translation, by means of my worthy friend Dr. Damen, the learned Professor of Mathematics in that university, and by that means to throw some light on this doubtful subject.

Since this was written, death has prematurely put an end to the useful labours of this ingenious and worthy successor of Gravesande.

After the publication of the books of Lucas de Burgo, the science of Algebra became more generally known, and improved, especially by many persons in Italy; and about this time, or soon after, namely about the year 1505, the first rule was there found out by Scipio Ferreus, for resolving one case of a compound cubic equation. But this science, as well as other branches of | Mathematics, was most of all cultivated and improved there by Hieronymus Cardan of Bononia, a very learned man, whose arithmetical writings were the next that appeared in print, namely in the year 1539, in 9 books, in the Latin language, at Milan, where he practised physic, and read public lectures on Mathematics; and in the year 1545 came out a 10th book, containing the whole doctrine of cubic equations, which had been in part revealed to him about the time of the publication of his first 9 books. And as it is only this 10th book which contains the new discoveries in Algebra, I shall here confine myself to it alone, as it will also afford sufficient occasion to speak of his manner of treating Algebra in general. This book is divided into 40 chapters, in which the whole science of cubic equations is most amply and ably treated. Chap. 1 treats of the nature, number and properties of the roots of equations, and particularly of single equations that have double roots. He begins with a few remarks on the invention and name of the art: calls it Ars Magna, or Cosa, or Rules of Algebra, after Lucas de Burgo and others: says it was invented by Mahomet, the son of one Moses an Arabian, as is testified by Leonardus Pisanus; and that he left four rules or cases, which perhaps only included quadratic equations: that afterwards three derivatives were added by an unknown author, though some think by Lucas Paciolus; and after that again three other derivatives, for the cube and 6th power, by another unknown author; all which were resolved like quadratics: that then Scipio Ferreus, Professor of Mathematics at Bononia, about 1505, found out the rule for the case cubum & rerum numero æqualium, or, as we now write it, , which he speaks of as a thing admirable: that the same thing was next afterwards found out, in 1535, by Tartalea, who revealed it to him, Cardan, after the most earnest intreaties: that, finally, by himself and his quondam pupil Lewis Ferrari, the cases are greatly augmented and extended, namely, by all that is not here expressly ascribed to others; and that all the demonstrations of the rules are his own, except only three adopted from Mahomet for the quadratics, and two of Ferrari for cubics.

He then delivers some remarks, shewing that all square numbers have two roots, the one positive, and the other negative, or, as he calls them, vera & ficta, true and fictitious or false; so the æstimatio rei, or root, of 9, is either 3 or - 3; of 16 it is 4 or - 4; the 4th root of 81 is 3 or - 3; and so on for all even denominations or powers. And the same is remarked on compound cases of even powers that are added together; as if , then the æstimatio x is=2 or-2; but that the form has four answers or roots, in real numbers, two plus and two minus, viz. 2 or - 2, and √ 3 or - √ 3; while the case has no real roots; and the case has two, namely 2 and - 2: and in like manner for other even powers. So that he includes both the positive and negative roots; but rejects what we now call imaginary ones. I here express the cases in our modern notation, for brevity sake, as he commonly expresses the terms by words at full length, calling the ablolute or known term the numero, the 1st power the res, the 2d the quadratum, the 3d the cubum, and so on, using no mark for the unknown quantity, and only the initials p and m for plus and minus, and ℞ for radix or root. The res he sometimes calls positio, and quantitas ignota; and in stating or setting down his equations, he, as well as Lucas de Burgo before him, sets down the terms on that side where they will be plus, and not minus.

On the other hand, he remarks that the odd denominations, or powers, have only one æstimatio, or root, and that true or positive, but none sictitious or negative, and for this reason, that no negative number raised to an odd power, will give a positive number; so of 2x=16, the root is 8 only; and if 2x3=16, the root is 2 only: and if there be ever so many odd denominations, added together, equal to a number, there will be only one æstimatio or root; as if , the only root is 2. But that when the signs of some of the terms are different as to plus and minus, they may have more roots; and he shews certain relations of the co-efficients, when they have two or more roots: so the equation has two æstimatios, the one true or 2, and the other fictitious or -4, which he observes is the same as the true æstimatio of the case , having only the sign of the absolute number changed from the former, the 3d root 2 being the same as the first, which therefore he does not count. He next shews what are the relations of the co-efficients when a cubic equation has three roots, of which two are true, and the 3d fictitious, which is always equal to the sum of the other two, and also equal to the true root of the same equation with the sign of the absolute number changed: thus, in the equation , the two true roots are 3 and √5 1/4 - 1 1/2, and the fictitious one is - √ 5 1/4 - 1 1/2, which last is the same as the true root of , viz. √ 5 1/4 + 1 1/2; and he here infers generally that the fictitious æstimatio of the case , always answers to the true root of . Cardan also shews what the relation of the co-efficients is, when the case has no true roots, but only one fictitious root, which is the same as the true root of the reciprocal case, formed by changing the sign of the absolute number. Thus, the case has no true root, and only one false root, viz. - 3, which is the same as the true root of : and he shews in general, that changing the sign of the absolute number in such cases as want the 2d term, or changing the signs of the even terms when it is not wanting, changes the signs of all the three roots, which he also illustrates by many examples; thus, the roots of , are + √40 - 4, and - 3, and - √40 - 4; and the roots of , are - √40 + 4, and + 3, and + √40 + 4.

And he further observes, that the sum of the three roots, or the difference between the true and sictitious roots, is equal to 11, the co-efficient of the 2d term. He also shews how certain cubic cases have one, or two, or three roots, according to circumstances: that the case has sometimes four roots, and sometimes none at all, that is, no real ones: that the case may have three true æquatios, or positive roots, but no fictitious or negative ones; and for this reason, that the odd powers of minus being minus, and the even powers plus, the two terms x3+bx would be negative, and equal to a positive sum ax2 + c, | which is absurd: and farther, that the case has three roots, one plus and two minus, which are the same, with the signs changed, as the roots of the case . He also shews the relation of the co-efficients when the equation has only one real root, in a variety of cases: but that the case has always one negative root, and either two positive roots, or none at all; the number of roots failing by pairs, or the impossible roots, as we now call them, being always in pairs. Of all these circumstances Cardan gives a great many particular examples in numeral co-efficients, and subjoins geometrical demonstrations of the properties here enumerated; such as, that the two corresponding or reciprocal cases have the same root or roots, but with different signs or affections; and how many true or positive roots each case has.

Upon the whole, it appears from this short chapter, that Cardan had discovered most of the principal properties of the roots of equations, and could point out the number and nature of the roots, partly from the signs of the terms, and partly from the magnitude and relations of the co-efficients. He shews in effect, that when the 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: that the co-efficient of the 2d term is equal to the sum of all the roots positive and negative collected together, and conseqnently that when the 2d term is wanting, the positive roots are equal to the negative ones: and that the signs of all the roots are changed, by changing only the signs of the even terms: with many other remarks concerning the nature of equations.

In chap. 2, Cardan enumerates all the cases of compound equations of the 2d and 3d order, namely, 3 quadratics, and 19 cubics; with 44 derivatives of these two, that is, of the same kind, with higher denominations.

In chap. 3 are treated the roots of simple cases, or simple equations, or at least that will reduce to such, having only two terms, the one equal to the other. He directs to depress the denominations equally, as much as they will, according to the height of the least; then divide by the number or co-efficient of the greatest; and lastly extract the root on both sides. So if 20x3 = 180x5, then 20 = 180x2, and 1/9=x2, and x = 1/3.

Chap. 4 treats of both general and particular roots, and contains various definitions and observations concerning them. It is here shewn that the several cases of quadratics and cubics have their roots of the following forms or kinds, namely that the case where the three parts √316, 2, √34, are in continual proportion.

Chap. 5 treats of the æstimatio of the lowest degree of compound cases, that is, affected quadratic equations; giving the rule for each of the three cases, which con- sists in completing the square, &c, as at present, and which it seems was the method given by the Arabians; and proving them by geometrical demonstrations from Eucl. I. 43, and II. 4 and 5, in which he makes some improvement of the demonstrations of Mahomet. And hence it appears that the work of this Arabian author was in being, and well known in Cardan's time.

Chap. 6, on the methods of finding new rules, contains some curious speculations concerning the squares and cubes of binomial and residual quantities, and the proportions of the terms of which they consist, shewn from geometrical demonstrations, with many curious remarks and properties, forming a foundation of principles for investigating the rules for cubic equations.

Chap. 7 is on the transmutation of equations, shewing how to change them from one form to another, by taking away certain terms out of them; as , to , &c. The rules are demonstrated geometrically; and a table is added, of the forms into which any given cases will reduce; which transformations are extended to equations of the 4th and 5th order. And hence it appears that Cardan knew how to take away any term out of an equation.

Chap. 8 shews generally how to find the root of any such equation as this , where m and n are any exponents whatever, but n the greater; and the rule is, to separate or divide the co-efficient a into two such parts z and a - z, as that the absolute number b shall be equal to ―(a - z).zm/(n-m), the product of the one part a - z, and the m/(n - m) power of the other part: then the root x is = z1/(n-m). The rule is general for quadratics, cubics, and all the higher powers; and could not have been formed without the knowledge of the composition of the terms from the roots of the equation.

Chap. 9 and 10 contain the resolution of various questions producing equations not higher than quadratics.

Chap. 11 is of the case or form . Cardan now comes to the actual resolution of the first case of cubic equations. He begins with relating a short history of the invention of it, observing that it was first found out, about 30 years before, by Scipio Ferreus of Bononia, and by him taught to Antonio Maria Florido of Venice, who having a contest afterwards with Nicolas Tartalea of Brescia, it gave occasion to Tartalea to find it out himself, who after great entreaties taught it to Cardan, but suppressed the demonstration. By help of the rule alone, however, Cardan of himself discovered the source or geometrical investigation, which he gives here at large, from Eucl. II. 4. In this process he makes use of the Greek letters a, <*>, g, d, &c, to denote certain indefinite numbers or quantities, to render the investigation general; which may be considered as the first instance of such literal notation in Algebra. He then gives the rule in words at length, which comes to this, ; illustrating it in a variety of examples; in the resolution | of which, he extracts the cubic roots of such of the binomials as will admit of it, by some rule which he had for that purpose; such as , which .

Chap. 12, of the case . This he treats exactly as the last, and finds the rule ; which he illustrates by many examples, as usual. But when b3 exceeds c2, which has since been called the irreducible case, he refers to another following book, called Aliza, for other rules of solution, to overcome this difficulty, about which he took insinite pains.

Chap. 13, of the case . This case, by a geometrical process, he reduces to the case in the last chapter: thus, find the æstimatio y of the case , having the same co-efficients as the given case ; then is , giving two roots. He shews also how to find the second root, when the first is known, independent of the foregoing case. From this relation of these two cases he deduces several corollaries, one of which is, that the æstimatio or root of the case , is equal to the sum of the roots of the case . As in the example , whose æstimatio is √(9 1/4 + 1 1/2), which is equal to the sum of 3 and √(9 1/4 - 1 1/2), the two roots of the case .

In chapters 14, 15, and 16, he treats of the three cases which contain the 2d and 3d powers, but wanting the first power, according to all the varieties of the signs; which he performs by exterminating the 2d term, or that which contains the 2d power of the unknown quantity x, by substituting y ± 1/3 the co-efficient of that term for x, and so reducing these cases to one of the former. In these chapters Cardan sometimes also gives other rules; thus, for the case , find first the æstimatio y of the case , then is : also for the case , first find the two roots of , then is x = (√34c2)/y the two values of x according to the two values of y. He here also gives another rule, by which a second æstimatio or root is found, when the first is known, namely, if e be the first estimatio or value of x in the case , then is the other value of .

In chapters 17, 18, 19, 20, 21, 22, 23, Cardan treats of the cases in which all the four terms of the equation are present; and this he always effects by taking away the 2d term out of the equation, and so reducing it to one of the foregoing cases which want that term, giving always geometrical investigations, and adding a great many examples of every case of the equations.

Chap. 24, of the 44 derivative cases; which are only higher powers of the forms of quadratics and cubics.

Chap. 25, of imperfect and special cases; containing many particular examples when the co-efficients have certain relations amongst them, with easy rules for finding the roots; also 8 other rules for the irreducible case .

Chap. 26, in like manner, contains easy rules for biquadratics, when the co-efficients have certain special relations.

Then the following chapters, from chap. 27 to chap. 38, contain a great number of questions and applications of various kinds, the titles of which are these: De transitu capituli specialis in capitulum speciale; De operationibus radicum pronicarum seu mixtarum & Allellarum; De regula modi; De regula Aurea; De regula Magna, or the method of finding out solutions to certain questions; De regula æqualis positionis, being a method of substituting for the half sum and half difference of two quantities, instead of the quantities themselves; De regula inæqualiter ponendi, seu proportionis; De regula medii; De regula aggregati; De regula liberæ positionis; De regula falsum ponendi, in which some quantities come out negative; Quomodo excidant partes & denominationes multiplicando. Among the foregoing collection of questions, which are chiefly about numbers, there are some geometrical ones, being the application of Algebra to Geometry, such as, In a rightangled triangle, given the sum of each leg and the adjacent segment of the hypotenuse, made by a perpendicular from the right angle, to determine the area &c; with other such geometrical questions, resolved algebraically.

Chap. 39, De regula qua pluribus positionibus invenimus ignotam quantitatem; which is employed on biquadratie equations. After some examples of his own, Cardan gives a rule of Lewis Ferrari's, for resolving all biquadratics, namely by means of a cubic equation, which Ferrari investigated at his request, and which Cardan here demonstrates, and applies in all its cases. The method is very general, and consists in forming three squares, thus: first, complete one side of the equation up to a square, by adding or subtracting some multiples or parts of some of its own terms on both sides, which it is always easy to do: 2d, supposing now the three terms of this square to be but one quantity, viz, the first term of another square to which this same side is to be completed, by annexing the square of a new and assumed indeterminate quantity, with double the product of the roots of both; which evidently forms the square of a binomial, consisting of the assumed indeterminate quantity and the root of the first square: 3d, the other side of the equation is then made to become the square of a binomial also, by supposing the product of its ist and 3d terms to be equal to the square of half its 2d term; for it consists of only three terms, or three different denominations of the original unknown quantity: then this equality will determine the value of the assumed indeterminate quantity, by means of a cubic equation, and from it, that of the original ignota, by the equal roots of the 2d and 3d squares. Here we have a notable example of the use of assuming a new indeterminate quantity to introduce into an equation, long before Des Cartes was born, who made use of a like assumption for a similar purpose. And this method is very general, and is here applied to all forms of biquadratics, either having all their terms, or wanting some of them. To illustrate this rule I shall here set down the process of one of his examples, which is this, . Now first sub- | tract 2x2 + 4x + 7 from both sides, then the first becomes a square, viz, . Next assume the indeterminate y, and subtract 2y (x2 - 1) - y2 from both sides, making the first side again a square, viz, . Of this latter side, make the product of the 1st and 3d terms equal to the square of half the 2d term, that is, , which reduces to ; the positive roots of which are y = 2 or √15; and hence, using 2 for y, the equation of equal squares becomes , the roots of which give ; and hence ; the two positive roots of which are √(3 + 1) and √(5 - 1), which are two of the values of x in the given equation . The other roots he leaves to be tried by the reader.

The 40th, or last, chap. is entitled, Of modes of general supposition relating to this art; with some rules of an unusual kind; and æstimatios or roots of a nature different from the foregoing ones. Some of these are as follow: If , and , and x : y :: c : d; then is .

Secondly, if , and , then is x + a : y - a :: y2 : x2.

Thirdly, when , the square will be taken away, by putting ; and then the equation becomes .

Cardan adds some other remarks concerning the solutions of certain cases and questions, all evincing the accuracy of his skill, and the extent of his practice; and then he concludes the book with a remark concerning a certain transformation of equations, which quite astonishes us to find that the same person who, through the whole work, has shewn such a profound and critical skill in the nature of equations, and the solution of problems, should yet be ignorant of one of the most obvious transmutations attending them, namely increasing or diminishing the roots in any proportion. Cardan having observed that the form may be changed into another similar one, viz, , of which the co-efficient of the term y is the quotient arising from the co-efficient of x divided by the absolute number of the first equation: and that the absolute number of the 2d equation is the root of the quotient of 1 divided by the said absolute number of the first; he then adds, that finding the æstimatio or root of the one equation from that of the other is very difficult, valde difficilis.

It is matter of wonder that Cardan, among so many transmutations, should never think of substituting instead of x in such equations, another positio or root, greater or less than the former in any indefinite proportion, that is, multiplied or divided by a given number; for this would have led him immediately to the same transformation as he makes above, and that by a way which would have shewn the constant proportion be- tween the two roots. Thus, instead of x in the given form , substitute dy, and it becomes ; and this divided by d3 becomes ; and here if d be taken = √c, it becomes ; which is the transformation in question, and in which it is evident that x is = y√c, and y = x/√c. Instead of this, Cardan gives the following strange way of finding the one root x from the other y, when this latter is by any means known; viz, Multiply the first given equation by y2x + 1, then add x2/4y2 to both sides, and lastly extract the roots of both, which can always be done, as they will always be both of them squares; and the roots will give the value of x by a quadratic equation.

Thus, multiplied by y2x + 1 gives ; and theroots are ; and this 2d side of the equation he says will always have a root also. It is indeed true that it will have an exact root; but the reason of it is not obvious, which is, because y is the root of the equation . Cardan has not shewn the reason why this happens; but I apprehend he made it out in this manner, viz, similar to the way in which he forms the last square in the case of biquadratic equations, namely, by making the product of the 1st and 3d terms equal to the square of half the 2d term: thus, in the present case, it is , which reduces to the equation in question. Therefore taking y the root of the equation , and substituting its value in the quantity , this will become a complete square.

Of Cardan's Libellus de Aliza Regula.

Subjoined to the above Treatise on cubic equations, is this Libellus de Aliza regula, or the algebraic logistics, in which the author treats of some of the abstruser parts of Arithmetic and Algebra, especially cubic-equations, with many more attempts on the irreducible case . This book is divided into 60 chapters; | but I shall only set down the titles of some few of them, whose contents require more particular notice.

Chap. 4. De modo redigendi quantitates omnes, quæ dicuntur latera prima ex decimo Euclidis in compendium. He treats here of all Euclid's irrational lines, as surd numbers, and persorms various operations with them.

Chap. 5. De consideratione binomiorum & recisorum, &c; ubi de æstimatione capitulorum. Contains various operations of multiplying compound numbers and surds.

Chap. 6. De operationibus p: & m: (i. e. + and -) secundum communem usum. Here it is shewn that, in multiplication and division, plus always gives the same signs, and minus gives the contrary signs. So also in addition, every quantity retains its own sign; but in subtraction they change the signs. That the √ +, or the square root of plus, is +; but the √ -, or the square root of minus, is nothing as to common use: (but of this below.) That √3 - is -; as √ - 8 is - 2. That a residual, composed of + and - may have a root also composed of + and -: So √(5-√24) is = √3-√2. The rules for the signs in multiplication and division are illustrated by this example; to divide 8 by 2 + √6 or √6 + 2. Take the two corresponding residuals 2 - √6 and √6 - 2, and by these multiply both the divisor and dividend; then the products are + and - respectively, and the quotients still both alike. Thus,

Divid.Divis.|Divid.Divis.
8√6 + 282 + √6
√6 - 2√6 - 22 - √62 - √6
√384 - 16 divide + 416 - √384 div. - 2
Quot. √96-8.Quot. √96 - 8.
And this method of performing division of compound surds, was fully taught before him, by Lucas de Burgo, namely, reducing the compound divisor to a simple quantity, by multiplying by the corresponding quantity, having the sign changed.

In chap. 11 and 18, and elsewhere, Cardan makes a general notation of a, b, c, d, e, f, for any indefinite quantities, and treats of them in a general way.

Cap. 2. De contemplatione p: & m: (or + and -), & quod m: in m: facit p: & de causis horum juxta veritatem. Cardan here demonstrates geometrically that, in multiplication and division, like signs give plus, and unlike signs give minus. And he illustrates this numerically, by squaring the quantity 8, or 6 + 2, or 10 - 2, which must all produce the same thing, namely 64.

Among many of the chapters which treat of the irreducible case , there is a peculiar kind of way given in chap. 31, which is entitled De æstimatione generali solida vocata, & operationibus ejus; in which he shews how to approximate to the root of that case, in a manner similar to approximating the square root and cube root of a number. The rule he uses for this purpose, is the 3d in chap. 25 of the last book, and it is this: Divide b into two parts, such that the sum of the products of each, multiplied by the square of the other, may be equal to (1/2)c; then the sum of the roots of these parts is the æstimatio or value of x required. So, of this equation ; the two parts are 9 and 1, and their roots 3 and 1, and their sum 4=x, as in the margin. Again, take . Here he invents a new notation to express the root or radix, which he calls solida, viz, x=√ solida 6 in 1/2, that is, the roots of the two parts of 6, so that each part multiplied by the root of the other, the two products may be 1/2 or (1/2)c. Then to free this from fractions, and make the operation easier, multiply that root by some number as suppose 4, that is the square part 6 by the square of 4, and the solid part 1/2 by the cube of 4; then x=1/4√ solida 96 in 32. Now, by a few trials, it is found that the parts are nearly 95 8/9 and 1/9, which give too much, or 95 9/10 and 1/10, which give too little, and thereof 95 17/19 and 2/19 are still nearer. Divide both by 42 or 16, then 5 151/152 and 1/152 are the quot. And the sum of their roots, or is nearly the value of the root x.

Cap. 42. De duplici æquatione comparanda in capitulo cubi & numeri æqualium rebus. Treats of the two positive roots of that case, neglecting the negative one; and shewing, not only that that case has two such roots, but that the same number may be the common root of innumerable equations.

Cap. 57. Detractatione æstimationis generalis capituli . Cardan here again resumes the consideration of the irreducible case, making ingenious observations upon it, but still without obtaining the root by a general rule. In this place also, as well as elsewhere, he shews how to form an equation in this case, that shall have a given binomial root, as suppose √m + n, where the equation will be , having √m + n for one root, namely the positive root. From which it appears that he was well acquainted with the composition of cubic equations from given roots.

Cap. 59. De ordine & exemplis in binomiis secudo & quinto. Contains a great many numeral forms of the same irreducible case , with their roots; from which are derived these following cases, with many curious remarks. When

Cap. 60. Demonstratio generalis capituli cubi æqualis rebus & numero. This demonstration of the irreducible case is geometrical, like all the rest. Some more ingenious remarks are again added, as if he reluctantly finished the book without perfectly overcoming the difficulty of the irreducible case. Cardan here also uses the letters a and b for any two indefinite numbers, in order to shew the form and manner of the arithmetical operations: thus a/b is the fraction for their quotient, also √a/b or √a/√b the square root of that quotient, and √3a/b or √3a/√3b the cube root of it, &c. |

Having considered the chief contents of Cardan's algebra, it will now be proper to sum them up, and set down a list of the improvements made by him, as collected from his writings:

And 1st, Tartalea having only communicated to him the rules for resolving these three cases of cubic equations, viz, having all their terms, or wanting any of them, and having all possible varieties of signs; demonstrating all these rules geometrically; and treating very fully of almost all sorts of transformations of equations, in a manner heretofore unknown.

2nd, It appears that he was well acquainted with all the roots of equations that are real, both positive and negative; or, as he calls them, true and fictitious; and that he made use of them both occasionally. He also shewed, that the even roots of positive quantities, are either positive or negative; that the odd roots of negative quantities, are real and negative; but that the even roots of them are impossible, or nothing as to common use. He was also acquainted with,

3d, The number and nature of the roots of an equation, and that partly from the signs of the terms, and partly from the magnitude and relation of the coefficients. He also knew,

4th, That the number of positive roots is equal to the number of changes of the signs of the terms.

5th, That the coefficient of the second term of the equation, is the difference between the positive and negative roots.

6th, That when the second term is wanting, the sum of the negative roots is equal to the sum of the positive roots.

7th, How to compose equations that shall have given roots.

8th, That, changing the signs of the even terms, changes the signs of all the roots.

9th, That the number of roots failed in pairs; or what we now call impossible roots were always in pairs.

10th, To change the equation from one form to another, by taking away any term out of it.

11th, To increase or diminish the roots by a given quantity. It appears also,

12th, That he had a rule for extracting the cube root of such binomials as admit of extraction.

13th, That he often used the literal notation a, b, c, d, &c.

14th, That he gave a rule for biquadratic equations, suiting all their cases; and that, in the investigation of that rule, he made use of an assumed indeterminate quantity, and afterwards found its value by the arbitrary assumption of a relation between the terms.

15th, That he applied Algebra to the resolution of geometrical problems. And

16th, That he was well acquainted with the difficulty of what is called the irreducible case, viz, , upon which he spent a great deal of time, in attempting to overcome it. And though he did not fully succeed in this case, any more than other persons have done since, he nevertheless made many ingenious observations about it, laying down rules for many particular forms of it, and shewing how to approximate very nearly to the root in all cases whatever.

OF TARTALEA.

Nicholas Tartalea, or Tartaglia, of Brescia, was contemporary with Cardan, and was probably older than he was, but I do not know of any book of Algebra published by him till the year 1546, the year after the date of Cardan's work on Cubic Equations, when he printed his Quesiti & Inventioni diverse, at Venice, where he resided as a public lecturer on mathematics. This work is dedicated to our king Henry the VIIIth of England, and consists of 9 books, containing answers to various questions which had been proposed to him at different times, concerning mechanics, statics, hydrostatics, &c.; but it is only the 9th, or last, that we shall have occasion to take notice of in this place, as it contains all those questions which relate to arithmetic and algebra. These are all set down in chronological order, forming a pretty collection of questions and solutions on those subjects, with a short account of the occasion of each of them. Among these, the correspondence between him and Cardan forms a remarkable part, as we have here the history of the invention of the rules for cubic equations, which he communicated to Cardan. under the promise, and indeed oath, to keep them secret, on the 25th of March 1539. But, notwithstanding his oath, finding that Cardan published them in 1545, as above related, it seems Tartalea published the correspondence between them in revenge for his breach of faith; and it elsewhere appears, that many other sharp bickerings passed between them on the same account, which only ended with the death of Tartalea, in the year 1557. It seems it was a common practice among the mathematicians, and others, of that time, to send to each other nice and difficult questions, as trials of skill, and to this cause it is that we owe the principal questions and discoveries in this collection, as well as many of the best discoveries of other authors. The collection now before us contains questions and solutions, with their dates, in a regular order, from the year 1521, and ending in .1541, in 42 dialogues, the last of which is with an English gentleman, namely, Mr. Richard Wentworth, who it seems was no mean mathematician, and who learned some algebra, &c, of Tartalea, while he resided at Venice. The questions at first are mostly very easy ones in arithmetic, but gradually become more difficult, and exercising simple and quadratic equations, with complex calculations of radical quantities: all shewing that he was well skilled in the art of Algebra as it then stood, and that he was very ingenious in applying it to the solutions of questions. Tartalea made no alteration in the notation or forms of expression used by Lucas de Burgo, calling the first power of the unknown quantity, in his language, cosa, the second power censa, the third cubo, &c, and writing the names of all the operations in words at length, without using any contractions, except the initial R for root or radicality. So that the only thing remarkable in this collection, is the discovery of the rules for cubic equations, with the curious circumstances attending the same. |

The first two of these were discovered by Tartalea in the year 1530, namely for the two cases , and , as appears by Quest. 14 and 25 of this collection, on occasion of a question then proposed to him by one Zuanni de Tonini da Coi or Colle, John Hill, who kept a school at Brescia. And from the 25th letter we learn, that he discovered the rules for the other two cases , and , on the 12th and 13th of February 1535, at Venice, where he had come to reside the year before. And the occasion of it was this: There was then at Venice one Antonio Maria Fiore or Florido, who, by his own account, had received from his preceptor Scipio Farreo, about thirty years before, a general rule for resolving the case . Being a captious man, and presuming on this discovery, Florido used to brave his contemporaries, and by his insults provoked Tartalea to enter into a wager with him, that each should propose to the other thirty different questions; and that he who soonest resolved those of his adversary, should win from him as many treats for himself and friends. These questions were to be proposed on a certain day at some weeks distance; and Tartalea made such good use of his time, that eight days before the time appointed for delivering the propositions, he discovered the rules both for the case , and the case . He therefore proposed several of his questions so as to fall either on this latter case, or on the cases of the cube and square, expecting that his adversary would propose his in the former. And what he suspected fell out accordingly; the consequence of which was, that on the day of meeting Tartalea resolved all his adversary's questions in the space of two hours, without receiving one answer from Florido in return; to whom, however, Tartalea generously remitted the forfeit of the thirty treats won of him.

Question 31 first brings us acquainted with the correspondence between Tartalea and Cardan. This correspondence is very curious, and would well deserve to be given at full length in their own words, if it were not too long for this place. I may enlarge farther upon it under the article Cubic Equations; but must here be content with a brief abstract only. Cardan was then a respectable physician, and lecturer in mathematics at Milan; and having nearly finished the printing of a large work on Arithmetic, Algebra, and Geometry, and having heard of Tartalea's discoveries in cubic equations, he was very desirous of drawing those rules from him, that he might add them to his book before it was finished. For this purpose he first applied to Tartalea, by means of a third person, a bookseller, whom he sent to him, in the beginning of the year 1539, with many flattering compliments, and offers of his services and friendship, &c, accompanied with some critical questions for him to resolve, according to the custom of the times. Tartalea however refused to disclose his rules to any one, as the knowledge of them gained him great reputation among all people, and gave him a great advantage over his competitors for fame, who were commonly afraid of him on account of those very rules. He only sent Cardan therefore, at his request, a copy of the thirty questions which had been proposed to him in the contest with Florido. Not to be rebuffed so easily, Cardan next applied, in the most urgent manner, by letter to Tartalea; which however procured from him only the solution of some other questions proposed by Cardan, with a few of the questions that had been proposed to Florido, but none of their solutions. Finding he could not thus prevail, with all his fair promises, Cardan then fell upon another scheme. There was at Milan a certain Marquis dal Valsto, a great patron of Cardan, and, it was said, of learned men in general. Cardan conceived the idea of making use of the influence of this nobleman to draw Tartalea to Milan, hoping that then, by personal intreaties, he should succeed in drawing the long-concealed rules from him. Accordingly he wrote a second letter to Tartalea, much in the same strain with the former, strongly inviting him to come and spend a few days in his house at Milan, and representing that, having often commended him in the highest terms to the marquis, this nobleman desired much to see him; for which reason Cardan advised him, as a friend, to come to visit them at Milan, as it might be greatly to his interest, the marquis being very liberal and bountiful; and he besides gave Tartalea to understand, that it might be dangerous to offend such a man by refusing to come, who might, in that case, take offence, and do him some injury. This manœuvre had the desired effect: Tartalea on this occasion laments to himself in these words, “By this I am reduced to a great dilemma; for if I go not to Milan, the marquis may take it amiss, and some evil may befal me on that account; I shall therefore go, although very unwillingly.” When he arrived at Milan however, the marquis was gone to Vigeveno, and Tartalea was prevailed on to stay three days with Cardan, in expectation of the marquis returning, at the end of which he set out from Milan, with a letter from Cardan, to go to Vigeveno to that nobleman. While Tartalea was at Milan the three days, Cardan plied him by all possible means to draw from him the rules for the cubic equations; and at length, just as Tartalea was about to depart from Milan, on the 25th of March 1539, he was overcome by the most solemn protestations of secrecy that could be made. Cardan says, “I shall swear to you on the holy evangelists, and by the honour of a gentleman, not only never to publish your inventions, if you reveal them to me; but I also promise to you, and pledge my faith as a true christian, to note them down in cyphers, so that after my death no other person may be able to understand them.” To this Tartalea replies, “If I refuse to give credit to these assurances, I should deservedly be accounted utterly void of belief. But as I intend to ride to Vigeveno, to see his excellency the marquis, as I have been here now these three days, and am weary of waiting so long; whenever I return therefore, I promise to shew you the whole.” Cardan answers, “Since you determine at any rate to go to Vigeveno, to the marquis, I shall give you a letter for his excellency, that he may know who you are. But now before you depart, I intreat you to shew me the rule for the equations, as you have promised.” “I am content,” says Tartalea: “But you must know, that to be able on all occasions to remember such operations, I have brought the rule into rhyme; for if I had not used that precaution, I should often have forgot it; and although my rhymes are not very good, I do not value that, as it is | sufficient that they serve to bring the rule to mind as often as I repeat them. I shall here write the rule with my own hand, that you may be sure I give you the discovery exactly.” These rude verses contain, in rather dark and enigmatical language, the rule for these three cases, viz. that their difference in the first case, and their sums in the 2d and 3d, may be equal to c the absolute number, and their product equal to the cube of 1/3 of b the coefficient of the less power; then the difference of their cube roots will be equal to x in the first case, and the sum of their cube roots equal to x in the 2d and 3d cases: that is, taking in the 1st case, or in the 2d and 3d, and ; then in the first case, and in the other two. At parting, T, fails not again to remind C. of his obligation: “Now your excellency will remember not to break your promised faith, for if unhappily you should insert these rules either in the work you are now printing, or in any other, although you should even give them under my name, and as of my invention, I promise and swear that I shall immediately print another work that will not be very pleasing to you.” “Doubt not, says C. but that I shall observe what I have promised: Go, and rest secure as to that point: and give this letter of mine to the marquis.” It should seem however that T. was much displeased at having suffered himself to be worried as it were out of his rules, for as soon as he quitted Milan, instead of going to wait upon the marquis, he turned his horse's head, and rode straight home to Venice, saying to himself, “By my faith I shall not go to Vigeveno, but shall return to Venice, come of it what will.”

After T's departure it seems C. applied himself immediately to resolving some examples in the cubic equations by the new rules, but not succeeding in them, for indeed he had mistaken the words, as it was very easy to do in such bad verses, having mistaken ((1/3)b)3 for (1/3)b3, or the cube of 1/3 of the coefficient, for 1/3 of the cube of the coefficient; accordingly we find him writing to T. in fourteen days after the above, blaming him much for his abrupt departure without seeing the marquis, who was so liberal a prince he said, and requesting T. to resolve him the example . This T. did to his satisfaction, rightly guessing at the nature of his mistake; and concludes his answer with these emphatical words, “Remember your promise.” On the 12th of May following C. returns him a letter of thanks, together with a copy of his book, saying, “As to my work, just finished, to remove your suspicion, I send you a copy, but unbound, as it is yet too fresh to be beaten. But as to the doubt you express lest I may print your inventions, my faith which I gave you with an oath should satisfy you; for as to the finishing of my book, that could be no security, as I could always add to it whenever I please. But on account of the dignity of the thing, I excuse you for not relying on that which you ought to have done, namely on the faith of a gentleman, instead of the finishing of a book, which might at any time be enlarged by the addition of new chapters; and there are besides a thousand other ways. But the security consists in this, that there is no greater treachery than to break one's faith, and to ag<*> grieve those who have given us pleasure. And when you shall try me, you will find whether I be your friend or not, and whether I shall make an ungrateful return for your friendship, and the satisfaction you have given me.”

It was within less than two months after this, however, that T. received the alarming news of Cardan's shewing some symptoms of breaking the faith he had so lately pledged to him; this was in a letter from a quondam pupil of his, in which he writes, “A friend of mine at Milan has written to me, that Dr. Cardano is composing another algebraical work, concerning some latelydiscovered rules; hence I imagine they may be those same rules which you told me you had taught him; so that I fear he will deceive you.” To which T. replies, “I am heartily grieved at the news you inform me of, concerning Dr. Cardano of Milan; for if it be true, they can be no other rules but those I gave him; and therefore the proverb truly says, ‘That which you wish not to be known, tell to nobody.’ Pray endeavour to learn more of this matter, and inform me of it.”

Tartalea, after this, kept on the reserve with Cardan, not answering several letters he sent him, till one written on the 4th of August the same year, 1539, complaining greatly of T's neglect of him, and farther requesting his assistance to clear up the difficulty of the irreducible case , which C. had thus early been embarrassed with: he says that when ((1/3)b)3 exceeds ((1/2)c)2, the rule cannot be applied to the equation in hand, because of the square root of the negative quantities. On this occasion T. turns the tables on C. and plays his own game back upon him; for being aware of the above difficulty, and unable to overcome it himself, he wanted to try if C. could be encouraged to accomplish it, by pretending that the case might be done, though in another way. He says thus to himself, “I have a good mind to give no answer to this letter, no more than to the other two. However I will answer it, if it be but to let him know what I have been told of him. And as I perceive that a suspicion has arisen concerning the difficulty or obstacle in the rule for the case . I have a mind to try if he can alter the data in hand, so as to remove the said obstacle, and to change the rule into another form, although I believe indeed that it cannot be done; however there is no harm in trying.”—“M. Hieronime, I have received your letter, in which you write that you understand the rule for the case , but that when ((1/3)b)3 exceeds ((1/2)c)2, you cannot resolve the equation by following the rule, and therefore you request me to give you the solution of this equation . To which I reply, that you have not used a good method in that case, and that your whole process is intirely false. And 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.” In Cardan's answer to this he says, “You have been misinformed as to my intention to | publish more on Algebra. But I suppose you have heard something about my work de mysteriis æternitatis, which you take for some Algebra I intend to publish. As to your repenting of having given me your rules, I am not to be moved from the faith I promised you for any thing you say.” To this, and many other things contained in the same letter, T. returned no answer, being still suspicious of Cardan's intentions, and declining any more correspondence with him. This however did not discourage C. for we find him writing again to T. on the 5th of January, 1540, to clear up another difficulty which had occurred in this business, namely to extract the cube root of the binomials, of which the two parts of the rule always consisted, and for which purpose it seems C. had not yet found out a rule. On this occasion he informs T. that his quondam competitor Zuanne Colle had come to Milan, where, in some contests between them, Colle gave Cardan to understand that he had found out the rules for the two cases , and , and farther that he had discovered a general rule for extracting the cube roots of all such binomials as can be extracted; and that, in particular, the cube root of √(108 + 10) is √(3 + 1), and that of √(108 - 10) is √(3 - 1), and consequently that . He then earnestly entreats T. to try to find out the rule, and the solution of certain other questions which had been proposed to him by Colle. By this letter T. is still more confirmed in his resolution of silence; so that, without returning any answer, he only sets down among his own memorandums some curious remarks on the contents of the letter, and then concludes to himself, “Wherefore I do not choose to answer him again, as I have no more affection for him than for M. Zuanne, and therefore I shall leave the matter between them.” Among those remarks he sets down a rule for extracting the cube root of such binomials as can be extracted, and that is done from either member of the binomial alone, thus: Take either term of the binomial, and divide it into two such parts that one of them may be a complete cube, and the other part exactly divisible by 3; then the cube root of the said cubic part will be one term of the required root, and the square root of the quotient arising from the division of 1/3 of the 2d part by the cube root of the first, will be the other member of the root sought. This rule will be better understood in characters thus: let m be one member of the given binomial, whose cube root is sought, and let it be divided into the two parts a3 and 3b, so that a3 + 3b be = m; then is a + √b/a the cube root required, if it have one. Thus in the quantity √(108+10), taking the term 10 for m, then 10 divides into 1 and 9, where a3=1 or a=1, and 3b=9 or b=3: therefore a + √b/a becomes 1 + √3 for the cube root of √(108 + 10). And taking the other member √108, this divides into the two equal parts √27 and √27, making a3 = √27, and 3b=√27; hence a=√3, and b=√3 also; consequently a + √b/a is = √3 + √3/3 or √(3 + 1) for the cube root of the binomial sought, the same as before. “And thus, he adds, we may know whether any proposed binomial or residual be a cube or a noncube; for if it be a cube, the same two terms for the root must arise from both the given terms separately; and if the two terms of the root cannot thus be brought to agree both ways, such binomial or residual will not be a cube.” And thus ends the correspondence between them, at least for this time. But it seems they had still more violent disputes when C. in violation of his faith so often pledged to the contrary, published his work on cubic equations 4 years afterwards, viz, in the year 1545, of which we have before given an account, which disputes, it is said, continued till the death of Tartalea in the year 1557.

The last article in the volume contains a dialogue on some other forms of the cubic equations, in the year 1541, between T. and a Mr. Richard Wentworth, an English gentleman, who it seems had resided some time at Venice, on some public service from England, as T. in the dedication of the volume to Henry VIII. king of England, makes mention of him as “a gentleman of his sacred majesty.” Mr. Wentworth had learned some mathematics of T. and being about to depart for England, requests T. to shew him his newly discovered rules for cubic equations, as a farewell-lesson; and it is worth while to note a few particulars in this conference, as they shew pretty nicely the limited knowledge of T. at that time, as to the nature and roots of such equations. T. had before, it seems, shewed Mr. W. the rules for the cases of the 3d and 1st powers, and now the latter desires him to do the same as to the three cases in which the 3d and 2d powers only are concerned. On this T. professes great gratitude to Mr. W. for many obligations, but desires to be excused from giving him the rules for these, because he says he intends soon to compose a new work on Arithmetic, Geometry, and Algebra, which he intends to dedicate to him, and in which he means to insert all his new discoveries. On Mr. W. urging him further, T. gives him the roots of some equations of that kind, as for instance: but not the rules for finding them.

In the course of the conversation T. tells him that “all such equations admit of two different answers, and perhaps more; and hence it follows that they have, or admit of, two different rules, and perhaps more, the one more difficult than the other.” And on Mr. W. expressing his wonder at this circumstance of a plurality of roots, T. replies, “It is however very true, though hardly to be believed, and indeed if experience had not confirmed it, I should scarcely have believed it myself.” He then commits a strange blunder in an example which he takes to illustrate this by, namely the equation , which, he says, it is evident has the number 2 for one of its roots; and yet, he adds, “whoever shall resolve the same equation by my rule, will find the value of x to be √3(7 + √50) + √3(7 - √50), which is proved to be a true root by sub- | stituting it in the equation for x. And therefore, continues he, it is manifest that the case x3 + bx = c admits of two rules, namely, one (as in the above example) which ought to give the value of x rational, viz 2, and the other is my rule, which gives the value of x irrational, as appears above; and there is reason to think that there may be such a rule as will give the value of x = 2, although our ancestors may not have found it out.”——“And these two different answers will be found not only in every equation of this form , when the value of x happens to be rational, as in the example above, but the same will also happen in all the other five forms of cubic equations: and therefore there is reason to think that they also admit of two different rules; and by certain circumstances attending some of them, I am almost certain that they admit of more than two rules, as, God willing, I shall soon demonstrate.” Now all this discourse shews a strange mixture of knowledge and ignorance: it is very probable that he had met with some equations which admit of a plurality of roots; indeed it was hardly possible for him to avoid it; but it seems he had no suspicion what the number of roots might be, nor that his reasoning in this instance was founded on an error of his own, mistaking the root , of the equation , for a different root from the number or root 2, when in reality it is the very same, as he might easily have found, if he had extracted the cube roots of the binomials by the rule which he himself had just given above for that purpose: for by that rule he would have found , and , and therefore their sum is 2 = x, the same root as the other, which T. thought had been different. And besides this root 2, the equation in hand, , admits of no other real roots. Nor does any equation of the same form, , admit of more than one real root.

It seems also they had not yet discovered that all cases belong to the rules and forms for quadratic equations, which have only two powers in them, in which the exponent of the one is just double of the exponent of the other, as ; but some particular cases only of this sort they had as yet ventured to refer to quadratics, as the case . But in the conclusion of this dialogue T. informs W. of another case of this sort which he had accomplished, as a notable discovery, in these words: “I well remember, says he, that in the year 1536, on the night of St. Martin, which was on a Saturday, meditating in bed when I could not sleep, I discovered the general rule for the case , and also for the other two, its accompanying cases, in the same night.” And then he directs that they are to be resolved like quadratics, by completing the square, &c. And in these resolutions it is remarkable that he uses only the positive roots, without taking any notice of the negative ones.

Tartalea also published at Venice, in 1556, &c, a very large work, in folio, on Arithmetic, Geometry, and Algebra. This is a very complete and curious work upon the first two branches; but that of Algebra is carried no farther than quadratic equations, called book the first, with which the work terminates. It is evidently incomplete, owing to the death of the author, which happened before this latter part of the work was printed, as appears by the dates, and by the prefaces. It appears also, from several parts of this work, that the author had many severe conflicts with Cardan and his friend Lewis Ferrari: and particularly, there was a public trial of skill between them, in the year 1547; in which it would seem that Tartalea had greatly the advantage, his questions mostly remaining unanswered by his antagonists.

OF MICHAEL STIFELIUS.

After the foregoing analysis of the works of the first algebraic writers in Italy, it will now be proper to consider those of their contemporaries in Germany; where, excepting for the discoveries in cubic equations, the art was in a more advanced state, and of a form approaching nearer to that of our modern Algebra; the state and circumstances indeed being so different, that one would almost be led to suppose they had derived their knowledge of it from a different origin.

Here Stifelius and Scheubelius were writers of the same time with Cardan and Tartalea, and even before their discoveries, or publication, concerning the rules for cubic equations, Stifelius's Arithmetica Integra was published at Norimberg in 1544, being the year before Cardan's work on cubic equations, and is an excellent treatise, both on Arithmetic and Algebra. The work is divided into three books, and is prefaced with an Introduction by the famous Melanchthon. The first book contains a complete and ample Treatise on Arithmetic, the second an Exposition of the 10th book of Euclid's Elements, and the third a Treatise of Algebra, and it is therefore properly the part with which we are at present concerned. In the dedication of this part, he ascribes the invention of Algebra to Geber, an Arabic Astronomer; and mentions besides, the authors Campanus, Christ. Rudolph, and Adam Ris, Risen, or Gigas, whose rules and examples he has chiefly given. In other parts of the book he speaks, and makes use also, of the works of Bretius, Campanus, Cardan (i. e. his Arithmetic published in 1539, before the work on cubic equations appeared), de Cusa, Euclid, Jordan, Milichius, Schonerus, and Stapulensis.

Chap. 1. Of the Rule of Algebra, and its parts. Stifelius here describes the notation and marks of powers, or denominations as he calls them, which marks for the several powers are thus:

1st,2d,3d,4th,5th,6th,&c.
,[dram],,[dram][dram],∫s,[dram] ,&c.
which are formed from the initials of the barbarous way in which the Germans pronounced and wrote the Latin and Italic names of the powers, namely, res or cosa, zensus, cubo, zensi-zensus, sursolid, zenfi-cubo, &c. And the coss or first power , he calls the radix or root, which is the first time that we meet with this word in the printed authors. He also here uses the signs or characters, + and -, for addition and subtraction, and the first of any that I know of: for in Italy they used none of these characters for a long time after. He has no mark however for equality, but makes use of the word itself. |

Chap. 2. Of the Parts of the Rule of Geber or Algebra: teaching the various reductions by addition, subtraction, multiplication, division, involution, and evolution, &c.

Chap. 3. Of the Algorithm of Cossic Numbers: teaching the usual operations of addition, subtraction, multiplication, division, involution, and extraction of roots, much the same as they are at present. Single terms, or powers, he calls simple quantities; but such as 1[dram] + 1 a composite or compound, and 2 - 8 a defective one. In multiplication and division, he proves that like signs give +, and unlike signs -. He shews that the powers 1, , [dram], , &c, form a geometrical progression from unity; and that the natural series of numbers 0, 1, 2, 3, &c, from 0, are the exponents of the cossic powers; and he, for the first time, expressly calls them exponents: thus,

Exponents,0,1,2,3,4,5,6,&c.
Powers,1,,[dram],,[dram] [dram],∫s,[dram] ,&c.
And he shews the use of the exponents, in multiplication, division, powers, and roots, as we do at present; viz, adding the exponents in multiplication, and subtracting them in division, &c. And these operations he demonstrates from the nature of arithmetical and geometrical progressions. It is remarkable that these compound denominations of the powers are formed from the simple ones according to the products of the exponents, while those of Diophantus are formed according to the sums of them; thus the 6th power here is [dram] or quadrato-cubi, but with Diophantus it is cubo-cubi; and so of others. Which is presumptive evidence that the Europeans had not taken their Algebra immediately from him, independent of other proofs.

Chap. 4. Of the extraction of the roots of cossic numbers. He here treats of quadratic equations, which he resolves by completing the square, from Euclid II. 4 &c. Also quadratics of the higher orders, shewing how to resolve them in all cases, whatever the height may be, provided the exponents be but in arithmetical progression, as &c; where it is plain that he always counts 0 for the exponent of the unknown quantity in the absolute term. 2, 1, 0 4, 2, 0 6, 3, 0 8, 4, 0

Chap. 5. Of irrational cossic numbers, and of surd or negative numbers. In this treatise of radicals, or irrationals, he first uses the character √ to denote a root, and sets after it the mark of the power whose root is intended; as √[dram] 20 for the square root of 20, and √ 20 for the cube root of the same, and so on. He treats here also of negative numbers, or what he calls surd or fictitious, or numbers less than 0. On which he takes occasion to observe, that when a geometrical progression is continued downwards below 1, then the exponents of the terms, or the arithmetical progression, will go below 0 into negative numhers, and will yet be the true exponents of the former; as in these,

Expon.-3-2-10123
Pow.1/81/41/21248
And he gives examples to shew that these negative exponents perform their office the same as the positive ones, in all the opesations.

Chap. 6. Of the perfection of the Rule of Algebra, and of Secondary Roots. In the reduction of equations he uses a more general rule than those who had preceded him, who detailed the rule in a multitude of cases; instead of which, he directs to multiply or divide the two sides equally, to transpose the terms with + or -, and lastly to extract such root as may be denoted by the exponent of the highest power.

As to secondary roots, Cardan treated of a 2d ignota or unknown, which he called quantitas, and denoted it by the initial q, to distinguish it from the first. But here Stifelius, for distinction sake, and to prevent one root from being mistaken for others, assigns literal marks to all of them, as A, B, C, D, &c, and then performs all the usual operations with them, joining them together as we do now, except that he subjoins the initial of the power, instead of its numeral exponent: thus, 3A into 9B makes 27AB, 3[dram] into 4B makes 12[dram]B, 2 into 4A[dram] makes 8 A[dram], 1A squared makes 1A[dram], 6 into 3C makes 18C, 2A[dram] into 5A makes 10A∫s, &c, &c. 8A[dram] divided by 4 makes 2A[dram], &c. The square root of 25A[dram] is 5A, &c. Also 2A added to 2 makes 2 + 2A, and 2A subtr. from 2 makes 2 - 2A. And he shews how to use the same, in questions concerning several unknown numbers; where he puts a different character for each of them, as, A, B, C, &c; he then makes out, from the conditions of the question, as many equations as there are characters; from these he finds the value of each letter, in terms of some one of the rest; and so, expelling them all but that one, reduces the whole to a final equation, as we do at present.

The remainder of the book is employed with the solutions of a great number of questions to exercise all the rules and methods; some of which are geometrical ones.

From this account of the state of Algebra in Stifelius, it appears that the improvements made by himself, or other Germans, beyond those of the Italians, as contained in Cardan's book of 1539, were as follow:

1st. He introduced the characters +, -, √, for plus, minus, and root, or radix, as he calls it.

2d. The initials , [dram], , &c. for the powers.

3d. He treated all the higher orders of quadratics by the same general rule.

4th. He introduced the numeral exponents of the powers, -3, -2, -1, 0, 1, 2, 3, &c, both positive and negative, so far as integral numbers, but not fractional ones; calling them by the name exponens, exponent: and he taught the general uses of the exponents, in the several operations of powers, as we now use them, or the logarithms.

5th. And lastly, he used the general literal notation A, B, C, D, &c, for so many different unknown or general quantities.

OF SCHEUBELIUS.

John Scheubelius published several books upon Arithmetic and Algebra. The one now before me, is intitled Algebræ Compendiosa Facilisque Descriptio, quâ depro- | muntur magna Arithmetices miracula. Authore Johanne Scheubelio Mathematicarum Professore in Academia Tubingensi. Parisiis 1552. But at the end of the book it is dated 1551. The work is most beautifully printed, and is a very clear and succinct treatise; and both in the form and matter much resembles a modern printed book. He says that the writers ascribe this art to Diophantus, which is the first time that I find this Greek author mentioned by the modern algebraists: he farther observes, that the Latins call it Regula Rei & Census, the rule of the thing and the square (or of the 1st and 2d power); and the Arabs, Algebra. His characters and operations are much the same as those of Stifelius, using the signs and characters +, -, √, and the powers , , [dram], , &c, where the character is used for 1 or unity, or a number, or the o power; prefixing also the numeral coefficients; thus 44∫[dram] + 11[dram] + 31 - 53. He uses also the exponents 0, 1, 2, 3, &c, of the powers, the same way as Stifelius, before him. He performs the algebraical calculations, first in integers, and then in fractions, much the same as we do at present. Then of equations, which he says may be of infinite degrees, though he treats only of two, namely the first and second orders, or what we call simple and quadratic equations, in the usual way, taking however only the positive roots of these; and adverting to all the higher orders of quadratics, namely, x4, ax2, b;

x6, ax3, b;
x8, ax4, b; &c.

Next follows a tract on surds, both simple and compound, quadratic, cubic, binomial, and residual. Here he first marks the notation, observing that the root is either denoted by the initial of the word, or, after some authors, by the mark √:, viz. the sq. root √:, the cube root w√:, and the 4th root, or root of the root thus v√:, which latter method he mostly uses. He then gives the Arithmetic of surds, in multiplication, division, addition, and subtraction. In these last two rules he squares the sum or difference of the surds, and then sets the root to the whole compound, which he calls radix collecti, what Cardan calls radix universalis. Thus √12 ± √20 is ra. col. 32 ± √960. But when the terms will reduce to a common surd, he then unites them into one number; as √27 + √12 is equal √75. Also of cubic surds, and 4th roots. In binomial and residual surds, he remarks the different kinds of them which answer to the several irrational lines in the 10th book of Euclid's elements; and then gives this general rule for extracting the root of any binomial or residual a ± b, where one or both parts are surds, and a the greater quantity, namely, that the square root of it is ; which he illustrates by many examples. This rule will only succeed however, so as to come out in simple terms, in certain cases, namely, either when a2-b2 is a square, or when a and √(a2 - b2) will reduce to a common surd, and unite: in all other cases the root is in two compound surds, instead of one. He gives also another rule, which comes however to the same thing as the former, though by the words of them they seem to be different.

Scheubelius wrote much about the time of Cardan and Stifelius. And as he takes no notice of cubic equations, it is probable he had neither seen nor heard any thing about them; which might very well happen, the one living in Italy, and the other in Germany. And, besides, I know not if this be the first edition of Scheubel's book: it is rather likely it is not, as it is printed at Paris, and he himself was professor of mathematics at Tubingen in Germany.

ROBERT RECORDE.

The first part of his Arithmetic was published in 1552; and the second part in 1557, under the title of, “The Whetstone of Witte, which is the seconde parte of Arithmetike: containing the Extraction of Rootes: The Cossike Practise, with the Rule of Equation: and the Workes of Surde Nombers.” The work is in dialogue between the master and scholar; and is nearly after the manner of the Germans, Stifelius and Scheubelius, but especially the latter, whom he often quotes, and takes examples from. The chief parts of the work are, 1st. The properties of abstract and figurate numbers. 2nd. The extraction of the square and cube roots, much the same as at present. Here, when the number is not an exact power, but having some remainder over, he either continues the root into decimals as far as he pleases, by adding to the remainders always periods of cyphers; or else makes a vulgar fraction for the remaining part of the root, by taking the remainder for the numerator, and double the root for the denominator, in the square root; but in the cube root he takes for the nominator either the triple square of the root, which is Cardan's rule, or the triple square and triple root, with one more, which is Scheubel's rule. 3d. Of Algebra, or “Cossike Nombers.” He uses the notation of powers with their exponents the same as Stifel, with all the operations in simple and compound quantities, or integers and fractions. And he gives also many examples of extracting the roots of compound algebraic quantities, even when the roots are from two to six terms, in imitation of the same process in numbers, just as we do at present; which is the first instance of this kind that I have observed. As of this quantity: 25[dram] + 80∫[dram]Square - 26[dram][dram] - 63[dram] (5 +Root. 8[dram] - 9.

4th. “The Rule of Equation, commonly called Algeber's Rule.” He here, first of any, introduces the character =, for brevity sake. His words are, “And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe lines of one lengthe, thus:=, bicause noe 2 thynges can be moare equalle.” He gives the rules for simple and quadratic equations, with many examples. He gives also some examples in higher compound equations, with a root for each of them, but gives no rule how to find it. 5th. “Of Surde Nombers.” This is a very ample treatise on surds, both simple and compound, and surds of various degrees, as square, cubic, and biquadratic, marking the roots in Scheubel's manner, thus: √, w√, v√. He here uses the names bimedial, binomial, and residual; but says they have been used by others before him, though this is the first place where I have observed the two latter.— Hence it appears that the things which chiefly are new in this author, are these three, viz. |

1. The extraction of the roots of compound algebraic quantities.

2. The use of the terms binomial and residual.

3. The use of the sign of equality, or =.

OF PELETARIUS.

The first edition of this author's algebra was printed in 4to at Paris, in 1558, under this title, Jacobi Peletarii Cenomani, de occulta parte Numerorum, quam Algebram vocant. Lib. duo.

In the preface he speaks of the supposed authors of Algebra, namely Geber, Mahomet the son of Moses, an Arabian, and Diophantus. But he thinks the art older, and mentions some of his contemporary writers, or a very little before him, as Cardan, Stifel, Scheubel, Chr. Januarius; and a little earlier again, Lucas Paciolus of Florence, and Stephen Villafrancus a Gaul.

Of the two books, into which the work is divided, the first is on rational, and the second on irrational or surd quantities; each being divided into many chapters. It will be sufficient to mention only the principal articles.

He calls the series of powers numeri creati, or derived numbers, or also radicals, because they are all raised from one root or radix. He names them thus, radix, quadratus cubus, quadrato-quadratus, or biquadratus, supersolidus, quadrato-cubus, &c; and marks them thus ℞, q, , qq, ∫s, q, b∫s, &c. Of these he gives the following series in numbers, having the common ratio 2, with their marks set over them, and the exponents set over these again, in an arithmctical series, beginning at 0, thus:

012345678
1qqqqb∫sqqq
1248163264128256&c.
And he shews the use of the exponents, the same as Stifelius and Scheubelius; like whom also he prefixes coefficients to quantities of all kinds, as also the radical √. But he does not follow them in the use of the signs + and -, but employs the initials p and m for the same purpose. After the operations of addition, &c, he performs involution and cvolution also much the same way as at present: thus, in powers, raise the coefficient to the power required, and multiply the exponent, or sign, as he calls it, by 2, or 3, or 4, &c, for the 2nd, 3d, 4th, &c, power; and the reverse for extraction: and hence he observes, if the number or coefficient will not exactly extract, or the sign do not exactly divide, the quantity is a surd.

After the operations of compound quantities, and fractions, and reduction of equations, namely, simple and quadratic equations, as usual, in chap. 16, De Inveniendis generatim Radicibus Denominatorum, he gives a method of finding the roots of equations among the divisors of the absolute number, when the root is rational, whether it be integral or fractional; for then, he observes, the root always lies hid in that number, and is some one of its divisors. This is exemplified in several instances, both of quadratic and cubic equations, and both for integral and fractional roots. And he here observes, that he knows not of any person who has yet given general rules for the solution of cubic equations; which shews that when he wrote this book, either Cardan's last book was not published, or else it had not yet come to his knowledge.

Chap. 17 contains, in a few words, directions for bringing questions to equations, and for reducing these. He here observes, that some authors call the unknown number res, and others the positio; but that he calls it radix, or root, and marks it thus ℞: hence the term, root of an equation. But it was before called radix by Stifelius.

Chap. 21 & seq. treat of secondary roots, or a plurality of roots, denoted by A, B, C, &c, after Stifelius.

The 2d book contains the like operations in surds, or irrational numbers, and is a very complete work on this subject indeed. He treats first of simple or single surds, then of binomial surds, and lastly of trinomial surds. He gives here the same rule for extracting the root of a binomial and residual as Scheubelius, viz, . Individing by a binomial or residual, he proceeds as all others before him had done, namely, reducing the divisor to a simple quantity, by multiplying it by the same two terms with the sign of one of them changed, that is by the binomial if it be a residual, or by the same residual if it be a binomial; and multiplying the dividend by the same thing: thus . And, in imitation of this method, in division by trinomial surds, he directs to reduce the trinomial divisor first to a binomial or residual, by multiplying it by the same trinomial with the sign of one term changed, and then to reduce this binomial or residual to a simple nomial as above; observing to multiply the dividend by the same quantities as the divisor. Thus, if the divisor be 4 + √2 - √3; multiplying this by 4 + √2 + √3, the product is 15 + 8√2; then this binomial multiplied by the residual 15 - 8√2, gives 225 - 128 or 97 for the simple divisor: and the dividend, whatever it is, must also be multiplied by the two 4 + √2 + √3 and 15 - 8√2. Or in general, if the divisor be a + √b - √c; multiply it by a + √b + √c, which gives ; then multiply this by a2 + b - c - 2a√b, and it gives (a2 + b - c)2 - 4a2b, which will be rational, and will all collect into one single term. But Tartalea must have been in possession of some such rule as this, as one of the questions he proposed to Florido was of this nature, namely to find such a quantity as multiplied by a given trinomial surd, shall make it rational: and it appears, from what is done above, that, the given trinomial being , the answer will be .

Chap. 24 shews the composition of the cube of a binomial or residual, and thence remarks on the root of the case or equation 1 p 3℞ eqnal to 10, which he seems to know something about, though he had not Cardan's rules.

Chap. 30, which is the last, treats of certain precepts relating to square and cubic numbers, with a table of such squares and cubes for all numbers to 140; also shewing how to compute them both, by adding always their differences. |

He then concludes with remarking that there are many curious properties of these numbers, one of which is this, that the sum of any number of the cubes, taken from the beginning, always makes a square number, the root of which is the sum of the roots of the cubes; so that the series of squares so formed, have for their roots — 1, 3, 6, 10, 15, 21, &c. whose diff. are the natural nos 1, 2, 3, 4, 5, 6, &c. Namely, , &c. Or in general, .

This work of Peletarius is a very ingenious and masterly composition, treating in an able manner of the several parts of the subject then known, excepting the cubic equations. But his real discoveries, or improvements, may be reduced to these three, viz.

1st. That the root of an eqnation, is one of the divisors of the absolute term.

2d. He taught how to reduce trinomials to simple terms, by multiplying them by compound factors.

3d. He taught eurious precepts and properties concerning square and cube numbers, and the method of constructing a series of each by addition only, namely by adding successively their several orders of differences.

RAMUS.

Peter Ramus wrote his arithmetic and algebra about the year 1560. His notation of the powers is thus, l, q, c, bq, being the initials of latus, quadratus, cubus, biquadratus. He treats only of simple and quadratic equations. And the only thing remarkable in his work, is the first article, on the names and invention of Algebra, which we have noticed at the beginning of this history.

BOMBELLI.

Raphael Bombelli's Algebra was published at Bologna in the year 1579, in the Italian language. It seems however it was written some time before, as the dedication is dated 1572. In a short, but neat, introduction, he first adverts, in a few words, to the great excellence and usefulness of arithmetic and algebra. He then laments that it had hitherto been treated in so imperfect and irregular a way; and declares it is his intention to remedy all defects, and to make the science and practice of it as easy and perfect as may be. And for this purpose he first resolved to procure and study all the former authors. He then mentions several of these, with a short history or character of them; as Mahomet the son of Moses, an Arabian; Leonard Pisano; Lucas de Burgo, the first printed author in Europe; Oroncius; Seribelius; Boglione Francesi; Stifelius in Germany; a certain Spaniard, doubtless meaning Nunez or Nonius; and lastly Cardan, Ferrari, and Tartalea; with some others since, whose names he omits. He then adds a curious paragraph concerning Diophantus: he says that some years since there had been found, in the Vatican library, a Greek work on this art, composed by a certain Diophantus, of Alexandria, a Greek author, who lived in the time of Antoninus Pius; which work having been shewn to him by Mr. Antonio Maria Pazzi Reggiano, public lecturer on mathematics at Rome; and sinding it to be a good work, these two formed the re- solution of giving it to the world, and he says that they had already translated five books, of the six which were then extant, being as yet hindered by other avocations from completing the work. He then adds the following strange circumstance, viz. that they had found that in the said work the Indian authors are often cited; by which they learned that this science was known among the Indians before the Arabians had it: a paragraph the more remarkable as I have never understood that asty other person could ever find, in Diophantus, any reference to Indian writers: and I have examined his work with some attention, for that purpose.

Bombelli's work is divided into three books. In the first, are laid down the definitions and operations of powers and roots, with various sorts of radicals, simple and compound, binomial, residual, &c; mostly aster the rules and manner of former writers, excepting in some few instances, which I shall here take notice of. And first of his rule for the cube root of binomials or residuals, which for the sake of brevity, may be expressed in modern notation as follows: let √b + a be the binomial, the term √b being greater than a; then the rule for the cube root of √(b + a) comes to this, . Which is a rule that can be of little or no use; for, in the first place, is the same as ―(P + Q)2; and , the original quantity first proposed. The next thing remarkable in this 1st book, is his method for the square roots of negative quantities, and his rule for the cube roots of such imaginary binomials as arise from the irreducible case in cubic equations. His words, translated, are these: “I have found another sort of cubic root, very different from the former, which arises from the case of the cube equal to the first power and a number, when the cube of the 1/3d part of the (coef. of the) 1st power, is greater than the square of half the absolute number, which sort of square root hath in its algorism, names and operations different from the others; for in that case, the excess cannot be called either plus or minus; I therefore call it plus of minus when it is to be added, and minus of minus when it is to be subtracted.” He then gives a set of rules for the signs when such roots are multiplied, and illustrates them by a great many examples. His rule for the cube roots of such binomials, viz. such as a + √- b, is this: First sind √3(a2 + b); then, by trials search out a number c, and a sq. root √d, such, that the sum of their squares c2 + d may be | and also ; then shall sought. Thus, to extract the cube root of 2 + √ - 121: here ; then taking c = 2, and d = 1, it is , and , as it ought; and therefore 2+√-1 is = the cube root of 2+√-121, as required.

The notation in this book, is the initial R for root, with q or c &c after it, for quadrate or cubic, &c root. Also p for plus, and m for minus.

In the 2d book, Bombelli treats of the algorism with unknown quantities, and the resolution of equations. He first gives the definitions and characters of the unknown quantity and its powers, in which he deviates from the former authors, but professes to imitate Diophantus. He calls the unknown quantity tanto, and marks it thus , Its square or 2d power potenza, , Its cube cubo, , and the higher names are compounded of these, and marked , &c, so that he denotes all the powers by their exponents set over the common character . And all these powers he calls by the general name dignita, dignity. He then performs all the algorism of these powers, by means of their exponents, as we do at present, viz, adding them in multiplication, subtracting in division, multiplying them by the index in involution, and dividing by the same in evolution.

In equations he goes regularly through all the cases, and varieties of the signs and terms; first all the simple or single powers, and then all the compound cases; demonstrating the rules geometrically, and illustrating them by many examples.

In compound quadratics, he gives two rules: the first is by freeing the potenza or square from its coefficient by division, and then completing the square, &c, in the usual way: and the 2d rule, when the first term has its coefficient, may be thus expressed; if , then . He takes only the positive root or roots; and in the case , which has two, he observes that the nature of the problem must shew which of the two is the proper one.

In the cubic equations, he gives the rules and transformations, &c, after the manner of Cardan; remarking that some of the cases have only one root, but others two or three, of which some are true, and others false or negative. And in one place he says that by means of the case he trisects or divides an angle into three equal parts.

When he arrives at biquadratic equations, and particularly to this case x4 + ax - b, he says, “Since I have seen Diophantus's work, I have always been of opinion that his chief intention was to come to this equation, because I observe he labours at finding always square numbers, and such, that adding some number to them, may make squares; and I believe that the six books, which are lost, may treat of this equation, &c.” —“But Lewis Ferrari,” he adds, “of this city, also laboured in this way, and found out a rule for such cases, which was a very fine invention, and therefore I shall here treat of it the best I can.” This he accordingly does, in all the cases of biquadratics, both with respect to the number of terms in the equation, and the signs of the terms, except I think this most general case only ; fully applying Ferrari's method in all cases. Which concludes the 2d book.

The 3d book consists only of the resolution of near 300 practical questions, as exercises in all the rules and equations, some of which are taken from Diophantus and other authors.

Upon the whole it appears that this is a plain, explicit, and very orderly treatise on algebra, in which are very well explained the rules and methods of former writers. But Bombelli does not produce much of improvement or invention of his own, except his notation, which varies from others, and is by means of one general character, with the numeral indices of Stifelius. He also first remarks that angles are trisected by a cubie equation. But I know not how to account for his assertion, that Diophantus often cites the Indian authors; which I think must be a mistake in Bombelli.

CLAVIUS.

Christopher Clavius wrote his Algebra about the year 1580, though it was not published till 1608, at Orleans. He mostly follows Stifelius and Scheubelius in his notation and method, &c, having scarcely any variations from them; nor does he treat of cubic equations. He mentions the names given to the art, and the opinions about its origin, in which he inclines to ascribe it to Diophantus, from what Diophantus says in his preface to Dyonisius.

STEVINUS.

The Arithmetic of Simon Stevin of Bruges, was published in 1585, and his Algebra a little afterwards. They were also printed in an edition of his works at Leyden in 1634, with some notes and additions of Albert Girard, who it seems died the year before, this edition being published for the benefit of Girrard's widow and children. The Algebra is an ingenious and original work. He denotes the res, or unknown quantity, in a way of his own, namely by a small circle ○, within which he places the numeral exponent of the power, as ○0, ○1, ○2, ○3, &c, which are the 0, 1, 2, 3, &c power of the quantity ○; where ○0, or the 0 power, is the beginning of quantity, or arithmetical unit. He also extends this notation to roots or fractional exponents, and even to radical ones.

Thus ○1/2, ○1/3, ○1/4, &c, are the sq. root, cube root, 4th root, &c;

and ○4/3 is the cube root of the square;

and ○3/2 is the sq. root of the cube. And so of others.

The first three powers, ○1, ○2, ○3, he also calls coste (side), quarre (square), cube (cube); and the first of them, ○1, the prime quantity, which he observes is also metaphorically called the racine or root, (the mark of which is also √), because it represents the root or origin from whence all other quantities spring or arise, called the potences or powers of it. He condemns the terms sursolids, and numbers absurd, irrational, irregular, inexplicable, or surd, and shews that all numbers are denoted the same way, and are all equally proper ex- | pressions of some length or magnitude, or some power of the same root. He also rejects all the compound expressions of square-squared, cube-squared, cube-cubed, &c, and shews that it is best to name them all by their exponents, as the 1st, 2d, 3d, 4th, 5th, 6th, &c power or quantity in the series. And on his extension of the new notation he justly observes that what was before obscure, laborious, and tiresome, will by these marks be clear, easy, and pleasant. He also makes the notation of algebraic quantities more general in their coefficients, including in them not only integers, as 3○1, but also fractions and radicals, as (3/4)○2, and √2○3, &c. He has various other peculiarities in his notations; all shewing an original and inventive mind. A quantity of several terms, he calls a multinomial, and also binomial, trinomial, &c, according to the number of the terms. He uses the signs + and -, and sometimes: for equality; also X for division of fractions, or to multiply crosswise thus, 5/7X2/3 : 15/14.

He teaches the generation of powers by means of the annexed table of numbers, which are the coefficients of all the terms except the first and last. And he makes use of the same numbers also for extracting all roots whatever: both which things had first been done by Stifelius. In extracting the roots of non-quadrate or non-cubic numbers, he has the same approximations as at present, viz, either to continue the extraction indefinitely in decimals, by adding periods of ciphers, or by making a fraction of the remainder in this manner, viz, nearly, and nearly; where n is the nearest exact root of N; which is Peletarius's rule, and which differs from Tartalea's rule, as this wants the 1 in the denominator. And in like manner he goes on to the roots of higher powers.

He then treats of equations, and their inventors, which according to him are thus: Mahomet, son of Moses, an Arabian, invented these ○1 egale à ○0, its derivatives, ○2 egale à ○1, ○0, And some unknown author, the derivatives of this. Some unknown author invented these ○3 egale à ○1 ○0, ○3 egale à ○2 ○0,

But afterwards he mentions Ferreus, Tartalea, Cardan, &c, as being also concerned in the invetion of them.

Lewis Ferrari invented ○4 egale à ○3 ○2 ○1 ○0.

He says also that Diophantus once resolves the case ○2 egale à ○1 Θ. In his reduction of equations, which is full and masterly, he always puts the highest power on one side alone, equal to all the other terms, set in their order, on the other side, whether they be + or -. And he demonstrates all the rules both arithmetically and geometrically. In cubics, he gives up the irreducible case, as hopeless: but says that Bombelli resolves it by plus of minus, and minus of minus; thus, if , then , that is, . He resolves biquadratics by means of cubics and quadratics. In quadratics, he takes both the two roots, but looks for no more than two in cubics or biquadratics. He gives also a general method of approaching indefinitely near, in decimals, to the root of any equation whatever: but it is very laborious, being little more than trying all numbers, one after another, finding thus the 1st figure, then the 2d, then the 3d, &c, among these ten characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. And finally he applies the rules in the resolution of a great many practical questions.

Although a general air of originality and improvement runs through the whole of Stevinus's work, yet his more remarkable or peculiar inventions, may be reduced to these few following: viz,

1st. He invented not only a new character for the unknown quantity, but greatly improved the notation of powers, by numeral indices, sirst given by Stifelius as to integral exponents; which Stevinus extended to fractional and all other sorts of exponents, thereby denoting all sorts of roots the same way as powers, by numeral exponents. A circumstance hitherto thought to be of much later invention.

2d. He improved and extended the use and notation of coefficients, including in them fractions and radicals, and all sorts of numbers in general.

3d. A quantity of several terms, he called generally a multinomial; and he denoted all nomials whatever by particular names expressing the number of their terms, binomial, trinomial, quadrinomial, &c.

4th. A numeral resolution of all equations whatever by one general method.

Besides which, he hints at some unknown author as the first inventor of the rules for cubic equations; by whom may probably be intended the author of the Arabic manuscript treatise on cubic equations, given to the library at Leyden by the celebrated Warner.

VIETA.

Most of Vieta's algebraical works were written about or a little before, the year 1600, but some of them were not published till after his death, which happened in the year 1603. And his whole mathematical works were collected together by Francis Schooten, and elegantly printed in a folio volume in 1646. Of these, the algebraical parts are as follow:

1. Isagoge in Artem Analyticam.

2. Ad Logisticen Speciosam Notæ priores.

3. Zeteticorum libri quinque.

4. De Æquationum Recognitione, & Emendatione.

5. De Numerosâ Potestatum ad Exegesin Resolutione.

Of all these I shall give a very minute account, especially in such parts as contain any discoveries, as we here meet with more improvements and inventions in the nature of equations, than in almost any former author. And first of the Isagoge or Introduction to the Analytic Art. In this short introduction Vieta lays down certain præcognita in this art, as definitions, axioms, notations, common precepts or operations of addition, subtraction, multiplication, and division, with rules for questions, &c. From which we find, 1st. That the names of his powers are latus, quadratum, cubus, quadrato-quadratum, quadrato-cubus, cubocubus, &c; in which he follows the method of Diophantus, and not that derived from the Arabians. | 2d. That he calls powers pure or adfected, and first here uses the terms coefficient, affirmative, negative, specious logistics or calculations, homogeneum comparationis, or the absolute known term of an equation, homogeneum adfectionis, or the 2d or other term which makes the equation adfected, &c. 3d. That he uses the capital letters to denote the known as well as unknown quantities, to render his rules and calculations general, namely, the vowels A, E, I, O, U, Y for the unknown quantities, and the consonants B, C, D, &c, for the known ones. 4th. That he uses the sign + between two terms for addition; — for subtraction, placing the grcater before the less; and when it is not known which term is the greater, he places = between them for the difference, as we now use <01>; thus A = B is the same as A <01> B; that he expresses division by placing the terms like a fraction, as at present; though he was not first in this. But that he uses no characters for multiplication or equality, but writes the words themselves, as well as the names of all the powers, as he uses no exponents, which causes much trouble and prolixity in the progress of his work; and the numeral coefficients set after the literal quantities, have a disagreeable effect.

II. Ad Logisticen Speciosam, Notæ Priores. These consist of various theorems concerning sums, differences, products, powers, proportionals, &c, with the genesis of powers from binomial and residual roots, and certain properties of rational right-angled triangles.

III. Zeteticorum libri quinque. The zetetics or questions in these 5 books are chiefly from Diophantus, but resolved more generally by literal arithmetic. And in these questions are also investigated rules for the resolution of quadratic and cubic equations. In these also Vieta first uses a line drawn over compound quantities, as a vinculum.

IV. De Æquationum Recognitione, & Emendatione. These two books, which contain Vieta's chief improvements in Algebra, were not published till the year 1615, by Alexander Anderson, a learned and ingenious Scotchman, with various corrections and additions. The 1st of these two books consists of 20 chapters. In the first six chapters, rules are drawn from the zetetics for the resolution of quadratic and cubic equations. These rules are by means of certain quantities in continued proportion, but in the resolution they come to the same thing as Cardan's rules. In the cubics, Vieta sometimes changes the negative roots into affirmative, as Cardan had done, but he finds only the affirmative roots. And he here refers the irreducible cafe to angular sections for a solution, a method which had been mentioned by Bombelli.

Chap. 7 treats of the general method of transforming equations, which is done either by changing the root in various ways, namely by substituting another instead of it which is either increased or diminished, or multiplied or divided by fome known number, or raised or depressed in some known proportion; or by retaining the same root, and equally multiplying all the terms. Which sorts of transformation, it is evident, are intended to make the equation become simpler, or more convenient for solution. And all or most of these reductions and transformations were also practised by Cardan.

Chap. 8 shews what purposes are answered by the foregoing transformations; such as taking away some of the terms out of an equation, and particularly the 2d term, which is done by increasing or diminishing the root by the coefficient of the 2d term divided by the index of the first: by which means also the affected quadratic is reduced to a simple one. And various other effects are produced.

Chap. 9 shews how to deduce compound quadratic equations from pure ones, which is done by increasing or diminishing the root by a given quantity, being one application of the foregoing reductions.

Chap. 10, the reduction of cubic equations affected with the 1st power, to such as are affected with the 2d power; by the same means.

In chap. 11, by the same means also, the 2d term is restored to such cubic equations as want it.

In chap. 12, quadratic and cubic equations are raised to higher degrees by substituting for the root, the square or cube of another root divided by a given quantity.

In chap. 13 affected biquadratic equations are deduced from affected quadratics in this manner, when expressed in the modern notation: If , then shall . For since , therefore , and its square is : but , therefore , or . And in like manner for the biquadratic affected with its other terms. And in a similar manner also, in chap. 14, affected cubic equations are deduced from the affected quadratics.

In chap. 15 it is shewn that the quadratic has two values of the root A, or has ambiguous roots, as he calls them; and also that the cubics, biquadratics, &c, which are raised or deduced from that quadratic, have also double roots.

Having, in the foregoing chapters, shewn how the coefficients of equations of the 3d and 4th degree are formed from those of the 2d degree, of the same root; and that certain quadratics, and others raised from them, have double roots; then in the 16th chap. Vieta shews what relation those two roots bear to the coefficients of the two lowest terms of an equation consisting of only three terms. Thus, And so on for the same terms with their signs variously changed. |

Chap. 17 contains several theorems concerning quantities in continued geometrical progression. Which are preparatory to what follows, concerning the double roots of equations, the nature of which he expounds by means of such properties of proportional quantities.

Chap. 18, Æquationum ancipitum constitutiva; treating of the nature of the double roots of equations. Thus, if a, b, c, d, &c, be quantities in continual progression; then, 1st, of equations affected with the first power, If ; then , Z=ab, & A=a or b. If ; then , , & A =a or c. And in general, if BA-An+1; then , , and A=a or k the first or last term. Where the number of termsan, bn, &c, in B is n + 1, and the number of terms in Z is n.

2d, For equations containing only the highest two powers. If ; then , the sum of all except the last, or sum of all except the first; where the number of terms in B is n+1, and the number of terms in Z is n.

3d. Of equations affected by the intermediate powers. If .

4th. Of the remaining cases. If ; then , and , and . If ; then , and ; and .

Chap. 19. Æqualitatum contradicentium constitutiva. Of the relation of equations of like terms, but the sign of one term different; containing these 5 theorems, viz,

Chap. 20. Æqualitatum inversarum constitutiva. Containing these six theorems, viz,

Chap. 21. Alia rursus æqualitatem inversarum constitutiva. In these two theorems:

Next follows the 2d of the pieces published by Alexander Anderson, namely,

De Emendatione Æquationum, in 14 chapters.

Chap. 1. Of preparing equations for their resolution in numbers, by taking away the 2d term; by which affected quadratics are reduced to pure ones, and cubia equations affected with the 2d term are reduced to such as are affected with the 3d only. Several examples of both sorts of equations are given. He here too remarks upon the method of taking away any other term out of an equation, when the highest power is combined with that other term only; and this Vieta effects by means of the coefficients, or, as he calls them, the unciæ of the power of a binomial. All which was also performed by Cardan for the same purpose.

Chap. 2. De transmutatione *prw_ton—e)\katon, quæ remedium est adversus vitium negationis. Concerning the transformations by changing the given root A for another root E, which is equal to the homogeneum | comparationis divided by the first root A; by which means negative terms are changed to affirmative, and radicals are taken out of the equation when they are contained in the homogeneum comparationis.

Chap. 3, De Anastrophe, shewing the relation between the roots of correlate equations; from whence, having given the root of the one equation, that of the other becomes known; and it consists of these following 8 theorems, mostly deduced from the last 4 chapters of the foregoing recognitio æquationum.

Chap. 4, De Isomæria, adversus vitium fractionis. To take away fractions out of an equation. Thus, if . Put A = E/D; then .

Chap. 5, De Symmetrica Climactismo adversus vitium asymmetriæ. To take away radicals or surds out of equations, by squaring &c the other side of the equation.

Chap. 6. To reduce biquadratic equations by means of cubics and quadratics, by methods which are small variations from those of Ferrari and Cardan.

Chap. 7. The resolution of cubic equations by rules which are the same with Cardan's.

Chap. 8. De Canonica æquationum transmutatione, ut coefficientes subgraduales sint quæ præscribuntur. To transmute the equation so that the coefficient of the lower term, or power, may be any given number, he changes the root in the given proportion, thus: Let A be the root of the equation given, E that of the transmuted equation, B the given coefficient, and X the required one; then take A = BE/X, which substitute in the given equation, and it is done.—He commonly changes it so, that X may be 1; which he does, that the numeral root of the equation may be the easier found; and this he here performs by trials, by taking the nearest root of the highest power alone; and if that does not turn out to be the root of the whole equation, he concludes that it has no rational root.

Chap. 9. To reduce certain peculiar forms of cubics to quadratics, or to simpler forms, much the same as Cardan had done. Thus, 1. If ; then is . 2. If ; then is . 3. If ; then is A=2B. 4. If ; then is A=B. 5. If ; then is A=B. 6. If ; then is A=D. 7. If ; then is A=B or=D. 8. If ; then is 9. If ; then is 10. If ; then is . 11. If ; then is .

Chap. 10. Similium reductionum continuatio. Being some more similar theorems, when the equation is affected with all the powers of the unknown quantity A.

Chap. 11, 12, 13 relate also to certain peculiar forms of equations, in which the root is one of the terms of a certain series of continued proportionals.

Chap. 14, which is the last in this tract, contains, in four theorems, the general relation between the roots of an equation and the coefficients of its terms, when all its roots are positive. Namely, 1. If ; then is A=B or D. 2. If ; then is A=B or D or G. 3. If ; then is A=B or D or G or H. 4. If ; then is A=B or D or G or H or K.

And from these last 4 theorems it appears that Vieta was acquainted with the composition of these equations, that is, when all their roots are positive, for he never adverts to negative roots; and from other parts of the work it appears that he was not aware that the same properties will obtain in all sorts of roots whatever. But it is not certain in what manner he obtained these theorems, as he has not given any account of the investigations, though that was usually his way on other occasions; but he here contents himself with barely announcing the theorems as above, and for this strange reason, that he might at length bring his work to a conclusion.

To this piece is added, by Alexander Anderson, an Appendix, containing the construction of the cubic equations by the trisection of an angle, and a demonstration of the property referred to by Vieta for this purpose.

De Numerosa Potestatum Purarum Resolutione. Vieta here gives some examples of extracting the roots of pure powers, in the way that had been long before practised, by pointing the number into periods of figures according to the index of the root to be extracted, and then proceeding from one period to another, in the usual way. |

De Numerosa Potestatum adsectarum Resolutione. And here, in close imitation of the above method for the roots of pure powers, Vieta extracts those of adfected ones; or finding the roots of affected equations, placing always the homogeneum comparationis, or absolute term, on one side, and all the terms affected with the unknown quantity, and their proper signs, on the other side. The method is very laborious, and is but little more than what was before done by Stevinus on this subject, depending not a little upon trials. The examples he uses are such as have either one or two roots, and indeed such as are affected commonly with only two powers of the unknown quantity, and which therefore admit only of those two varieties as to the number of roots, namely according as the higher of the two powers is affirmative or negative, the homogeneum comparationis, on the other side of the equation, being always affirmative; and he remarks this general rule, if the higher power be negative, the equation has two roots; otherwise, only one; that is, affirmative roots; for as to negative and imaginary ones, Vieta knew nothing about them, or at least he takes no notice of them. By the foregoing extraction, Vieta finds both the greater and less root of the two that are contained in the equation, and either of them that he pleases; having first, for this purpose, laid down some observations concerning the limits within which the two roots are contained. Also, having found one of the roots, he shews how the other root may be found by means of another equation, which is a degree lower than the given one; though not by depressing the given equation, by dividing it as is now done; but from the nature of proportionals, and the theorems relating to equations, as given in the former tracts, he finds the terms of another equation, different from that last mentioned, from the root &c of which, the 2d root of the original equation may be obtained.

In the course of this work, Vieta makes also some observations on equations that are ambiguous, or have three roots; namely, that the equation , or as we write it is ambiguous, when the 2d term is negative, and the 3d term affirmative, and when 1/3 of the square of 6 the coefficient of the 2d term, exceeds 11, the coefficient of the 3d term, and has then three roots. Or in general, if , and (1/3)a2>b, the equation is ambiguous, and has three roots. He shews also, from the relation of the coefficients, how to sind whether the roots are in arithmetical progression or not, and how far the middle root differs from the extremes, by means of a cubic equation of this form . In all or most of which remarks he was preceded by Cardan.—Vieta also remarks that the case , has three roots by the same rule, viz, 2, 2, 5, but that two of them are equal. And farther, that when (1/3)a2 is=b, then all the three roots are equal, as in the case , the three roots of which are 2, 2, 2. But when (1/3)a2 is less than b, the case is not ambiguous, having but one root. And when ab=c, then a=x is one root itself.

Many curious notes are added at the end, with remarks on the method of finding the approximate roots, when they are not rational, which is done in two ways, in imitation of the same thing in the extraction of pure powers, viz, the one by forming a fraction of the remainder after all the figures of the homogeneum comparationis are exhausted; the other by increasing the root of the equation in a 10 fold, or 100 fold, &c, proportion, and then dividing the root which results by 10, or 100, &c: and this is a decimal approximation. AndVieta observes that the roots will be increased 10 or 100 fold, &c, by adding the corresponding number of ciphers to the coefficient of the 2d term, double that number to the 3d, triple the same number to the 4th, and so on. So if the equation were , then will have its root 10 fold, and will have it 100 fold.

Besides the foregoing algebraical works, Vieta gave various constructions of equations by means of circles and right lines, and angular sections, which may be considered as an algebraical tract, or a method of exhibiting the roots of certain equations having all their roots affirmative, and by means of which he resolved the celebrated equation of 45 powers, proposed to all the world by Adrianus Romanus.

Having now delivered a particular analysis of Vieta's algebraical writings, it will be proper, as with other authors, to collect into one view the particulars of his more remarkable peculiarities, inventions, and improvements.

And first it may be observed, that his writings shew great originality of genius and invention, and that he made alterations and improvements in most parts of algebra; though in other parts and respects his method is inferior to some of his predecessors; as, for instance, where he neglects to avail himself of the negative roots of Cardan; the numeral exponents of Stifelius, instead of which he uses the names of the powers themselves; or the fractional exponents of Stevinus; or the commodious way of presixing the coefficient before the quantity or factor; and such like circumstances; the want of which gives his Algebra the appearance of an age much earlier than its ownBut his real inventions of things before not known, may be reduced to the following particulars.

1st. Vieta introduced the general use of the letters of the alphabet to denote indefinite given quantities; which had only been done on some particular occasions before his time. But the general use of letters for the unknown quantities was before pretty common with Stifelius and his successors. Vieta uses the vowels A, E, I, O, U, Y for the unknown quantities, and the consonants B, C, D, &c, for known ones.

2d. He invented, and introduced many expressions or terms, several of which are in use to this day: such as coefficient, affirmative and negative, pure and adsected or affected, unciæ, homogeneum adfectionis, homogeneum comparationis, the line or vinculum over compound quantities thus ―(A + B). And his method of setting down his equations, is to place the homogeneum comparationis, or absolute known term, on the righthand side alone, and on the other side all the terms which contain the unknown quantity, with their proper signs.

3d. In most of the rules and reductions for cubic and | other equations, he made some improvements, and variations in the modes.

4th. He shewed how to change the root of an equation in a given proportion.

5. He derived or raised the cubic and biquadratic, &c equations, from quadratics; but not by composition in Harriot's way, but by squaring and otherwise multiplying certain parts of the quadratic. And as some quadratic equations have two roots, therefore the cubics and others raised from them, have also the same two roots, and no more. And hence he comes to know what relation these two roots bear to the coefficients of the two lowest terms of cubic and other equations, when they have only 3 terms, namely, by comparing them with similar equations so raised from quadratics. And, on the contrary, what the roots are, in terms of such coefficients.

6. He made some observations on the limits of the two roots of certain equations.

7. He stated the general relation between the roots of certain equations and the coefficients of its terms, when the terms are alternately plus and minus, and none of them are wanting, or the roots all positive.

8. He extracted the roots of affected equations, by a method of approximation similar to that for pure powers.

9. He gave the construction of certain equations, and exhibited their roots by means of angular sections; before adverted to by Bombelli.

OF ALBERT GIRARD.

Albert Girard was an ingenious Dutch or Flemish mathematician, who died about the year 1633. He published an edition of Stevinus's Arithmetic in 1625, augmented with many notes; and the year after his death was published by his widow, an edition of the whole works of Stevinus, in the same manner, which Girard had left ready for the press. But the work which entitles him to a particular notice in this history, is his “Invention Nouvelle en l' Algebre, tant pour la sobution des equations, que pour recognoistre le nombre des solutions qu'elles reçoivent, avec plusieurs choses qui sont necessaires a la perfection de ceste divine science;” which was printed at Amsterdam 1629, in small quarto in 63 pages, viz, 49 pages on Arithmetic and Algebra, and the rest on the measure of the supersicies of spherical triangles and polygons, by him then lately discovered.

In this work Girard first premises a short tract on Arithmetic; in the notation of which he has something peculiar, viz, dividing the numbers into the ranks of millions, billions, trillions, &c.

He next delivers the common rules of Algebra, both in integers, fractions and radicals; with the notation of the quantities and signs. In this part he uses sometimes the letters A, B, C, &c, after the manner of Vieta, but more commonly the characters of Stevinus, viz, ○0, ○1, ○2, ○3, &c, for the powers of the unknown quantity, with their roots ○5/2, ○1/2, ○1/4, ○2/3, ○<*>/4, &c, used by Stevinus; and sometimes the more usual marks of the roots as, √ or √2, √3, √4, &c; presixing the coefficients, as 6○2, or 3√532, or 2○1/2. In the signs he follows his predecessors so far as to have + for plus, - or ÷ for minus, = for general or indefinite difference, A + B for the sum, A - B or A = B for the difference, AB the product, and A/B for the quotient of A and B. He uses the parentheses ( ) for the vinculum or bond of compound quantities, as is now commonly practised on the continent; as A(AB+Bq), or √3 (A cub. - 3AqB); and he introduces the new characters ff for greater than, and § for less than; but he uses no character for equality, only the word itself.

Girard gives a new rule for extracting the cube root of binomials, which however is in a good measure tentative, and which he explains thus: To extract the cube root of 72 + √5120.

The squares of the terms{5184
5120
their difference64,and its
cube root 4. Which shews that the difference between the squares of the terms required is 4; and the rational part 72 being the greater, the greater term of the root will be rational also; and farther, that the greater terms of the power and root are commensurable, as also the two less terms. Then having made a table as in the
2 + √0
3 + √5
4 + √12
5 + √21
margin, where the square of the rational term always exceeds that of the other, by the number 4 above mentioned, one of these binomials must be the cubic root sought, if the given quantity have such a root, and it must be one of these four forms, for it is known to be carried far enough by observing that the cube root of 72 is less than 5, and the cube root of 5120 less than 21; indeed, this being the case, the last binomial is excluded, as evidently too great; and the first is excluded because one of its terms is 0; therefore the root must be either 3+√5 or 4+√12. And to know whether of these two it must be, try which of them has its two terms exact divisors of the corresponding terms of the given quantity; then it is found that 3 and 4 are both divisors of 72, but that only 5, and not 12, is a divisor of 5120; therefore 3+√5 is the root sought, which upon trial is found to answer. It is remarkable here that Girard uses 4+√20 instead of 4+√12, and 5+√29 instead of 5+√20, contrary to his own rule.

Girard then gives distinct and plain rules for bringing questions to equations, and for the reduction of those equations to their simplest form, for solution, by the usual modes, and also by the way called by Vieta Isomeria, multiplying the terms of the equation by the terms of a geometrical progression, by which means the roots are altered in the proportion of 1 to the ratio of the progression. He then treats of the methods of finding the roots of the several sorts of equations, quadratic, cubic, &c; and adds remarks on the proper number of conditions or equations for limiting questions. The quadratics are resolved by completing the square, and both the positive and negative roots are taken; and he observes that sometimes the equation is impossible, as 2<*> equ. 6○1 - 25, whose roots, he adds, are 3 + √-16 and 3 - √-16.

The cubic equations he resolves by Cardan's rule, except the irreducible case, which he the first of any resolves by a table of sines; the other cases also he resolves by tables of sines and tangents; and adds geometrical constructions by means of the hyperbola or the | trisection of angles. He next adds a particular mode of resolving all sorts of equations, that have rational roots, upon the principle of the roots being divisors of the last or absolute term, as before mentioned by Peletarius; and then gives the method of approximating to other roots that are not rational, much the same way as Stevinus.

Having found one root of an equation, by any of the former methods, by means of it he depresses the equation one degree lower, then finds another root, and so on till they are all found; for he shews that every algebraic equation admits of as many solutions or roots, as there are units in the index of the highest power, which roots may be either positive or negative, or imaginary, or, as he calls them, greater than nothing, or less than nothing, or involved; so the roots of the equation 1○3 equ. 7○1 - 6, are 2, 1, and - 3; and the roots of the equation 1○4 equ. 4○1 - 3 are

1,
1,
-1 + √-2,
-1 - √-2.

In depressing an equation to lower degrees, he does not use the method of resolution of Harriot, but that which is derived from the general relation of the roots and coefficients of the terms, which he here fully and universally states, viz, that the coefficient of the 2d term is equal to the sum of all the roots; that of the 3d term equal to the sum of all the products of the roots, taken two by two; that of the 4th term, the sum of the products, taken three by three; and so on, to the last or absolute term, which is the continual product of all the roots; a property which was before stated by Vieta, as to the equations that have all their roots positive; and here extended by Girard to all sorts of roots whatever: but how either Vieta or he came by this property, no where appears that I know of. From this general property, among other deductions, Girard shews how to find the sums of the powers of the roots of an equation; thus, let A, B, C, D, &c, be the 1st, 2d, 3d, 4th, &c, coefficient, after the first term, or the sums of the products taken one by one, two by two, three by three, &c; then, in all sorts of equations,

A}will be the sum of the{roots,
Aq-2Bsquares,
A cub.-3AB+3Ccubes,
Aqq-4AqB+4AC+2Bq-4Dbiquadrates.

Girard next explains the use of negative roots in Geometry, shewing that they represent lines only drawn in a direction contrary to those representing the positive roots; and he remarks that this is a thing hitherto unknown. He then terminates the Algebra by some questions having two or more unknown quantities; and subjoins to the whole a tract on the mensuration of the surfaces of spherical triangles and polygons, by him lately discovered.

From the foregoing account it appears that,

1st, He was the first person who understood the general doctrine of the formation of the coefficients of the powers, from the sums of their roots, and their products, &c.

2d, He was the first who understood the use of negative roots in the solution of geometrical problems.

3d, He was the first who spoke of the imaginary roots, and understood that every equation might have as many roots real and imaginary, and no more, as there are units in the index of the highest power. And he was the first who gave the whimsical name of quantities less than nothing to the negative. And,

4th, He was the first who discovered the rules for summing the powers of the roots of any equation.

OF HARRIOT.

Thomas Harriot, a celebrated astronomer, philosopher, and mathematician, flourished about the year 1610, about which time it is probable he wrote his Algebra, as he was then, and had been for many years before, celebrated for his mathematical and astronomical labours. In that year he made observations on the spots in the sun, and on Jupiter's satellites, the same year also in which Galileo first observed them: he left many other curious astronomical observations, and amongst them, some on the remarkable comets of the years 1607 and 1618. His Algebra was left behind him unpublished, as well as those other papers, at his death, which happened in the year 1621, being then 60 years of age, and but six years after the first publication of the principal parts of Vieta's Algebra by Alexander Anderson; so that it is probable that Harriot's Algebra was written before this time, and indeed that he had never seen these pieces. Harriot's Algebra was published by his friend Walter Warner, in the year 1631: and it would doubtless be highly grateful to the learned in these sciences, if his other curious algebraical and astronomical works were published from his original papers in the possession of the Earl of Egremont, to whom they have descended from Henry Percy, the Earl of Northumberland, that noble Mæcenas of his day. The book is in folio, and intitled Artis Analyticæ Praxis, ad Æquationes Algebraicas nova, expedita, & generali methodo, resolvendas; a work in all parts of it shewing marks of great genius and originality, and is the first instance of the modern form of Algebra in which it has ever since appeared. It is prefaced by 18 definitions, which are these: 1st, Logistica Speciosa; 2d, Equation; 3d, Synthesis; 4, Analysis; 5, Composition and Resolution; 6, Forming an Equation; 7, Reduction of an Equation; 8, Verification; 9, Numerosa & Speciosa; 10, Excogitata; 11, Resolution; 12, Roots; 13 and 14, The kinds and generation of equations by multiplication, from binomial roots or factors, called original equations,

asa + b=aa+ ba
a - c- ca - bc,
ora + b=aaa+baa+bca
a + c+caa-bda
a - d-daa-cda-bcd,
where he puts a for the unknown quantity, and the small consonants, b, c, d, &c, for its literal values or roots; 15, The first form of canonical equations, which are derived from the above originals, by transposing the homogeneum, or absolute term,
thus aa+ ba
- ca= + bc, &c;
16, The secondary canonicals, formed from the primary by expelling the 2d term,
thus aa= + bb,
or aaa- bba
- bca
- cca =+ bbc
+ bcc;
| 17, That these are called canonicals, because they are adapted to canons or rules for finding the numeral roots, &c. 18, Reciprocal equations, in which the homogeneum is the product of the coefficients of the other terms, and the first term, or highest power of the root, is equal to the product of the powers in the other terms, as .

After these definitions, the work is divided into two principal parts; 1st, of various generations, reductions, and preparations of equations for their resolution in the 2d part. The former is divided into 6 sections as follows.

Sect. 1. Logistices Speciosæ, exemplified in the 4 operations of addition, subtraction, multiplication, and division; as also the reduction of algebraic fractions, and the ordinary reduction of irregular equations to the form proper for the resolution of them, namely, so that all the unknown terms be on one side of the equation, and the known term on the other, the powers in the terms ranged in order, the greatest first, and the first or highest power made positive, and freed from its coefficient; as ,

or .
In this part he explains some unusual characters which he introduces, namely
= for equality, as a = b.
> for majority, as a > b,
< for minority, as a < b
; but the first had been before introduced by Robert Recorde.

Sect. 2. The generation of original equations from binomial factors or roots, and the deducing of canonicals from the originals. He supposes that every equation has as many roots as dimensions in its highest power; then supposing the values of the unknown letter a in any equation to be b, c, d, f, &c, that is a=b, and a=c, and a=d, &c; by transposition, or equal subtraction, these become , and , and , &c, or the same letters with contrary signs, for negative values or roots; then two of these binomial factors multiplied together, gives a quadratic equation, three of them a cubic, four of them a biquadratic, and so on, with all the terms on one side of the equation, and 0 on the other side, since, every binomial factor being = 0, the continual product of all of them must also be = 0. Thus,

a + b= aaa+ baa + bca
a + c+ caa - bda
a - d- daa - cda-bcd = 0
an original equation, and
aaa+ baa + bca
+ caa - bda
- daa - cda= + bcd
its canonical, deduced from it. And these operations are carried through all the cases of the 2d, 3d and 4th powers, as to the varieties of the signs + and -, and the proportions of the roots as to equal and unequal, with the reciprocals, &c. From which are made evident, at one glance of the eye, all the relations and properties between the roots of equations, and the coefficients of the terms.

Sect. 3. Æquationum canonicarum secundariarum a primariis reductio per gradus alicujus parodici sublatiouem radice supposititia invariata manente. Containing a great many examples of preparing equations by taking away the 2d, 3d, or any other of the intermediate terms, which is done by making the positive coefficients in that term, equal to the negative ones, by which means the whole term vanishes, or becomes equal to nothing.

They are extended as far as equations of the 5th degree; and at the end are collected, and placed in regular order, all the secondary canonicals, so reduced, so that by the uniform law which is visible through them all, the series may be continued to the higher degrees as far as we please.

Sect. 4. Æquationum canonicarum tam primariarum, quam secundariarum, radicum designatio. A great many literal equations are here set down, and their roots assigned from the form of the equation, that is all their positive roots; for their negative roots are not noticed here; and it is every where proved that they cannot have any more positive roots than these, and consequently the rest are negative. That those are roots, he proves by substituting them instead of the unknown letter a in the equation, when they make all the terms on one side come to the same thing as the homogeneum on the other side.

Sect. 5, In qua æquationum communium per canonicarum æquipollentiam, radicum numerus determinatur. On the number of the roots of common equations, that is the positive roots. This Harriot determines by comparing them with the like cases found among his canonical forms, which two equations, having the same number of terms with the same signs, and the relations of the coefficients and homogeneum correspondent, he calls equipollents. And whatever was the number of positive roots used in the composition of the canonical, the same, he infers, is the number in the proposed common equation. It is remarkable that in all the examples here used, the number of positive roots is just equal to the number of the changes in the signs from + to and from - to +, which is a circumstance, though not here expressly mentioned, that could not escape the observation, or the eye, of any one, much less of so clear and comprehensive a sight as that of Harriot. In this section are contained many ingenious disquisitions concerning the limits and magnitudes of quantities, with several curious lemmas laid down to demonstrate the propositions by, which lemmas are themselves demonstrated in a pure mathematical way, from the magnitudes themselves, independent of geometrical sigures; such as, 1, If a quantity be divided into any two unequal parts, the square of half the line will be greater than the product of the two unequal parts. 2, In three continued proportionals, the sum of the extremes is greater than double the mean. 3, In four continued proportionals, the sum of the extremes is greater than the sum of the two means. 4, In any two quantities, one-fourth the square of the sum of the cubes, is greater than the cube of the product of the two quantities. 5, Of any two quantities q and r, then (1/27)(qq + qr + rr)3 > 1/4 (qqr+qrr)2. 6, If any quantity be divided into three unequal parts, the square of 1/3 of the whole quantity is greater than 1/3 of the sum of the three products made of the three unequal parts. 7, Also the cube of the 1/3 part of the whole, is greater than the solid or continual product of the three unequal parts. |

Sect. 6. Æquationum communium reductio per gradus alicujus parodici exclusionem & radicis supposititiæ mutationeni. Here are a great many examples of reducing and transforming equations of the 2d, 3d, and 4th degrees; chiefly either by multiplying the roots of equations in any proportion, as was done by Vieta, or increasing or diminishing the root by a given quantity, after the manner of Cardan. The former of these reductions is performed by multiplying the terms of the equation by the corresponding terms of a geometrical progression, the 1st term being 1, and the 2d term the quantity by which the root is to be multiplied. And the other reduction, or transforming to another root, which may be greater or less than the given root by a given quantity, is performed commonly by substituting e + or - b for the given root a, by which the equation is reduced to a simpler form. Other modes of substitution are also used; one of which is this, viz, substituting (ee ± bb)/e or e ± bb/e for the root a in the given equation by which it reduces to this quadratic form , from whence Cardan's forms are immediately deduced; namely , and therefore ; where he denotes the cube or 3d root thus √3), but without any vinculum over the compound quantities.

In this section, Harriot makes various remarks as they occur: thus he remarks, and demonstrates, that eee - 3.bbe = -ccc -2.bbb is an impossible equation, or has no affirmative root. He remarks also that the three cases of the equation aaa - 3.bba = + 2.ccc are similar to the three conic sections; namely to the hyperbola when c > b, to the parabola when c = b, or to the ellipsis when c < b, and for which reason this case is not generally resoluble in species.

Having thus shewn how to simplify equations, and prepare them for solution, Harriot enters next upon the second part of his work, being the

Exegetice Numerosa,

or the numeral resolution of all sorts of equations by a general method, which is exemplified in a great number of equations, both simple and affected as far as the 5th power inclusive; and they are commonly prepared, by the foregoing parts, by freeing them from their 2d term, &c. These extractions are explained and performed in a way different from that of Vieta; and the examples are first in perfect or terminate roots, and afterwards for irrational or interminate ones, to which Harriot approximates by adding always periods of ciphers to the given number or resolvend, as far as necessary in decimals, which are continued and set down as such, but with their proper denominator 10, or 100, or 1000, &c.

He then concludes the work with

Canones Directorii,

which form a collection of the cases or theorems for making the foregoing numeral extractions, ready arranged for use, under the various forms of equations, with the factors necessary to form the several resolvends and subtrahends.

And from a review of the whole work, it appears that Harriot's inventions, peculiarities, and improvements in algebra, may be comprehended in the following particulars.

1st. He introduced the uniform use of the small letters a, b, c, d, &c, viz, the vowels a, e, &c for unknown quantities, and the consonants b, c, d, f, &c for the known ones; which he joins together like the letters of a word, to represent the multiplication or product of any number of these literal quantities, and prefixing the numeral coefficient as we do at present, except only separated by a point, thus 5.bbc. For a root he set the index of the root after the mark √; as √3) for the cube root. He also introduced the characters > and < for greater and less; and in the reduction of equations, he arranged the operations in separate steps or lines, setting the explanations in the margin on the left hand, for each line. By which, and other means, he may be considered as the introducer of the modern state of Algebra, which quite changed its form under his hands.

2d. He shewed the universal generation of all the compound or affected equations, by the continual multiplication of so many simple ones, or binomial roots; thereby plainly exhibiting to the eye the whole circumstances of the nature, mystery and number of the roots of equations; with the composition and relations of the coefficients of the terms; and from which many of the most important properties have since been deduced.

3d. He greatly improved the numeral exegesis, or extraction of the roots of all equations, by clear and explicit rules and methods, drawn from the foregoing generation or composition of affected equations of all degrees.

OF OUGHTRED'S CLAVIS.

Oughtred was contemporary with Harriot, but lived a long time after him. His Clavis was first published in 1631, the same year in which Harriot's Algebra was published by his friend Warner. In this work, Oughtred chiefly follows Vieta, in the notation by the capitals A, B, C, D, &c, in the designation of products, powers, and roots, though with some few variations. His work may be comprehended under the following particulars.

1. Notation. This extends to both Algebra and Arithmetic, vulgar and decimal. The Algebra chiefly after the manner of Vieta, as abovesaid. And he separates the decimals from the integers thus, 21<03>56, which is the first time I have observed such a separation, and the decimals set down without their denominator.

2. The common rules or operations of Arithmetic and Algebra. In algebraic multiplication, he either joins the letters together like a word, or connects them by the mark X, which is the first introduction of this character of multiplication: thus A X A or AA or Aq. But omitting the vinculum over compound factors, used by Vieta. He introduces here many neat and useful contractions in multiplication and division of decimals: as that common one of inverting the multiplier, to have fewer decimals, and abridge the work; that of omitting always one figure at a time, of the divisor, for the same purpose; dividing by the component factors of a number instead of the number itself; as 4 and 6 for 24; and many other neat contractions. He | states his proportions thus 7.9 :: 28.36, and denotes continued proportion thus <04>; which is the first time I have observed these characters.

3. Invents and describes various symbolical marks or abbreviations, which are not now used.

4. The genesis and analysis of powers. Denotes powers like Vieta, and also roots, thus √q6, √c20, √qq24, &c; and much in his manner too performs the numeral extraction of roots. He here gives a table of the powers of the binomial A + E as far as the 10th power, with all their terms and coefficients, or unciæ as he calls them, after Vieta.

5. Equations. He here gives express and particular directions for the several sorts of reductions, according as the form of the equation may require. And he uses the letter u after √, for universal, instead of the vinculum of Vieta. And observes that the signs of all the terms of the powers of A + E are positive, but those of A - E are alternately positive and negative.

6. Next follow many properties of triangles and other geometrical sigures; and the first instance of applying Algebra to Geometry, so as to investigate new geometrical properties; and after the algebraical resolution of each problem, he commonly deduces and gives a geometrical construction adapted to it. He gives also a good tract on angular sections.

7. The work concludes with the numeral resolution of affected equations, in which he follows the manner of Vieta, but he is more explicit.

OF DESCARTES.

Descartes's Geometry was first published in 1637, being six years after the publication of Harriot's Algebra. That work was rather an application of Algebra to Geometry, than the science either of Algebra or Geometry itself, purely and properly so called. And yet he made improvements in both. We must observe however, that all the properties of equations, &c, which he sets down, are not to be considered as even meant by himself for new inventions or discoveries; but as statements and enumerations of properties, before known and taught by other authors, which he is about to make some use or application of, and for which reason it is that he mentions those properties.

Descartes's Geometry consists of three books. The sirst of these is, De Problematibus, quæ construi possunt, adhibendo tantum rectas lineas & circulos. He here accommodates or performs arithmetical operations by Geometry, supposing some line to represent unity, and then, by means of proportionals, shewing how to multiply, divide, and extract roots by lines. He next describes the notation he uses, but not because it is a new one, for it is the same as had been used by former authors, viz, a + b for the addition of a and b, also a - b for their subtraction, ab multiplication, a/b division, aa or a2 the square of a, a3 its cube, &c: also √(a2 + b2) for the square root of a2 + b2, and for the cube root, &c. He then observes, after Stifelius, that there must be as many equations as there are unknown lines or quantities; and that they must be reduced all to one final equation, by exterminating all the unknown letters except one; when the final equation will appear like these, Where he uses for = or equality, setting the highest term or power alone on one side of the equation, and all the other terms on the other side, with their proper signs.

Descartes next defines plane problems, namely, such as can be resolved by right lines and circles, described on a plane superficies; and then the final equation rises only to the 2d power of the unknown letter. He then constructs such equations, viz. quadratics, by the circle, thus finding geometrically the root or roots, that is, the positive ones. But when the lines, by which the roots are determined, neither cut nor touch, he observes that the equation has then no possible root, or that the problem is impossible. He then concludes this book with the algebraical solution of the celebrated problem, before treated of by the antients, namely, to sind a point, or the locus of all the points, from whence a line being drawn to meet any number of given lines in given angles, the product of the segments of some of them shall have a given ratio to that of the rest.

Lib. 2. De Natura Linearum Curvarum. This is a good algebraical treatise on curve lines in general, and the first of the kind that has been produced by the moderns. Here the nature of the curve is expressed by an equation containing two unknown or variable lines, and others that are known or constant, as yz cy - cxy/b + ay - ac. But, not relating to pure Algebra, the particulars will be most properly placed under the article of curve lines, and other terms relating to them. Only one discovery, among many ingenious applications of Algebra to Geometry, may here be particularly noticed, as it may be considered as the first step towards the arithmetic of infinites; and that is the method of tangents, here given, or, which comes to the same thing, of drawing a line perpendicular to a curve at any point, which is an ingenious application of the general form of an equation, generated in Harriot's way, that has two equal roots, to the equation of the curve. Of which a particular account will be given at the article Tangents.

Lib. 3. De Constructione Problematum Solidorum, et Solida excedentium. Descartes begins this book with remarks on the nature and roots of equations, observing that they have as many roots as dimensions, which he shews, after Harriot, by multiplying a certain number of simple binomial equations together, as x - 2 0, and x - 3 0, and x - 4 0, producing x3 - 9xx + 26x - 24 0. He here remarks that equations may sometimes have their roots false, or what we call negative, which he opposes to those that are positive, or as he calls them true, as Cardan had done before. As a natural deduction from the generation or composition of equations, by multiplication, he infers their resolution, or depression, or decomposition, namely, dividing them by the binomial factors which were multiplied to produce the equation: and he observes that by this operation it is known that this divisor is one of the binomial roots, and that there can be no more roots than dimensions, or than those which form with the unknown letter | x, binomials that will exactly divide the equation, as Harriot had shewn before. Descartes adverts to several other properties, mostly known before, which he has occasion to make use of in the progress of his work; such as, that equations may have as many true roots as the terms have changes of the signs + and -, and as many false ones as successions of the same signs: which number and nature of the roots had before been partly shewn by Cardan and Vieta, from the relation of the coefficients, and their signs, and more fully by Harriot in his 5th section. And hence Descartes infers the method of changing the true roots to false, and the false to true, namely by changing the signs of the even terms only, as Cardan had taught before. Descartes then adverts to other reductions and transmutations which had been taught by Cardan, Vieta, and Harriot, such as, To increase or diminish the roots by any quantity; To take away the 2d term: To alter the roots in any proportion, and thence to free the equation from fractions and radicals.

Descartes next remarks that the roots of equations, whether true or false, may be either real or imaginary; as in the equation x3 - 6xx + 13x - 10 0, which has only one real root, namely 2. The imaginary roots were first noticed by Albert Girard, as before mentioned. He then treats of the depression of a cubic equation to a quadratic, or plane problem, that it may be constructed by the circle, by dividing it by some one of the binomial factors, which, in Harriot's way, compose the equation. Peletarius having shewn that the simple root is one of the divisors of the known term of the equation, and Harriot that that term is the continual product of all the roots, Descartes therefore tries all the simple divisors of that term, till he finds one of them which, connected with the unknown letter x, by + or -, will exactly divide the equation. And the process is the same for higher powers than the cube. But when a divisor cannot be thus found, for depressing a biquadratic equation to a cubic, he gives another rule, which is a new one, for dissolving it into two quadratics, by means of a cubic equation, in this manner: Let the given biqu. be + x4* . pxx. qx. r 0; where the sign of (1/2)p in the two quadratics must be the same as the sign of p in the given equation, and in the 1st quadratic the sign of q/2y must be the same as the the sign of q, but in the 2d quadratic the contrary. Then if there be found the root yy of this cubic equation y6 . 2py4 + (+pp)/(.4r)yy-qq 0, where the sign of 2p is the same as of p in the given biquadratic, but the sign of 4r contrary to that of r in the same: Then the value of y, hence deduced, being substituted for it in the two quadratic equations, and their two pairs of roots taken, they will be the four roots of the proposed biquadratic. And thus also, he hints, may equations of the 6th power be reduced to those of the 5th, and those of the 8th power to those of the 7th, and so on. Descartes does not give the investigation of this rule; but it has evidently been done, by assuming indeterminate quantities, after the manner of Ferrari and Cardan, as coefficients of the terms of the two quadratic equations, and, after multiplying the two together, determining their values by comparing the resulting terms with those of the proposed biquadratic equation.

After these reductions, which are only mentioned for the sake of the geometrical constructions which follow, by simplifying and depressing the equations as much as they will admit, Descartes then gives the construction of solid and other higher problems, or of cubic and higher equations, by means of parabolas and circles; where he observes that the false roots are denoted by the ordinates to the parabola lying on the contrary side of the axis to the true roots. Finally, these constructions are illustrated by various problems concerning the trisecting of an angle, and the finding of two or four mean proportionals; which concludes this ingenious work.

From the foregoing analysis may easily be collected the real inventions and improvements made in algebra by Descartes. His work, as has been observed before, is not algebra itself, but the application of algebra to geometry, and the algebraical doctrine of curve lines, expressing and explaining their nature by algebraical equations, and on the contrary, constructing and explaining equations by means of the curve lines. What respects the geometrical parts of this tract we shall have occasion to advert to elsewhere; and therefore shall here only enumerate the circumstances which belong more peculiarly to the science of Algebra, which I shall distinguish into the two heads of improvements and inventions. And

1st. Of his improvements. That he might fit equations the better for their application in the construction of problems, Descartes mentions, as it were by-the-bye, many things concerning the nature and reduction of equations, without troubling himself about the first inventors of them, stating them in his own terms and manner, which is commonly more clear and explicit, and osten with improvements of his own. And under this head we sind that he chiefly followed Cardan, Vieta, and Harriot, but especially the last, and explains some of their rules and discoveries more distinctly, and varies but a little in the notation, putting the first letters of the alphabet for the known, and the latter letters for the unknown quantities; also x3 for aaa, &c; and for =. But Herigone used the numeral exponents in the same manner two years before. Descartes explained or improved most parts of the reductions of equations, in their various transmutations, the number and nature of their roots, true and false, real and what he calls imaginary, called involved by Girard; and the depression of equations to lower degrees.

2d. As to his inventions and discoveries in algebra, they may be comprehended in these particulars, namely, the application of algebra to the geometry of curve lines, the constructing equations of the higher orders, and a rule for resolving biquadratic equations by means of a cubic and two quadratics.

Having now traced the science of Algebra from its origin and rude state, down to its modern and more polished form, in which it has ever since continued, with very little variation; having analysed all or most | of the principal authors, in a chronological order, and deduced the inventions and improvements made by each of them; from this time the authors both become too numerous, and their improvements too inconsiderable, to merit a detail in the same minute and circumstantial way: and besides, these will be better explained in a particular manner under the word or article to which each of them severally belongs. It may therefore now suffice to enumerate, or announce only in a cursory manner, the chief improvements and authors on algebra down to the present time.

After the publication of the Geometry of Descartes, a great many other ingenious men followed the same course, applying themselves to algebra and the new geometry, to the mutual improvement of them both; which was done chiefly by reasoning on the nature and forms of equations, as generated and composed by Harriot. Before proceeding upon these however, it is but proper to take notice here of Fermat, a learned and ingenious mathematician, who was contemporary and a competitor of Descartes for his brightest discoveries, which he was in possession of before the geometry of Descartes appeared. Namely, the application of algebra to curve lines, which he expressed by an algebraical equation, and by them constructing equations of the 3d and 4th orders; also a method of tangents, and a method de maximis et minimis, which approach very near to the method of Fluxions or Increments, which they strikingly resemble both in the manner of treating the problems, and in the algebraic notation and process. The particulars of which, see under their proper heads. Besides these, Fermat was deeply learned in the Diophantine problems, and the best edition of Diophantus's Arithmetic, is that which contains the notes of Fermat on that ingenious work.

But to return to the successors of Descartes. His geometry having been published in Holland, several learned and ingenious mathematicians of that country, presently applied themselves to cultivate and improve it; as Schooten, Hudde, Van-Heuraet, De Witte, Slusius, Huygens, &c; besides M. de Beaune, and perhaps some others in France.

Francis Schooten, professor of mathematics in the university of Leyden, was one of the first cultivators of the new geometry. He translated Descartes's Geometry out of French into Latin, and published it in 1649, with his commentary upon it, as also Brief Notes of M. de Beaune; both of them containing many ingenious and useful things. And in 1659 he gave a new edition of the same in two volumes, with the addition of several other ingenious pieces: as two posthumous tracts of de Beaune, the one on the nature and constitution, the other on the limits of equations, shewing how to assign the limits between which are contained the greatest and least roots of equations, extended and completed by Erasmus Bartholine: two letters of M. Hudde on the reduction of equations, and on the maxima and minima of quantities, containing many ingenious rules; among which are some concerning the drawing of tangents, and on the equal roots of equations, which he determines by multiplying the terms of the equation by the terms of any arithmetical progression, <*> being one of the terms, the equation is commonly depressed one degree lower: also a tract of Van Heuraet on the rectifi- cation of curve lines; the elements of curves by De Witte; Schooten's principles of universal mathematics, or introduction to Descartes's geometry, which had before been published by itself in 1651; and to the end of the work is added a posthumous piece of Schooten's (for he died while the 2d vol. was printing) intitled Tractatus de concinnandis demonstrationibus geometricis ex calculo algebraico. Schooten also published, in 1657, Exercitationes Mathematicæ, in which are contained many curious algebraical and analytical pieces, amongst others of a geometrical nature.

An elaborate commentary on Descartes's Geometry was also published by F. Rabuel, a Jesuit; and James Bernoulli, enriched with notes, an edition of the same, printed at Basil in 169—.

The celebrated Huygens also, among his great discoveries, very much cultivated the algebraical analysis: and he is often cited by Schooten, who relates divers inventions of his, while he was his pupil.

Slusius, a canon of Liege, published in 1659, Mesolabum, seu deæ mediæ propor. per circulum & ellips. vel hyperb. infinitis modis exhibitæ; by which, any solid problem may be constructed by infinite different ways. And in 1668 he gave a second edition of the same, with the addition of the analysis, and a miscellaneous collection of curious and important problems, relating to spirals, centres of gravity, maxima and minima, points of inflexion, and some Diophantine problems; all shewing him deeply skilled in Algebra and Geometry.

There have been a great number of other writers and improvers of Algebra, of which it may suffice slightly to mention the chief part, as in the following catalogue.

Peter Nonius, or Nunez, a Spaniard, wrote about the time of Cardan, or soon after.

In 1619 several pieces of Van Collen, or Ceulen, were translated out of Dutch into Latin, and published at Leyden by W. Snell; among which are contained a particular treatise on surds, and his proportion of the circumference of a circle, to its diameter.

In 1621 Bachet published, in Greek and Latin, an edition of Diophantus, with many notes. And another edition of the same was published in 1670, with additions by Fermat.

In 1624 Bachet's Problemes Plaisans et Delectables, being curious problems in mathematical recreations.

In 1634 Herigone published, at Paris, the first course of mathematics, in 5 vols. 8vo; in the 2d of which is contained a good treatise on Algebra; in which he uses the notation by small letters, introduced by the Algebra of Harriot, which was published three years before, though the rest of it does not resemble that work, and one would suspect that Herigone had not seen it. The whole of this piece bears evident marks of originality and ingenuity. Besides + for plus, he uses <01> for minus, and | for equality, with several other usful abbrevations and marks of his own. In the notation of powers, he does not repeat the letters like Harriot, but subjoins the numeral exponents, to the letter, as Descartes did two years afterwards. And Herigone uses the same numeral exponents for roots, as √3 for the cube root.

In 1635 Cavalerius published his Indivisibles; which proved a new æra in analytics, and gave rise to other new modes of computation in analytics. |

About 1640, et seq. Roberval made several notable improvements in analytics, which are published in the early volumes of the Memoirs of the Academy of Sciences; as, 1. A tract on the composition of motion, and a method of tangents. 2, De recognitione æquationum. 3, De geometrica planarum & cubicarum æquationum resolutione. 4, A treatise on indivisibles, &c.

In 1643 De Billy published Nova Geometriæ Clavis Algebra. And in 1670 Diophantus Redivivus. He was an author particularly well skilled in Diophantine problems.

In 1644 Renaldine published, in 4to, Opus Algebraicum, both ancient and modern, with mathematical resolution and composition. And in 1665, in folio, the same, greatly enlarged, or rather a new work, which is very heavy and tedious. In this work Renaldine uses the parentheses (a2+b2) as a vinculum, instead of the line over, as ―(a2 + b2).

In 1655 was published Wallis's Arithmetica Infinitorum, being a new method of reasoning on quantities, or a great improvement on the Indivisibles of Cavalerius, and which in a great measure led the way to infinite series, the binomial theorem, and the method of fluxions. Wallis here treats ingeniously of quadratures and many other problems, and gives the sirst expression for the quadrature of the circle by an insinite series. Another series is here added for the same purpose, by the Lord Brouncker.

In 1659 was published Algebra Rhonii Germanice; which was in 1668, translated into English by Mr. Thomas Brancker, with additions and alterations by Dr. John Pell.

In 1661 was published in Dutch, a neat piece of Algebra by Mr. Kinckhuysen; which Sir I. Newton, while he was professor of mathematics at Cambridge, made use of and improved, and he meant to republish it, with the addition of his method of fluxions and infinite series; but he was prevented by the accidental burning of some of his papers.

In 1665 or 1666 Sir Isaac Newton made several of his brightest discoveries, though they were not published till afterwards: such as the binomial theorem; the method of fluxions and infinite series; the quadrature, rectification, &c of curves; to find the roots of all sorts of equations, both numeral and literal, in infinite converging series; the reversion of series, &c. Of each of which a particular account may be seen in their proper articles.

In 1666 M. Frenicle gave several curious tracts concerning combinations, magic squares, triangular numbers, &c; which were printed in the early volumes of Memoirs of the Academy of Sciences.

In 1668 Thomas Brancker published a translation of Rhonius's Algebra, with many additions by Dr. John Pell, who used a peculiar method of registering the steps in any algebraical process, by means of marks and abbreviations in a small column drawn down the margin, by which each line, or step, is clearly explained, as was before done by Harriot in words at length.

In 1668 Mercator published his Logarithmotechnia, or method of constructing logarithms; in which he gives the quadrature of the hyperbola, by means of an infinite series of algebraical terms, found by dividing a simple algebraic quantity by a compound one, and for the first time that this operation was given to the public, though Newton had before that expanded all sorte of compound algebraical quantities into infinite series.

In the same year was published James Gregory's Exercitationes Geometricæ, containing, among other things, a demonstration of Mercator's quadrature of the hyperbola, by the same series.

And in the same year was published, in the Philosophical Transactions, Lord Brouncker's quadrature of the hyperbola by another infinite series of simple rational terms, which he had been in possession of since the year 1657, when it was announced to the public by Dr. Wallis. Lord Brouncker's series for the quadrature of the circle, had been published by Wallis in his Arithmetic of Infinites.

In 1669 Dr. Isaac Barrow published his Optical and Geometrical Lectures, abounding with profound researches on the dimensions and properties of curve lines; but particularly to be noticed here for his method of tangents, by a mode of calculation similar to that of Fluxions, or Increments, from which these differ but little, except in the notation.

In 1673 was published, in 2 vols. folio, Elements of Algebra, by John Kersey; a very ample and complete work, in which Diophantus's problems are fully explained.

In 1675 were published Nouveaux Elemens des Mathematiques, par J. Prestet, prêtre: a prolix and tedious work, which he presumptuously dedicated to God Almighty.

About 1677 Leibnitz discovered his Methodus Differrentialis, or else made a variation in Newton's Fluxions, or an extension of Barrow's method, for it is not certain which. He gave the first instance of it in the Leipsic Acts for the year 1684. He also improved infinite series, and gave a simple one for the quadrature os the circle, in the same acts for 1682.

In 1682 Ismael Bulliald published, in folio, his Opus Novum ad Arithmeticam Infinitorum, being a large amplification of Wallis's Arithmetic of Insinites.

In 1683 Tschirnausen gave a memoir, in the Leipsic Acts, concerning the extraction of the roots of all equations in a general way; in which he promised too much, as the method did not succeed.

In 1684 came out, in English and Latin, 4to, Thomas Baker's Geometrical Key, or Gate of Equations Unlock'd; being an improvement of Descartes's construction of all equations under the 5th degree, by means of a circle and only one and the same parabola for all equations, using any diameter instead of the axis of the parabola.

In 1685 was published, in folio, Wallis's Treatise of Algebra, both Historical and Practical, with the addition of several other pieces; shewing the origin, progress, and advancement of that science, from time to time. It cannot be denied that, in this work, Wallis has shewn too much partiality to the Algebra of Harriot. Yet, on the other hand, it is as true, that M. de Gua, in his account of it, in the Memoirs of the Academy of Sciences for 1741, has run at least as far into the same extreme on the contrary side, with respect to the discoveries of Vieta; and both these I believe from the same cause, namely, the want of examining the works of all former writers on Algebra, and specifying their several discoveries; as has been done in the course of this article. |

In 1687 Dr. Halley gave, in the Philos. Trans. the construction of cubic and biquadratic equations, by a parabola and circle; with improvements on what had been done by Descartes, Baker, &c. Also, in the same Transactions, a memoir on the number of the roots of equations, with their limits and signs.

In 1690 was published, in 4to, by M. Rolle, Traité d' Algébre; in 1699 Une Methode pour la Resolution des Problemes indeterminés; and in 1704 Memoires sur Pinverse des tangents; and other pieces.

In 1690 Joseph Raphson published Analysis Æquationum Universalis; being a general method of approximating to the roots of equations in numbers. And in 1715 he published the History of Fluxious, both in English and Latin.

In 1690 was also published, in 4 vols 4to, Dechale's Cursus seu mundus mathematicus; in which is a piece of algebra, of a very old-fashioned sort, considering the time when it was written.

About 1692, and at different times afterwards, De Lagny published many pieces on the resolution of equations in numbers, with many theorems and rules for that purpose.

In 1693 was published, in a neat little volume, Synopsis Algebraica, opus posthumum Johannis Alexandri.

In 1694, Dr. Halley gave, in the Philos. Trans. an ingenious tract on the numeral extraction of all roots, without any previous reduction. And this tract is also added to some editions of Newton's Universal Arithmetic.

In 1695 Mr. John Ward, of Chester, published, in 8vo, A Compendium of Algebra, containing plain, easy, and concise rules, with examples in an easy and clear way. And in 1706 he published the first edition of his Young Mathematician's Guide, or a plain and easy introduction to the mathematics: a book which is still in great request, especially with beginners, and which has been ever since the ordinary introduction of the greatest part of the mathematicians of this country.

In 1696 the Marquis de l'Hòpital published his Analyse des insiniment petits. And gave several papers to the Leipsic Acts and the Memoires of the Academy of Sciences. He left behind him also an ingenious treatise, which was published in 1707, intitled Traité analytique des Sections Coniques, et de la construction des lieux geometriques.

In 1697, and several other years, Mr. Ab. Demoivre gave various papers, in the Philos. Trans. containing improvements in Algebra: viz. in 1697, A method of raising an infinite multinomial to any power, or extracting any root of the same. In 1698, The extraction of the root of an infinite equation. In 1707, Analytical solution of certain equations of the 3d, 5th, 7th, &c degree. In 1722, Of algebraic fractions and recurring series. In 1738, The reduction of radicals into more simple forms. Also in 1730, he published Miscellanea analytica de seriebus & quadraturis, containing great improvements in series, &c.

In 169, Mr. Richard Sault published, in 4to, A New Treatise of Algebra, apply'd to numeral questions and geometry. With a converging series for all manner of Adfected Equations. The series here alluded to, is Mr. Raphson's method of approximation, which had been lately published.

In 1699 Hyac. Christopher published at Naples, in 4to, De constructione æquationum.

In 1702 was published Ozanam's Algebra; which is chiefly remarkable for the Diophantine analysis. He had published his mathematical dictionary in 1691, and in 1693 his course of mathematics, in 5 vols 8vo, containing also a piece of algebra.

In 1704, Dr. John Harris published his Lexicon Technicum, the first dictionary of arts and sciences: a very plain and useful book, especially in the mathematical articles. And in 1705 a neat little piece on algebra and fluxions.

In 1705 M. Guisnée published, in 4to, his Application de l'algebre a la geometrie: a useful book.

In 1706 Mr. William Jones published his Synopsis Palmariorum Matheseos, or a new introduction to the mathematics: a very useful compendium in the mathematical sciences. And in 1711 he published, in 4to, a collection of Sir Isaac Newton's papers, intitled Analysis per quantitatum series, fluxiones, ac differentias: cum enumeratione linearum tertii ordinis.

In 1707 was published by Mr. Whiston, the first edition of Sir Isaac Newton's Arithmetica Universalis: sive de compositione et resolutione arithmetica liber: and many editions have been published since. This work was the text book used by our great author in his lectures, while he was professor of mathematics in the university of Cambridge. And although it was never intended for publication, it contains many and great improvements in analytics; particularly in the nature and transmutation of equations; the limits of the roots of equations; the number of impossible roots; the invention of divisors, both surd and rational; the resolution of problems, arithmetical and geometrical; the linear construction of equations; approximating to the roots of all equations, &c. To the later editions of the book is commonly subjoined Dr. Halley's method of finding the roots of equations. As the principal parts of this work are not adapted to the circumstance os beginners, there have been published commentaries upon it by several persons, as s'Gravesande, Castilion, Wilder, &c.

In 1708 M. Reyneau published his Analyse Demontrée, in 2 vols 4to. And in 1714 La Science du Calcul, &c.

In 1709 was published an English translation of Alexander's algebra. With an ingenious appendix by Humphry Ditton.

In 1715 Dr. Brooke Taylor published his Methodus Incrementorum: an ingenious and learned work. And in the Philos. Trans. for 1718, An improvement of the method of approximating to the roots of equations in numbers.

In 1717 M. Nicole gave, in the memoirs of the academy of sciences, a tract on the calculation of finite differences. And in several following years, he gave various other tracts on the same subject, and on the resolution of equations of the 3d degree, and particularly on the irreducible case in cubic equations.

Also in 1717 was published a treatise on Algebra by Philip Ronayne.

Also in 1717 Mr. James Sterling published Lineæ tertii Ordinis; an ingenious work, containing good improvements in analytics. Also in 1730 Methodus Differentialis: sive tractatus de summatione et interpolatione serierum insinitarum: with great improvements on infinite series. |

In 1726 and 1729 Maclaurin gave, in the Philos. Trans. tracts on the imaginary roots of equations. And afterwards was published, from his posthumous papers, his treatise on Algebra, with its application to curve lines.

In 1727 came out s'Gravesande's Algebra, with a specimen of a commentary on Newton's universal arithmetic.

In 1728 Mr. Campbell gave, in the Philos. Trans. an ingenious paper on the number of impossible roots of equations.

In 1732 was published Wolsius's Algebra, in his course of mathematics, in 5 vols. 4to.

In 1735 Mr. John Kirkby published his arithmetic and algebra. And in 1748 his doctrine of ultimators.

In 1740 were published Mr. Thomas Simpson's Essays; in 1743 his Dissertations, and in 1757 his Tracts; in all which are contained several improvements in series and other parts of Algebra. As also in his algebra, first printed in 1745, and in his Select Exercises, in 1752.

Also in 1740 was published professor Saunderson's Elements of Algebra, in 2 vols. 4to.

In 1741 M. de la Caille published leçons de mathematiques; ou elemens d'algebre & de geometrie.

Also in 1741, in the memoirs of the academy of sciences, were given two articles by M. de Gua, on the number of positive, negative, and imaginary roots of equations. With an historical account of the improvements in Algebra; in which he severely censures Wallis for his partiality; a circumstance in which he himself is not less faulty.

In 1746 M. Clairaut published his Elemens d'algebre, in which are contained several improvements, especially on the irreducible case in cubic equations. He has also several good papers on different parts of analytics, in the memoirs of the academy of sciences.

In 1747 M. Fontaine gave, in the memoirs of the academy of sciences, a paper on the resolution of equations. Besides some analytical papers in the memoirs of other years.

In 1761 M. Castillion published, in 2 vols 4to, Newton's universal arithmetic, with a large commentary.

In 1763 Mr. Emerson published his Increments. In 1764 his Algebra, &c.

In 1764 Mr. Landen published his Residual Analysis. In 1765 his Mathematical Lucubrations. And in 1780 his Mathematical Memoirs. All containing good improvements in infinite series, &c.

In 1770 was published, in the German language, Elements of Algebra by M. Euler. And in 1774 a French translation of the same. The memoirs of the Berlin and Petersburgh academies also abound with various improvements on series and other branches of analysis by this great man.

In 1775 was published at Bologna, in 2 vols 4to, Compendio d'Analisi di Girolamo Saladini.

Besides the soregoing, there have been many other authors who have given treatises on Algebra, or who have made improvements on series and other parts of Algebra; as Schonerus, Coignet, Salignac, Laloubere, Hemischius, Degraave, Mescher, Henischins, Roberval, the Bernoullis, Malbranche, Agnesi, Wells, Dodson, Manfredi, Regnault, Rowning, Maseres, Waring, Lorg- na, de la Grange, de la Place, Bertrand, Kuhnius' Hales, and many others.

Algebra

, numeral, is that which is chiefly concerned in the solution of numeral problems, and in which all the given quantities are expressed by numbers only. As used by the more early authors, Diophantus, Paciolus, Stifelius, &c.

Algebra

, specious, or literal, is that commonly used by the moderns, in which all the quantities, both known and unknown, are represented or expressed by species or general characters, as the letters of the alphabet, &c; in consequence of which general designation, all the conclusions become universal theorems for performing every operation of the like kind. There are specimens of this method from Cardan and others about his time, but it was more generally employed and introduced by Vieta.

Algebraical

, something relating to algebra.

Thus we say algebraical solutions, curves, characters or symbols, &c.

Algebraical Curve, is a curve in which the general relation between the abscisses and ordinates can be expressed by a common algebraical equation.

These are also called geometrical lines or curves, in contradistinction to mechanical or transcendental ones.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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ALDHAFERA
ALEMBERT (John le Rond D')
ALFECCA
ALFRAGAN
ALGAROTI
* ALGEBRA
ALGEBRAIST
ALGENEB
ALGOL
ALGORAB
ALGORISM